Central Forces and Secular Perihelion Motion

8/14/2019 Central Forces and Secular Perihelion Motion

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Central Forces and Secular Perihelion Motion

Maurizio M. DEliseo

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

[email protected]

Can.J.Phys. 85(10):1045-1054 (2007)

Abstract: The first-order orbital equation and its perturbed version in the form of an integro-differential equation provides a new method to study, respectively, the elliptical orbit and the secularmotion of a planetary perihelion due to the perturbing action of an important category of centralforces: the inverse-power radial forces. For this, a general formula is found. The related prob-lems of the gravitational bending of fast particles and light rays are studied using the same methods.

PACS Nos.: 02.60.Nm. 04.25.-g. 46.15.Ff. 45.50.Pk

Resume: Lequation orbitale du premier ordre et sa version perturbee sous la forme duneequation integro-differentielle nous fournissent une nouvelle methode pour etudier respectivementlorbite elliptique et le mouvement seculaire du perihelie planetaire du a leffet perturbateurdune importante categories de forces perturbatives centrales en puissance negative de r. Nousobtenons une formule generale pour ces effets. La meme methode permet detudier la deformationgravitationelle de la trajectoire de particules rapides et de rayons lumineux.

Introduction

The determination of the motion of two interacting bodies is a fundamental problem of celestial mechanics that canbe handled by many different methods. Here, with the help of the Laplace and area integrals, and of the complex-variable formalism, we deduce the first-order orbital equation, which furnishes a new way to obtain the conic solution.A modification of this equation under the form of a nonlinear integro-differential equation gives a simple methodof dealing with an important category of perturbing central forces: inverse power radial forces. The search for anadmissible solution of this equation clarifies the physical origins of the secular perihelion motion of a perturbed ellipse,and provides a general expression for it. By the same formalism, it is easy to study the bending of the paths of fastparticles and of light rays, thus extending the applicability range of the equations to unbounded systems. The problemstreated here, because of the spherical symmetry of the forces involved and of the ensuing angular momentum integral,

are essentially bi-dimensional, so that the three-dimensional formalism of the ordinary vector calculus is redundant.We look instead at framework of complex variables, and identify the plane of motion of the gravitational two-bodyproblem with the complex plane1. An object of mass M is at the origin. The position x, y of the second object ofmass m is given by the complex quantity r = x + iy, and the equation of motion can be expressed as

r = rr3

= ei

r2

, (1)

where = G(m+M), r = rei = r(cos +i sin ), r = rei is the complex conjugate ofr, r = |r| = rr =

x2 + y2,while the polar angle = (t) is the true longitude, that is, the point (x, y) has the polar form rei, where r is themodulus and is its argument. From r = rei we have by time-differentiation

r = r + i r

ei, r = r. (2)

The solution of (1) requires knowledge of the functions r() and (t), but here, we are interested only in the determi-nation of the function r(), which describes the shape of the orbit.

We denote with Re (r) and with Im (r) the real and the imaginary part of r, respectively. Thus Re (r) = ( r+ r)/2 =x = rcos , Im (r) = (r r)/2i = y = rsin , Re (ir) = Im (r), and Im(ir) = Re(r). It is useful to considera complex number z = zei as an oriented vector starting from the origin, of length z and making an angle withthe real axis, so that we will use boldface letters to denote complex quantities, with the exception of the complexexponential ei.


