Orbital Averages

7/29/2019 Orbital Averages

1/13

Orbital averages

and the secular variation of the orbits

Maurizio M. DEliseo

Osservatorio S.Elmo - Via A.Caccavello 22, 80129 Napoli Italy

Orbital averages are employed to compute the secular variation of the elliptical planetary elementsin the orbital plane in presence of perturbing forces of various kinds. They are also useful as an

aid in the computation of certain complex integrals. An extensive list of computed integrals is given.

Keywords: Unperturbed motion, perturbation equations, third-body perturbations, otherforce laws, orbital averages, loop integrals.

Introduction

We employ orbital averages for the analytical and nu-merical determination of the secular part of the variationof the elements a, e, (semi-major axis, eccentricity, ar-gument of perihelion) of an elliptic orbit due to perturb-ing forces. The derivations are developed thorough an

averaging process of the first-order equations arising fromthe method of variation of the arbitrary constants. Weshall use the formalism of complex variables, as we con-sider only the perturbations acting in the orbital plane.As a byproduct of our work we deduce some useful meth-ods for the computation of certain awkward integrals re-lated to the geometry of the ellipse.

I. UNPERTURBED MOTION

We begin with a short review, in the complex nota-tion, of the principal formulas and results of the two-body

problem employed in this work.The position of a planet on the orbital plane we sup-

pose lying on the complex plane, is given by the vari-able r = r(t), where r = r(t) = x(t) + iy(t). Thenr2 = rr, r being the complex conjugate of r. The realand the imaginary parts of a complex number r are de-noted by R(r) = (r + r)/2 and by I(r) = i(r r)/2.Then R(r) = R(r) = I(ir), I(r) = I(r) = R(ir).

In polar coordinates r = rei = r(cos + i sin ), being the true longitude, measured from the arbitraryfixed axis x. The function r(t) will be known as soonas we found the time dependence of , so that r(t) =r[(t)]exp i(t); we have also

r =

1

r

dr

dt+ i

r. (1)

If we write = k2(M + m) k2M, where M is Sunsmass which we take as unity, k is Gauss gravitationalconstant, the initial value problem

Z r + rr3

= 0, r(0) = r0, r(0) = r0, (2)

is solved if are known four independent integrals of themotion, that can be found introducing the following in-

tegral operations on Z

C(Z)

dt I(rZ) = 0, (3)

H(Z)

dt R( rZ) = 0, (4)

E(Z) dt I(rr) i Z = 0. (5)

We so easily obtain three constant functions, two realand one complex. They are

1. Area integral: C(Z) = 0 I(rr) = i( rr rr)/2 = r2 = c = real const. It follows: dt =r2d/c .

2. Energy integral: H(Z) = 0 |r|2/2 /r= h,h = /2a, r r = (2/r 1/a).

3. Eccentricity vector: E(Z) = 0 r =(i/c)(r/r+ e), e eexp(i), the eccentricity vec-tor, is a complex constant, e is the eccentricity and

is the argument of perihelion.

4. Elliptic orbit in terms of the true longitude:I(rr) = c r = rei = (c2/) ei[1 + e cos( )]1, a is the semi-major axis, c2 = a(1 e2), f is the true anomaly, ei =eieif, df = d. Elliptic orbit in terms of the ec-centric anomaly : r = r() = (a cos + ib sin ae) ei r=

rr = a(1 ecos ) (origin of coordi-

nates at the center of the ellipse), rd = andt.

5. Third law: cT =

T

0c dt = I

T

0rr dt

=

I (rdr)

=I2

0 r

() dr

()

= 2ab =T2/a3 = 42/.

6. Other definitions: b = a

(1 e2), n2 /a3(n is the mean motion), T = 2/n is the period ofmotion. The mean longitude is defined as nt.

II. ORBITAL AVERAGES

In presence of perturbations, each orbital element Ei a, c, e, becomes a function of time, and the perturbation


2/13

2

equations are first-order equations for the elements of theform

Ei = g(r, r, r, r, t

). (6)

To obtain the secular part of the perturbations of theelements we average with respect to time the right-handside of the perturbation equations, and obtain so the sec-

ular values

Ei

. For this, we need to know the meanvalue of some orbital variables and functions over the un-perturbed motion, where with the word orbital we meanperiodicity sharing the same period of the elliptical mo-tion. By definition, the temporal average of a periodicfunction g(t) over the periodicity interval (0, T) is

g(t)av g(t) = 1T

T0

g(t) dt. (7)

If g(t) is a total derivative of a periodic function h(t),

g(t) =

dh(t)

dt , then g(t)av =

h(T)

h(0)

T = 0,

because of the periodicity of motion. If g is a constant,then g = g. The temporal averages can be also calcu-lated by means of angular variables in the range [0, 2]employing the following relations:

dt

T=

ndt

2=

rd

2 a=

r2df

2 a b=

d

2, (8)

applied respectively to the functions g(t), g(), g(f), g().In the following we shall give properties, methods of

calculation and averages of the orbital functions we need

to know. We shall use the notation Fav to indicate theaverage force exerted by the perturbing planet, while thenotation Fav is reserved to the successive average withrespect to the orbit of the perturbed planet.

III. THE PLANETARY EQUATIONS

When the motion is perturbed, the equation of motionbecomes

r = rr3

+ F, (9)

where F = F(r, r, r, r, t) is the perturbing force in theplane xy. The force F is central ifF = g(r) r, where g(r)is a real function of r. It is assumed that F is of smallmagnitude as compared to the Keplerian term. There-fore, the planet moves on a weakly perturbed elliptic or-bit. The time scales of variation of its elements are afew orders of magnitude longer than the orbital period.Hence, one might perform the averaging of the quantitiesof interest over fast evolution, the mean anomaly, or anyother angular variable according to the relations (8). Theplanet will move along a variable orbit which at every in-stant t can be described as an osculating ellipse, in which

the orbital elements are supposed slowly changing withthe time. Mathematically this concept can be treatedwith the method of variation of arbitrary constants thatcan be reduced to action, on the generic integral of themotion Ei = Ei(r, r, r, r), of the differential operatord/dt

Ei =

dEidt

Ei

r

dr

dt +

Ei

r

d r

dt = F

Ei

r +

F

Ei

r ,(10)

which means to consider each element as variable and toperform the ordinary time derivatives of the integrals ofthe motion with the convention that1

dr

dt= 0,

dr

dt= F, (11)

for the perturbed motion, evidencing so only the acceler-ations produced by the perturbing forces. Thus we find,from the integrals

c = I( rr), (12)

1a

=1

r r +

2

r, (13)

e = i c

r r

r

, (14)

the following expressions of the planetary equations inthe plane

c = I(Fr), (15)

a =2 a2

R(F r), (16)

e = i

cF + r I(Fr)

, (17)

were we can put

r =i n a2

b

r

r+ e

, r = i n a

2

b

r

r+ e

.

