Central Forces And Secular Perihelion Motion

Central Forces and Secular Perihelion Motion Maurizio M. D’Eliseo 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 integrodifferential equation provides a new method to study, respectively, the elliptical orbit and the secular motion of a planetary perihelion due to the perturbing action of an important category of central forces: the inverse-power radial forces. For this, a general formula is found. The related problems 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 Resum´ e: L’´equation orbitale du premier ordre et sa version perturb´ee sous la forme d’une ´equation int´egro-diff´erentielle nous fournissent une nouvelle m´ethode pour ´etudier respectivement l’orbite elliptique et le mouvement s´eculaire du p´erih´elie plan´etaire dˆ u ` a l’effet perturbateur d’une importante cat´egories de forces perturbatives centrales en puissance n´egative de r. Nous obtenons une formule g´en´erale pour ces effets. La mˆeme m´ethode permet d’´etudier la d´eformation gravitationelle 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 can be handled by many different methods. Here, with the help of the Laplace and area integrals, and of the complexvariable 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 method of dealing with an important category of perturbing central forces: inverse power radial forces. The search for an admissible 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 fast particles and of light rays, thus extending the applicability range of the equations to unbounded systems. The problems treated 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-body problem with the complex plane1 . An object of mass M is at the origin. The position x, y of the second object of mass m is given by the complex quantity r = x + iy, and the equation of motion can be expressed as r¨ = −µ

r eiθ = −µ 2 , 3 r r

(1)

p √ where µ = G(m+M ), r = reiθ = r(cos θ +i sin θ), r¯ = re−iθ is the complex conjugate of r, r = |r| = r¯ r = x2 + y 2 , while the polar angle θ = θ(t) is the true longitude, that is, the point (x, y) has the polar form reiθ , where r is the modulus and θ is its argument. From r = reiθ we have by time-differentiation ³ ´ r˙ = r˙ + i r θ˙ eiθ , r¯˙ = r¯˙ . (2) The solution of (1) requires knowledge of the functions r(θ) and θ(t), but here, we are interested only in the determination of the function r(θ), 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 = r cos θ, Im (r) = (r − r¯)/2i = y = r sin θ, Re (ir) = −Im (r), and Im (ir) = Re (r). It is useful to consider a complex number z = zeiθ as an oriented vector starting from the origin, of length z and making an angle θ with the real axis, so that we will use boldface letters to denote complex quantities, with the exception of the complex exponential eiθ .

2 The first-order orbital equation

If we multiply Eq. (1) by r¯, we have r¨ r¯ = −µ

r r¯ r2 µ = −µ =− . r3 r3 r

(3)

If we take the imaginary part of both sides of (3), we find Im (¨ r r¯) = −Im

³µ´ r

= 0.

(4)

It is easy to verify that d Im (r˙ r¯) = Im (¨ r r¯) + Im (r˙ r¯˙ ). dt

(5)

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

(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) eiθ = f (r)(r/r). We substitute (2) ˙ which can be cast in three equivalent forms into (7) and find the fundamental relation ` = r2 θ, 1 θ˙ = r2 ` r2 dt = dθ ` d ` d = 2 . dt r dθ

(8b)

µ ˙ iθ iµ d iθ r¨ = − θe = e , ` ` dt

(9)

d³ iµ ´ r˙ − eiθ = 0. dt `

(10)

(8a)

(8c)

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

or

The expression in parentheses is a complex constant, which for later convenience we denote as (iµe)/`. We have thus deduced the Laplace integral r˙ =

iµ iθ (e + e), `

(11)

where r˙ is the orbital velocity, e = e exp(iω) is a complex constant that we will call the eccentricity vector, and e is the (scalar) eccentricity. The vector e is directed toward the perihelion, the point on the orbit of nearest approach to the center of force, and ω is the argument of the perihelion. To keep the customary notation we use the same letter e 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 θ derivative. By (8c) we find r˙ =

iµ ` 0 r = (eiθ + e), 2 r `

(12)

3 where a prime denotes differentiation with respect to θ, and we get r 0 = (reiθ )0 = (r0 + ir)eiθ =

iµ iθ (e + e)r2 . `2

(13)

If we multiply (13) by e−iθ , we obtain the complex Bernoulli equation5 r0 + ir =

iµ (1 + e e−iθ )r2 . `2

(14)

By taking the imaginary and the real parts of both sides of (14), we obtain the orbit r(θ) and its derivative r0 (θ), respectively. However, we find it convenient to work with the reciprocal of r, so we divide both sides by r2 , make the variable change r(θ) → 1/u(θ), multiply by i, and convert (14) into an equation in u u + iu0 =

µ (1 + e e−iθ ). `2

(15)