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The first-order orbital equation

If we multiply Eq. (1) by r, we have

r r = r rr3

= r2

r3

= r

. (3)

If we take the imaginary part of both sides of (3), we find

Im (r r) = Im

r

= 0. (4)

It is easy to verify that

d

dtIm (r r) = Im (r r) + Im (r r). (5)

We have Im (r r) = 0 because the term in brackets is the square of the module |r|, a real quantity. Then from (5)and (4) we write

d

dtIm (r r) = 0. (6)

Equation (6) implies that Im (r r) is time independent. We denote its real value by

Im (r r) =r r r r

2i= (7)

Equation (7) is the area integral. This derivation holds for any central force f(r) ei = f(r)(r/r). We substitute (2)

into (7) and find the fundamental relation = r2, which can be cast in three equivalent forms

1

r2

=

(8a)

dt =r2

d (8b)

ddt

= r2

dd

. (8c)

We can rewrite (1) using (8a) as

r =

ei =i

d

dtei, (9)

or

d

dt

r i

ei

= 0. (10)

The expression in parentheses is a complex constant, which for later convenience we denote as ( ie)/. We have thusdeduced the Laplace integral

r =i

(ei + e), (11)

where r is the orbital velocity, e = e exp(i) is a complex constant that we will call the eccentricity vector, and e isthe (scalar) eccentricity. The vector e is directed toward the perihelion, the point on the orbit of nearest approach tothe center of force, and is the argument of the perihelion. To keep the customary notation we use the same lettere for the eccentricity and for the complex exponential. The Laplace integral can be found in ref. 2.To find the orbit, in (11) we transform the time derivative into a derivative. By (8c) we find

r =

r2

r =

i

(ei + e), (12)


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where a prime denotes differentiation with respect to , and we get

r = (rei) = (r + ir)ei =

i

2(ei + e)r2. (13)

If we multiply (13) by ei, we obtain the complex Bernoulli equation5

r + ir =

i

2(1 + e ei)r2. (14)

By taking the imaginary and the real parts of both sides of (14), we obtain the orbit r() and its derivative r(),respectively. However, we find it convenient to work with the reciprocal of r, so we divide both sides by r2, make thevariable change r() 1/u(), multiply by i, and convert (14) into an equation in u

u + iu =

2(1 + e ei). (15)

Despite its heterogeneous nature, it is convenient to write the left-hand side of (15) in terms of the complex quantityu = u() u() + iu(), so that we have

u =

2(1 + e ei), (16)

which we call the first-order orbital equation.

From (16), we can immediately deduce the orbit and its apsidal points. The orbit is given by the real part ofu

Re (u) =1

r

=

2

1 + Re (e ei)

=

2

1 + e cos( ). (17)Equation (17) is the polar equation of a conic section of eccentricity e with a focus at the origin. If the orbit is anellipse (0 < e < 1), we have the relation 2/ = a(1 e2), where a is the semi-major axis (which lies on the apse line).The reader will find the standard treatment of the two-body problem with vectorial methods in ref. 6. Thus, thenames given earlier to |e| = e and are justified. The apsidal points are determined from the condition Im (u) = 0because r() = 0 at these points. If Im (u) = 0 for every value of , then e = 0, and we have a circular orbit withu = Re (u) = /2. I f 0 < |e| < 1, then (16) gives the position of the two apsidal points rmin and rmax. At thesepoints we have

Im (u) =

2

e sin(

) = 0, (18)

so that, by considering the derivative [Im (u)] = u(), we find rmin when +2n = and rmax when +(2n+1) = ,where n = 0, 1, 2, . . . .

Perturbing central forces

Consider the gravitational motion of a body perturbed by a central force of the type F = r(n+2)ei where issmall and n 1 is a positive integer. The equation of motion is

r = ei

r2 e

i

rn+2

. (19)

Formally integrating the above equation with respect to t, we obtain

r =

ei

r2

dt

ei

rn+2

dt,

and, by means of the area integral we get by Eq. (8b)

r =i

ei + k

ei

rn

d, (20)

where k is a complex quantity. Because the function r() is unknown at this stage, we cannot do the last integral, andwe must stop here. So, contrarily to the unperturbed case, with the help of the area integral this expression cannotbe reduced to a total derivative, that is to an integral of the motion.


