Calculation of Apsidal Precession Via Perturbation Theory

Calculation of apsidal precession via perturbation theory

L. Barbieri, F. Talamucci1,∗

1DIMAI – Dipartimento di Matematica e Informatica, University of Florence, Italy

Email address

federico.talamucci@uniﬁ.it (F. Talamucci)

∗Corresponding author

Abstract

The calculus of apsidal precession frequencies of the planets is developed by means of a perturbation thecnique. A model of concentric rings (ring model), suitable for improving calculations, is introduced. Conclusive remarks concerning a comparison between the theoretical, the calculated and the observed data of the precession frequencies are performed.

Keywords

Orbital Mechanics, apsidal precession frequencies, ring model

1 Introduction 2 Apsidal precession

According to [3], we will discuss the effect of a force F added to the Keplerian force. We consider a mass m particel sub- The solar system is a gravitationally bound system encom- jected to a total force which is the sum of a force F and the passing the Sun, the planets and many other celestial bod- Keplerian force: ies. As it is known, apsidal precession consists in the ro- k tation of a planet apsidal line, which is the line passing ˆ Ftot = − 2 r + F. through aphelion and perihelion. The precession of each r planet is caused by the gravity effects of celestial bodies (in In the Keplerian motion F = 0 the Lenz vector A = particular other planets) or by a relativistic effect. We will mp × L − mkˆr (where p is the momentum and L = r × p consider only classical effects and we will shortly refer to is the angular momentum) is a constant of motion whose the relativistic ones only at the end of the work. direction defines the apsidal line and the magnitude is pro- portional to the eccentricity. Because of the perturbation We will focus on the simplified system formed by the Sun introduced by the force F the Lenz vector A is no longer a and the planets, disregarding the rest of the bodies. The constant of motion and it changes in direction and in mag- Keplerian model is able to reproduce the revolution mo- nitude. The change in direction causes the line precession tion whenever the apsidal line is fixed: the starting point and the change in magnitude produces variation in time of consists in showing how the apsidal line precedes and the the eccentricity. An important property of the apsidal pre- eccentricity varies, if a force F, ascribable, as an example cession motion can be deduced by writing the precession the presence of other planets, is added to the Keplerian frequency A × A˙ force. ν = (1) ||A||2 In the second part we make use of the “ring model” in where A˙ = m(2(r˙ · F)r − (r · r˙)F − (r · F)r˙). As we can see, order to calculate the apsidal precession frequencies. The (1) is linear with respect to F, hence if the perturbation F basic assumption of such model is that the precession pe- is the sum of two forces, then the precession frequency due riod of every single planet is much greater than any other to F is the sum of the two frequencies of the singular contri- revolution period. In this way, the model deals with the bution due to the first and the second force. We are going planets alterating the revolution motion as concentric rings now to analyse the remarkable case where the perturbative arXiv:1802.07115v1 [physics.class-ph] 20 Feb 2018 centered in the Sun with a uniform mass distribution. In force is particular, by means of suitable geometrical properties, it H ˆr F(r) = is possible to consider orbits as coplanar and circular. Such r3 r approximation let us enter the central force formalism and where H is a negative number. In the next Section we will the perturbation theory is applied directly to the orbit find that this is a for the gravitational perturbation on the equation. revolution motion of planets. The effective potential energy is: 2 Secondly, in order to calculate the frequency of precession, k ||L|| + mH Ueff = − + (2) we make use of the perturbation theory performed in [1]. r 2mr2 mH We will calculate the precession frequencies of each planet In most significant cases the ratio ||L||2 is much smaller and we will comment the differences between theoretical then 1. As we can see from (2), the effective potential en- frequencies, calculated frequencies and the observed data. ergy is the same as the Keplerian case, so that the orbit is