(18)

In the right-hand side of these equations, in the first-order of approximation, all the elements are considered asconstants. The equation for c is useful in the treatmentof central perturbing forces, because then c = I(Fr) = 0,

whereby the first equation for e is simplified to

e =

e

e+ i

e = i c

F = e

e+ i = i c

eF. (19)

Equating separately the real and the imaginary part weget

e = c eR

i F

e

=

c e

I

F

e

, (20)

= cI

iF

e

= c

R

F

e

. (21)


3/13

A Third-body perturbations 3

If the force F is central, then F = Ke, with K realconstant. From this we deduce that

e = c eI

Fe

= c e

I(K) = 0, (22)

so that we can write

e = i , = c

e F = c

K. (23)

In this circumstance the eccentricity vector e rotates uni-formly about the origin in the r-plane, with a constantlength. In general we can write, by averaging,

a = 2 a2

TR

F dr

=

n a2

R

F dr

, (24)

e = icF i

T

I(Fr) dr

= icF i n

2

I(Fr) dr, (25)

in which there appear closed contour or loop integrals2over the unperturbed ellipse. From Eq. (24) we see that ifF is central then a doesnt have secular terms, becausethe integral is a pure imaginary number.

IV. SECULAR PERTURBATIONS

A. Third-body perturbations

Suppose that another planet P is moving on a copla-nar orbit around the Sun in the hypothesis that the sys-tem P, P be non-resonant, so that the respective meanmotions are non-commensurable. If we denote with r, r

the position vectors of P and P, and with the quan-tity k2m, being m the mass of P, the perturbing forceon P is given by35

F =

r r

|r r|3 r

r3

. (26)

The first term is the direct force of P on P, while thesecond term is the inertial indirect force due to the choiceof the Sun as origin of the reference frame.5 The deter-mination of the secular perturbations requires a doubleaveraging process: first, we average the perturbing force

with respect to P

and obtain thus Fav, after we averagerespect to P the perturbation equations for a, e after thesubstitution ofF with Fav, obtaining thus a , e. Thisprocedure is allowed for the first-order perturbations, be-cause r does not contain terms depending from P. No-tice that the indirect term of the force gives a null con-tribute to any secular perturbation, since

rr3

= 0.

We have, by definition

Fav =

2ab

20

r r

|r r|3 r2 df =

T

T0

r r

|r r|3 dt,(27)

and it follows Gauss theorem, because with

d = r2 df

2 a b=

dt

T, (28)

we can write

Fav =

r r

|r r|3 d,

where the integral is taken along the ellipse of the dis-turbing body in the direction of motion. So the problemis reduced to that of determining the secular orbital ef-fects of a massive ring whose elementary distribution ofmass has the measure given by Eq. (28).68 For the ana-lytical determination of this force, we must approximatethe irrational factor

|r r|3 = (r2 + r2 2rr cos( ))3/2= r3

1 +

r2

r2 2 r

r

cos( )3/2

r33/21 , r > r, (29)

r3

3/22 , r > r

, (30)

where 3/22 is as

3/21 , but with rand r

interchanged.We develop these expressions in powers of the ratios

r/r or r/r. This requires, when applied to planetaryperturbations, a great number of terms for an acceptableconvergence, but the hard work is done by computer alge-

bra. We can also write 3/2i =

1i

1/2i and expand

in powers ofr/r or r/ronly the second factor. These twochoices, when applied to the same problem, are equiva-lent, but they codify differently the information on theperturbing force in their succession of terms. For ourpurposes it is more advantageous to use the first optionin the later treatment of an elliptical perturbing orbit,

while for the circular orbit we shall use the second one,that has the advantage to give more compact expressions.

B. Average force of a planet on a circular orbit

Let be a planet P of mass m in a circular orbit, sothat r = aei

= aeint, where n is the constant mean

motion. Consider the force Fe on the point r = rei ofthe orbit of an internal planet P lying in the same plane.With the notation

= r/a < 1, = 1 + 2 2 cos , (31)

=

, d = d

, (32)we have

Fe =

aei

r

|aei r|3 =

ei ei

a2 r

a3

1

3/2

. (33)

Therefore, averaging with respect to P

Feav =1

2ab

20

Fexr2d =1

2

20

Fex d

=

2

20

d

3/2

r

r

ei

a2 r

a3

. (34)


4/13

B Average force of a planet on a circular orbit 4

In this integral we write

Feav =

2

20

d

r

r

ei

a2 r

a3

1

1/2

2a2r

r

20

d

ei

1 + cos +

2

4

+3

4

2 cos2 + . . . . (35)From the formula2

0

ein

=

20

cos n

=

2n

1 2 , (36)

with < 0, n = 0, 1, 2, . . . we see that to order 5 inthe numerator only the following terms contribute to thetotal mean force

Feav =

2a2r

r

20

d

[1 3

82 5

644

cos

+ 2

3

4 cos2 + 3

8

2 cos3

2

3

16 3

1285]

(37)

so that, after the integration we have

Feav =

a2

(

64 + 82 + 34)

128 (1 2)

r

r

(38)

=

128 a2

75

(1 2) 11 3 3

r

r

(39)

=

128 a

75 r

(a2 r2) 11r

a2 3r

3

a4

r

r

(40)

