Methods of Celestial Mechanics - Brouwer and Clemence

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M E T H O D S O F 
Celestial Mechanics 
DIRK BROUWER GERALD M. CLEMENCE 
Director of the Observatory Scientific Director 
Yale University United States Naval Observatory 
New Haven, Connecticut Washington, D.C. 
Academic Press 
N E W Y O R K A N D L O N D O N • 1961 
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PREFACE 
T h i s volume on celestial mechanics is in tended to provide a compre -
hensive background for practical applications. I t is suitable for advanced 
undergradua te and graduate s tudents as well as for engineers and others 
desiring to acquire some working knowledge of the subject. I t may 
be used also as a reference volume by specialists. T h e presentat ion and 
ar rangement have been designed to facilitate numerical work as m u c h 
as possible. But we have aimed at much more than a handbook of com-
putat ion. W e have developed the subject logically from first principles, 
and have tried to show the motivation for the processes employed, 
and to lay the groundwork for future independen t investigations by 
others . 
Celestial mechanics is the branch of as t ronomy that is devoted to 
the motions of celestial bodies. T h e principal force governing these 
motions, at least those within our own galaxy, is gravitation, and tha t 
is the only force dealt with in this book. In nearly all cases it is believed 
that the observed mot ions can be completely described wi thout recourse 
to non-gravitat ional phenomena , and that the numerous discrepancies 
between observation and theory are for the most par t due to the in-
completeness of the existing theoretical deve lopments ; dur ing the past 
two centuries, as a rule, the precision of observations has increased 
faster than the precision of the calculations. But from a strictly pragmat ic 
viewpoint, this belief is founded on nothing more substantial than 
previous successes in dealing with the relatively rough observational 
material then available. T h e vast quanti t ies of calculations requi red for 
a stricter examination of the subject have deterred many investigators 
from taking it up . T h e recent development of bet ter calculating machines 
has already led to renewed activity in the field, in which exciting dis-
coveries doubtless remain to be made , and a desire to facilitate such 
work has been our chief motive in wri t ing this book. 
T h e outs tanding examples of impor tan t non-gravitat ional forces now 
known are those that cause the irregular and seasonal changes in the 
speed of rotation of the earth, those tha t affect the mot ions of satellites 
within the ear th ' s a tmosphere , and those affecting the mot ions of comets 
and the components of close binary systems. T h e only section in this 
book that deals with any of these topics is tha t on a tmospher ic drag as it 
affects the motion of an artificial satellite (Chapter X V I I , Section 1 4 ) . 
V 
vi PREFACE 
We restrict ourselves to the inverse-square law of gravitation. T h e 
cases where it is known to be inadequate are the motions of the lines 
of apsides of the inner planets, and they can be very simply dealt wi th 
by adding small corrections in accordance with the principles of general 
relativity. For them the reader is referred to any general work on 
relativity. 
Celestial mechanics may be s tudied as a mental discipline, to gain a 
general appreciation of principles and formal relations, or it may be 
s tudied with a view to practical applications. I t is the latter that we have 
had exclusively in view. We give all of the principal methods tha t have 
been used most extensively for numerical calculations and none tha t 
are not suitable for the purpose . 
A good working knowledge of differential and integral calculus is 
presupposed. In addition, some knowledge of college algebra, the theory 
of equations, and ordinary differential equations is necessary for full 
unders tanding of some of the derivations, bu t lack of it will not prevent 
practical use of the methods . 
T h e organization of the book is not in logical sequence, so that it 
mus t be read th rough from beginning to end, bu t rather in more or less 
independent chapte rs ; the order of chapters might be rearranged in 
various ways wi thout hamper ing a proper unders tanding . T h e exceptions 
are Chapters I and I I , which belong before X I - X V I I . Also, Chapter IV 
logically precedes V, and Chapters VI - IX are connected with one another. 
Chapters I I I and X - X V I I are each virtually self-contained. T h u s a 
considerable variety of courses of s tudy may be laid out by making a 
judic ious selection of chapters . 
Chapter IV is given chiefly as prerequisi te to Chapte r V, in order 
that the latter may be unders tood wi thout a full course in finite differ-
ences, and also in order to make precepts for interpolation readily 
available. W e have included in Chapters V I - I X an account of those 
parts of spherical and practical as t ronomy that are essential for dealing 
with astronomical observations. Aberrat ion has been t reated more 
completely than the other parts , on account of the casual way in which 
it has been passed over by other authors . 
T h e methods themselves are all given in Chapters X I - X V I I . In the 
final chapter we have stressed the use of canonical variables for the 
quanti tat ive solution of problems in celestial mechanics. W e have thought 
to serve this purpose best by present ing a single unified approach, 
leaving aside all tha t is not specifically needed. Perhaps conspicuous by 
its absence is the partial differential equation of Hami l ton-Jacobi in 
its usual form, which is of part icular value for the solution of problems 
that permi t separation of the variables. W e prefer to emphasize the 
advantages of the more general form of the equation, due to von Zeipel, 
PREFACE vii 
which was specially designed to serve as the basis of a powerful method 
of per turbat ion theory. I t contains Delaunay 's method as a special case. 
Thos e interested in other aspects of the subject will find a wealth of 
additional information in E. T . Whi t taker ' s Analytical Dynamics and 
other treatises. 
Notat ion in celestial mechanics presents many difficulties, mostly 
owing to the great n u m b e r of symbols required. I t is often impossible 
to avoid using one symbol in two or more different senses wi thout 
in t roducing notations that are excessively cumbersome, or so novel as 
to confuse those who already have some familiarity with the subject. 
W e have generally preferred the notat ions most commonly used by 
those who have wri t ten most on a part icular topic, even to the extent 
of occasionally using two symbols in different chapters to mean the 
same thing. W e have, however, made an exception for the most commonly 
occurr ing quanti t ies , such as mean anomaly, for which a single symbol 
is used th roughout . In this way we have hoped to make it as easy as 
possible for the reader to consult the original l i terature, wi thout imposing 
an u n d u e burden on one who restricts himself to this book. 
I t is our pleasant duty to express our cordial thanks to Dr . Gen- ichi ro 
Hori of the depar tment of astronomy of Tokyo Universi ty. T h e first 
year of his sojourn at the Yale Observatory coincided with the period 
when the manuscr ip t was receiving its finishing touches. T o his careful 
reading of the entire manuscr ip t we owe many corrections and improve-
ments . Dr . Hor i hasbeen equally helpful in reading the proofs. 
DIRK BROUWER 
G. M . CLEMENCE 
February, 1961 
CHAPTER I 
ELLIPTIC MOTION 
1. H i s t o r i c a l i n t r o d u c t i o n . T h e subject of celestial mechanics 
may be said to have had its beginning in the publication by Sir Isaac 
Newton of his Philosophiae Naturalis Principia Mathematica in 1687, 
usually referred to as the Principia. I n this celebrated work Newton 
formulated the laws of mot ion and the law of universal gravitation; and 
he derived some of the most significant propert ies of planetary and 
satellite motion. 
T h e derivation of the famous three laws of Keple r had preceded 
Newton ' s work by many years, and the historical way of present ing the 
introduct ion to celestial mechanics is therefore that chosen by Newton 
in Book I, sections II and I I I , where he enquires what information 
concerning the force acting upon a planet may be derived from Kepler ' s 
laws, which may be stated as follows: 
I . T h e orbit of a planet lies in a plane that passes th rough the sun, 
and the area swept over by the line joining the sun and the planet is 
proport ional to the t ime elapsed. 
I I . T h e orbit of a planet is an ellipse of which the sun occupies one 
focus. 
I I I . T h e ratio between the square of the period of revolution and the 
cube of the semi-major axis is the same for all planets revolving around 
the sun. 
F r o m the s ta tements that the orbit of a planet lies in a plane that 
passes th rough the sun and that the area swept over by the radius vector 
is proport ional to the t ime, it may be concluded that the force acting 
upon a planet mus t be directed along the line joining it wi th the sun. 
Th i s force mus t be attractive rather than repulsive since the path of a 
planet is concave toward the sun. F r o m the s ta tement that the orbit of 
a planet is an ellipse with the sun in one focus it then follows that the 
force that keeps a planet in its orbit mus t vary inversely as the square 
of its distance from the sun. Finally, Kepler ' s th i rd law is found to 
require that the forces acting upon different planets mus t be directly 
proport ional to the planetary masses. Actually, since the orbits of the 
principal planets lie nearly in the same plane, this reasoning cannot 
1 
2 ELLIPTIC MOTION 
fully prove the isotropic character of gravitational attraction. T h e argu-
ment is s t rengthened appreciably by the consideration of cometary 
orbits , many of which have large inclinations relative to the orbital 
planes of the principal planets. Th i s , also, was considered by 
Newton . 
T h e s tudy of visual binary stars toward the end of the eighteenth 
century enhanced the interest in the quest ion concerning the universality 
of the law of gravitation. In most cases of visual binary stars nothing is 
known about the motion in the line of sight, and the only information 
about the orbit is to be obtained from the observed fact that the projec-
tion of the t rue relative orbit upon the plane tangent to the sky is an 
ellipse in which the law of areas is obeyed but in which the focus is 
not occupied by the central star. T h e question therefore arises what 
law of gravitational attraction would produce orbits having this proper ty . 
T h e answer is that the inverse square law is the only isotropic law that 
satisfies the observations, bu t that other more complicated and noniso-
tropic laws would also be compatible with the observed facts. Investiga-
tions of this type were made before the int roduct ion of the spectrograph 
for measurements of radial velocities. Wi th the aid of information 
concerning the motion in the line of sight the answer might be made 
more positive. 
In Newton ' s t ime there was ample justification for proceeding first 
with the establ ishment of the basic principles, by induct ion, from 
observed phenomena . Even now an analysis of the problem along these 
lines is of considerable interest and instructive. In this volume we 
assume that the laws of mot ion and the law of gravitation are available 
as the basic rules of celestial mechanics, and derive the consequences 
of these assumpt ions . Th i s amounts to following from the beginning 
the principle of the t ime-honored and fruitful method of scientific 
investigation: the validity of the basic assumptions is tested by the 
comparison of observation with theory. Newton stated this principle 
in his "Rules of Reasoning in Phi losophy" (Principia, Book I I I ) as 
follows: " I n experimental philosophy we are to look upon proposit ions 
inferred by general induct ion from phenomena as accurately or very 
nearly t rue, notwi ths tanding any contrary hypothesis that may be 
imagined, till such t ime as other phenomena occur, by which they may 
either be made more accurate, or liable to except ions." 
I t is well known that observations of the motions of the heavenly 
bodies have confirmed the validity of these basic assumptions to a 
remarkably high degree of precision. T h e full significance of the preci-
sion of the quanti tat ive agreement between theory and observation is 
evidently not realized by numerous writers of letters and pamphle ts 
who year after year advocate changes in the basis of celestial mechanics . 
LAWS OF MOTION 3 
T h e r e is one significant exception: the in t roduct ion of the general 
theory of relativity has made it necessary to look upon the Newtonian 
basis of celestial mechanics as an approximat ion to a more fundamental 
set of rules. T h e difficulty of the relativistic formulation and the high 
degree of approximat ion achieved with the Newtonian laws are the 
justification for proceeding on the basis of the classical theory. In the 
few cases where modifications are required by the general theory of 
relativity they may be in t roduced as small corrections to the results of 
the analysis. 
T h e most significant observed deviation from Newtonian theory is 
the excess in the advance of the perihelion of Mercury , which is so 
well established that this observational confirmation of general relativity 
can no longer be seriously quest ioned. A similar bu t m u c h smaller 
excess in the motion of the perihelion of the ear th ' s orbit is now fairly 
well confirmed observationally, bu t not with the same s t rength as in 
the case of Mercury ' s perihelion. 
2* T h e l a w s of m o t i o n a n d the l a w of g r a v i t a t i o n . T h e three 
laws of mot ion may be stated: 
I. Every body continues in its state of rest, or of uniform mot ion 
in a straight line, unless compelled by an impressed force to change that 
state. 
I I . T h e rate of change of m o m e n t u m is proport ional to the impressed 
force, and takes place in the direction in which the force acts. 
I I I . T o every action corresponds an equal and opposite reaction. 
T h e Newtonian law of gravitational at traction is: Every two particles 
in the universe attract each other with a force that is directly proport ional 
to the p roduc t of their masses, and inversely proport ional to the square 
of the distance between them. 
Before under taking to apply these laws to the simplest p rob lem of 
celestial mechanics, a few comments may be made . 
(a) I t is assumed that coordinate systems exist in which the laws of 
motion apply. Such a coordinate system is called a Newton ian frame 
of reference. 
(b) T h e velocity of a particle of mass m is defined by the differential 
quot ient dsjdt—the limiting value of As/At as At approaches zero, As 
being the distance traveled in the interval At. 
(c) T h e m o m e n t u m is defined by m dsjdt. 
(d) Velocity, m o m e n t u m , and force are vectors wi th components . 
In a Cartesian coordinate system, 
4 ELLIPTIC MOTION 
Force Fr9 F. 
For brevity, derivatives with respect to the t ime will frequently be 
denoted by a dot. Hence the velocity components will be wri t ten xy y, z 
and the components of the acceleration x, y> z. 
(f) Newton ' s law of gravitationis stated for particles of mass only, 
and does not apply to bodies of finite dimensions. I t can be shown, 
however, (see Chapter I I I ) that bodies with spherical symmetry attract 
each other as if their masses were concentrated in their respective centers. 
As will be shown in Chapter I I I , for bodies whose distr ibution of mass 
differs from spherical symmetry, if the distances between the bodies 
are large compared with their dimensions, the mutua l attraction ap-
proaches that which would apply if their masses were concentrated 
in their respective centers of mass. 
In the solar system the sun and the principal planets are spheroids 
with relatively small degrees of oblateness, and their mutua l distances 
are large compared with their dimensions. Hence, the attraction between 
the sun and planets not a t tended by satellites is found to be very nearly 
the same as if the bodies were point masses, that is, their masses were 
concentrated in their centers . For planets a t tended by satellites, a first 
approximation will give the motion of the centers of mass of these 
systems. T h e motion of the ea r th -moon system is in several respects the 
most interesting of these. One might first obtain the motion of the center 
of mass of the ea r th -moon system on the assumption that the mass of 
this system were concentrated in the center of mass. T h e next step 
would be to obtain the small correction to this mot ion due to the 
distr ibution of mass within this system. T h e complete solution of this 
problem requires a knowledge of the principal features of the motion 
of the moon around the earth. 
= Fz. ( 1 ) F F m -a-
m~d 
m 
Velocity 
Momentum 
(e) T h e second law of motion for a particle of mass may then be 
stated 
F = F 
Since m will be considered a constant, these may be wri t ten 
EQUATIONS OF MOTION 5 
T h e smaller bodies in the solar system, the minor planets and meteo-
rites, may have very irregular shapes, bu t on account of their small 
dimensions compared with their distances from the sun and the principal 
planets, it is permit ted , well within the observational accuracy, to treat 
these bodies as point masses. For comets the approximat ion may appear 
to be less justified on account of their large dimensions in some instances. 
However , even for comets there is good evidence of a s t rong concentra-
tion of mass in the nuclei. N o well-established deviations in cometary 
orbits that may be ascribed to nonspherical mass distr ibution are known. 
A m o n g the planets a t tended by satellites there are several examples 
of satellites with distances from their pr imaries that are only a few t imes 
the radius of the pr imary. T h e smallest known ratio is tha t of Jupi te r ' s 
fifth satellite with the ratio 2.54 to 1. In such cases the effect of the 
nonspherici ty of the planet upon the satellite's motion is not at all 
negligible. Th i s problem is s tudied in detail in Chapters I I I and X V I I , 
also in relation to orbits of artificial satellites for which a more elaborate 
solution is required than for natural satellites in the solar system. 
(g) T h e r e is one further circumstance that simplifies to a considerable 
extent the s tudy of the motions of bodies in the solar system. Compared 
with the sun, the masses of the principal planets are small. T h e largest 
mass is that of Jupi ter , about 1/1000 that of the sun. T h e next largest 
mass is that of Saturn, about 1/3500 that of the sun. Hence , except in 
the case of a body close to a principal planet, the attraction by the sun 
is always overwhelmingly m u c h greater than the attractions by the 
principal planets . T h e masses of the minor planets, comets, and meteo-
rites are known to be very small. For example, the total mass of all the 
minor planets combined has been est imated to be less than 1/500 the 
mass of the ear th. 
I n view of these circumstances a useful first approximation to the 
motion of a principal planet, minor planet, comet, or meteoric particle 
may be obtained by considering only the mutua l at traction between the 
sun and the body whose motion is to be studied, t reat ing both bodies 
as point masses. T h i s is known as the two-body problem, which will 
be studied in the following sections. 
3 . E q u a t i o n s o f m o t i o n for the t w o - b o d y p r o b l e m . Consider 
two point masses, ma and mbi with coordinates in a Cartesian coordinate 
system fa, rjaj £a; £&, r)bi £ 6 . W h e n motions in the solar system are to be 
treated, the coordinate system is chosen to be a r ight -handed system. 
Denote the distance between the two masses by r. T h e force acting 
between the two masses is then 
p __ fr2 mamb 
r2 ' 
6 ELLIPTIC MOTION 
and the components of the force acting upon mb are 
Fte = k*mamb Fbn = k*mamb -^=^», FH = ^F^-
It is seen that Fa$ = — J F 5 | , etc. , as should be the case in view of the 
third law of motion. If the expressions for Fa§, Fa , ... Fh^ are taken as 
the r ight -hand members of the equations of motion (1), the explicit 
equations of motion are found to be 
maL = k2mamb 
1 F. Tisserand, "Mecanique Celeste," Gauthier-Villars, Paris, 1889. 
k2mamb 
F = F 
1 at] — x 
( 2 ) 
maL = k2mamb 
Hence the components of the force acting upon ma are 
k2maml 
the value of k depending upon the units of mass, t ime, and distance 
chosen. For numerical calculations the choice of uni ts is commonly 
determined by those employed in the national ephemerides and in 
auxiliary tables. For theoretical investigations it is common to adopt 
uni ts such that k2 = 1 or to eliminate k from the equations. By paying 
proper at tention to the dimensions of derived quanti t ies it will always 
be possible to restore k in the final results of the analysis if desired. 
Especially in view of numerical applications, it is advantageous to choose 
the factor k2 in the expressions for F rather than a factor / (such as 
used by T i s se rand 1 ) . 
T h e line mamb will have the direct ion cosines ' 
MOTION OF CENTER OF MASS 7 
with 
r2 = (L - Lf + (va - i»T + (L - Q2-
In this problem the equations of mot ion could be wri t ten down almost 
at once, and the problem is thereby reduced to that of a system of six 
differential equations, each of the second order. T h e complete integrat ion 
of this system requires the introduct ion of twelve constants of integra-
t ion. 
4 . M o t i o n of the c e n t e r o f m a s s . Six constants of integrat ion 
may be obtained by making use of the fact that the r igh t -hand member s 
of the corresponding equat ions of (2) and (3) are equal bu t opposi te . 
Hence , by adding in pairs, there result: 
maL + = 0, 
WaVa + mbVb = 0, (4) 
ma£a + mbtb = 0. 
U p o n integrat ing these once, we obtain 
™aL + Wbkb = «i, 
™arja + nihr)b = a2, (5) 
™aL + mbtb = <*3> 
ocly oc2, and oc3 being arbi trary constants . One further integration gives 
MaL + Mb€b = *it + ft, 
MaVa + MbVb = OC2t + ft, ( 6 ) 
"*aL + ™>blb = O^f + ft. 
ft, ft, and ft being three additional arbi t rary constants . 
™>bL = k2mamb
 l-
(3) mB7]H = k2mamb
 J 
mbtb = k2mam 
8 ELLIPTIC MOTION 
Another form for these expressions may be obtained by introducing 
£g> Vgy ta as the coordinates of the center of mass. T h e n 
MaL + mb€b = (™a + ™b)i;gi 
WaVa + ™bVb = (ma + ^b)Vg> (7) 
™ala + Wblb = (ma + mb)£g. 
Hence 
2 - i - a? '3 
I t is impor tant to have a clear unders t and ing of the meaning of con-
stants of integration. Some confusion may persist unless both the ma the -
matical and the physical meanings are fully grasped. In this particular 
case, the interpretat ion is immediately clear. F r o m a mathematical 
point of view the meaning of the constants of integration is s imply 
that , whatever values be assigned to ocl9 a 2, a 3 , fSl9 j82, j83, the solution of 
the problem in which these constants appear will satisfy the original 
equations. F r o m a physical point of view the solution will usually be 
employed to describethe motion of a particular physical system. In 
this case, a particular set of values of the constants of integration will 
serve to identify the system. 
As an illustration, suppose the solar system to be limited to two 
bodies, the sun and the earth. T h e constants fil9 jS2, ^83, divided by the 
sum of the masses, furnish the coordinates of the center of mass of the 
system at t = 0. T h e constants ocl9 a 2 , a 3 , divided by the sum of the 
(8) 
which state that the center of mass of the system has a uniform motion 
in a straight line, the coordinates of the center of mass at t = 0 being 
N(ma + ™>b), j8 2/(m a + j8 3/(m a + mb\ and t h e ' c o n s t a n t velocity 
being given by 
T h e direction cosines of the velocity are evidently 
EQUATIONS FOR CENTER OF MASS 9 
masses, furnish the velocity components of the center of mass of the 
system. 
I t is known tha t the velocity of the solar system, if de termined relative 
to the nearer stars, is 19 km/sec in the direction right ascension 1 8 h , 
declination + 3 0 ° , approximately. T h e direction cosines of the apex, in 
an equatorial coordinate system with the £ axis toward the nor th celestial 
pole, are 
0.000, - 0.866, + 0.500, 
and the velocity components of the motion of the solar system in km/sec 
0.0, - 16.5, + 9.5. 
These are evidently the values in km/sec of 
T h e integration constants ocv a 2 , a 3 , ft, ft, ft are therefore significant 
only for the description of the location of the center of mass of the system 
of two bodies; they are irrelevant if the motion within the system is to 
be studied. 
Wi th the aid of (6) it is possible to reduce the system (2), (3) from 
one of the twelfth order to one of the sixth order by either of two 
methods . 
5* E q u a t i o n s of m o t i o n a b o u t the c e n t e r of m a s s . In t roduce 
x a ~ £a £gy J a = Va ^gy %a = £ a %>gy 
* 6 = £b — igy yb = Vb~ Vgy *b = £& ~ £</• 
Since 
— L = x b — xay Vb — Va = Jb ~ Jay £& ~ £ a = %b ~ Za> 
and since 
lg = 0, 7 ) , = 0, £, = 0, 
Eqs . (2) and (3) may be wri t ten: 
ma xa = k2mamh 
maya = k2mamb 
ma za = k2mamb 
(9) 
10 ELLIPTIC MOTION 
(10) 
with 
mb zb = k2mamb -
r2 = (*« - xbf + (ya - ybf + (za - zbf 
T h i s amounts to using the moving center of mass as origin, and com-
parison of Eqs . (2), (3) with (9), (10) shows that the equations are not 
changed by this t ransformation. I t would be simple to show quite 
generally tha t the equat ions remain unchanged if the new origin has any 
arbitrary uniform motion with reference to the original coordinate 
system and that they also remain unchanged regardless of the orientation 
of the coordinate system. 
Equat ions (7) now become 
maxa + mbxb = 0, 
WaJa + ™>bjb = °> (11) 
maza + mbzb = 0. 
Hence xb, yb, zb can be eliminated from (9) and xa,yayza from (10). 
Th i s requires expressing r in te rms of xa> yai za alone or in te rms of 
x b j Vb> z b alone. Since 
mbJb = k2mamb'-
Mb
 x b = klmamb 
xa xb -
( 1 2 ) 
z a z b ~ 
it follows that 
EQUATIONS FOR RELATIVE MOTION 11 
or, if 
rl = x2
a+yl + z2
ai xb 
(13) 
Mo k2 
Eqs . (9) and (10) may be wri t ten 
ya = (14) 
(15) 
Evidently, the solution of either (14) or (15) is all that is required. If, 
by the complete solution of (14), xay yay za are obtained as functions of 
the t ime and six constants of integration, the expressions for xby yby zb 
can be wri t ten down with the aid of (11). Hence by the procedure 
followed the reduct ion to a set of equations of the sixth order has been 
achieved. These equations will not be used for the integration, bu t they 
will be referred to in a later section. 
6. E q u a t i o n s for the re la t ive m o t i o n . A more common procedure 
is to obtain the motion of mb relative to ma. I n order to obtain these 
equations, pu t 
x = xb - xa, y = y b - y a , z = zb - za. 
By subtract ing Eqs . (9), divided by may from the corresponding equa-
tions (10), divided by mby with 
r2 = x2 + y2 + # 2 , /x = k2 (ma + mb\ 
the result ing equations are: 
Xb —-
X — y -- z - (16) 
T h e equations (16) represent the new system, reduced to one of the 
sixth order. If x, y, z are obtained by integration as functions of t and 
six constants of integration, the expressions for xay xby yay yby zay zb may 
be obtained from 
12 ELLIPTIC MOTION 
which may be derived with the aid of (11). 
7 . T h e i n t e g r a l s of area* I t is seen that if the first equat ion 
of (16) is mult ipl ied by y> and the second by x, the r ight -hand m e m -
bers become identical. Subtract ion gives 
xy — yx = 0. 
T h e three equations (16) can be combined by pairs in three different 
ways, as follows: 
yz — zy = 0, 
zx — xz = 0, (18) 
xy — yx = 0. 
These equations may be integrated, yielding 
yz — zy = cx, 
zx — xz — c2, (19) 
xy — yx = c3 . 
These are the three integrals of area, cly c2, cs being arbitrary constants. 
I t is seen by subst i tut ion of (19) that 
cxx + c2y + csz = 0. (20) 
T h i s is the equat ion of a plane passing th rough the origin, which the 
coordinates in the relative orbit mus t satisfy. Hence: T h e relative orbit 
of a mass mb about ma lies in a fixed plane passing th rough ma. T h i s 
plane is the orbital plane. T h e orientation of the orbital plane is defined 
by the ratios between cl9 c2) cs. 
Put , for brevity, 
G = Vcl + c\ + 4 
(17) 
Xb = 
jb 
AREA INTEGRALS 13 
G always to be taken positively. T h e direction cosines of the normal 
to the orbital plane (20) are then 
C\ C2 C3 
G * G ' G ' 
T h e orbital plane is usually defined geometrically by the longitude of the 
ascending node and the inclination. If the motion in the orbit is direct, 
tha t is, counterclockwise viewed from the positive z direction, / is taken 
to be in the first quadran t and the normal is taken to be on the same side 
of the xy plane as the positive z axis. If the motion in the orbit is re t ro-
grade, / is taken in the second quadrant , and the normal is taken on 
the side of the xy plane that contains the negative z axis. Again, viewed 
from this side the motion in the orbit is counterclockwise, bu t viewed 
from the positive z axis the motion is clockwise. 
Z 
F I G . 1. Orientation of an orbital plane with its normal OQ. 
Imagine a sphere (Fig. 1) with center at the origin of coordinates. 
T h e orbital plane will intersect this sphere in a great circle. T h e point 
N is the intersection of this circle with the xy plane, where the body 
crosses to that side of the xy plane that contains the positive z axis. 
Let the normal intersect the sphere in Q. Consider the spherical triangles 
QXN and QYN. T h e cosine formula gives 
^ c i 
cos XQ = = sin Q sin / , 
cos YQ = -Q — — cos Q sin / . 
Also 
/\ c3 
cos ZQ = — — cos / . 
14 ELLIPTIC MOTION 
I t will now be shown that \cly \c2y \cz represent the projections on the 
yz, zx9 and xy planes, respectively, of the area swept over by the radius 
vector in a uni t of t ime. Consider first the motion in the plane of the 
orbit, and let polar coordinates ry ip be used to specify the position of mb 
relative to ma. T h e origin of the angular coordinate may be chosen in 
F I G . 2 . Differential area. 
dA/dt being the area swept over in the uni t of t ime. Now let rectangular 
coordinates in the orbital plane be defined by 
(#) — r cos ipy (y) = r sin ip. 
T h e n 
(x) = — rip sin ip + f cos ip> (y) = rip cos ip -\- f sin ipy 
and it is found that 
(*) C P ) - 0 0 (*) = '¥• ( 2 2 ) 
Let the projection of r on the yz plane be denoted by rly and the projection 
of Aip by Aipx\ similarly, let the projections on the zx plane be r 2 , Aip2y 
and on the xy plane r 3 , Aip3, the angles ipv ip2y 0 3 being counted from the 
yy zy and x axes respectively. I t follows then that the projections of the 
area of the triangle OPPx are, with the omission of the terms containing 
the products ArjAipjy 
any fixed direction in the plane of the orbit . In Fig. 2 , let P be the posi-
tion of mb att ime t and P x that at t ime t -f At. T h e area of the triangle 
OPP1 is then 
\r (r +Ar)AiP = %r2AiP + ^rArAip. 
By taking the limiting value for At —• 0, there is obtained 
( 2 1 ) 
A change to rectangular coordinates may be made with the t ransforma-
tions 
y — rY cos , z — r2 cos </r2 , x = r3 cos </r3 , 
z = ^ sin , x = r 2 sin i/r2 > ^ — r3 s i n 03 > 
which finally give, similarly to (22), 
yz — zy = r 2 ^ , 
zx — xz = r\^i2, (24) 
xy — yx = rlifj. 
3 
T h e sum of the squares of the three equat ions (19) is 
G2 = C2 + c l + c l 
— y2%2 z2y2 _J_ z2£2 _|_ X2Z2 _|_ 2̂̂ ,2 _|_ ^2^2 _ 2j,2'jte — 2ZXZX — Ixyxy 
— (r2 — x2) x2 + (r2 — y2) y2 + (y2 — z2) z2 — Tyzyz — 2zxzx — 2xyxy, 
with the final expression 
G2 = r2 (x2 + y2 + s 2 ) - (** + yy + **) 2 . (25) 
In this form, since 
xx -\- yy -\- zz = rr, 
it is seen that G is independent of the orientation of the coordinate 
system. In the particular case that 7 = 0, G = c3 is twice the area 
swept over by the radius vector in the orbital plane. In view of the 
independence of G of the orientation of the coordinate system this 
must hold generally, and hence 
G = r2i(;. 
AREA INTEGRALS 15 
(23) 
Hence, from (24), (23), and (19), it follows tha t 
T h e projections of the areas swept over by the radius vector in the uni t 
of t ime on the three planes are, therefore, 
16 ELLIPTIC MOTION 
T h e constant of areas for the relative orbit may be represented as a 
vector having the length G and directed along the normal to the orbital 
plane. T h e components cly c2y cs of this vector represent the constants 
of areas for the planes perpendicular to the three coordinate axes. 
8. T h e v i s v i v a i n t e g r a l . T h e latin words vis viva mean living 
force, often translated kinetic energy. Equat ions (16), mult ipl ied suc-
cessively by the factors xy yy z and added, give 
xx + yy + zz • 
T h i s may be wri t ten 
[x2 + y2 + z2) 
U p o n integration this gives 
(26) x2 + y2 + z2 = 2 
C being a constant of integration. T h e left-hand member of (26) is the 
square of the relative velocity. T h e constant C is, therefore, independent 
of the orientation of the coordinate system. Equat ion (26) states that 
the square of the velocity of one body relative to the other is inversely 
proport ional to the distance between them, with the exception of an 
additive constant . 
I t is useful to write this integral in a different form. Put t ing U = ybjry 
Eqs. (16) may be wri t ten 
(27) 
Since U is a function of the coordinates xy yy z only, i.e., not of the 
velocity components xy yy zy while also the independent variable t does 
not occur explicitly in U, we have 
MOTION IN PLANE 1 7 
dt 
Integrat ion gives 
±(*«+^+**) = U + C. (28) 
Th i s may be wri t ten 
i> V2 — U = C. 
For a further discussion of the integral, see Section 15 of this chapter . 
9. M o t i o n in the o r b i t a l p l a n e . Since the motion takes place in 
a plane, it is permissible to introduce a coordinate system x, y, z such 
that the xy plane coincides with the orbital plane. T h e n z = 0, and the 
equations of mot ion become 
Now, int roduce polar coordinates by 
x = r cos ip> y = r sin ijj. 
T h e n 
x2 + y2 = r2 + r2ifj2. 
These equations are a system of the second order, bu t the presence of 
two constants of integration renders them fully equivalent to the system 
(29), which is of the fourth order. 
(29) 
with 
x — y = 
r2 = x2 + y2. 
Consequently, the integral of area and the vis viva integral may be 
writ ten 
rV = G, 
(30) 
from which it follows that , if the three equat ions (27) are mult ipl ied 
successively by x, y, z and added, 
dU 
x x + yy + zz = 
18 ELLIPTIC MOTION 
T h e variable I/J occurs th rough its derivative only. Elimination of 
if) yields 
or 
(31) 
T h u s (31) may be wri t ten 
= - V ^ f . (33) 
gives 
Th i s equation will be useful later on. In order to obtain the equation 
for r in te rms of the angular coordinate ipy it is convenient to eliminate 
t between (31) and the first equation of (30). T h e result is 
dr dr dib 
dt di/j dt 
dr G 
difj r2 
(32) 
T h e expression unde r the radical may be expressed as 
Hence, if 
(32) becomes 
One further transformation, 
or 
MOTION IN PLANE 19 
or 
Th i s may be integrated: 
I/J = arc cos a + y , (34) 
y being a constant of integration. T h e expression (34) is equivalent to 
a = cos (if; — y ) . (35) 
But 
Hence, if this expression for o is in t roduced into (35) there results 
(36) 
Th i s may be compared with the equation for r in polar coordinates if 
the orbit is an ellipse with the origin at the focus. T h e n 
(37) 
co being the longitude of the perihelion, a the semi-major axis, and e the 
eccentricity. 
Wi th one slight further modification (36) becomes 
1 cos (0 - y ) . (38) 
Comparison of (37) and (38) then gives 
y = co, 
h 1 - e1. 
(39) 
20 ELLIPTIC MOTION 
G = y W l - e2) 
(40) 
= V£p, 
where p is the parameter , 
p = -
Subst i tut ion of this expression for G into the last equat ion of (39) gives 
C = ~ T a - <41> 
With the aid of the new constants of integration a, e> the integrals (30) 
may be writ ten 
r2ifj = VJip = v W l — e1), 
T h e area of the entire ellipse is -nab, where b is the semi-minor axis, or 
b == a V l — e2. 
Let the period of revolution in the ellipse be P. T h e mean motion n is 
defined as the mean rate at which the angle I/J increases dur ing a per iod; 
f2 + f 2 iA 2 — \^ 
10. Kep ler ' s th i rd l a w . Since G = 2dA/dt, the area swept over 
by the radius vector in the uni t of t ime is given by 
(42) 
n 
Consequently, the areal velocity is 
Th i s mus t equal the r ight -hand member of (42), from which 
na2 = V / ^ > 
F r o m these results it follows that the value of the integration constant 
G in terms of the common constants a, e is 
ECCENTRIC ANOMALY 21 
or, with /x = n2az, 
ndt = a—v* 
{1 + ecos(0 - c o ) ) 2 ' 
the integration of which would furnish f as a function of I/J — to, from 
which I/J as a function of t could be obtained. Al though this procedure is 
possible, in order to accomplish our purpose it is bet ter to re turn to 
Eq. (31) which, with the aid of (40) and (41) may be wri t ten 
lia(l -e2)+2^r 
With /x = a 3 « 2 this becomes 
V - a2 + a2e2 + 2ar - r2 
'a2e2 - (r - a) 2 , 
V ^ ( l — * 2) = 
(44) 
if / , the t rue anomaly, is in t roduced here by 
/ = tfj - d > . 
W e have not yet obtained an expression giving I/J or / as a function of t. 
A direct attack upon this problem would be the subst i tut ion of (37) 
into the first equat ion of (30), giving 
or 
/x = nW. (43) 
Thi s is the analytical expression for Kepler ' s th i rd law. 
11. T h e e c c e n t r i c a n o m a l y . So far, only three of the four 
constants required for the complete integration of the Eqs . (29) have 
been obtained. These three are ei ther 
w, C, G, 
or 
to, a, e. 
T h e equation for the orbit has been found in the form 
T h e integration of this expression is elementary. Pu t 
r — a — — ae cos u, (45) 
u being an auxiliary variable named the eccentric anomaly. W e have 
dr = + ae sin u du (46) 
and 
a2e2 — (r — a)2 — a2e2 sin 2 u. 
Hence, the result of the transformation is 
n dt = (1 — e cos u) du. 
Thi s furnishes upon integration 
u — e sin u — n (t — T), (47) 
T being a constant of integration. 
12. T h e m e a n a n o m a l y . F r o m Equat ion (44) for r in te rms of 
the t rue anomaly, it follows that at perihelion 
/ = 0 , r = a(l-e)\ 
at aphelion 
/ = 7T, r = a (1 + e). 
F rom (45) it follows also that 
for u = 0, r — a (1 — e); 
for u = TT, r = a (1 + e). 
Hence the angles w and / are equal for 
u = f = 0 and w = f = TT. 
In (47) let 7 1 correspond to the instant of perihelion passage, and put 
n(t-T) = I. (48) 
Th i s defines the mean anomaly /, which by definition is zero at perihelion. 
22 ELLIPTIC MOTION 
or 
na dt = 
POSITION IN PLANE 23 
For u = 77 it follows that / = TT.Hence all three anomalies are zero 
at perihelion and equal 77 at aphelion. Equat ion (47) may now be wri t ten 
u — e sin u = /, (49) 
which is Kepler ' s equat ion. 
T h e constant T occurr ing in (47) is the fourth constant of integration 
required to define the motion in the orbital plane. In this form it would 
be necessary to give the t ime of one perihelion passage. T h i s da tum, 
together with the values of the constants of integration a, e, & and the 
parameter / x , would define the motion in the orbit for any t ime. T h i s 
form is frequently used for cometary orbits . For planetary orbits it is 
more common to use as the fourth constant of integration the value 
of / at the zero epoch. Let this epoch be t = t0. T h e n , if / = l0 for 
t = t0, the expression for the mean anomaly is 
l=l0 + n(t- g , (50) 
n being given by (43) as a function of a and / x . 
13. F o r m u l a s for o b t a i n i n g the p o s i t i o n in the o r b i t a l p l a n e . 
In order to compute the position in the orbit at any t ime, it is necessary 
to solve u from Kepler ' s equation, which is t ranscendental in u. After 
u has been found, the radius vector may be obtained from (45), 
r = a (1 — e cos u). (51) 
In order to obtain / it is useful first to obtain r c o s / from (44), which 
yields 
r cos / = 
cos / = 
T h e n it is easily seen that 
1 + cos / : 
1 — cos / = 
a (cos u — e), 
24 ELLIPTIC MOTION 
These may be wri t ten 
T h e expressions (51) and (52) thus furnish the radius vector and the 
t rue anomaly if the eccentric anomaly u is known. T h e r e is never any 
question concerning the quadran t of / . T h e quadran t of f/2 is evidently 
the same as that of u/2 since u and / go th rough 0 and IT s imultaneously. 
Formula (52) is particularly useful for logarithmic calculation. T h e 
factor (1 -f e)* (1 — e)~* rnay be simplified by introducing cp by 
e = sin 99. 
For elliptic orbits, therefore, 
°<<P<2-
T h e n , if <p' = (TT/2) — cp, 
1 + e I + sing? 
= tan 
Hence 
For calculations with calculating machines rectangular coordinates 
are frequently more convenient to use than polar coordinates. In order 
to obtain the desired expression, introduce a coordinate system xy lying 
in the orbital plane with the x axis in the direction of the perihelion. 
T h e quotient of these gives 
tan^ (52) 
V 1 — e V 1 — sin 9 
= cot = cot ( 
(53) 
MOTION ABOUT CENTER OF MASS 25 
T h e n , in t e rms of the eccentric anomaly, 
x = r c o s / = a (cos u — e), (54) 
y — r sin / = a V1 — e2 sin uy (55) 
the former of which was derived above. T h e latter is found at once from 
r2 — r2 c o s 2 / = a2 (1 — 2e cos u + e2 cos 2 u — cos 2 w + 2e cos w — e2)y 
r2 sin 2 / = a 2 (1 — e2) sin 2 w. 
T h e square root of the two members of this equat ion leads to no ambi -
guity of sign because 
0 < u < 77 requires 0 < / < TT, 
7T < U < 2TT requires n < / < 27r. 
14. M o t i o n a b o u t the c e n t e r of m a s s . I t is desirable at this 
stage to re turn to some of the equations that were obtained before those 
for the relative motion (16) upon which the discussion in Sections 7 
th rough 13 is based. I t was remarked in section 5 that the integration 
of the two-body problem might be based upon either of Eqs . (14) or (15). 
Compar ison of these equat ions with those of (16) shows tha t their 
integration should lead to the same form of solution as tha t obtained 
by the integration of (16). T h e essential difference is that /x should be 
replaced by /xa for (14) and by FIB for (15). 
F r o m (17) it appears that in order to express xa, ya, zay it is necessary to 
replace a in the expression for x> y> z by 
aa = H ; a> 
ma + mb 
and to change & to 6J + ?r. I n order to express xbi ybi zb) it is necessary 
to replace a in the expression for xy yy zy by 
a = _j a 
ma + rnb 
leaving all the other constants of integration unchanged. I n particular, 
n remains unchanged. T h i s should give 
n* al = H>a-
n2 d\ = pb 
which are in agreement with the values for / x , / x a , / x b in t roduced. 
26 ELLIPTIC MOTION 
with similar expressions for the variables rj and £. Hence, if 
U* = ^ » , (56) 
Eqs . (2) and (3) may be wri t ten 
™a L = 
dU* 
Mb I& -
dm 
Mb I& -
^b 
a t / * e m 
mbr)b = 
dVb 
dU* 
nib L = 
dU* 
Ha 9 
nib L = 
Kb 
m a Va = -^r> m b Vb = -j^r' (57) 
™a L 
I t is impor tan t to note that [/* is a function of the six dependent 
variables £ a , rjaj £a, £b, rjb, £ 6 , i.e., C7* is not a function of the velocity 
components and the independent variable, ty is not explicitly present 
in the expression for U*. As a consequence of these propert ies , the 
following relation will hold: 
• (58) 
By mult iplying the six equations (57) successively by £ a , rjay £a, £6, 
£ 6, and adding, the r ight -hand member of the sum will equal the r ight-
hand member of (58). Hence , the result may be wri t ten 
™A (IJA + VaVa + LATA) + m b (£Jb + VbVb + TBTB) 
Thi s equation may be integrated, giving 
*« (t + Vl + O + \mb + v l + (I) =U* + C*, (59) 
15. T h e e n e r g y i n t e g r a l . We now re turn to Eqs . (2) and (3), the 
original equations for the problem of two bodies, with the purpose of 
obtaining the vis viva integral and the integrals of area for these equa-
tions. Since in these equat ions 
r2 = (ib - L ) 2 + (Vb - Va)2 + (£* - U 2 , 
it follows that 
POTENTIAL ENERGY 
C* being a constant of integration. Wri t ing 
27 
et + vi + ti = vt2> 
the kinetic energy of the system, 7 1*, may be wri t ten 
Hence the integral (59) may be wri t ten 
T* — U* = C*. (60) 
T h i s is the symbolic form of the energy equation, stating that the sum 
of the kinetic energy, T*, and the potential energy, — [/*, is a constant . 
T h e potential energy is counted from zero for infinite separation of 
the two masses. 
16. T h e p o t e n t i a l e n e r g y . In order to verify the preceding 
statement , consider the amount of work that mus t be done in order to 
increase the distance between two masses, ma and m&, from r 7 to r n 
(rn > ri)- T h e work done if the distance is increased by an amoun t dr 
is given by 
D W DR. 
r2 
U p o n integration this becomes 
which equals £/* for r = r 7 . Hence the potential energy is zero for 
r = oo and is negative for a finite distance between the two bodies. 
T h e function C7* is also called the force function of the system, 
because the partial derivatives of [/* with respect to the coordinates 
equal the components of the forces acting on the masses. 
- k2mamb 
For rn — oo the work done is 
28 ELLIPTIC MOTION 
A third name for £/* is the potential . A word of caution on the use 
of the word potential is in order, since it is occasionally defined with 
the opposite sign, especially in Ge rman works on physics. 
17. C h a n g e to a c o o r d i n a t e s y s t e m w i t h the o r i g i n a t the 
c e n t e r of m a s s . T h e energy integral obtained applies to any arbi-
trary coordinate system in which the Newtonian laws of mechanics are 
valid. Suppose now that the origin be placed at the center of mass and 
that therefore Eqs . (9) and (10) are used, with 
x a — £a £g> J a ~ Va ên z a = Ca tg> 
Xb ~ £b ig> Jb = Vb Vo' Zb ~ £g> 
and 
I t follows that 
If now 
we obtain 
[<h* + A ) , 
- (oc2t + /?2), Vg 
lg ( a 3 * + AO-
£a — xa 
Va = ya 
£b — xb ' 
i b = yb-
U = zb • £ a — z a 
vi = *l + yl + K> 
y*2 — J / 2 j 
a a (*lxa + a 2 J a + a 3 * a ) 
- ( a ^ & + oc2yb + OC3ZB) -f 
b 
"X2 K + AL + a 3 ) > 
_ ( a 2 + a 2 + a 2 ) < 
ORIGIN AT CENTER OF MASS 29 
Hence, 
T*= T** + l K + m & ) F 2 . (62) 
T h e energy integral for the system having the origin at the center 
of mass will be 
T** - U* = C**, (63) 
since £/* is independent of the origin. Compar ison of T* and 7 1 ** 
by (61) or (62) gives 
C* = C * * + i K + m 6 ) F 2 , 
showing that the total energy in the system £, 17, £ differs from tha t in 
the system having the origin in the center of mass by one half thesum 
of the masses mult ipl ied by the square of the velocity of the center of 
mass. 
I t is of interest to compare the integral (63) with that obtained for 
the relative motion (26). T h e relations (17) give, upon differentiation, 
Hence , 
T h e n , if 
T** = imaVl + lmtVl, 
it follows that 
r = ^ + 2 K k ) K H H 4 (61) 
the second par t of the expressions for F * 2 , F * 2 canceling each other 
because 
maxa + mbxb = 0, maya + mbyb = 0, maza + mbzb = 0. 
If Vg is the velocity of the center of mass in the RJY £ coordinate system, 
we have 
30 ELLIPTIC MOTION 
If this is in t roducted into (63) the result is 
while (26) gave 
Th i s , then, is the total energy in the motion in a coordinate system the 
origin of which coincides with the center of mass. 
c . 
I t is seen that the two expressions are identical, provided that 
Since the integration of the equations of the relative mot ion gave 
it follows that 
18. T h e i n t e g r a l s of a r e a . If the first of Eqs . (2) is mult iplied 
by — rja and the second by +£ f f l , .we obtain, by addition, 
= k2mamh (64) 
Similarly, if the first of Eqs . (3) is mult ipl ied by — rjb and the second 
by we obtain, by addit ion, 
= k2mamb (65) 
T h e r ight -hand members of the two equat ions, (64) and (65), are 
opposite. Hence, by addition, 
MaiLVa — VaL) + ^bUbVb ~ Vbtb) = 0. 
REFERENCE TO ECLIPTIC 31 
T h i s equat ion may be integrated: 
MaiLVa — yJa) + ^bi^blb ~ Vb€b) = Cz • (66) 
By the same procedure , two similar integrals are obtained: 
Jla) + MbiVbtb - Lib) = C* 1 > 
(67) 
(68) 
In this form the integrals (66), (67), and (68) express the constancy 
of the moment s of m o m e n t u m about the three coordinate axes. Again, 
it may be emphasized that these integrals hold in an arbitrary coordinate 
system in which the Newtonian laws of mot ion are valid. I t is left to 
the reader to examine the constants CFY CGY C£ in a coordinate system 
having its origin in the center of mass, and also to compare the integrals 
that apply in this case with the integrals of area (19) obtained in the 
problem of relative motion. 
T h e formulas derived in the preceding sections permi t the calculation 
of the position in the relative orbit at any t ime in a coordinate system 
the xy plane of which coincides with the plane of the orbit , the x axis 
being directed toward the perihelion. I t is assumed that the three 
elements ay e, l0 are known, and that in addit ion the parameter / x = 
k2{ma + mh) is given. 
T h e requi rements of practical calculation go beyond this . W e mus t 
develop a procedure for obtaining the position in a coordinate system 
having the same origin, may as the xy system, bu t an arbi trary orientation. 
T w o part icular cases occur frequently in problems of planetary motion. 
19. C o o r d i n a t e s r e f e r r e d to the e c l i p t i c . Let the coordinates 
be designated by # ( e ) , y i e \ z(e), and let the x{e)yie) plane coincide with 
the ecliptic for a given date, say 1950.0, and the xie) axis be directed 
toward the mean vernal equinox of that same date. 
For convenience of later applications we shall consider the more 
general case in which the coordinate z is not necessarily zero. T h e change 
from x, y, zy to x(e), y{e\ z l e ) can be accomplished by a linear t ransforma-
tion 
^(E) = pie)x + Qle)y + Rle)g 
X ~ X X 
y ,(E) = P I E ) * + Q(e)y + R(e)S 
Y 1 & Y J 1 Y ' 
(69) 
•(E) = p(e)x + Q(e)y _|_ P ( E ) 5 . 
32 ELLIPTIC MOTION 
T h e nine vectorial orbital constants for the ecliptic are to be found as 
functions of the three elements, / , Q> <JJ that define the orientation of 
the xyz system relative to that of the xie)y(e)zie) system. T h e nine con-
stants are the nine direction cosines of the axes x> y, z relative to the 
axes in the ecliptic sys tem; for example, the three direction cosines of 
the x axis are Px™, Py
{e\ P 2
( e ) . 
F I G . 3. Direction cosines. 
F r o m this relationship, it follows at once that the relation may be 
inverted: 
X = P^x(e) + P<eye) _|_ Pl&zie), 
y = Qfx™ + Q{
y
e)yie) + 0 < e , * ( e \ 
X 1 Y - / 1 z 7 
and that the following relations will hold, 
P(e)2 i Q(e)2 i P(e)2 _ 1 
X ' X^X 1 X • 
p(e)2 + p(e)2 + p(e)2 = 1 ? 
P(e)Q(e) _L p(e)Q(e) i p(e)Q(e) — Q 
a; ^ 03 ' 1/ ^ 2 / 1 z ~ z ' 
P(e)P(e) i 0 ( e ) 0 ( e ) 4 - Rie)Rie)
 = 0 
(70) 
(71) 
as well as all the other similar relations that may be writ ten down by 
a cyclical change of x, y, z, P , Q, R. 
T h e direction cosines may be obtained by spherical t r igonometry by 
considering the intersections of the coordinate planes on a uni t sphere 
with mn as center. 
REFERENCE TO EQUATOR 33 
P(e) _ 
X 
cos CO cos Q — sin co sin Q cos / , 
n(e) = 
X- X 
— sin co cos Q — cos co sin Q cos / , 
n e ) = sin Q sin / , 
pie) — 
y 
cos co sin Q + sin co cos Q cos / , 
Qie) = 
X-> y — sin co sin .Q + cos co cos Q cos / , 
V 
— cos Q sin / , 
p(e) _ 
z 
sin co sin / , 
QT = cos co sin / , 
i ? « e . = cos / . 
20 . C o o r d i n a t e s r e f e r r e d to the e q u a t o r . Le t the coordinates 
be designated by xy yy z and let the xy plane coincide with the mean 
equator for a given date, and let the x axis be directed toward the mean 
vernal equinox of the same date. T h e s tandard equator and equinox 
of 1950.0 are used extensively for calculations of orbits of minor planets 
and comets. 
If the orbital elements Q'y CD' are referred directly to the s tandard 
equator and equinox, the vectorial orbital constants for the equator , 
(72) 
Suppose that we wish to obtain the direction cosines of the xy yy z axes 
relative to the x{e) axis. 
I n the spherical triangle xie)Nxy x{e)N = Qy Nx = co, <£ xie)Nx = 
180° — / . T h e cosine formula gives 
A 
pie) — c o s x(e)x — c o s ^ c o s Q — s i n w $'mQ cos / . 
x 
I n the spherical tr iangle xie)Nyy x{e)N = Q, Ny = 90° + co, # ( e ) iVy = 
180° — / , and the cosine formula gives 
Q[x = c o s x ( e ) y = — s u i 6 0 c o s ^ — c o s 6 0 s ^ n * ^ c o s I* 
In the spherical triangle x{e)Nzy x{e)N = Qy Nz = 90°, <£ * ( e ) JV2 = 
90° — / , the cosine formula gives 
== cos # ( e ) # — sin 42 sin / . 
X 
T h e remaining six direction cosines may be obtained similarly. T h e 
complete set of nine vectorial orbital constants for the ecliptic is 
34 ELLIPTIC MOTION 
Px, Qx, Rx> etc., may be obtained from the formula for Px
e\ Qx
e), Rx
e\ 
etc., by merely subst i tut ing / ' for / , Q' for Q, and a/ for a>. I t is common, 
however, to give the elements / , £?, OJ referred to the ecliptic. If this is 
the ecliptic of the same date as that of the equator and equinox of the 
coordinate system x9 y, z, it is evident that the change from xie\ yie), z{e) 
F I G . 4 . Ecliptic and equatorial coordinates. 
z(e) 
F I G . 5. Orbital and ecliptic coordinates. 
to x9 y, z may be obtained by a rotation of the x{e)yie)zie) system about 
the x{e) axis, clockwise by the angle e, the obliquity of the ecliptic. Th i s 
transformation is 
x — x^^ 
y == ye) c o s e (̂e) s j n e > 
z = yie) sin e + z(e) cos e. 