1 limited between two values of r. The orbit equation is Internal field Expanding the argument of (4) in powers of r and calculating the integral, one can obtain the d2ρ mH d2ρ mk R + 1 + ρ = + Ω2ρ = formula (see [2]) dθ2 ||L||2 dθ2 ||L||2 +∞ 2 2n Gm X (2n − 1)!! r corresponding to the equation of a driven harmonic oscil- V (r) = − 1 + . mk R (2n)!! R n=1 lator with a driving force given by ||L||2 . The solution can be easily computed in terms of the inverse radius as We can calculate the gravitational field in the case r < R: 1 mk +∞ 2 2n−1 ρ = = (1 + e cos(Ω(θ − ω))) (3) Gm X (2n − 1)!! r 2 2 g (r) = (2n) r ||L|| Ω int R2 (2n)!! R n=1 where e and ω are two integration constant values. In the +∞ X 2n−1 case of interest the factor Ω is close to 1 and smaller than = αnr (5) 1: the curve (3) is sketched in Figure 1 in the case Ω = 0.9. n=1 As we can see, the effect is a rotation of the Keplerian orbit In order to simplify the latter expression, we follow a pro- in its plane. cedure showed in [4] and based on a comparison technique 90 between series. Let r1 and r2 be respectively the radial 120 60 distance of the aphelion and the perihelion of a planet sub- 4 jected to the field (5) and calculate the constant values A 3 150 30 and B such that 2  A + B = P+∞ α r2n−1 1  r2 r3 n=1 n 1 1 1 (6) 180 0 0 A B P+∞ 2n−1  2 + 3 = n=1 αnr2 r2 r2

By virtue of the low eccentricity assumption we have r1 ≈ 210 330 r2 ≈ r0 ≈ r, where r0 is the mean radius of an orbit. We can therefore solve system (6) by considering the ﬁrst order 240 300 270 terms in (5): 3 5 7 A = 4α1r0 + 6α2r0 + 8α3r0 + ... 4 6 8 (7) Figure 1: the orbit (3), Ω = 0.9. B = −3α1r0 − 5α2r0 − 7α3r0 − ...

From (7) we achieve A and B in powers of r0:  P+∞ 2n+1 P+∞ 3 The ring model  A = n=1(2(n + 1))αnr0 = n=1 An (8) P+∞ 2(n+1) P+∞ We introduce now a model based on the assumption that  B = − n=1(2n + 1)αnr0 = − n=1 Bn the precession period of every single planet is much greater Finally, using condition r1 ≈ r2 ≈ r, we can write gint(r) than any other revolution period. As a consequence, we can as study the precession motion of a specific planet by treating A B g (r) = + . (9) the rest of the planets as uniform concentric rings, cen- int r2 r3 tered in the Sun, whose masses are equal to the respective We see that this method produces a reduction of the series planetary masses. Furthermore, each radius is equal to the of radial functions (5) into the sum of two radial functions respective mean orbital radius. (9). Furthermore, the series of functions is simplified into On the basis of this assumption, we simplify the geometry the two numerical series (8). From a physical point of view, of the problem by using specific properties of the solar sys- the two contributions in the central field (9) have to be re- tem. By virtue of the small eccentricities, the rings can be lated respectively to the modification of the form of the approximated as circles. On the other hand, the small incli- Keplerian orbit and to the onset of the precession motion. nation of the orbital planes from the ecliptic plane allow us As we can see, the approximation (9) corresponds to the to approximate the orbits as coplanar orbits (central force effect of rotation of the Keplerian orbit discussed in the approximation). previous Section. By employing this model we calculated the precession frequencies using perturbation techniques illustrated in Chap- External field In this case, by expanding the inte- R ter 4 of [1] . The perturbation is caused by the gravity of the grand function in (4) in powers of r and by remarking the other planets (treated as rings) which alter the Keplerian invariance of the same formula with respect to the replace- motion of the single planet. ment r ↔ R, we obtain