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To proceed, we observe that when the perturbation is switched off ( = 0), we have k = e. If suddenly at theinstant t = 0 ( = 0) we introduce it, we can suppose that, since is small, for continuity reasons in the followingmotion k will be near e, and that however, starting from this value, it will change slowly. So, in absence of a validalternative, we tentatively put k = e. In these conditions, we could call Eq. (20) the Laplace quasi-integral. Theutility of this concept is implicit in the consequences of its use.

By means of the same developments used to obtain the first-order orbital equation, from the Laplace quasi-integralit is a simple matter to deduce the following perturbed first-order orbital equation, which is formally the nonlinearintegro-differential equation

u =

2

1 + e ei

+i ei

2

uneid. (21)

Since we have here introduced the same constant e of the unperturbed motion when = 0, we will be forced to makesome maintenance work to avoid unwanted solutions when >> 0 (t >> 0).

To find u() we take the real part of Eq. (21) and, because of the smallness of, we can use an iterative perturbationscheme to obtain ever more approximate values of the function Re (u) = u(). As starting value we put, under theintegral sign, the unperturbed elliptical solution u0. Then in the first approximation we can write Eq. (21) under theform

u =

2

1 + e

1 +

i

e

un0e

id

ei

. (22)

Since we are only interested in non-periodical variations of the orbit, in the product

un0ei =

n

2n

1 +

e

2ei +

e

2ein

ei, (23)

we will consider only the constant part that, after integrating, gives a term linear in that produces a cumulativesecular perturbation in u and hence a spiraling path of the body. The other terms produce only tiny periodical effectson the motion and they do not play any further role here. Let us see now how the mathematics can help our physicalintuition, allowing the possibility of a bounded orbital motion.

We denote with (n/2n)ne the value of the integral, where n, when n > 2, is a polynomial in e with rationalcoefficients, but we postpone to the Appendix its explicit determination. With this notation Eq. (22) becomes

u =

2

1 + e

1 + i

n1n2n

ei

. (24)

Without prejudice to the accuracy of the solution we can remove the non-periodical -term by considering that anexpression as 1 i , when > 0 is a very small quantity, represents an infinitesimal rotation ei in the complexplane of amplitude around the origin. Here we follow the hypothesis that a secular term could be in reality thefirst term of a series expansion of a periodic function of very long period. So Eq. (24) can be written under the form

u =

2

1 + e ei

, (25)

where

1 1 n1n2n

.

The apsidal points are given by Im (u) = 0 and by

1

r

=

2

1 + e ei()

when

ei() = 1

,

in particular we have rmin when

= [ + (1 ) ] = ( + ) = 0Thus we can define a new variable eccentricity vector given by

e+ e ei() e ei(0+ ) = e ei , (26)


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that should be introduced from the beginning in Eq. (21) and in the expression of un0 to avoid the occurrence ofsecular terms. Then Eq. (25) formally becomes the orbital equation of the rotating ellipse

u =

2

1 + e+ei

, (27)

from which

u = Re (u) =

2

[1 + e cos(

)] .

Thus the tentative introduction ofe in the integro-differential equation has b een compelling, for physical reasons, tothe introduction of the revolving ellipse as a first approximation to the true motion.

This behavior is analogous to that of an heavy spinning top whose tilted axis precesses around the vertical to avoidthe fall. Likewise, in order to avoid a disruptive inward or outward spiraling motion, an elliptic orbit steadily rotatesunder the action of a perturbing central force. In Eq. (26) the perigee argument () 0 + represents an angularvariable linearly dependent from that reduces to = 0 when = 0 or when = 0 (absence of perturbations), andfrom Eq. (22) and Eq. (27) its analytical expression is given by

() =

e

un0e

id. (28)

From Eq. (28) we find the general formula of the secular perihelion shift due to small perturbing inverse-power radial

forces after a revolution

=

e

20

un0 eid =

2n1n2n

=2n

an(1 e2)n . (29)