REFERENCE TO EQUATOR 35 
p x 
= n e ) . 
= QR> 
Rx = ^ e ) . 
Py = P ( e ) cos e 
y 
— P ( e ) sin e, 
Qy = Q < e ) cos e - Qif s i n € j 
Ry = R(e) COS 6 
y 
— P ( e ) sin e, 
2 7 
= P ( e ) sin e 
y 
+ P< e ) COS 6, 
Qz = sin e + COS €, 
= # l e > sin e 
1/ 
+ P ( e ) COS €. 
' 2 
T h e complete formulas are, therefore, 
p« cos co cos Q — sin co sin Q cos 7, 
= — sin co cos Q — cos co sin Q cos 7, 
= sin Q sin 7, 
Py = (cos CD sin Q + sin co cos Q cos 7) cos e — sin co sin 7 sin e, 
Qy = (— sin co sin Q + cos co cos Q cos 7) cos € — cos co sin 7 sin e, 
Ry = — cos .Q sin 7 cos e — cos 7 sin e, 
PZ = (cos co SIN Q + sin to cos Q cos 7) sin € + sin co sin 7 cos e, 
Q* (— sin co sin Q + cos co cos Q cos 7) sin e + cos co sin 7 cos e, 
= — cos sin 7 sin e + cos 7 cos e. 
(73) 
Since in an elliptic orbit z = 0, the actual formulas needed for the 
calculation of a position are 
x = Pjc+ Qxy, 
y = Pyx+Qyy, (74) 
= Pzx + 
I t is not u n c o m m o n to introduce the eccentric anomaly directly into 
these expressions. If we define 
Ax = aPx, Bx = a VT^~e*Qx, 
Av = aPv, Bv = a V T ^ Q V , (75) 
A, = aPz, Bz = a W ^ Q Z , 
Consequently, 
36 ELLIPTIC MOTION 
the rectangular equatorial coordinates may also be obtained from 
x = Ax (cos u — e) + Bx sin w, 
y — Ay (cos u — e) + By sin uy (76) 
z — Az (cos w — e) + £ 3 sin u. 
Since the equatorial vectorial constants are direction cosines, the 
same relations (71) hold among these constants as among the ecliptic 
constants . Among the A's and B ' s the corresponding relations are 
Al + A\ + A\ = a\ 
Bl + Bl + Bl=a*{\-e% (77) 
AXBX + AvBy + ,4 A = 0. 
I t is sometimes impor tant to have expressions similar to (76) for the 
derivatives. T h i s requires the expression of x, y in te rms of the eccentric 
anomaly. Wi th the aid of 
an n 
u = — = --, 
r 1 — e cos u 
it follows that 
a2n . an sin a 
x = — a sin uu = sin u = — 
r 1 — e cos u ' 
y = a \/\ — e2 cos uu — \ / l — e2 cos w = 
an Vl — ^2 cos w 
1 — e cos w 
Hence 
x = ^ (— Ax sin w + ^ cos w), 
j> == ^ (— Ay sin w + By cos w), (78) 
z = — (— sin w + cos u). 
I t is hardly necessary to emphasize that n should be expressed in radians 
per uni t of t ime , and that the uni t of t ime used in expressing the 
derivatives should be the same as that used in n. 
MATRICES 37 
2 1 . I n t r o d u c t i o n of m a t r i c e s . T h e expressions for the vectorial 
orbital constants in t e rms of the elements show so m u c h regularity in 
s t ructure that it is not difficult to devise an effective scheme for the 
calculation. Nevertheless , there are advantages in a different procedure 
tha t avoids the use of spherical t r igonometry altogether. T h i s procedure 
involves the use of matrices, in t roduced into the mathematical l i terature 
by Cayley. Matr ix algebra is an impor tan t b ranch of mathemat ics . W e 
shall not treat it as such, bu t shall give here only so m u c h as is necessary 
for the convenient t ransformation of coordinates. 
For our purpose , a matr ix is to be considered as a rectangular array 
of number s (denoted here by symbols) which is operated upon by the 
use of simple rules. 
Le t two matrices be given: 
< < 4 *I * I 
a = < < K K 
< < < K K 
the superscr ipt of any e lement denot ing the row, the subscript denot ing 
the column. 
T h e produc t ab is defined by 
C = ab = 
a\b\ + a\b\ + aft a\b\ + a\b\ + a\b\ a\b\ + «ft + 
a\b\ + ^bl + albl a\b\ + a\bl + alb\ aft + aft + a® 
a\b\+a\b\ + alb\ a\b\ + a\b\ + a\b\+a\bl + albl 
tha t is, if the general e lement of a is denoted by a[, that of b by b{9 
of C by c[9 the general e lement of ab is 
(79) 
T h e essential feature is that the rows of a are mult ipl ied by the columns 
of b. 
I t is not necessary that two matrices to be mult ipl ied each have the 
same n u m b e r of rows and co lumns , bu t it is necessary that a has as 
many columns as b has rows. T h u s , 
a\ a\ 
b\ 
a\b\ + a\b\ 
2 2 1 
b\ 
= a\b\ + a\b\ 
< < 1 a\b\ + a\b\ 
38 ELLIPTIC MOTION 
the product ab having invariably as many rows as a has, and as many 
columns as b has. If the n u m b e r of columns of a is different from the 
n u m b e r of rows of b> the p roduc t ab is not defined. So it happens easily 
that the product ab may be defined, while ba is not. Even when ab and 
ba are bo th defined, they are generally different from each o ther ; tha t 
is, multiplication of matrices is in general not commutat ive . 
If the rows and columns of a matr ix are interchanged, the new matr ix 
is said to be the transpose of the old one. T h e transpose of a will be 
denoted by a'. T h u s , if 
a = 
1 1 
a2 
< -\ 
< -\ 
then 
< < < 
< < < 
Although multiplication of matrices is not in general commutat ive , 
it is associative, as in ordinary algebra. T h a t is, 
abc = (ab)c = a(bc). 
22 . C h a n g e of o r d e r in a p r o d u c t of m a t r i c e s . I n the applica-
tions to follow it will often be convenient to change the order in a 
p roduc t of matrices. I t may be verified by means of the rule for mul t i -
plication that 
ab = (b'aj, (80) 
abc = (c'b'a'Y = (b'a')'c = a(c'b')' = [(bc)'a']' = [c'(ab)r\\ (81) 
By application of these rules, matrices may be mult ipl ied in any order 
that proves convenient, provided of course that the number s of rows 
and columns are such as to make the multiplication possible. 
23 . R o t a t i o n m a t r i c e s . Let two Cartesian systems x, y, z and 
x\ y\ z\ having the same origin be given, and suppose the direction 
cosines are known. T h e n 
x' = P'xx+Q'J + R'xz, 
yf = P'yx+0'yy + R'yz> (82) 
z' = P'zx+Q'J + R'zz. 
ROTATION MATRICES 39 
Assume that the system x\y\ z' is rotated counterclockwise about 
the x axis through an angle a, as seen from the positive end of the x' 
axis. T h e new coordinates being x'\ y", z", we may write 
x" = P'±x+Q'£ + R'fr 
y" = P'y'x+Q'y'y + R'y'z, (83) 
z" = P'z'x+Qf
z'y + R'z'z. 
T h e relations between the coordinates x , y', z' and x'\ y'\ z" are 
y" = y' cos OL + z' sin a, 
z" = — y' sin a + z cos a. 
If this is subst i tu ted into the left-hand member of (83), there results 
with the aid of (82), 
P" = P', 
X X' 
P'' = P' cos OL + P' sin a, 
V 
P" P' sin a 4 - P ' cos a, 
= Q'y C 0 S a + S i n a> 
0 * = - Q ; s i n a +Q'zcos a, 
i ? " = R' cos a + R! sin a, 
2/ ^ ' Z ' 
i?' sin a + i?' cos a. 
Thi s transformation may be performed by the multiplication of two 
matrices: 
P" P" P" 
x y z 
Qx Qy Qz R" R" R" 
X V Z 
P' P' 
x y 
PL 
Q' Q' Q' 
x z~y *~ z 
R' R' R' 
1 0 0 
0 cos oc — sin OL 
0 sin OL cos OL 
40 ELLIPTIC MOTION 
Also, the transformation from x', y\ z to x'\ y'\ z" is given by 
x y 
1 0 0 
0 cos a — sin a 
0 sin a cos a 
or equivalently, 
1 0 0 x' 
/ ' = 0 cos a sin a / 
0 — sin a cos a z' 
T h e first of these alternative forms is the one we shall use in what 
follows. With this unders tanding, that the coordinates to be transformed 
are wri t ten to the left of the matr ix effecting the transformation, we 
shall denote the latter by p(oc): 
p(oc) = 
1 0 0 
0 cos a — sin a 
0 sin a cos a 
which corresponds to a positive (counterclockwise as seen from the 
positive end of the axis) rotation about the x' axis th rough an angle a. 
Similarly, a positive rotation about the y' axis corresponds to the 
transformation 
x = x cos a — z sin a, 
y" = y\ 
z" = x' sin a + z cos a, 
or 
x" y" z" 
cos a 0 sin a 
0 1 0 
— sin a 0 cos a 
and we put 
q(oc) = 
cos OL 0 sin a 
0 1 0 
— sin a 0 cos a 
II * ' / * ' ii 
| x" y" z' z' 
GENERAL ROTATIONS 41 
Again, a positive rotation about the z' axis corresponds to the t rans-
formation 
x" = x' cos a + y' sin a, 
y" = — sin a + y' cos a, 
z" = z\ 
or 
# y z x y z 
cos a — sin a 0 
sin a cos a 0 
0 0 1 
and we pu t 
»•(<*) = 
cos a — sin a 0 
sin a cos a 0 
0 0 1 
I t is seen that 
P ( ~ <*) = />'(«)> = tf'(°0> r (— a) = r ' (a). 
24. G e n e r a l r o t a t i o n s of c o o r d i n a t e s y s t e m s . Any rotation 
whatever of a coordinate system may be decomposed into a succession 
of rotations about the various axes. Consider, for example, the t rans-
formation of coordinates in the system xy y, z to the equatorial system 
Xy y, z tha t was treated in Section 20. T h i s t ransformation may be 
accomplished as follows: 
1. A rotation — co about the z axis, r (— co), 
2. A rotation — I about the x' axis, p (— 7), 
3. A rotation — Q about the z" axis, R (— Q)> 
4. A rotation — e about the # ( e ) axis, p (— e). 
T h e transformation is then 
\\xyz\\ = 
\xyz\\ 
cos co sin co 0 
-sin co cos co 0 
0 0 1 
1 0 0 
0 cos I sin I 
0 -sin I cos / 
cos Q sin Q 0 
-sin Q cos .Q 0 
0 0 1 
1 0 0 
0 cos € sin e 
0 -sin e cos e 
42 ELLIPTIC MOTION 
In a numerical application, the last four matrices would be multiplied 
first, yielding the vectorial constants in matrixform, which would then 
be used as required. T h u s , 
P P P 
Qx Qy Qz 
Rx Ry Rz 
= r(-u>)p(-I)r(-Q)p(-e), 
x y z \\ = \\ x y z 
P P P 
x x *• y z 
Qx Qy Qz Rx Ry Rz 
T h e computat ion of a search ephemeris is given as an example of the 
use of these rotation matrices. 
Computation of an Ephemeris 
I t is required to calculate a search ephemeris for Periodic Comet 
Harr ington, using the orbital elements given in I .A.U. Circular N o . 1713, 
as follows: 
T 1960 June 28.8327 co 232?8391 
e 0.559273 1211 9?1327 
a 3.590373 / 8?6838 
n = ka 3/: 
0.01720209895 
(3.590373) 3 / 2 
n° = 57?29577951 x n = 0.°1448754 
e° = 57P29577951 x e = 32?0440 
Equinox 1950.0 
= 0.002528553 
a V l - e* = 3.590373 x 0.828984 = 2.976360 
p p 
x x x y 
Q* Qy Qz 
Rx Ry Rz 
cos co sin co 0 
-sin co cos co 0 
0 0 1 
1 0 0 
0 cos I sin I 
0 -sin / c o s / 
cos Q sin Q 0 
-sin Q cos Q 0 
0 0 1 
1 0 0 
0 cos € sin e 
0 -sin e cos € 
GENERAL ROTATIONS 43 
-0 .60406 -0 .79694 0 
+0.79694 -0 .60406 0 
0 0 
x 
1 
1 0 0 
0 +0.98854 +0.15098 
0 -0 .15098 +0.98854 
-0.48683 +0.87349 0 
-0.87349 -0 .48683 0 
0 
-0 .60406 -0 .78781 -0 .12032 
+0.79604 -0 .59714 -0 .09120 
0 -0 .15098 +0.98854 
0 1 
1 0 0 
0 +0.91744 +0.39788 
0 -0 .39788 +0.91744 
-0 .48683 +0.87349 0 
-0 .87349 -0 .48683 0 
0 
1 
0 1 
0 0 
+0.98222 -0.14411 -0 .12032 
+0.13362 +0.93682 -0 .09120 
+0.13188 +0.07350 +0.98854 
+0.98222 -0 .08434 -0 .16772 
+0.13362 +0.94163 +0.30897 
+0.11388 -0 .32589 +0.93617 
Check: 
1 0 
0 +0.91744 +0.39788 
0 -0 .39788 +0.91744 
0 
0 +0.91744 +0.39788 
0 -0 .39788 +0.91744 
'\\x,y, * l l = I I * , y, 0|| 
ZP2 = ZQ* = ZR2 = 1 
ZPQ = ZQR = ZRP = 0 
P P P 
Qx Qy Q, 
R~ R„ R~ 
= \\a (cos u — e), a\/l — e2 sin w, 0|| 
P P P 
x x y M z 
Qx Qy Q, 
Rx Ry Rz 
44 
|cos u — e, sin u, 0| | 
= ||cos u — e, sin u, 0| | 
ELLIPTIC MOTION 
aP„ aP„ aP, 
a VT^QX aVV^Qy aVY^Q, 
+3.52653 -0 .30280 -0 .60220 
+0.39768 +2.80266 +0.91960 
-q, £|| = \\p cos a cos 8, p sin a cos 8, p sin 8|| 
= \\x,y,*\\ + \\X,Y,Z\\, 
where X, Y, Z, the geocentric equatorial coordinates of the sun, may 
be taken from the Astronomical Ephemeris. Then 
tan a — VIE 
tan 8 = 
P = Vie + V2 + H 
r = V(x2 +y2 + z2) 
= a (1 — e cos u). 
t 1960 June 5.0 June 15.0 June 25.0 July 5.0 July 15.0 July 25.0 
t - T - 2 3 . 8 3 3 - 1 3 . 8 3 3 - 3 . 8 3 3 + 6.167 + 16.167 + 26.167 
I — 3.4528 - 2.0041 - 0 . 5 5 5 3 + 0 . 8 9 3 4 + 2.3422 + 3.7910 
u — 7.803 - 4 .540 - 1 . 2 5 9 + 2.028 + 5.306 + 8.563 
cos u + 0.9907 + 0.9969 4-0.9998 + 0 . 9 9 9 4 + 0.9957 + 0.9889 
cos u — e + 0.4314 + 0.4376 + 0.4405 + 0.4401 + 0.4364 + 0.4296 
sin u - 0.1358 - 0.0792 - 0 . 0 2 2 0 + 0.0354 + 0.0925 + 0.1489 
X + 1.4673 + 1.5117 + 1.5447 + 1.5661 + 1.5758 + 1.5742 
Y — 0.5112 - 0.3545 - 0 . 1 9 5 0 - 0 . 0 3 4 1 + 0.1271 + 0.2872 
z - 0.3847 - 0.3364 - 0 . 2 8 5 5 - 0 . 2 3 2 5 - 0.1777 - 0.1218 
X 0.2764 + 0.1106 - 0 . 0 5 8 4 - 0 . 2 2 5 7 — 0.3866 — 0.5368 
Y 0.8957 + 0.9264 + 0 . 9 3 1 1 + 0.9095 + 0.8625 + 0.7911 
Z 0.3884 + 0.4018 + 0.4038 + 0 . 3 9 4 4 + 0.3740 + 0.3431 
t + 1.7437 + 1.6223 + 1.4863 + 1.3404 + 1.1892 + 1.0374 
+ 0.3845 + 0.5719 + 0 . 7 3 6 1 + 0.8754 + 0.9896 + 1.0783 
i + 0.0037 + 0.0654 + 0.1183 + 0 . 1 6 1 9 + 0.1963 + 0.2213 
a 0 M 9 . m 6 l M 7 . r a 6 l M 5 . m 4 2 M 2 ! n 6 2 * 3 9 . m l 3 h 0 4 . m 4 
8 4 - 0 ° 0 7 4 -2° l l ' + 4°05' + 5°46' + 7°14' + 8°25' 
P 1.7856 1.7214 1.6628 1.6091 1.5595 1.5126 
r 1.6007 1.5887 1.5829 1.5836 1.5909 1.6048 
At perihelion r = q = a(l — e) = 1.582376. 
POLAR COORDINATES 45 
25 . U s e of p o l a r c o o r d i n a t e s . For analytical developments it is 
useful to obtain expressions for xie), y(e), z i e ) in t e rms of the polar 
coordinates in the orbital plane, r a n d / . In these expressions the orienta-
tional e lements co, Q, and I will be retained. W e begin with 
*<e> = p<e> r cos / + Qx
e) r s in/ , 
ye) = pie) r cos / + Q{
y
e) r s in / , 
z(e) = p(e) r c o s / + r s in/ , 
in which the explicit expression (72) for the P ' s and Q's mus t be sub -
st i tuted. I t is then found that 
xie) — (j cos /cos co — r s i n / s i n to) cos Q 
— (r cos / sin co + r s i n / cos co) sin Q cos 7, 
ye) — ( r cos /cos co — r sin /s inco) sin Q 
+ (r cos / sin co + r sin / cos co) cos Q cos 7, 
#<e) = ( r cos / s in co -f- Y sin / c o s co) sin I. 
These may be wri t ten 
#(e) = r cos ( / + co) cos Q — Y sin ( / + co) sin Q cos 7, 
ye) = r cos ( / + co) sin Q + r sin ( / + co) cos £? cos 7, (84) 
(̂e) _ r s i n _|_ ^ sin I. 
In x(e) and y e ) a further combinat ion of t e rms can be made by making 
use of 
cos 2 - + sin 2 - = 1, cos 2 - — sin 2 ^ = cos 7, 
which produces 
x(e) __ r c o s 2 ^ c o s (y -f- ,Q) -)_ r sin 2 ^cos ( / + to — Q), 
ye) = r cos 2 ^ sin ( / + to + Q) — Y sin 2 ^ sin ( / + co — 13), 
#(e) _ R s i n / sin ( / + co). 
I t is frequently useful to express the a rguments in t e rms of 
v = / + co + Qy the true orbital longitude; 
to = co + Qy the longitude of the perihelion; 
Q , the longitude of the ascending node. 
46 ELLIPTIC MOTION 
T h e expressions become 
y 
*<e> 
.(e) 
.(e) r cos 2 - cos v + r sin 2 - cos (v — 2Q), 
r cos 2 ^ sin ^ — r sin 2 ^ sin — 2Q)y 
r sin / sin (v — Q). 
(85) 
T h e a rgument occurr ing in £ ( e ) , / + OJ = v — Q, is called the a rgument 
of the lat i tude. 
If B is the lati tude of the planet above the xie)yie) plane, then 
in agreement with the expression for zie) in formula (85). 
T h e angle v is counted as the sum of the arcs xie)N + NP, a slight 
inconvenience which could be remedied by adopt ing a depar ture point O ' 
in the orbital plane such that ON = x{e)N = Q. T h e use of a depar ture 
point in the orbital plane is common in certain forms of planetary 
theory. However, in such cases the orbital plane is in constant motion, 
and the definition of a depar ture point is not as simple a mat te r as 
equat ing the arcs O'N and x{e)N. I t will be found to depend upon the 
solution of a differential equation. 
26. R e d u c t i o n to the e c l i p t i c . Finally, it may be desired to 
obtain t rue polar coordinates, i.e., the position of the planet may be 
defined by the three coordinates r, V9 By in which V is the arc xie)Q. 
If the inclination is small, it is evident that at all t imes V — v is small. 
Th i s quanti ty is known as the reduct ion to the ecliptic. An analytical 
solution of this problem is obtained from the formula in the spherical 
triangle NPQ, 
z(e) — r s j n ft 
T h e spherical triangle NPQ on the uni t sphere gives 
sin B = sin / sin ( / + co) 
= sin / sin (v — Q), 
tan(V-Q) 
For brevity, pu t V — Q = p, v 
cos / tan (v — Q) 
• Q = q, and pu t 
J3 = 
cos / or 
CALCULATION OF ELEMENTS 47 
T h e equation then has the form 
1 - / 3 
tan/) = Y^Tptm 
which is t reated in chapter II and has the solution 
p - q = — (j8 sin 2q - \ fi2 sin 4q + ^ jS3 sin 6q — ...) 
or, after subst i tut ion of the original quanti t ies for p> q, /?, 
V-v = - tan 2 !- sin 2(v -Q)+\ tan 4 ^s in 4(v - Q) - \ tan 6 ~ sin 6(v -Q) + ... 
2 2 3 2 (86) 
For small values of I this formula for the reduction to the ecliptic 
is rapidly convergent and in many applications only one or two te rms 
of the expansion are required. 
27. C a l c u l a t i o n of the e l e m e n t s f r o m the c o o r d i n a t e s a n d 
v e l o c i t y c o m p o n e n t s at a g i v e n t i m e . In Section 20 the rectangular 
coordinates and velocity components have been obtained from the 
orbital e lements . T h e reverse problem mus t frequently be solved: 
given y, z, x, y, z for a particular epoch, find the elements of the 
orbit. 
In order to simplify the calculations, it is desirable to eliminate FI 
from the formulas. By inspection of Eqs . (16) it is seen that this can 
be accomplished by using, instead of T> a variable R such that 
V / x DT = DR. 
Equat ions (16) then become 
(87) 
and / x cannot appear in any of the resultsderived from these equat ions. 
Since the change from T to R as independent variable is equivalent to 
pu t t ing FX = 1, all the formulas that apply to the relative motion in the 
problem of two bodies derived in Sections 6 th rough 13 apply to Eqs . (87) 
provided the subst i tut ion / x = 1 is made . 
For orbits of asteroids and comets ma represents the mass of the 
sun, which is taken as unity, and mh is pu t equal to zero; consequently 
/ x = k2. 
If the uni t of t ime in which T is expressed is the mean solar day, the 
introduct ion of the variable r amounts to the use of a uni t of K ' 1 mean 
48 ELLIPTIC MOTION 
solar days. T h e numerical value of k is given in Section 31 . If the 
velocity components were obtained with the uni t of t ime w mean solar 
days, these values mus t first be divided by wk or, more generally, by 
w\/Jx. In the following it is assumed that xy y, z have been divided by 
the proper factor, and are therefore expressed in the uni t of t ime 
required to eliminate / x from the formulas. 
T h e problem is conveniently resolved into two parts: 
(a) T h e calculation of the elements a, e, / 0 , which define the position 
in the orbital plane, relative to the perihelion of the orbit . T h e first 
step in the calculation is to compute for the given date, t> 
r2 = x2 + y2 + z2
y 
V2 = x2 + y2 + z2, 
rf = xx + yy + zz. 
T h e following relations are then used: 
- = - - V\ (88) 
a r y 
e sin u = a Kf> (89) 
e cos u = rV2 — 1, (90) 
/ = u — e sin u. (91) 
T h e mean daily motion is found from 
n = Vv<a*>2 (92) 
in the general case, or 
n = ka-*12 (93) 
for minor planets and comets. Finally, 
l0 = l - n ( t - tQ) (94) 
furnishes the mean anomaly at t — t0. 
Equat ion (88) is the vis viva integral (26) with JJL — 1, C = — \i\2a. 
Equat ion (89) follows from (46); (90) results from (45) which may be 
writ ten 
r f cos u = 1 — 
a 
in which r/a is replaced by 
r 
a 
which follows from (83). 
= 2 - rV2, 
file:///i/2a
CALCULATION OF ELEMENTS 49 
(96) 
— Ax 
= x (a/r) cos u — xaz/2 sin w, 
aPy = Ay = y (a/r) cos u — yazl2 sin w, 
aPz = A = z (a/r) cos u — zaz/2 sin w, 
bQx 
= BX = x (a/r) sin u + xa3/2 (cos u 
bQy = By — y ( A L R ) s m u + ya*/2 (cos u -e), 
= BZ = z (a/r) sin u + zazl2 (cos u 
(97) 
with 
b = a cos <f> = a V l — e2. 
These formulas follow from (76) and (78). For example, the first of 
each set gives 
Ax (cos u — e) + Bx sin u = xy 
— Ax sin u -\- Bx cos u = xra1/2
y 
if in the expression for x the factor r/an is replaced by ral12. Solution 
of Axy Bx gives 
Ax (1 — e cos u) = x cos u — xra1/2 sin w, 
Bx (1 — e cos u) = x sin u + tfra1/2 (cos w — e)y 
which may then be b rough t into the form of Eqs . (96), (97). 
Finally, the orbital elements co, Qy I may be computed from the vec-
torial orbital constants by the formulas 
sin I sin co = — Py sin e + Pz cos 6, 
sin I cos co = — Qy sin e + £?z cos e, 
which give 7 and co, and 
sin Q = ( P y cos co — sin co) sec e, 
cos Q = Px cos to — (Jz sin co, 
I t may be remarked that the integral of area (25), which with /x = 1 
becomes 
a (1 - e2) = r2V2 - (rf)2
y (95) 
could be used, with (88), to obtain a and e. T h e disadvantage is that 
this would lead to an evaluation of e from 1 — e2
y which gives a weak 
determinat ion of e if the eccentricity is small. T h e use of (89) and (90) 
for the determinat ion of e and u avoids this loss of significant figures. 
(b) T h e calculation of the orientational elements co, Qy L T h e con-
stants AxyAyyAzy BxyByyBz and subsequent ly the vectorial orbital 
constants Pxy Pyy Pzy Qxy Qyy Qz are obtained from 
50 ELLIPTIC MOTION 
which give Q. T h e s e relations follow from the explicit expressions (73) 
for the vectorial orbital constants . 
T h e relations (77) may be used as a check on the constants A and B 
or the corresponding relations (71) on the P ' s and Qys. Of these the most 
useful is 
PxQx + PyQy + PzQt = 0. 
A complete check of the elements a>, Q> I would be obtained by com-
put ing the vectorial constants in matr ix form (see Section 24) after 
these angles have been found. 
28. A c c u r a c y of the e l e m e n t s . I t is of interest to examine the 
determinat ion of e and u from Eqs . (89) and (90). T h e r e is an apparent 
loss of accuracy due to the circumstance that bo th these equations have 
the eccentricity as a factor. T h e max imum numerical value of the r ight-
hand members of these equations is e. Hence , if e is small, these r ight-
hand members will have fewer significant figures than the coordinates 
and velocity components used in obtaining them. T h e loss of significant 
figures is t ransmit ted th rough tan u to u. 
Suppose that the r igh t -hand members are computed to six decimal 
places. The i r max imum error is, let us say, e = 5 X 10~ 7 . T h e max imum 
error in tan u and in u is therefore e - 1 e . For example, if e = 0 .01, 
(89) and (90) give tan u to only four decimal places, and the uncer-
tainty in u becomes 5 X 10~ 5 radians = 0?003 — 10", while the 
uncer ta inty that would normally be expected with the use of six decimal 
places is of the order 0 " 1 . 
T h e uncer ta inty in w, of the order e~xe, is carried forward into / and 
affects any angle counted from the perihelion. I t is a reflection of the 
fact that the position of the perihelion is indeterminate in the case of 
circular orbits, and that the determinacy of the perihelion may be said 
to be proport ional to the eccentricity of the orbit . T h i s does not, how-
ever, affect the accuracy with which the position of the planet is deter-
mined in space, as may be verified from the expressions 
u = I + e sin «, rja = 1 — e cos w, 
which show that the deviation from uniform circular mot ion is expressed 
by t r igonometr ic functions of u that have the eccentricity as a factor. 
I t is important , however, to use u in the evaluation of / in (91) and in 
the calculation of the orientational elements (96), (97) to the full 
n u m b e r of decimal places desired in the elements. 
A similar uncer ta inty occurs in the evalution of co from (98) if / is 
small, and this is carried forward into the evaluation of Q by (99). 
CONSTANTS FOR EQUATOR 51 
If either e or I or bo th are small, the use of the full numbers of significant 
figures in u and co will assure that the sum of the angles Q + to + I, 
the mean orbital longitude, is accurately determined, regardless of the 
fact that all three of the angles, Q, co, and /, are individually known with 
considerably lower accuracy. T h e significance of these considerations 
may be illustrated by the expressions in polar coordinates (85). 
29. C o n s t a n t s for the e q u a t o r . F r o m (84) the corresponding 
expressions for the rectangular equatorial coordinates are readily 
obtained: 
x = r cos ( / + co) cos Q — r sin ( / + co) sin Q cos 7, 
y = r cos ( / + co) sin Q cos e + r sin ( / + co) (cos Q cos I cos e — sin I sin e), 
z = r cos ( / + co) sin Q sin e + f sin ( / + co) (cos Q cos I sin e + sin I cos e). 
These formulas may be pu t into the form 
x = r sin a sin (A + / + co), 
y = r sin b sin (B + / + co), 
z = r sin c sin (C + / + co), 
if sin a, sin b, sin c and A, B, C are de termined as follows: 
sin a sin A = cos Q, 
sin a cos A = — sin Q cos 7, 
sin 6 sin B = sin .Q cos e, 
sin 6 cos B = cos cos 7 cos e — sin 7 sin e, 
sin c sin C = sin ,Q sin e, 
sin c cos C = cos i3 cos 7 sin e + sin 7 cos e. 
Finally, if 
A' = A +co , 
75' = £ +co , 
C - - - C + CO, 
the expressions become 
x = r sin # sin (^4' + / ) , 
y = r sin b sin (7?' + / ) , 
z = r s'm c sin (C" + / ) . 
These formulas have been used extensively in the past for logari thmic 
calculations. T h e constants sin a, sin b> sin c, A\ B\ C are referred to 
as the constants for the equator . 
52 ELLIPTIC MOTION 
30. E x p r e s s i o n s in t e r m s of in i t ia l c o o r d i n a t e s a n d v e l o c i t y 
c o m p o n e n t s . Let x0,y0,z0 be the coordinatesand x0yy0,z0 the 
velocity components at t = t0 in the relative motion in the two-body 
problem. T h e solution of the problem is 
x = x (*), y = y (*), z = z (t). 
T h e object of this section is to find such expressions with x0, y0, z0> x0)yQy 
z0 as constants of integration. 
For short intervals of t ime it may be expected that these functions of 
the t ime may be developed in Taylor ' s series 
x (t) = x0 + x0(t — t0) + ^xQ (t — tQf + ... 
and similar expressions for y and z. 
Let the differential equations 
x = — jjixr*3, y = — fJLyr~3, z = — / x ^ r - 3 , 
be wri t ten 
x = Fxxy y = Fxyy z = Fxz. 
T h e n , by successive differentiation, 
x — Fx x + Fx x = F2 x + G2 x 
"x = F2x + (F2 + G2)x + G2x 
= (F2 + G2FX) x + (F2 + G2)x=Fzx + G,x. 
T h i s may be cont inued indefinitely. T h e same functions FP9 Gp arise 
with y and z. Subst i tu ted in the Taylor ' s series there results: 
x = f(t) x0 +g(t)xQi 
y=f(t)y0+g(t)y0, (100) 
* = f(t) * o + * ( 0 *o-
F r o m their formation it is evident that f(t)y g(t) are obtained as functions 
of the t ime in the form 
00 
1=0 
(101) 
00 
g(t) = 2 W - foY> 
3=0 
in which the coefficients a j y bj are originally expressed as functions of 
INITIAL CONDITIONS 53 
/xr" 3 and its successive derivatives, all for t = t0. By differentiating the 
expressions (100) twice it is found tha t 
x=f*o+ 8xo 
= ~ J
r l x = - 7 s (/*o + £*<))• 
T h u s / and g mus t satisfy the equations 
/ + /x/r- 3 = 0, g+iipr* = 0. 
T h e y are, therefore, part icular solutions of these differential equat ions 
with the initial condit ions, at t = tQi 
/(0) = + 1, g(0) = 0, 
/ (0) = 0, g(0) = + 1, ( 1 0 2 ) 
which follow from the expressions (ICO) and their first derivatives. 
A simple me thod for obtaining the expansions of / and g in power 
series in t — t0 requires obtaining first the coefficients ck in the expansion 
00 
4 - = X c * e - '«)*• 
T h e coefficients ck are obtained by repeated differentiation of /xr~~3 
at t = t0. T h e derivatives may all be expressed in t e rms of r 0 , r0 by 
making use of 
iw = i(**+# + **) 
= + J'2 + z2 + + + zz 
= — —. 
T h e energy integral gives 
whence 
T h e result ing expressions for ck are 
' 1 = ~ 3 / x r - 5 ( r 0 r 0 ) , 
54 ELLIPTIC MOTION 
a0 = + 1, 
ai = 
a2 = 
0, 
1 3 
aB = 
<*4 = 
a5 = 
3 7 
+ 8 ^ 0 " 
1 2 7-
a6 = 
7 -1 2 1 7 -
M i + + ^ 3 - 2 + ...]» (103) 
and similarly for g> 
- + q ^ V i + ^ , - 2 + . - ] . (104) B3+2 
T h e values of a0, aiy b0, bx follow immediately from the initial condi-
tions (102) of / , gy and their first derivatives. T h e results are 
<2 = + y F o W - j m V + 
35 15 
= - y / ^ W + W ( V o ) - y A o 7 f l _ 1 ( ¥ o ) . 
<4 = + y F o n ( ¥ o ) 4 - y / i V W + ^ V ^ V o ) 2 
31 15 
+ - —nhfcr1 + —fih-7a-2. 
If this series is subst i tu ted into 
/ + /Lt/r-8 = 0, i ' + j ^ = 0, 
replacing / , £ by the series (101), there results 
X 0' + *) 0' + 2)*, + a (t - *0)' - - X - *o)fc X < l ~ *oY> 
Equat ing the coefficients of (t — t0)
j on both sides gives 
INITIAL CONDITIONS 55 
An example of the usefulness of the series expansions of the functions 
/ and g arises in the problem of determining the orbit of a newly dis-
covered object. T h e calculations yield the heliocentric coordinates and 
velocity components for a date near the mean date of the observations. 
T h e next step might be the calculation of the elements, such as by the 
me thod in Section 27 of this chapter . However , frequently the most 
immediate need is the calculation of an ephemeris for a period of a 
few weeks or mon ths . I t is evident that the series expressions for the 
functions / a n d g may serve to advantage for this purpose . T h i s calcula-
tion can be made before the evaluation of the elements other than the 
semi-major axis, which is present in the expressions for the coefficients. 
Another use of the series is in the start of a numerical integration. 
T h e form in which the coefficients a and b are given renders the 
series for / and g applicable to any conic section. If the series are applied 
to an elliptic orbit , a modified form may have advantages. If the nota-
tion D for djndt is in t roduced, the differential equations for / and g 
may be wri t ten 
T h e Taylor ' s series expansions become power series in / — / 0 ; i.e., 
increments in the mean anomaly instead of in t — t0. I n particular, the 
expansion of # 3 / r 3 becomes 
DJ + ^f=0, D*g + £g = 0. 
If now 
1 P •• 
h = 0, 
b. = 0, 
K 
f* 'o ' 
/ * V o -
56 ELLIPTIC MOTION 
C 0 = +P3, 
Cl = - *P% 
CT= + \P\I + W ) - \ P \ 
C3= - ^ 6 9 ( 3 + 7^) + 8/»'?> 
C 4 = + y /»'(! + 14? 2 + 21q*) - ±/>8(31 + 233?*) + 2p\ 
These te rms can easily be verified by not ing that 
Dp* = —jp'+1q, 
Dq> = P - pq* + p*). 
Precisely the same recurrence relations hold between the coefficients 
A, B, C in the expansions of / and g as were found for a, b, c. T h e 
final result is 
/ = 1 - M - HF - p*q(l - l0f 
P*(l+5q*) pV(3 + 7?') 
, / > 9 ] ( ' - ' o ) M 08(19 + 140?2) -;i + i v + 2 
g = i - h •p\i-hf-
j/>5(l + 5? 2) ^ ( 3 + 7^) ,P7qW-lo)«-
For circular orbits p = 1 and q = 0, and the expansions reduce to 
expansions of cos (/ — / 0 ) and sin (/ — / 0 ) respectively. 
I t is necessary to obtain in this form, in addit ion to a, n = f j L l l 2 a ~ s l 2 . 
Closed forms for the functions / and g may be derived by making 
use of the fact that these functions are applicable also to the 
coordinate system X, Y in the orbital plane, with the X axis directed 
toward the perihelion. T h e n 
X(t)=f(t)X0+g(t)X0, 
Y(t)=f(t)Y0+g(t)Y0. 
GAUSSIAN CONSTANT 57 
T h e solution of / and g from these equat ions is 
/ ( / ) = G - W o - YXJ9 
g(t) = G~\YX, - XY0), 
in which 
G — X0Y0 — Y0X0i 
= |>ifl(l - e 2 ) ] 1 / 2 
= na\\ - e2)1'2 
— nab. 
I t is now convenient to express / , g in t e rms of the eccentric anomaly 
by us ing 
X = #(cos u — e)y X = — an sin w(l — e cos w) - 1 , 
Y = a ( l — e 2 ) 1 / 2 sin w, y = + <ro(l — e 2 ) 1 / 2 cos w(l - e cos w)"1. 
Le t w 0 be the eccentric anomaly at t = £ 0, w that at t = t. T h e n it is 
easily found that 
XY0 — y X 0 = abn[cos (u — u0) — e cos u0] (1 — 6 cos « 0)
_ 1> 
y Z 0 — Z y 0 == 0&[sin (u — w0) — d?(sin u — sin w0)]. 
T h e result is 
/ = 
= 1 -
g = 
3 1 . T h e G a u s s i a n c o n s t a n t . I t was remarked in Section 2 that 
the numerical value of the gravitational constant depends on the uni ts 
of mass, t ime, and distance chosen. Keple r ' s th i rd law, 
k2(ma + mh) = n2a3, 
or, with n = ITTJP, 
a? 
k\ma + mb) = 4TT2^ , 
58 ELLIPTIC MOTION 
is a simple relation between mass, t ime, and distance that may con-
veniently be used in practical applications of elliptic motion to find 
any one of these three quantit ies when the other two are known. T h e 
units of mass, t ime, and distance might be arbitrarily chosen, and then 
k might be determined by observation. In particular, it might seem 
logical to adopt for these uni ts the gram, centimeter, and the ephemeris 
second, as is done in physics. Such a choice of uni ts would be excessively 
inconvenient, however. If the gram were taken as uni t of mass, the 
mass of the earth would first have to be determined in grams (which is 
a geophysical, not an astronomical, problem), and then transferred to 
other bodies. T h e result would run to a large n u m b e r of figures, and 
no beneficial result would ensue; there is no practical problem in celestial 
mechanics that requires knowledge of the mass of any body in grams. 
Greater difficulty would be encountered with the uni t of distance. 
Distances between bodies in the solar system can not be measured 
directly in centimeters with m u c h precision. For example, the distance 
from the earth to the sun in centimeters is known to only three or four 
significant figures, and not as the result of a direct measurement b u t 
of derivation by complicated processes. 
Purely as a mat ter of convenience, therefore, theunits are chosen 
such that the masses, t imes, and distances involved in astronomical 
problems can be easily measured and expressed. In many applications 
of celestial mechanics to the motions of bodies in the solar system, the 
uni t of mass chosen is the mass of the sun, and all other masses are 
expressed as fractions of it. T h u s , if ma is the mass of the sun, we adopt 
ma = 1 and write m = mb/ma. T h e n 
k2(l + m) = n2a\ 
T h e unit of t ime chosen is the ephemeris day, because the periods 
of bodies in the solar system can be easily and precisely measured in 
ephemeris days. T h u s , in the expression of Kepler ' s third law, P is 
commonly measured in ephemeris days and n in radians per ephemeris 
day. 
Fur the rmore , owing to the difficulty of measur ing distances, distance 
is not defined in absolute physical uni ts or any arbitrary units , bu t k is 
chosen arbitrarily instead. T h e value of k is 0.017 202 098 950 000 000, 
and this n u m b e r is known as the Gaussian constant . T h e uni t of distance 
is therefore a derived one, obtained from the Gaussian constant and 
the units of mass and t ime. I t is called the astronomical unit , often ab-
breviated to a.u. T h e dimensions of k are then M ~ 1 / 2 L 3 / 2 7 1 - 1 . 
I t is impor tant to r emember that the definition of the astronomical 
uni t is given by Kepler ' s th i rd law with the stated numerical value of ky 
the sun 's mass as the uni t of mass, and the ephemeris day as the uni t 
NOTES AND REFERENCES 59 
of t ime. It is often stated that the astronomical uni t is the distance from 
the sun to a body of infinitesimal mass revolving in a circular orbit 
with a period of 3 6 5 . 2 4 + ephemeris days, bu t this is a description 
only, not a definition. Even as a description it leaves something to be 
desired. 
As a mat ter of historical interest it may be noted that Gauss in tended 
to choose k such that the uni t of distance would tu rn out to be half 
the major axis of the ear th 's orbit . Actually, because Gauss used a 
value for the ear th 's mass that we know to be too small by about 
7 % , it tu rns out that half the major axis of the ear th 's orbit , as derived 
from Kepler ' s th i rd law with Gauss 's value of k, is 1.000 000 03 a.u. 
M u c h confusion has been caused by careless usage of the words semi-
major axis and mean distance in astronomical l i terature. In fact, Kepler ' s 
th i rd law loses its geometrical significance when a th i rd body of appre-
ciable mass is in t roduced in the system. In such cases the law should 
be looked upon as what it is in fact, merely a definition of the uni t of 
distance. For example, owing to the dis turbing influence of the other 
planets, an ellipse that has a semi-major axis of about 1.000 000 2 a.u. 
gives a bet ter approximation to the ear th 's actual orbit than does the 
one derived from Kepler ' s th i rd law. 
N O T E S A N D R E F E R E N C E S 
T h e epicyc l ica l t h e o r y o f H i p p a r c h u s w a s t h e first a t t e m p t t o represent t h e 
m o t i o n s of the p lane t s b y a p u r e l y m a t h e m a t i c a l theory , w i t h o u t re ference to 
phys ica l causes ; in f o r m it r e s e m b l e s the d e v e l o p m e n t s in t r i g o n o m e t r i c ser ies , 
w h i c h are still i n d i s p e n s a b l e to t h e subjec t , and w h i c h o c c u p y m u c h of th is book . 
Y e t there is a vast d i f ference b e t w e e n t h e empir ica l representa t ion b y H i p p a r c h u s 
and his s u c c e s s o r s a n d the m o d e r n representa t ion b y t r i g o n o m e t r i c ser ies that 
satisfy t h e e q u a t i o n s of m o t i o n . In t h e f o r m e r t h e a m p l i t u d e s a n d p h a s e c o n s t a n t s 
of all the e p i c y c l e s w e r e o b t a i n e d i n d e p e n d e n t l y f r o m observat ion; in t h e latter 
the n u m b e r of avai lable cons tant s is s tr ict ly l i m i t e d . U n t i l the t i m e of K e p l e r 
it w a s c u s t o m a r y to represent the annual parallax of a p lanet b y o n e e p i c y c l e a n d 
the eccentr i c i ty of the orbit b y another . In K e p l e r ' s tabulae Rudolphinae, 1621, the 
e q u a t i o n of the cen ter is d e r i v e d for the first t i m e b y an el l ipt ic formula , a n d the 
m o d e r n w a y o f t rans forming f r o m h e l i o c e n t r i c to g e o c e n t r i c p o s i t i o n s w a s in tro -
d u c e d . 
K e p l e r w a s led to his t h e o r y of e l l ipt ic m o t i o n b y analys is o f observa t ions of 
M a r s , the o n e p lanet w i t h a suff ic iently large eccentr i c i ty and rapid m o t i o n to 
require , in h i s day, m o r e than o n e e p i c y c l e to represent the eccentr ic i ty . H i s 
f a m o u s e q u a t i o n has rece ived m o r e a t tent ion t h a n any o ther in a s t r o n o m y . 
T h e first ful ly a d e q u a t e t r e a t m e n t of e l l ipt ic m o t i o n w a s b y N e w t o n in t h e 
Principia, 1687, e m p l o y i n g geometr i ca l m e t h o d s . A deta i l ed analytical t r e a t m e n t 
w a s g i v e n b y E u l e r in Theoria motuum planetarum et cometarum, \1AA, w h o also 
under took the first ca lculat ion of the m u t u a l per turbat ions of Jupi ter and S a t u r n 
in accordance w i t h N e w t o n ' s law of gravi tat ion. 
CHAPTER I I 
EXPANSIONS IN ELLIPTIC MOTION 
1. I n t r o d u c t i o n . T h e methods presented in Chapter I suffice for the 
calculation of the coordinates of a planet in an elliptic orbit at any t ime 
from the elements of the orbit . For various applications in celestial 
mechanics it is desirable to have available methods that will permi t the 
expansion of the coordinates and functions of the coordinates in an 
elliptic orbit in periodic series. In the movement in an ellipse all finite 
and continuous functions of the coordinates re turn to the same values 
after a complete revolution. Such functions are therefore developable 
in periodic series in t e rms of any continuously increasing angular 
variable that increases by 2TT dur ing a complete revolution. T h e angular 
variables that are of particular interest in this connection are the mean 
anomaly, /, the eccentric anomaly, w, and the t rue anomaly, / . These 
are not the only arguments that may be considered; other arguments 
have actually been used in some applications. T h e functions that present 
themselves most naturally are either even or odd periodic functions of 
these variables, giving rise to either cosine series or sine series. Since 
it is frequently more convenient to operate with exponential series than 
with tr igonometric series, it is useful to be familiar with the develop-
ments in the exponential form. 
2 . E x p a n s i o n s in a F o u r i e r s e r i e s . Let f(x) be a periodic function 
with the period 2-TT in x, the expansion being 
00 OC 
(1) 
1 1 
T h e n 
(2) 
f(x) sinpx dxy p = + 1, + 2 ... oo. 
60 
f(x) cos px dxy 
In 
f(x) dx} 
FOURIER SERIES 61 
a0 -- f(x) dx, 
o 
71 
f(x) cospx dx, p = + 1, + 2 ... 0 0 . 
(3) 
If f(x) is an odd periodic function, 
oot 
f(x) = 2 ^ p sin px, 
i 
(4) 
f(x) sinpx dx, p = + 1, + 2 ... 0 0 . 
o 
In the exponential form, let 
+00 
/ (*) = 2 A » E v i x ' 
—oo 
(5) 
where E is the base of natural logari thms, a n d i2 = — 1; then 
*/(*) £ - 3 , i a : ^ , £ = — 0 0 .» + °°. (6) 
T h e integral expressions for the coefficients Av, as applied to real 
functions / ( # ) , are to be looked upon merely as an abbreviation for the 
corresponding expression 
271 
f(x) (cos px — i sin px) dx. 
o 
Hence 
A0 — ciQi 
Ap = av ibp, 
A-v = ap + ibp, p = + 1, + 2, ... °°. 
(7) 
If the periodic series is expressed in the form 
+00 +00 
/ (*) = X a p c ° s + X b'psin p X y (g) 
—oo —oo 
If f(x) is an even periodic function, 
00 
f(x) = a0 + 2 X av
 c o s Px> 
i 
62 ELLIPTIC EXPANSIONS 
one single relation, 
*LP = a'P = <V (9) 
b'_p = - b'v = - bv, 
A , = "P-*',> (10) 
applies for all values of p> positive,negative, and zero. 
Expansions in terms of the exponentials may be writ ten in a par t i -
cularly convenient form by the introduct ion of 
E" = A, E™ = Y", E*' = 0 , (11) 
so that 
2 cos / - A + A~\ 2 cos u = Y + Y~\ 2 c o s / = 0 + &-\ 
2i sin / = A — A-1, 2i sin u = Y — Y'1, 2i sin / = 0 — 
3 . T h e true a n o m a l y e x p r e s s e d in t e r m s of the e c c e n t r i c 
a n o m a l y . T h e equation relating the t rue anomaly to the eccentric 
anomaly is 
so that 
and (12) becomes 
Since /? as a function of e is obtained as the root of a quadrat ic equation, 
there are two solutions, the second of which, fi' = e - 1 ( l + V I — e2)> 
tan ; (12) 
For the development of the solution it is desirable to introduce /? by 
e = sin cp = (13) 
(14) 
(15) 
with 
TRUE ANOMALY 63 
|8 
If (12) is wri t ten in the exponential form and the notation (11) is 
used, the equation becomes 
(16) 
(17) 
(18) 
F r o m this equat ion <J> may be expressed in te rms of fi and Y by 
which may be wri t ten 
By taking the natural logarithm of both sides of (18) there results 
if = iu + In (1 - ^ y - 1 ) - l n (1 - j8Y) 
= iu - OSY-1 4 
f = u + 2 (p sin u sin 2u 4 sin 3M + ... j . (19) 
will not be considered. I t is seen that if e is small, /? is approximately 
\e, while /?' = is approximately 2 e _ 1 . 
For further applications the following expansions of powers of j8 in 
powers of e are useful. 
64 ELLIPTIC EXPANSIONS 
Equat ions similar to (12) or (15) occur also in quite different fields. 
For example, the result obtained in this section was used in the derivation 
of the development in series of the reduction to the ecliptic in Section 26 
of Chapter I . 
4 . T h e m e a n a n o m a l y e x p r e s s e d in t e r m s of the t rue a n o m a l y . 
T h e expression of u in te rms of / , or of y in te rms of 0y may be subst i tuted 
into Kepler ' s equat ion in order to obtain / expressed in te rms of / . 
Kepler ' s equation, 
may be wri t ten 
/ = u — e sin w, (22) 
I — u — 
F r o m (21) we obtain 
and developed with the binomial theorem as 
Y - r - 1 = (1 - /32) (0 - p&2 + )8 2 0 3 - fi*0* + ... 
- 0-1 + P0-* - £ 2 0 - 3 + £ 3 0 - 4 - ...) 
= 2/(1 - £ 2 ) (sin / - p sin 2 / + jS2 sin 3 / - ...). 
(23) 
(Y - Y-1) 
Thi s may be writ ten 
Y - r - 1 = (1 - / S 2 ) 
= ( l - £ 2 ) -
4 
y — y - 1 = — 
(Y- y - 1 ) . 
Th i s relation may be inverted since (15) is not changed if fi is changed 
to — j8 while u a n d / a r e interchanged. Consequently, (19) becomes 
u =f~2 ( jSsin/ (20) 
T h e relation corresponding to (18) is 
(21) 
the final result may be writ ten 
/ = / - 2[fi(l + coscp) s i n / - fi2(\ + coscp) sin If + j3 3(^ + coscp) sin 3 / - . . . ] . 
(25) 
5. I n t r o d u c t i o n of B e s s e l f u n c t i o n s . Al though the relations 
obtained in the preceding sections are of considerable usefulness, of 
more general importance is the development of functions in periodic 
series in te rms of the mean anomaly. T h e first problem to be considered 
is the solution of Kepler ' s equation in the form of an expression for u 
as a Fourier series in te rms of the mean anomaly. 
F r o m 
u — I = e sin u\ 
it is evident that u — / is an odd periodic function of u and therefore 
of /. I t can therefore be expanded in a Fourier series 
00 
e sin u = 2 ^ 6S sin si, 
I 
so that 
- ± r . 
7 7 J 0 
By integrating by parts , it is found that 
b„ = f — e sin u cos si -\- — [ cos si die sin «)1 
L STT ^STT) v y J u = 0 
T h e first part is zero and the second part may be writ ten, by subst i tut ion 
of u — / for e sin w, 
I f * . 
b. = — e sin u sin si dL 
BESSEL FUNCTIONS 65 
Subst i tut ion into (23) gives 
(fi s i n / - £ 2 sin If + £ 3 sin 3 / - ...). (24) 
In this expression the expansion (20) of u in te rms of / m a y be subst i tu ted . 
Since 
: cos<p = V l — £2> 
cos J/ -f cos <iw. 
66 ELLIPTIC EXPANSIONS 
so that 
and 
(28) 
OO j 
e sin u = 2 ^ - Js(
se) s m SU (29) 
CO i 
u = I + 2 X - JsM sin (30) 
i s 
I t is necessary to s tudy in some detail the propert ies of the Bessel 
functions of the first kind, defined by (27); these may be writ ten 
71 
cos (stp — x sin cp) dep. (31) 
Consider the integral 
cos (scp - x sin cp) dtp -
271 
sin (scp - x sin cp) dep. 
I t is seen tha t for cp — 2TT — cp', 
sin (scp — x sin 99) = sin (2sn — scp' -{- x sin 9 ' ) 
= — sin (scp' — x sin cp'). 