It easily seen that also to higher approximations Feavretains the same general structure with the same firstterm and more terms with higher odd powers of the ratior/a. This averaged perturbing force is central, so thatwe can employ the simplified perturbation equation for .As value of the radius a of the circular orbit we take thesemi-major axis of the true elliptical orbit of P becausethe average value of the modulus of the radius vector r

is a to order e. We find

r = 32

a e A e, (41)

rr2

= 5

8a3 (4 + 3 e

2

) e B e, (42)r

(a2 r2)

= a(1e2)(A B) a(A +B)+ 2AB2a e2 AB

e

(43)

C e, (44)

where

A

(a + a)2 a2e2, B

(a a)2 a2e2.(45)

A is real for every positive value of a, while B is realfor a outside a definite interval, that, for the Earth, is

(0.9833..., 1.0167...), where the formula is meaningless.We have then to order 5

Feav =

128

75

aC 11

a3A 3

a5B

e, (46)

and so

= c

e Fe

av , (47)in which is a numerical factor for the conversion fromradian/day to arcsec/century given by

=365.25 100 180 3600

7.533822048 109.

(48)

Then, recalling that / = m, the centennial preces-sion rate is

= mc

128 a

11

aA+

3

a2B+

75

a4C

. (49)

If we expand Feav in powers of , we have

Feav = r

1

2 a3+

9 r2

16 a5+

75 r4

128 a7+ . . .

, (50)

the first term it is proportional only to the position vec-tor r. Thus an external planet exerts an approximatelylinear repulsive force, directed away from the center, ona particle located somewhere near the center of the orbit.We have to push the approximation as far as the termrr10 because of the relative greatness of the ratio r/a forthe internal planets Mars, Earth, Venus and Mercury.In the literature9 was obtained with another method the

approximation (see Eq. (38))

Feav =

2ar

(a2 r2) , (51)

for a numerical evaluation of the classical part of themotion of Mercurys perihelion. To this regard we notethat the computation can be done analytically since weknow that in this case by Eqs. (43), (47) we get

= c e

Feav = c

2

i

iaCi, (52)

with the obvious meaning of the symbolism. We find sothe value of 532.53, very near to the exact value ob-tained by treating the problem in its full generality. Ob-viously this is only a coincidence, due to an almost exactcompensation among the various planets, but it leavesthe false impression that the omitted terms do not de-stroy this excellent agreement. With the more completeexpression of Feav we obtain the value of 553.97. Thedifference is due to the fact that we have neglected theellipticity of the orbits of the disturbing planets, otherthat the mutual inclination of orbits, which in this caseis rather significant.


5/13

C Average force of a planet in an elliptical orbit 5

We consider now the situation in which the orbit of P

is internal to that of P. With

= 1 + 2 2cos , = 1 = a/r < 1, (53)

we have, at the order 3

Fiav =

2

r

r3

2

0

d

3/2

e

i 1=

2

r

r3

20

d

ei 1

11/2

2

r

r3

20

d

ei 1

1 + cos +

2

4

2

r

r3

20

2/4 1

d = r

r3

2 44(1 2)

= rr3

a2 4r24(a2 r2) =

r

r3

(a2 r2) 3r24(a2 r2)

=

4

r

r3

+3

4

r

r

1

(a2 r2) . (54)

By developing in powers of we find

Fiav = r

r3+

3

4a2

r

r5+ . . . . (55)

Now we have (m /)

= c e

Fiav

= 3

4mc

ei

a2 r2

, (56)

whereei

a2 r2

=a(1 e2)(A + B) a(A B)

2a ae2 ABe D e,

(57)

with A, B defined as before, so that

int = 3

4mcD. (58)

C. Average force of a planet in an elliptical orbit

Let us suppose the orbit of P elliptical. Then Favdepends also from the mutual geometrical disposition ofthe two orbits, i.e. from the angular distance ofthe respective perihelia.

If P is external, with = f

f+

, = r/r, we

have

Fe = r r

|r r|3 =(r r)

(r2 + r2 2 rr cos )3/2

=(r r)

r3 (1 + 2 2cos )3/2

(r r)

r3

1 +

9

42

+225

644 + + 3cos + 15

42 cos2 + . . .

= (r r)j

Hj cosj, j = 0, 1, 2 . . . (59)

where the coefficients Hj are power series in , given,with = 3/2, by

H0 =1

r3

(1 + 22 + 24 + . . .

), (60)

H1 =2

r3

( + 3 + 5 + . . .

), (61)

H2 =3

r3 (2 + 4 + 4 + . . . ) , (62)

H3 = . . . (63)

with

=(+ 1)

2!, =

( + 1)(+ 2)

3!, . . . (64)

By developing the above expression of Fe after the sub-stitution

cosj =rj

rj + rjrj

2rjrj, (65)

we can write the force as the sum

Fe

= F0 + F1 + F2 + F3 + . . . , (66)

where

F0 =j

h0j rr2j 1

r2j+3

, F1 =j

h1j r2j r

r2j+3

, (67)

F2 =j

h2j r2

r2j r

r2j+5

, F3 =j

h3j rr2j r

2

r2j+5

, (68)

F4 =j

h4j r3

r2j r

2

r2j+7

, F5 =j

h5j r2

r2j r

3

r2j+7

, (69)

and so on. The coefficients hij (i, h = 0, 1, 2 . . . ) arerational numbers. After the average with respect to P,Feav will have terms proportional to

ek

a2j+nkrir2j ,

ek

a2j+nkrir2j.

where n = 3, 5, 7 . . . . From this we are lead to thefollowing conclusions as regard to Feav:

All terms go to zero as the ratio ai+2ja2j+nk

fori,j,k,n .

F0av is a force of the central type that coincides(with e = 0) with the expression of Feav alreadyfound for the circular orbit.

F1av gives the contributes of order e, while all thesuccessive terms give contributes containing theproduct of powers of e, e.

F2av and F3av, F4av and F5av, and in general Fiav andFi+1av give contributes of the same order of great-ness, so that they must be calculated always to-gether.