+∞ 2 2n Gm X (2n − 1)!! R V (r) = − 1 + . 3.1 Ring’s gravity field r (2n)!! r n=1 Consider a uniform ring of mass m and radius R and a Thus the gravitational field g for r > R is polar coordinate system (r, θ) in the plane of the ring and centered in its center. The gravitational potential is dV (r) Gm R 2 g (r) = − = − Gm Z 2π Rdθ ext 2 V (r) = − √ (4) dr R r 2πR r2 + R2 − 2rRcosθ +∞ 2 2(n+1) 0 X (2n − 1)!! R + (2n + 1) that is a complete elliptical integral of the first kind. In (2n)!! r order to exhibit the solution, we have to make a distinction n=1 +∞ between the case of internal field (r < R) and external field β1 X βn+1 = + . (10) (r > R). r2 r2(n+1) n=1

2 Using the same method employed for the internal field, one which comes under the standard form in perturbation the- can write (10) as ory. The solutions of the unperturbed problem = 0 are the ones of the Kepler problem: C D gext(r) = + (11)  r2 r3 ρ(θ) = Mk (1 + e cos(θ − ω)) = Mk (1 + e cos(f))  ||L||2 ||L||2 where the coefficients are  Mk  s(θ) = − ||L||2 e sin(f)  C = β − P+∞ (2n − 1) βn+1 = C − P+∞ C  1 n=1 r2n 0 n=1 n 0 (12) (we set f = θ − ω). P+∞ βn+1 P+∞  D = n=1(2n) 2n−1 = − n=1 Dn. Applying the method of variation of constants, we replace r0 the just written unperturbed solutions in (16). The inte- The result is formally the same as in the case of internal gration constants ω and e of the unperturbed problem now field and we have only to take into account apart the dif- depend on the variable θ. Straightforward calculations lead ferent values of the coefficients. to the following differential system in ω and e:

 de dω dθ cos f + e dθ sin f = 0 3.2 Calculation of precession frequency  de dω dθ sin f + e 1 − dθ cos f (17)  H Mk We move forward now to the calculation of precession fre- = e cos f + 1 + Φ ||L||2 (1 + e cos f) quency, by using the techniques and the results performed so far. The total force acting on a planet of mass M is given Combining the first equation in (17) with the second one, by the sum of gravitational forces exerted by the Sun (with we get mass M hereafter) and by the rings, representing the rest dω H Mk = − cos f 1 + (1 + e cos f) . (18) of planets. We have to consider that there are some rings dθ e Φ ||L||2 inside planetary orbit and others outside, so that the total radial force is At this point we can employ the first order perturbation theory: by integrating (18) over an entire revolution period, we achieve GM M F (r) = − r2 ∆ω = ω(2π) − ω(0) X X Z 2π + M gint,j (r) + gext,j (r) H Mk = − cos f 1 + 2 (1 + e cos f) dθ. j,internal j,external 0 e Φ ||L|| where the first sum is extended over the internal rings and Having in mind that e and ω assume at the first order the the second over the external ones. Setting k = GM M and constant unperturbed values, we have using (8),(9), (11) and (12) we obtain πH Mk ∆ω = − P P 2 −k + M( Cj + Aj ) Φ ||L|| F (r) = j,internal j,external r2 P P and, recalling (15): M( Dj + Bj ) + j,internal j,external (13) r3 HMπ HM 2 ∆ω = − = − Triv. (19) −k + Φ H ||L||2 4πM 2r4 = + . 0 r2 r3 The result depends on H but not on Φ, consistently with H H the fact that the term bringing about precession is 3 . We As we can see, only the second term r3 is responsible for r Φ also remark in (19) the inverse proportionality with respect precession, while the first one r2 causes simply a change in eccentricity. Thus, the ring model is able to explain to the square of the moment of inertia: the fact is physically not only the apsidal precession but also the secular motion reasonable, since the moment of inertia reveals the body’s of eccentricity variation. By using (13) the orbit equation opposition to the motion of rotation, in this case the rota- takes the form tion of the apsidal line. Furthermore, the proportionality 2 with respect to Triv can be explained by considering that d2ρ kM ΦM HMρ + ρ = − − (14) the greater is the revolution period, the longer is the time dθ2 ||L||2 ||L||2 ||L||2 of exposure to the perturbation in a complete revolution orbit. By means of (19) and assuming the slow advance of 1 where ρ = r . In order to introduce a perturbation approach the apsidal line, we can write the precession frequency as we define P7 1 3600 s Φ i=1 mi −3 = v v 10 (15) ω˙ th = (∆ω)|arcsec 360(365 · 24 · 3600) (20) k M Triv 2π year where the mass mi is the mass of a ring (i. e. of a planet). where the numerical factors let us convert the result from In this way, equation (14) is written as rad arcsec s to year . d2ρ kM kM H + ρ = − 1 + ρ dθ2 ||L||2 ||L||2 Φ 3.3 Planets precession frequencies We now make use of (20) in order to calculate the preces- which is in turn equivalent to the following system: sion frequencies of each planet. The calculation procedure  takes into account both that the term H in (19) varies from  ds = kM − ρ − kM 1 + H ρ dθ ||L||2 ||L||2 Φ (16) one planet to another and that it contains the specific val-  dρ ues of each ring: actually, if we focus on the perturbation s = dθ