Hence the value of the integrand of the second integral for <p' = 2n — cp 
is equal but opposite to the value of the integrand for cp. T h e integration 
between 0 and TT is therefore canceled by that between 77 and 277. Conse-
quent ly an expression equivalent to (31) is 
571 
£-i(s<p-x shop) Jy, (32) 
Again the first integral disappears, and the second integral, upon 
subst i tut ion of u — e sin u for /, becomes 
1 C31 
bs — — cos (su — se sin u) du. (26) 
SIT J o 
In accordance with the conventional notation for Bessel functions, let 
1 CN 
Js(se) = — cos (su — se sin u) du, (27) 
7 7 J o 
it is seen from (33) that Js(x) is the coefficient of Esi<p in the Four ier 
expansion 
+00 
£ix sin <p = ^ JJ^ £ s t > (34) 
s=—oo 
If now 5: is in t roduced by 
so that 
2i sin g> = z , 
T z 
(34) may be wri t ten 
+00 
E(xl2Hz-l/z) = ^ ^ ^ (35) 
s=—00 
Consequently, / s ( # ) is the coefficient of # s in the expansion of 
F(x, z) = £<*/2><*-i/*> (36) 
in a series of positive and negative powers of z. 
Thi s series for F{x, z) may be obtained by the multiplication of two 
exponential series: 
F(x, z) = Exz/2 • E~x/2z 
T h e subst i tut ion 
gives 
OC - j3 = s 
(37) 
(38) 
F(x, z) 
BESSEL FUNCTIONS 67 
(33) £-si<p . Eixsin<P d(p, 
If this is wri t ten 
68 ELLIPTIC EXPANSIONS 
has only positive powers of x/2, as is evident from the two exponential 
series that were mult ipl ied in order to obtain F(x, z). 
For application to the expansions in elliptic motion x may be considered 
real and s an integer. 
Since F(x, z) does not change if x is changed to — x while z is changed 
to —zy it follows from (38) that 
U - x) = ( - 1)' Js(x). (40) 
Similarly, F(xy z) does no t change if z is changed to — z'1, which 
furnishes 
J-S{x) = ( - 1)' Js(x). (41) 
T h e combinat ion of (40) and (41) gives 
/_s(- x) = Js(x). (42) 
F r o m these relations it follows that only positive values of the index s 
need be considered. For positive values of s the series (39) may be 
writ ten 
the numerical value of which is less than uni ty regardless or the value 
of Xy provided that /3 is chosen sufficiently large. Hence the series (43) 
for Js is absolutely convergent for all values of x. I t may also be concluded 
that the series (38) for F(xy z) is absolutely convergent regardless of 
the value of xt for all values of z excluding 0 and oo. Th i s follows from 
(43) 
T h e ratio of any te rm to the preceding one is 
In the double series (37) the lower limit of summat ion for s has been 
chosen as — oo instead of that corresponding to a = 0, which would 
be equivalent to s + /? = 0. Since for negative values of s + j8 the 
factorial (s + /?)! is infinite, it is seen that extending the summat ion 
to s = — oo is permissible. T h i s absence of t e rms for which s + P is 
negative is seen to be equivalent to the fact that the series in powers 
of x/2y 
(39) 
BESSEL FUNCTIONS 69 
if the notation 
is in t roduced. Equat ing the coefficients of zs on both sides results in 
\ L/.-i(*) - = ;;(*)• (45) 
F r o m (44) and (45) it is seen that 
Js-i(x) = £ /,(*) + /;(*), 
7«+i(*) = ^ /«(*) - J » -
(46) 
(47) 
Changing s to s — 1 and s + 1, respectively, in (44) gives 
J six) + J8+i(x) 
or, with (46), 
Js-4*) = 
Differentiation of (35) with respect to x gives 
the fact that F(x, z) has no poles in the complex plane except for z = 0 
and z = oo. 
Differentiation of (35) with respect to z gives 
(
1 +00 +00 
1 +7i)2/ s(*)* s = 2*./sM*s-1-
* ' -00 -00 
By equat ing the coefficients of z8"1 on both sides of this equality thereresults 
\ Us-M + Jm(*)] = * Ux). (44) 
70 ELLIPTIC EXPANSIONS 
Thi s procedure may be cont inued, and leads to the conclusion that 
any Bessel function with a rgument x and with index s - f k> k being an 
arbitrary positive or negative integer, may be expressed as a linear 
combination of Js(x) and J's(x) with coefficients that are functions of the 
a rgument x. An equally impor tant proper ty is that Js(x) satisfies a 
linear differential equat ion of the second order with variable coefficients. 
Th i s differential equation may be obtained by differentiating (45) and 
eliminating / s - i ( * ) a n d J*+i(x)y 
W i t h the aid of (45), this may be writ ten 
[/s_2(*) - 2Js(x) + / s + 2 ( x ) ] , 
and, with (47), 
Js(x) - U*) -
or 
/;(*) - Js(x) = 0. (48) 
Th i s differential equation may be used to define the Bessel functions. 
T h e mathematical theory shows that , in addit ion to the Bessel function 
of the first kind that was int roduced in the preceding section, another 
class of functions satisfies this differential equation. For the applications 
in this chapter only the Bessel functions of the first kind with real 
a rgument x and with integral index s are needed. 
An interesting proper ty of the functions Js(x) may be obtained by 
multiplying the two series 
£<*/2><«-i/,> = ; o W + + jz(x)z* + ; 3 ( ^ 3 + _ 
- J^z-i + Mx)*r* - Mx)ar* + ... 
and 
+ Ux)^1 + A W * " 2 + / s W * " 8 + 
the coefficients of which have been expressed as Bessel functions of the 
argument + x and positive index with the aid of (40), (41), and (42). 
APPLICATIONS 71 
T h e product of these two series is found to be 
1 = t / o W ] 2 + 2[ / , (*)] 2 + 2[J2{x)Y + 2[Ux)f + ... 
» (49) 
= t / o W l 2 + 2 % [Js{x)Y. 
1 
For a real a rgument x all functions / s ( ^ ) for integral values of the 
index are real. Hence it follows that no te rm in the r ight -hand member 
of (49) can exceed unity. Consequent ly , 
l / 0 ( * ) l < i ; < 1/V5T ^ o . 
6* A p p l i c a t i o n of B e s s e l f u n c t i o n s . T h e function Epiu, developed 
in an imaginary exponential series in te rms of /, will give rise to the 
expansion 
Eviu = ^ AsEsiK (50) 
—oo 
T h e coefficients As follow from 
(51) 
Eviu d{E~sil). 
]^{se\ s ^ 0. (52) 
For s = 0, (51) gives 
71 Eviu dl 
£viu £-8x1 ft A* = 
Integrat ion by parts gives 
T h e first part in the r ight -hand member disappears, and the second part 
may be wri t ten 
]?-i[(s-p)u-se s i n u ] fiu 
By the definition (32) this may be wri t ten 
T h e exponentials in the integrand give rise to contr ibut ions different 
from zero only if / > , / > + 1, or/> — 1 is zero. T h e following different 
cases mus t therefore be considered: 
P = 0, AQ = l; 
P = ±h A0=-el2; (53) 
P ^ 0 , ^ ± l , Ao=0. 
T h e results (52), (53), giving 
+oo t -J 
E«* = A0 + p% - JrJse) E«l, 
—CO 
or 
(54) 
the symbol Z' indicating that the te rm for s = 0 is omit ted, may be 
applied to t r igonometric expansions. Since (50) may be wri t ten 
+00 +00 
cos pu + i sin pu = ^ As cos si + i ^ As sin sly 
— CO —00 
it follows that 
+00 
cos pu = ^ As cos si 
— 00 
+oo f 2 
= A o + P X ' 7 Js-p(se)cos sl> (55) 
-oo 
the constant te rm of which is determined by (53). T h e terms with equal 
arguments of opposite sign may be combined: 
Y* = AQ + p -Js.v(se)A\ 
[Js-P(se) - Ja+9(se)] cos si, (56) 
cos pu — A0 + p cos si 
72 ELLIPTIC EXPANSION 
4 . = 
) i u _ £ d > - l ) t u j d U t 
or, since 
dl = (1 — e cos u) du 
APPLICATIONS 73 
A0 being given by (53). Also 
+00 
sin pu = 2) As sin si 
—oo 
+oo f | 
= P X 7 s i n 5/> ( 5 ? ) 
-oo A 
which may be writ ten 
0 0 rl 1 l 
smpu = p | j / 8 - J , (w) - — j JS-V(- sin 5 / 
= * X 7 + sin rf. (58) 
1 A 
For p — \ the special cases are 
£ + 0 0 , j 
COS ft = — - + ̂ ~ Js-l(s*) C0S ̂ 
—00 
00 1 
L X S 
which, by (45), becomes 
e ^ l 
C O S M = — - + 2 2 ^ - J's(
se) c o s ^ ; (59) 
z 1 ^ 
also, 
+00 j 
sin w = 2) _ Js-i(se) s m sl 
-00 ^ 
= X 7 Us-i(se) + Js+i(se)] sin 5 / , 
1 A 
which, by (44), may be writ ten 
2 0 0 1 
sin u = - 2) ~ /s(^) s i n (60) 
These results could have been obtained, of course, by int roducing 
instead of (50) the t r igonometr ic Fourier series for cos pu> sin pu. 
T h e expansions of Eviu, cos pu, and sin pu are of great importance 
in applications to problems in celestial mechanics. M a n y functions of 
the coordinates in elliptic motion are readily expressed in periodic 
74 ELLIPTIC EXPANSIONS 
A0 = B0 
(63) 
Similarly, by (56), 
00 
f(u) = a0 + 2 2) « p c o s P w (64) 
gives 
00 
f(u) = c0 + 2 cs cos ^/ (65) 
with 
1 0 0 
c* = ~ X Pav[J*-v(se) - J8+P(se)l 
C0 = aQ — eaV (66) 
Also, by (58), 
oc 
f(u) = 2y£bp sin />u (67) 
gives 
00 
/(«) = 2 X d»sin rf» (68) 
= 7 X PhUs-vW + / , + » ] . (69) 
A + S-x). 
series in terms of the eccentric anomaly. T h e series (54), (56), (58) can 
then be used to change to series expressed in terms of the mean anomaly. 
For example, consider 
+00 +00 
/(«) = X B,E»» = %B9Y>. (61) 
p=—00 —00 
T h i s can be expanded, by application of (54), as 
+ 00 +00 
/(«)=£ AsE
sil =2^yl8> (62) 
s=—00 —00 
with 
s ^ 
APPLICATIONS 75 
/„(*) = i -
m -
m -
m 
m -
m -
m = 
m -
T h e following developments obtained from the general series (43) will 
be useful for further applications. 
76 ELLIPTIC EXPANSIONS 
(b) T h e ratio between the radius vector and the semi-major axis, 
with (59) becomes 
= 1 — e cos uf 
I sin 3/ 
(71) 
sin 5/ sin 41 
or 
J's(se) cos sly (72) 
cos / 
cos 3/ 
(73) 
cos 5/ 
T h e two cases jus t treated are examples of developments in which 
the coefficients may be obtained as finite expressions in te rms of the 
Bessel functions. Th i s is t rue for only a very limited n u m b e r of functions 
of the coordinates in elliptic motion. Among those for which infinite 
series of Bessel functions appear as coefficients, one of the most impor tan t 
is the development of / in terms of /. 
A few examples will now be worked out in de t a i l : 
(a) T h e solution of Kepler ' s equation was found by (30): 
00 j 
u = / + 2 X - JM s i n si. (70) 
1 s 
With the expansions for Js(x) given in the present section it is then found 
that , to the seventh power of e, 
APPLICATIONS 77 
In order to expand this logari thm in positive and negative powers of Yy 
let C and x be in t roduced by 
1 _ * y _ * y - i = q i - * Y ) ( 1 -xY-1) 
= C(l + x2) - CxY - CxY-\ 
Thi s gives 
Cx = ^ C(l+x2) = \f 
f=l + 
| sin 31 
(75) 
?6 sin 61 e1 sin 11. 
(d) T h e logari thm of the radius vector. T h e problem is the expression 
of 
= In 
(c) T h e equation of the center. T h e expression (19) f o r / i n t e rms of 
u gives by application of (67) 
CO -I 00 
f = u + 2%-X PvUs-vM + Js+vM] sin si. 
S = l * p=l 
If the expression for u in t e rms of / is in t roduced, the series for / becomes 
/ = 1 + 2 X 7 + X PUrJ.") + J«JL«)]\ (74) 
S = l * ' P=l ) 
Wri t ten more explicitly this becomes 
/ = / + \ Ui(e) + fl/oto + JM] + W - i t o + JM + •••} sin / 
+ \ (U2e) + RU2e) + J3(2e)] + p[Jtfe) + Jt(2e)] + ...} sin 2/,etc. 
In this series the Bessel functions and the various powers of f3 may be 
replaced by expressions in powers of e. T h e result is, to the seventh 
power in e, 
78 ELLIPTIC EXPANSIONS 
1 + * 2 2 " 
Th i s is the same relation as that found between /3 and e in Eq . (13). 
Hence 
= i ( i + vT=l») . 
T h e problem is then reduced to the expansion of 
In - = - In (1 + £ 2 ) + In (1 - j3Y) + In (1 - ^SV-1) 
In 
(76) 
- l n ( l + J 8
2 ) - / 3 r 
= - l n ( l + £ 2 ) - 2 cos pu 
W i t h the aid of ( 6 4 ) this becomes 
In - In (1 + |32) + ej8 
CO j 00 
s=l
 s
 p=i 
T h e constant t e rm in this expression, in te rms of e, is 
T h e series may be expanded and yields 
ln^ 
cos / 
cos 3/ 
(77) 
or x e 
APPLICATIONS 79 
An impor tant proper ty of the four functions that have been developed 
in Fourier series with multiples of / as a rguments and power seriesof e 
as coefficients is that the lowest power of e occurr ing in the coefficient 
of a sine or cosine t e rm equals the mult iple of / in the argument . T h e 
power series further progress in powers of e2 so that in a coefficient of 
the cosine or sine of an odd argument only odd powers of e occur, and 
in the coefficient of a te rm with an even a rgument only even powers of e 
occur. Th i s proper ty is closely related to the propert ies of the expansions 
of the Bessel functions. I t was first emphasized by d 'Alembert . For this 
reason it was called by E. W . Brown the d 'Alember t characteristic. 
I t is seen that this same proper ty will also apply to functions of the 
types 
(f) Expansion of (r/a) s i n / = yja — V l — e2 sin u. T h e series (60) 
for sin u mus t be mult ipl ied by 
£im(f-l) cos m(f — /), sin m(f — /), 
n and m being positive or negative integers. On the other hand, in the 
development of Fourier series in the mean anomaly of functions of the 
types 
Eimi\ cos mj sin m/, 
the relation between the lowest power of e in the coefficient and the 
multiple of / in the a rgument will be destroyed. T h e relations that 
exist can however be discovered, as may be shown in the following 
examples: 
(e) Expansion of (r/a) cos / = x/a = cos u — e. T h e series (59) for 
cos u gives, neglecting powers of e above the seventh, 
cos / = 
V l - e2 = 1 
Extensive tabulat ions of series in elliptic motion were given by 
U . J. J . Leverr ier . 2 
Leverr ier ' s basic series are the series in mult iples of the mean anomaly 
for 
x = r--h y=f-l, 
from which he derives the series for xp> yq; p, q = 0, 1, ... 7, p + q < 7. 
These are used in t u rn for obtaining series for xp sin hy, xp cos hy, 
with coefficients expressed as polynomials in h. 
Based upon this work by Leverr ier are the extremely useful tables of 
the developments of functions in the theory of elliptic mot ion of A. 
Cay ley. 3 T o the seventh power of e Cayley gives 
x* cos;/ , x* sin;/ , ; , p = 0, 1, . . . 7, 
80 ELLIPTIC EXPANSIONS 
- sin / = 
i J 
sin 3/ 
j sin 5/ 
5 sin 11 
In cos;/ , s i n ; / 
j = 0, 1, 2, ... 5; q = - 5, - 4 , . . . - 1, + 1, . . . + 4. 
Newcomb ' s deve lopments 4 are in effect based on the series for 
p = lr -N=I(F-I), 
expressed in positive and negative powers of A = Eil. T h e n 
E?P = 1 + fa • 
EM = 1 + yrj 
2 U. J. J. LEVERRIER, Ann. Obs. Paris 1, 343-357 (1855). 
3 A. CAYLEY, Mem. Roy. Astron. Soc. 29, 191-306 (1861). 
4 S. NEWCOMB, Astron. Papers, 5, 1-48 (1895). 
T h e result is, again to the seventh power of e> 
CALCULATION 81 
and, finally, 
m = 0 , l , . . . o o , w - \q\ = 0 , 2 , 4 , .... 
T h e coefficients i7^(j8, y) are polynomials in j8 and y of degree m in 
both /? and y . T h e y are impor tant in Newcomb ' s me thod for developing 
the dis turbing function. See Chapter X I I I . 
7 . C a l c u l a t i o n of the B e s s e l f u n c t i o n s . In principle, formula (44) 
can be used to obtain the Bessel functions of all orders required for a 
given argument , once any two of t hem are known. But this process, 
a l though it is algebraically exact, leads to the rapid accumulat ion of end 
figure errors if it is applied several t imes in succession. In numerical 
work it is preferable either to apply formula (43) directly, or to interpolate 
the functions from published tables of the Bessel functions. Still bet ter 
for systematic calculation, especially with automatic calculating machines, 
is to obtain J0(x) in either of the two ways ment ioned, and then to cal-
culate Js(x) by successive mult ipl icat ion of J0(x) by factors tha t are 
very easily calculated. 
Denot ing the ratio of Js{x) to / s _ i (# ) by ps, we have from Eq. (44) 
whence we obtain the cont inued fraction 
T o make use of this cont inued fraction in calculation, derive ps from 
1 
where 
or 
82 ELLIPTIC EXPANSIONS 
starting with so large a value of s that for the first application it is 
permit ted to pu t ps+1 = 0, and working down to pv T h e n 
7 iW = Pi /oM. 
J2(x) = p2 J^x), 
Js(x) = Ps h(*\ etc-> 
and formula (49) may be used as a check. T h e largest value of s required 
for the first application may be determined by trial, using the largest 
value of x that is to be employed in the problem; the same value of s 
will then suffice for all smaller values of x. 
As an example we give in Table 1 the figures necessary for the 
calculation of J8(e), e being the eccentricity of Mars , 0.0932 6685. By 
application of (43) it is found that J0(e) = 0.99782 65057. 
T A B L E 1 
Calculation of Js(e) 
s rs = 2s/e JS(e) 
1 1 2 3 5 . 8 8 2 3 0 9 7 0 . 0 
1 0 2 1 4 . 4 3 8 4 6 3 4 0 . 0 0 4 2 3 9 4 — 
9 1 9 2 . 9 9 4 6 1 7 1 0 . 0 0 4 6 6 3 4 — 
8 1 7 1 . 5 5 0 7 7 0 7 0 . 0 0 5 1 8 1 6 1 6 9 — 
7 1 5 0 . 1 0 6 9 2 4 4 0 . 0 0 5 8 2 9 3 5 4 2 — 
6 1 2 8 . 6 6 3 0 7 8 0 0 . 0 0 6 6 6 2 1 7 6 6 0 . 0 0 0 0 0 0 0 0 0 0 
5 1 0 7 . 2 1 9 2 3 1 7 0 . 0 0 7 7 7 2 6 4 0 0 0 . 0 0 0 0 0 0 0 0 1 8 
4 8 5 . 7 7 5 3 8 5 3 6 0 . 0 0 9 3 2 7 3 6 1 2 0 . 0 0 0 0 0 0 1 9 7 0 
3 6 4 . 3 3 1 5 3 9 0 2 0 . 0 1 1 6 5 9 6 2 4 1 0 . 0 0 0 0 1 6 8 9 2 9 
2 4 2 . 8 8 7 6 9 2 6 8 0 . 0 1 5 5 4 7 2 9 2 8 0 . 0 0 1 0 8 6 5 5 0 2 
1 2 1 . 4 4 3 8 4 6 3 4 0 . 0 2 3 3 2 5 1 6 8 1 0 . 0 4 6 5 8 2 7 3 7 0 
0 — 0 . 0 4 6 6 8 4 2 0 4 9 0 . 9 9 7 8 2 6 5 0 5 7 
A partial check is obtained by use of (49), writ ten in the form 
[J0(X)Y . [1 + 2{PI + P\PI + PIPIPI + ...)] : = 1. 
T h e particular example furnishes 
1 - 1.00000 00000 0 
p \ = 0.00217 94149 9 2pl = 0.00435 88299 8 
p \ = 0.00054 40634 7 2p\p\ = 23714 8 
CALCULATION 83 
Pi = 0.00024 17183 1 2p\p\pl = 5 7 
Pl = 0.00013 59468 3 2p\p\plp\ = 0 
Sum = [Jo(e)]~2 = 1.00436 12020 3 
Me) = 0.99782 65057, 
in agreement to the ten th decimal place with the value obtained from 
the series. 
In most applications it is necessary also to compute ]s(2e), Js(3e), etc. 
T h e numerical convergence of the series for J0 and of the calculation 
of the values of ps remains satisfactory even for large values of the 
argument that arise with high mult iples of e. 
Following is an example of the use of Bessel functions in the t rans-
formation of a rguments from the eccentric anomaly to the mean anomaly. 
In the development of the series for a'/A, where 2a' is the major axis 
of the orbit of Jupi ter and A is the value of the distance between Jupi ter 
and Mars (assuming elliptic motion for both planets), the following 
terms are found, together with the appropria te Bessel functions, cal-
culated with the eccentricity of Mars 0.09326685, that are needed in 
order to calculate the coefficient of cos (/' — /) by formulas (55), (57). 
All the number s are expressed in uni ts of the eighth decimal (Table 2). 
T A B L E 2 
Terms in a'jA Bessel functions Products 
+ 396 cos (/' + 2M), -2J_3(-e) = - 3379, o, 
+ 41206 cos (/' + u), - /_,(- e) = - 108655, - 45 , 
+ 2879796 cos /', -OJ^i-e) = 0, 0, 
+ 23572402 cos (/' - M), Jo(-e) = -f- 99782651, + 23521168, 
- 108643 cos (/' - 2M), 2M-e) = 
- 9316547, + 10122, 
+ 1677 cos (/' - 3M), 3 / , ( - e ) = 4- 325965, + 5, 
— 17 cos (/' — 4w), 4 / . ( - e ) = - 6757, 0. 
In these expressions the pr imed symbols relate to Jupi ter and the others 
to Mars . Mult ip lying the pairs of numbers on a line, and adding the 
products at the right, we obtain + 0.23531250 for the coefficient of 
cos (/' — /). T h e same Bessel functions will suffice for the calculation 
of the coefficients of sin (JV — /) and cos (JLF — I), J being any integer 
or zero. For other multiples of /, different Bessel functions will be 
needed. 
84 ELLIPTIC EXPANSIONS 
8. S o l u t i o n of K e p l e r ' s e q u a t i o n . T h e numerical solution of 
Kepler ' s equation 
I = u — € sin u 
for u, given e and /, received m u c h attention, as was natural for one of 
the earliest t ranscendental equations in as t ronomy requir ing frequent 
solution. H u n d r e d s of methods have been published. Equat ion (70) is 
practicablefor numerical work when the eccentricity is small. Here we 
shall describe two general methods that suffice for all the needs of the 
practical computer . T h e first is the method of differential correction, 
which is especially useful when no calculating machine is at hand . 
F r o m an approximate value of u corresponding to the given values 
of e and /, a more accurate value is found by successive approximations. 
We denote the approximate value of u by u0 and the initial correction 
to it by Au0f so that u0 + Au0 is a bet ter approximation to u than u0. 
Calculate the value of / corresponding to u0 from Kepler ' s equation 
and, calling it / 0 , denote the difference / — / 0 by Al0. W e have, rigorously 
l0 = u0 — e sin u0. 
T h e n by Taylor ' s theorem, omit t ing the square and higher powers of 
Au0> we have the approximate equation 
*l« = Hr0
Au° 
= (1 — e cos u0) AuQi 
or 
Aun = 0 
u 1 — e cos u0 
Now put ux = u0 + AUQ\ calculate the value of / corresponding to uv 
and calling it lx pu t / — / x = Alv T h e n , by the same process as before, 
we have 
A
 J / l 
Aux = 1 1 — e cos ux' 
and the procedure may be repeated as often as is desired. I t will be 
found that the successive values of AUJ diminish rapidly, so that only 
a few repetit ions are necessary. 
For the first approximation u0 it is convenient to use the following 
abridged form of (70) or (71): 
u = I + e sin / + \ e2 sin 21. 
KEPLER S EQUATION 85 
As a numerical example we show the figures for the calculation of 
u when / == 30° and e = 0.3 precisely. For convenience in using the 
t r igonometr ic tables we mult iply the equat ions by the n u m b e r of 
degrees in a radian, thus using 17.18873 instead of e and 2.57831 instead 
of \ e2. But in forming the divisor of the expression for Au it is necessary 
to use e for itself. So we have 
40?82725 
41.35818 
41.35756 
0.6537806 
0.6607642 
0.6607560 
K 
29?58959 
30.00048 
30.00000 
+ 0?41041 
- 0 . 0 0 0 4 8 
1 
0.77300 
0.775 
+ 0?53093 
- 0 . 0 0 0 6 2 
W h e n a desk calculating machine is at hand it is much more expedi-
tious to use another me thod that does not require any figures to be 
wri t ten down if a table of sines with a rgument in decimals of a degree 
or in radians is at hand . If decimals of a degree are used, mult iply e 
by the n u m b e r of degrees in a radian; all angles will t hen be expressed 
in degrees and decimals. 
Wr i t e the formula 
« = / + e sin u. 
Inser t / in the products dial and e in the keyboard. T h e n , by repeated 
addit ions and subtract ions various mult iples of e are added to or s u b -
tracted from / unt i l the condit ion is fulfilled tha t the n u m b e r appear ing 
in the mult iplier dial, which is an approximat ion to sin u, is equal to 
the sine of the angle in the products dial, which is an approximat ion 
to u. At each stage it is necessary to look u p the sine of the angle that 
is in the products dial, and to increase or decrease the n u m b e r in the 
mult ipl ier dial in the direction indicated. For the preceding example, 
the number s s tanding in the machine at successive stages may be as 
follows. 
Multiplier dial 
0.0 
0.5 
0.62 
0.652 
0.6588 
Products dial 
30.00000 
38.59437 
40.65701 
41.20705 
41.32394 
Multiplier dial 
0.6603 
0.6607 
0.66075 
0.6607560 
Products dial 
41.34972 
41.35659 
41.35745 
41.35756 
In practice the result will be approached much more rapidly than the 
figures above would indicate. T h e compute r soon learns to "overshoot 
sin Uj - COS Uj 
86 ELLIPTIC EXPANSIONS 
the m a r k " in making the successive approximations, so that the actual 
n u m b e r of approximations may be reduced to three or four. 
Ei ther of the methods described is applicable to a high-speed auto-
matic calculating machine, the t r igonometr ic functions being calculated 
in the machine instead of being read from pr in ted tables. As with most 
applications of such machines, the n u m b e r of successive approximations 
required depends m u c h on the ingenuity of the computer . 
9. S o l u t i o n of the e q u a t i o n s of m o t i o n in t e r m s of the m e a n 
a n o m a l y . T h e expansions of the coordinates in elliptic motion in 
the preceding sections have been obtained as Fourier series in which the 
arguments are multiples of the mean anomaly and the coefficients are 
power series in the eccentricity. We shall now treat expansions of the 
coordinates that can be obtained directly from the equations of motion. 
T h e procedure will be to obtain the coordinates as power series in the 
eccentricity, the coefficients of which are Fourier series in the mean 
anomaly. 
T h e equations of mot ion of a planet in an orbit , the plane of which 
coincides with the plane of reference, may be writ ten in polar coordinates: 
These equations are satisfied by a circular orbit with radius a. T h e 
angular motion is then uniform and v = A = nt + A0. T h e two equa-
tions give 
if Gc is the value of the integral of area in the circular orbit with radius a. 
Next, consider noncircular orbits that have the same period, i.e., 
orbits with mean motion n and semi-major axis a. In t roduce 
(78) 
Gc = a2n, 
(79) 
r = a{\ + p\ 
v — A = (80) 
G = G c ( l + y ) , 
and use as independent variable 
A = nt + A0. 
SOLUTION OF EQUATIONS OF MOTION 87 
T h e equat ions, wri t ten as 
may, if the middle t e rm in the left-hand member of the first equat ion 
is wri t ten 
be t ransformed into 
(82) 
For the circular orbit these equations are satisfied by 
p = y = 0. 
Expansion in powers of p and y gives 
g + p = 3 p 2 _ 6pp + l 0 p i __ ... 
+ Y(2-6P + l2P*-20p* + ...) (83) 
+ y\\ - 3P + 6P* - ...) 
and, if v — A = < f , the equat ion of the center, 
= - 2p + 3P
2 - V + 5p 4 - ... 
+ y(l - 2p + 3p* - V + ...). 
(84) 
Consider p and y as small quantit ies of the first order. Ignor ing the 
second powers and products of these quantit ies in the r ight -hand 
members , pu t 
+ PI = 2yi> 
= YI — 2PV 
(81) 
8 8 ELLIPTIC EXPANSIONS 
from which we obtain successively: 
px = Zy1 — e cos (A — o>j, 
— 3y x + 2e cos (A — ou), 
— 3yxA + cx + 2e sin (A — # ) . (86) 
(85) 
Th i s solution is entirely general; e and o> are the two necessary constants 
of integration for the equat ion for pv I t is seen that yx = 0 is necessary 
in order to avoid a t e rm proport ional to A in the difference vx — A = 
for which a periodic function of A is desired. T h e constant cx is taken 
to be zero, so that the mean value of $ may be zero. For brevity we 
introduce / for A — d>. 
T h e next step is to subst i tute p = px + p2> 7 = 72 m t r i e r ight -hand 
members , and to retain te rms of the order of the square of p± and of 
the first power of p2 and y 2 . T h e result is 
: 3P? +72 — 1p2. 
(87) 
T h e value 
p1 = — e cos / 
is subst i tuted in the te rms 3pl appearing in the r ight -hand members , 
giving 
- P2 = + 2y 2 
e2 cos 2/, 
- 72 — 2p2 + e2
 - f e2 cos 2/. 
Integrat ion of the first equat ion gives 
Subst i tut ion of this value of p2 into the second equation gives 
— 3y 2 - e2 cos 2/, 
<?2 = - 3 (y 2 
2 sin 2/, 
a constant c 2 having been omit ted in the expression for S2. 
SOLUTION OF EQUATIONS OF MOTION 89 
In order to avoid a secular t e rm in i 2 it is necessary to choose 
Hence 
I t would have been permissible to add a t e rm a cos / + b sin / to the 
solution for p2. However, this would have dis turbed the d 'Alember t 
character of the solution. Moreover , it is seen that the e in t roduced as 
a constant of integration in px corresponds to the eccentricity, so tha t 
at perihelion (/ = 0), 
p = — e, P l = — e, p2 = 0; 
at aphelion (/ = 7r), 
p = + e, P l = + e, p2 = 0. 
Any addit ion to p2 of the form a cos / would have modified the meaning 
of e> and the addit ion of a sine te rm, b sin /, would have modified the 
meaning of to. 
Proceeding now to the determinat ion of p 3 , <^3, the equations are foundto be 
D2P 
~J%T + Ps = 6PiP* - 6p\ - 6y2/>! + 2y 3, 
DS 
-ffi = - 2Ps + 6pip2 - 4p\ - 2V2pi + 73-
T h e calculation may be arranged as in Tables 3 and 4. 
T A B L E 3 
Calculation for p 3 
(89) 
e3 cos / e3 cos 3/ 
+ 
+ 2 y 3 = + 2 y 3 
3 
2 
^ 2 
- 3 
4- * + 2 
4- ^ + 2 
dX2 + Pz 
Pi 
= + 2 y 3 
= + 2 y 3 4- xe3 cos / 
+ 3tf3 cos 3/ 
- | e z cos 3/ 
e2 cos 2/, 
e 2 sin 21. 
72 = 
(88) 
90 ELLIPTIC EXPANSIONS 
£ 3 cos / e 3 cos 3 / 
- 2/>3 + y 3 = 
+ f>P\P% = 
— 3 y 3 
— 2x 
3 
"2 
+ 3 
+ 4 
+ 2 
+ 1 
— 2y2pl = - 1 — 
- 3 y 3 + <*- 2jc) e 3 cos / + ^ e 3 cos 3/ 
$ 3 = - 3 y 3 A + <*- 2x) e3 sin / 
A constant cz is omit ted in the expression for cfz. 
T h e expression for cf3 requires y 3 = 0. T h e n , in order to obtain 
p s = 0 at perihelion and at aphelion, we take x = + F - T h e final 
result is then 
(90) 
P 3 = 
3 3 = 
?3 cos / - £3 cos 3/, 
ez sin 3/. j £3 sin / 
T h e general character of the solution has now become apparent : 
only even powers of e will occur in the series for y, while odd-order 
contr ibutions to p may require additional terms with a rgument cos / 
in order to satisfy the condition that all contr ibut ions to p beyond p 1 
vanish at perihelion and aphelion. I t may be remarked that these 
addit ions to p of te rms of odd order are necessary only to make e 
correspond to the eccentricity. Suppose the addit ion had not been made 
to p s . Designate the e so int roduced by ev T h e expression for p would 
then have become 
(A) 
instead of 
P = e2 + (—«i + 0 e 3 ) cos/ e\ cos 21 - e\ cos 3/ 
(B) P — e2 + (-e- ez) cos / - e 2 cos 21 - e3 cos 3/, 
T A B L E 4 
CALCULATION FOR @ Z 
SOLUTION OF EQUATIONS OF MOTION 91 
and the transformation from (A) to (B), correct to te rms of the th i rd 
order in e, would have been made afterwards by the subst i tut ion 
3 3 
Finally, the expansion could have been made by subst i tut ing in (82) 
P = Pi + P2 + Ps + P4 + •••> 
y = 72 + r 4 + 
in which use is made of the circumstance that only te rms of even order 
appear in y. By collecting t e rms of corresponding order the equations 
could have been writ ten, at the beginning of the integration, to any 
order desired: 
dX2 
dX2 P2 = ?>p\ + 2y 2, 
d ^ + Pa = fyiPi ~ 6Pi — fyzPv dX2 
d2o 
4 + Pt = GpiPa + 3pa - Wplp*. + 10p 4 + 2y 4 — 6y2/>2 + 12y2p? + y\, dX2 
~dX ~ ~ 2 p i ' 
— = — 2p2 + 3pr + y2, 
— 2p 3 + 6pYp2 — 4p\ — 2y2pv 
= — 2p 4 + 3p2
2 + 6 ^ 3 — \2o\p2 + 5p* + y 4 — 2y 2 p 2 + 3y 2 p
2 . 
T h e calculation of the te rms of the fourth order proceeds as in Tables 5 
and 6. 
dto± 
dX 
d£« 
dX 
92 E L L I P T I C E X P A N S I O N S 
T A B L E 5 
Calculation for /d 4 
e4 e4 cos 21 e* cos 4/ 
+ 3p° 
+ i o r f 
- 6y2p2 
+ \1YA 
+ y\ 
+ 2y« 
= 
+ 2y4 
9 
8 
-4- — 
+ 8 
9 
4 
+ 1 5 
+ 4 
+ 2 
- 3 
+ i 
_ 3 
2 
+ 5 
3 
2 
- 3 
+ 8 
+ 8 
4- ^ 
+ 4 
4- ^ 
+ 4 
= + 2y4 + i * 
— e4 cos 2/ 4- 5e4 cos 4/ 
P4 + 2y4 + *«• + | « 4 cos 2/ — -| g4 cos 4/ 
T A B L E 6 
Calculation for 
e4 e4 cos 21 e4 cos 4/ 
+ M 
+ 6pjp3 
- 12pJP2 
+ 5PJ 
- 2y2p2 
+ 3 W J 
= 
- 4 y 4 
+ y* 
1 
2 
+ 8 
9 
8 
3 
2 
3 
4 
2 
3 
3 
2 
+ 2 
1 
2 
3 
4 
4- ^ 
+ 3 
+ 1 
+ 1 
4 - * + 2 
4- ^ 
+ 8 
= - 3 y 4 8 E — e4 cos 2/ 4- J ^ c 4 c o s 4 / 
4̂ - 3 ( y 4 + 1 9 ~ 6 ~ S I N 4 1 
which would have been integrated independent ly of the equat ion for £ . 
T h e disadvantage of this procedure is that some of the features that 
arose in the course of the integration wi th regard to the evaluation of 
the successive te rms in y are eliminated. These features are essential 
in applications to problems in which the constant of integration corres-
ponding to y is not be available at the start. 
I t is worth not ing that the expansion 
y = y 2*
2 + y 4*
4 + . . . 
illustrates the development of a constant of integration, y, in te rms of 
the constant e which is in t roduced in the course of the solution. T h i s 
procedure was required in order to satisfy the conditions that $ shall 
be a periodic function of / wi thout linear te rm. If a linear t e rm in A 
had been permi t ted in & the solution would have been that of a motion 
in an orbit with a mean motion different from the one prescribed. 
T h e equations of elliptic motion in the plane of the orbit form a 
system of the fourth order. Hence four constants are required for the 
general solution. T h e expansion obtained in this section contains four 
constants, a> e, to, and A0. T h e constant a appears as the constant giving 
the size of the orbit and ( through n) the period, bu t the expansions of 
p and $ are independent of the value of a. Hence the developments 
obtained represent, by permit t ing a to vary, the complete solution of the 
differential equat ions. 
SOLUTION OF EQUATIONS OF MOTION 9 
T h i s requires 
y 4 = 
T h e final results are therefore 
P4 = 
4 cos 21 -
e* sin 21 
?4 cos 4/, 
- e4 sin 4/. 
(91) 
T h e r e is no constant t e rm in /o 4 , and no further constant te rms should 
appear in the higher approximations since the constant t e rm of r/a is 
1 + 2e2> a s obtained in Section 6. 
T h e calculation would have been simplified somewhat by in t roducing 
for G its value 
G = a2n Vl - e\ 
yielding for the equat ion for p 
94 ELLIPTIC EXPANSIONS 
10* T h e r o t a t i n g c o o r d i n a t e s y s t e m . I t was found that the 
expansion of the rectangular coordinates with the aid of the Bessel 
functions is considerably simpler than that of the equation of the 
center. Th i s suggests that perhaps the same condit ion may apply to 
the direct expansion of the solution from the differential equations. 
T h e use of rectangular coordinates also opens up the possibility of 
int roducing imaginary exponentials instead of t r igonometr ic functions, 
which may simplify the operations. In order to obtain coordinates that 
are closely related to p and <f, which are zero in the case of circular 
motion, it is suggested to introduce a rectangular coordinate system 
that rotates uniformly with an angular velocity n, the mean motion in 
a circular orbit from which the deviations are to be obtained. 
T h e original equations in rectangular coordinates for an orbit whose 
plane coincides with the xy plane are 
In t roduce a coordinate system XY rotat ing with an angular velocity n 
in the direction of the motion of the planet. If 
x = 
A = nt + A0, 
then 
X 
Y 
x cos A + y sin A, 
— x sin A + y cos A, 
(92) 
y 
X X cos A — Y sin A, 
X sin A + Y cos A. 
(93) 
Successive differentiation gives 
X — x cos A + y sin A — nx sin A + ny cos A = x cos A + y sin A + nYy 
Y = — x sin A + y cos A — nx cos A — ny sin A = — x sin A + y cos A — nX, 
X = x cos A + y sin A — nx sin A + ny cos A + nY, 
Y = — x sin A + y cos A — nx cos A — ny sin A — nX. 
By subst i tut ing 
— nx sin A + ny cos A = nY + n2X, 
— nx cos A — ny sin A = — nX + # 2 F , 
r2 = x2 + y2' 
ROTATING COORDINATES 95 
(94) 
Now 
But these expressions are identical with the r ight -hand members of 
the equations (94). T h e latter may then be wri t ten 
with 
F = — , r2 = X2 + Y 2 . 
T h e th i rd te rms in the left-hand members may be absorbed in the 
partial differential quotients in the r ight -hand members by put t ing 
F' =F + \n\X2+Y% 
X-2nY = ^ , (96) 
V + 2nX=3£. 
X - 2nY - n2X 
Y + 2nX - n2Y = 
(95) 
X - 2nY - n2X = cos A -f - sin A, 
Y + 2nX - n2Y - sin A cos A. 
- cos A -f • sin A, 
we obtain 
X = x cos A + y1 sin A + 2nY + n2X, 
Y = — x sin A + y cos A — 2wX + n2Y. 
T h u s the equations may be wri t ten 
96 ELLIPTIC EXPANSIONS 
(98) 
Hence, if 
(99) 
the equations are 
(100) 
a£ — lanrj 
(97) 
aij + 2an£ -
If, further, the mean longitude A is used as independent variable the 
equations become 
Suppose we wish to deal with elliptic orbits with mean motion n. 
T h e equations may be satisfied by the circular orbit with radius a> 
related to n by 
^ = a?n2. 
If the constant A0 is chosen properly, theX axis is directed constantly 
toward the planet , and in this circular orbit we have 
X = ay 7 = 0. 
I n order to s tudy the deviations from circular mot ion in elliptic orbits, 
we may introduce 
X = a(l+ I ) , Y = ar,9 
and use £ and rj as our new variables. T h e subst i tut ion into Eqs . (96) gives 
ROTATING COORDINATES 97 
X = r cos (v — A) = r cos $ = a(l + p) cos 
y = r sin (z; — A) = r sin < f = <z(l + p) sin 
Hence 
_ l = (l + p ) C o s « f - 1 
or, conversely, 
(1 + p ) 2 = (l + t ? + v)\ 
P = Vl+2£ + e + V2- i 
tan 6 
(101) 
(102) 
(103) 
= v — £v + £2v — i v3 — £*y + dy* + ••• • 
W e note that at perihelion and at aphelion $ = 0, so that for those 
points of the orbit f = p, rj = 0, and hence at perihelion (/ = 0), 
€ = — e, V = 0, 
at aphelion (/ = 7r), 
f = + e% rj = 0. 
= (1 + p ) s i n « f 
= ^ + p<̂ 
I t is of interest to express the variables £ and rj in t e rms of the quant i -
ties p and $ in t roduced in Section 9. Since 
x — r cos v, y = r sin v, 
it follows from (92) that 
98 ELLIPTIC EXPANSIONS 
For the development of the r ight -hand members of Eqs . (100), say 
to the fourth power in £ and 77, advantage may be taken of the c i rcum-
stance that these r ight -hand members are the partial differential quotients 
of a single function Q. Hence it will be sufficient to develop Q in powers 
of £ and r] to the fifth power and take the derivatives with respect to 
£ and 77 afterwards. W e may write 
We now subst i tute 
f = f 1 + f 2 + & + f * 
V = Vi + V2 + Vs + t?4> 
and collect te rms of corresponding order in the equations. T h e results are 
= 1 - f + e - e + f4 - f6 
In order to obtain jQ, the te rms 
should be added. Hence, 
ROTATING COORDINATES 99 
I t is seen that the equations for each order have the same left-hand 
members , while the r ight -hand members are again functions of 
and rj's tha t have been obtained in the preceding approximat ions . 
T h e equations for gl9 rj1 may be solved as follows. A first integral of 
the second equation is 
If this is mult ipl ied by 2 and added to the first equation there results 
the equat ion 
% + t1 = 2Cl, 
with the integral 
& = 2C1 + A cos (A - B\ 
A and B being constants of integration. Subst i tut ion into the equation 
for yields 
% = - 3CX - 2A cos (A - B) 
aA 
V l = - 3QA + q - 2A sin (A - B). 
We wish a solution in which rj1 is purely periodic with mean value zero. 
Th i s requires 
q = 0, q = o. 
; ^ 2 + 1 0 | I \ - Y ^ i 3 . 
f 3 ^ 3 + 3 ^ 2 + 3 £ 3 7 H - 6 ^ 2 — 1 2 ^ ^ ! 
- 1 2 ^ 2 - 5 ^ + 1 5 ^ -
— 3f 4 = - 6 ^ 3 — 3 | 2
2 + 3T ? 1 T ? 3 - rj2 + 1 2 ^ 2 — 6^2rjl 
+ + 3 ^ i - 6&? : 
- 3 ^ 2 H - 3£ 2 = -
100 ELLIPTIC EXPANSIONS 
Fur ther , the perihelion and aphelion conditions for fx require 
A = — e, B = GJ. 
In t roducing the mean anomaly / = A — <2, we finally obtain 
£i = — e cos /, rj1 = -\- 2e sin /. 
In all further approximations the aim will be to have, at perihelion 
and aphelion, £ 2 = 0, £ 3 = 0, etc. 
For any successive approximation, say of £j9 rjj9 the equations will be 
where Xjf Yj are the r ight -hand members which will be found to be 
cosine series and sine series, respectively, in /. Th i s may be seen at 
once for X2, Y2 by subst i tut ion of the results for gl9 rjv Hence Yj will 
have no constant t e rm and the integral of the second equation gives 
(104) -2f, = C , + J Y,d\, 
Cj being a constant of integration. By adding twice this integral to the 
equation for ^ we obtain 
r - 6 = 2C, + * , + 2 j Y,d\. (105) 
T h e constant t e rm of ^ obtained from this equation is 
f,o = 2Cj + Xi0, (106) 
in which Xj0 denotes the constant te rm of Xj. Subst i tut ion into (104) 
gives a constant te rm in drjj/dX, 
— 3C3- — 2Xj0. 
Since this must be zero, Cj is determined by 
Cj = — § Xj0. 
If this is subst i tuted into the r ight -hand member of Eq. (106) it is 
seen thaj the constant te rm in this r ight -hand member , and therefore 
the constant te rm in becomes 
= (107) 
ROTATING COORDINATES 101 
For the th i rd approximat ion we obtain 
e3 cos / e* cos 3 / 
3 3 
3 3 
4 + 3171̂ 2 = + 4 
+ 4 g = - 3 - 1 
- 6TLVL = + 6 - 6 
e* sin I e3 sin 3/ 
+ 3 ^ 2 = 
9 3 
+ = — 2 + 2 
- fiffo = - 3 - 3 
+ = + 9 - 3 
9 39 
F 3 = + q e 3 sin / — £ 3 sin 3/. 
o o 
Y2 
Y2dX = 
e 2 sin 2/, 
e 2 cos 2/, 
e 2 cos 2/ 
X2 + 2 J F 2 ^ A = £2 cos 2/, 
e 2 cos 2/, 
e2 cos 2/, 
• e2 sin 2/. 
X3 = e3 cos / - ez cos 3/, 
Th i s will hold for every approximation, and it becomes therefore u n -
necessary to obtain C, explicitly. T h e procedure is s imply to determine 
the constant in ^ by (107) and to ignore the constant that arises in 
DRJT/DX by (104). 
For the second approximat ion we obtain 
For the determinat ion of | 4 , ry4 we have Tab les 7 and 8. 
T A B L E 7 
C a l c u l a t i o n o f AT4 
e4 e* cos 21 e* cos 4 / 
= 9 
8 4-- a 
^ 8 
9 
8 + 2 
3 
8 
+ 317x773 = 9 
8 
+ 2 7 
8 
_L 3 2 
+ A 
3 
64 
+ 1 2 # a 
3 
2 + 2 
+ 9 — 12 + 3 
— 12^172 4- -3-+ 2 
3 
2 
15 
8 
5 
2 
5 
8 
+ 1 5 ^ + 2 
15 
2 
_ 1 5 4 
8 î 
= 45 
4 4~ 15 15 
4 
4- — e4 
^ 64 * + 4e 4 cos 21 — W * 4 c o s 4 / 
102 ELLIPTIC EXPANSIONS 
£3 cos / - - e3 cos 3/, 
Xz + 2jY3dX = — 3 ez cos 3/, 
^3 = e 3 cos / H e 3 cos 31. 
T h e te rm — § e3 cos / has been added in order to make f3 zero at 
perihelion and aphelion. 
C3 - 2f 8 H y 3 = e 3 cos / + e 3 cos 3/, 
*?3 = £ 3 sin / H - £3 sin 3/. 
ROTATING COORDINATES 103 
ex sin 21 e4, sin 41 
+ i 
7 
1 6 
3 
8 ^ 1 6 
+ = 9 
4 + 8 
- 6 ^ 2 = 3 
4 
3 
8 
- n^Uii = - 6 + 3 
I 9 2 _ 
+ 2 — + 4 
9 
8 
+ mim = - 5 5 
2 
1 5 > 3 _ + 15 1 5 
~ 2 
Y, = + 3e4 sin 2/ - ^g1 e4 sin 41, 
- f e4 cos 2/ + f|- e4 cos 41, 
- f e*> cos 2/ - -W* 4 c o s 4/> 
— J e4 cos 2/ + _ 6 _ 7 _ , 4 c o s 4 / > 
— f e4 cos 21 + ff e4 cos 4/, 
- <?4 sin 21 + M *4 s i n 
I t is a useful exercise to verify that the results for £ and rj obtained 
by this method agree with the series for (r/a) cos / and (r/a) sin / given 
in Section 6 of this chapter . 
T h e calculation of the fourth-order t e rms by subst i tut ion of the 
previous solutions in their t r igonometr ic form in the r ight -hand members 
X^y Y 4 is, at least for some of the terms, rather tedious. T h e use of the 
imaginary exponentials is recommended . For example, if A = exp il, 
£ 3 = + ^-*{A* - A - A-* + A-*). 
T h e n , by multiplication, 
- 6 ^ 3 = e* (A* - 2 + A-*) 
T A B L E 8 
CALCULATION OF F 4 
11. C o m p l e x r e c t a n g u l a r c o o r d i n a t e s . T h e greater ease in 
performing operations with imaginary exponentials than with t r igono-
metr ic functions suggests the use of imaginary variables in the differen-
tial equations. T h e variables that appear to have particular advantages 
are 
u = £ + irj = - 1, 
s = £ - i-q = r-E-i(f-l) - 1, 
so that 
(108) 
dhi du dQ 
% 
+ i 8 Q 
dt] 
dh 8Q . dQ 
d\* dr/ 
T h e derivatives of Q with respect to u and s may easily be expressed: 
dQ dQ d£ , dQ drj 
du d£ du drj du 
1 dQ , \ dQ 
2 d£ 1 2i dV 
1 / dQ . dQ\ 
2\d£ drj ) 
Equat ions (100) may then be wri t ten 
By adding and subtract ing these equations it follows that 
104 E L L I P T I C E X P A N S I O N S 
Also, 
(1A* -9A + 9A~l - 7A-*), 
(1A* - 16 A2 + 18 - 16A-2 + 7A-*) 
- 2 cos 21 — cos 4/j . 
C O M P L E X C O O R D I N A T E S 105 
8s 2 \ 0 ? dr) J 
T h i s gives for the equat ions (108) 
Since 
it follows that 
£2 + = us, 
T h u s 
= I (1 + «) (1 + ,) + (1 + u)^'2 (1 + s)~v\ 
Following G. W . Hill we introduce the notat ion 
(109) 
(1 + + I2 = 1 + % + ¥ + V2 
= 1 + u+s + us (110 
= (1 + « ) ( ! + « ) . 
( I l l ) 
(112) 
T h e principal advantage of this form is that the imaginary uni t does not 
appear, and will not have to appear in the solution if the developments 
are made with imaginary exponentials. An impor tan t feature of the 
variables u and s is that they are conjugate complex. Hence it is necessary 
only to obtain the solution for one of the two;the other can be found by 
taking the conjugate. 
and, similarly, 
3Q _ 1 / _ a Q . 3Q\ 
so that 
T h e equat ions become 
D2u + 2Du 
D2s - 2Ds = 
106 ELLIPTIC EXPANSIONS 
T h e n 
Th i s expression is subst i tuted into the first of Eqs . (112), and the equa-
tion is split u p into separate equations for the te rms of the first order, 
second order, etc., by int roducing 
u = ux + u2 + us + ... 
s = *i + s2 + *3 + •••• 
T h e resulting equations are 
D2ux + 2Dux -
D2u2 + 2Du2 (u2 + s2) = 
T h e corresponding equations, 
D\ - 2Dsx -
D\ — 2Ds2 -f 
(«2 +
 S2) = etc., 
may be writ ten down by merely interchanging u and s everywhere in 
the r ight -hand members . 
T h e solution desired for W ,̂ S-^y IS of the form 
u, = aA + bA~\ 
(113) 
s1 = bA + aA-1, 
in which A = Eil, I — A — co and a and b are real factors to be deter-
mined. 
T h e definition of the differential operator D shows that 
DE**X = pEivX, D2EivX = p2EipX. 
Chapter X I V contains an illustration of the use of complex rectangular 
coordinates in the problem of the moon ' s motion. A detailed application 
to the problem of elliptic motion is therefore omit ted, and only the 
first steps are indicated. 
Expansion of Q, given by (111), in powers of u and s gives 
COMPLEX COORDINATES 107 
D 2tt x 
2Du, 
= aA+ bA-1 
= 2aA - 2bA-1 
2 { U l = \{a + b)A + + b)A~* 
0 - & . + \>)A +(! \a+\*)A-> 
D\ 
IDs, 
= bA+ aA-1 
= - 2bA + 2aA - 1 
= j(a + b)A + 
2 > 
+ b)A~i 
In order to satisfy the equations, either a or b may be chosen arbitrarily, 
but the two constants must satisfy the relation 
3a + b = 0. 