6/13

C Average force of a planet in an elliptical orbit 6

As regard to a, the demonstration that at the first orderthis element does not have secular terms arising from theplanetary perturbing force here is immediate only for thecentral part F0av. The other forces F

iav will give little

positive and negative contributes, that in the long runcounteract themselves.

Now we insert Feav in the perturbation equations andwe get, for the secular part,

a = a2n

R

Feav dr

, (70)

e = ic

Feav i n

2

I(Feav r) dr, (71)

and, dividing by e,

ee

+ i = ic e

Feav i n2 e

I(Feavr) dr, (72)

so that we must find Feav and loop integrals of the form

rir2jdr,

rir2jdr, (73)

I(ri1r2j+2

)dr,

I(ri+1r2j

)dr. (74)

Considerations of the same type can be made when theorbit of P is internal. Then we have the force

Fi

= FO + FI + FII + FIII + . . . (75)

FO =

jkoj

r

r2j+3

r2j , FI =

jkIj

1

r2j+3

rr2j,

(76)

FII =j

kIIj1

r2j+3

rr2j, FIII=

j

kIIIjr2

r2j+5

rr2j ,

(77)

FIV =j

kIVjr

r2j+5

r2

r2j , FV =

j

kVjr2

r2j+7

r3

r2j ,

(78)

and so on, where kNj are rational numbers. Then the

typical terms of Fiav are

ek

a2j+k r

i

r2j+ni , ek

a2j+k r

i

r2j+ni . (79)

All terms go to zero as the ratio ai+2ja2j+nk

for fori,j,k,n .

FOav is a force of the central type that coincides(with e = 0) with the expression of Fiav alreadyfound for the circular orbit.

FIav gives the contributes of order e, while all thesuccessive terms give contributes containing theproduct of powers of e, e.

FIIav and FIIIav , FIVav and FVav, and in general FNav andFNIav give contributes of the same order of greatness,so that they must be calculated always together.

At last, we must find

Fiav

and loop integrals of the form

ri dr

r2j+n1,

ri dr

r2j+n1, (80)

I

ri1

r2j+n3

dr,

I

ri+1

r2j+n1

dr. (81)

V. APPLICATIONS

After the formal development of the planetary perturb-ing force, we are left with the practical application of theformulas. For the secular perturbations in the planetaryproblem,10,11 we have some possibilities, each dependingfrom the concrete problem at hand. So, after the choiceof the order of approximation, we can proceed first sym-bolically, and then we shall have the characteristic struc-

ture of each of the particular terms considered, and afterthe successive numerical determination we shall have thecontribution to the secular variation of the same term.All this it is possible because we are in a linear environ-ment: first-order perturbations, integrations of a sum ofelementary terms, real and imaginary parts determina-tions. We can verify the work done with a direct de-termination of a and e. For this it is required thenumerical computation of the double integrals

a m

22ab b

20

20

R(F r) r2r2df df, (82)

i m

4 2a ab b 2

0 2

0 r I(rr)

|r

r

|3

+c (r r)

|r

r

|3

r2r2df df,

(83)

where

r =i

c

r

r+ e

. (84)

At last we get, for a century,

asec = 36525 a , (85)

esec = 36525 e R

e

e

, (86)

sec =

I e

e . (87)

For a numerical verification of the method applied to themotion of the perihelion, we preferred to consider theEarth instead of the usual Mercury, because the copla-narity of the orbits involved is best verified for the formerplanet. The results for the Earth to order 5 are given inthe following tables, where we have employed the plane-tary elements given in the Appendix for the epoch Jan-uary 1, 2000. In the second table, relative to the Earthsperihelion in arcsec/century, the first column is referredto the full classical approximation, while the others giverespectively the contributions, computed by our method,of the circular (by Eq. (50)) and elliptical parts, and their


7/13

A Other force laws 7

sum. The diction Theory is referred to complete pub-lished calculations, which however make reference to aslightly different epoch.12

Planet Theory(A) e = 0 e = 0 Total (B) (B-A)

Mercury -13.75 3.30 -15.01 -11.71 2.04

Venus 345.49 508.73 -151.97 356.76 +11.27

Mars 97.69 21.05 75.52 96.57 -1.12

Jupiter 696.85 709.79 -12.95 696.84 -0.01

Saturn 18.74 32.82 -12.85 19.97 +1.23

Uranus 0.57 0.60 -0.03 0.57 0.00

Neptune 0.18 0.18 0.00 0.18 0.00

Total 1145.77 1276.47 -117.29 1159.18 +13.41

Secular motion of the Earth perihelion

We have a rather good agreement between the two sets ofresults. The discrepancies are mainly due to: 1) to havingneglected the non-coplanarity of planetary orbits, 2) tohaving used elements related to different epochs, 3) theorder of approximation considered. As a further example

we consider the secular advance of the perihelion of Mars.Here Theory refers to the computation of Doolittle.13

Planet Theory(A) e = 0 e = 0 Total (B) (B-A)

Mercury 0.62 0.64 -18.01 -14.71 -0.96

Venus 49.48 47.52 -2.2 45.32 -2.2

Earth 229.03 208.73 6.03 214.76 -14.27

Jupiter 1247.24 1468.28 -214.38 1253.90 +6.66

,Saturn 66.77 63.29 3.40 66.69 -0.08

Uranus 1.20 1.13 0.07 1.20 0.00

Neptune 0.34 0.34 0.00 0.34 0.00

Total 1594.67 1789.90 -207.75 1582.15 -12.52

Secular motion of the Mars perihelion

with the same restrictions as in the previous table. Inconclusion, are worth noting the significant correctionsto the secular motion of the perihelion due to the pres-ence of ellipticity of the perturbing planets and to theneglect of the relative inclinations of the orbits. It is alsoevident that Venus and the Earth, for their respectiveproximity to Mercury and Mars, would require the intro-duction of more higher-order terms in the developmentof their disturbing force.