3 which the planet with mass M (i) undergoes, the factor H external ring at the n − order. In order to calculate the is precession frequencies using the expression (20) we cut the ring series at a certain order, depending on the accuracy H(i) = − which the precession frequencies are measured at. The pre- +∞ +∞ (i) X X (i) X X (i) cession frequencies data are acquired from [5] and plotted M B + D (21) j,n j,n in Table 2. In the same text the frequencies are claimed to j,internal n=1 j,external n=1 be with no error, thus we decide to take as errors on the (i) (i) where the Dj,n and Bj,n are respectively the contributions frequencies, one of the last digits declared: for example the of the j − th internal ring at the n − order and of the j − th case of Mercury is treated as (ω¯˙ ± ∆ω¯˙ )obs = 5.75 ± 0.01.

Mass Mean radius(r0) Revolution period(Triv) (Kg)(m)(year) Mercury 3.302 · 1023 5.79 · 1010 0.241 Venus 4.868 · 1024 1.082 · 1011 0.615 Earth 5.974 · 1024 1.496 · 1011 1 Mars 6.418 · 1023 2.279 · 1011 1.88 Jupiter 1.899 · 1027 7.783 · 1011 11.86 Saturn 5.685 · 1026 1.429 · 1012 29.46 Uranus 8.682 · 1025 2.871 · 1012 84.10 Neptune 1.024 · 1026 4.498 · 1012 164.86

Table 1: Mass, mean radius and revolution period of the planets, according to the data stated in [6].

˙ ∆ω¯˙ ˙ ∆ω¯˙ Terrestrial ω¯obs ω¯˙ obs Jovian ω¯obs ω¯˙ obs arcsec arcsec planets ( year ) planets ( year ) Mercury 5.75 0.0017 Jupiter 6.55 0.0015 Venus 2.05 0.005 Saturn 19.50 0.0005 Earth 11.45 0.0009 Uranus 3.44 0.003 Mars 16.28 0.0006 Neptune 0.36 0.03

Table 2: Observed precession frequencies with relative precision (the data are taken from [5]).

Concerning the theoretical calculation of the frequencies calculate the planets precession frequencies through and the cutting in the series (21), it is necessary to make the precision equal or as near as possible to the precision ω˙ (i) = 1 T (i) P P+∞ B(i) th 4πr (i)4 riv j,internal n=1 j,n ∆ω¯˙ 0 ω¯˙ of the measured frequencies. Denoting by ω˙ 1 an obs + P P+∞ D(i) N expressions of ω˙ which has k terms and with ω˙ and ω˙ j,external n=1 j,n th 2 3 (23) (i) two expressions of ω˙ th which have respectively k + 1 and (i) (2n−1)!! 2 r0 2n+1 (i) B = Gmj (2n + 1)(2n) r k+2 terms, our criterion consists in checking if the following j,n (2n)!! Rj 0 conditions are respected: (i) (2n−1)!! 2 Rj 2n−1 Dj,n = Gmj (2n + 1)(2n) (2n)!! (i) Rj . r0