We adopt 
a = + - e, b = — -e, 
so that the solution becomes 
ux = + ±eA - | e^l"1, 
I t is seen that this solution corresponds to 
(A + A-1) = — e cos 7, 
(114) 
{A - A-1) = +2e sin /, 
and is therefore in agreement with the requi rement that e shall be the 
Hence 
DA* = pA*>, D2Ap = p2Av. 
Subst i tut ion of the solution into the equations for ux and s1 gives 
108 ELLIPTIC EXPANSIONS 
eccentricity of the orbit . In order to assure this for the higher approxi-
mations, conditions mus t be in t roduced at perihelion (/ = 0), 
A = Ei0 = + 1, u = £ = — e, s = g = — e, 
and at aphelion (/ = 7r), 
A = Ein = — 1, u = g = + e, s = £ = + e. 
Since the first approximat ion gives at perihelion, 
ux = — e, s1 = — ey 
and at aphelion, 
ux = + ey sx = + e, 
it is necessary that , at perihelion and at aphelion, respectively, 
« 2 = W 3 = W 4 = - = °> 
^2 = ^3 = ^4 = ••• = ^ ' 
These conditions are sufficient for the evaluation of the coefficients of 
A0 in the even approximations and A1, A-1 in the odd approximations. 
12. E x p a n s i o n s b y h a r m o n i c a n a l y s i s . In any practical applica-
tion of celestial mechanics the aim is finally to obtain results in numerical 
form. I t is always possible, at least in principle, to solve a specific problem 
requir ing extensive calculations by retaining some or all of the para-
meters involved (say the elements of an elliptic orbit) in literal form 
until the final s tep. Since the invention of calculating machines and their 
gradual increase in power and efficiency, it has become more efficient 
to introduce numerical developments at earlier stages of a problem, and 
sometimes at the very beginning, as with the numerical integration of 
orbits. I t is impor tant for the worker in the field to be familiar with 
both literal and numerical methods , so that he can use the one best 
adapted to the case in hand; it often happens that a mixture of the two 
is useful. A common example is expansion in Fourier series where the 
arguments are retained as literal quantit ies while the coefficients are 
expressed as number s . 
T h e numerical coefficients of the series given in preceding sections 
of this chapter may be calculated easily if the eccentricity is small enough, 
bu t in cases where it is much greater than 0.1 or 0.2 the numerical 
coefficients may be found more readily by harmonic analysis. Also, 
H A R M O N I C A N A L Y S I S 109 
the series developments of such functions as rm cos"/ , m and n being 
any integers, may be m u c h facilitated by harmonic analysis. 
Le t 8 be any independent angular variable, and F a periodic function 
of 8 which it is desired to express in the form 
F = \ c0 + q cos 8 + c2 cos 28 + ... + \ cn cos nd 
+ Si sin 8 + s2 sin 26 + ... + sn sin nd> 
the coefficients being number s . If 2n + 1 special values of 8 are chosen, 
and if for each special value the value of F is calculated as well as the 
values of the t r igonometr ic functions, then there result 2n + 1 linear 
equat ions from which, if they are independent , the coefficients may be 
calculated. T o render the equat ions independent it is sufficient to divide 
the period of 8 into an even n u m b e r of equal par ts . If the initial value of 
8 is taken as zero it is found that 2n parts suffice to determine all the 
coefficients wi th the exception of sn. 
If F(8) = F(— 6), then the series consists exclusively of a constant 
and cosine te rms, in which case only the first n + 1 special values of 
F, F(0) and F(7T) included, are distinct from the remaining ones. If 
F(6) = — F(— 0 ) , then the series consists exclusively of sine te rms, 
F(0) = F(TT) = 0, and only the first n — 1 special values of F, F(0) 
and ^ ( 7 7 ) excepted, are distinct from the remaining ones. 
In the general cases, where both cosines and sines are present , denote 
the angle 2ir\2n by a, and take the special values of 8 equal to 0, a, 2a, 
... noc,(2n — l)oc in tu rn . T h e n the solution of the equat ions is given by 
1 x^ 1 
ck = - 2j F(J a ) cos kjot, k = 0, 1, 2 , « , 
1 2 ^ : 1 
sk = -Z/ FU0C) s i n * = 1. 2 , n — 1. 
71 3=1 
I t will be noticed that since sin noc = 0, there is no formula for sn. 
Also it will be noticed that since cos noc = — 1, cos 2noc = + 1, etc., 
the formula for cn reduces to the sum of the special values taken with 
alternating signs, divided by n. 
T h e foregoing formulas, al though they are very systematic and 
therefore adaptable to automatic calculating machines, are not well 
suited to hand calculation because they lead to needless repeti t ion of 
the same operat ions. For hand calculation it is easy to derive a m u c h 
less laborious scheme for any part icular value of 2n. As an example 
we give here a scheme for 2n = 8. Deno te the special values of 
110 ELLIPTIC EXPANSIONS 
F by F0y Fv F29 F7. T h e n the coefficients are given by the following 
equations: 
0.4 + ^ 4 , 
0 
4 = *"o -FA, 
1.5 + F» 
1 
5 = *"i - F » 
2.6 + FE, 
2 
6 = F* ~F» 
3.7 = ^ 3 + F„ 
3 
7 
= FS 
Thi s check cannot be relied upon to the last decimal since the end-
figure errors of the coefficients are increased by the multiplications. 
Both checks should always be applied. 
T h e series given by a harmonic analysis may be regarded as an inter-
polation formula suitable for obtaining values of the function inter-
mediate between those specially calculated. As with any interpolation 
formula, the accuracy of an interpolated value increases as the n u m b e r 
of specially calculated values of the function that are taken into account 
0.2 = 0.4 + 2.6, 
1.3 = 1.5 + 3.7, 
4 * 0 = 0 . 2 + 1.3, 2 ( * 1 + * 8 ) = j , 
4* 4 = 0.2 - 1.3, 2fo - r 3 ) = | cos 45°, 
4c2 = 0.4 — 2.6, 2(s1 + s3) = sin 45°, 
4s 2 = 1.5 — 3.7, 2(sx — s9) 
A valuable check on the results may be obtained for the cosine 
coefficients by the condit ion 
-^0 = \CQ ~t~ Cl + C2 H~ ••• ~f~ \Cn' 
A complete check would consist in verifying that every special value of 
F is correctly given by the series. 
A valuable check on the sine coefficients may be obtained by finding 
the value of dFjdd for 0 = 0. T h e n 
s1 + 2s2 + 3*3 + . . . + (n — 1) sn-.v 
HARMONIC ANALYSIS 111 
increases. T h e coefficients obtained by harmonic analysis are only 
approximations, which can be made as accurate as we please by taking 
n sufficiently great. I t can be shown that thecoefficient we have denoted 
by ciy i < n, would, if the series were indefinitely extended, be equal to 
and the value of cn is 
cn ~T~ cZn ~f~ cbn "f" •••• 
Hence, the lower the order of a cosine coefficient the higher the 
accuracy with which it is found, with the exception of cny which is more 
accurate than any of the preceding ones. 
Similarly, if the series were indefinitely extended, st would represent 
and again it is seen that the coefficients of lower order are de termined 
with higher accuracy. 
As a good working rule the n u m b e r of special values should be taken 
large enough so that cn¥1 and sn may be regarded as having zero values. 
I t is sometimes difficult to decide in advance how many special values 
of the function will yield results of the required accuracy without 
unnecessary labor. If only a few analyses are to be made, it is possible 
to begin with eight special values, and to start the analysis by calculating 
cn and sn_x. If either of these has appreciable values, the n u m b e r of 
special values may then be doubled, t hus saving the ones already 
calculated, and the test repeated. If many analyses having similar 
convergence are to be performed, it will be preferable to exper iment 
with an intermediate n u m b e r of special values. 
Special schemes of analysis may be devised that make it possible to 
increase the n u m b e r of special values only slightly, and still to save 
most of the work already done. Examples of such schemes are given 
by Brown and Brouwer . 4 
As an example of harmonic analysis we calculate the Fourier expansion 
of the function 
F = + [1 - 0.6 cos (0 + 30°)] 1 / 2 
with eight special values, to five decimals. 
F0 = 0.69310 F± = 0.91908 F2 = 1.14018 F3 = 1.25680 
F 4 = 1.23273 F5 = 1.07484 Fe = 0.83666 F 7 = 0.64842 
4 E . W . B R O W N AND D . BROUWER, Tables for the development of the disturbing function 
with schedules for harmonic analysis. Trans. Yale Univ. Obs. 6, Pt. 5 . , 1 4 3 ( 1 9 3 2 ) . 
112 ELLIPTIC EXPANSIONS 
0.4 = + 1.92583 °r = - 0.53963 
1.5 = + 1.99392 0.2 = + 3.90267 ^ = - 0.15576 
2.6 = + 1.97684 1.3 = + 3.89914 ? = + 0.30352 
3.7 = + 1.90522 I = + 0.60838 
4c0 = + 7.80181 2 (« i + c3) = - 0.53963 
4c 4 = + 0.00353 2(^1 - c3) = - 0.54033 
4c 2 = - 0.05101 2(h + h) = + 0.32005 
4s 2 = + 0.08870 2 («i - * 3 ) = + 0.30352 
1 , _ 
2 c 0 — 
+ 0.97523 
- 0.26999 = + 0.15589 
C 2 = - 0.01275 ^2 = + 0.02218 
c 3 = + 0.00018 *3 = + 0.00413 
1, _ 
2 C 4 — + 0.00044 
+ 0.69311 + 0.21264 
Check: + 0.69310 Check: + 0.21642 
I t is evident from the convergence of the coefficients that the cosine 
coefficients can be relied upon to not more than four decimals and the 
sine coefficients to not more than three . In other words, an interpolated 
value of the function calculated from the series would be accurate to 
not more than three decimals. T o obtain more accurate results it would 
be necessary to repeat the analysis with a greater n u m b e r of special 
values. 
Whenever a function of two independent angular variables a and j3 
can be expressed in the form 
F(a, /?) = 2 (Ajtk cos jot cos kfi + Bj fc sin joc sin k/3 + C3 k cosja sin fyS 
+ Dj k sin joc cos 
where J, K = 0, 1, 2, the numerical calculation of the coefficients 
can be reduced to a double application of harmonic analysis for a single 
variable. 
H A R M O N I C A N A L Y S I S 113 
Suppose the period of a to be divided into 2m parts and the period 
of fi into 2n par ts . T h e n Amn special values of F are required. Beginning 
with the 2m special values corresponding to fi = 0 calculate the m + 1 
cosine coefficients corresponding to j = 0, 1, 2, m> and the m — 1 
sine coefficients corresponding to j = 1, 2, m — 1. T h e n repeat 
the process for the 2m special values of F corresponding to another 
special value of /J, then do the same for the remaining special values 
of F. In this way 2n special values of the coefficients of cos joc, sin joc 
will be obtained. Choosing any set of these corresponding to a single 
value of j , say the 2n special values of cos 2a, analyze these with respect 
to fi; the results are the coefficients A2 fc, C2 k. T h e other coefficients 
are obtained in similar fashion. 
T h e analysis being completed, it will usually be desirable to express 
the result as a conventional double Fourier series, by replacing products 
of sines and cosines by sums and differences. T h e following formulas 
suffice for the purpose . 
A cos joc cos kfi = + \ A cos (joc + kfi) + \ A cos (joc — A/?), 
B sin joc sin kfi = — \ B cos (joc + kfi) + \ B cos (joc — kfi), 
C cos joc sin kfi = + ^ C sin (joc + kfi) — \ C sin (joc — kfi), 
D sin joc cos kfi — + J D sin (/a + kfi) + ^ Z) sin (;a — fyS). 
W h e n either j or & is zero or both are zero, the formulas are not used, 
the te rms already being in the desired form. I t is wor th not ing that 
B0 k and Bj 0 do not appear. Also, if the analysis is made first with 
respect to a, then D0 k does not appear, and in any case C 0 0 and D0 0 
are absent. I t is also worth not ing that when j = k, both differing from 
zero, the formulas yield formal values for coefficients of sin 0; these 
are fictitious and should never be wri t ten down because they may lead 
to errors in subsequent operations. 
I t is always possible and desirable to arrange the series so that either 
j or k does not take on negative values or, alternatively, so that one of 
t h e m does not take on positive values, using the relations 
+ \A cos (joc + kfi) = + \A cos ( — joc — kfi), 
+ \ C sin (j<x + kfi) = — ijr C sin (-joc - kfi). 
W h e n this has been done, then supposing j not to be negative, the 
series should be arranged in blocks, so that within each block j is constant 
and k ranges from its largest positive to its largest negative value. 
114 ELLIPTIC EXPANSIONS 
N O T E S A N D R E F E R E N C E S 
E . W . B r o w n a n d C . A . S h o o k , Planetary Theory, C h a p t e r s II a n d I I I , C a m b r i d g e 
U n i v . P r e s s , L o n d o n a n d N e w York , 1933 , c o n t a i n s m u c h u s e f u l in format ion 
o n t h e d e v e l o p m e n t of f u n c t i o n s in e l l ipt ic m o t i o n . V a r i o u s s c h e d u l e s for h a r m o n i c 
analys is are c o n t a i n e d in A p p e n d i x A of Planetary Theory, as w e l l as in E . W . B r o w n 
a n d D i r k B r o u w e r , Tables for the Development of the Disturbing Function, C a m -
b r i d g e , U n i v . Pres s , L o n d o n a n d N e w York , 1933 , a n d Trans. Yale Univ. Obser-
vatory 6, P t . 5 (1932) ; 
CHAPTER I I I 
GRAVITATIONAL ATTRACTION BETWEEN BODIES 
OF FINITE DIMENSIONS 
1. I n t r o d u c t i o n . T h e law of gravitational attraction is valid for 
two material particles, not for bodies of finite dimensions and with 
arbitrary distr ibution of mass. I t will be shown, however, that spherical 
bodies with a dis tr ibut ion of mass such that layers of equal density are 
concentric spheres, attract each other as if the mass were concentrated 
in their centers. Moreover , it will be shown that if the distance between 
two bodies is large compared with their dimensions, the attraction 
between them is sensibly as if their mass were concentrated in their 
centers of mass. These results make it possible in many cases to disregard 
the dimensions and the distr ibution of mass, and to treat the gravitational 
attraction between two bodies as if they were material particles. 
Nevertheless, there are cases in the solar system and in systems of binary 
stars in which deviations from sphericity have impor tant effects. Hence 
it is necessary to examine the case of gravitational at traction between 
two finite bodies each having an arbi trary distr ibution of mass. Th i s 
problem presents considerable complications. It is much easier to treat 
the attraction between a body of finite dimensions and a material particle. 
Th i s simplified problem applies to many cases that present themselves 
in astronomy,and will be treated first. 
2 . A t t r a c t i o n of a par t i c l e b y a b o d y of f inite d i m e n s i o n s 
a n d a r b i t r a r y d i s t r i b u t i o n of m a s s . In a Cartesian coordinate 
system, fixed in the sense of Newtonian mechanics , let X0y F 0 , Z 0 be 
the coordinates of the center of mass E of a body M , and let X, Y, Z 
be the coordinates of a particle at P with mass m. Consider at Q an 
element of mass dM of the body M, and let rj, £ be the coordinates 
of Q in a coordinate system with axes parallel to those of the fixed system 
bu t with its origin at E. Pu t 
x = X — X0, y = Y — y 0 , z = Z — Z 0 , 
so that x, y, z are the coordinates of m relative to the center of mass 
of M. T h e distance A between P and Q is obtained by 
A' = (* - f ) 2 +{y- jf + (z- if. (1) 
115 
These integrals may be pu t in a more precise form by introducing the 
density, K> which mus t be considered a finite function of the coordinates 
£, r), £. T h e n 
dM = K(g9ri9Z)d£dvdZ, 
and the components of the force are expressed as triple integrals, the 
integration to be extended over the whole mass M. T h e components 
then become 
116 BODIES OF FINITE DIMENSIONS 
Hence the three components of the force due to the attraction by dM are 
Integrat ion over the entire body M yields the components of the force 
Fx=fm 
(M) 
d£ drj d£ 
(2) b\ =fm 
(M) 
Fz =fm 
J (M) 
d£ drj dl 
In this form three triple integrations are required. I t is possible to 
obtain expressions in which only one triple integral occurs. Th i s form 
is obtained by considering the partial derivatives of J 2 with respect to 
x9 y, z. I t follows from (1) that 
AdA = (x — £) dx + (y — rj) dy + (z — £) dzy 
or 
T h e force acting upon the particle m, due to the attraction by the 
element of mass dMy is fm dMjA2 in the direction PQ. T h e components 
of the force along the three coordinate axes are obtained by mult iplying 
the force by the direction cosines of the direction PQ. Since the 
coordinates of Q relative to P are £ — x> rj — y, £ — zy it appears, with 
the aid of (1), that the direction cosines are 
ATTRACTION BY FINITE BODY 117 
In these integrals x, y> z are to be regarded as parameters that mus t 
be t reated as constants in the integrations. Also, the limits of integration 
are independent of these parameters . I t is permissible, in this case, to 
interchange the order of differentiation with respect to a parameter and 
integration, provided that everywhere within the range of integration 
A 7^ 0. Th i s condit ion merely states that the particle m should not be 
a part of the mass M\ it should be either exterior to M or within a cavity 
in it. T h e case that the particle is a par t of M would require special 
investigation. 
Now define the potential U by 
A (6) 
Similarly, the forces acting upon the different e lements of M may be 
transferred to the center of mass of M, and the resultant force will be 
equal and opposite to the force acting upon m. Hence the components 
may be writ ten 
T h e left-hand member may be writ ten as d(A 1). Hence 
(3) 
Subst i tut ion of (3) into (2) gives for the components of the force 
(4) 
Fx=fm 
(M) 
(M) 
Fv=fmj 
F% =fin 
) KM) ^ 
d£ drj 0%. 
I t follows from (4) that the components of the force acting upon 
m may be wri t ten 
(7) 
(8) FT F m -
1 V ~ 
- m Ff 
u = 
' (M) 
d£ drj 0%, (5) 
which, for brevity, will usually be wri t ten 
dM 
118 BODIES OF FINITE DIMENSIONS 
where r is defined by 
U = 
r2 = x2 + y2 
T h e methods used in the theory of the potential are of great importance 
in the theory of equi l ibr ium of rotating fluids. I ts application to the 
figure of celestial bodies is properly a province of dynamical astronomy. 
Th i s subject, however, lies outside the scope of th is book. 
3* L c g c n d r e p o l y n o m i a l s . For the subjects treated in this 
volume it is sufficient to obtain a method of expressing U as a function 
of x, y, z that is generally applicable when the distance between m and 
the center of mass of M is large compared with the dimensions of the 
body M. On this assumption a development in series is available that 
will be presented in the following sections. 
For later applications it is desirable to re turn to the original fixed 
Cartesian coordinate system. T h e n U mus t be considered a function 
of the differences X — X0i Y — Y 0 , Z — Z 0 , and the components of the 
force on m become 
(9) 
T h e components of the force that determines the motion of the center 
of mass of M are 
(10) 
T h e problem of finding the mutua l attraction between M and m has 
been reduced to that of obtaining U9 defined by (5) as a function of 
xy yy z. General propert ies of U are s tudied in the theory of the 
Newtonian potential . T h e most fundamental of these is that for all points 
not belonging to the mass M , U satisfies the partial differential equation 
of the second order: 
d2u , d2u , d2u 
8x2 1 By2 1 dz2 
This equation was first given by Laplace. I t is obtained immediately 
by partial differentiation of (3). Wi th the aid of this equation it may 
be shown that the potential for a body M with spherical symmetry 
becomes 
LEGENDRE POLYNOMIALS 119 
A2 = r2-2(x£+yrj+zQ+p2 
For brevity, pu t 
(12) 
: q — cos S, (13) 
so that 
A2 = r2 (1 - Iqoc + a 2). (14) 
I t is evident from (13) that S is the angle POQ and that , therefore, 
I ? I < i-
T h e integrand of 
(15) 
will now be developed as a power series in oc with the aid of the poly-
nomials of Legendre , Pn{q), defined by 
= P 0 + Pi* + P2oc2 + ... + Pnoc« + ... (16) 
in which Pn are polynomials in q of degree n. 
T h e general expression for Pn(q) is obtained by the binomial develop-
ment of the left-hand member of (16) as 
[1 - {Iqoc - oc2)}-1'2 (17) 
where the usukl notation is employed for the binomial coefficients 
Let p be the distance of an element of mass dM from the center of 
mass of the body M. T h e n 
e + v2 + z2
 = p2- (ii) 
I t is then unders tood that for any point of the body My p/r< 1. T h e 
expression (1) for A2 may now be writ ten 
120 BODIES OF FINITE DIMENSIONS 
Since the r ight-hand members of (16) and (17) should be identical, the 
general expression for Pn(q) is obtained by finding the coefficient of a n 
in the r ight-hand member of (17). Th i s is a double series in a and q, 
and in the r igh t -hand member of (16) this double series is arranged as a 
single series in a with coefficients that are functions, polynomials in this 
case, of q. 
Collection of the coefficients of an in the r ight-hand member of (17) 
gives for the general expression of Pn(q) 
Po(q) = l, 
= 
PM = \ W -i) . 
P*{q) 3?), 
PM) - 30<?2 + 3), 
P*{q) - 7 0 9
3 + 15?), 
P*{q) =4^( 
' - 315? 4 + 105?2 - 5). 
These polynomials will be of frequent usefulness. 
Another form of development is obtained by pu t t ing 
then, in view of (13), 
2q = or + a~1. 
Also, 
2 cos nS = o-n + <j" n, 
1 - 2qoc + oc2 = (1 - oca) (1 - aa" 1 ) , 
(1 - Iqoi + a 2 ) - 1 / 2 = (1 - oca)-1'2 (1 - a a " 1 ) " 1 7 2 
(18) 
F rom this expression it appears that Pn is a polynomial of degree n 
with only odd powers of q if n is odd and with only even powers if n 
is even. Subst i tut ion of part icular values of n gives the following 
polynomials of Legendre of order 0 th rough 6: 
(19) 
POTENTIAL—PRINCIPAL PARTS 121 
2 cos (n - 4) S + .... 
T h e two binomial series of which the produc t is formed in the r ight-
hand member of (19) are absolutely convergent for |a| < 1 since |a| = 1. 
Hence the double series obtained is absolutely convergent for |a| < 1. 
T h e following propert ies of Pn will be useful for our immediate needs: 
(a) Pn(+ 1) = + 1, P«(- 1) = ( - I f . (21) 
Th i s proper ty follows at once by put t ing q = ± 1 in (14) which then 
reduces to the special cases 
1 = 1 ± oc + a
2 ± a 3 + ...; 
1 Toe 
(b) \Pn(q)\ < 1 if - K q < + L (22) 
Since the coefficients in the r ight -hand member of (20) are all positive,the max imum value of Pn for real values of S is the sum of the 
coefficients. I t is reached for S = 0, q = + 1; this, in view of (21), 
proves (22). 
T h e importance of (22) for our present purpose is that it shows that 
the series (16) is absolutely convergent for |a| < 1 and all values of q 
that belong to real values of S. T h e proof is obtained by comparing 
(16) with the absolutely convergent series for |a| < 1, 
= 1 + oc + oc2 + ... + ocn + .... (23) 
1 - ( 
T h e absolute values of the coefficients in the r ight -hand member of (16) 
being at most equal to the corresponding coefficients in (23), the former 
mus t also converge absolutely. 
4 . T h e p r i n c i p a l p a r t s of U. Return ing now to the integral (15) 
for U, subst i tut ion of (16) yields an absolutely convergent series, for 
oc < 1. I t is therefore permit ted to write 
T h e two series are mult ipl ied, and the coefficient of ocn yields: 
• 2 cos (n — 2) S 
(20) 
Pn(cosS) 2 cos nS 
dM. (24) 
122 BODIES OF FINITE DIMENSIONS 
Let this integral be writ ten 
u=u0 + u1 + u2 + us + .... 
T h e individual parts are to be evaluated by restoring | , rj, £ in the 
integrands by using qp = (x£ + yy + #£)/r-
T h e various parts of U are as follows: 
U0=£(dM = f^. (25) 
Th i s is the result that would be obtained if all the mass of the body 
M were concentrated in its center of mass. 
But these three integrals are zero since rj9 £ are measured in a 
coordinate system that has the center of mass of M as origin. Hence 
f/x = 0. (26) 
T h e expression for U2 is 
which may be writ ten 
(27) 
T h e six integrals that appear in this expression are moments of the 
second order which are related to the moments of inertia of the body M. 
In order to arrive at a more suitable form, let I be the moment of inertia 
about an axis through the center of mass, and let a, /?, y be the direction 
cosines of this axis. T h e n , if 6 is the angle between this axis and the line 
connecting an element dM with coordinates £, rj, £, 
I = J p2 sin 2 BdM, 
POTENTIAL—PRINCIPAL PARTS 123 
X = Y : Z = X2 + Y 2 + Z 2 
if X, Y, Z are the coordinates of any point on the surface of the ellipsoid 
and oc, fi, y the direction cosines of the line joining this point with the 
center. T h e last of these shows that the distance of any point X, Y, Z 
from the center equals 1 / V I , which proves that the surface has no 
real points at infinity since the moments of inertia of a three-dimensional 
body about any axis are different from zero. T h e surface is therefore 
an ellipsoid and no other form of quadrat ic surface. 
By a proper choice of axes the equation of an ellipsoid may be put 
in the form 
AX2 + BY2 + CZ2 = + 1. (30) 
where p 2 = £ 2 + rf -f £ 2 as before. T h i s integral may be developed by 
making use of 
p cos 6 = & + r)p + £y. 
Hence 
/ = J 0 2 - ( f « + ^ + ^ ) 2 ] d M , 
which may be writ ten as 
J = J[(L 2 + *?2 + £ 2 ) ( « 2 + P2 + R 2 ) - (F* + + Cy)2] dM 
= /[ft2 + i2) «2 + (C2 + A ]8 2 + ( £ 2 + V2) J2 ~ 2ritPy - lUyoc-l^ocfi] dM. 
T h e moments of inertia A , B , C about the X, Y, Z axes, respectively, 
and the products of inertia D , E, F are defined by the integrals 
A = $(rj2 + £2) dM, D = fr)£ dM, 
B = Stt2
 + £2)dM, E = jtfdM, 
C = J (£ 2 + T)2) dM, F = fijr) dM. 
T h e momen t of inertia about an axis th rough the center of mass with 
direction cosines a, j3, y may, therefore, be writ ten 
I = Aoc2 + Bfi2 + Cy 2 - IDfiy - 2£ya: - IFocfi. (28) 
Now consider the surface of the second degree having the origin at 
its center, 
AX2 + BY2 + CZ2 - 2DYZ - 2EZX - 2FXY = + 1. (29) 
Th i s is called the ellipsoid of inertia of the body M. Comparison between 
(28) and (29) shows that the following relations will hold: 
124 BODIES OF FINITE DIMENSIONS 
T h e reduction from the general form (28) to that of (30) requires the 
solution of an algebraic equation of the third degree. Th i s solution 
locates the direction of the axes bu t leaves complete freedom in the 
assignment of the XY YY and Z axes. As a rule there is a mechanical 
preference for a particular choice. If the positive directions of the X 
and Z axes have been decided upon, that of the Y axis follows wi thout 
ambiguity if the coordinates are to form, say, a r ight -handed system. 
Let it be assumed that the axes are so chosen that the equation of 
the ellipsoid has the form (30). These axes are said to be the principal 
axes of the body. Wi th this choice of axes we have D = E = F = 0. 
T h e importance in connection with our present problem is that in this 
case the last three integrals in the expression (27) for U2 vanish. T h e 
three remaining integrals may be expressed in terms of A y By and C. 
F rom the definition of A y By C it follows that 
{ ? + y]* + t*)dM 
£2dM 
rfdM = 
l2dM = 
Thi s gives after simple reduction, 
\ 
2 
A + B - C 
This expression for t / 2 must be looked upon as merely a convenient 
2 
C + A - B 
B + C - A 
A + B +C 
2 
Hence 
(31) 
For a moment , let a, j8, y be the direction cosines of the line joining 
the point x, y, z with the origin. T h e n 
= Aoc2 + Bp2 + c y 2 , 
which, in view of (28) and the choice of the principal axes as coordinate 
axes, is the moment of inertia of the body about the line joining its center 
of mass with the point x, yy z. If this is denoted by / , then (31) may 
be writ ten 
(32) 
POLAR COORDINATES 125 
short form. In all applications it is necessary to obtain for the potential 
U an expression in which the coordinates of the point m appear explicitly 
so that partial differential quotients with respect to these coordinates 
may be readily obtained. Th i s requi rement is satisfied by the expression 
(31). 
A more special case is that of the spheroid, which, in addit ion to being 
rotationally symmetrical about the Z axis, is also symmetrical about an 
equatorial plane perpendicular to this axis. I t will be noted that the 
expression (34) for U2 is the same regardless of whether the body 
possesses an equatorial plane of symmetry . T h e same expression holds 
for a spheroid as for a pear-shaped body that is rotationally symmetrical 
about the Z axis. 
T h i s simplified form of U2 may be used for the attraction of planets 
upon their satellites. I t is applicable to binary star systems if one of 
the components is of such small dimensions that it may be t reated as 
a particle of mass. In very close pairs the tidal deformation may become 
significant, destroying the rotational symmetry and requir ing the use 
of the general formula (33). 
5. I n t r o d u c t i o n of p o l a r c o o r d i n a t e s . Let polar coordinates be 
in t roduced by 
x = r cos ip cos 
y = r sin I/J cos /?, 
z = r sin /?, 
I/J being the longitude and j8 the lati tude, the latter to be dist inguished 
from the direction cosine j8 used previously. T h e expression for U2 may 
then be wri t ten 
(A cos 2 I/J + B sin 2 I/J) cos 2 p - C sin 2 p\ 
Csin2£J, [A+B + C) 
(A+B + C)-
giving finally 
(33) 
N o w suppose that the body M is rotationally symmetrical about the 
Z axis. T h e n A = B, and (33) reduces to 
(34) 
126 BODIES OF FINITE DIMENSIONS 
6. T h e e x p r e s s i o n for U3. T h e next part of the potential is 
T h e development will give rise to ten different integrals of the type 
J f y d M with j + k + l = 3, 
j , ky and / being zero or positive integers 1, 2, or 3. 
Suppose the body is symmetrical about the XY plane and about the 
ZX plane. In this case only three of the ten integrals are different from 
zero, and the expression for U3 becomes 
An example of a body to which this expression is applicable is a pear-
shaped body with the X axis as axis of the pear. 
If the body is symmetrical about all three coordinate planes, as is, 
for example, a homogeneous ellipsoid with three unequal axes, all of 
the ten integrals will vanish. In this case the moments of odd order 
will vanish generally and U3 = 0, U5 — 0, etc. 
7* T h e e x p r e s si o n for £/ 4 . T h e general expression (24) gives 
(37) 
In the most general case there will be fifteen different integrals of 
moments of the fourth order of the form J^jrjk^ldM. T h e expression 
(37) can readily be writ ten in te rms of these integrals. T h e development 
will be given here only for the case of a body symmetrical about all 
three coordinate planes. In this case only six moments of the fourth 
order will be different from zero, namely, those in which all three of 
the exponents j , k> I are even. Th i s t rea tment will apply, for example, 
to a homogeneous ellipsoid with three unequal axes. T h e n 
•3(P + V2 + P)]dM. (35) 
(36) 
- 3 (e + v2
 + £ 2) 2] dM. 
In the more special case of a spheroid, let the Z axis be the axis of 
rotational symmetry . T h e n 
\^dM = J V dM, J f £ 2 dM = fr]2£2 dM. 
In addit ion there should be a relation between j^2rj2dM and j^dM. 
Thi s relation can easily be discovered as follows. Let 
P\ = ? + v2, 
and introduce cylindrical coordinates p x , ip, £. T h e density K will, on 
account of the rotational symmetry, be a function of p 1 and £ but 
independent of ip. Hence, wi th 
dM = K di px dpx dip, 
| £ Y dM = | | * < / £ P J </Pl I'" sin 2 0 cos 2 ifj d^ 
77" /* * 
4 
T h e limits of the integrations with respect to P l , £ mus t be int roduced 
to correspond to the particular body being considered. Similarly, 
| (i2 + V
2)2 dM = | | * d£ pt dPl diP 
Hence 
= 2TT II K d£ p\ dpv 
POTENTIAL—FOURTH ORDER 127 
£ 2f 2 dM 
f y </m]. 
(38) 
rfdM 
128 BODIES OF FINITE DIMENSIONS 
I t then follows at once that 
J ? m = J yfdM = | J (£2 + iffiM. 
After introduct ion of these relations and further simplification, the 
expression (38) for C74 for a spheroid reduces to 
# 4 = r 4 ( s in 2 £ - ^sin 22j8), 
the expression becomes 
A rather unexpected simplification is that , whereas the various integrals 
in (38) have different functions of the coordinates of m as factor, in 
the spheroidal case one common function of these coordinates appears. 
Th i s feature could be obtained from the general properties of the 
Newtonian potential, which will not be examined here . 
8. T h e p o t e n t i a l of a s p h e r o i d . T h e results obtained for a 
spheroid may be collected in the form 
dM. 
(40) 
with 
] dM, 
] dM. 
If then polar coordinates are in t roduced by 
dM. 
(39) 
z2 = r2 sin 2 jS, 
POTENTIAL OF SPHEROID 129 
T h e constants M , / , K tha t determine the potential of the spheroid 
mus t be obtained from observational data, i.e., from the gravitational 
attraction of the spheroid upon another body. 
For the earth the parameter / may be derived from measurements 
of the acceleration of gravity at the ear th 's surface. T h e parameter K 
for the earth has tu rned out to be small, of the same order of magni tude 
as coefficients of harmonics of higher order that appear in the potential 
of a body with as irregular a surface s t ructure as the earth has. 
In the moon ' s motion the parameter / is impor tant bu t the solar effect 
on the moon ' s mot ion is so m u c h greater than the effect of the ear th 's 
oblateness that the evaluation of / from the moon ' s motion presents 
peculiar difficulties. 
An interest ing method of obtaining / for the earth is by writ ing / 
as the produc t of two factors, 
3C-A C 
J ~ 2 C ' MR2 * 
T h e ratio (C — A)/C may be found from the observed values of the 
constant of precession, while the ratio C/MR2 may be secured from the 
theory of hydrostat ic equi l ibr ium of the earth. T h i s method was used 
with much success by G. H . Darwin and further developed by W. de 
Si t ter ; its l imitation depends on the degree of applicability of the theory 
of hydrostat ic equi l ibr ium to the earth. 
T h e s tudy of motions of artificial satellites has opened up the oppor t -
unity of evaluating the constants / and K, and other parameters that 
enter into the ear th 's potential , by a method free from the complications 
present in the older methods . 
Evidently, B2 is of dimension [L 2 ] , J5 4 of dimension [L 4 ] . T h e more 
commonly used parameters are / and K, related to B2i J5 4 by 
in which R is the equatorial radius of the spheroid. T h e n 
U - . . ] . (41) 
Instead of K the parameter D defined as - 8 - BJt~* is frequently used. 
T h e n the K t e rm in (41) becomes 
s i n 2 ^ + sin 4 p\. 
130 BODIES OF FINITE DIMENSIONS 
With the availability of artificial satellites it is possible to apply to 
the earth the method that has been available for the s tudy of the potential 
of Saturn from its natural satellites. T h e inner six satellites of this planet, 
from Mimas to Ti tan , have mean distances ranging from 3.11 R to 
20.48 R. T h e motions of the nodes and pericenters of these satellites 
are determined to a large extent by the oblateness of the pr imary. F rom 
the observed motions for two or more satellites at different distances 
from the pr imary the values of / and K (or D) may be obtained. T h e 
application by H . Jeffreys to the Saturn sys tem 5 is recommended for 
further reading. For the motions of the pericenter and node of a close 
satellite, see Chapter X V I I . 
T h e form of the potential of a spheroid given in Eq . (40) corresponds 
to a general result obtained by Laplace. For a body with rotational 
symmetry the general expression is 
in which Pk are Legendre polynomials . In the case of the spheroid, 
on account of the symmetry with respect to the equatorial plane, only 
even te rms are present in the series. For bodies wi thout rotational 
symmetry a generalized expansion, also given by Laplace, is available. 6 
9. P o t e n t i a l for t w o b o d i e s of f inite d i m e n s i o n s . T h e develop-
ment of the potential as given in the preceding sections is applicable 
to the great majority of problems in dynamical astronomy. T h e r e are 
some problems, however, that require the development of the potential 
for two bodies each of finite dimensions. T h e function U mus t then 
be writ ten 
where A is now the distance between the two elements of mass dM of 
the body M and dM' of the body M'. T h e integrations are to be extended 
over the masses of both bodies. 
Let O be the center of mass of M , O ' that of M ' , and let O be the 
origin of a Cartesian coordinate system with coordinates 77, £, and O' 
that of a coordinate system with parallel axes with coordinates rj\ 
5 H . JEFFREYS, Monthly Notices Roy. Astron. Soc, 113, 8 1 ( 1 9 5 3 ) ; 114, 4 3 3 ( 1 9 5 4 ) . 
6 F . TISSERAND, "Mecanique Celeste," Vol. 2 , Chaps. X V I - X I X . Gauthier Villars, 
Paris, 1 8 9 2 . 
POTENTIAL FOR TWO FINITE BODIES 131 
Fur ther , let O O ' coincide with the Ox axis. If r is the distance O O ' , 
77, £ the coordinates of dM, and 77', £' the coordinates of d M ' , 
J 2 = ( r + f _ £ ) 2 + _ ^ 2 + _ 
Hence, if 
= r 2 ( ! _ 2qoc + a 2), 
the reciprocal of A may be developed by (16), giving 
l+P1(q)'CL+P2(q)'cfi+J9 (42) 
and, similar to the expansion used in preceding sections, 
(43) 
since 
J £ dM =-- 0 , $£'dM'=0. 
U2 = 
)-2£?+riV'+ U']dMdM'. 
K f - e)2
 + - V ) 2 + (£ - D 2 ] | dM dM' 
T h e product terms yield nothing on account of the choice of origins. 
Hence U2 may be writ ten 
[ f 2 + V 2 + £ ' 2 ( V 2 + £' 2)] ««f'. 
132 BODIES OF FINITE DIMENSIONS 
Regardless of the orientation of the coordinate systems, 
J(£ 2 +r!* + P)dM = l(A+B + C), 
J ( f 2 + rj'2 + Cf2) dM' = \(A' + B' + C ) , 
if A, Bf C, A'yB', C are the moments of inertia about the principal axes. 
Moreover, 
are the moments of inertia of the two bodies about the line OO'. Hence , 
/ = Aa* + Bb* + Cc\ 
I' = A'a'2 + B'b'* + C'c'\ 
if a, b, c are the direction cosines of OO' relative to the principal axes 
of M, and a', b', c' those relative to the principal axes of M'. T h e 
resulting expression for U2 is 
fM' rl 
g ( ^ + B + C)-| (Aa* + Bb* + Cc2)] (44) 
] • 
which may be compared with (31), (32). 
Th i s expression for U2 is not yetdefinitive, since the mot ion of the 
center of mass of M will be determined by the components of the force 
and that of M' by the components 
where X, Y, Z are the coordinates of M and X\ Y\ Z' those of M' 
in a fixed coordinate system. Hence the coordinates of O ' relative to 
O are X' - Xy Y' — Y, Z' - Z, and 
r 2 = (x - xy + (Y - Yy + ( z - zy. 
T h e direction cosines that appear in (44) must now be expressed in 
terms of these coordinates. Let the principal axes of M have the direction 
NOTES AND REFERENCES 133 
b2 = 
Y') + - z'W 
r* 
Y') + v2(Z 
-x')+npr- Y') + v3(Z - z'W 
r2 
Similar expressions may be wri t ten for a'2
y b'2
y c'2 by introducing 
A/, \i{y v^y etc., for the direction cosines of the principal axes of M'. 
T h e r e is no difficulty in principle concerning the development of the 
higher por t ions in the expression (42) for U. T h e procedure is the same 
as that followed for the at traction of a finite body upon a material 
particle. In the most general case these higher port ions soon consist of 
so numerous te rms that the development becomes difficult. For most 
astronomical problems, the developments given in this chapter suffice. 
If additional te rms are needed for any particular problem they may 
readily be obtained by extending the procedure followed. In all such 
cases it is desirable to take advantage of all the simplifications due to 
symmetry and particular orientation that each individual problem may 
present . 
N O T E S A N D R E F E R E N C E S 
F . T i s s e r a n d ' s Traite de Mecanique Celeste, V o l . 2 , c o n t a i n s an e x h a u s t i v e 
t r e a t m e n t of t h e subjec t c o v e r e d i n th i s chapter . A use fu l addi t ional re ference is 
T h o m s o n a n d T a i t ' s Treatise on Natural Philosophy, Part I I , n e w e d i t i o n , e s p e -
cial ly C h a p t e r V I , C a m b r i d g e U n i v . P r e s s , L o n d o n a n d N e w York , 1 8 8 3 . 
a2 = 
c2 --
cosines Xly fily vly A2, /x2, v2y A3, /z 3, vs relative to this fixed coordinate 
system. T h e n 
CHAPTER I V 
CALCULUS OF FINITE DIFFERENCES 
1. R e p r e s e n t a t i o n of f u n c t i o n s . A function is often represented 
by an analytic expression in te rms of one or more independent variables, 
indicated by symbols, which may be imagined to vary continuously over 
a range of numerical values, infinite or finite. Th i s formula explicitly 
prescribes a set of mathematical operations on the variables, by means 
of which the function is defined for any particular values of the variables. 
T h e infinitesimal calculus is concerned with the differentiation and 
integration of such expressions. An alternative form for the representa-
tion of functions is the numerical form, in which numerical values of 
the function are given for certain definite values of its independent 
variable or variables. T h e values of the independent variable, when there 
is only one, are usually wri t ten in a column, and beside each one is 
placed the corresponding value of the function. Such an exhibit is called 
a table. T h e independent variable is called the argument. T h e argument 
is usually, bu t not always, given at equal intervals; the difference between 
two consecutive arguments taken without regard to sign is called the 
tabular interval, the interval of the argument , or simply the interval. 
W h e n there are two independent variables the values of one (called the 
vertical a rgument) may be wri t ten along the left margin and the other 
(the horizontal a rgument ) across the top, the values of the function 
forming a rectangular array which is known as a table of double entry. 
Tables with one independent variable are called tables of single entry. 
A third form in which functions may be represented is the graphical, 
in which, unlike the numerical , the representation is cont inuous. Th i s 
form is not, however, much used in celestial mechanics, for it is not 
compatible with great precision. 
T h e numerical form of a function may be preferable to the analytic 
for any of a number of reasons. Tables of logari thms and of t r igonometr ic 
functions are made available as a t ime-saving device because the calcula-
tion of individual values by substi tut ion of numbers into infinite series 
is too laborious for the computer . T h e ephemerides of the sun, moon, 
and planets are tabulated in the annual volumes of the American Ephe-
meris and similar publications because their calculation from the 
expressions that form the basis of the ephemerides is not practicable 
134 
DIFFERENCES 135 
for many who require these data. As a th i rd example we have tables 
that represent numerical solutions of differential equations obtained 
directly by numerical processes, that is, wi thout initial expression of 
the solution analytically. 
T h e more common tables of logari thms and of t r igonometric functions 
are furnished with such a small tabular interval that interpolation is 
rendered very easy, the process being known as linear interpolation. 
Such extended tabulat ion is not always practicable, even for tables that 
mus t be used frequently, and it is therefore necessary to have more 
general methods of interpolation than the linear method, applicable 
when linear interpolation would lead to inaccuracy. I t is useful also to 
be able to differentiate and integrate functions expressed in the numerical 
form, particularly to integrate those functions that cannot be integrated 
analytically, or whose analytical development would require great labor. 
These three operations—interpolation, numerical differentiation, and 
numerical integrat ion—consti tute the calculus of finite differences, 
sometimes called the finite calculus. 
T h e advantage of analytic methods is that they are the only ones 
that can lead to perfectly general results, that is, results known to be 
valid for all values of the independent variables. T h e advantage of 
numerical methods is that they are always available. Again, numerical 
methods are often less laborious and less liable to error than analytic 
ones, and they are capable of yielding equally great accuracy. Moreover, 
celestial mechanics, like any other branch of applied mathematics , aims 
at practical applications. Here numerical results are always needed, and 
for many practical applications intermediary analytic developments are 
entirely superfluous. 
2. D i f f e r e n c e s . Table 1 gives values of the polynomial x3 — 
3x — 23 for some integral values of x, with their differences. 
T h e numbers f1 are obtained by subtract ing each value of / from the 
one immediately below it, in succession, and the other columns are 
calculated similarly. T h e number s fl are called the first differences, the 
f11 are the second differences, etc. I t will be seen later that the successive 
orders of differences are closely related to the successive derivatives of 
the funct ion; in fact it may be noticed that the second, third, and fourth 
differences are precisely equal to the corresponding derivatives in this 
particular example, bu t this is not t rue in general. T h e differences of 
any order are in general smaller than those of the order immediately 
preceding, that is, the differences converge. If the function and all of 
its derivatives are cont inuous and do not become infinite within the 
range of the table, the differences converge more rapidly the smaller 
136 FINITE DIFFERENCES 
T A B L E 1 
POLYNOMIAL 
X / / /" 
— 3 — 4 1 
+ 1 6 
— 2 — 2 5 
+ 4 
— 1 2 
+ 6 
— 1 — 2 1 
— 2 
— 6 
+ 6 
0 
0 — 2 3 
— 2 
0 
+ 6 
0 
+ 1 — 2 5 
+ 4 
+ 6 
+ 6 
0 
+ 2 — 2 1 
+ 1 6 
+ 1 2 
+ 6 
0 
+ 3 — 5 
+ 3 4 
+ 1 8 
+ 6 
0 
+ 4 + 2 9 
+ 5 8 
+ 2 4 
+ 5 + 8 7 
the tabular interval. In fact we may always make the convergence as 
rapid as we please by taking the interval sufficiently small (which in 
this case would of course require nonintegral values of x). 
A polynomial is a special kind offunction exhibit ing properties not 
ordinarily encountered in celestial mechanics. A special proper ty of 
Table 1 above is that the representation is exact, without any errors. 
Table 2, giving the sun ' s declination as calculated on the assumption 
of elliptic motion for the earth, is more representative of the functions 
to be dealt with later. 
T h e differences in this table converge but slowly, and only up to a 
certain order, after which they begin to alternate in sign. If further 
orders of differences are calculated they will be found to diverge. Th i s 
behavior is caused by the errors of rounding present in / . Each value 
of / , al though calculated with all possible care, is necessarily affected 
by an error that may be of any size between + 0?00005 and — 0?00005. 
These errors build up in the successive orders of differences until they 
assume the governing role at the order where the differences alternate 
in sign. T h e r e is never any advantage in carrying the differences beyond 
this point, which in this example would be / I X . 
Th i s property of bui lding up , possessed by the errors of rounding, 
is exhibited in Tab le 3, which presents the worst possible case of 
divergence of errors that can arise when the functions are cont inuous 
D I F F E R E N C E S 
T A B L E 2 
DECLINATION OF THE SUN 
137 
1966 / fl /" 
[V y V I I l 
Feb. 19.75 — l l ! 2 4 4 1 
+ 3.6921 
Mar. 1.75 — 7.5520 + 1812 
+ 3.8733 — 1075 
11.75 — 3.6787 + 737 + 51 
+ 3.9470 — 1024 — 14 
21.75 + 0.2683 — 287 + 37 + 8 
+ 3.9183 — 987 — 6 — 9 
31.75 + 4.1866 — 1274 + 31 — 1 + 23 
+ 3.7909 — 956 — 7 + 14 
Apr. 10.75 + 7.9775 - ^ 2 2 3 0 + 24 + 13 — 11 
+ 3.5679 — 932 + 6 + 3 
20.75 + 11.5454 — 3162 + 30 + 16 — 2 
+ 3.2517 — 902 + 22 + 1 
30.75 + 14.7971 — 4064 + 52 + 17 — 4 
+ 2.8453 — 850 + 39 — 3 
May 10.75 + 17.6424 — 4914 + 91 + 14 — 5 
+ 2.3539 — 759 + 53 — 8 
20.75 + 19.9963 — 5673 + 144 + 6 
+ 1.7866 — 615 + 59 
30.75 + 21.7829 — 6288 + 203 
+ 1-1578 — 412 
June 9.75 + 22.9407 — 6700 
+ 0.4878 
19.75 + 23.4285 
and are accurately calculated, and when the tabular interval is small 
enough for good convergence. T h e hypothetical errors in the function 
are tabulated, instead of the function itself. 
Th i s array suggests a rule that can often be used to advantage in 
checking calculated functions. Difference the function to the order 
where the differences alternate in sign, and call this the mth order. 
T h e n if any mth difference exceeds 2 m _ 1 in absolute magni tude there 
is certainly an error in the function greater than half a uni t of the last 
decimal. T h e converse is not t r u e ; an error greater than half a uni t 
does not necessarily produce such a large effect, as will be seen in the 
following section. 
An exception to the rule occurs when the tabular interval is taken 
so great that the differences do not converge, as for example if cos x 
is tabulated at an interval of 180° beginning at 0°, bu t such cases can 
138 FINITE DIFFERENCES 
I II III IV v VI 
€ € € € € € € 
+ i 
— 1 
- 1 
+ 1 
+ 1 
— 1 
I 
— s 
+ 1 
+ 1 
— 1 
+ 1 
+ i 
always be easily avoided by trial, or by sufficient knowledge of the 
behavior of the function. 
In extended calculations it is seldom practicable, and never efficient, 
to limit the errors to half a uni t of the last decimal, bu t the rule is easily 
modified to apply to such cases. If, for example, errors of 3 uni ts are 
permit ted, then no mth difference should exceed 3 X 2 m , in general, 
for errors of p units , the mth differences should not exceed p X 2m. 
3* D e t e c t i o n of c a s u a l e r r o r s * By a casual error is meant an 
error that affects only isolated values of the function, such as a typo-
graphical error or an error of calculation. Such errors, if they are not 
too numerous , can easily be detected by examination of the differences. 
In order to show the effect of such an error, the polynomial tabulated 
in Tab le 1 is given again in Table 4 with the value of / corresponding 
to x = + 1 altered by one unit . 
T h e fourth differences, instead of vanishing as they did before, exhibit 
a pa t te rn that is easily recognized as the binomial coefficients in the 
expansion of (a — 6) 4 . If the fifth differences are wri t ten it will be seen 
that they correspond to the coefficients in the expansion of (a — b)h. 
If the error in the function had been p un i t s instead of 1 the pat tern 
would still be easily recognized; each of the fourth differences would 
merely be mult ipl ied by p. I n general, an isolated error of p units in 
the function produces a characteristic pat tern in the differences of the 
mth order, which corresponds to the binomial coefficients in the ex-
+ 2 
— 4 
— 2 + 8 
+ 4 — 1 6 
+ 2 — 8 + 3 2 
— 4 + 1 6 
— 2 + 8 
+ 4 
+ 2 
T A B L E 3 
DIVERGENCE OF ERRORS 
CASUAL ERRORS 139 
T A B L E 4 
POLYNOMIAL WITH AN ERROR 
X / z 1 /" 
— 3 — 4 1 
+ 1 6 
— 2 — 2 5 
+ 4 
— 1 2 
+ 6 
— 1 — 2 1 
— 2 
— 6 
+ 7 
+ 1 
0 — 2 3 
— 1 
+ 1 
+ 3 
— 4 
+ 1 — 2 4 
+ 3 
+ 4 
+ 9 
+ 6 
+ 2 — 2 1 
+ 1 6 
+ 1 3 
+ 5 
— 4 
+ 3 — 5 
+ 3 4 
+ 1 8 
+ 6 
+ 1 
+ 4 + 2 9 
+ 5 8 
+ 2 4 
+ 5 + 8 7 
pansion of p(a — b)m. For more general types of functions than poly-
nomials the pat tern is likely to be partly obscured by the effects of errors 
of rounding, which may make it difficult to detect errors of small 
magni tude , that is, errors only a little larger than the errors of round ing ; 
in general, the pat tern becomes more difficult to discern the more 
slowly the differences converge. 
T h e rule for correcting casual errors in a function is as follows: W h e n 
the characteristic pat tern is detected in the differences of the mth order, 
then if m is even take the largest mth difference, divide it by the value 
taken from the following table, and add the quot ient (nearest whole 
number ) to the function in the same l ine; if m is odd take the mean, 
without regard to sign, of the two largest consecutive mth differences, 
prefix the sign of the lower, divide by the value in the table below and 
add the quotient (nearest whole number ) to the function on the inter-
mediate line. These number s are, except for sign, the largest coefficients 
in the binomial expansions of order m. T h e y may usefully be compared 
with the numbers 2 m _ 1 , of the preceding section. 
m 
Divisor 
2 3 4 5 6 7 8 9 
+ 2 - 3 - 6 + 1 0 + 2 0 - 3 5 - 7 0 +126 
140 FINITE DIFFERENCES 
Table 5 is given to show the application of this rule. Th i s is a table 
of five-place logari thms, with an error of a uni t purposely in t roduced 
into log 50. T h e tabular interval has been chosen large enough to make 
the differences slowly convergent. T h e effect of the errors of rounding 
is so large in the eighth differences as almost to obscure the characteristic 
pat tern, bu t the two salient features remain. T h e differences gradually 
diminish for some distance above and below the largest one, and they 
alternate in sign. Dividing the largest difference, + 94, by the value 
from the table, — 70, gives — 1 as the correction to be applied to the 
function 69898. If the table is now constructed anew it will be seen 
that the new eighth differences are much smoother than before; the largest 
in the region where they alternate in sign is 34. 