A. Other force laws

We examine now the effects of some other forcelaws,14,15 beginning from:

General relativity (GR). The general relativisticcorrection to the gravitational law in the first approxi-mation can be obtained introducing modifications to theclassical equation of the motion. If we put = |r|/c,where c is the speed of light in vacuum (173.144 AU/day),

and set

1 2, GR requires the following correc-

tions to t, m, r:16

t t01 t0

1 +1

22

= t

1 +

r

, (88)

01 0

1 +1

22

= 0

1 +

r

, (89)

r

r0

r0 1

1

2

2 = r0 1

r , (90)

to the order 2, where the last is a pure general rela-tivistic effect because involves the radial distance r, andwhere we have put 2 |r|2/c2 = (2)/(c2) = 2/r,with /c2 defining the gravitational radius of themass M.

We make these substitutions in the equation of motionby dropping the zero suffixes

r + r

r3 r(

1 + r

)2 + (

1 + r

)r

r3(

1 r

)3 = 0, (91)and, to the O() we have:

r + rr3

1 + 2

r

1 +

r

1 + 3

r

= 0, (92)

r +r

r3= 6r

r4= F. (93)

The perturbing force is central, withrr4

= e/(2 b3),

so that

= c e

F = 3 cb3

= 3 n a b

c2b3

= 3 n

c2 a(1 e2) (94)

For the centennial motion we have

=

T

0

3 nc2a(1 e2) dt =

6 c2a(1 e2) . (95)

We see from the above developments that the variationswith the speed of the planet of time, mass and radialdistance give a respective contribution of 1/3, 1/6 and1/2 to the perihelion precession.

Almost inverse-square law. Let us suppose thatthe gravitational law goes as r(2+), where 0 < 1.Then by expanding in powers of

1

r(2+) 1

r2 Ln(r)

r2+ . . . , (96)

and we have the perturbing central force

F = Ln(r)r2

r

r. (97)

The secular perihelions motion is given by

av = c

eF = nab

e

Ln(r)

r2ei

,

= 2ab

nabei

e

20

Ln(r)

r2r2eifdf (98)

= n

2e

20

Ln(r) eifdf (99)


8/13

A Other force laws 8

Now

Ln(r) Ln(a) + e cos f e2

5

4+

cos2f

4

+ ...,

(100)

so

n

1

2+

1

8e2 +

1

16e4 +

5

128e6 +

7

256e8 + ...

.

(101)Let us calculate for a tentative explanation of the non-

classical perihelion shift of Mercury. We find

42.95 = 2.71937 108 = = 1.579 107. (102)Webers Law. Now we briefly study a proposed alter-

native to the Newtons law, Webers law,17 and apply itto Mercury perihelion. This law it is interesting, otherthan for historical reasons, also because it introduces ad-ditional terms, containing the temporal derivatives of r,to the inverse-square law. By equating the two expres-sions of r we find

dr

dt + i

c

r =

i

c +

i

c e ei

, (103)

and, taking the real part

dr

dt=

c

R(

i e ei)

=

c

esin( ) = c

esin f,

(104)

d2r

dt2=

r2ecos( ) =

r2ecos f, (105)

because, from the area integral

d

dt=

c

r2d

df. (106)

Webers law is

F =

1 +1

c2

dr

dt

2 2

c2r

d2r

dt2

r

r3, (107)

where c is the speed of light. Substituting for the deriva-tive and introducing the true anomaly f = we have

F = e 2

c2r2

e

2c2 2

rcos f +

e

2c2cos2f

eif,

(108)

and we find

Fav =

e

2ab

2

0

2

c2 e

2c22

r

cos f+e

2c2cos2feifdf

= 2

c2b3e, (109)

a complex constant. The secular perihelion motion isgiven by

= c e

Fav = n c2a(1 e2) , (110)

and numerically for Mercury

= n

c2a(1 e2) = 14.32. (111)

B. The lunar apse

As last application of the method of the averages, weapply them to the derivation at order m3 of the partof the motion of the lunar perigee independent from theeccentricity, where m = n/n is the ratio of the meanmotions of the Sun and the Moon, in the hypothesis ofthe main lunar problem (the Sun in a circular orbit inthe same plane of Moons orbit). The perturbing force

F =

r r

|r r|3 r

r3

, (112)

with r = aei

= aei

becomes, with /a3 = n3 andneglecting the solar parallax,

F = n2(r r)

1 + 3r

acos( ) + . . .

n2r

n2r + 3 n2r ra

cos( )

= n2r + 32

n2ei

r

ei(

) + ei()

= 1

2n2r + 3

2n2re2i

= 12

n2r + 32

n2re2i

. (113)

In the unperturbed motion at order e, we have for theMoons orbit in terms of the mean longitude

r0 = aei 3

2a e +

1

2a e e2i. (114)

In the first approximation, by solving the perturbationequations, we find the evection, given by the followingterms18

r = 4516

a e m e2i

+15

16a e m e2(

). (115)

In the second approximation we put r = r0 + r in theequation

e = i

{I(rr)F + r I(Fr)

}, (116)

with r = ndr/d. We cannot use Fav in this equation,because in r, r are present terms containing . We find,at order e, the following constant terms

in2a3

I(rr)F = i

3

4m2 +

135

32m3

n e, (117)

i

n2a3r I(Fr) = i

45

16m3n e, (118)

and periodical functions of , that give a zero contribu-tion to the double average. Thus

e = 14 2

20

20

e d d = i

3

4m2 +

225

32m3

n e,

(119)

so that from Eq. (23)

e = 0, =

3

4m2 +

225

32m3

n. (120)


9/13

A Analytical Methods 9

The m3 term is that found for the first time numericallyby Clairaut and algebraically by DAlembert in their re-spective theories of lunar motion, and this solved an ap-parent problematic aspect of the lunar orbit evidencedsince the publication of the Principia of Newton.

VI. ORBITAL AVERAGES - B

A. Analytical Methods

Some averages are immediate. So

r = 1T

T0

dr

dtdt =

1

T

dr = 0, (121)

because this is a contour integral over a closed orbit.In general for the analyticity of the integrands we havernr = 0, n 0. From the orbital expression of r wehave at once

r

r

=

ei

= ei

eif

= e eif = e,(122)

rn

r

= rn1 e, (123)

rn

rn+2

=

ein

2ab

20

einfdf = 0, n = 1, 2 . . . .