ω˙ 2−ω˙ 1 ∆ω ˙ ∆ω¯˙ In (23) the coefficient N is just needed for converting the re- = & ˙ ω˙ 1 th ω˙ 1,th ω¯ oss sult into arcsec units. We point out the following quantities ω˙ 3−ω˙ 2 = ∆ω ˙ < ∆ω¯˙ year ω˙ 2 th ω˙ 2,th ω¯˙ oss which play a significant role in calculating the precession frequency: and in that case we set ω˙ th =ω ˙ 1,th. Calculations can be • Ring mass mj : as we can see from the second and simplified if we recall (19) and (20) and we write the third equation of (23), the precession frequency scales linearly with respect to the ring mass, so that at the lower orders the most massive rings give a ω˙ − ω˙ H − H 2 1 = 2 1 (22) contribution of the same importance as the closest ω˙ 1 th H1 ones.

• The ratio between the mean radius of the planet r0 where H1 and H2 are achieved from (21), by considering and a ring radius Rj . The smaller is such ratio the respectively k and k + 1 terms. On the fround of (8), (12) closer are the orders of a ring series; moreover, sev- and (19)–(21) we obtain the deﬁnitive formula which we eral orders of this ring series appear in the precession

4 frequency. in (22).

(1) (1) (1) (1) (1) (1) (1) (1) • Ring radius r0 of the planet: the first equation in H = − M (B5,1 + B2,1 + B2,2 + B3,1 + B2,3 + B3,2 (23) shows that the precession frequency scales as + B(1) + B(1) + B(1) + B(1) + B(1) + B(1)) 1 : the reason is related with our comment on the 2,4 6,1 2,5 3,3 4,1 5,2 (i)4 4 r0 kg · m = −5.17 · 1048 , (24) moment of inertia, made at the end of Section . s2 So, much closer is the planet to the sun then much H(2) = − M (2)(B(2) + B(2) + B(2) + B(2) + B(2) + B(2) smaller is the opposition to the apsidal line preces- 5,1 3,1 3,2 3,3 3,4 3,5 (2) (2) (2) (2) (2) (2) sion. + B3,6 + B3,7 + B6,1 + B5,2 + B3,8 + B3,9 ) kg · m4 • Revolution period T : again the first of (23) ex- = −7.71 · 1050 , rev s2 hibits a linear dependence of the precession fre- H(3) = − M (3)(B(3) + D(3) + D(3) + D(3) + D(3) + B(3) quency on Trev; an explication in this respect is pro- 5,1 2,1 2,2 2,3 2,4 5,2 vided at the end of the Section . (3) (3) (3) (3) (3) (3) + D2,5 + B6,1 + D2,6 + D2,7 + D2,8 + B4,1 + B(3) + D(3) + B(3) + D(3) + B(3) + B(3) We are going to calculate the precession frequencies of each 4,2 2,9 4,3 2,10 4,4 5,3 4 planet by employing the method illustrated so far and we (3) (3) 51 kg · m + D1,1 + D2,11) = −2.33 · 10 , (25) will explain the difference among planets according to the s2 (4) (4) (4) (4) (4) (4) (4) (4) scaling factors. H = − M (B5,1 + B5,2 + D3,1 + B6,1 + D3,2 + D3,3 (4) (4) (4) The numerical values of the coefficients Bj,n and Dj,n for + D2,1 + B5,3 + D3,4) the planets are listed in the addendum. The data concern- kg · m4 ing the Solar System are achieved from Table 1. = −9.72 · 1050 . (26) s2 The terms are sorted by ascending order in magnitude. An overview on the latter expression shows that the major con- Terrestrial planets Concerning Mercury, Venus, tribution to the precession motion comes from the closest Hearth and Mars, it is necessary to hold many terms of planets and from the most massive ones (as Jupiter and Sat- (21) in order to get close the values of Table 2 it was nec- urn). In particular, the first order of Jupiter’s ring gives the essary holding many terms in (21). largest contribution in all the computed expressions H(1)– The following expressions, listed according to increasing H(4). In table 3 we list the precession frequencies calculated distance of the planet from the Sun, indicate (21) for terres- with the ring model (ω˙ teo) in comparison with the observed trial planets, calculated via the approximation formulated data.