Despite its heterogeneous nature, it is convenient to write the left-hand side of (15) in terms of the complex quantity u = u(θ) ≡ u(θ) + iu0 (θ), so that we have u=

µ (1 + e e−iθ ), `2

(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 of u Re (u) =

¤ ¤ µ£ 1 µ£ = 2 1 + Re (e e−iθ ) = 2 1 + e cos(θ − ω) . r ` `

(17)

Equation (17) is the polar equation of a conic section of eccentricity e with a focus at the origin. If the orbit is an ellipse (0 < e < 1), we have the relation `2 /µ = 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, the names given earlier to |e| = e and ω are justified. The apsidal points are determined from the condition Im (u) = 0 because r0 (θ) = 0 at these points. If Im (u) = 0 for every value of θ, then e = 0, and we have a circular orbit with u = Re (u) = µ/`2 . If 0 < |e| < 1, then (16) gives the position of the two apsidal points rmin and rmax . At these points we have Im (u) = −

µ e sin(θ − ω) = 0, `2

(18)

so that, by considering the derivative [Im (u)]0 = u00 (θ), we find rmin when θ+2nπ = ω and rmax when θ+(2n+1)π = ω, where n = 0, 1, 2, . . . . Perturbing central forces

Consider the gravitational motion of a body perturbed by a central force of the type F = −κ r−(n+2) eiθ where κ is small and n ≥ 1 is a positive integer. The equation of motion is r¨ = −µ

eiθ eiθ − κ n+2 . 2 r r

(19)

Formally integrating the above equation with respect to t, we obtain Z iθ Z iθ e e r˙ = −µ dt − κ n+2 dt, r2 r and, by means of the area integral we get by Eq. (8b) ¢ κ iµ ¡ iθ r˙ = e +k − ` `

Z

eiθ dθ, rn

(20)

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

4 To proceed, we observe that when the perturbation is switched off (κ = 0), we have k = e. If suddenly at the instant t = 0 (θ = 0) we introduce it, we can suppose that, since κ is small, for continuity reasons in the following motion k will be near e, and that however, starting from this value, it will change slowly. So, in absence of a valid alternative, we tentatively put k = e. In these conditions, we could call Eq. (20) the Laplace quasi-integral. The utility 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-integral it is a simple matter to deduce the following perturbed first-order orbital equation, which is formally the nonlinear integro-differential equation Z ¢ iκ e−iθ µ ¡ u = 2 1 + e e−iθ + un eiθ dθ. (21) ` `2 Since we have here introduced the same constant e of the unperturbed motion when θ = 0, we will be forced to make some maintenance work to avoid unwanted solutions when θ >> 0 (t >> 0). To find u(θ) we take the real part of Eq. (21) and, because of the smallness of κ, we can use an iterative perturbation scheme to obtain ever more approximate values of the function Re (u) = u(θ). As starting value we put, under the integral sign, the unperturbed elliptical solution u0 . Then in the first approximation we can write Eq. (21) under the form · µ ¶ ¸ Z iκ µ u= 2 1+e 1+ un0 eiθ dθ e−iθ . (22) ` µe Since we are only interested in non-periodical variations of the orbit, in the product µ ¶n ¯ iθ µn e −iθ e n iθ u0 e = 2n 1 + e + e eiθ , ` 2 2

(23)

we will consider only the constant part that, after integrating, gives a term linear in θ that produces a cumulative secular perturbation in u and hence a spiraling path of the body. The other terms produce only tiny periodical effects on the motion and they do not play any further role here. Let us see now how the mathematics can help our physical intuition, allowing the possibility of a bounded orbital motion. We denote with (µn /`2n )λn e θ the value of the integral, where λn , when n > 2, is a polynomial in e with rational coefficients, but we postpone to the Appendix its explicit determination. With this notation Eq. (22) becomes · µ ¶ ¸ µ κµn−1 λn −iθ u= 2 1+e 1+i θ e . (24) ` `2n Without prejudice to the accuracy of the solution we can remove the non-periodical θ-term by considering that an expression as 1 ± iν θ, when ν > 0 is a very small quantity, represents an infinitesimal rotation e±iνθ in the complex plane of amplitude ± ν θ around the origin. Here we follow the hypothesis that a secular term could be in reality the first term of a series expansion of a periodic function of very long period. So Eq. (24) can be written under the form u=

¢ µ ¡ 1 + e e−iσθ , `2

(25)

where σ ≡1−ν ≡1−

κµn−1 λn . `2n

The apsidal points are given by Im (u) = 0 and by ´ ³ ´ 1 µ ³ = 2 1 + e e−i(σθ−ω) when e−i(σθ−ω) = ±1 , r ` in particular we have rmin when σθ − ω = θ − [ ω + (1 − σ) θ ] = θ − ( ω + ν θ ) = 0 Thus we can define a new variable eccentricity vector given by e+ ≡ e eiω(θ) ≈ e ei(ω0 +ν θ) = e eiν θ ,