(124)

Many averages we must compute are of the type

rmrn m, n , (125)with m, n 0 positive integers. We can limit ourselvesto take m 0, because every combination of r, r canbe reduced to one of the precedent type employing theequality r1 = r/r2:

rm

rn = rmrmrmrn = rmrn2m =rmrn2m.

(126)

An important property of these average is that they areof the form

rmrn

= Kem, (127)

where K is real, because we havermrn

=

e eim

2ab e

g(r) eimfdf, (128)

where g(r) is a periodical even function of f, and the in-tegral of the imaginary part of g(r) eimf is zero, becausethis function is odd. In particular, ifF is ofcentral type,F = g(r)r, we have F = Ke.

It is also evident that in generalrmrn

= eim

eimfrmn

emamn. (129)

The calculations of an average will be done by adoptingthe more convenient variable for the situation at hand.

If n = 0, it will be used the eccentric anomaly

m, 0 = ameim

2

20

(1ecos )(cos e+i

1e2 sin )md.

If m = 0, it will be convenient employ the eccentricanomaly for n > 0, and the true anomaly for |n| < 2. Sowe have, for (0, n) and for (0,

n)

rn = 12a

20

rn+1d =

an

2

20

(1 ecos )n+1d,rn = 1

2ab

20

df

rn2

=an3

2b2n3

20

(1+e cos f)n2df.

For m + n + 2 > 0 the computation with respect to thetrue anomaly requires the integral

eim

2ab

20

eimfrm+n+2df, (130)

with r = a(1

e2)/(1 + e cos f), that it can be done by

means of the repeated use of Cauchy integral formula2for the derivatives. If we substitute in the integral

eif s, ecos f e(s2 + 1)

2s, df ds

is, (131)

we obtain

1

2iab

[2a(1e2)

e

]m+n+2 |s|=1

s2m+n+1

[(sp)(sq)]m+n+2 (132)

where |s| = 1 is the unitary circle centered at the originand where

p =

1 e

2

1e , q =

1 e2

+ 1e ,

(p q) = 2

1 e2e

. (133)

are the solutions of the equation in s

s2 +2

es +

1

e= 0. (134)

The pole within the circle |s| = 1 is p, of order m + n + 2.Then by Cauchy integral formula

|s|=1

f(s) ds

(s p)k=

2i f(k1)(p)

(k 1)!, (135)

we can write

(m, n) =1

ab

[2a(1 e2)

e

]m+n+2f(m+n+1) (p)

(m + n + 1)!,

(136)

with

f(s) =s2m+n+1

(s q)m+n+2 . (137)


10/13


This formula fails for m + n + 1 < 0, and in this circum-stance we resort to the calculation of

ein

2ab

20

df einf[

1 + ecos f

a(1 e2)](m+n+2)

, (138)

for m n + 2 > 0. Some results:

n,

n

= (

1 e2 1)

n

(n

1 e2 + 1)e2n

en, (139)

n,(n + 1) =(

1 e2 1)nae2n

en, (140)

n,(n + 2) = ein

2ab

20

einfdf =

{1ab n = 0

0 n > 0,

(141)

(142)

1,4 = e2b3

, (143)

1,5 = a eb5

, (144)

1,6 =(12 + 3e2)a2e

8b7 , (145)

1,7 = (4 + 3e2)a3e

2b9, (146)

0,1 = 1a

(147)

0,2 = 1a2(1 e2)1/2 , (148)

0,3 = 1a3(1 e2)3/2 , (149)

0,4 = e2 + 2

2a4(1 e2)5/2 , (150)

0,

5

=

3e2 + 2

2a5

(1 e2

)7/2

, (151)

0,7 = 15e4 + 40e2 + 8

8a7(1 e2)11/2 , (152)

0,9 = 35e6 + 210e4 + 168e2 + 16

16a9(1 e2)15/2 , (153)

0,11 = 315e8 + 3360e6 + 6048e4 + 2304e2 + 128

128a11(1 e2)19/2 ,(154)

r =

rei

=

reif

ei, (155)

rcos f = rcos( ) = Rr ei

= 3

2a e,

(156)

rsin f = rsin( ) = Ir ei

= 0. (157)

Sometimes it is possible to obtain the same result moreeasily by means of a clever use of already known relations.So, from the immediate averages

1

r

=

1

2a

20

d =1

a, (158)

r

r

=

1

T

T0

dr

dt

dt

r

=1

T

dr

r

=2i

T= ni,

(159)

we have from the orbital expression of r

1

r

= 1

e

1

r

ic

e

r

r

= 1a e

+cn

e=

1 e2 1

a e. (160)

From the expression

rmrn = ei

2 a

20

[a cos +ia

1e2 sin a e

]m

[a(1 e cos )]n+1 d (161)

with m, n 0, we find

(0, 1) =a (2 + e2)

2, (162)

(0, 2) =a2 (2 + 3e2)

2, (163)

(1, 0) =

3

2a e, (164)

(1, 1) = a2(4 + e2)

2e, (165)

(1, 2) = 5a3(3e2 + 4)

8e, (166)

(1, 3) = 3a4(8 + 12e2 + e4)

8e, (167)

(1, 4) = 7a5(5e4 + 20e2 + 8)

16e, (168)

(1, 6) = 9a7(35e6 + 280e4 + 336e2 + 64)

128e, (169)

(1, 8) = 11a9(63e8 + 840e6 + 2016e4 + 1152e2 + 128)

256e.