T A B L E 5 
LOGARITHMS WITH AN ERROR 
X / f /" in 
f 
VI / V i i i 
1 5 1 7 6 0 9 
+ 1 2 4 9 4 
2 0 3 0 1 0 3 
+ 9 6 9 1 
— 2 8 0 3 
+ 1 0 3 0 
2 5 3 9 7 9 4 
+ 7 9 1 8 
— 1 7 7 3 
+ 5 5 0 
— 4 8 0 
+ 2 5 7 
3 0 4 7 7 1 2 
+ 6 6 9 5 
— 1 2 2 3 
+ 3 2 7 
— 2 2 3 
+ 1 0 8 
— 1 4 9 
+ 9 0 
3 5 5 4 4 0 7 
+ 5 7 9 9 
— 8 9 6 
+ 2 1 2 
— 1 1 5 
+ 4 9 
— 5 9 
+ 2 9 
— 6 1 
4 0 6 0 2 0 6 
+ 5 1 1 5 
— 6 8 4 
+ 1 4 6 
— 6 6 
+ 1 9 
— 3 0 
+ 3 9 
+ 1 0 
4 5 6 5 3 2 1 
+ 4 5 7 7 
— 5 3 8 
+ 9 9 
— 4 7 
+ 2 8 
+ 9 
— 4 2 
— 8 1 
5 0 6 9 8 9 8 
+ 4 1 3 8 
— 4 3 9 
+ 8 0 
— 1 9 
— 5 
— 3 3+ 5 2 
+ 9 4 
5 5 7 4 0 3 6 
+ 3 7 7 9 
— 3 5 9 
+ 5 6 
— 2 4 
+ 1 4 
+ 1 9 
— 3 5 
— 8 7 
6 0 7 7 8 1 5 
+ 3 4 7 6 
— 3 0 3 
+ 4 6 
— 1 0 
— 2 
— 1 6 
+ 2 6 
+ 6 1 
6 5 8 1 2 9 1 
+ 3 2 1 9 
— 2 5 7 
+ 3 4 
— 1 2 
+ 8 
+ 1 0 
— 2 1 
— 4 7 
7 0 8 4 5 1 0 
+ 2 9 9 6 
— 2 2 3 
+ 3 0 
— 4 
— 3 
— 1 1 
7 5 8 7 5 0 6 
+ 2 8 0 3 
— 1 9 3 
+ 2 3 
— 7 
8 0 9 0 3 0 9 
+ 2 6 3 3 
— 1 7 0 
8 5 9 2 9 4 2 
D I R E C T I N T E R P O L A T I O N 141 
Th i s example exhibits well the power of this method of detecting 
casual errors . In most cases arising in practice the differences will 
converge more rapidly, and the errors are likely to be larger; both of 
these features conduce to ease in applying the method . T w o cautions 
are needed. If the errors are so numerous that the characteristic pat terns 
overlap one another the method becomes difficult if not impossible to 
apply, though the large size and " raggedness" (lack of smoothness) of 
the differences may nevertheless indicate that errors are present . T h e 
method can be used only to detect casual e r rors ; systematic errors, such 
as those caused by omit t ing significant te rms of a series expansion, 
cannot be detected. 
4 . D i r e c t i n t e r p o l a t i o n . Direct interpolat ion is the process of 
numerical ly evaluating a function by means of a table of it, for a value 
of the independent variable between two of those tabulated. Tab le 6 
shows the symbolic notation used. 
T A B L E 6 
NOTATION 
Arg. / f f"1 
*-• 8 L 2 
8 4 - 2 
S - 3 / 2 
* - i 8 * i 
8 - l / 2 
*o /. S 2 
0 
S 4 
0 
8 
1/2 1/2 
*i A 8 2 i 8 t 
8 3 / 2 8 3 / 2 
* 2 /. K 
In addit ion /x8x is used to mean \ ( § 3 / 2 + §1/2)? means \ ( 8 i 1 / 2 + 8i / 2 ) , etc. 
T h e tabular interval is denoted by w, that is, w = xx — x0, if the interval 
is constant. I t is usually, bu t not always, considered that the a rgument 
for which the interpolated value of / is wanted lies between x0 and xv 
Call this a rgument x. T h e n n — (x — x0)/w is called the interpolating 
factor, and the value of / for a rgument x may be denoted by / n . In 
many of the tables in common use w is a power of 10 and it may be 
expressed in any uni ts whatever, such as 1 day, 0.00001 day, 0.01 
142 FINITE DIFFERENCES 
degree, etc. I t is to be noted that n, on the contrary, is always a pure 
number . In linear interpolation, commonly used in t r igonometr ic tables, 
it is assumed that 
/ n = / o + w8 1 / 2 . (1) 
Since § 1 / 2 = / i — / 0 , this is equivalent to the equation 
fn = / o + n(fi ~fo) 
(2) 
= (1 - n)fo + nfv 
If / is plotted as ordinate against x as abscissa, then, since this is a 
linear equation in n, and since for n = 0, fn = / 0 , and for n = 1, 
fn == fly the equation represents a straight line th rough the points f0 
and fv T h u s the process of linear interpolation subst i tutes for the 
function a series of straight-line segments joining consecutive tabular 
values. If the tabular interval is so small that the function has no 
appreciable curvature between two consecutive tabular values, the result 
of a linear interpolation is sufficiently accurate, otherwise not . Funct ions 
that are in widespread use, such as five-place sines and tangents , can 
be efficiently tabulated at such small intervals, bu t this is not practicable 
for functions having a restricted application, or when many significant 
figures are needed; in such cases, instead of construct ing very extensive 
tables it is easier to use more general methods of interpolation. 
Since two points on a curve can furnish no information about its 
curvature, it is evident that more than two points are needed for a more 
general method of interpolation than the linear. Wha t is done is to pass 
a smooth curve through 3, 4, 5, or more consecutive tabular values, 
and to assume that the curve represents the function in between the 
tabular values. A smooth curve may be passed through a finite n u m b e r 
of points in infinitely many ways; it is desirable to choose a way that 
will give a simple analytic approximation to the function. T h e commonest 
procedure is to subst i tute for the function a polynomial in powers of n. 
If three points are used the polynomial contains a constant te rm and 
the first and second powers of n; if four points are used the th i rd power 
is needed, etc. T h e result resembles the familiar Taylor series expansion 
of a function, bu t instead of the successive derivatives appearing in the 
formula, the successive orders of differences occur. As an example of 
the process the case of three points, or second-order interpolation, is 
examined in detail. 
Let the three tabular values of f(x) be f_l9 / 0 , and fx. A polynomial 
of the second degree has the form A + Bx + Cx2, where A> B, and C 
are constants. T h e y are to be determined by the condition that the curve 
DIRECT INTERPOLATION 143 
must pass th rough f_ly / 0 , and fv For x — 0 we have immediately 
A = f0. For x = + 1 and x = — 1 we have 
A+B + C=fl9 
A - B + C=f-V 
Subtract ing the second equation from the first gives 
2 # = fx —f-i =A —f0 + / 0 — / - l = §1 /2 + 8 - i / 2 -
But since 
S - l / 2 
B 
Adding the two equations gives 
2A+2C=f1 +/_,. 
Subst i tut ing for A its value / 0 , 
2 C = / 1 - 2 / 0 + / _ 1 
= ( / 1 - / „ ) - ( / o - / _ 1 ) 
= 8 l / 2 - S - l / 2 
Hence 
/ ( * ) = / o + ( s 1 / 2 - i ^ ) * + i § ^ 2 
= / 0 + S 1 / 2 * + S ^ ( * - l ) / 2 , 
or, when x is equal to «, 
/„ = / 0 + S 1 / 2 « + S ^ ( « - l ) / 2 . 
Th i s is one example of a formula for interpolation with second 
differences. A formula including the effect of th i rd or higher differences 
could be found by a similar method. T h e formula may be used in its 
present form or it may be rearranged in numerous ways, making use 
of the definitions of 8 and 8 2 in terms of / , or collecting terms in various 
ways. One form that is particularly convenient for mental calculation is 
/ n = / o + [ 8 i / 2 - i ( l - ^ ) S 0
2 ] n. 
= 8 l / 2 80> 
— °l/2 2 °0* 
144 FINITE DIFFERENCES 
Another form, known as Lagrange 's formula, is 
fn = / - i ( i « 2 - i » ) + / o ( i - « 2 ) + / i ( i « 2 + *»)• 
If w = ^ this takes the simple form 
5. E v e r e t t ' s a n d B e s s e l ' s f o r m u l a s . Once the general principles 
by which interpolation formulas may be constructed have been grasped, 
it is not necessary to examine the detailed derivations of all the formulas 
likely to be used in practice. Of these the most generally useful are 
Evere t t ' s and Bessel's. 
Everet t ' s formula is 
/„=/«. + » 8!/2 + E l 8 o + E\ S 2 + El SJ + E\ S* + 
where 
Bessel's formula is 
fn = fo + » 8 ! / 2 + + 8 D + B 3 8 ?/ 2 + fi4(§0 + 8 l ) + & 8V2 + 
Lagrange 's formula has the apparent advantage that the differences 
are not required, bu t whether this is a real advantage is quest ionable. 
Every manuscr ipt table, and many pr inted ones, ought to be differenced 
before use in order to verify the accuracy. Once the differences have 
been formed it is usually efficient to use them in interpolation, because 
the functions of n are needed to fewer significant figures than n itself, 
whereas with Lagrange 's formula all the numerical coefficients are 
needed to the same accuracy as the function to be interpolated. Fu r the r -
more, without knowing the size of the differences it is impossible to 
tell how many values of / a r e needed in any part icular case, as will appear 
later. 
E V E R E T T A N D B E S S E L 145 
T h e n u m b e r of te rms that mus t be used in any part icular application 
of an interpolation formula depends on the size of the error that can 
be tolerated in the result , and on the size of the differences. In most 
work it is considered permissible to cut off the formula at the point 
where the size of the largest neglected te rm does not exceed half a uni t 
of the last decimal of the function. T h e extreme values of the Besselian 
coefficientsare, approximately, 
F r o m these are easily deduced the following convenient l imit ing values 
of the differences that may be neglected. 
Difference neglected: second third fourth fifth 
Largest numerical value: 4 60 20 500 
T h u s , if the fourth differences are larger than 20 and the fifth differences 
smaller than 500, the first five t e rms of Bessel 's formula should be used. 
T h e use of Everet t ' s formula is l imited to an even n u m b e r of t e rms 
because E0 and Ex are of the same order of magni tude . T h e accuracy 
of any even n u m b e r of te rms of Everet t ' s formula is precisely equal 
to that of the same n u m b e r of te rms of Bessel's. T h u s the first four 
te rms of Everet t ' s formula include the effect of th i rd differences, and 
the first six include the effect of fifth differences. In general, Everet t ' s 
formula is to be preferred when the highest order of differences taken 
into account is odd, and Bessel's when it is even; the fifth te rm of 
Bessel's formula may, however, be used together with the first four of 
Everet t ' s when fourth differences mus t be taken into account, fifth 
differences being negligible. 
W h e n fourth differences are not negligible either formula may 
nevertheless often be limited to its first four t e rms by utilizing a powerful 
device known as the throwback, said to have been invented by Lagrange. 
where 
146 FINITE DIFFERENCES 
It will be observed that B*/B2 = (n + l)(n — 2)/12, and this function 
varies over a small range for n between 0 and 1, with an average value 
of about — 0 . 1 8 4 . If we put 
M2 = S 2 - 0 . 1 8 4 S 4 
then M2 may be used instead of S 2 in Everet t ' s or Bessel's formula, 
with an error of less than half a uni t if S 4 is less than 1000. In such 
cases the throwback should nearly always be used. 
Bessel's and Everet t ' s coefficients are somewhat laborious to calculate, 
especially for the higher orders . T h e y have therefore been tabulated 
extensively as functions of n. In practical calculation m u c h labor may 
be wasted by carrying more figures in the coefficients than are actually 
needed. As a general rule, the interpolation coefficients are needed to 
the same n u m b e r of decimals as the n u m b e r of significant figures in the 
differences they mult iply. 
6. N e w t o n ' s f o r m u l a . Everet t ' s and Bessel's formulas are known 
as central-difference formulas because the differences involved all lie near 
the center of the port ion of the table used. Another type of formula, 
due to Newton, makes use of the diagonal differences. I t may be writ ten 
which is known as the backward formula. 
T h e forward formula is sometimes useful for interpolating near the 
beginning of a table, where the central differences may not be available, 
and the backward formula may be used near the end of a table. T h e 
fn = / o + w 8 l / 2 
which is known as the forward formula, or 
U-m/2 
INVERSE INTERPOLATION 147 
chief value of the formula is, however, for extrapolation in connection 
with s tep-by-step integration (Section 12). I t may also be used for 
extending a table, by put t ing n = — 1 in the forward formula or n = 1 
in the backward one, bu t this should not usually be done for more than 
one step if at all, owing to the rapid accumulation of error. T h e use 
of Newton ' s formula for extending a table is exactly equivalent to 
assuming the highest order of differences taken into account to be 
constant. T h e result can be obtained simply by copying the value of the 
highest ordered difference immediately below, and extending the table 
down and to the left by successive addit ions. 
7. L a g r a n g e ' s f o r m u l a for i n t e r p o l a t i o n to h a l v e s . Al though 
Lagrange 's formula is not recommended for general use there is one 
application where its convenience may outweigh its disadvantages, that 
is, extensive interpolation to halves with a calculating machine. T h e 
four-point formula, to be used when fourth differences are negligible, 
takes the very simple form 
/ i / 2 = ^ ( - / - i + 9 / „ + 9 / 1 - / 2 ) , 
and the six-point formula, to be used when sixth differences are 
negligible, is 
fxn = 2^6 ( 3 / - * ~ 2 5 / - i + 1 5 0 / o + 1 5 0 / i " 2 5 / 2 + 3 / 3 } -
8. Inverse i n t e r p o l a t i o n . Th i s name is given to the process of 
finding the numerical value of the independent variable for a given value 
of the function. T h e problem is often one of great difficulty if analytical 
methods are employed, bu t with numerical methods it is only slightly 
more t roublesome than direct interpolation. A formula may be found 
simply by solving any interpolation formula for n in te rms of / w , / 0 , 
and the differences, bu t if the formula is cut off at any order of 
differences, say the 7 t h , it will be found that n has to be determined 
by solving an equation of the jih degree, which may be very laborious. 
It is better to proceed by reiteration, a special case of successive approxima-
tions. In this powerful numerical method the formula is arranged in such 
a way that by subst i tut ing an approximate value of the result into it 
a more accurate value is obtained, and the process may be repeated 
as often as necessary. In the present problem an approximate value of n 
is found by linear inverse interpolation. Th i s value of n is used to 
obtain approximate values of the interpolation coefficients, which can 
148 FINITE DIFFERENCES 
then be used to yield a more accurate value of n. For example, Bessel's 
formula may be used in the form 
n 3 1 / 2 =fn -f0 - B2 (8» + o2) - B3 S 3
/ 2 , 
where S 2 + S* should be replaced by Ml + M\ if the fourth differences 
are appreciable and not too large to be th rown back. By neglecting the 
te rms involving B2 and B3 in the first approximation the equation can 
easily be solved for n. T h i s value of n being used to find B2 and B3> 
the equation may again be solved for w, and this new value of n may 
be used to get improved values of B2 and B3. T h e process should be 
repeated unti l the value of n is not changed by further repetit ions. 
An impor tant caution is necessary in inverse interpolation. T h e value 
of n can be obtained to no more significant figures than are present 
in 8 1 / 2 . Additional figures, if writ ten down, are likely to deceive the 
computer into thinking that he has more accuracy than he really does. 
One must always be on guard against "manufactur ing decimals, ' ' which 
is justifiable only in very unusual instances. A familiar example of the 
difficulty arises when a small angle is sought to be determined from 
its cosine. One remedy is to obtain the first difference to more decimals, 
but it is usually preferable to avoid the difficulty by a suitable t rans-
formation. Instead of the cosine, for example, the versed sine or the 
haversine may be used. 
9. E r r o r of a n i n t e r p o l a t e d q u a n t i t y . I t has been pointed 
out that the max imum error of a value of a function that has been 
calculated with all possible care is ± 0 . 5 uni t of the last decimal. 
Fur thermore , all values of the error between these limits are equally 
likely, which means that about half of the errors in an extended series 
of values lie between the limits ± 0.25, and half lie outside these limits. 
Th i s fact may be expressed by the s tatement that the probable error 
is ± 0.25. A further error is in t roduced by the process of interpolation. 
If second and higher differences are zero, and if the interpolated quanti ty 
is rounded to the same n u m b e r of decimals as the tabular value, another 
similar error is introduced. In this case the max imum error of the 
interpolate is ± 1 . 0 and the probable error is about ± 0 . 3 . Slightly 
greater accuracy may be obtained by keeping additional figures in the 
interpolate. If this is done the error of the interpolate is only very 
slightly greater than that of the tabular value. In general, however,the 
additional figures are not wor th the t rouble of writ ing them down. 
T h e y are not significant figures in any sense of the word ; the first extra 
decimal is subject to an error of 5 units and the second is entirely 
fictitious. 
DIFFERENTIATION 149 
In practice the second differences are usually different from zero, and 
if they are neglected an additional error is introduced, which may 
amount at max imum to 1/8 of the second difference. Since it is the usual 
practice to neglect contr ibut ions of the second and higher-ordered 
differences if they are less than 0.5, it may be stated as a general rule 
that the max imum error of an interpolate is somewhat greater than 
± 1 . 5 uni ts , and its probable error approaches ± 0 . 5 unit . 
T h e error of a quant i ty obtained by inverse interpolation is governed 
by different rules. If the errors of the tabular functions are kept within 
+ 0.5, the error of any first difference is within + 1 . 0 and the error 
of the result will lie in the range + 1/8, if the contr ibut ion of differences 
beyond the first is inappreciable. 
10. N u m e r i c a l d i f f e r e n t i a t i o n . It has been seen in Section 2 
that the successive orders of differences of a function are closely related 
to the successive derivatives. T o find the precise expression for the 
derivatives in te rms of the differences it is necessary only to differentiate 
any of the interpolation formulas as many t imes as required. We have 
bu t 
and hence 
Similarly, 
df dfdn 
dn 1 
... 4f df 
o d2f d2f 
w2 -M = -J-. e t c . 
Other forms may be obtained by rearranging terms and by simple 
substi tut ions. Usually a derivative is wanted either for a point of 
tabulation or for a halfway point. Some of the most useful formulas 
follow. 
dx dn dx ' 
dx w 
dx dn' 
dx* an' 
For example, differentiating Bessel's formula gives 
T A B L E 7 
SIN x 
X / = s i n x fl /" 
— 3 0 ° — 0 . 5 0 0 0 0 
+ 1 5 7 9 8 
— 2 0 — 0 . 3 4 2 0 2 
+ 1 6 8 3 7 
+ 1 0 3 9 
— 5 1 1 
— 1 0 — 0 . 1 7 3 6 5 
+ 1 7 3 6 5 
+ 5 2 8 
— 5 2 8 
0 0 . 0 0 0 0 0 
+ 1 7 3 6 5 
0 
— 5 2 8 
+ 1 0 + 0 . 1 7 3 6 5 
+ 1 6 8 3 7 
— 5 2 8 
— 5 1 1 
+ 2 0 + 0 . 3 4 2 0 2 
+ 1 5 7 9 8 
— 1 0 3 9 
+ 3 0 + 0 . 5 0 0 0 0 
For x = 0 and n = 0 we have 
+ 10° £ = + 0.17365 + 0.00088 = + 0.17453. 
In the s tudy of the infinitesimal calculus it is usually not necessary 
to consider in what units a function or its independent variable may 
be expressed, bu t in numerical work it is necessary to keep the uni ts 
in mind at all t imes. A derivative of a function takes on different 
numerical values according to the uni t employed. In the present case x 
has been expressed in degrees. Hence we may say that dfjdx at x = 0 
is 0.17453 per 10 degrees, or 0.017453 per degree, or 1.00000 per radian 
( remembering that one radian is 57.29578 degrees), or 1.00000 for 
brevity. T h e last s ta tement is already familiar. 
150 FINITE DIFFERENCES 
..,(3) 
Table 7 illustrates the principle of numerical differentiation. 
(4) 
(5) 
(6) 
INTEGRATION 151 
12. N u m e r i c a l i n t e g r a t i o n . In celestial mechanics it is necessary 
to consider the numerical solution of ordinary differential equations of 
the second order. In the most usual form, to which the succeeding 
chapter is devoted, three s imultaneous equations of the second order 
are integrated to give the rectangular coordinates of a planet or comet 
as functions of the t ime. T h i s section is devoted to the consideration 
of two simpler examples. 
T h e method used is precisely the opposite of numerical differentiation, 
where the differences of a function are used to calculate the derivatives, 
and formulas have been given for the purpose. In the present problem 
the derivatives of a function are calculated first, and differenced as a 
check. T h e resulting table is then extended to the left by summation, 
that is, by successively adding consecutive values of the derivative to 
a starting value. Finally the values of the integrals are obtained from 
the summat ion by means of a formula. Formulas for the purpose can 
be derived by integrating an interpolation formula, considering n as the 
independent variable. Those needed are given below, where the first 
summat ion is denoted by and the second by n / . 
(9) 
(10) 
As a first example consider y = J(3x2 — 3) dx. I t is known that the 
result is y = xs — 3x + C, C being an arbitrary constant. T h e word 
arbitrary means that no matter what constant is chosen the result is 
mathematically correct. It will, however, be noticed that no numerical 
value of y can be stated unless C is a known, definite quant i ty . In physical 
problems the constants of integration are determined by the initial 
conditions, which amount in this example simply to the numerical value 
11. S p e c i a l f o r m u l a s . A particularly useful form of (3) may be 
obtained by subst i tut ing for the differences their values in t e rms of the 
functions. T h u s , if the fifth differences are less than 15, 
n w ( ^ ) 0 = f ~ 2 - ^ i + ^ - f v (7) 
and if the seventh differences are less than 70, 
- / _ 3 + 9/_ 2 - 45f_± + 45/i - 9/ 2 + fy (8) 60w (-
152 FINITE DIFFERENCES 
of y for some particular value of x. (For a double integration either 
two values of y or one value of y and one of its first derivative are needed.) 
In the present problem the initial condition is at our disposal; we shall 
choose it to give the polynomial of Section 2, xz — 3x — 23 . 
First are calculated some values of 3x2 — 3, which are differenced. 
T h e results are in Tab le 8, where the column lf is yet to be filled in. 
T A B L E 8 
NUMERICAL INTEGRATION 
y / f1 /" y 
i 
y 
i i 
y 
in 
y 
— 3 
— 3 0 . 5 
+ 2 4 
- \ 5 — 3 1 . 1 2 5 
— 2 
— 2 1 . 5 
+ 9 
— 9 
+ 6 
— 2 1 . 8 7 5 
+ 9 . 2 5 
— 9 
— 1 
— 2 1 . 5 
0 
— 3 
+ 6 
— 2 1 . 6 2 5 
+ 0 . 2 5 
— 3 
+ 6 
0 
— 2 4 . 5 
— 3 
+ 3 
+ 6 
— 2 4 . 3 7 5 
— 2 . 7 5 
+ 3 
+ 6 
+ 1 
— 2 4 . 5 
0 
+ 9 
+ 6 
— 2 4 . 1 2 5 
+ 0 . 2 5 
+ 9 
+ 6 
+ 2 
— 1 5 . 5 
+ 9 
+ 1 5 
+ 6 
— 1 4 . 8 7 5 
+ 9 . 2 5 
+ 1 5 
+ 6 
+ 3 
+ 8 . 5 
+ 2 4 
+ 2 1 
+ 6 
+ 9 . 3 7 5 
+ 2 4 . 2 5 
+ 2 1 
+ 6 
+ 4 
+ 5 3 . 5 
+ 4 5 
+ 2 7 
+ 6 
+ 5 4 . 6 2 5 
+ 4 5 . 2 5 
+ 5 + 7 2 
I t is evident that the values of lf obtained by summat ion will be on 
the half-lines and the star t ing value is therefore needed for some half-line 
value of the argument . T h e value of y for x = + 0.5, de termined by 
direct substi tut ion, is — 24.375. T h e value of lf at the same point is 
determined by formula (9). T h u s 
- 24.375 = w | " / 1 / I + l ( + 3 ) j , 
and since we have w = 1, lf1/2 = — 2 4 . 5 . Inser t ing this value in the 
table, it is now easy to extend the table as far as we please, in either 
direction. F rom these values of lf the corresponding values of y may 
be found by the formula. T h e y are given to the right on Table 8, with 
their differences. If this table of y is interpolated to halves, the results 
will be found to agree with the table in Section 2. 
INTEGRATION 153 
t I u cos u sin u x y 
— 3 0 — 1 0 . 4 5 3 9 — 1 3 . 0 3 9 3 + 0 . 9 7 4 2 1 6 — 0 . 2 2 5 6 2 0 + 1 . 5 4 8 4 3 — 0 . 4 4 2 1 2 
— 2 0 — 6 . 9 6 9 3 — 8 . 7 0 3 3 + 0 . 9 8 8 4 8 5 — 0 . 1 5 1 3 1 8 + 1 . 5 7 6 9 7 — 0 . 2 9 6 5 2 
— 1 0 — 3 . 4 8 4 6 — 4 . 3 5 4 7 + 0 . 9 9 7 1 1 3 — 0 . 0 7 5 9 3 1 + 1 . 5 9 4 2 3 — 0 . 1 4 8 7 9 
0 0 . 0 0 0 0 0 . 0 0 0 0 + 1 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 + 1 . 6 0 0 0 0 0 . 0 0 0 0 0 
+ 1 0 + 3 . 4 8 4 6 + 4 . 3 5 4 7 + 0 . 9 9 7 1 1 3 + 0 . 0 7 5 9 3 1 + 1 . 5 9 4 2 3 + 0 . 1 4 8 7 9 
+ 2 0 + 6 . 9 6 9 3 + 8 . 7 0 3 3 + 0 . 9 8 8 4 8 5 + 0 . 1 5 1 3 1 8 + 1 . 5 7 6 9 7 + 0 . 2 9 6 5 2 
+ 3 0 + 1 0 . 4 5 3 9 + 1 3 . 0 3 9 3 + 0 . 9 7 4 2 1 6 + 0 . 2 2 5 6 2 0 + 1 . 5 4 8 4 3 + 0 . 4 4 2 1 2 
T h i s first example differs from the integration of a planetary or 
cometary or satellite orbit intwo impor tant respects. First , the results 
are exact, no errors of rounding being present . Second, the values of 
the function to be integrated could all be calculated in advance. In the 
integration of orbits this cannot be done, for the function to be integrated 
can be determined only after the integral is known, and it is necessary 
to proceed from step to step by extrapolation, with successive approxim-
ations at each step. At any stage in the calculation the table of n / i s extra-
polated one step, by assuming some order of differences (usually about 
the sixth) to be constant . T h e corresponding value of y is then calculated 
and tested to see if it yields the extrapolated value of / ; if not, it is 
adjusted. T o illustrate this process we integrate the orbit of a hypothetical 
planet with negligible mass, moving under the attraction of the sun 
alone from perihelion to aphelion. I t is known that the orbit is an ellipse. 
Let its semi-major axis be 2 astronomical uni ts and its eccentricity 0.2. 
Take the plane of the orbit as the reference plane, with the X axis directed 
toward the perihelion. T h e equations of motion are 
d2x __ L 2 x 
where r2 = x2 + y2, and k is the Gaussian constant , 0.01720209895. 
I t is known that these equations are t rue , with the stated value of 
ky if Xy yy and r are measured in astronomical uni ts and t in ephemeris 
days. I t has been found by experience that for an orbit of this size and 
shape a tabular interval of 20 days will give the requisite degree of con-
vergence to the differences, bu t in this example an interval of 10 days 
is used to save space in pr int ing the differences. T h u s we shall have 
dt2 r 3 : 
d2y ¥n y 
dt2 r 3 ' 
T A B L E 9 
STARTING VALUES 
154 FINITE DIFFERENCES 
w2 = 100 for the factor necessary in convert ing the second summat ions 
of x and y into x and y. But it is somewhat inconvenient to mult iply 
— x/rs and — y / r 3 by the small factor k2
y and afterward their second 
summat ions by 100. I t is better to keep 11 fx and 11 fy of the same order 
of magni tude as x and yy which is easily accomplished by taking for 
the equat ions to be integrated, 
& = (12) 
where the factor w2k2 is 0.0295 9122 for a 10-day interval. 
T h e origin of t ime being at our disposal, we take it to be the instant 
of perihelion passage. T w o of the constants of integration are obtained 
immediately from the dimensions and orientation of the orbit . W e have 
xo = ro = a(\ — e) = 1-6 a.u. and y0 = 0. For the other two constants 
we take zvkx0 and wky0\ wkx0 = 0 since x0 = 0, and wky0 = 
wkamr-\\ — e2)112 = 0.1489745. T o start the integration it is convenient 
to calculate seven consecutive values of fx and fy. T h e mean motion in 
degrees per day is 57°2957795 ka~312 = 0°3484 64933, from which are 
derived the values of the mean anomaly. T h e eccentric anomaly is 
obtained by Kepler ' s equation, and x and y by 
x = <z(cos u — e)y 
y = <z(l — e2)112 sin u. 
From these values of x and y we first find ry then 1/r3, then w2k2/rz
y 
and finally fx and fy. F r o m these the first seven lines of Tables 10 and 11 
are constructed, leaving llf and lf to be filled in later. 
T h e start ing values uf0 and J / 1 / 2 are obtained by the formulas 
(13) 
(14) 
remember ing that /JLS0 = ^ ( 8 _ ] / 2 + S 1 / 2 ) , with two similar equations for 
y and yy after which the tables may be extended by inspection down 
and to the left to t = 40. 
T h e next step is to calculate fx and fy for t = 40. For this the values 
of x and y are needed. Provisional values may be obtained by means 
of formula (10) if the necessary values in the tables are extrapolated 
by copying the value of / v on the two lines next below. T h e values 
INTEGRATION 155 
t "/. y . A J X fl 
— 30 — 109729 
— 3220 
— 20 — 112949 
— 1975 
+ 1245 
+ 63 
— 10 — 114924 
— 667 
+ 1308 
+ 26 
— 37 
— 15 
0 + 1.6009638 
— 57796 
— 115591 
+ 667 
+ 1334 
— 26 
— 52 
+ 15 
10 + 1.5951842 
— 172720 
— 114924 
+ 1975 
+ 1308 
— 63 
— 37 
— 7 
20 + 1.5779122 
— 285669 
— 112949 
+ 3220 
+ 1245 
— 107 
— 44 
+ 15 
30 + 1.5493453 
— 395398 
— 109729 
+ 4358 
+ 1138 
— 136 
— 29 
+ 8 
40 + 1.5098055 
— 500769 
— 105371 
+ 5360 
+ 1002 
— 157 
— 21 
+ 8 
50 + 1.4597286 
— 600780 
100011 
+ 6205 
+ 845 
— 170 
— 13 
+ 8 
500 — 2.3933823 
— 59327 
+ 51314 
+ 51 
— 45 
+ 1 
+ 1 
— 3 
510 — 2.3993150 
— 7962 
+ 51365 
+ 7 
— 44 
— 1 
— 2 
520 — 2.4001112 
-f 43410 
+ 51372 
— 38 
— 45 
530 — 2.3957702 
+ 94744 
+ 51334 
540 — 2.3862958 + 51253 
of x and y obtained in this way are sufficiently accurate, bu t this will 
not be t rue of the differences wri t ten in by inspection, which will have 
to be revised after fx and fy become known, and it is more convenient 
to use instead of (10) a formula that will give provisional values of x 
and y in te rms of the diagonal differences already writ ten down. Such 
a formula may be obtained by integrating Newton ' s backward inter-
polation formula twice. After a little manipulat ion the result is, wi th 
an accuracy that will suffice for most practical applications, 
*o = "/o + iV-i + S - 3 / 2 + 0.0791667 8% + 0.075 S ! 5 / 2 
1 (15) 
+ 0.07135 84_3 + 0.0682 S 5_ 7 / 2 + 0.065 8 ^ + 0.06 S7_9/2. 
T A B L E 1 0 
INTEGRATION FOR X 
156 FINITE DIFFERENCES 
T A B L E 1 1 
INTEGRATION FOR y 
t fv /!? A11 fl 
— 3 0 + 3 1 3 3 1 
1 0 0 9 3 
— 2 0 + 2 1 2 3 8 
1 0 5 1 2 
— 4 1 9 
+ 2 0 5 
— 1 0 + 1 0 7 2 6 
1 0 7 2 6 
— 2 1 4 
+ 2 1 4 
+ 9 
— 9 
0 0 . 0 0 0 0 0 0 0 
+ 1 4 8 8 8 4 8 
0 
1 0 7 2 6 
0 
+ 2 1 4 
0 
— 9 
1 0 + 0 . 1 4 8 8 8 4 8 
+ 1 4 7 8 1 2 2 
— 1 0 7 2 6 
1 0 5 1 2 
+ 2 1 4 
+ 2 0 5 
— 9 
— 1 5 
2 0 + 0 . 2 9 6 6 9 7 0 
+ 1 4 5 6 8 8 4 
— 2 1 2 3 8 
1 0 0 9 3 
+ 4 1 9 
+ 1 8 1 
— 2 4 
— 3 
3 0 + 0 . 4 4 2 3 8 5 4 
+ 1 4 2 5 5 5 3 
— 3 1 3 3 1 
9 4 9 3 
+ 6 0 0 
+ 1 5 4 
— 2 7 
— 1 1 
4 0 + 0 . 5 8 4 9 4 0 7 
+ 1 3 8 4 7 2 9 
— 4 0 8 2 4 
8 7 3 9 
+ 7 5 4 
+ 1 1 6 
— 3 8 
+ 4 
5 0 + 0 . 7 2 3 4 1 3 6 
+ 1 3 3 5 1 6 6 
— 4 9 5 6 3 
— 7 8 6 9 
+ 8 7 0 
+ 8 2 
— 3 4 
— 4 
5 0 0 + 0 . 1 6 4 2 4 5 4 
— 9 9 1 8 3 6 
— 3 5 2 1 
+ 2 1 2 8 
— 7 
+ 5 
+ 3 
— 4 
5 1 0 + 0 . 0 6 5 0 6 1 8 
— 9 9 3 2 2 9 
— 1 3 9 3 
+ 2 1 2 6 
— 2 
+ 4 
— 1 
5 2 0 — 0 . 0 3 4 2 6 1 1 
— 9 9 2 4 9 6 
+ 7 3 3 
+ 2 1 2 8 
+ 2 
5 3 0 — 0 . 1 3 3 5 1 0 7 
— 9 8 9 6 3 5 
+ 2 8 6 1 
5 4 0 — 0 . 2 3 2 4 7 4 2 + 4 9 9 3 
T h e forward formula, used for integrat ing backward, may be obtained 
from this simply by changing the signs of all the subscripts, and of the 
coefficients of the odd-ordered te rms . T h e results of the application of 
this formula are used to start a third table, which contains x, y> and 
r 2 . If a table is available giving 1/r3 with a rgument r 2 nothing else need 
be writ ten down; 1/r3 may be placed on the keyboard of a calculating 
machine, multiplied by w2k2> the result transferred to the keyboard, and 
multiplied first by — x and then by — y, giving fx and fy, which may 
be entered in the integration tables. T h e computer will soon learn by 
experience whether it is worthwhile to write down the value of 1/r3, 
and to difference these quantit ies as it goes along, for a check. I t may 
even be desirable to difference x and y if the computer is prone to make 
INTEGRATION 157 
- 0.0031 8 5 _ 5 / 2 - 0.003 8I3. 
If the new values of x and y differ from the provisional ones by half 
a un i t or more they should be corrected and the values of fx and fy 
examined to ascertain whether they also need correction. If so, the 
corrected values of fx and fy should be entered in the integration table, 
the differences corrected, and formula (16) reapplied. In general the 
tabular interval should be chosen so that the provisional values of x 
and y may need to be corrected by a few units of the last decimal, and 
small enough that the provisional values of fx and fy need be altered 
bu t seldom.W h e n the final values of fx and fy have been obtained, the tables may 
be extended to the left, giving llfx and ufy for t = 50, and the first 
step of the integration is completed. T h e second step is like the first; 
and the rules for any step can be reduced to the following: 
1. Having a diagonal of the integration tables complete, find p ro -
visional values of x and y by formula (15). 
2. Calculate fx and fy from the differential equations and write them 
in the tables, filling in the differences above and to the right. 
3. Tes t x and y by formula (16); correct them if necessary, and 
re-examine fx and fy. T h e n complete the diagonal of the tables. 
In the present example the tabular interval is so small that step 3 
is hardly necessary. Th i s makes for convenience in the calculation of 
a single step, bu t more steps than necessary are required and there is 
no net saving in labor. Moreover , the accumulat ion of end-figure errors 
is greater than necessary. 
T h e complete tables are not pr inted, bu t only a few lines at the 
beginning and end of the integration. 
many mistakes. In any case the only check on the work as a whole is 
given by the differences in the integration tables ; great care mus t be 
taken to write t hem in correctly. Owing to the accumulat ion of errors 
of rounding, the last decimal of x and y becomes uncertain after a few 
steps of the integration, and it is usually permissible to cut it off. 
Wi th fx and fy for t = 40 entered in the integration tables, the diagonal 
differences above and to the right may be filled in, bu t before filling 
in those to the left the provisional values of x and y mus t be tested 
by the more accurate formula 
(16) 
158 FINITE DIFFERENCES 
13. A c c u m u l a t i o n of e r r o r s in n u m e r i c a l i n t e g r a t i o n . In 
general it is not possible to determine how much error is accumulated 
in an integration, bu t it can be determined by the theory of errors what 
probable error is to be expected after any n u m b e r of steps. For the 
example of the preceding section, however, the theory of elliptic motion 
makes feasible exact determinat ion of the error. T h e t rue coordinates 
and velocities at aphelion are found to be x — — 2.4 a.u., y = 0, 
x = 0, y = — 0.5773503 a.u. per ephemeris day. T h e theory also gives 
the t ime of aphelion passage as t = 516.551259. Formulas (13) and (14) 
may be used to obtain the coordinates and velocity components for 
several dates, which are afterward interpolated to this t ime. T h e results 
are 
x = - 2.3999888, wkx = + 0.0000007, x = + 0.0000041, 
y = _ 0.0000047, why = - 0.0993166, y = - 0.5773516. 
I t is seen that the errors in x and y are greater than those in x and y. 
Thi s might have been expected, because an error in / a t any step produces 
an equal error in all the values of lf that follow, whereas the corresponding 
error in 11 f increases by this amount at every step. 
T h e general theory of the accumulation of errors in numerical 
integration shows that after n steps the probable error of a double integral 
is 0.1124 7z 3 / 2 (in units of the last decimal), which means that in an 
extended n u m b e r of examples about half the errors should be larger 
than this, and half smaller. In the present example n = 52, which gives 
42 for the probable error of x and y. Both of the actual errors are larger 
than the probable error, and this is not to be wondered at, partly because 
the general theory applies only when the integration is carried through 
a number of revolutions of the planet, and partly because two examples 
are not sufficient for a test of any theory of errors. 
Use of the above expression for the probable error will enable the 
programmer to decide before starting an integration how many decimals 
must be carried th roughout the work. Suppose for example that an 
asteroid is to be integrated for 10 years with a 10-day interval, that 
is, for 365 steps, or 183 each way if the integration is commenced in 
the middle of the arc. T h e probable error after 183 steps is about 280. 
Suppose further that a probable error of 0.1 second or arc, or 0.0000005 
radian, is the largest that can be tolerated. Compar ing this with 280 shows 
that nine decimals are necessary in the calculations. 
Th i s test should always be applied before starting any integration. 
Otherwise the programmer will either have intolerable errors in his 
work or he will expend much useless labor manipula t ing superfluous 
decimals. 
SYMBOLIC OPERATORS 159 
T h e general theory of the accumulat ion of errors has an impor tan t 
application to the errors of the osculating Kepler ian elements of an 
orbit obtained by numerical integration. I t has been shown that whereas 
the mean error of the mean orbital longitude is proportional to the three-
halves power of the number of steps, the mean errors of the other five 
e lements are proport ional to the square root of the n u m b e r of steps. 
Th i s is t rue , whether the elements are given directly by the integration 
or whether they are obtained afterwards by t ransforming the rectan-
gular coordinates and velocity components . 
14. S y m b o l i c o p e r a t o r s . All of the formulas given so far in 
this chapter , as well as many others, may be derived in a very elegant 
and simple manner by means of symbolic operators , which we discuss 
in this section. First , however, we make a few remarks about inter-
polation from a more general s tandpoint than before. 
Let the values of any function f(x) be given for n discrete values of 
OC • 0C~^ y OC 2) • • • j OC* y 
which need not be in ari thmetical progression bu t may 
be any values of x whatever. Consider the polynomial 
(x — X2) (x — ff3) ... (x — Xn) ^ ^ (x — X}) (x — ff3) ... (x — xn) 
(xn x±) (xn x2)... (xn xn-i) 
Evidently, cp(x) = / ( # i ) for x = xl9 since all the coefficients vanish 
except the one factored by / f a ) , which coefficient becomes uni ty. 
Similarly, <p{x) = f(x2) for x = x2, etc. T h e polynomial cp(x) is called 
Lagrange 's interpolation formula; it is a generalized form of the formula 
previously denoted by the same name. 
T h e quot ient 
/ f a + i ) 
<P(X) 
fa - x2) fa - * 3 ) . . . (x1 - x n ) J (x2 - xj fa - x3)... fa - xn) 
(X - X±) (X - X2) ... (X — ^ n -x) rt \ 
+ - + (x.. - x,\ (x- - x.) ... (x- - x—*YKn)' 
; / f a ) 
xk+l Xk 
is called the first divided difference of / f a ; we denote it for brevity 
by the symbolic notation [k, k + 1]. T h e second divided difference is 
denoted [k, k + 1, k + 2] and is given by 
[*,* + ! , * + 2] = 
[* + ! , * + 2 ] - [ * , * + !] 
xk+2 xk 
Similarly the pth divided difference is given by 
[k,k + 1 , k + p] 
[* + 1, A + 2 , k + p] - [k, k + 1 , k + p - l ] 
xk+p xk 
160 FINITE DIFFERENCES 
[ X >
 1 ] = ^ - * 
/ = / i + [*. 
whence 
Similarly, 
whence 
[x, 1] = [1, 2 ] + [*, 1,2] ( * - * , ) , 
and in general, 
/(*) = A + (* ~ xi) [h 2] + (x - * x ) <* - * 2 ) [1, 2, 3] + ... 
+ (x — xj (x — x2) (x — xn^) [1, 2 , w ] +R(x)9 (17) 
where 
JR(^) = (# — ^ ) (# — x2) ••• (^ — #n) [x9 1, 2, »]. 
Lagrange 's interpolation polynomial has the same values as f(x) for 
and therefore has the same divided differences at 
these points. Now cp(x) is a polynomial of the (n— l) th degree, and 
the wth divided difference of such a polynomial is zero, as may be shown 
by induction. Hence the same is t rue of the nth divided difference of 
f(x)y and 
/(*) = cp(x) + R(x). 
Since f(x) = cp(x) for x — xl9 x2, xny then at these values of x9 
R(x) = 0. Hence the (n— l) th derivative of R(x) is zero for at least 
one value of x in the interval xx to xn. 
Now, differentiating n — 1 t imes, 
f(n-i) W = („ _ 1)! [1, 2, n] + R<»-u (x). 
If | is the value of x for which i ? ( n _ 1 ) ( ^ ) = 0, then 
/ ( n - l ) ( £ ) = ( „ _ ! ) ! [ i , 2 , . . . , » ] , 
whence 
[1,2 it] = 
W e may extend the meaning of divided differences to include the 
case where x is not one of the discretevalues xl9 x2> xn. W e may 
then write 
L*' 1 J ~ * , - X ' 
or, with yet more brevity, 
fx-f 
S Y M B O L I C O P E R A T O R S 161 
and 
[*, 1, 2, n] = f
{n)(£) 
nl 
For any value of x there will be some value of £ for which this equation 
holds. 
W e may now write for R(x)y 
f{n)(i) 
R(x) = —^r (x — xi) (x - x2)... ( * - * n ) > 
and this equation shows that if the values xly x2, xn are in ari thmetical 
progression, R(x) has a m i n i m u m value when x lies in the middle of 
the interval x1 to xn. Fu r the rmore , for any values of xv x2> xny R(x) 
is smaller if x is within the interval xx to xn than if it is outside. 
If we confine ourselves to represent ing the function f(x) over the 
interval xx to xn by a polynomial of the (n — l ) th degree, then we use 
formula (17) dropping the final t e rm R(x), and we have Newton ' s 
interpolation formula. 
If the function is tabulated for values of x tha t are in ari thmetical 
progression, which is the case commonly found in astronomy, then there 
are three conventional notations for the differences, which are shown 
in Tab le 12. 
T A B L E 12 
NOTATIONS FOR DIFFERENCES 
Central 
differences 
Forward 
differences 
Backward 
differences 
/-
S - 5 / 2 - 3 
A2 
- 3 
/ - 2 ^ x 
8 - 3 / 2 
8 3 , 
-3/2 
A 
- 2 - 8 F-x 
f-x 
8 2 
- i f-x ^ 2 f-x 
8 - l / 2 
8 3 , 
-1/2 
-X 
A3 
- 2 P . K 
/ » K /. A2 
- i fo K 
\„ h3, 
1/2 
J 
o 
J 3 
—1 I 7 ! K 
fx 
\„ 
K 
h3, 
1/2 
fx 0 fx ' 2 
S 3 /2 
8\ 
3/2 
A 
i 
J 3 
o ^ 2 K 3 
/. 8 2 
2 
8\ 
3/2 
/. J 2 
i ^ 3 
S 5/2 2 
A3 
i I 7 , K 
K 
f3 A2 
2 
/ a K 
S 7/2 
A 
3 
/ < /. 
162 FINITE DIFFERENCES 
For the central differences the subscripts are equal along a horizontal 
line, for the forward differences along the forward diagonal, and. for 
the backward differences along the backward diagonal. In previous 
sections we have used only the notation for central differences. W e have, 
for example, 
and, as earlier in this chapter, 
§1/2 — A ~~/o> 
8 0
 = S l / 2 ~ 8 - 1 / 2 = f l - 2fo +/-!> 
/ x 8 0 = i ( 8 l / 2 + 8 - l / 2 ) -
If as before we denote the interval of the a rgument by w, n consecutive 
tabular a rguments by and any n u m b e r within the range 
x1 to xn by x, then the general interpolat ing factor 9 is defined by 
x = xx + 9w, 
where 9 is not necessarily between 0 and 1. Also, 
x — x1 = 9wy 
x — x2 = (9 — I) w, etc., 
and, using our symbolic notation, 
[ U ] = ( / 8 - / ^ = ^ , 
[1,2, 3] =A\\2w\ 
[1, 2, 3, 4] = J s /3 ! w3, etc. 
Subst i tu t ing these expressions into (17), omit t ing the remainder R, 
f(Xl + 6w) =f(Xl) 4 
(18) 
which is known as Gregory 's formula. Except for notation, it is the same 
as Newton ' s formula of Section 6 . 
I t mus t be remembered that As
y for example, does not mean the cube 
of A bu t instead the thi rd forward difference of / f a ) . I t is only in 
consequence of the particular symbolic notat ion we have chosen that 
(19) may be derived from (18) by treat ing (1 + A)6 as if (1 + A) were 
being raised to the power 9 . T h e symbol A as used here is said to be 
an operator. 
We now introduce the new operator Ey such that E operat ing on f(x) 
gives f(x + w). Symbolically, 
Ef(x) = f(x + w). 
Also, we have the operators 
Af(x)=f(x + w)-f(x), 
§/(*) =f(x + \ w ) - f ( x - \w\ 
/ * / ( * ) = £ [ / ( * + i « ) + / ( * - i « 0 ] -
Now since 
we have 
Ef(x) = / (* + «,) = (! +A)f(x), 
E=l+A (20) 
Also, if all derivatives off(x) exist, we may write by Taylor ' s theorem 
/ ( * + W ) = / ( * ) + £ ^/<«>(x). (21) 
Since e, the base of natural logari thms, raised to any power, say y, 
is given by 
^ = i + y + ^ + f r + - . 
Eq. (21) may be writ ten symbolically as 
f(x + zv)= e™Df(x). (22) 
SYMBOLIC OPERATORS 163 
Symbolically we may write (18) as 
f(Xl + 0tv) = {l (19) 
since if (1 + A)6 is expanded by the binomial theorem the result is 
164 FINITE DIFFERENCES 
I t is t rue that in general we do not know whether the first n derivatives 
of f(x) exist since we have only the tabulated values to refer to, bu t 
for purposes of interpolation we rely on the representation of f(x) by 
a polynomial, the derivatives of which are known to exist. 
W e may now write 
Ef(x) =f(x + zv) = (l +A)f(x) = e"*>fix\ 
whence, symbolically, 
E= 1 + A = ewD. (23) 
For backward differences we have 
TO = / ( * ) - / ( * - » ) • 
Operat ing on both sides with E, 
EVf(x) =/(* + w) -/(*) = Af{x\ 
whence 
EP =A = E - l , (24) 
or 
E = rh?- (25) 
N o w we can express f(x± + 8w) in te rms of the backward differences as 
f(Xl + dw) = (i - ?)-7(*i) 
0(6 + 1) (6+ 2)... (e + n-2)_n_, 
(26) 
(» - 1)! 
which is known as Gregory 's backward formula. 
Fur the r we have 
»/(*) =/(* + iw)-/(*-i») 
= (i + j ) i / 2 / ( * ) _ ( i +j)-i/vw, 
whence 
S = ( 1 + J ) i / 2 _ ( i + J ) - i / « . 
F rom this we find 
= 1 + ^ 
E = I +A = (27) 
= 2 Ak 8 */(*,) = 2 Bk | .8*/(*i), (29) 
in which S 0 / ^ ) = f(x^) and /x8°/(#i) = Hf(xi)i a n d -4 and denote the 
coefficients in the expansion of the lines above. W e write 
S Ak 8 = 2 A'k 8* / (* i ) + ^ K » V ( * i ) , 
in which ^4' and fi' correspond to odd values of k, and ^4" and B" 
to even values. Now, since 
[|-8 + (1 + ^ S 2 ) 1 / 2 ] - 2 0 - [ | 8 - (1 + i S 2 ) 1 / 2 ] 2 0 = [— i S + (1 + i S 2 ) 1 ' 2 ] 2 0 , 
it follows that A' and B' are odd functions of 0 and ^4" and S " even 
functions of 0 , and that 
i:̂ 8V(*1)=̂ B;M8V(*,), 
^^'8*/(*I)=Zfi;>8V(*i). 
Hence we may also write 
+ to) = 8*/(*i) + ZB't'ii&ftxJ 
= ZB'k M s */(*!) + £ ^ ' s*/(*i). ( 3 0 ) 
If in the first of (30) we pu t xx -f- for xlt we have 
/(*! + \w + 6w)=SA'k 8*/(*i + \w) + 2B'k'fi8
kf(x1 + \w), 
which is Bessel 's formula of Section 5. T h e second of (30) is known 
as the Newton-Cotes formula. In Bessel's formula we may change 
8/(*i + iw) to / ( * , ) — / ( * , ) and Ssf(Xl + \w) to S 2 / (* 2 ) — 8*f(Xl), etc., 
obtaining Everet t ' s formula of Section 5. 
S Y M B O L I C O P E R A T O R S 165 
and, similarly, 
,t = (1 + | 8 2 ) 1 / 2 - (28) 
T h e n 
/(*! + da) = E»/(Xl) 
= [ i s + ( i + i W « » / ( * i ) 
166 FINITE DIFFERENCES 
T h e formulas for numerical differentiation may be obtained from the 
equation 
1 + J = ewD = [i-S + (1 + i -8 2 ) 1 ' 2 ] 2 , 
which can be used to express D and its powers in terms of the diagonal 
differences A or V or the central differences S. For example, 
A2 AZ An 
WD = In (1 + A) = A - y + y - ... + ( ~ l )*" 1 - , 
whence 
Formulas in te rms of central differences may be obtained using 
wD = 2 In [|-S + (1 + i S 2 ) 1 ' 2 ] = j ?C f c 8*, 
:27G f r Sfc. 
or 
Formulas for numerical integration may be obtained with equal 
facility. Symbolically we may write D~1f(x) for a function of which the 
derivative is f(x), so that 
S - V ( * ) = Sf(*)dx, 
D~2f(x) = $$f(x) dx\ etc. 