ω˙ teo−ω¯˙ oss Planets ω¯˙ oss ω˙ teo ω¯˙ oss arcsec arcsec ( year )( year ) Mercury 5.75 5.48 0.05 Venus 2.05 11.61 Earth 11.45 12.68 0.11 Mars 16.28 17.23 0.06

Table 3: Theorical precession frequencies compared to observed data

The calculated values are in agreement with the ob- and Mercury. This is the reason why many pertur- served data – eleven percent at worst – except for Venus: bation terms of Venus appear in the expression (25) as to the latter case, the model does not provide consistent concerning the Earth, and many perturbation terms results (actually the relative error has been omitted) ow- of the Earth appear in the expression (26) of Mars. ing to a low eccentricity of that planet (considerably lesser Furthermore, all the terms which appearing (25) and than the other ones) producing a more perturbability of the in (26), are larger than the perturbation terms com- perihelion. perturbations. For such a case, the inadequacy ing form Venus in the expression of Mercury (24). of the calculated values in present also in [5]. Concerning • The ratio between the mean radii of Earth and the rest of the planets, the differences among Mercury,Earth RT Jupiter R = 0.19 and the ratio between the mean and Mars is strong and they can be explained on the ground G radii of Mars and Jupiter RM = 0.29 are larger of the scaling factors introduced in the previous Section: RG than the ratio between the mean radius of Mercury and Jupiter Rm = 0.07: actually RT = 2.7 Rm and • The ratio between the mean radii of Venus and Earth RG RG RG R R RV M m is = 0.72 and it is the largest ratio registered R = 4.1 R . Therefore, Jupiter contributes to per- RT G G among internal or external planets. Immediately af- turbing Earth and Mars to a greater extent than ter the largest ratio is RT = 0.66 between Earth Mercury. RM and Mars. The mass of Venus and of the Earth are • The external field due to the rings is stronger than an order of magnitude greater than those of Mars the internal field, for the same radial distance from