(170)

(1,1) = e (171)

(1,2) =

1 e2 1ae2

e, (172)

(1,3) = 0, (173)(1,4) = e

2 b3, (174)

(1,5) = a eb5

, (175)

(1,6) = 3 a2(4 + e2)e

8 b7, (176)

(1,

7) =

a3(

4 + 3 e2)e

2 b9

, (177)

(1,9) = 3 a5(

5 e4 + 20 e2 + 8)e

8 b13, (178)

(1,11) = a7(

35 e6 + 280 e4 + 336 e2 + 64)e

16 b17, (179)

(2, 0) =5

2a e2, (180)

(2,5) = 0, (181)

(2,7) = 3 e2

4a5(1 e2)7/2 , (182)

(2,9) = 5(e2 + 2)e2

4a7(1 e2)11/2 . (183)


11/13


ei

a + r

=

a(1 e2) A + aa e2 A

e, (184)

ei

a r

=

a(1 e2) + B aa e2 B

e, (185)

r

a2 r2

=1

2

ei

a r ei

a + r

(186)

=a(1 e2)(A B) a(A + B) + 2AB

2a e2 ABe,

(187)ei

a2 r2

=1

2a

ei

a r +ei

a + r

(188)

=a(1 e2)(A + B) a(A B)

2a ae2 ABe, (189)

with

A

(a + a)2 a2e2, B

(a a)2 a2e2. (190)

Loop integrals of the type

f(r, r) dr and

f(r, r) drover an ellipse in the complex plane are present whenthe expression to be averaged contains r or r. This is

accomplished by using the identity r

dt = dr

in the av-erages f(r, r) r and f(r, r) dr/dt, together with theexpressions

T =2

n, r =

i

c

r

r

+i

ce, r r =

2

r

+1

a

,

(191)

dr

dt=

1

2r

(r r + r r

)=R(r dr)

rdt= ic

r

+i

c+

i

c

r

r

e.

(192)

These are easily computed when is is possible touse, in the integrand, the eccentric anomaly in orbits

parametrization. We put

r = (a cos + ib sin ae) ei, (193)r= a(1 e cos ), (194)

dr = (a sin ib cos ) eid. (195)

It is immediate to see that, when the force F is central,the integral

F dr is a pure imaginary number, because

F dr =

g(r) r dr =

20 g(

r

)[ (b2

a2)

2 sin2a2

esin i(abecos +a b)]d,(196)

and, since g(r) is an even function of , the terms con-taining sin , sin2 are zero in the integration. In generalwe have, with n relative integer,

rmrndr am+n+1em+1 m 0, (197)

rmrndr am+n+1em1 m 1. (198)

Some examples:r

dr

dt

=

1

T

r dr =

1

2T

r2

r

dr +1

2T

rdr,

(199)r dr = i a b e,

rdr = i a b e, (200)

r2

r

dr = 2 r dr rdr = 3 i a b e, (201)and

r

dr

dt

=

1

2n i a b e. (202)

Curve rectification. From

|dr|dt

=

2

r

a, (203)

|dr| = 2n

2a3

20

2 ar r2 d

= a 2

01

e2 cos2 d = ellipse lenght.

(204)

Some other integrals:dr

r

=2i(

1 e2 + e2 1)e2

1 e2 e, (205)dr

rn

=ni

[a(1 e2)]n1 e, n = 2, 3. (206)rn

rn

dr = (e)n+1 2nia(1

1e2)n((1e2)3/2+ e21)e2n+2

1 e2 .

(207)

In these expressions we may assume that = 0, e = e,i.e. that the semi-major axis of the ellipse lies on the realaxis.

rrdr = i a2b(

e2 2) , (208)rr

2dr = i a3b(

e2 2) , (209)rr

4dr =1

4 i a5b

(9e4 8e2 8) , (210)

rr6dr =

1

8 i a7b

(25e6 + 30e4 72e2 16) , (211)

r2dr = 4 i a2b e, (212)

r3dr = 15

2 i a3b e2, (213)

r4dr = 14 i a4b e3, (214)

r5dr = 105

4 i a5b e4, (215)

r2dr = 2ia2b e, (216)

r4dr = ia4b (3e2 + 4) e, (217)

rndr = 0 because

T0

rn

rdt =

[rn+1

n + 1

]T0

= 0.

(218)


12/13


Loop integrals of the formrndr

rm,

rndr

rm, (219)

can easily be transformed to an orbital average in thetrue anomaly we already have considered. For example,

rnrm d

r =T0

rn

rm r dt =

1

c20

rnr

rm2 df

=i

c2

20

rn

rm2

r

r+ e

df

=i

c2

20

rn+1

rm1df +

i e

c2

20

rn

rm2df.

(220)

The independent variable f of each of these integrals canbe transformed, if more convenient, in by means of therelation df = (b/r) d. This is an interesting by-productof the calculus of orbital averages: the possibility of alter-nate computations of certain integrals. Since an angularaverage of a finite two-body expression can be done in twoways, with f, as independent variables, we can choicethe most convenient for the calculation, and automati-cally we have the result of the corresponding integral inthe other variable. We denote as dual the two correlatedexpressions. In practice, the method is the following:given the integral of an expression derived from a two-body orbital function, we can interpret it as an average inf or , and after the computation, made with the moreconvenient variable, we have also the value of the dualintegral in or f obtained by using the relations

df =

b

r d, d =

r

b df. (221)

We give here only the simplest example, which non re-quires any integration at all. With > , / = e < 12

0

d

( + cos )2=

1

2

20

d

(1 + e cos )2

=1

2a2(1 e2)220

r2d =2a2

1 e2

2(1 e2)2 1

=2

2(1 e2)3/2 =2

(2 2)3 . (222)

Last, the Greens theorem on the complex planeD

f

rdxdy =

1

2 i

D

f(r, r) dr, (223)

where D is the simply connected domain bounded bythe unperturbed elliptic orbit D, gives immediately, foreach computed value of a line integral, the value of asurface integral over the domain and viceversa, given afunction g(r, r), we first integrate it

g(r, r) dr = f(r, r), (224)

and after we calculate the line integral of f.Some examples:

D

dxdy =

D

r

rdxdy =

1

2i

D

r dr = a b ,

(225)

Dr

r3

dxdy =

dr

r

= (1 e2 1 e2)

e

1

e2

.

(226)

All these results are purely geometrical, without refer-ence to the underlying dynamical problem we employ assolution device.