Also, we may write for a function whose first differences are 
f(x). T h e n t 
N O T E S A N D R E F E R E N C E S 
A g o o d b o o k o n n u m e r i c a l ca lcu lat ion is A n d r e w D . B o o t h , Numerical Methods 
( 2 n d ed . ) A c a d e m i c Pres s , N e w York , 1957. In it are p r o v e d m o s t of the f o r m u l a s 
that are g i v e n h e r e , as w e l l as m a n y o t h e r s , and also g i v e n are re ferences to t h e 
l i terature sufficient to e n a b l e t h e reader to p u r s u e any b r a n c h of t h e subjec t as 
far as h e w i s h e s . 
A n e x h a u s t i v e treatise o n the ca lcu lus of finite d i f ferences is H e n r i M i n e u r , 
Techniques de Calcul Numerique, Librair ie P o l y t e c h n i q u e C h . Beranger , Paris , 1952 . 
A n e x c e l l e n t co l l ec t ion of aids to in terpo la t ion , w i t h p r e c e p t s for the ir u s e , is 
Interpolation and Allied Tables, prepared b y H . M . N a u t i c a l A l m a n a c Office, 
H . M . S ta t ionery Office, L o n d o n , 1956. It a lso conta ins use fu l s e c t i o n s o n n u m e -
rical di f ferent iat ion a n din tegrat ion , differential e q u a t i o n s , and prec i s ion of 
ca lculat ion . A c o m p a n i o n book le t o n Subtabulation w a s i s s u e d in 1958 . 
CHAPTER V 
NUMERICAL INTEGRATION OF ORBITS 
1. I n t r o d u c t i o n . T h e method of numerical integration is the most 
powerful known in celestial mechanics for calculating the motion of any 
body in the solar system for a few revolutions a round its pr imary, with 
all the precision demanded by modern observations. For obtaining the 
orbit for many revolutions experience indicates that analytic methods 
are likely to be more efficient, except for orbits of large eccentricity 
where analytic methods become progressively more difficult. T h u s , 
numerical methods have been used for most comets and for many minor 
planets, while analytic methods have been applied to the eight principal 
planets, to the moon, and to most other satellites and a n u m b e r of 
minor planets as well. Whe the r this condition will long persist cannot 
be foretold. T h e recent advances in comput ing with punched cards and 
the experiments with electronic techniques now in progress will certainly 
render both numerical and analytic methods much more efficient than 
they have been in the past. It cannot be known as yet whether either 
method will gain at the expense of the other, bu t it is certain that the 
practical celestial mechanician will always profit by the use of a judicious 
combination of numerical and analytic methods . 
In this chapter are discussed in detail the two most common methods 
of numerical integration of orbits, Cowell 's and Encke 's . These methods 
owe their populari ty partly to the widespread availability of calculating 
machines and partly to the availability of the rectangular coordinates 
of the seven largest planets up to 1980, which have been publ ished in 
a convenient arrangement with auxiliary tables by the British Nautical 
Almanac Office; without these two aids these methods would probably 
be less efficient than others. 
In Cowell 's method no explicit use is made of a conic section as the 
first approximation to the orbit . T h e equations of motion in rectangular 
coordinates are integrated directly, giving the rectangular coordinates 
of the dis turbed body. T h e process is analogous to that used in 
Chapter IV, Section 12, with the difference that three coordinates are 
needed instead of two, and at each step the attractions of the d is turbing 
planets are added to that of the sun. T h e origin is usually taken at the 
pr imary but this restriction is not necessary, and the center of mass 
167 
168 NUMERICAL INTEGRATION 
of the system or of any dis turbing body may be used. T h e only restriction 
is that the motions of all bodies exerting appreciable effects are supposed 
to be already known relative to the chosen origin. Since no use is made 
of the conic section as a first approximation the method could be applied 
in systems not dominated by a single mass, such as the motion of a 
satellite of a binary star. T h e only practical disadvantage of the me thod 
is that the integrals contain many significant figures and change rapidly 
with the t ime. In consequence the integration tables are slowly con-
vergent, which compels the use of a small tabular interval. 
In Encke 's method the coordinates are not obtained directly, bu t 
instead the integration gives the difference between the actual coor-
dinates and the coordinates in the osculating orbit, that is, the position 
the body would have if it cont inued to move in the conic section 
corresponding to the coordinates and velocity components at a particular 
instant called the epoch of osculation. T h e departures from the osculating 
orbit are called per turbat ions . T h e y vanish at the epoch of osculation. 
T h e advantage of this method is that for dates near the epoch the 
perturbat ions are small ; they can be expressed by a few significant 
figures, which permits a larger tabular interval than with Cowell 's 
method. T h e disadvantage is that in the course of t ime the perturbat ions 
increase to a large size, which requires occasional rectification of the 
orbit. T h e coordinates and velocities are determined at a new epoch 
and the integration recommenced. I t seems that the difficulty might be 
avoided by taking for the first approximation a conic section that 
approximates the actual motion over a longer interval of t ime than does 
the osculating orbit , bu t ordinarily this is not practicable for lack of 
sufficient information. 
A method much used dur ing the nineteenth century, and still of value, 
is that of the variation of elements. In this method the quantities obtained 
by the integration are the six osculating elements. T h e y change 
comparatively slowly, which means that a rather large tabular interval 
can be used, bu t the differential equations are more complicated in form 
than the equations in rectangular coordinates. 
Probably the method most elegant in conception is that of Hansen. 
He starts with the osculating ellipse as Encke does, bu t instead of 
perturbat ions of the rectangular coordinates he integrates the per turba-
tions of three other quantit ies. T h e fundamental one, the per turbat ion 
of the mean anomaly, is given by a double integral. T h i s is the quanti ty 
to be added to the mean anomaly in the ellipse osculating at the 
fundamental epoch, the result being used with the other osculating 
elements to calculate the longitude, latitude, and radius vector. T h e 
longitude thus obtained is the actual longitude, bu t the radius vector 
and the latitude require small corrections, obtained by integrating two 
COWELL'S METHOD 169 
equat ions of the first order. T h u s it appears that the complete solution 
involves only four constants of integration, bu t this is not strictly t rue . 
T h e complete solution involves three other variables, which are closely 
related to the per turbat ions of the inclination and node of the osculating 
ellipse bu t which are m u c h smaller in magni tude , becoming appreciable 
only after many revolutions of the dis turbed body. T h e advantage of 
the me thod is that it is usually possible to neglect these last three 
quanti t ies (but if not, they may be obtained very simply) and that the 
other three per turbat ions are smaller in magni tude than in any other 
known method . T h e disadvantage is that several quanti t ies are needed 
that have not been extensively tabulated (as have the rectangular coor-
dinates of the d is turbing planets), which requires several rather laborious 
transformations. N o extensive application of the method has ever been 
made, and it is likely that with special tables to facilitate the t rans-
formations it might be very efficient. 
2 . E q u a t i o n s f o r C o w c l l ' s m e t h o d . T h e equat ions of mot ion of 
two point-masses ma and mb unde r their mutua l action have been given 
in Section 3 of Chapter I. T h e f - componen t s are given by 
In t roduce additional point-masses mv m 2 , m3> ... into the system, and 
denote any one of these by rrij. Evidently the at tract ions of mj on ma 
and mb are given by equations similar to (1), and the total accelerations 
of ma and mb be obtained by s u m m i n g all these attractions, giving 
™a£a = k2mamb
 b ^ a , mh£b = k2mbma (1) 
k2mbma 
k2mamb 
(3) 
(2) 
with similar equations for rj and £, and 
Pla = (L - W + (va - v,Y + (£« - Q\ 
Let the origin of coordinates be taken at ma, which is equivalent to 
the linear t ransformation 
with similar equat ions for rj and £, and 
r2 = (L ~ to)2 + (Va ~ V,)2 + (£a - £&)
2. 
£b f a —
 x> £j f a —
 xj> 
170 NUMERICAL INTEGRATION 
T h i s and similar equations for y and z are the fundamental equations 
in CoweH's method. If ma represents the sun then it is evident that 
xy yy z are the heliocentric coordinates of the body whose motion is 
sought, and xjy yjy zj are the heliocentric coordinates of any otherbodies 
acting on m. F rom the way in which the equations have been derived 
it is seen that in (5) the first t e rm represents the action of the sun on my 
the first t e rm in the parentheses represents the action of m3- on my and 
the second represents the action of ntj on the sun. 
T h e equations may be used equally well for satellite motion by taking 
the origin at the pr imary and one of the ntj for the sun. 
If the body whose motion is to be determined is an asteroid or comet, 
m is pu t equal to zero. In this case the m3 are all small compared with 
uni ty and the xjy yjy z3- may usually be regarded as completely known, 
which permits the solution to be obtained by successive approximations. 
At each step of the integration approximate coordinates of the body 
are obtained by extrapolation, and these coordinates are used to calculate 
w2xy w2yy w2z by a process similar to that used in the preceding chapter 
except that the port ions contr ibuted by the dis turbing planets mus t be 
calculated separately for each planet and added to the port ion contr ibuted 
by the sun. W h e n the integration extends over a very few revolutions 
the body will not depart very far from the orbit given by the ellipse 
at the epoch of osculation, and unless the body approaches very close 
to a major planet the planetary attractions may be calculated in advance, 
from which follows 
£ j £b ~ X3 X> 
and put 
r) = x2 + y2 + #2, 
p2 = (Xj - x)2 + {y3 - y)2 + (zj - zf. 
Divide Eq. (2) by ma and Eq . (3) by mb, and subtract the first from 
the second. T h e result is the equation of motion of mb relative to ma, 
(4) x = — k2 (ma + mb) 
with similar equat ions for y and z. 
Suppose that ma is taken as the uni t of mass, and that all other masses 
are measured in this unit . T h e n we may pu t ma = 1 and drop the 
subscript from mby and Eq. (4) may be wri t ten 
(5) x = — k\\ + m) 
COWELL'S METHOD 171 
using the position of the body in the osculating orbit instead of in its 
actual orbit . In this way much labor may be saved. W h e n the integration 
extends over many revolutions, however, such an approximation is not 
permissible; in such cases it is sometimes possible to use the results 
of a previous integration in calculating the planetary attractions. 
3* N u m e r i c a l a p p l i c a t i o n of C o w e l l ' s m e t h o d . T o illustrate the 
numerical processes described in the preceding section, we set up an 
integration scheme for the asteroid Ceres. T h e elements referred to the 
ecliptic and mean equinox 1950.0, which are chosen for the basis of 
the work, are 
Epoch of osculation, 
1941 January 6.0 E.T. = J.D. 243 0000.5, 
Mean anomaly, 
/ = 75°46 ,11794 = 75?76998, 
Arc from the node on the ecliptic to the perihelion, 
OJ = 71°4 ,5. , ,06, 
Longi tude of node on the ecliptic, 
ft = 80048'5o:'7i, 
Inclination to the ecliptic, 
/ = 10o35'49."00, 
Eccentricity, 
e = 0.0794 2668, 
Semi-major axis, 
a = 2.7672 3786 a.u. 
F rom these are obtained 
sin / = + 0.1838 9893, sin ft = + 0.9871 7554, 
cos / = + 0.9829 4516, cos ft = + 0.1596 3850, 
sin g o = +0 .9459 0471, cos cp = 0.9968 4071, 
cos = + 0.3244 4457, 
and from the obliquity of the ecliptic at 1950.0, 
sin e = + 0.3978 8118, cos e = + 0.9174 3695, 
172 NUMERICAL INTEGRATION 
Ax = - 2.3965 7958, Ay = + 0.9984 2280, A2 = + 0.9576 8656, 
Bx = - 1.2849 7379, By = - 2.2997 8838, 5 , = - 0.8179 9285. 
Check: AXBX + AyBy + J 2 5 , = + 0.0000 0003. 
In deriving the mean motion from the semi-major axis by Kepler ' s 
th i rd law we add the mass of Mercury to that of the sun, for a reason 
to be described in Section 7, which gives the mass of the sun -f- Mercury 
= 1.0000 00167, whence for a 10-day interval wk = 0.1720 2101 82, 
and therefore the mean motion is 0.2141 0874 66 degrees per day. Next 
are calculated the coordinates and velocity components for the osculating 
date, to eight decimal places. 
/ = 75°46 , l i : , 94 = 1^3224 35683, u = 114007 16333, 
sin u = + 0.9855 71232, cos u = + 0.1692 61179, 
x = - 1.4817 2875, r 2 = 7.4530 9385, wkx = + 0.0812 3006, 
y = - 2.1769 1244, r = 2.7300 3550, wky = - 0.0520 1752, 
z = - 0.7201 5692, —T= = 0.0378 78256, wkz = - 0.0409 9650. 
r\a 
T h e most convenient way to start the integration is to calculate from 
the osculating elements the coordinates and attractions for three dates 
on each side of the epoch. These may require slight correction later 
when the planetary at tractions are taken into account (Table 1). 
T A B L E 1 
STARTING DATA 
Julian Date / u sin u cos u 
242 9970.5 69.°34672 73P71495 + 0.959 8785 + 0.280 4162 
242 9980.5 71.48781 75.90155 + 0.969 8785 + 0.243 5888 
242 9990.5 73.62889 78.08160 + 0.978 4427 + 0.206 5184 
243 0000.5 75.76998 80.25513 + 0.985 5712 + 0.169 2612 
243 0010.5 77.91107 82.42214 + 0.991 2665 + 0.131 8732 
243 0020.5 80.05215 84.58264 + 0.995 5334 + 0.094 4100 
243 0030.5 82.19324 86.73667 + 0.998 3785 + 0.056 9250 
whence 
COWELL'S METHOD 173 
— w2k2jrz 
1 . 7 1 5 1 0 6 
• 1 . 6 3 9 6 9 6 
• 1 . 5 6 1 8 5 9 
• 1 . 4 8 1 7 2 9 
• 1 . 3 9 9 4 4 4 
• 1 . 3 1 5 1 4 3 
1 . 2 2 8 9 6 3 
- 2 . 0 0 6 8 4 5 
- 2 . 0 6 6 6 1 2 
- 2 . 1 2 3 3 2 0 
- 2 . 1 7 6 9 1 2 
- 2 . 2 2 7 3 3 9 
- 2 . 2 7 4 5 5 6 
- 2 . 3 1 8 5 2 5 
- 0 . 5 9 2 6 8 9 
- 0 . 6 3 6 1 3 8 
- 0 . 6 7 8 6 4 5 
- 0 . 7 2 0 1 5 7 
- 0 . 7 6 0 6 2 2 
- 0 . 7 9 9 9 9 0 
- 0 . 8 3 8 2 1 6 
7 . 3 2 0 2 9 6 
7 . 3 6 4 1 6 0 
7 . 4 0 8 4 5 0 
7 . 4 5 3 0 9 3 
7 . 4 9 8 0 2 8 
7 . 5 4 3 1 9 0 
7 . 5 8 8 5 1 4 
- 0 . 0 0 1 4 
- 0 . 0 0 1 4 
- 0 . 0 0 1 4 
- 0 . 0 0 1 4 
- 0 . 0 0 1 4 
- 0 . 0 0 1 4 
- 0 . 0 0 1 4 
9 4 0 6 7 
8 0 7 3 8 
6 7 4 7 9 
5 4 3 1 4 
4 1 2 6 0 
2 8 3 3 7 
1 5 5 5 9 
Since eight decimals are to be used in the integration, the last co lumn 
is needed to nine so tha t the eighth will be accurate after mult iplying 
by the coordinates. Next , the planetary attractions are calculated for 
the same seven dates, with an extra decimal so that after being added 
to the solar at traction the sum may be accurately rounded to eight 
decimals. In Tab le 2, the results are given for each planet, with their 
sum, the three quanti t ies for each date being the x, y, and z components . 
T A B L E 2 
PLANETARY ATTRACTIONS ON CERES 
Julian Date Venus Earth Mars Jupiter Saturn S u m 
+ 1 4 . 6 — 2 . 1 + 0 . 4 — 4 9 . 0 — 3 . 1 — 3 9 . 2 
2 4 2 9 9 7 0 . 5 + 0 . 7 — 7 . 7 + 0 . 5 — 4 0 . 6 — 2 . 4 — 4 9 . 5 
— 0 . 7 — 3 . 4 + 0 . 2 — 1 7 . 6 — 0 . 9 — 2 2 . 4 
+ 1 4 . 3 — 0 . 5 + 0 . 4 — 4 8 . 6 — 3 . 1 — 3 7 . 5 
2 4 2 9 9 8 0 . 5 + 4 . 4 — 7 . 9 + 0 . 6 — 4 1 . 1 — 2 . 4 — 4 6 . 4 
+ 1 . 0 — 3 . 5 + 0 . 2 — 1 7 . 8 — 0 . 9 — 2 1 . 0 
+ 1 3 . 0 + 1.1 + 0 . 4 — 4 8 . 1 — 3 . 1 — 3 6 . 7 
2 4 2 9 9 9 0 . 5 + 8 . 0 — 7 . 9 + 0 . 6 — 4 1 . 6 — 2 . 4 — 4 3 . 3 
+ 2 . 6 — 3 . 5 + 0 . 2 — 1 7 . 9 — 0 . 9 — 1 9 . 5 
+ 1 0 . 7 + 2 . 7 + 0 . 4 — 4 7 . 6 — 3 . 1 — 3 6 . 9 
2 4 3 0 0 0 0 . 5 + 1 0 . 8 — 7 . 5 + 0 . 7 — 4 2 . 1 — 2 . 3 — 4 0 . 4 
+ 4 . 0 — 3 . 3 + 0 . 2 — 1 8 . 0 — 0 . 9 — 1 8 . 0 
+ 7 . 7 + 4 . 2 + 0 . 4 — 4 7 . 2 — 3 . 1 — 3 8 . 0 
2 4 3 0 0 1 0 . 5 + 1 2 . 7 — 7 . 0 + 0 . 7 — 4 2 . 5 — 2 . 3 — 3 8 . 4 
+ 5 . 1 — 3 . 1 + 0 . 2 — 1 8 . 1 — 0 . 9 — 1 6 . 8 
+ 4 . 1 + 5 . 6 + 0 . 4 — 4 6 . 7 — 3 . 1 — 3 9 . 7 
2 4 3 0 0 2 0 . 5 + 1 3 . 7 — 6 . 1 + 0 . 8 — 4 3 . 0 — 2 . 3 — 3 6 . 9 
+ 5 . 8 — 2 . 7 + 0 . 3 — 1 8 . 2 — 0 . 9 — 1 5 . 7 
+ 0 . 3 + 6 . 8 + 0 . 4 — 4 6 . 2 — 3 . 2 — 4 1 . 9 
2 4 3 0 0 3 0 . 5 + 1 3 . 7 — 5 . 1 + 0 . 8 — 4 3 . 5 — 2 . 3 — 3 6 . 4 
+ 6 . 1 — 2 . 2 + 0 . 3 — 1 8 . 3 — 0 . 9 — 1 5 . 0 
x y z r2 
174 NUMERICAL INTEGRATION 
These data together with the starting values calculated by the formulas 
of Chapter IV give the first seven lines of the Integrat ion Tables 3 ,4 , 
and 5. Only enough figures of the Jul ian Date are given to suffice for 
identification. 
T A B L E 3 
INTEGRATION FOR X 
Date y fx / 
997 + 256209 
— 13450 
998 + 242759 
— 13596— 146 
+ 32 
999 + 229163 
— 13710 
— 114 
+ 29 
— 3 
000 — 1.48190830 
+ 8229586 
+ 215453 
— 13795 
— 85 
+ 29 
0 
001 — 1.39961244 
+ 8431244 
+ 201658 
— 13851 
— 56 
+ 25 
— 4 
002 — 1.31530000 
+ 8619051 
+ 187807 
— 13882 
— 31 
003 — 1.22910949 + 173925 
T A B L E 4 
INTEGRATION FOR y 
Date y fy z 1 /" / 1 V 
997 + 299787 
+ 6178 
998 + 305965 
+ 5584 
— 594 
+ 12 
999 + 311549 
+ 5002 
— 582 
+ 8 
— 4 
000 — 2.17717626 
— 5043084 
+ 316551 
+ 4428 
— 574 
+ 13 
4- 5 
001 — 2.22760710 
— 4722105 
+ 320979 
+ 3867 
— 561 
+ 12 
— 1 
002 — 2.27482815 
— 4397259 
+ 324846 
+ 3318 
— 549 
003 — 2.31880074 + 328164 
COWELL'S METHOD 175 
T A B L E 5 
INTEGRATION FOR Z 
Date y /. / /" r 
9 9 7 + 8 8 5 2 9 
+ 5 6 4 5 
9 9 8 + 9 4 1 7 4 
+ 5 3 9 6 
— 2 4 9 
— 2 
9 9 9 + 9 9 5 7 0 
+ 5 1 4 5 
— 2 5 1 
0 
0 0 0 — 0 . 7 2 0 2 4 4 1 9 
— 4 0 4 6 8 7 4 
+ 1 0 4 7 1 5 
+ 4 8 9 4 
— 2 5 1 
— 2 
0 0 1 — 0 . 7 6 0 7 1 2 9 3 
— 3 9 3 7 2 6 5 
+ 1 0 9 6 0 9 
+ 4 6 4 1 
— 2 5 3 
+ 1 
0 0 2 — 0 . 8 0 0 0 8 5 5 8 
— 3 8 2 3 0 1 5 
+ 1 1 4 2 5 0 
+ 4 3 8 9 
— 2 5 2 
0 0 3 — 0 . 8 3 8 3 1 5 7 3 + 1 1 8 6 3 9 
= + 0.0822 9586, 
and similarly for y and z. 
T h e next step is to recalculate fx,fyyfz for dates 001 , 002, 003, using 
values of x9 y9 z obtained from the integration tables. I t is found that 
the provisional values of x9 y9 z require small corrections, bu t that the 
values of / are not changed. T h e integration now proceeds as in the 
example of the last chapter . For purposes of the integration the values 
of x9 y, z obtained at each step need to be wri t ten to only six decimals, 
bu t if the integration orbit is afterward to be compared with observation 
it will be desirable to keep seven decimals, and to make the small 
corrections needed to the extrapolated values at each s tep ; the values 
of x9 y, z will then be definitive, and may be used for comparison with 
observations. 
Fo r the x coordinate 
! / i / 2 = wk*Q + ^ 
176 N U M E R I C A L I N T E G R A T I O N 
4 . E q u a t i o n s f o r E n c k e ' s m e t h o d . Let xQy yQy z0 be the helio-
centric rectangular coordinates of a point-mass m moving under the 
attraction of the sun alone. I t is known that the orbit is given by the 
equations 
Xo=-k%l+m)%, y0 = -k*(l+m)%, z0=-k%l+m)^, (6) 
ro ro ro 
with 
rl = 4+yl + *o-
Let £, 77, £ represent the increments of xQy yQy z0 produced by the 
attractions of the planets. T h e n the actual coordinates x, yy z of m at 
any t ime are 
x = x0 + f, y = y0 + 77, z = z0 + £, 
and the actual equations of motion are Eq. (5), 
x = - k2(l + m) (7) 
and similar equations for y and z. 
Subtract ing Eq. (6) from (7) gives 
(8) x — x0 = £ = k2(l + m) 
with similar equations for ij and £. 
T h e per turbat ions £, 17, J might be determined by direct integration 
of Eq. (8). T h e term x0/rl might be calculated for every step of the 
integration in advance by the laws of elliptic motion, the term x\rz 
determined at each step by extrapolating £ and adding it to x0 to give xy 
etc., bu t this way of proceeding would not be convenient in practice. 
Since £ is a small quanti ty, # 0 /
r o * s nearly equal to x/r*y and these two 
terms would have to be calculated to many more significant figures than 
are needed in their difference. Encke was therefore led to seek a t rans-
formation that avoids this difficulty. 
Trea t ing only the equation for £, since those for ij and £ are exactly 
similar, we have 
But 
whence 
If we put 
we have 
ENCKE'S METHOD 177 
1 + 2 ? , 
(1 + 2q)-™, 
If TJ, I are so small in comparison with x0, y0, z0 that their squares 
may be neglected, then we may write 
g = « r f + j w + « b C t ( 1 0 ) 
which is an easily calculated function, whereas 1 — (1 + 2q)~sl2 is 
not . Tables might be constructed giving this latter function with 
a rgument q, bu t such tables would have to be rather extensive and would 
be rather inconvenient to interpolate. If q is small compared with 
unity, then an approximate value of 1 — ( 1 + 2q)~312 will be given 
by the first few te rms of the binomial expansion. W e have approximately 
1 - ( 1 +2q)-^2 = 3q-~q2. 
If we now define a function / (not to be confused with the / of the 
integration tables) to be 
( 9 ) 
r2 = x2 + y2 + z2 
= (*O + £) 2 +(JO + ^ ) 2 + (*O + 0 2 
= r« + 2* 0£ + 2y07, + 2z^ + ? + r,* + t,\ 
178 NUMERICAL INTEGRATION 
which are the equations to be integrated in Encke 's method . T h e solution 
involves six constants of integration, which are chosen so that the 
coordinates and velocity components in the undis turbed orbit (xQy yQy z0y 
x0y j>0, z0) are the same as those in the actual orbit at a part icular date, 
that is, f, 77, £, £ , 7], £ are all zero. T h e date at which this is the case 
is called the epoch of osculation. 
Equat ions (11) are rigorous provided that q is calculated by formula 
(9). In practice formula (10) will be used on starting the integration. 
T h e per turbat ions f, 77, £ will gradually increase in size and when their 
squares become appreciable formula (9) will be introduced. 
Since xy yy z differ from x09 yQ9 z0 by the small quanti t ies f, 77, £, and 
since mi is small, we may subst i tute x0y y0y z0 for x9 y, z in formulas 
(11) and the error thus caused will be of the order of mult ipl ied 
by its attraction on the asteroid or comet, that is, of the order of mf. 
Thi s subst i tu t ion is usually made, and it permits the planetary attractions 
to be calculated for a large n u m b e r of steps in advance. T h e result ing 
per turbat ions are said to be accurate to the first order of dis turbing 
forces. If the integration covers a sufficiently long t ime interval the 
per turbat ions will become so large that the error commit ted becomes 
appreciable, and then the planetary attractions may be calculated by 
the rigorous formulas. T h e necessity for this may, however, be avoided 
by an operation called rectifying the orbit . T h e coordinates of the 
asteroid or comet are obtained at a new date by calculating accurate 
values of xQy y0y z0 and adding them to f, 77, £, and the velocity components 
are obtained by adding wx09 wy09 wz0 to values of w£y wr\y wt derived 
from f, 77, £ by numerical differentiation. T h e values of a9 e9 ny and the 
vectorial constants of the ellipse corresponding to the coordinates and 
velocity components are calculated and the integration is recommenced 
at the new epoch of osculation. In general it is preferable to rectify 
the orbit rather than to use the rigorous formulas for the planetary 
attractions, because the labor is less. 
then when q is small / w i l l be close to 3, and since / changes much less 
rapidly than q9 it is easy to interpolate in a table giving / as a function of q. 
Equat ions (8) now become, th rough multiplication by w2 
w2£ = w2k2 
W2TJ = w2k2(l + m) -
w2t = w2k2{\ + m) -
(11) 
ENCKE'S METHOD 179 
5. N u m e r i c a l a p p l i c a t i o n of E n c k e ' s m e t h o d . T h e example 
given for Cowell 's method is used, bu t with a 20-day interval, which 
makes w2k2 = 0.1183 6492. We begin by calculating the coordinates 
from the osculating orbit for one date before the epoch and for a number 
of dates afterward (Table 6). 
T A B L E 6 
J . D . / u sin u cos u 
2 4 2 9 9 8 0 . 5 7 1 . 4 8 8 7 5 . 9 0 2 4- 0 . 9 6 9 8 8 + 0 . 2 4 3 5 9 
2 4 3 0 0 0 0 . 5 7 5 . 7 7 0 8 0 . 2 5 5 + 0 . 9 8 5 5 7 + 0 . 1 6 9 2 6 
2 4 3 0 0 2 0 . 5 8 0 . 0 5 2 8 4 . 5 8 3 + 0 . 9 9 5 5 3 + 0 . 0 9 4 4 1 
2 4 3 0 0 4 0 . 5 8 4 . 3 3 4 8 8 . 8 8 4 + 0 . 9 9 9 8 1 + 0 . 0 1 9 4 8 
2 4 3 0 0 6 0 . 5 8 8 . 6 1 7 9 3 . 1 6 1 + 0 . 9 9 8 4 8 — 0 . 0 5 5 1 4 
2 4 3 0 0 8 0 . 5 9 2 . 8 9 9 9 7 . 4 1 2 + 0 . 9 9 1 6 4 — 0 . 1 2 9 0 0 
2 4 3 0 1 0 0 . 5 9 7 . 1 8 1 1 0 1 . 6 3 8 + 0 . 9 7 9 4 4 — 0 . 2 0 1 7 3 
rl w2k2lr3
0 
— 1 . 6 3 9 7 — 2 . 0 6 6 6 — 0 . 6 3 6 1 7 . 3 6 4 4- 0 . 0 0 5 9 2 3 
— 1 . 4 8 1 7 — 2 . 1 7 6 9 — 0 . 7 2 0 2 7 . 45 3 + 0 . 0 0 5 8 1 7 
— 1 . 3 1 5 1 — 2 . 2 7 4 6 — 0 . 8 0 0 0 7 . 5 4 3 4- 0 . 0 0 5 7 1 3 
— 1 . 1 4 1 1 — 2 . 3 5 9 2 — 0 . 8 7 5 2 7 . 6 3 4 4- 0 . 0 0 5 6 1 1 
— 0 . 9 6 0 5 — 2 . 4 3 0 7 — 0 . 9 4 5 6 7 . 7 2 5 4- 0 . 0 0 5 5 1 3 
— 0 . 7 7 4 7 — 2 . 4 8 8 7 — 1 . 0 1 0 8 7 . 8 1 6 4- 0 . 0 0 5 4 1 7 
— 0 . 5 8 4 7 — 2 . 5 3 3 2 — 1 . 0 7 0 4 7 . 9 0 5 4- 0 . 0 0 5 3 2 5 
T h e planetary attractions for the first three dates may be obtained 
by multiplyini y those in the previous example by 4. T h e addit ional 
values needed are given in Table 7 in the same form as before. 
T A B L E 7 
PLANETARY ATTRACTIONS ON CERES 
Julian Date Venus Earth Mars Jupiter Saturn S u m 
2 4 3 0 0 4 0 . 5 1 4 . 0 4- 3 0 . 8 + 1 .2 — 1 8 2 . 8 — 1 2 . 6 — 1 7 7 . 4 
4- 5 1 . 2 — 1 5 . 2 + 3 . 2 — 1 7 5 . 6 — 9 . 3 — 1 4 5 . 7 
4- 2 3 . 6 — 6 . 8 4- 1 . 2 — 7 3 . 6 — 3 . 4 — 5 9 . 0 
— 3 9 . 6 + 3 5 . 4 + 1 .3 — 1 7 8 . 6 — 1 2 . 7 — 1 9 4 . 2 
2 4 3 0 0 6 0 . 5 + 3 4 . 3 — 3 . 9 + 3 . 5 — 1 7 9 . 0 — 9 . 2 — 1 5 4 . 3 
+ 1 7 . 8 — 1 .9 4- 1 .3 — 7 4 . 2 — 3 . 3 — 6 0 . 3 
— 5 2 . 8 4- 3 5 . 4 + 1.1 — 1 7 4 . 3 — 1 2 . 7 — 2 0 3 . 3 
2 4 3 0 0 8 0 . 5 + 8 . 5 4- 8 . 4 4- 3 . 5 — 1 8 2 . 3 — 9 . 0 — 1 7 0 . 9 
4- 7 .1 4- 3 . 5 4- 1 . 4 — 7 4 . 9 — 3 . 1 — 6 6 . 0 
— 4 9 . 7 4- 3 1 . 2 + 0 . 8 — 1 6 9 . 9 — 1 2 . 8 — 2 0 0 . 4 
2 4 3 0 1 0 0 . 5 — 1 9 . 1 + 2 0 . 2 + 3 . 3 — 1 8 5 . 4 — 8 . 9 — 1 8 9 . 9 
— 5 . 5 + 8 . 7 4- 1 . 4 — 7 5 . 4 — 3 . 0 — 7 3 . 8 
180 NUMERICAL INTEGRATION 
W e find the starting values by the equations of Chapter IV, imposing 
the condition that the per turbat ions and their first derivatives must be 
zero at the epoch, and we then have sufficient information to fill in the 
first three lines of the integration Tables 8, 9, and 10 which are shown 
with several steps of integration added. 
T A B L E 8 
INTEGRATION FOR X 
Date "/ y f1 /" f m 
980 — 150 
+ 2 
000 + 12 — 148 — 13 
— 74 — 11 + 4 
020 — 62 — 159 — 9 + 8 
— 233 — 20 + 12 
040 — 295 — 179 + 3 — 6 
— 412 — 17 + 6 
060 — 707 — 196 + 9 0 
— 608 — 8 + 6 
080 — 1315 — 204 + 15 
— 812 + 7 
100 — 2127 — 197 
— 1009 
120 — 3136 
T A B L E 9 
INTEGRATION FOR y 
Date "/ y f /" 
980 — 186 
+ 24 
000 + 13 — 162 — 11 
— 79 + 13 — 3 
020 — 66 — 149 — 14 + 4 
— 228 — 1 + 1 
040 — 294 — 150 — 13 + 2 
— 378 — 14 + 3 
060 — 672 — 164 — 10 + 3 
— 542 — 24 + 6 
080 — 1214 — 188 — 4 
— 730 — 28 
100 — 1944 — 216 
— 946 
120 — 2 8 9 0 
ENCKE'S METHOD 181 
Date " / y / 
980 — 84 
+ 12 
000 + 6 
— 35 
— 72 — 3 
+ 9 — 4 
020 — 29 
— 98 
— 63 — 7 
+ 2 
+ 6 
+ 2 
040 — 127 
— 159 
— 61 — 5 
— 3 
— 3 
— 1 
060 — 286 
— 223 
— 64 — 6 
— 9 
+ 4 
+ 3 
080 — 509 
— 296 
— 73 — 3 
— 12 
100 — 805 
— 381 
— 85 
120 — 1186 
T h e prel iminary values of / on the first three lines are s imply the 
planetary at tractions. T h e n , for the f coordinate, 
Jo 
I f j L _ / ] 
12 y ° ^ 2 4 0 y c 
1 = 0 — — (-
1 1 2 1 - 1 4 8 > + 25><-
13) = + 12, 
Yl/2 = 6 + 2 /o -r 1 2 / o = 0 + j ( --148) + j j ( - 4 ) = - 7 4 , 
and similarly for rj and £. 
T h e first three lines of the integration tables being filled in, the 
attractions are recomputed for the dates preceding and following the 
osculating date . I t is found that some of them require corrections of 
a unit , which are made and the differences corrected. T h e integration 
now proceeds in the usual way. First the values of £, 77, £ are calculated 
for date 040 by formula (15) of Chapter IV, then q and fq are obtained 
and finally the attractions, which are added to the planetary attractions 
and the results writ ten in the tables. Next , 77, £ are calculated again 
by the more accurate formula (16) of Chapter IV and the former values 
are corrected if necessary. I t is convenient to form a table of 77, £, 
q, and fq, as they are obta ined step by s tep. Tab le 11 is given as a 
example with the necessary corrections to five of the provisional per-
turbat ions already int roduced. In this example no corrections to the 
provisional values of fq were found to be necessary. 
T A B L E 10 
INTEGRATION FOR Z 
182 NUMERICAL INTEGRATION 
T A B L E 1 1 
ENCKE PERTURBATIONS 
Date i V £ q fq 
0 0 0 0 0 0 0 0 
0 2 0 — 7 5 — 7 8 — 3 4 + 4 0 + 1 2 0 
0 4 0 — 3 1 0 — 3 0 6 — 1 3 2 + 1 5 6 + 4 6 8 
0 6 0 — 7 2 3 — 6 8 6 — 2 9 1 + 3 4 1 + 1 0 2 3 
0 8 0 — 1 3 3 2 — 1 2 3 0 — 5 1 5 + 5 9 0 + 1 7 7 0 
1 0 0 — 2 1 4 3 — 1 9 6 2 — 8 1 2 + 8 9 7 + 2 6 9 1 
X 
pa 
Vi - y 
Zj - z 
P) 
T h e data in Table 11 and in the integration tables are all in uni ts of the 
eighth decimal. 
6. E q u a t i o n s w i t h o r i g i n at the c e n t e r of m a s s . Equat ions (9) 
of Chapter I may be extended to any n u m b e r of dis turbing point-masses 
simply by writ ing 
# = X k2m3 
3 
i 
z = ^ k2m3-
3 
with pf = (Xj — x)2 + (y3 — y)2 + (z3- — z)2, in which the origin of 
coordinates is at the center of mass of the whole system. These equations 
are simpler in form than (5) owing to the absence of the te rms that 
express the action of the dis turbing planets on the sun, and they would 
be more convenient to use than (5) if the coordinates of the sun and 
planets referred to the center of mass of the solar system were readily 
available, which is not the case. T h e general theories that give the 
coordinates of the major planets as functions of the t ime are all referred 
to the sun as origin. T h e slight complication of these theories, and of 
the equations of motion for another body, that arises from this choice 
of origin is more than compensated by the advantage that a separate 
theory of the sun 's motion is rendered unnecessary. 
Notwi ths tanding the inconvenience to the computer of having to 
calculate the coordinates of the sun and planets referred to the center 
of mass, it is sometimes advantageous to use this origin when integrat ing 
the orbit of a comet or asteroid. W h e n the sun is used as center and 
such a body is very distant from the sun, the te rm x3/r* of equations (5) 
ORIGIN AT CENTER OF MASS 183 
* = *,fV^ + S*^iV£' (I3) 
Pa j Pj 
and similar equations hold for y and z. These are the equations for 
Cowell 's me thod referred to the barycenter. In applying them it is 
necessary to obtain the barycentric coordinates xS9 yS9 zsy xj9 yjy zi from 
the heliocentric coordinates of the planets. If the heliocentric coordinates 
are denoted by xj9 yjy zjy then 
1 
Xjy 
1 + m} (14) 
with similar equations for the other coordinates. 
T h e coordinates and velocity components at the epoch of osculation 
differ from those referred to the sun. If they have been obtained from 
heliocentric elements it is necessary to make the following t ransforma-
t ions before calculating the starting values for the integration. T h e 
coordinates are 
(15) 
with similar equations for y and z. T h e velocity components referred 
to the barycenter may be obtained from 
(16) 
with similar equations for y and z9 where the second part of the cor-
rections may be conveniently calculated by numerical differentiation of 
the second part of (15) calculated for several dates. 
and (11) may become much greater than (Xj — x)lp)> In the case of 
the action of Jupi ter on Pluto, for example, the ratio of these two te rms 
averages about 4 0 2 / 5 2 or 64. Consequent ly the former of these two 
terms produces an additional significant figure in the attractions, which 
shortens the tabular interval needed for good convergence of the differ-
ences. T h e use of the center of mass as origin avoids this difficulty and 
permits a larger tabular interval to be used. 
Separat ing the solar t e rm from the others, as was done in Eq. (5) 
and denoting the barycentric (referred to the center of mass) coordinates 
of the sun by xS9 ysy zsy Eq . (12) for x becomes 
184 NUMERICAL INTEGRATION 
Thi s is equivalent to assuming, so far as the action of mv is concerned, 
that the asteroid or comet moves in an elliptic orbit around the center 
of mass ofthe sun and mp. In applying Eqs. (14) and (15) the coordinates 
of mp may be included, but it often happens that even this is not necessary 
and that throughout the whole problem the sun 's mass may be augmented 
by mp and no further at tention paid to it. For example, the center of 
mass of Mercury and the sun is about 0.0000 0007 a.u. from the center 
of the sun, and in most asteroid orbits so small a correction to the 
coordinates of the disturbed body may be neglected. 
Upon completion of the integration it will often be desired to refer 
the coordinates again to the sun, which may be done by 
with similar equations for y and z. 
T h e equations for Encke 's method referred to the barycenter are 
(17) 
and similar equations for rj and £, where q and r\ are obtained from 
and the x0- and xs are derived by (14). 
7. I n t e g r a t i o n w i t h a u g m e n t e d m a s s of the s u n . When the 
disturbed body is so far from the sun that, in Eqs . ( i2 ) , one or more 
of the pj is nearly equal to psy then, denot ing such a p}- by pp and the 
corresponding dis turbing mass by mpy if mp is sufficiently small we may 
have 
(18) 
to the same n u m b e r of significant figures as are needed in the attractions. 
Th i s is commonly the case with the action of Mercury on an asteroid, 
or with the action of the four inner planets on Pluto. In such cases, 
instead of (13) we may write 
(19) 
AUGMENTED MASS OF SUN 185 
An augmented mass of the sun may be used in Eqs . (5) or (11) equally 
as well as in (13). In such cases the te rm — xjrp is neglected in the 
integration, and afterward included if necessary by the use of Eqs . 
(14) and (15). 
Whenever an augmented mass of the sun is used in an integration 
the same augmented mass must be used in calculating the start ing values 
for the integration. T h e way in which the augmented mass is to be 
used depends on the elements that are chosen to correspond to the 
constants of integration. Wi th either Encke 's or Cowell 's me thod the 
actual constants of integration come from the coordinates and velocity 
components at the epoch of osculation, and these constants are not the 
same for the augmented mass of the sun as for the unaugmented mass. 
F rom the way in which the coordinates and velocities were derived from 
the elements in Chapter I it is evident that the only effect is to change 
the relation between n, the mean motion, and a, the semi-major axis; 
in applying Kepler ' s th i rd law, k2 (1 + m -f- mp) is to be subst i tuted 
for k2(l + m). 
I t is indifferent w rhether, on commencing operations, n is derived from 
a or a from n, bu t once the decision is made all subsequent operations 
must be consistent with it. If the equations of chapter I, section 27 
are to be used at the end of the integration it will be more convenient 
to regard a as the fundamental element. T h e n the procedure will be 
governed by the initial data available. If a alone is given the course is 
obvious. If a and n are both given, reject n and use the value given by 
If only n is given it is to be assumed, lacking any s tatement to the 
contrary, that it is referred to the sun alone. Therefore derive a from 
then, using this value of <z, obtain the corrected value of n from (20). 
In using the equations of Chapter I, Section 27 at the end of the 
integration the augmented mass should not be used if, as is usual, the 
elements are to be referred to the sun. 
8 . R e l a t i v e a d v a n t a g e s of C o w e l l ' s a n d E n c k e ' s m e t h o d s . In 
general it may be said that neither method is clearly superior to the 
other when the work is carried out with a desk calculating machine. 
Encke's method permits the use of a larger tabular interval bu t each 
step requires more t ime than in Cowell 's method. For comets it is often 
recommended to use Encke 's method when the comet is near the sun 
k2(l + m + mD) = n2a3. (20) 
k2(l + m) = n2a3, (21) 
186 NUMERICAL INTEGRATION 
and Cowell 's when it is far from the sun. W h e n close approaches occur 
the Encke per turbat ions increase in size very rapidly, necessitating a 
small tabular interval, and then the method loses all its advantages. T h e 
change from either method to the other may be made without difficulty 
by calculating coordinates and velocity components for a new epoch of 
osculation and starting the integration anew at the new epoch. 
Wi th modern large-scale calculating machinery, where the process of 
integration may be carried out by operators who have little knowledge 
of the art of computat ion, or it may even be made entirely automatic, 
Cowell 's method is clearly superior to Encke 's . Wi th Encke 's method 
the necessity for making judgmen t s regarding the formula to be used 
for q, and the desirability of rectification, as well as the continually 
increasing size of the per turbat ions , are complications that are easily 
dealt with by a computer as they arise, bu t they cannot be easily foreseen 
in advance, as is necessary with automatic methods . 
N O T E S A N D R E F E R E N C E S 
C o w e l l ' s m e t h o d w a s first u s e d in p r e d i c t i n g the re turn of H a l l e y ' s c o m e t 
in 1910 , P . H . C o w e l l , A . C . D . C r o m m e l i n , " I n v e s t i g a t i o n of t h e m o t i o n of 
H a l l e y ' s C o m e t f r o m 1 7 5 9 - 1 9 1 0 , " A p p e n d i x to Greenwich Observations 1909, 
N e i l l , B e l l e v u e , E n g l a n d , 1910. In th is appl icat ion t h e authors ca lculate t h e 
s e c o n d di f ferences of the c o o r d i n a t e s , w h i c h are t h e n d irect ly o b t a i n e d b y t h e 
d o u b l e s u m m a t i o n , ins tead of s u m m i n g the s e c o n d der ivat ives a n d a p p l y i n g t h e 
in tegrat ion formula to t h e s u m m a t i o n s , as is d o n e in th i s chapter . H o w e v e r , 
t h e y d o r e c o m m e n d u s e of the latter p r o c e d u r e , w h i c h is assoc ia ted w i t h a smal ler 
a c c u m u l a t i o n of errors o f r o u n d i n g . 
E n c k e ' s m e t h o d is d e s c r i b e d b y h i m in the Berliner jfahrbuch, 1857 . 
J a m e s C. W a t s o n , Theoretical Astronomy, L i p p i n c o t t , 1900 , C h a p t e r V I I I , 
descr ibes several m e t h o d s of specia l p e r t u r b a t i o n s , w i t h n u m e r i c a l e x a m p l e s . 
E x a m p l e s o f a n u m b e r of m e t h o d s are g i v e n in Planetary Coordinates for the 
years 1960-1980, L o n d o n , H . M . S ta t ionery Office. 
T h e a c c u m u l a t i o n of errors in n u m e r i c a l in tegra t ion , for t h e case w h e r e the 
o n l y errors arise f r o m r o u n d i n g the e n d - f i g u r e s , w a s first treated b y D i r k B r o u w e r 
in Astron. J. 46, 149 (1937 ) . 
T h e a d v e n t of a u t o m a t i c h i g h - s p e e d ca lcu la t ing m a c h i n e s has p r o d u c e d w i d e -
spread interest in m e t h o d s of n u m e r i c a l in tegrat ion . M a n y n e w formulas have 
b e e n i n t r o d u c e d , and C o w e l l ' s m e t h o d has b e e n red i scovered several t i m e s , 
b e i n g n o w k n o w n b y a var ie ty of n a m e s . B u t it d o e s n o t appear that any m e t h o d 
is super ior to C o w e l l ' s and E n c k e ' s for general u s e in the n u m e r i c a l in tegrat ion 
of orbi ts . T h e a c c u m u l a t i o n of error m a y b e r e d u c e d to an abso lu te m i n i m u m 
b y caus ing the s tep- in terva l to b e automat ica l ly var ied , a c c o r d i n g to the s ize of 
the dif ferences of h i g h e s t order reta ined. 
CHAPTER V I 
ABERRATION 
1• I n t r o d u c t i o n . Because of the finite velocity of light, the apparent 
position of any celestial body depends upon the motions both of this 
body and of the earth dur ing the interval of t ime required for light 
to travel from the body to the observer. 
While the light is travelling from a celestial body to an observer on 
the moving earth, bo th the body and the earth move awayfrom the 
positions they occupied in space at the instant the light left the body. 
T h e ray that is received at the observer is one that was emitted, at 
some t ime previous to the instant of observation, in the direction toward 
the position that the earth was later to occupy; consequently, by the 
t ime the ray is received at the observer, the body is no longer located 
in the direction from which the ray comes. Moreover , because the 
earth is in motion, the direction whence the ray is coming when it 
arrives at the observer differs from the direction whence it appears 
to be coming; the latter (which is the direction in which the body is 
actually seen) is the direction of motion of the light relative to 
the observer, not its actual geometrical direction in space, and this 
relative direction depends on the velocity of the observer at the 
instant. 
Consequently, the apparent direction in which a body is observed 
on the celestial sphere is neither the actual geometric direction at the 
instant of observation, nor the direction where the body was geo-
metrically located at the t ime the light left it. 
T h e apparent displacement of celestial bodies from their actual 
geometric directions that is caused by the progressive motion of light 
in combination with the motions of the observer and of the bodies 
themselves is called aberration. T h e displacement of the observed 
apparent position from the actual geometric position at the instant of 
observation is known as planetary aberration. T h e displacement of the 
observed position from the position where the body was geometrically 
located at the instant when the light left it is called stellar aberration. 
Planetary aberration may be regarded as the resultant of two effects: 
the stellar aberration due to the instantaneous velocity of the observer 
at the instant of observation and the geometric displacement of the body 
187 
188 ABERRATION 
in space due to its motion dur ing the interval while the light was 
travelling to the earth. 
In calculating the position in which a body will appear at a particular 
t ime, from the geometric positions of the body and of the earth as derived 
from gravitational theory, allowance mus t be made for these two effects. 
Conversely, in deriving from observed positions the geometric positions 
in space, appropriate corrections mus t be applied. 
F I G . 1 . Aberration 
In Fig. 1 let ET and PT denote the actual geometric positions of the 
earth and a celestial body at the instant T when an observation is made, 
and let Et and Pt denote their actual positions at an earlier instant t 
when the ray that reaches the observer at t ime T left the body. T h e 
geometrical path of the light-ray is along PtET. Denote by V the velocity 
vector of the actual light-ray, and by v the velocity vector of the earth 
at the instant T. T h e relative velocity of the light ray is represented 
by Vr, the side of a parallelogram having v for its other side and V 
for its diagonal. N o w draw a line from Pt parallel to v, cut t ing Vr 
produced at Pa. T h e n Pa is the apparent position of the body at the 
instant T. W e have further PtPaIPtET = v/V, and PtET = V(T — t), 
whence PtPa — v(T — t), which is an impor tant principle often of value 
in treating the stellar aberration. I t may be stated: the apparent position 
of an object seen by an observer in motion is displaced from the position 
in which it would be seen if the observer were at rest by an angular 
amount rigorously equivalent to a linear displacement of the object 
parallel to the observer 's direction of instantaneous motion, and equal 
in amount to v{T — t), where v is the observer 's instantaneous velocity, 
and T — / is the t ime required for light to travel from the object to 
the observer. 
T h e principles jus t stated are completely rigorous and general, bu t 
owing to our imperfect knowledge of the distances and motions of the 
STELLAR ABERRATION 189 
stars, they are never rigorously applied to stellar positions in practice. 
T h e distances and velocities of the stars are known with such low 
accuracy that it is not practicable to calculate their motions dur ing the 
interval T — t. Hence the planetary aberrat ion is never calculated for 
the stars, bu t only the stellar aberrat ion. Moreover, in calculating the 
stellar aberrat ion the motion of the solar system th rough space is 
neglected. In consequence the positions of stars given in star catalogues 
are not their t rue positions, and the proper motions are not the t rue 
proper motions, bu t these discrepancies are of no consequence to 
celestial mechanics ; stellar positions are used in celestial mechanics only 
for defining a reference system of coordinates, and the catalogue positions 
(with one small exception noted on page 197 ) are as useful for this 
purpose as the t rue positions would be. 
2 . S t e l l a r a b e r r a t i o n . T h e amount of the apparent displacement 
due to stellar aberrat ion depends on the instantaneous velocity of the 
observer at the instant the light reaches h im. In a r ight -handed 
rectangular coordinate system with fixed origin and axes in fixed direc-
tions in space, the components of the velocity vector of the observer, 
xy y, Zy are each the sum of the linear velocity due to the diurnal rotation 
of the earth, the orbital motion of the earth relative to the sun (including 
lunar and planetary per turbat ions as well as the elliptic motion) , and 
the motion of the sun relative to the center of mass of the solar system 
(but excluding the motion of the solar system among the stars). 
Relative to parallel axes with origin at the observer, let L, B denote 
longitudinal and latitudinal spherical coordinates referred to the xy plane. 
T h e direction cosines of the line toward Pt whence the light actually 
comes are 
where L and B are the spherical coordinates of the point Pt. T h e 
direction cosines of the apparent position Lay Ba are similarly 
£ a = cos Ba cos L a , rja = cos Ba sin L a , £ a = sin Ba. 
T h e points Pty Pa are diametrically opposi te to the points toward 
which V and Vr are directed. Hence 
£ = cos B cos L, rj = cos B sin L, £ — sin By 
vjv, v = - vjv, - vjv, 
with similar equations for £ a , rja, £a. 
190 ABERRATION 
W e have also 
VJa =V£ + x, Varja = Vy]+y, Va£a = VC+z, 
where Va£a and V£ represent direction components of the lines toward 
Pa and Pt. Therefore 
V x 
cos Ba cos La = cos B cos L + — , 
V . y 
cos Ba sin La = cos B sin L + — , (1) 
-^r sin £ a = sin B + — . 
Mult iplying the first of these equations by cos L, the second by sin L, 
and adding, then mult iplying the first by sin L and the second by 
cos L and subtracting, we obtain 
V 1 
-—- cos Ba cos (La — L) = cos B + — (x cos L + y sin L), 
F 1 
- ~ cos Ba sin (L a — L) = — — (x sin L — y cos L), 
(2) 
whence 
Mult iplying the first of (2) by cos \ (La — L), the second by sin \ 
(La — L), and adding, an equation for cos Ba is obtained, which being 
combined with the preceding equation for sin Ba gives after some 
manipulat ion 
tan (Ba - B) (4) 
zcosB —ixcosL + j ) s i n L + t a n \ ( L a —L) (ycosL — xsmL)\sinB 
tan (La - L) , 
— sec B(x sin L — y cos L) 
V + sec B{x cos L + y sin L)' (3) 
V + 2r sin B + cos L + j sin ^ + tan \ (La— L) (y cos L — x sin L)} cos 5* 
T h e values for L a — L and i ? a — B so obtained are completely 
rigorous and general. F rom them may be developed practical formulas 
for the actual calculation of the stellar aberration in any system of 
coordinates, to any desired degree of accuracy. Formulas (3) and (4) 
are independent of the distance of the object, and they, or formulas 
derived from them, are convenient to use when the distance of the 
observed object is unknown. 