Before go on, it is worthwhile to observe that if we apply the differential operator (1 iD), D d/d, to both sidesof Eq. (21), we obtain the Binets form of the equation of motion (19)

(1 iD)u = u + u = 2

+un

2(30)

This equation transforms into the general relativistic form[5] if we put n = 2, = 3 2 in (30), where = /c2 GM/c

2

is the gravitational radius of the central body, and c is the speed of light. Then (30) becomes

u + u =

2+ 3u2 (31)

The last term of (31) is known as Einstein correction term in the weak-field approximation. It corresponds to aneffective r4 force law. From Eq. (29) with n = 2, = 3 2, 2 = 1 we obtain the standard result[6]

= 2 n (n1)

2n= 62

4=

6

a(1 e2) .

From Eq. (26) we can immediately deduce other important properties of the secular perturbed motion. Themagnitude of the perturbed eccentricity is constant and equal to that of the undisturbed motion, since |e+| = |e| = e,but this implies the correspondent invariability of the semi-major axis a, because a = 2(1 e2)/. Thus the motionhappens in the annulus a(1 e) r a(1 + e).

Fast particles and light rays

Lets consider now the situation in which a particle moves with a very high velocity v near a star, and fix thecoordinate axes so that, at the closest distance R from the star ( = = 0), its velocity is r = iv, and R = v. Then

the area integral is given by = r2 = vR.In this geometric arrangement the expression of the orbital velocity (11) at = 0 is, with /v2 < R/2,

iv =iv

R(1 + e) , hence e =

R

1 > 1. (32)


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The orbital equation (16) with /2 = /R2 and e = e becomes

u =

R2

1 +

R

1

ei

, (33)

whose solution is the hyperbola

u0 = Re (u) =1

r

= A + B cos

R2

+R

R2

cos , (34)

so that e = B/A. For r, u0 0 and we have

cos = AB

= 1e

.

If we denote with the angle between the two asymptotes, we have = /2 + /2, and, by developing cos , wehave

sin

2=

1

e.

Since is supposed small, sin(/2) /2 at O(2), and we get

=2

e =2A

B =2

R 2

R +22

R2 + . . .

Now we consider the additional effect of a radial perturbing force ei/rn+2, with n 1. The path equation, from(21), is

u = A + B ei +iAei

uneid. (35)

The first-order solution is quickly found from

Re (u) =1

r

= A + B cos A

Im

ei

un0 e

id

,

where it is not necessary introduce any arbitrary constant, because the initial conditions are already present in thestructure of the equation. We write it for n = 2

1

r

= A + B cos +A2B (cos + sin )

+A

3A2 + 2B2 B2 cos2

3,

and for r we have, by omitting the 2 term

=2A

B+

A2 +

4

3AB +

2A3

B

. (36)

It is interesting to obtain this bending also as expression of a rotation of the vector e. For our system Eq. (28)

becomes, with e = e = B/A and with u = u0 = A + B cos

= =A

B

+/2/2

un0 eid. (37)

If we put n = 2 we find again the perturbation part of Eq. (36). So the bending may be visualized as due to a-rotation of the unperturbed hyperbola as a whole around its focus. We calculate now this effect on light rays ingeneral relativity. The relativistic null-geodetic equation of light rays is equivalent to formally put = in theBinets orbit equation (31),7 and in our formalism this recipe requires the computation of the limit, starting from (21)

u = lim

2+

e

2ei +

i ei

2

uneid

. (38)


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We can do this limit by noting that in the unperturbed ( = 0) context the second term in the right-hand side ofEq. (38) must be different from zero and to produce the straight-line solution. This can happen only if simultaneouslye approach infinity so as to make the unperturbed hyperbola a straight line. Under these conditions we can pute R/ (see Eq. (32)), and we find

e

2=

R

v2R2=

1

R. (39)

Then for = 0 Eq. (38), in the relativistic limit for light rays, gives the straight line equation

u =ei

R u0 = Re (u) = cos

R.