VII. DISCUSSION

We have given a rather consistent number of exam-ples of application of the method of the orbital averages,so it is now possible to draw conclusions about its prosand cons. From the theoretical point of view, it reduces

the secular perturbation of two gravitationally interact-ing bodies in complex geometric situations to a succes-sion of ever smaller simple forces each of which provides acontribution that can be computed exactly. Having avail-able a literal expression deepens our knowledge about thevarious factors that help produce the final result. An im-portant point to emphasize is that, with the symbolicformulas, the work can be done once and for all becausein actual cases will suffice replacing the various symbolswith the numerical values of the orbital parameters. Wehave also harnessed the power of complex analysis to ob-tain our results in an elegant way. This also paradoxicallyconstitutes the major drawback of the method: it works

under conditions of coplanarity or, at most, in situationsof almost coplanarity, but we believe that in its scope itallows to obtain interesting results, especially in the cal-culation of the effects of gravitational forces arising fromalternative theories to general relativity. In conclusion,we think that this method represent a useful working toolthat can provide valuable services to the researcher in awide field of study.

VIII. APPENDIX

Tables of the planetary orbital elements.? The figures

are rounded to the fourth decimal.

Jan 1, 2000 a b e (rad.)

Mercury 0.3871 0.3788 0.2056 1.351870079

Venus 0.7233 0.7233 0.0067 2.295683576

Earth 1.0000 0.9999 0.0167 1.796767421

Mars 1.5237 1.5170 0.0934 5.865019079

Jupiter 5.2034 5.1973 0.0484 0.257503259

Saturn 9.5371 9.5231 0.0541 1.613241687

Uranus 19.1913 19.1699 0.0472 2.983888891

Neptune 30.0690 30.0679 0.0086 0.784898126


13/13

13

Jan 1, 2000 m = / c n

Mercury 1.67 107 0.010473 7.142 102Venus 2.44 106 0.014629 2.796 102Earth 3.01 106 0.017199 1.720 102Mars 3.31 107 0.021142 9.143 103Jupiter 9.59 104 0.039236 1.470 103Saturn 2.87 104 0.053046 5.88 104Uranus 4.37

105 0.075179 2.08

104

Neptune 5.18 105 0.094527 1.05 104

Electronic address: [email protected] Battin R.H. An Introduction to the Mathematics and Meth-

ods of Astrodynamics. Revised Edition, Aiaa Education Se-ries, (1999).

2 Lang S. Complex Analysis. Fourth Ed., Springer, (2000).3 Moulton F.R. An Introduction to Celestial Mechanics. 2nd

Ed., Art. 197, p.352, Dover Publications, (1984).4 Murray C.D., Dermott S.F. Solar System Dynamics.

C.U.P. (1999).5 Brouwer D., Clemence G. Methods of Celestial Mechanics.

Academic Press, (1961).6 Boccaletti, D., Pucacco, G. Theory of Orbits. Vol. 2,

Springer, (1998).7 Vashkovyak, M.A. and Vashkovyak, S.N., Force function of

a slightly elliptical Gaussian ring and its generalization toa nearly coplanar system of rings. Solar System Research,Volume 46, Number 1, Pages 69-77, (2012).

8 Iorio, L., Orbital Perturbations Due to Massive Rings.Earth, Moon, and Planets, Online First, 8 May 2012.

9 Price M.P., Rush W.F. Nonrelativistic contribution to Mer-curys perihelion precession. Am. J. Phys. 47, 531 (1979).

10 Brumberg, V.A., Evdokimova, L. S. and Skripnichenko,V.I., Secular perturbations in general planetary theory. Ce-lestial Mechanics and Dynamical Astronomy, Volume 11,

Number 1, Pages 131-138, (1975)11 Migaszewski, C., Gozdziewski, K., A secular theory of

coplanar, non-resonant planetary system. Mon. Not. R. As-tron. Soc. Volume 388, Issue 2, pages 789802, (2008).

12 Clemence G.M., Reviews of Modern Physics 19, 361(1947).

13 Doolittle, E., The secular variations of the elements of thefour inner planets computed for the epoch 1850.0 G.M.T.Transactions of the American Philosophical Society, NewSeries, Vol. 22, No. 2 (1912).

14 Baryshev, Y. and Teerikorpi, P., Predictions of GravityTheories. Astrophysics and Space Science Library, 1, Vol-ume 383, Fundamental Questions of Practical Cosmology,Pages 111-130, (2012).

15 Lin-Sen Li, Parameterized post-Newtonian orbital effects inextrasolar planets. Astrophysics and Space Science, OnlineFirst, 28 April 2012.

16 Ney P., Electromagnetism and Relativity. Harper & Row,New York, (1962).

17 Roseveare, Mercurys Perihelion from Leverrier to Ein-stein. Clarendon, Oxford (1982).

18 DEliseo M.M. The eccentricity vector of the Moon. Chi-nese J.Phys. 3, (2012)

19 Stewart, M.G. Precession of the perihelion of Mercurys

orbit. Am. J. Phys. 73, 730 (2005).20 Van Laerhoven, C. and Greenberg, R., Characterizing

multi-planet systems with classical secular theory. Celes-tial Mechanics and Dynamical Astronomy, Online First,23 April 2012 April 2012.

21 Bombardelli, C., Bau, G. and Pelaez, J., Asymptotic so-lution for the two-body problem with constant tangential

thrust acceleration. Celestial Mechanics and Dynamical As-tronomy, Volume 110, (2011).

22 Lin-Sen Li, Influence of the gravitational radiation damp-ing on the time of periastron passage of binary stars. Astro-physics and Space Science, Volume 334, Number 1, Pages125-130, Number 3, Pages 239-256, (2011).

23 Lara, M., Palacin, J.F. and Russell, R.P. Mission de-

sign through averaging of perturbed Keplerian systems: theparadigm of an Enceladus orbiter. Celestial Mechanics andDynamical Astronomy, Volume 108, Number 1, Pages 1-22, (2010).

24 Dulliev, A.M., Evolution of Almost Circular Orbits ofSatellites under the Action of Noncentral GravitationalField of the Earth and Lunisolar Perturbations. ISSN00109525, Cosmic Research, Vol. 49, No. 1, pp. 7281.Pleiades Publishing, Ltd., (2011).

25 Seidelmann K.P. Ed., Explanatory Supplement to the As-tronomical Almanac. p.316 (Table 5.8.1), University Sci-ence Books, Mill Valley, California, (1992).

Orbital Averages

Documents