If the distance is known with sufficient precision, however, it is 
sometimes more convenient to treat the stellar aberration as equivalent 
PLANETARY ABERRATION 191 
to a linear displacement of the object. If Xu Yu Zu Rt arethe t rue 
geometric rectangular coordinates and distance of the object at the 
instant t when light left the body, Xay Ya, Za its apparent coordinates 
at the instant of observation T, and xy y, z the coordinates of the observer 
at t ime T, then we have directly, from the principles of the preceding 
section, with complete rigor and generality, 
Xa — Xt = — Rt, 
Y a - Y t = ^Rt, (5) 
Za Zt = — Rt, 
in which, for perfect rigor, the origin of coordinates mus t be taken at 
the center of mass of the solar system. If the travel-t ime of the light, 
RtlVy is denoted by r, Eqs . (5) become 
Xa — Xt = XTy 
Y a - Y t = y T i (6) 
Za — Zt = ZT. 
3 . P l a n e t a r y a b e r r a t i o n . T h e stellar aberration, due to the 
velocity of the observer, is only the displacement of the apparent position 
(Pa in the diagram) from the geometr ic position Pt which the body had 
when the light left it. By means of a further correction for the motion 
of the body from Pt to PT dur ing the l ight-t ime, if this motion can 
be determined, the geometric position of the body at the instant of 
observation can be calculated. T h e combinat ion of the two effects, that 
is, the displacement of the apparent position of the body from its t rue 
position at the instant of observation Ty is the planetary aberration, 
angle PTETPa in the diagram. Denot ing the actual coordinates of the 
body at the instant of observation by X T , Y T , Z T , and developing the 
quantit ies XT — Xu YT — Yu ZT — Zt in powers of r by Taylor ' s 
theorem, we have 
Y T - Y t = Ytr -\ 
ZT — Zt — ZtT -j 
(7) 
XT — Xt — Xtr + 
192 ABERRATION 
Subtract ing Eqs. (7) from (6) gives rigorous expressions for the 
rectangular components of the planetary aberration, 
— Xj> — {x - Xt)r -
Ya 
- Y t ) r - \ YtT* - i YtT* -
— ZT = (z -Zt)r ~ \ l :tr* - \zy -
It is especially impor tan t to notice that in these equations the origin 
of coordinates is not arbitrary, bu t must be taken at the center of mass 
of the solar system if perfect rigor is aimed at. 
Equat ions (8) are not suited for practical use, owing to the difficulty 
of calculating the derivatives appearing therein. An equally rigorous and 
more convenient method of calculating the planetary aberration is to 
treat its two parts separately. Knowing XTy YTy ZTand the corresponding 
distance RT we take rx — RT/V as the first approximation to r and 
calculate Xv Yly Zly R1 for the instant T — r v T h e second approxima-
tion to r is r 2 = Ri/V> which yields a second approximation to the 
coordinates, X2y Y2> Z 2 , R2. Proceeding in this way it will be found 
after two or three approximations that r does not change. At this stage 
we have Xty Yti Zh and the stellar aberration may then be calculated 
by Eqs. (6) or Eqs. (3) and ( 4 ) , or approximately by some other method. 
If the motion of the observed body is sensibly rectilinear dur ing the 
t ime interval r , then in Eqs. (8) the terms involving T 2 and higher 
powers of r become negligibly small, and fur thermore we have sensibly 
Xt — XTy Yt = YTy Zt — ZT, whence 
- X T = (x- XT) T , 
YA YT — {y YT) T , 
zn 
— ZT = (z — ZT) T . 
These equations may be yet further simplified for calculation by 
writing rx for T , and by taking the sun instead of the center of mass 
of the solar system for the origin of coordinates. In this shape they are 
often used for practical calculations; in the case of comets and planets 
the errors int roduced by the approximations are generally less than 
O'/Ol, but for satellites the simplified equations should be used with 
caution. 
T h e r ight-hand members of Eqs. (9), simplified as jus t described, 
express the change in the coordinates of the observed body relative to 
the moving observer as origin, during the period in which the light 
(8) 
(9) 
DIURNAL ABERRATION 193 
passes from the body to the observer. T h e planetary aberrat ion is 
therefore, unde r the assumptions noted, equal to this change. Hence , 
under the same assumptions similar equat ions will hold for any system of 
coordinates with origin at the moving observer. For example, the planet-
ary aberrat ion in r ight ascension, a, and declination, S, is given by 
AOL = — a T j , 
M = - 8 r v
 ( 1 0 ) 
which are to be applied to the geometrical place. If a geometric ephemeris 
giving the right ascension and declination of the object as functions of 
the t ime is available, the derivatives d and S may easily be found by 
numerical differentiation. 
4 . D i u r n a l a b e r r a t i o n . In the expressions of the preceding 
sections the velocity components x, y, z are, in the complete theory, 
those of the observer. I t is customary, however, for convenience, to 
separate the stellar aberration into two parts , and to treat the port ion 
depending on the orbital motion of the earth as distinct from the port ion 
due to the rotation of the earth on its axis. T h e latter port ion is 
called the diurnal aberra t ion; when this is t reated separately, the 
remaining port ion, called the annual aberration, may be obtained 
wi thout loss of rigor by using in expressions (3), (4), (6) the velocity 
components of the center of the earth instead of those of the observer. 
In consequence of the ear th ' s rotation on its axis, the observer is 
continuously carried toward the east point of the horizon with a speed, 
in meters per second. 
v = 464 p cos 9?' , 
<p' being his geocentric lat i tude and p the radius of the earth at his 
station in te rms of the equatorial radius. T h e constant 464 is the speed 
of a point on the equator . 
T h e effect of this motion is to produce a displacement of the apparent 
positions of all celestial bodies toward the east point of the horizon, 
on great circles passing th rough this point, expressed by 
v • a 
s = — sin 0 , 
6 being the angular distance of the body from the east point . Pu t t ing 
for v and V their numerical values and dividing by arc 1", 
s = 0''319 p cos cp' sin 0. 
194 ABERRATION 
T o find the effect of the displacement on the right ascension and 
declination of a body, consider the spherical triangle PES formed by 
the pole P , the east point of the horizon E, and the body S. T h e n 6 
is the side ES; and if we put q for the angle at 5 , and h for the hour 
angle of the body, the aberrat ion in right ascension and declination will 
be 
cos SAoc = s sin q, 
AS — s cos q. 
W e have also 
sin 6 sin q = cos A, 
sin 9 cos q = sin 8 sin h, 
whence 
AOL = 0"319 p cos cp' cos h sec S, 
AS = 0"319 p cos cp' sin h sin S. 
In using these expressions we may pu t p = 1 and the astronomical 
latitude for cp' wi thout appreciable error. 
T h e diurnal aberrat ion is of no consequence in differential astro-
metric work such as photographic work, because it affects all bodies 
in the same region by nearly the same amount , and the small residual 
effects are eliminated by the plate constants. I t must , however, be taken 
into account in all measurements of the relative positions of widely 
separated bodies, as in meridian astronomy. 
5. C a l c u l a t i o n of the a n n u a l a b e r r a t i o n . W h e n a numerical 
accuracy approaching O'.'OOl or bet ter is aimed at, the velocity 
components x> y> z may be most conveniently obtained by numerical 
differentiation of the ear th 's rectangular equatorial coordinates referred 
to the center of mass of the solar system, and to a fixed equinox and 
equator. T h e coordinates of the earth relative to the sun may be obtained 
from the American Ephemeris simply by changing the signs of the sun ' s 
rectangular equatorial coordinates, and the coordinates of the sun 
relative to the center of mass may be derived from the data in Planetary 
Coordinates. 
In practice, however, this procedure has seldom been followed in the 
past. Instead, the velocity of the sun relative to the center of mass was 
neglected, and the earth was assumed to move in an elliptic orbit a round 
the sun. It is still customary to use differentialformulas instead of the 
rigorous expressions (3) and (4). 
CALCULATION 195 
: cos B cos Z> — sin B cos LJZ? — cos B sin L J L , 
: cos B sin L — sin B sin LzJi? + cos B cos L J L , (12) 
: sin B + cos B AB. 
Mult iplying the first of these equations by — sin L and the second 
by cos L, and adding, 
--cos BAL. (13) 
Mult iplying the first of (12) by — sin B cos L, the second by — sin B 
sin L, the third by cos By and adding, 
— sin B cos L — sin B sin L - cos 5 — AB. (14) 
Subst i tut ing for Zl l ( e», AY™, AZ™ their values (5) in (13) and (14) 
we have 
(15) 
cos BAL = 
where i ( e ) , y { e \ z(e) are referred to the ecliptic. 
In order to express i ( e ) , y i e \ i ( e ) in te rms of the elliptic elements of 
the ear th 's orbit , let us define 
ifj, the longitude of the earth in its orbit, 
r, the earth's radius vector, 
co, the longitude of the earth's perihelion, 
e = sin 99, the eccentricity of its orbit 
its mean angular motion, 
a, its semi-major axis. 
If we put L for the ecliptic longitude of any body, B for its lat i tude, 
and R for its distance, we have 
Z ( e ) = R cos B cosL, 
7<e> = R cos 5 sin L, (11) 
Z<*> = sin 5 . 
Differentiating, 
196 ABERRATION 
We then have, neglecting the earth 's lati tude, 
= r cos 
ye) = r sin 
= 0, 
(17) 
Subst i tut ing these values of ift and f in (16) we find 
xie) = — an sec cp (sin if; + e sin co), 
ye) _ _|_ a w s e c ^ ( c o s ifj -\- e cos to) . 
Subst i tut ing these values in (15) and put t ing for brevity 
_ an sec cp 
K — y , 
cos BAL = K (sin ifj + e sin co) sin L + /c (cos I/J + e cos to) cos L 
= /c cos (0 — L) + £/C cos (to — L), (18) 
= AC sin 5 sin (ip — L) + e/c sin J5 sin (to — L). 
T h e last te rms of these equations are independent of the earth 's 
longitude. For a particular star they are nearly constant dur ing several 
centuries, and by convention they are omit ted from the formulas, being 
included in the coordinates of the stars as determined by observation. 
In the usual formulas we put O = <A— 180°, the t rue longitude of 
the sun, and obtain 
cos BAL = — K cos ( O — L), 
AB = — K sin B sin ( O — L). 
T h e quanti ty K is the constant of aberration, and its conventional 
value is 20' '47. 
and, by differentiation, 
x i e ) = f cos I/J — rijj sin xjj, 
y(e) — f sin ifj + rip cos I/J> (1 
= 0. 
By the laws of elliptic mot ion we have 
n cos (pt 
C A L C U L A T I O N 197 
In equatorial coordinates we have, by a similar development , 
x 
— = K (sin O — e sin to), 
y 
-p: = — K cos e (cos O — e cos co), 
-p: = — /c sin e (cos O — £ cos co), 
and , again omit t ing the te rms involving co, 
cos SAoc = — K cos e cos O cos a — K sin O sin a, (20) 
J 8 = K cos 8 cos O sin S sin a — /<: sin Q sin 8 cos a — K sin e cos Q cos S. 
If we put for brevity 
C = — K cos e cos O , D = — /c sin O, 
c = cos a sec S, d = sin a sec S, 
c' — tan e cos 8 — sin a sin 8, d' = cos a sin 8, 
Eqs . (20) become 
Aoc = Cc + Dd, 
AS = Cc' + Dd\ 
(21) 
which is Bessel's form, often used in calculating the annual aberrat ion 
of the stars. T h e quanti t ies C and D are tabulated for every day in the 
American Ephemeris. 
T h e terms of the stellar aberrat ion involving GO, which affect the 
catalogue mean places of stars, are of no consequence in meridian 
astronomy, because the position of any object observed with a transit 
circle is its apparent position, and it is usually compared with an apparent 
ephemeris in which the planetary aberrat ion has been calculated by Eqs . 
(10); these equat ions automatically include the effect of the elliptic term, as 
it is called. On the other hand, photographic positions of celestial bodies 
are obtained by referring the image of the body to the catalogue mean 
places of stars in the immediate neighborhood. Such observations are 
affected by the elliptic port ion of the aberrat ion and, unless due p re -
cautions are taken, appreciable errors will be produced in the elements 
of the orbit. One remedy is to apply a special correction to every 
published photographic observation, before making any use of it. T h e 
corrections are given, in seconds of arc, by the matr ix product 
cos a — sin a sin 8 I 
sin a cos oc sin 8 
0 cos 8 
(22) |cos SAccASW = ||-f- 061 - 0."336 + 0'/027| 
II 
198 ABERRATION 
T h e matrix on the right is usually needed for another purpose, which 
makes this calculation an easy one. 
T h e approximate differential methods commonly used in calculating 
the annual aberration suffice for the ordinary purposes of celestial 
mechanics, the order of accuracy being about O'.'Ol. T h e neglect of the 
square of the displacement in the differential formulas produces errors 
in the zodiacal region of the sky not exceeding O'.'Ol, provided that 
the sun 's longitude and the coordinates of the object are referred to 
the same equinox. T h e chief error in t roduced by the assumption that 
the earth moves around the sun in an elliptic orbit arises from the 
velocity of the earth around the center of mass of the ear th-moon 
system. For a particular region of the sky this error takes the form of 
a periodic error with period one mon th and coefficient 0'.'008. Another 
similar error of period twelve years and coefficient 0'.'008 arises from 
neglecting the velocity of the sun around the center of mass of the sun 
and Jupi ter . T h e corresponding error involving Saturn has a coefficient 
of 0'.'002. In the discussion of photographic observations of objects in 
the solar system errors of this size may usually be neglected; the elliptic 
port ion of the stellar aberration is the only port ion of it that needs to 
be taken into account. 
6. E p h e m e r i d e s * In refined work the positions that are observed 
are never used directly, but instead the differences between the observed 
positions and positions calculated from some theory of the motion (in 
the sense observed position minus calculated position, or O — C) are 
discussed. An ephemeris is a table of calculated positions with the t ime 
as a rgument ; with some looseness the te rm may be applied to any set 
of calculated positions. In calculating a precise ephemeris it is desirable 
to treat the aberration in the same way as the observations are affected, 
so as to make the two directly comparable, and different methods mus t 
be used for different classes of observations. 
A geometric ephemeris gives the actual position of the body at the 
times indicated. I t is impossible to observe t rue geometric positions, 
and hence a geometric ephemeris is useless for any precise work. I t is 
chiefly employed when only a low degree of accuracy is aimed at, as 
when the only purpose is to find the body. 
An apparent ephemeris may be obtained by applying the planetary 
aberration to a geometric ephemeris . Positions observed with a transit 
circle are apparent positions. Hence ephemerides of the sun, moon, and 
principal planets are usually apparent ephemerides . Phenomena such as 
eclipses of the sun or of Jupi ter ' s satellites depend on the apparent 
positions of the bodies concerned, and therefore apparent ephemerides 
are useful in this connection also. 
SPECIAL CASES 199 
Observed positions obtained by compar ing an object with catalogue 
places of stars in the immediate neighborhood are neither geometric 
positions nor apparent positions, bu t of an intermediate class that may 
be te rmed astrometric positions. T h e y are free from the principal parts 
of the stellar aberration, that is, of the diurnal aberrat ion and of the 
circular port ion of the annual aberration, but they are affected by the 
barycentric motion of the observed object in the t ime taken for light 
to travel from it to the observer, and by the elliptic port ion of the annual 
aberrat ion. An astrometric ephemeris may therefore be obtained by 
antedat ing the heliocentric position of the object (which may be sub -
sti tuted for the barycentr ic position with an error not exceeding O'.'Ol)for the l ight-t ime by successive approximations in the way described 
in Section 3, and by then subtract ing the results of (22) from the geo-
centric ephemeris . Alternatively, the apparent ephemeris may first be 
calculated and the circular port ion of the annual aberrat ion then sub -
tracted. Th i s latter method is not perfectly rigorous if the abridged 
formulas are used for the planetary aberration, bu t is often sufficiently 
accurate. 
A fourth kind of ephemeris , called an astrographic ephemeris , has 
been used to some extent. I t is easier to calculate than the astrometric 
ephemeris , from which it differs only by the elliptic port ion of the annual 
aberration. Before comparing it with a photographic observation the 
expression (22) must be either added to the observed position or sub -
tracted from the ephemeris . 
7. S p e c i a l c a s e s of a b e r r a t i o n . 
(a) T h e sun. Dur ing the t ime required for light to pass from the sun 
to the earth, the motion of the sun around the center of mass of the solar 
system is virtually zero. T h u s , of the two parts making up the planetary 
aberration one vanishes and only the stellar aberration remains. I t is 
worthy of note that, contrary to the practice for the stars, the complete 
annual aberrat ion is applied to the solar ephemeris . Were this not done, 
observations of the sun 's apparent place would disagree with the ephe-
meris, the discrepancy in the longitude having an annual period with 
a coefficient of 0'.'343. 
(b) T h e moon. Dur ing the t ime required for light to pass from the 
moon to the earth, the moon moves approximately 0'.'7 in geocentric 
orbital longitude, with a variation of about 0"05 due to the eccentricity 
of its orbit . Th i s is the full amount of the planetary aberration, the 
annual aberration being canceled by the heliocentric motion of the moon 
during the l ight-t ime. In fact, the aberration is not applied to the lunar 
ephemeris at all; the tables of the moon are constructed so as to give 
200 ABERRATION 
the apparent position of the moon directly, by slightly modifying the 
geometric elements of the moon ' s orbit. In some applications it is im-
portant to remember that the lunar parallax tabulated in the ephemeris 
is not, strictly speaking, the t rue parallax at the t ime indicated. 
(c) A satellite. W h e n a satellite is observed by photographing it 
against the background of stars, the derived position is the astrometric 
position. When it is observed by measur ing its distance and position 
angle from its primary, the stellar aberration, being virtually the same 
for both bodies, may be neglected, or if it is appreciable it may be applied 
differentially. Leaving this aside, the observation connects the position 
of the satellite when light left it with the position of the pr imary when 
light left the primary. T o a very high degree of approximation the 
satellite partakes of the heliocentric motion of its pr imary dur ing the 
l ight-time of the latter, and therefore it is necessary in practice only 
to correct the observation for the motion of the satellite relative to its 
pr imary dur ing the t ime required for light to pass from the satellite 
to the observer. 
N O T E S A N D R E F E R E N C E S 
T h e aberrat ion of l ight w a s d i s c o v e r e d i n 1725 b y J a m e s Brad ley , a f terwards 
A s t r o n o m e r Roya l , d u r i n g a ser ies o f a t t e m p t s b y several a s t r o n o m e r s to m e a s u r e 
t h e annual parallax of a star, an enterpr i se u n d e r t a k e n w i t h a v i e w to e s t a b l i s h i n g 
observat iona l ly that t h e earth w a s n o t at t h e c e n t e r of t h e solar s y s t e m . Brad ley 
o b s e r v e d o n D e c e m b e r 17 that g a m m a D r a c o n i s w a s at a m o r e s o u t h e r l y d e c l i n a -
t ion than w a s e x p e c t e d , a n d o n D e c e m b e r 2 0 f o u n d it t o b e still farther s o u t h , 
t h e m o t i o n b e i n g o p p o s i t e in d irec t ion f r o m t h e m o t i o n that w o u l d h a v e r e s u l t e d 
f r o m parallax. H e o b t a i n e d a t e l e s c o p e for t h e p u r p o s e of s t u d y i n g t h e effect , 
and o b s e r v e d a n u m b e r of br ight stars t h r o u g h o u t a year , 1 7 2 7 - 2 8 , after w h i c h , 
b y analysis of t h e observat iona l mater ia l , h e ascer ta ined t h e character o f the 
p h e n o m e n o n a n d e x p l a i n e d it correct ly . 
CHAPTER VII 
COMPARISON OF OBSERVATION A N D THEORY 
1. I n t r o d u c t i o n . T h e word theory is used in celestial mechanics 
to mean any mathematical expression from which can be derived the 
coordinates of a body as functions of the t ime. Theor ies are of two types, 
special and general. A special theory is one that gives the coordinates 
for special values of the t ime only; a numerical integration of the equa-
tions of heliocentric mot ion of a comet or planet is an example of a 
special theory. In a general theory the t ime is represented by a symbol 
for which any value may be subst i tu ted at will and the coordinates 
obtained for the corresponding da te ; therefore a general theory cannot 
be entirely numerical in form. I t may be entirely analytic, as the lunar 
theory of C . E. Delaunay, which expresses the coordinates as functions 
of seven symbols corresponding to the six elements of the orbit and 
the t ime ; or it may be part ly analytic and part ly numerical , as E. W . 
Brown's lunar theory, in which numerical values are subst i tu ted for 
some of the elements . T h e r e are also general theories in which numerical 
values are subst i tu ted for all of the elements, the only quant i ty denoted 
by a symbol being the t ime, such as G. W . Hill 's theory of Jup i te r ; 
such theories are commonly, though somewhat inexactly, called n u m e r -
ical general theories. 
Al though any value of the t ime may be subst i tuted in a general 
theory, it does not necessarily follow that the result has any physical 
meaning. General planetary theories usually have contained t r igono-
metr ic functions factored by the t ime, the values of which increase 
wi thout limit as the t ime increases wi thout l imi t ; this characterist ic 
prevents their being valid for more than a few centuries. T h e lunar 
theory is free from this drawback and in form is valid for all t ime. But 
the elements of the orbit and the masses of the dis turbing bodies still 
have to be de termined by observation. T h e observations being limited 
in n u m b e r and of limited accuracy, the theory necessarily departs further 
and further from the t ru th for t imes further and further removed from 
the fundamental epoch. 
A theory is of no use unti l it is compared with observation, since 
this is the only way of testing its validity. Fu r the rmore , no theory has 
ever been constructed which was at first as accurate as the existing 
201 
202 OBSERVATION AND THEORY 
observations, because the elements of the orbit are not sufficiently well 
known at the beginning. Therefore , the initial comparison of any theory 
with observation is primari ly for the purpose of improving the provisional 
elements on which the theory was based; often at the same t ime the value 
of some dis turbing mass or of some other astronomical constant may 
be determined. 
A theory usually gives the coordinates of a body referred to the center 
of its pr imary as origin. T h u s , the lunar theory gives the geocentric 
coordinates of the moon, the theory of Jupi ter the heliocentric coor-
dinates of Jupi ter , and the theory of Hyper ion the Saturnicentr ic 
coordinates of this satellite. T h e observations, however, are always 
referred to the earth as origin, and not to the center of the earth bu t 
to the point where the observer stands. A comparison of observation 
and theory necessarily entails one or more transformations of coordinates, 
and these transformations are the subject of the present chapter. 
2 . M o t i o n sof the p l a n e s of r e f e r e n c e . T h e planes of reference 
most commonly used in celestial mechanics are the plane of the ecliptic 
and the plane of the equator . T h e great circles in the sky where these 
two planes cut the celestial sphere are the ecliptic and equator. T h e 
vernal equinox (or for brevity, the equinox) is the one of the two 
intersections of the ecliptic and equator that the sun passes about 
March 21 . T h e ecliptic, equator, and equinox are all in continuous 
motion, and hence the lat i tude, longitude, declination, and right 
ascension of any celestial body all change continuously. Most of the 
changes are different in different regions of the sky. W h e n they are 
developed analytically in the way commonly used, it is found that two 
kinds of terms appear. The re are periodic terms, which have in their 
a rguments certain elements of the orbits of the earth and m o o n ; these 
are called nutational te rms. T h e r e are also secular terms, which contain 
powers of the t ime and are independent of the instantaneous positions 
of the earth and moon ; these are the precessional te rms. It is convenient 
to treat these two classes of terms separately. 
If the declination and right ascension of any body are measured with 
a transit circle the declination is referred to the instantaneous equator 
and the right ascension to the instantaneous equinox. T h e coordinates 
are termed apparent declination and apparent right ascension. These 
coordinates are affected by planetary aberration, and if they are freed 
from the effects of aberration they are then called t rue declination and 
t rue right ascension. T h e effects of the nutat ion may be removed from 
the t rue coordinates, and the coordinates are then said to be referred 
to the mean equator and mean equinox of date. In addition, the effects 
PRECESSION 203 
of the precession over a specified interval of t ime may be removed. T h e 
interval of t ime is usually taken to be that since the beginning of some 
tropical year, as 1950.0, and the coordinates are then said to be referred 
to the equator and equinox 1950.0. ( T h e beginning of the tropical year 
is the instant when the sun 's mean longitude is 280°, and it mus t be 
carefully dist inguished from the beginning of the calendar year. T h e 
instant 1950.0, for example, is 1950 January 0.923 E .T . ) 
3 . P r e c e s s i o n . T h e instantaneous rates of change per tropical 
century of the right ascension and declination of any body, due to 
precession, are given by 
pa = m + n sin a tan S, ^ 
pd = n cos a, 
where a and S are the instantaneous right ascension and declination 
referred to the mean equinox and equator of date, and the conventionally 
adopted values of m and n are 
m = 4608'/50 + 2'/79 T 
= 307?233 + 0?186 T, (2) 
n = 2004'/68 - 0"85 Ty 
T being reckoned in tropical centuries from 1900.0. 
T h e total change of the right ascension and declination of any body, 
due to precession, between any two dates t0 and tl9 is given by 
Aoc = [hp*dTy 
J (3) 
AS = I xp6dT. 1 
Equat ions (3) although writ ten as definite in tegra ls , are actually 
differential equations. T h e y might be solved by s tep-by-s tep numerical 
integration as described in Chapter IV, but this would be rather laborious; 
it is easier to proceed by successive approximations if tx — £0 is less than 
a century or so, unless the body is within a few degrees of the pole. 
It will usually be sufficient to evaluate m and n at the t ime 
T = \{t1 + t0)y and then to subst i tute in the approximate equat ions 
Aoc = (t1 — t0) (m + n s in a 0 tan 8 0 ) , 
AS = (tx — t0) (n cos a 0 ) , 
(4) 
204 OBSERVATION AND THEORY 
a 0 and S 0 being the values of oc and 8 at t ime t0. T h e first approximation is 
a i = a o + ^ a > ^ 
S, = S0+A8. 
Now recalculate the expressions (4), using oc1 and 8L instead of oc0 and S 0 . 
T h e new values of and AS will differ very little from the first 
approximation, and the mean of the two results may usually be 
adopted. 
In practice it will usually be possible to avoid all calculation beyond 
a little interpolation, by using the special tables provided in the American 
Ephemeris or the British Astronomical Ephemeris. 
4 . N u t a t i o n . T h e effects of nutat ion on the right ascension and 
declination of a body, unless it is wi th in a few degrees of the pole, may 
be calculated by 
Aoc — (cos e + sin e sin oc tan 8) AJJ — cos oc tan 8 Ae> 
(6) 
AS = sin e cos ocAip + sin ocAe, 
where oc and 8 are the right ascension and declination of the body, e 
is the obliquity of the ecliptic, and AI/J and Ae are the nutat ion in longitude 
and obliquity given in the American Ephemeris for every day of the 
year. Equat ions (6) are not rigorous, being derived from the differential 
relations among the parts of a spherical triangle. T h e quantit ies Aoc and 
AS cannot exceed 2 0 " on a great circle, and hence the equations suffice 
to give these quantit ies accurate to 0"01 unless tan 8 is very large. 
Owing to the smallness of Aoc and AS the tr igonometric functions are 
needed to only three or four significant figures, and hence it is not 
necessary to define a, 8, and e r igorously; either the apparent values 
or the values at the beginning of the tropical year in which the 
observation is made may be used. 
In practice, right ascensions and declinations are seldom if ever 
referred to the mean equinox and equator of date. Transit-circle ob-
servers usually publish apparent positions, while micrometric and 
photographic observers usually publish positions referred to the mean 
equinox and equator of the beginning of the tropical year of observation 
(occasionally a s tandard equinox and equator such as 1900.0 or 1950.0 
is used). I t is therefore convenient to combine Eqs . (6) with the reduction 
for precession dur ing a fraction of a year; any additional reduction for 
precession is then made for a whole n u m b e r of tropical years. 
NUTATION 205 
p cos e — X = m, 
p sin s = n. 
T h e n if we pu t for brevity 
F = cos s + sin s sin oc tan S, 
it is seen from (8) that 
m + n sin oc tan 8 = pF — A, 
whence 
m + n sin a tan 8 A 
r = . 
p p 
T h e n the sum of Eqs . (6) and (7) may be writ ten 
(8) 
Aoc = (T H j (m + n sin oc tan 8) + A —— — As cos a tan 8, 
' ' (9) AI/J 
P 
If we put 
Aip 
A = r H — , a = m + « sin a tan 8, P 
B = — As, 
b = cos a tan 8, 
a = n cos a, 
E = A —— , = — sin a, 
Eqs. (9) become 
= + 56 + E, 
AS = Aa' + Bb\ 
(10) 
Wi th the elapsed fraction of the tropical year denoted by r , Eqs . (4) 
for the precession become 
Aoc = r(m + n sin oc tan 8), 
(7) 
AS — rn cos oc, 
where the subscripts of oc and S may be dropped. T h e reduct ion for 
precession and nutat ion combined is obtained by adding Eqs . (6) to 
Eqs . (7). T h e resulting expressions may be simplified by a t rans -
formation. 
Let us define two quantit ies p and A by the equations 
206 OBSERVATION AND THEORY 
T h e quantit ies A, B, E may be obtained from the American Ephemeris 
or the British Astronomical Ephemeris, while a, b,'a\ V may be calculated 
from the approximate position of the body. 
I t is customary to express Aoc in seconds of t ime, by dividing a, b, 
and E by 15. 
It is convenient to combine the reduct ion for nutat ion and precession 
since the beginning of the year with the reduction for annual aberration, 
by adding Eqs . (10) to Eqs . (21) of Chapter VI . T h e quantit ies obtained, 
when added to the astrometr ic place at the beginning of the year, 
give the apparent place. Conversely they may be subtracted from the 
apparent place to obtain the astrometric place at the beginning of the 
tropical year. In all ordinary cases the quanti t ies a, 6, cy d, a', b\ c\ d' 
may be calculated wi th the use of either the astrometric place or 
the apparent place of the object. 
5. G e o c e n t r i c p a r a l l a x . T h e horizontal parallax of any body at 
a particular instant is defined as theangle at the center of the body 
subtended by the ear th 's equatorial radius. I t is nearly equal to the 
apparent vertical displacement of the body against the background of 
stars when it is rising or setting. T h e constant of the solar parallax is 
the angle subtended by the earth 's equatorial radius at a distance of 
one astronomical unit , and its adopted value is 8780. Hence if the 
geocentric distance to any body, expressed in astronomical units , is 
denoted by r, then the horizontal parallax is given by 
sin 77 = sin 8780/r. (11) 
Unless the body is very near the earth the simpler formula 
77 = 8780/r 
may be subst i tuted for (11). T h e error in this approximation is less than 
0701 if r is greater than 0.007 a.u. 
W h e n a body is observed on the meridian, as is the case always with 
the transit circle and sometimes with photography, the parallactic 
displacement is in declination only, the right ascension not being affected. 
T h e correction to be applied to the observed declination in order to 
obtain the t rue declination is 
AS = pir sin (<pf — S), 
where p is the radius of the earth at the observer 's lati tude, expressed 
in units of the equatorial radius, and q>' is the geocentric lat i tude. 
GEOCENTRIC PARALLAX 207 
~~ sin y 
where 
tan y = tan <p '/cos hy 
and h is the hour angle. T h e hour angle is found from the local sidereal 
t ime, 8y by 
h = 6 - oc. 
Formulas (12) suffice for correcting any observation for parallax, 
unless the body is very near the earth. I t is, however, seldom necessary 
to use them. I t is the practice of most observers to publ ish so-called 
parallax factors with their observations. These factors may take either 
of two forms. T h e y may be the components in right ascension and 
declination corresponding to a distance of 1 a.u., in which case they 
need only to be divided by the actual distance, or they may be the com-
ponents corresponding to a parallax of 1", in which case they are to 
be mult ipl ied by the actual horizontal parallax. 
T h e application of parallax corrections to the observed coordinates 
of an object reduces them to the values that they would have if the 
observations had been made from the center of the earth, and hence 
this way of proceeding is appropr ia te when the observations are to be 
compared with a geocentric ephemeris . I t often happens , however, that 
no precise ephemeris exists, and then the observations may be compared 
with theoretical positions especially calculated for the purpose . In such 
cases it is advantageous, instead of correcting the observations for 
parallax, to refer the calculated positions to the observer by the use of 
topocentric coordinates. 
Let xy yy z be the calculated equatorial rectangular heliocentric 
coordinates of the observed object at the instant, preceding the t ime 
of observation, when light left i t ; let Xy Yy Z be the geocentric coor-
dinates of the sun at the instant of observation referred to the same 
axes; and let JC, yy z be the topocentr ic coordinates of the center of the 
earth. T h e n the calculated distance, right ascension, and declination 
appropriate for direct comparison with observation are given by 
p cos S cos oc = x + X + x, 
p cos § sin oc = y + Y + y, (13) 
p sin § = z + Z + z-
If a body is observed off the meridian, the corrections to the observed 
right ascension and declination are 
. pTT cos cp' sin h 
Aoc = - ^ , cos 6 
J § = P77 s i n s i n ~ s) 
(12) 
208 OBSERVATION AND THEORY 
T h e coordinates x, yy z are functions of the observer 's lat i tude, and 
of the local sidereal t ime, given in uni ts of the seventh decimal by 
A = 426.6 cos 9 ' , 
x = — A cos 0, 
y = — A sin 0, 
Z = — 426.6 sin cp'. 
T h e sidereal t ime 9, referred to the equinox 1950.0, may be found in 
decimals of a circumference by 
0.79115 + 1.0027 37803 (J.D. — 243 0000.0) - A, 
A being the west longitude of the observer. 
6. P r e c e p t s . Chapters VI and VII contain all the principles that 
are necessary for comparing observations with theory. T h e procedure 
for the more common cases arising in practice may be described as 
follows. 
Case 1. Observed place apparent topocent r ic—Computed place 
apparent geocentric. Correct the observed place for parallax by (12) or 
by using the parallax factors. 
Case 2. Observed place apparent topocent r ic—Computed place 
astrographic topocentric . Free the observation from the circular port ion 
of the annual aberration, from the nutat ion, and from the precession 
since the beginning of the current year by subtract ing the results of 
(10) and Chapter VI, Eq. (21). T h e n reduce the observation for precession 
to the same equinox and equator as the computed place by expressions 
(4) and (5) or by special tables. Free the observation from the elliptic 
term of the aberration by adding the results of Chapter VI, Eq. (22). 
Case 3. Observed place astrometric topocent r ic—Computed place 
astrographic topocentric . Reduce the observation to the same equinox 
and equator as the computed place, and add the results of Chapter VI, 
Eq . (22). 
Case 4. Observed place astrometric topocen t r ic—Computed place 
astrometric geocentric. Reduce the observation to the same equinox and 
equator as the computed place, and correct the observation for parallax 
by (12) or by the parallax factors. 
NOTES AND REFERENCES 209 
N O T E S A N D R E F E R E N C E S 
T h e correc t ion of a s t ronomica l o b s e r v a t i o n s for t h e effects of p r e c e s s i o n , p r o p e r 
m o t i o n , n u t a t i o n , aberrat ion , and paral lax b e l o n g s to the subjec t of spher ica l 
a s t r o n o m y , of w h i c h o n l y the barest out l ine can be g i v e n here . A va luab le treat ise 
o n the subjec t is W . C h a u v e n e t , Manual of Spherical and Practical Astronomy, 
2 V o l s . , L i p p i n c o t t , P h i l a d e l p h i a , 1855, repr in ted b y D o v e r P u b l i c a t i o n s , N e w 
York , 1960 . P e r h a p s e v e n m o r e va luable is S i m o n N e w c o m b , A Compendium of 
Spherical Astronomy, M a c m i l l a n , N e w York , 1906; a p h o t o g r a p h i c reprint has 
b e e n p r o d u c e d b y U n i v e r s i t y M i c r o f i l m s , A n n A r b o r , M i c h i g a n . T h e m o s t 
m o d e r n work o n the subjec t is W . M . S m a r t , Spherical Astronomy, C a m b r i d g e 
U n i v . Pres s , L o n d o n and N e w York , 1944 , re i s sued C a m b . U n i v . Press , 1 9 6 0 . 
CHAPTER V I I I 
THE METHOD OF LEAST SQUARES 
1. I n t r o d u c t i o n . In any theory of the motion of a body, either 
around another body or about its axis of rotation, certain constants 
appear that have to be determined by observation. W h e n nothing about 
these constants is known in advance, as is the case with the elements 
of the orbit of a newly discovered object, the determinat ion of them 
may be difficult; in the case ment ioned an extensive literature exists 
for this specific purpose . W h e n approximate values of the constants are 
known, however, they may be incorporated into a theory of the motion, 
which may then be used to calculate theoretical positions of the object. 
A comparison of the theory with observations will show that the theory 
does not represent the observations exactly. Each observation furnishes 
a residual, observed place minus computed place, O — C, which is due 
to three causes, and to no others. Firs t , the theory may be inadequate . 
Second, the observations are affected by errors. Th i rd , the errors of 
the approximate constants used in determining the computed posit ions 
contr ibute to the residuals. In this chapter are discussed the two classes 
of errors last ment ioned, and it is shown how the approximate values 
of the constants may be improved by an analysis of the discrepancies 
between observation and theory. 
T h e method of analysis may be used to obtain not only more accurate 
values of the orbital elements of a body, but also improved values of 
any other parameter on which the observations depend. As examples 
may be ment ioned the e lements of the ear th 's orbit, the masses of 
dis turbing planets, the solar parallax, the constant of nutat ion, and other 
astronomical constants . In any particular case it is necessary that the 
observations be such that an error in the assumed value of the constant 
will exert an appreciable influence. 
2 . F r e q u e n c y d i s t r i b u t i o n of e r r o r s of o b s e r v a t i o n s . T h e 
errors of observations are due to various causes, among which are the 
optical defects of telescopes, fluctuations of the density of the at-
mosphere , and imperfections of the screws used in measuring pho to-
graphic plates. T h e errors of a series of observations are of different 
210 
DISTRIBUTION OF ERRORS 211 
sizes. If we imagine the size of an error to be plot ted as ordinate against 
the number of observations having that size of error as abscissa, the 
resulting curve is called a frequency dis t r ibut ion curve or probabil i ty 
distr ibution curve of errors . I n general such a curve is shaped like a 
bell that is more or less symmetrical about the point of zero error , 
showing that positive and negative errors occur wi th equal frequency 
and tha t small errors are more numerous than large ones. 
In most cases dealt with in as t ronomy the frequency distr ibut ion of 
errors is not known in advance, bu t can be de te rmined only after a 
very careful investigation of the observational material , if at all. I t is, 
however, possible to make an assumption about the dis t r ibut ion that 
will be adequate if the errors are each the sum of a n u m b e r of 
simultaneously occurring errors, each one of different origin. Such errors 
are often called accidental errors as dist inguished from systematic errors, 
which may have a single origin. In practice astronomical observations 
are usually affected by errors of bo th kinds, and the presence of 
systematic errors may invalidate the conclusions drawn. W e are, however, 
forced to proceed as if the errors were accidental, at least in the first 
instance, since there is no mathematical theory of systematic errors . 
Regarding accidental errors, there is a theorem in the theory of 
probabil i ty stating that the frequency distr ibution of a sum of r andom 
variables, each wi th its own arbitrary frequency distr ibution, asym-
totically approaches the function 
/(*) dx = hu-1'2 exp { - h2x2} dx (1) 
if the individual random variables are independent of one another . In 
this expression f(x) dx is the relative n u m b e r of cases where the r andom 
variable has a value between x and x + dx, TT is the familiar constant, 
the expression exp {— h2x2} means the base of natural logari thms 
raised to the power — h2x2, and h is called the modulus of precision. 
T h e factor hrr~l12 is in t roduced in order to make 
f(x) dx = 1. 
Instead of h we may introduce the mean of the squares of the random 
variable x. T h e square root of this mean is called the s tandard deviation, 
c j , or the r .m.s . ( root-mean-square) error. T h e relation between cr and h is 
a2 = hrr-1'2 x2 exp {— h2x2}d* 
212 LEAST SQUARES 
T A B L E 1 
DISTRIBUTION OF ERRORS 
Multiple of p.e. N u m b e r of larger errors 
1 5 0 0 0 or 1 in 2 
2 1 7 7 4 or 1 in 6 
3 4 3 0 or 1 in 2 3 
4 7 0 or 1 in 1 4 0 
5 8 or 1 in 1 2 0 0 
6 1 or 1 in 1 0 , 0 0 0 
Any marked deviation from the figures in the table is an indication 
that the dis tr ibut ion is not Gaussian. 
If we have a linear combinat ion of two random variables 
x = axxx + a2x2i (2) 
in which ax and a2 are constants , while xx and x2 have normal 
distr ibutions with s tandard deviations G 1 and cr 2, the distr ibution of x 
can be found as follows. 
or 
and 
T h e frequency dis tr ibut ion (1) is said to be a normal dis tr ibut ion or 
a Gaussian distr ibution. If the errors have such a dis tr ibut ion the 
s tandard deviation a is called the mean error or the s tandard error. 
Another parameter , m u c h used in astronomy, is the probable error, 
which is defined as the quant i ty tha t exceeds half of the errors in absolute 
magni tude , and is exceeded by half of them. T h e probable error (p.e.) 
as thus defined has a definite meaning no mat ter what the distr ibution 
of errors is. If, and only if, the distr ibution is Gaussian, then it may 
be shown that the probable error is connected with the s tandard deviation 
by the relation 
p.e. = 0.6745a. 
Table 1 shows, for a Gaussian dis tr ibut ion of 10,000 errors, the 
n u m b e r that exceed various mult iples of the probable error. 
in which 
a* = a\a\ + a\a\. (4) 
Hence the distr ibution of a linear combinat ion of two random variables 
each with a normal distr ibution is a normal distr ibution, with a s tandard 
deviation given by (4). T h i s principle can be extended to a linear 
combinat ion of any n u m b e r of r andom variables. 
3 . T h e p r e f e r r e d v a l u e of a m e a s u r e d q u a n t i t y . Let us consider 
n measurements xly x2y xn of a quanti ty, and let us denote the 
t rue value of the quant i ty by a. T h e n the error e,- of any measurement 
Xj is Sj = Xj — a. If the errors have a normal dis t r ibut ion the probabil i ty 
of any measurement xi is 
^ = e x p { - ( * , - a ) V 2 o * } , 
and the joint probabil i ty for the n measurements is 
fefefexp i ~ i [ { x i - a ) 2 + ( * 2 ~ a ) 2 + - + ( * « ~ a ) 2 ] ! - ( 5 ) 
T h e preferred estimate of a is defined as tha t which makes (5) a 
max imum or 
J_ 
2o* 
-Z{Xj-af (6) 
a min imum. T o make (6) a m i n i m u m it is sufficient to have 
dZ 
whence 
= 0 or 227 (*, - A) = 0, 
a = -ZXj. (7) 
PREFERRED VALUE 213 
| ^ </#2. (3) 
I t can be shown that for 
x < axxx + tfgtfg < # + dxy 
T h e s imultaneous or joint probabili ty that xx has a value between 
and xx + d ^ , and also x2 between x2 and ,x:2 -f rf#2 is 
214 LEAST SQUARES 
Note that in (8) a denotes the t rue (unknown) value of the measured 
quanti ty, and not the preferred estimate of it given by (7). 
If the frequency distr ibution of the errors is not a normal one we 
can still find preferred estimates for a and a if the form of the distr ibution 
is known. Let the probabili ty of any single measurement be f(a, xly x2, 
xn)y where / is any known function. T h e n the joint probabil i ty is 
nf(Xj - a), 
and the preferred estimates can be found from 
T h e mean error of the ari thmetic mean (7), as dist inguished from 
the expression (8), may be determined as follows. Let there be n observa-
tions Xj,j = 1, 2, w, with t rue errors ejy normally distr ibuted. T h e 
error of the ar i thmetic mean is 
1 y 
n J 
which is a linear combinat ion of n random variables. If a is the mean 
error of one observation, then by (4), 
a2 = na2/n2 = o2\n, 
whence o~e = on~ll2
y or the mean error of the a r i thmet ic mean is 
the mean error of one observation divided by the square root of the 
number of observations. 
We have seen that if the t rue value of the measured quanti ty were 
known, the mean error of one observation would be given by (8). T h e 
t rue value is, however, not known; we have only the estimate given 
by (7). After this estimate is obtained we can subtract it from each of 
the measured values; the differences are called the residuals v. Let us 
T h u s the preferred est imate of a is the ari thmetic mean of the 
observations. T h e preferred estimate of a may be found from 
whence 
Ufa -a)2. (8) 
and 
WEIGHTS 215 
whence 
x = a -
and 
Vj = 8j 
Now if av is the s tandard deviation of the residuals Vj and oE of the 
errors ejt we have 
Since a\ = [^2]/w, it follows that 
or 
T h e n , denoting the mean error of the ar i thmetic mean by fi, 
(9) 
4 . W e i g h t s of o b s e r v a t i o n s . Usually we are compelled to act as 
if the quality of different observations were the same, bu t in general 
it is not. Assume that the mean errorof an observation xx is a1 and 
that of x2 is cr 2 . T h e n the joint probabil i ty is given by 
T h e function to be made a m i n i m u m is 
denote the ar i thmetic mean of the x3- by x> as before the t rue error of 
Xj by ej9 and the t rue value of the measured quant i ty by <z. T h e n 
Vj = Xj — x, 
8j = Xj — a. 
Denot ing by [x] the sum of the xjy and by [e] the sum of the 8 j y 
[e] = [x] — na = nx — na, 
T h e quant i ty p, is said to be the weight of the observation xs. 
5. I n d i r e c t m e a s u r e m e n t s . As was ment ioned in Section 1, the 
usual cases in as t ronomy are not those where the quant i ty sought is 
measured directly; more commonly the quant i ty measured is some 
known function of the quant i ty sought. T h e quant i ty sought is usually 
a constant whose value is already approximately known. In the simplest 
case only one quant i ty is sought ; let us call this quant i ty x and the 
measured quanti ty y = f(x). Suppose we have n observations y j y 
j = 1, 2, n9 from which it is required to deduce the preferred value 
of x. Denote the approximately known value of x by xc, the value to 
be determined from the observations by x0> and the difference x0 — xc 
by Ax. T h e procedure is to calculate y from xc for each observation, 
obtaining y c y and then to form the differences A5y = yi—yc. Since 
in general y is a function of other parameters as well as of x9 the quant i ty 
y c varies from observation to observat ion; to show the variation we may 
write Aij = yj—yic. 
Now Ay may be expressed as a function of Ax by T ay lo r ' s expansion 
(id 
216 LEAST SQUARES 
(10) 
If we pu t px = \\o\ and p2 = \ja\y then 
or, for n observations,, 
I t is assumed that Ax is so small that its square and higher powers 
are negligible, whence, with sufficient accuracy, 
so we may find for each observation an equat ion of the form 
whence 
EQUATIONS OF CONDITION 217 
where the quant i ty in parentheses is a known number . Equat ion (11) 
is called an equat ion of condition, and Ax is called an unknown. 
For brevity let us wri te (11) 
aAx = Ay. (12) 
As in previous sections, the condit ion for the preferred value of Ax is 
d 
dAx 
or 
E(a0Ax-Aoyf = 0, 
[a2Ax] = [a Ay], (13) 
whence 
(14) 
and finally x0 is obta ined by 
x0 = xc+ Ax. 
Suppose tha t of the n equations of the form (12), supposed of equal 
weight, it happens that for q of them, a is a constant . T h e n these q 
equat ions contr ibute to (13) a por t ion 
qa2Ax = a[Ay] = Vq a[Ay]/Vq-
Comparison with (12) shows that we may write an equation of condi t ion 
Vq aAx = [Ay]/Vq = Vq [Ay]/q, 
or 
Vq aAx = Vq Ay, (15) 
which shows that the q equations may be replaced by the single equat ion 
(15), Ay being the mean of the q values of Ay. I n this case the equat ion 
aAx = Ay 
is said to have the weight q, and it is to be mul t ip l ied by V~k before 
use wi th the other n — q equat ions to obtain (13) and (14). 
6. E q u a t i o n s o f c o n d i t i o n . In the general case it is desired to 
determine not one bu t n unknowns from a series of observations. Le t 
the observed quant i ty , such as a spherical coordinate of a body, be 
denoted by x, and let Ax denote the discrepancy observed x minus 
computed x (O — C) between any observation and the corresponding 
Ax = 
218 LEAST SQUARES 
Assume for the momen t tha t Ax is not affected by errors of the first 
two classes ment ioned in Section 1; then it is composed entirely of 
contr ibut ions from {jjm T h e partial derivatives appearing as coefficients 
in (16) can be calculated and expressed as n u m b e r s ; three significant 
figures are almost always sufficient. T h e unknowns of the equations are 
the which are sought to be determined. I t is known from the theory 
of linear equations that n equations of the type (16) are a necessary 
and sufficient n u m b e r for the determinat ion of the provided that 
the equations are all independent , tha t is, provided that the determinant 
of the coefficients on the left-hand side does not vanish. If the equations 
are independent , values of the ^ can be found that will exactly satisfy 
the n equations, and if the observations were not affected by errors 
nothing more would be required. T h e presence of accidental errors in 
the r ight -hand members of the equations, however, prevents the t rue 
values of the ^ from being de te rmined; only approximate values can 
be found, which will be nearer to or further from the t ru th according 
as the errors of the observations are smaller or greater. T h e effect of 
the accidental errors may be diminished by increasing the n u m b e r of 
observations, and the n u m b e r of equat ions. In astronomical problems 
the number of equat ions used is seldom less than 2n and it is often 
m u c h greater ; when the highest precision is wanted several hundreds , 
or even thousands , of observations may be employed for the determina-
tion of a few unknowns . 
W h e n the n u m b e r of equat ions is greater than the n u m b e r of un -
knowns, then in view of the errors of the observations, no exact solution 
exists; it is not possible to find values of the ^ that exactly satisfy all 
of the equations. If any particular values of the ijj are subst i tuted into 
the left-hand members of the equations and the results severally sub-
t racted from the Ax, the remaining number s are called residuals "after 
the solut ion." As has been seen earlier in this chapter the values of the 
to be preferred are those that render the sum of the squares of these 
residuals a m i n i m u m ; this is the principle of least squares. 
I t is not immediately obvious how the equations are to be solved in 
order to obtain the desired m i n i m u m sum of the squares of the residuals; 
computed value. Let ch j = . 1, 2, n denote the assumed values 
of the constants used in calculating the computed values, and let the 
corrections to the assumed values of the constants be denoted by 
so that the corrected value of any constant is c, + T h e n the equat ions 
of condit ion, derived in the same way as (11), may be writ ten 
L=Ax- (16) 
NORMAL EQUATIONS 219 
the remainder of this chapter is devoted to the exposition of an 
expeditious method and, equally important , to a method for es t imat ing 
the uncer ta inty in the values of the unknowns as they are finally obtained. 
7* W e i g h t s of e q u a t i o n s . Every equation of condit ion may be the 
result of a single observat ion; in this case if the several observat ions 
are assumed to be of equal precision the equations are said to be of 
equal weight. On the other hand an equat ion may express the result 
of several observations, or the precision of the different observations 
may be known to vary; in either of these cases it may be desirable to 
assign different weights to the equat ions. If the precision of the different 
observations is the same, then the weight of an equat ion may be taken 
as proport ional to the n u m b e r of observations that it depends on. If 
the probable errors of the observations are known varying quanti t ies , 
then the weight of a single observation should be taken to be inversely 
proport ional to the square of the error. T h u s the w e i g h t y of an equat ion 
based on q observations having probable errors el9 e2, eqy is given by 
where k is a constant , to be chosen so as to make the weights of the 
different equations vary over a convenient range, say from 1 to 9, or 
from 0.1 to 0.9. I t is seldom justified to keep more than one significant 
figure in p , and almost never more than two. 
T h e value of k being fixed and the e3- known, then the probable error 
of uni t weight, often wri t ten p.e. 1, is k1/2. 
I t often happens , however, that the probable error of an observation 
is not known in advance bu t mus t be derived from the equations 
themselves. In such cases it may be convenient to assign a weight of 
unity to each observation, in which case the weightof an equat ion is 
equal to the number of observations that it depends on. 
W h e n the equations of condition are of unequal weight, then the result 
to be preferred is not that which renders the sum of the squares of the 
residuals a m i n i m u m ; instead it is required to minimize the sum of 
the squares of the numbers obtained by mul t ip ly ing each residual by 
the square root of its weight. 
8. F o r m a t i o n of n o r m a l e q u a t i o n s . Suppose that there are 
m equations of condition involving n unknown quanti t ies , m > n\ then 
the least-square criterion may be satisfied by combining the m equat ions 
into n normal equat ions as was indicated in Section 5, and solving the 
220 LEAST SQUARES 
normal equations. Instead of mult iplying each equat ion by the square 
root of its weight, it may be preferable to operate with the weight 
itself, organizing the calculation so as to produce the same result as 
if the square root had been used. I n this case, mult iply each equation 
by its weight, and calculate the sum of all the numerical quanti t ies 
entering into each equation. T h e result may be wri t ten according to 
the scheme of Table 2, where for convenience the literal quanti t ies and 
the sign of equality are placed at the top instead of in the equations. 