(26)

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

¢ µ ¡ 1 + e+ e−iθ , `2

(27)

from which u = Re (u) =

µ [1 + e cos(σθ − ω)] . `2

Thus the tentative introduction of e in the integro-differential equation has been compelling, for physical reasons, to the 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 avoid the fall. Likewise, in order to avoid a disruptive inward or outward spiraling motion, an elliptic orbit steadily rotates under the action of a perturbing central force. In Eq. (26) the perigee argument ω(θ) ≈ ω0 + ν θ represents an angular variable linearly dependent from θ that reduces to ω = ω0 when θ = 0 or when ν = 0 (absence of perturbations), and from Eq. (22) and Eq. (27) its analytical expression is given by Z κ ω(θ) = un0 eiθ dθ. (28) µe From Eq. (28) we find the general formula of the secular perihelion shift due to small perturbing inverse-power radial forces after a revolution Z 2π κ 2πκµn−1 λn ∆ω = un0 eiθ dθ = µe 0 `2n 2πκλn = . (29) µan (1 − e2 )n Before go on, it is worthwhile to observe that if we apply the differential operator (1 − iD), D ≡ d/dθ, to both sides of Eq. (21), we obtain the Binet’s form of the equation of motion (19) (1 − iD)u = u00 + u =

µ κun + `2 `2

(30)

This equation transforms into the general relativistic form[5] if we put n = 2, κ = 3 α`2 in (30), where α = µ/c2 ≡ GM/c2 is the gravitational radius of the central body, and c is the speed of light. Then (30) becomes u00 + u =

µ + 3αu2 `2

(31)

The last term of (31) is known as Einstein correction term in the weak-field approximation. It corresponds to an effective r−4 force law. From Eq. (29) with n = 2, κ = 3 α`2 , λ2 = 1 we obtain the standard result[6] ∆ω = 2πκ ·

λn µ(n−1) µ 6πα = 6πα`2 · 4 = . `2n ` a(1 − e2 )

From Eq. (26) we can immediately deduce other important properties of the secular perturbed motion. The magnitude 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 )/µ. Thus the motion happens in the annulus a(1 − e) ≤ r ≤ a(1 + e). Fast particles and light rays

Let’s consider now the situation in which a particle moves with a very high velocity v near a star, and fix the coordinate axes so that, at the closest distance R from the star (θ = ω = 0), its velocity is r˙ = iv, and Rθ˙ = v. Then the area integral is given by ` = r2 θ˙ = vR. In this geometric arrangement the expression of the orbital velocity (11) at θ = 0 is, with γ ≡ µ/v 2 < R/2, iv =

iγv (1 + e) , R

hence

e=

R − 1 > 1. γ

(32)

6 The orbital equation (16) with µ/`2 = γ/R2 and e = e becomes · µ ¶ ¸ γ R −iθ u= 2 1+ −1 e , R γ

(33)

whose solution is the hyperbola u0 = Re (u) =

1 γ R−γ = A + B cos θ ≡ 2 + cos θ, r R R2

(34)

so that e = B/A. For r → ∞, u0 → 0 and we have cos θ∞ = −

A 1 =− . B e

If we denote with δ the angle between the two asymptotes, we have θ∞ = π/2 + δ/2, and, by developing cos θ∞ , we have sin

δ 1 = . 2 e

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

2A 2γ 2γ 2γ 2 2 = = ≈ + 2 + ... e B R−γ R R

Now we consider the additional effect of a radial perturbing force κ eiθ /rn+2 , with n ≥ 1. The path equation, from (21), is Z iκA e−iθ −iθ un eiθ dθ. (35) u=A+Be + µ The first-order solution is quickly found from Re (u) =

¶ µ Z 1 κA = A + B cos θ − Im e−iθ un0 eiθ dθ , r µ

where it is not necessary introduce any arbitrary constant, because the initial conditions are already present in the structure of the equation. We write it for n = 2 1 κA2 B (cos θ + θ sin θ) = A + B cos θ + r µ ¡ 2 ¢ κA 3A + 2B 2 − B 2 cos2 θ + , 3µ and for r → ∞ we have, by omitting the δ 2 term 2A κ δ= + B µ