Equation (38) becomes, by inserting the general relativistic values = 32, n = 2

u =ei

R+ 3 i ei

u20 e

id,

with solution

Re (u) =1

r

=cos

R 3

R2Im

ei

cos2 eid

=cos

R+ 2 cos

2R2

. (40)

For = we have in Eq. (40) cos /2, from which we find

=4

R+ O

3

R3

, (41)

the general relativistic result.8

In the precedent analysis we have seen that in our formalism the general relativity requires a finite limit for theratio e/2 (or for its inverse). We can use this knowledge for obtaining an alternate derivation of the light bendingas a perihelion shift of the path from the perturbation equation for e.

From Eq. (37), with u0 = cos /R we have

= =

e Rn

+/2

/2

ei cosnd. (42)

For the Einstein correction term for which n = 2, = 32 we get from Eq. (39)

= =32

e R2

+/2/2

ei cos2 d

=3

R

+/2/2

ei cos2 d =4

R, (43)

which is general-relativistic value (41) at the lowest order.

Conclusions

We have seen the utility of the first-order orbital equation and of the related integro-differential equation in thetreatment of the motions due to an important class of perturbing forces. These can be solved by alternate methods,so that our analysis can be also considered as a test of the validity of the first-order equation. Besides, with ourinvestigation we have gained a better understanding of the interplay between physics and mathematics in thissubject. We need to be always looking at what we already know in new ways, and solve old problems with newmethods. This makes the subject accessible to readers of all mathematical tastes. Not only does each approach offera different view, but the combination of viewpoints yields insights not available otherwise, and sometimes new resultsare found in a rather easy manner. So the first-order equation helped us to easily deduce a new general formula forthe secular angular perihelion motion. It is not difficult to find other interesting applications of our equations, forexample, gravitational lensing, gravitational friction in stellar systems, and scattering problems of various type.


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Appendix

The general algebraic expression of n can be obtained from the constant part of the product (23) by means of arepeated application of the binomial theorem as follows. We have, by expanding the power in the integrand,

un0 =n

2n

1 +

1

2

e ei + e ei

n

= n

2n

np=0

np

e e

i

+ e eip

2p,

e ei + e ei

p=

pq=0

pq

e

pqe

q e(p2q)i,

so that we can write

un0ei =

n

2n

np=0

pq=0

np

pq

e

pqe

q1e

2pe(p2q+1)i.

This expression is constant when the exponential becomes equal to the unity, that is when p = 2q 1, so that at lastwe have

n

2nn e =

n

2n

nq=0

n! e2(q1) e

22q1 q! (q 1)!(n 2q + 1)! . (44)

The first few values of n, n = 1, 2 . . . , are

1 =1

2, 2 = 1, 3 =

3

2+

3

8e2, 4 = 2 +

3

2e2.

1

T.W.Gamelin, Complex Analysis. Springer - N.Y. - 2001.2 P. S. Laplace, Ouvres. Gauthier-Villars, Paris, 1878), Tome 1, p. 181, formula P.3 R. E. Williamson. Introduction to Differential Equations. McGraw-Hill, New York, 1997, p. 84.4 V.R.Bond and M.C.Allman, Modern Astrodynamics. Princeton University Press,Princeton, - New Jersey. 1998.5 W.Rindler, Relativity. OUP. New York. 2001. p.242, Eq.(11.45).6 W.Rindler, Relativity. OUP. New York. 2001. p.243.7 W.Rindler, Relativity. OUP. New York. 2001. p.248; R. Adler, M. Bazin, M. Shiffer. Introduction to General Relativity 2nd

ed. McGraw-Hill, New York. 1975 , p. 216. Eq. (6.149).8 W.Rindler, Relativity. OUP. New York. 2001. pp. 248-2499 D.R.Brill and D.Goel, Am.J.Phys 67, 316 (1999).

Central Forces and Secular Perihelion Motion

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