All of the symbols below the line denote n u m b e r s ; the partial derivatives, 
denoted by a for brevity, are pure numbers , while the Ax of Eq. (16), 
denoted by C, will usually be expressed in seconds of arc, degrees, or 
radians. T h e mixing of dimensions in the 2 will not cause any difficulty. 
T A B L E 2 
SCHEME FOR C O N D I T I O N A L EQUATIONS 
u = C z 
a l n Cx * 1 
Pia12 Pl&lZ Pl^ln PA 
a 2 1 a 2 2 a 2 n c 2 
p2a21 p2a22 p2a2Z P2a2n p2C2 P2%2 
aml a m n cm 
Pmaml Pmam1 Pmam3 Pmamn Pm^m 
T h e first normal equation, called the normal equation in £jl9 is obtained 
by mult iplying the first of the original equations by pxall9 the second 
by p2a2v t n e t h i rd by p3a31) etc., and forming the sum of the resulting 
m equations. T h e normal equation in £ 2 is obtained by mult iplying the 
first of the original equations by pid129 the second by p2^22> e t c - > a n d 
summing the result ing m equat ions. Repeating this process n t imes 
yields the n normal equat ions. T h e coefficients in the normal equations 
should actually be calculated one at a t ime, by accumulat ing the partial 
products . T h u s , the coefficient of £ 2
 m the normal equation in £ x is 
P\AWa\2 + P2a2\a22 + — + Pm^ml^m2> 
and the absolute te rm in the normal equation in | 3 is 
Plal3Cl + P2a2^2 + — + PmamBCm' 
NORMAL EQUATIONS 221 
I t will be observed that the quantit ies to the left of the sign of equality 
in the normal equations form a square symmetr ic array about the 
diagonal from upper left to lower r ight . T h u s the number s below and 
to the left of the pr incipal diagonal need not be calculated. 
T h e same operat ions performed on the summat ion as on the other 
number s const i tute a valuable check on the formation of the normal 
equations. Each result should be the sum of the n u m b e r s formed by 
proceeding to the left as far as the principal diagonal, then u p the 
column. If the equat ions of condi t ion are more than ten in n u m b e r it 
will usually be worthwhile to carry one more decimal in the normal 
equations than was present in the equations of condi t ion; if of the order 
of a hundred , two extra decimals may be retained. 
9. N o r m a l e q u a t i o n s . T h e normal equations may be wri t ten as 
in Table 3, where the same notat ion as was used for the equations of 
condi t ion is employed for the new number s . 
T A B L E 3 
N O R M A L EQUATIONS 
tl £ 3 
L = c 2 
011 alz aln c t 
« 2 n c 2 
033 aZn c 3 £ 3 
c„ 
T h e r e are now n equat ions in n unknowns to be solved, and the 
symmetry produced by the method used to obtain t hem simplifies their 
solution. 
I t is known from the theory of equations tha t a system of n independent 
linear equations in n unknowns has a un ique solution if, and only if, 
the de terminant of the coefficients is different from zero. I n this system 
of equations the de terminant of the coefficients is 
a i \
 a\2 a i z ••• a m 
a21 a22 a23 ••' a2n 
a31 a32 a33 a3n 
anl an2 an3 ••• ann 
222 LEAST SQUARES 
and a.u = aH. T h e value of any unknown £t- is the quotient of two 
determinants . In this quot ient the denominator is the de terminant of 
the coefficients, wr i t ten above, and the numera tor is a de terminant that 
may be obtained from the other by subst i tut ing for the zth column the 
column of C 's . T h e problem is to find the most efficient method of 
evaluating the quot ients of these various determinants . 
Al though in a strict mathematical sense a solution may always be 
obtained if the denominator determinant is different from zero, it often 
happens in practice that this de te rminant , while not zero, is a very 
small quanti ty, that is, small in comparison with the product of the 
quantit ies in the principal diagonal of the de terminant . For example, 
if the principal diagonal consists of numbers of the order of unity, then 
0.01 is a small quant i ty . If this is the case, and if the numerators are 
not also small, the physical interpretat ion is simply that the original 
observations are not well suited to determine the quantit ies sought, as 
for example when a small angle is sought to be de te rmined from its 
cosine. The re will be a loss of significant figures in the results, and the 
computer ' s first impulse will be to overcome this defect by carrying 
additional decimals in the equations. By this means a formal solution 
may be obtained to any desired n u m b e r of significant figures, bu t the 
labor is likely to be wasted so far as any physical significance is concerned. 
Th i s will be indicated by the large probable errors of the unknowns . 
T h e computer will find it worthwhile to estimate the errors before 
increasing the n u m b e r of significant figures in the equat ions, and 
thereby he will often save himself m u c h useless work. 
Another condition frequently arising in practice is that bo th the 
denominator de terminant and some or all of the numera tor determinants 
are small quanti t ies. In this case the solution is said to be nearly 
indeterminate , and this happens when two or more of the unknowns 
are strongly correlated; that is, a certain range of solution exists, and 
the equations will be nearly satisfied for a considerable range of values 
of the correlated unknowns , provided that these values have a certain 
linear relation to each other. I t will generally be advisable in such cases 
not to a t tempt to solve for the correlated unknowns separately, bu t to 
solve instead for the linear combinat ion of them, and for only one of 
a pair of the correlated unknowns themselves. N o real information is 
lost by this procedure, and the computer will be saved a great deal of 
labor. In addition, the physical significance of the resul ts is much more 
readily seen. 
These refinements of the work are discussed later. Firs t is given a 
formal solution of the normal equations. 
FORMAL SOLUTION 223 
10. F o r m a l s o l u t i o n . T h e solution of normal equations has 
received the attention of many writers, as is natural for a subject of 
widespread application. T h e theory is simple and easily unders tood, and 
will be found in almost any book on the theory of equations. T h e problem 
is the arrangement of the actual calculations in such a form as to require 
few numbers to be wri t ten, to make accessible all of the per t inent 
information contained implicitly in the equations, and to provide 
adequate checks on the numerical operat ions. T h e method given here 
is the result of the jo int efforts of many expert computers at different 
t imes. I t seems first to have been conceived in principle by M . H . 
Doolittle about 1878, bu t the details have beengreatly altered and it 
is not possible to credit the method as it now stands to any one person. 
In accordance with the principle tha t no solution is of m u c h value 
without an estimate of the errors involved, no a t tent ion is given to any 
method lacking this quality. T o avoid cumbrous notat ion, a literal 
solution is given in T a b l e 4 for the case of four unknowns . T h i s is 
followed by a numerical example. T h e principle, once apprehended, 
can easily be extended to any number of unknowns . 
T A B L E 4 
SOLUTION OF N O R M A L EQUATIONS 
d iz £ 4 
c 2 c 3 
c 4 z 
"ll «12 «13 014 1 0 0 0 Si 
N2 <*22 «23 «24 0 1 0 0 St 
033 034 0 0 1 0 
Nt 044 0 0 0 1 
Ni 013 014 1 0 0 0 Si 
Ei 1 6» &13 ^14 1/011 0 0 0 Zilan 
0*23 ^24 1 0 0 
E, 1 &23 &24 «l/<*22 1/^22 0 0 ^2lld22 
0*33 0*34 fx ft 1 0 £ 3 1 
E3 
1 &34 fi/dn fz/dn 1/0*33 0 ^zifd^z 
di4t gi £ 3 1 
Et 
1 gi/du £3/0*44 1/0*44 ^41/0*44 
sa 
1 $41 $42 $43 s4i ^44 
s3 
1 $31 $32 $33 s3i 
1 $21 $22 $23 $24 ^24 
224 LEAST SQUARES 
First write the normal equations in order, bu t wri te the absolute 
terms along the top in a row instead of down the column. Below each 
absolute te rm write 1 on the line containing the equation to which it 
belongs, and 0 on the other lines. Wri te the check-sums in the r ight -hand 
column. T h e y are different from those obtained in forming the normal 
equations, and are 
2\ = aU + #12 + #13 + au + 1> 
E2 = a12 + a22 + a23 + a24t + 1, 
^ 3 = #13 + #23 + #33 + #34 + 
^ 4 = #14 + #24 + #34 + #44 + ! • 
1. Wri te the first normal equation in the next row. 
2. Mult iply every n u m b e r in row N1 by l/an to get row Ev and 
verify the check-sum, placing a check-mark beside it. Th i s is the 
equation by which £ x will be determined after £ 2 , £4 a r e known; it 
is often called the elimination equat ion in 
3. Cover up the upper line Nx wi th a strip of paper and calculate 
#*22 = #22 ^12#12» e i ~ ^12» 
^23 = #23 ^12#13> 221 = E2 b12Uv 
^24 — #24 ^12#14> 
verifying the check-sum. 
4. Mult iply every number in the row just wri t ten by l/d22 to get row 
Z?2, and verify the check-sum. 
5. Cover up lines N1 and iV 2 and calculate 
^33 = #33 ^13#13 ^23#*23> f l = = ~ b13 ^23^1 > 
^34 = #34 ^13#14 ^23#*24» j2 = ^23> 
^ 3 1 = 2Z — b13E1 — &23^21> 
verifying the check-sum. 
6. Mult iply every number in the row jus t wri t ten by l/d33 to get row 
E3, and verify the check-sum. 
7. Cover up lines Nlf N2, and iV3, and calculate 
du = au — buau — b2id2^ — bMdM, 
Si = — b u — b2Aei — bzJl> 
g2 = — b2A — h\f* 
274 1 = 274 buEx bME21 b3AE3V 
F O R M A L S O L U T I O N 225 
8. Mul t ip ly every n u m b e r in the row jus t wri t ten by l / d 4 4 to get row 
E4 and verify the check-sum. 
T h e steps from 1 th rough 8 consti tute what is known as the forward 
solution. I t will be observed that line £ 4 gives the value of £ 4 in t e rms 
of Cl9 C 2 , C 3 , and C 4 . I n the back solution similar expressions are 
obtained for the other unknowns . In the scheme given, the line £ 4 is 
copied on the line below and is designated as 5 4 , bu t this is done here 
for convenience in notation, and in practice no advantage will be gained 
by wri t ing the line twice. 
9. T h e line S3 is obtained by 
S31 = flld33 — ^41^34* 
S32 = fJd33 — ^42^34» 
S33 = I M T T — ̂43^34* 
^34 = — ^44^34» 
^34 - 2 ^ 
10. T h e line S2 is obtained by 
^21 — eild22 %^23 %^24> 
^22 = V ^ 2 2 — 532^23 542^24> 
% = — ^33^23 %^24> 
$24 = — ^34^23 ^44^24* 
^24 = S21jd22 -- ^34^23 "~ ^44^24 
11. T h e line Sx is obtained by 
hi = l/an - %^12 ~~ •̂ 31̂ 13 %^ 1 4, 
^12 = — ^22^12 ^32^13 ,y42^14> 
S13 = — 523^12 — •̂ 33̂ 13 ^43^14* 
SU = — •̂ 24̂ 12 •̂ 34̂ 13 ^44^14? 
^14 - ^24^12 " ~ ^34^13 - ^ 4 4 < 
I t will be observed that the quanti t ies s 3 4 , s 2 3 , su, s12, s 1 3 , s 1 4 are 
severally identical with s 4 3 , s32, s 4 2 , s21, s 3 1 , sA1. Hence they need not be 
calculated, bu t may be filled in by inspection. T h e check-sums should 
be verified at each stage of the calculations. 
226 LEAST SQUARES 
12. I t now remains to calculate the values of the unknowns from 
and to tabulate t hem with the factors which, being mult ipl ied by the 
probable error of uni t weight, give the probable errors of the unknowns . 
These factors are severally s]1*, s^ 2 , s^, s1^. 
13. T h e values of the unknowns should now be subst i tuted into the 
normal equations, which should be exactly satisfied, except for errors 
of rounding. 
14. Next subst i tute the values of the unknowns into the original 
equations and calculate the residuals Vj. If the v5 have a Gaussian 
distr ibution, then the probable error of uni t weight is given by 
and the probable error of any unknown is obtained by mult iplying 
by the factor previously calculated. 
If the law of distr ibution of the residuals is not Gaussian, this is good 
evidence that the observations were affected by some systematic effect 
in addit ion to those designated ijly £2, gn. I n this case the probable 
errors calculated as jus t described may be entirely illusory, and the 
physical interpretat ion of the unknowns themselves is open to doubt . 
T h e doubt will be the more serious, the greater the depar ture of the 
residuals from a Gaussian distr ibution. 
I t will be noticed that if the n u m b e r of equations of condition is 
equal to the n u m b e r of unknowns (m = n), the probable error cannot 
be calculated; the equations will be exactly satisfied and p.e. 1 will be 
of the form 0/0. If m exceeds n by a factor of less than 2 then the 
probable error is bu t weakly determined, and is entitled to little con-
fidence. 
11. N u m e r i c a l e x a m p l e . T h e following example is given as an 
illustration of the content of this chapter . I t should be worked th rough 
as an exercise by any computer who is using the scheme for the first t ime. 
A set of equations of condition, with their weights, is given in Table 5. 
T h e v's are the residuals determined after the solution. 
T h e first thing to be noted is that the coefficients are of different 
orders of magni tude, those of £ 4 being about ten t imes the others. T h e 
Sll^l S12^2 S13^3 ~t~ ̂ 14^4» 
^21^1 ^22^2 ^23^3 ~f~ ^24^4> 
S31^1 " I " S32^2 "f" S33^3 ~f~ ^34^4* 
$41 ̂ 1 ~f~ $42^2 H~ $43^3 "f" J44^4> 
p.e. 1 = 0.6745 
E X A M P L E 227 
p fi £ 2 £ 3 £4 = C V 
1 + 1.155 + 1.507 + 2.622 — 12.29 + 9'.'16 + 0f'79 
2 + 2.417 + 0.833 + 4.702 + 14.62 + 12.43 — 0.44 
1 + 0.931 + 1.904 4- 1.554 + 6.10 + 5.48 + 0.41 
1 + 2.955 + 0.725 + 3.324 — 10.48 + 6.98 - j - 0.44 
1 4- 1.164 + 1.820 + 0.949 + 11.75 + 1.60 — 0.24 
3 + 1.188 + 0.828 — 0.016 + 11.48 — 2.37 + 0.44 
3 + 1.199 + 1.735 — 0.647 — 12.11 — 4.49 — 0.40 
1 + 0.947 + 1.854 — 1.318 — 6.38 — 6.30 — 0.10 
numerical work to follow will be simplified if they are reduced to 
approximately the same size. Accordingly the several columns are 
mult ipled by the factors 
0.34 0.53 0.21 0.068 0.080, 
each number being approximately the reciprocal of the largest n u m b e r 
in the corresponding column above; this amounts to the in t roduct ion 
of new unknowns r]v rj2y rj3y rjA such that 
f! = 4.25 r]ly £ 3 = 2.62 773, 
£ 2 = 6.62 t ? 2 , £ 4 = 0.85 t ? 4 . 
T h e new equat ions of condition are given in Tab le 6. T h e product 
of each equation by its weight, whenever the weight differs from uni ty, 
T A B L E 6 
TRANSFORMED EQUATIONS OF C O N D I T I O N 
^3 C 2 
+ 0.393 + 0.799 + 0.551 — 0.836 + 0.781 4- 1.688 
+ 0.822 + 0.441 + 0.987 + 0.994 + 0.994 4- 4.238 
+ 1.644 + 0.882 + 1.974 + 1.988 + 1.988 4- 8.476 
+ 0.317 + 1.009 + 0.326 + 0.415 4- 0.438 + 2.505 
4- 1.005 + 0.384 + 0.698 — 0.713 + 0.558 4- 1.932 
+ 0.396 + 0.965 + 0.199 4- 0.799 + 0.128 4- 2.487 
+ 0.404 + 0.439 — 0.003 + 0.781 — 0.190 + 1.431 
+ 1.212 + 1.317 — 0.009 + 2.343 — 0.570 4- 4.293 
4- 0.408 + 0.920 — 0.136 — 0.823 — 0.359 + 0.010 
+ 1.224 + 2.760 — 0.408 — 2.469 — 1.077 + 0.030 
+ 0.322 + 0.983 — 0.277 — 0.434 — 0.504 4- 0.090T A B L E 5 
EQUATIONS OF C O N D I T I O N 
228 LEAST SQUARES 
is given immediately below the equation, and the check-sums have been 
added. 
T h e normal equations and their solution appear in Tab le 7, according 
to the scheme described. T h e check-sums used to verify the formation 
of the normal equations are not pr inted. 
T A B L E 7 
N O R M A L EQUATIONS AND SOLUTION 
Cx c2 c 3 c 4 
+ 1 . 8 5 9 + 0 . 5 4 4 + 3 . 2 3 8 + 1 . 8 6 9 2 
Nx + 3 . 8 6 6 + 4 . 1 0 2 + 2 . 4 6 4 + 0 . 8 3 6 1 0 0 0 + 1 2 . 2 6 8 
N2 + 7 . 2 0 8 + 1 . 4 4 8 — 0 . 5 4 5 0 1 0 0 + 1 3 . 2 1 3 
N, + 3 . 0 1 7 + 1 . 7 4 7 0 0 1 0 + 9 . 6 7 6 
N* + 8 . 0 4 4 0 0 0 1 + 1 1 . 0 8 2 
Nx + 3 . 8 6 6 + 4 . 1 0 2 + 2 . 4 6 4 + 0 . 8 3 6 1 0 0 0 + 1 2 . 2 6 8 
Ex 1 + 1 . 0 6 1 + 0 . 6 3 7 + 0 . 2 1 6 + 0 . 2 5 9 0 0 0 + 3 . 1 7 3 
+ 2 . 8 5 6 — 1 . 1 6 6 — 1 . 4 3 2 — 1 . 0 6 1 1 0 0 + 0 . 1 9 7 
E2 1 — 0 . 4 0 8 — 0 . 5 0 1 — 0 . 3 7 1 + 0 . 3 5 0 0 0 + 0 . 0 6 9 
+ 0 . 9 7 2 + 0 . 6 3 0 — 1 . 0 7 0 + 0 . 4 0 8 1 0 + 1 . 9 4 2 
E, 1 + 0 . 6 4 8 — 1 . 1 0 1 + 0 . 4 2 0 + 1 . 0 2 9 0 + 1 . 9 9 8 
+ 6 . 7 3 8 — 0 . 0 5 4 + 0 . 2 3 7 — 0 . 6 4 8 1 + 7 . 2 7 2 
E, = S, 1 — 0 . 0 0 8 + 0 . 0 3 5 — 0 . 0 9 6 + 0 . 1 4 8 + 1 . 0 7 9 
s, 1 — 1 . 0 9 6 + 0 . 3 9 7 + 1 . 0 9 1 — 0 . 0 9 6 + 1 . 2 9 9 
st 
1 — 0 . 8 2 2 + 0 . 5 3 0 + 0 . 3 9 7 + 0 . 0 3 5 + 1 . 1 4 0 
Sx 1 + 1 . 8 3 1 — 0 . 8 2 3 — 1 . 0 9 5 — 0 . 0 0 8 + 0 . 9 0 3 
rj1 = - 0.604 ± 1.35 p.e. 1„, ^ = - 2757 ± 0.46 p.e. 1^, 
V 2 = + 0.111 ± 0.73 p.e. 1„, £ 2 = + 0:'73 ± 0.39 p.e. 1^, 
773 = + 1.532 ± 1.04 p.e. 1„, f3 = + 4701 ± 0.22 p.e. 1 € > 
^ 4 = _ 0.030 ± 0.38 p.e. lv, f4 = - 0.'026 ± 0.026 p.e. l s t . 
T h e values of the ?/s are subst i tuted into the normal equat ions, which 
are found to be satisfied within the errors of rounding. T h e values of 
the f's are then subst i tuted into the original equations of condition, 
giving the residuals v. T h e sum of the squares of the z/s, each square 
being multiplied by its weight, is 2.50, whence 
p.e. l f i = 0.6745 (2.50/4) 1 ' 2 = ± 0. , ,53. 
COMBINATIONS OF UNKNOWNS 229 
Of the eight residuals it is seen that seven are smaller than the probable 
error of uni t weight, which is not significant, because the residuals are 
not all of uni t weight. T h e probable size of a residual of weight p is 
± W h e n the residuals are compared wi th their probable sizes 
it is found that four are larger and four smaller in absolute value. In 
addit ion half of the residuals are positive and half negative, and none 
is as large as twice its probable size. All of these condi t ions are more 
favorable than are often encountered, and they indicate that the results 
are entit led to m u c h confidence. 
T h e final results are 
fx = - 2."57 ± 0:'24, & = + 4"01 ± ( H 2 , 
£ 2 = + o:73 ± 0:'21, f4 = - 07026 ± 0:'014. 
12. C o m b i n a t i o n s of t h e u n k n o w n s . I t is not infrequently 
desired, after making the solution, to combine two or more of the 
unknowns . For example, in correcting a nearly circular orbit , it may 
be advantageous to use as unknowns A (e sin &) and A (e cos co), e and 
w being the eccentricity and longitude of per icenter ; and it is desired 
afterward to find Ae and Acoy wi th their probable errors . N o difficulty 
is likely to occur in combining the unknowns themselves, bu t the 
method of obtaining the probable errors is not immediately obvious. 
Le t it be required to determine the probable errors of two quanti t ies , 
u and v, which are known functions of £ x and f2. T h e probable error 
of u is given by 
and the probable error of v by a similar expression. 
Often the probable error of a linear combinat ion of the unknowns 
is required. For example, suppose 
" = *lf 1 + * 2 & + 
the k being constants . T h e n the probable error of u is given by 
p . C 1 | + $22^2 H~ ^33^3 2$i2^i^2 ^-S13^l^S H~~ 2 $ 2 3 ^ 2 ^ 3 ) 1 / 2 > 
I t may occasionally be desired to find what linear combinat ion of two 
unknowns , say £ x and £ 2 , is de termined with the smallest probable e r ror ; 
that is, it is desired to find values of kx and k2 that will render the 
expression snk\ + s 2 2& 2 + 2s12k1k2 a m i n i m u m . T h e problem as stated 
is indeterminate , because kx and k2 may always be chosen so as to make 
230 LEAST SQUARES 
the probable error of + k2t;2 as small as we please. However, if 
either of the k is fixed, then the problem becomes determinate . Suppose 
kx — 1. T h e n the expression to be minimized is S-Q -\- S22k2 ~f~" 2s 1 2 ^2> 
which is a m i n i m u m when k2 = — s12/s22. 
In the preceding numerical example let us find what combinat ion of 
£ x and | 2 is de termined with the smallest probable error. Taking k2 = 1, 
and working first with the rj's, we have kx = 0.823/1.831 = 0.45 and 
the combinat ion is 0 . 4 5 ^ + rj2 = — 0 . 1 6 1 . T h e probable error of 
this combination is 
p.e. 1, [1.831 (0.45)2 + 0.530 ( l ) 2 - 1.645 (0.45)] 1 / 2 = p.e. l r / (0.40). 
W e now have 
0.45 i h + i ? a = 0 . 1 0 6 f 1 + 0 . 1 5 1 f a , 
which is determined with a probable error of 
p.e. ltj (0.40) = p.e. l f (0.40) (0.080) = p.e. 1,(0.0320), 
whence 0 . 7 0 ^ + f 2 = — 1.07 is determined with a probable error 
of p.e. 1 | (0.21) or 0711, which is considerably less than the probable 
errors of either £ x or £ 2 . 
13 . C o r r e l a t i o n s . Whenever a linear combinat ion of two unknowns 
can be determined with a smaller probable error than either one of t hem 
separately, the unknowns are said to be correlated. T w o unknowns £ x 
and £ 2 are correlated whenever the cross t e rm s12 is different from zero, 
or equivalently in most cases, when the cross te rm a12 of the normal 
equations is different from zero. Correlations are the rule rather than 
the exception in least-square solut ions; in the numerical example of this 
chapter correlations exist between every pair of unknowns . 
Correlations are of different degrees of magni tude . T h e stronger the 
correlation, the larger the cross te rm a{j relative to ( a , ^ ^ ) 1 / 2 . F r o m the 
way in which the normal equations are formed it is not possible for 
to exceed (auajj)1,2
9 bu t it may approach it very closely. In such 
cases difficulty will be encountered with the solution, which is nearly 
indeterminate ; dur ing the numerical work it will be found that one of 
the numbers d22, rf33, du becomes very small, which causes a loss of 
significant figures in the elimination equations. T h e computer ' s first 
impulse may be to repeat the whole calculation with additional decimals, 
and a formal solution may always be obtained in this way, often at 
t remendous expense of t ime and energy. Both the correlated unknowns 
will, however, have large probable errors and therefore little physical 
C O R R E L A T I O N S 231 
meaning. I t is often possible to avoid the difficulty, wi thout the need 
for extra decimals, by solving for only one of the correlated unknowns 
directly, subst i tu t ing for the other a linear combinat ion of the two. T h e 
best combinat ion (that is, the one determined with the smallest probable 
error) cannot be de termined in advance of making the solution, and when 
possible it is usually preferable to make the solution first, then to 
determine the best combination to use, and to repeat the solution with 
the new unknowns . A fair approximation to the best combinat ion can 
often be obtained by such a choice that the cross t e rm in the normal 
equations vanishes, which may be done before any solution is made, 
in the following way. If the unknowns in question are £ x and £ 2 , 
subst i tute for £ x in the solution £ x + a12/an | 2 . T h i s subst i tut ion 
requires all of the coefficients in the normal equat ion for £ 2 to be ap-
propriately modified by the following subst i tut ions: 
for a12 write 0, 
for a22 write a22 — {al2fjally 
for a23 write a 2 3 — (a12/au)a13, 
for a 2 4 write tf24 — (a12lan)aUi 
for C2 write C 2 — (d12lan)Clyfor E2 write Z2 ~ (a12/au)I]v 
T h e normal equation for ^ remains unchanged, with the exception 
of the coefficient a12, and it becomes the normal equat ion for 
Applying this principle to r)1 and rj2 of the numerical example, we 
have as the combinat ion to be subst i tuted for rjly 
Vi + ^(4.102/3.866) = V l + l.Oftfc. 
Alternatively, we may subst i tute for rj2y 
77X4.102/7.208) + V 2 = 0 .57^ + V t . 
I t is to be noted that neither of these combinat ions is the one 
determined with the smallest probable error, which can be determined 
only after the solution is made ; in the present example nothing would 
be gained by such a subst i tut ion, because the correlations are not very 
strong. 
232 LEAST SQUARES 
14. N o r m a l p l a c e s . If several observations are taken dur ing a 
sufficiently short interval of t ime the equations of condit ion will vary 
bu t little over the interval. In such cases it is advantageous to average 
the separate residuals O — C, and to form a single equation of condit ion 
for the mid-epoch of the observations, giving it a weight equal to the 
number of observations. A residual O — C, obtained from several 
observations so combined, is called a normal place. N o explicit rule 
can be given for the formation of normal places; experience and j udge -
ment mus t play a part . Such a combinat ion of observations is, however, 
always permissible if the residuals in question can be regarded as varying 
uniformly with the t ime. In addition, if the observations are very 
numerous it may be worthwhile to fit a parabola to the residuals, and 
to use the constant t e rm of the parabolic expression as the absolute 
t e rm of an equat ion of condit ion. 
N O T E S A N D R E F E R E N C E S 
T h e m e t h o d of least squares b e l o n g s to t h e m o r e genera l s u b j e c t o f t h e ca lcu lus 
of observat ions . A g o o d e l e m e n t a r y b o o k is D a v i d B r u n t , The Combination of 
Observations, C a m b r i d g e U n i v . P r e s s , L o n d o n a n d N e w York , 1 9 3 1 . A s tandard 
treatise is E . T . W h i t t a k e r a n d G. R o b i n s o n , The Calculus of Observations, B lackie , 
L o n d o n , 1924 . 
T h e m e t h o d of s o l v i n g the n o r m a l e q u a t i o n s g i v e n i n th i s c h a p t e r is o f t e n ca l l ed 
an e l i m i n a t i o n m e t h o d , b e c a u s e in t h e forward s o l u t i o n t h e u n k n o w n s are s u c c e s s -
ive ly e l i m i n a t e d to t h e p o i n t w h e r e o n e e q u a t i o n in o n e u n k n o w n r e m a i n s . O t h e r 
m e t h o d s e n j o y i n g p o p u l a r i t y i n s o m e appl i ca t ions are k n o w n as s q u a r e - r o o t 
m e t h o d s a n d re laxat ion m e t h o d s . T h e y appear to p o s s e s s n o advantages for t h e 
t y p e of app l i ca t ion d i s c u s s e d h e r e , a n d re laxat ion m e t h o d s in part icular m a y 
lead to w o r t h l e s s resul t s in s o l u t i o n s w h e r e s o m e of the u n k n o w n s are s t rong ly 
corre lated . C o n s i d e r for e x a m p l e a case w h e r e t w o u n k n o w n s , say # a n d y, are s t r o n g -
ly correlated. T h e e q u a t i o n s wi l l b e satisfied over a cons iderab le range of va lues o f 
x a n d y, b u t n o t if x a n d y are a l l o w e d to vary o v e r the ir ranges i n d e p e n d e n t l y 
of e a c h other; c o r r e s p o n d i n g to a n y p e r m i s s i b l e v a l u e of x there is o n e a n d o n l y 
o n e va lue of y sat i s fy ing t h e e q u a t i o n s . T h e e l i m i n a t i o n m e t h o d , b y d e t e r m i n i n g 
the u n k n o w n s in s u c c e s s i o n , m a k e s t h e v a l u e of x, say, d e p e n d o n t h e p r e v i o u s l y 
a s s igned va lue of y a n d preserves t h e neces sary relat ion b e t w e e n t h e m , w h e r e a s 
t h e relat ion is n o t p r e s e r v e d w i t h re laxat ion m e t h o d s ; w i t h t h e m it m a y b e f o u n d 
at the e n d that the resul ts d o n o t satisfy t h e e q u a t i o n s . 
T h e p r e c e d i n g remarks are s o m e t i m e s o b j e c t e d t o o n t h e g r o u n d that c o n s i s t e n t 
va lues of x and y h a v e n o m o r e "phys i ca l real i ty" t h a n t h e i n c o n s i s t e n t o n e s 
o b t a i n e d b y re laxat ion. T h e o b j e c t i o n has s o m e w e i g h t . W h a t is o v e r l o o k e d 
is that t h e c o n s i s t e n t v a l u e s of x a n d y are b e t t e r s u i t e d t h a n o thers for u s e in 
forecas t ing future o b s e r v a t i o n s , a n d also for ver i fy ing t h e a r i t h m e t i c o f t h e 
so lu t ion . 
CHAPTER I X 
THE DIFFERENTIAL CORRECTION OF ORBITS 
1. I n t r o d u c t i o n . T h e methods presented in Chap te r s I and I I 
serve to calculate the position at any t ime of a body moving in an elliptic 
orbit the elements of which are known. For a body in the solar system, 
the observed data from which the knowledge of its location is obtained 
are 
t, oc, 8, 
i.e., the t ime of the observation and the geocentric spherical coordinates, 
r ight ascension and declination. I t should further be specified to what 
equator and equinox these coordinates are referred. 
Let x, y, z be the heliocentric coordinates of the object and X, Y, Z 
the geocentric coordinates of the sun, referred to the mean equator and 
equinox of 1950.0. T h e geocentric coordinates of the object are 
x + X, y + Y, z+Z. 
Let p be the geocentric distance. T h e geocentric right ascension oc and 
declination S are related to the geocentric equatorial coordinates by the 
formulas 
x + X = p cos oc cos 8, 
y + Y = p sin a cos 8, 
z + Z = p sin 8. 
T h e observed a, 8 at t ime t are not immediately comparable with the 
geocentric oc, 8 that appear in these formulas ; appropriate corrections 
for geocentric parallax, aberration, etc., mus t be applied to the 
observations or to the rectangular coordinates (see Chapters VI and VII ) . 
Le t Ax, Ay, Az be small corrections to the heliocentric coordinates 
of the object, AX, AY, AZ those to the geocentric coordinates of the 
s u n ; then 
Ax + AX = — p sin oc cos SAoc — p cos oc sin SJS + cos oc cos SAp, 
Ay -{-AY = + p cos oc cos SAoc — p sin oc sin SJS + sin oc cos SAp, 
Az +AZ = + p cos SAS + sin SAp, 
233 
234 ORBIT CORRECTION 
from which it is found that 
A* = — l — 1 r [ - sin oc(Ax + AX) + cos oc(Ay + AY)], 
p COS 8 
AS = i [ - cos a sin S(J* + AX) — sin a sin S(Ay + AY) + cos S(J# + AZ)]. 
If it may be assumed tha t X, Y, Z are not in need of correction, 
the relations between Aoc, AS and the corrections Ac3- to any set of six 
elliptic elements are 
. ^\ doc dx . -%r\ doc dy . 
J« = X ^ ^ + X^^> 
A * ^\ dS dx A , ^ dS dy A , ^\ dS dz A 
= X r x + X % + X 
or 
doc _ doc dx doc dy dS _ dS dx dS dy dS dz 
dCj dx dCj dy dc3-
 9 dc3 dx dc3- dy dc3- dz dc3 
I t is more common to deal with cos 8Aoc. 
In matrix form these expressions are 
dx dy dz 
dc1 dc± dcx 
dx dy dz 
dco dc2 dco 
dx dy dz 
dc$ dc6 dc% 
For numerical calculations this plan recommends itself. T h e r e is an 
obvious advantage in spli t t ing the calculation into two par ts and obtaining 
the final coefficients by a matrix multiplication. T h e partial derivatives 
dx/ dcjy dy/ dc3, dz/ dc3- depend on the heliocentric coordinates and 
velocity components of the observed object alone, while the elements 
of the second matr ix may be wri t ten explicitly 
— p~x sin oc — p~x cos a sin 8 
+ p~x cos oc — p~x sin a sin S 
0 + p'1 cos 8 
doc ~ do 
— cos d — 
ox dx 
doc ~ dS 
— cos 8 — 
dy dy 
0 
as 
dz 
doc ~ 
—— cos o 
dc± 
dc 
doc 
dCa 
cos 8 
db 
dc^ 
doc ~ dS 
cos o —— 
dc9 
_as_ 
dCR 
(1) 
doc 
cos 8 
dS 
dx 
cos 
dx 
doc 
8 
dS 
COS 8 
dy 
COS 
dy 
0 
dS 
0 0 
dz 
RECTANGULAR COORDINATES 235 
T h e y can easily be computed from an approximate geocentric 
ephemeris provided the geocentric distance is also furnished. T h e latter 
is required in any case in order to compute the correction foraberrat ion 
and in some methods for the parallax. 
2 . U s e of r e c t a n g u l a r e q u a t o r i a l c o o r d i n a t e s . One of the most 
common cases is tha t a table of the heliocentric rectangular equatorial 
coordinates at equal intervals is available. F r o m this table the values 
of xy y, z and their derivatives for any date may be obtained by numerical 
processes explained in Chapter IV. Also, let the elements a, e be available. 
Let the t ime interval of the table for x, yy z be w days, and let the 
derivatives of x, y, z wi th the uni t of t ime w days be denoted x, y, z. 
Fur ther , let n be the mean motion in radians in w days, and t the t ime 
in uni ts of w days counted from an epoch chosen arbitrarily. Now 
consider how the variations in the coordinates depend on the 
variations of the elements. I t will be assumed that these variations are 
small so that their squares and products may be ignored. 
(a) A change in the mean anomaly l0 at the initial epoch. T h i s is 
equivalent to an advance 
At = Aljn 
of the ephemeris . Hence 
Ax = xAt = xAlJn, 
Ay = yAt = yAlJn, 
Az = zAt = zAl0/n. 
(b) A change in the orientation of the orbit. T h i s may be thought of 
as the resultant of rotations of the orbit, AI/J1 about the x axis, AI/J2 about 
the y axis, AI)JS about the z axis. A positive rotation is considered to 
be a rotation that is counterclockwise as seen from the positive direction 
of the axis about which the rotation is performed. T h e n 
Ax = zAifj2 — yAipz, 
Ay = xAip3 — ZAI/JV 
Az = yA\jj1 — XAI/J2. 
(c) A change in the semi-major axis. T h e effect of such a change is 
twofold: a change in scale and a change in the mean anomaly proport ional 
to the t ime elapsed since the epoch. T h e first of these is obtained from 
Ax _ Ay _ Az _ Aa 
x y z a 
236 ORBIT CORRECTION 
and 
x y z 2 a 
By combining the two parts , the changes in the coordinates are found 
to be 
Ax 3 .v Aa 
a 
Ay = (y -\ty) 
Aa 
a 
Az = (* - | » ) 
Aa 
a 
(d) A change in the eccentricity. T h e most convenient way of obtaining 
these coefficients is to pu t 
~ = Hx + Kx, 
in which H and K are independent of the directions of the coordinate axes. 
If xy y are the coordinates of the planet in its orbital plane with the 
x axis directed to the perihelion, these expressions require 
8i = Hx + Kx, 
% = W + KS. 
T h e solution of H and K from these equations is 
y(dx/de) — 5c(dyjde) 
H 
K = 
xy — yx 
x(dy/de) — y(8x/8e) 
xy — yx 
Ax _ Ay _ Az _ 3 Aa_ 
T h e second follows from n2az = constant, which may be expressed by 
2 ^ + 3 ^ = 0. 
n a 
Hence the change in the mean anomaly is 
3 Aa 
tAn = — -tn , 
2 a 
R E C T A N G U L A R C O O R D I N A T E S 237 
G = a2n yl—e2 = a2n cos 9?, 
if e = sin 9 . For the evaluation of the numera tors , expressions in 
te rms of the eccentric anomaly are useful. Wi th 
, du sin u . n 
u — e sin u = /, 
T h u s 
cos u + e sin u(2 — e2 — e cos u) 
H = : -— , A. = 1 - e2 ' n(l - e2) 
F r o m these expressions u may be eliminated by making use of 
(x2 + y2 + z2)1/2 = r — a(l — e cos #), 
+ yy + = rr — e sin w, 
or 
T h u s 
de I — e cos w ' 1 — e cos u ' 
# = «(cos u — e), y = a cos cp sin w, 
it follows that 
y = an cos cp -
Multipl icat ion of the appropriate r ight -hand members gives 
dx ± dy a2n — cos u e2 cos w — e + e cos 2 u 
y de X de cos 99 1 — e cos w 
a 2 n i 1 \ = (cos w + £), 
cos cp 
(2 — e2 — e cos w). 
_ dy _dx a2 sin a 
X de ^ de cos 99 
cos u sin w 
T h e denominator in these expressions is twice the areal velocity in 
the orbital plane in w days, which may be wri t ten 
2 3 8 O R B I T C O R R E C T I O N 
Both H and K contain e as a factor in the denominator . T h u s if e 
is small, there is a loss of accuracy in the numerical calculation. T h i s 
is not a serious disadvantage. I t merely requires the calculation of rr 
and of r/a(l — e2) to more significant figures than are required in H 
and K. I t depends on the smallness of e how many extra significant 
figures are to be used. T h e coordinates x, y, z and their derivatives 
can as a rule be obtained with the necessary n u m b e r of significant 
figures, and the additional amount of calculation required is negligible. 
T h e combinat ion of these results gives the following expressions: 
Set I 
Al0 *h 4k # 3 Aaja Ae 
Ax = x/n 0 + z - y 
3 , . 
x - - t x Hx + Kx 
Ay = y/n — z 0 + x 
3 , . 
y ~ i t y 
Hy + Ky 
Az = z/n + y — X 0 
3 
z -~tz Hz + Kz 
These are to be subst i tuted into the left-hand matrix of the matrix 
product (1). 
T h e small rotations AI/J19 AI/J2, AI/J3 may be considered to be the three 
components in the equatorial coordinate system of a single rotation 
vector Aip. Let J p , Aq, Ar be the components of the vector Aif; in a 
coordinate system xyzy the xy plane of which coincides wi th the orbital 
plane with the x axis directed toward the perihelion. T h e relations among 
the two sets of rotation components are evidently 
= P J B J p + & J q + * i B J r > 
Aifj2=PyAp+QyAq + RyAr) ( 2 ) 
Ai/j3=PzAp+QzAq + RzAr. 
T h e rotation components J p , Aq, AT can easily be expressed in t e rms 
of corrections to the elements / , £?, co. Let y be the origin of longi-
tude, N be the ascending node of the orbit with the plane of the 
ecliptic, A W = 90°, NX = co, the angular distance of the perihelion 
from the node (see Fig. 1). Rotations Ap, Aq about OX and OY r e -
spectively are equivalent to 
Ap cos co — Aq sin co about ON, 
Ap sin co -f- Aq cos co about ON'. 
T h e rotation about ON equals AI, while the diagram shows that the 
rotation about ON' equals sin IAQ. As a result of the rotation about 
RECTANGULAR COORDINATES 239 
s i n i <a / I 
F I G . 1. Differential correction of the elements 7, & and CJ. 
I t should be noted that these formulas are independent of the choice 
of the plane of reference. Hence, if Q,' oo' represent the equatorial 
elements, i.e., the inclination relative to the equatorial plane, the longi-
tude of the node on the equator, and the angle from this node to the 
perihelion, respectively, the relations are 
AI' = Ap cos co' — Aq sin co ' , 
sin FAQ' = Ap sin co' + Aq cos co' , 
d c o ' + cos FAQ' = d r . 
Other useful relations may be obtained with the aid of Chapte r I, 
Sections 23, 24. For example: 
11AF s i n FAQ' d c o ' + cos FAQ' 11 
= | | d p d q d r || r ( — co' ) 
] cos co' + sin co' 0 
= || d p d q d r | | — s i n co' c o s c o ' 0 
0 0 1 
= | | d p cos co' — d q sin co' d p sin co' + d q cos co' d r | | . 
ON' the new node will be at B. T h e rotation d r about the z axis is 
given by the difference between the arcs AX' and NX, if OX' is the 
new x axis. But BX' — NX = d c o , AB = cos IAQ. Since AX' = 
+ AX' — NX = BX' — NX+AB, or d r - + cos IAQ. 
T h u s , finally 
AI = Zip cos co — Aq sin co, 
sin IAQ = Ap sin co + Aq cos co, (3) 
d c o + cos IAQ = Av. 
240 ORBIT CORRECTION 
Finally, 
| | J & A*p2 A^\\ 
= | | J p Aq Ar\\r(-co')p(-r)r(-Q') 
II 1 0 
= \\Ar sin FAQ' Aco' + cos / ' A Q ' 1 1 
0 
= \\AF — sin VAOJ' AQ' + cos / ' J o / , 
0 cos / ' + sin / ' 
0 — sin / ' cos / ' 
cos Q' + sin Q' 0 
— sin Q' cos Q' 0 
0 0 1 
cos Q' + sin Q' 0 
— sin Q' cos Q' 0 
0 0 1 
cos Q'AI' + sin Q' sin / ' A t / 
sin Q'AI' — cos £?' sin I'Aaj' 
cos / ' J o / + AQ' 
where the t ransposed vector has been wri t ten to save space across the 
page. T h u s 
AF = cos Q'Aifj1 + sin Q'Aifj2i 
sin I'AOJ' = sin Q'A\\s1 — cos Q'Aifj2, (4) 
cos FAOJ' + AQ' = AI/J3. 
If the eccentricity is small, the position of the perihelion will be poorly 
determined. Hence there will also be a degree of indeterminateness in 
both Al0 and in Aip3. Th i s is apparent from the coefficients of Al0 and 
AI/JZ in Ax and Ay. For a circular orbit that coincides with the equatorial 
plane, 
x/y = — y/x 
while z = 0. T h u s J / 0 and Aip3 cannot be separately obtained in a 
solution in which these unknowns are used. T h e difficulty may be 
avoided by using Al0+ Aip3 and Aip3 as unknowns . Hence: 
^ Al0 - yAifj3 = -n (Al0 + A^J3) + ( - 5 - AI/J3) 
^Al0 + xA^3 = J ( J / 0 + J<£3) + ( ~ \ + *) 
\ A \ , + 0 ^ 3 = J {Ak+m + ( - « ) ^ a -
R E C T A N G U L A R C O O R D I N A T E S 241 
— z 0 
— x 
Hx + Kx 
Hy + Ky 
Hz + Kz 
While this choice of unknowns is suitable for most cases of moderately 
small eccentricities, the determinateness may be increased by choosing 
d p , d q , and d r as unknowns instead of Aifjly di/r 2, d ^ 3 . Subst i tu t ion of 
(2) into the te rms involving these unknowns in Set I , gives: 
+ zAfa -yA^3 = (Pyz - Pzy)Ap + (Qyz - Qzy)Aq + (R* - Rzy)Ar, 
- ZAI/J, + XAI/Js = (Pzx - Pxz)Ap + (Qzx - Qxz)Aq + (Rzx - Rxz)Av, 
+ yAfa - xAi/;2 = (Pxy - Pyx) d p + (Qxy - Qyx) d q + (Rxy - Ryx) d r . 
If then the unknown d / 0 + d r is in t roduced instead of d / 0 , and eAv 
instead of d r , the final expressions become: 
Set III 
d / 0 + d r d p d q eAx Aaja Ae 
Ax= -n Pyz-Pzy Qyz-Qzy
 l-(-^ + Ryz-Rzy^ x - l t x Hx + Kx 
Ay= £ Pzx-Pxz Qzx-Qxz l ^ + R ^ - R ^ y-3-ty Hy + Ky 
Az= -n Pxy-Pyx Qxy-Qyx )-(-^ +Rxy-RyX} z - \ t z Hz + Kz 
T h i s form requires somewhat more calculation than Set I I , bu t 
it has the advantage of yielding a determinate solution regardless of the 
values of eccentricity or inclination. I t is r ecommended for general 
application. 
A X 
Ax = -
n 
n 
Az = -
n +y 
N o w — x j n — y and — y j n + x will be small , of the order of e. 
T h u s the unknown di/r 3 will appear wi th small coefficients, a difficulty 
in numerical applications. T h i s is avoided by int roducing eAtfjs as 
unknown, the coefficients of which are obtained by dividing those of 
AI/JZ by e. Consequent ly , the expressions become: 
Set II 
Al0 + Aifj3 A\\sx AI/J2 eAi/;3 Aaja Ae 
242 ORBIT CORRECTION 
I t will also be found that the coefficients of the unknowns in the 
observation equations for Aoc, AS will all be of the same order of 
magni tude, with one exception. T h i s exception is the coefficient of A a/a. 
Thi s coefficient depends upon the interval covered by the observations. 
T h e longer the interval, the more significant the par t tha t has t as a 
factor will become. 
T h e vectorial orbital constants that occur in the coefficients of Set I I I 
define the orientation of a plane xy tha t is to be chosen to coincide as 
nearly as may be with the osculating orbital plane of the planet, with 
the x axis in the direction of the perihelion. I t follows that 
Py PZ 
Ax Ay Az\\ = \\Ax Ay Az\\ Q* Qy Q, 
Rx Ry 
Thi s yields the remarkably simple form. 
Set IV 
AIQ+AT Ap Aq eAr Aa/a Ae 
Ax = - 0 0 - ( — - — y) x— Hx + K5c 
n e \ n J 1 2 
Ay= y- 0 0 + y~\t] Hy + Kf 
Az = 0 +y - x 0 0 0 
In some applications it will be advantageous to use this form. T h e 
coefficients may be expressed with the aid of series in te rms of the mean 
anomaly and, if the n u m b e r of observations is very great, tabulat ion 
as functions of the mean anomaly may be desirable. T h e coefficient of 
Aaja requires special t rea tment on account of the presence of the factor t. 
T h e equivalent form given by 
X 
3 • 
= X -In-l )-
y 
3 • 
-2*y 
= y 
may be used, / — l0 being expressed in radians. I t is obvious that whole 
revolutions must be retained in the value of / — / 0 . 
Tab le 1 gives expansions in te rms of the mean anomaly of the coeffi-
cients of Set IV. T o simplify the typography a common factor a has 
been omit ted. T h i s factor must be applied before the coefficients are used. 
RECTANGULAR COORDINATES 243 
T h e series expansions in Table 1 are generally given to the th i rd 
power of the eccentricity, with the exception that the parts of the coeffi-
cients of Aaja that have / — / 0 as a factor are complete to e 4 . 
T A B L E I 
COEFFICIENTS FOR S E T I V EXPANDED IN T E R M S OF THE M E A N A N O M A L Y 
Coefficients of Al0 -f- Ar 
Coefficients of Aa/a 
Ax 
in — 
a 
in 
a 
| sin / -
Coefficients of eAv 
\ sin 3/ + e2 sin 4. 
j cos 3/ — e2 cos 41 -
e3 cos 5/ 
Coefficients of Ae 
f- e2 cos 41 + ez cos 5/ 
244 ORBIT CORRECTION 
Coefficient of Ap 
Obviously, the same procedure may be applied to the solar coordinates; 
AX, AY, AZ may be expressed in te rms of corrections to the elements 
of the earth 's orbit . In this case, however, several impor tant special 
considerations enter. In the first place, the unknown Aaja should be 
omitted. T h e astronomical uni t is used as the uni t of distance for all 
measurements in the solar system, and the dimensions of the ear th 's 
orbit in terms of this uni t are so well known that nothing can be gained 
by the introduct ion of an unknown to correct the scale of the ear th 's 
orbit . In the second place, the corrections AI/J are closely related to the 
correction to the obliquity of the ecliptic and the equinox correction. 
Let Ae be the correction to the obliquity of the ecliptic used in the 
calculation of the equatorial rectangular coordinates of the sun, and AE 
the equinox correction, i.e., the common correction that should be 
applied to all r ight ascensions. F rom (4) with Q' = 0 it is evident that 
Atp1 = Ae, 
AI/J2 = — sin eAoj', 
J03 = — AE + cos eAco'. 
Instead of using Aipv Aift2, AI/JZ it may be advantageous to use 
|| A+[ J0 2' 4 $ || = || d ^ J 0 2 4 £ 3 II 
1 0 0 
= || AI/J1 Aip2 Atp3 || 0 cos e — sin e 
0 sin e cos e 
Coefficient of Aq 
| cos 21 - e2 cos 3/— e3 cos 41 
R E C T A N G U L A R C O O R D I N A T E S 245 
T h u s 
or 
= cos eAip2 + sin eAi/j3y 
= — sin eAip2 + cos SAI/J.. 
= Ae, 
= — sin eAE, 
= — cos eAE + Aw. 
I t will be noted that , as Q' = 0, Aa>' is identical with Zla/. 
These represent rotations about the vernal equinox, the summer 
solstice, and the nor th pole of the ecliptic, respectively. T h e principal 
advantage of AI/J2', Aifj3 over Aifj2y Aifj3 is the bet ter separation between 
Aw and AE. In AI/J2, AE appears with the factor — sin ey while in 
Aifj3 it appears in linear combinat ion with cos eAw. 
T h e relations to be used are evidently 
Set V 
Al'0 + AI/J3 Aifjx AI/J2 e!Aifj3 Ae' 
AX = 4 0 + Z \ ( - X - - Y ) H'X + K'X 
Y 1 / Y \ 
AY = 4 -Z 0 ±l-JL+x) H'Y + K'Y 
n e \ n J 
AZ = ~ +Y -X \ ( - — \ H'Z + K'Z 
If Aifj2'y AI/J3 are introduced, we obtain 
Set VI 
Al'0+A^ Ah A^ e'J0 8 Ae' 
AX= 4 0 - Y sin e + Z cos e \- (- 4 - Y cos e - Z sin e) H'X + K'X 
n e \ n I 
AY = 4 — Z + X s i n £ J - ( _ Z + ^ cos e\ H'Y + K't 
n e \ n 1 
AZ = ^ + y - Z c o s e i.(-4 + ^ 8 i n e ) H'Z + K'Z 
n e \ n I 
246 O R B I T C O R R E C T I O N 
However, if ecliptic rectangular coordinates are in t roduced by the 
relations 
Xe — X, 
Ye = Y cos e + Z sin e, 
Z e = — Y sin g + Z cos e, 
and if Ze = 0, it is seen that AX may be simplified (see AXe in Set VI I 
below). If then AYe, AZe are int roduced, the result ing expressions are 
Set VII 
Al^+A+'3 A^ e'A^ Ae' 
AXe = 
AYe = 
AZt, = 
n' 
x± 
n' 
0 
0 
Y P -X, 0 
dXe 
Be 
dYP_ 
0 
In Set VII the coordinates Xe, Ye are the solar coordinates referred 
to the plane of the ecliptic, with the Xe axis in the direction of the 
vernal equinox. For the purpose of calculating the coefficients of the 
observation equations, these coefficients could be conveniently tabulated 
as a function of the day of the year. 
T h e complication arising from the motions of the vernal equinox 
and the sun 's perigee mus t be given proper at tention if observations 
covering a long interval of t ime are to be used. In many applications 
the use of the s tandard equinox and equator 1950.0 is desirable. If Set V 
or VI is used, and the equatorial coordinates and velocity components 
are derived from the solar coordinates tabulated for this s tandard 
equinox and equator, no at tention to this problem is necessary. Set VII , 
with Ze = 0, implies the use of the ecliptic of date. I t is almost necessary 
then also to use the mean equinox of date. If the eight coefficients 
arising in this set are tabulated as a function of the day of the year