µ ¶ 4 2A3 2 πA + AB + . 3 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 θ Z κA +π/2 n iθ u e dθ. (37) δ = ∆ω = µ B −π/2 0 If we put n = 2 we find again the perturbation part of Eq. (36). So the bending may be visualized as due to a δ-rotation of the unperturbed hyperbola as a whole around its focus. We calculate now this effect on light rays in general relativity. The relativistic null-geodetic equation of light rays is equivalent to formally put ` = ∞ in the Binet’s orbit equation (31),7 and in our formalism this recipe requires the computation of the limit, starting from (21) ¸ · Z µ e −iθ iκ e−iθ µ n iθ + 2 e + u e dθ . (38) u = lim `→∞ `2 ` `2

7 We can do this limit by noting that in the unperturbed (κ = 0) context the second term in the right-hand side of Eq. (38) must be different from zero and to produce the straight-line solution. This can happen only if simultaneously e approach infinity so as to make the unperturbed hyperbola a straight line. Under these conditions we can put e ≈ R/γ (see Eq. (32)), and we find µe µR 1 = 2 2 = . `2 γv R R

(39)

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

e−iθ cos θ −→ u0 = Re (u) = . R R

Equation (38) becomes, by inserting the general relativistic values κ = 3α`2 , n = 2 Z e−iθ u= + 3 i α e−iθ u20 eiθ dθ, R with solution

µ Z ¶ 1 cos θ 3α −iθ 2 iθ Re (u) = = − 2 Im e cos θ e dθ r R R ¡ ¢ 2 − cos2 θ α cos θ + . = R R2

For θ = θ∞ we have in Eq. (40) cos θ∞ ≈ −δ/2, from which we find µ 3¶ 4α α δ= + O , R R3

(40)

(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 the ratio µe/`2 (or for its inverse). We can use this knowledge for obtaining an alternate derivation of the light bending as a perihelion shift of the path from the perturbation equation for e. From Eq. (37), with u0 = cos θ/R we have Z +π/2 κ δ = ∆ω = eiθ cosn θ dθ. (42) µ e Rn −π/2 For the Einstein correction term for which n = 2, κ = 3α`2 we get from Eq. (39) Z +π/2 3α`2 δ = ∆ω = eiθ cos2 θ dθ µ e R2 −π/2 Z 4α 3α +π/2 iθ e cos2 θ dθ = , = R −π/2 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 the treatment 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 our investigation we have gained a better understanding of the interplay between physics and mathematics in this subject. We need to be always looking at what we already know in new ways, and solve old problems with new methods. This makes the subject accessible to readers of all mathematical tastes. Not only does each approach offer a different view, but the combination of viewpoints yields insights not available otherwise, and sometimes new results are found in a rather easy manner. So the first-order equation helped us to easily deduce a new general formula for the secular angular perihelion motion. It is not difficult to find other interesting applications of our equations, for example, gravitational lensing, gravitational friction in stellar systems, and scattering problems of various type.

8 Appendix

The general algebraic expression of λn can be obtained from the constant part of the product (23) by means of a repeated application of the binomial theorem as follows. We have, by expanding the power in the integrand, · ¸ ¢ n µn 1 ¡ iθ ¯ e + e e−iθ un0 = 2n 1 + e ` 2 ¢p ¶ ¡ iθ n µ n X ¯ e + e e−iθ e µ n = 2n , ` p=0 p 2p ¶ p µ ¡ iθ ¢p X p ¯ e + e e−iθ = ¯p−q eq e(p−2q)iθ , e e q q=0

so that we can write un0 eiθ =

¶µ ¶ p−q q−1 p µ n ¯ µn X X n e e (p−2q+1)iθ p e e . q `2n p=0 q=0 p 2p

This expression is constant when the exponential becomes equal to the unity, that is when p = 2q − 1, so that at last we have n µn µn X n! e2(q−1) e λ e = . n 2n 2n 2q−1 ` ` q=0 2 q! (q − 1)! (n − 2q + 1)!

(44)

The first few values of λn , n = 1, 2 . . . , are λ1 =

1 2 3 4 5 6 7 8 9

1 , 2

λ2 = 1,

λ3 =

3 3 2 + e , 2 8

λ4 = 2 +

3 2 e . 2

T.W.Gamelin, Complex Analysis. Springer - N.Y. - 2001. P. S. Laplace, Ouvres. Gauthier-Villars, Paris, 1878), Tome 1, p. 181, formula P . R. E. Williamson. Introduction to Differential Equations. McGraw-Hill, New York, 1997, p. 84. V.R.Bond and M.C.Allman, Modern Astrodynamics. Princeton University Press,Princeton, - New Jersey. 1998. W.Rindler, Relativity. OUP. New York. 2001. p.242, Eq.(11.45). W.Rindler, Relativity. OUP. New York. 2001. p.243. 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). W.Rindler, Relativity. OUP. New York. 2001. pp. 248-249 D.R.Brill and D.Goel, Am.J.Phys 67, 316 (1999).