5 a point of the ring. may improve the model by introducing a non zero eccen- • The revolution periods of Earth and Mars are greater tricity for the orbit. Secondly, the assumption of compla- than the revolution period of Mercury (third Ke- nar orbits can be released too, in order to make way for a pler’s law). model providing more realistic data. In line with this think- ing, one can set the problem of motion by employing the All these factors help us explain the disparities in the pre- classical mechanics which encompasses eccentricity and rel- cession frequencies of the Terrestrial planets. ative inclination of the orbit planes and at the same time by still making use of the rings assumption to formulate Jovian planets In order to achieve the degree of accu- perturbation. The geometrical features of the new problem racy listed in Table 2 it is necessary to hold many terms in are more complex and a system of Keplerian coordinates, (21). The following expressions show (21) for each Jovian whose details can be found in [3], are needed. planet, and the stopping criterion is the one we explained It can be seen that the improvement produced by the cor- above. As before, the values of (H(i) are listed by increasing rection leads to a better accuracy and to a concrete close- distance from the Sun: ness of the calculated precession frequencies to the observed data, except for Mercury. In this latter case, the theorical H(5) = − M (5)(B(5) + B(5) + B(5) + B(5) + B(5) + B(5) 6,1 6,2 6,3 6,4 6,5 7,1 value is lower than the one calculated by the not corrected 4 (5) (5) kg · m model and the difference from the observed data is more + B + B ) = −2.66 · 1055 , 6,6 8,1 s2 remarkable: more precisely, the theorical precession fre- (6) (6) (6) (6) (6) (6) (6) (6) arcsec H = − M (D + D + D + D + D + B quency of Mercury becomes 5.32 year versus the observed 5,1 5,2 5,3 5,4 5,5 7,1 arcsec (6) (6) (6) (6) (6) (6) datum 5.75 year ([7]). The disparity can be explained by + D5,6 + B7,2 + B8,1 + D5,8 + B7,3 + D5,9) the proximity to the Sun and the non negligible curvature kg · m4 of the space–time, so that the relavistic effect gives rise to = −9.06 · 1055 , s2 an additional term in the total force: (7) (7) (7) (7) (7) (7) (7) k r 3GM r H = − M (D6,1 + D5,1 + D6,2 + B8,1 + D6,3 F = − + F − tot r2 ||r|| rings c2r4 ||r|| + B(7) + D(7) + B(7) + D(7) + B(7) + B(7) 8,2 5,2 8,3 6,4 8,4 8,5 where the first term is the contribution of the central Kep- kg · m4 + D(7)) = −1.17 · 1055 , lerian force and the second term takes account of the effects 6,5 s2 due to the planets. The latter one is still a classical term (8) (8) (8) (8) (8) (8) (8) H = − M (D6,1 + D5,1 + D7,1 + D7,2 + D7,3 but no longer of central type, owing to the more complex geometrical structure of the problem. The last term is the kg · m4 + D(8)) = −9.38 · 1054 . relativistic contribution of space–time curvature caused by 6,2 s2 the Sun mass. Hence, the total perturbation to the equa- Again, the terms in the sums are sorted by ascending order tion of motion is of magnitude. Concerning the Jovian planets the contribute 3GM r F = F − (27) to the precession mainly comes from the mutual interaction rings c2r4 ||r|| among themselves, since they are much more massive than the terrestrial planets. As a consequence, the values of H(i) As shown above, the precession frequency induced merely are much larger than those computed for the terrestrial set. by the first and the second term in (27) is given by the The precession frequencies for the Jovian planets calculated sum of the two frequencies corresponding to each of the via the ring model are listed in Table 4 and compared to contributions. Once calculations have been carried out, it the observed data. can be seen that the relativistic term is able to explain the difference between the theorical value and the observed one 0.43 arcsec . As one can remark, the attained results fit with the ob- year served data –at worst twenty percent of precision – with the exception of Neptune, which exhibits a precision of seventy 5 Conclusions percent. Furthermore, the differences among the precession frequencies are remarkable. Such differences can be The Lenz vector shows that the ordinary reason for the explained by considering once again the scaling factors: precession motion is the perturbation to the keplerian mo- • The largest ratio that the mean radius of Jupiter tion produced by planets. Actually, the Lenz vector is a forms with any other planet is the one with Saturn constant of motion for the Keplerian motion. For a single RJ is the greatest ratio between internal and exter- planet, the other ones have the effect of an additional force RS nal planets including Jupiter; on the other hand, the acting on it and this entails that Lenz vector is no longer mass of Saturn is the second one in the entire solar a constant of motion but it precedes around a given axis system. These two facts explain the maximum factor according to a specific frequency. On that basis, we for- (21) for Saturn in the overall Solar system. mulated the ring model in order to calculate the planets • The huge mean radii of the two external planets precession frequencies by means of a perturbation method. The model consists of a central force plus a term H Uranus and Neptune produce the lowest precession r3 frequencies. which causes the rotation of the apsidal lines in the revolution orbit plane. The ring model provide consistent results for any planet except for Venus: actually Venus is much 4 Corrections to the ring model more susceptible to perturbations because of its eccentricity, so that the method is not quite reliable for describ- We will give here an explication of the disparity between ing the precession motion. The model shows which are the the theorical results obtained via the ring model and the prevalent factors in the precession frequency: the ring mass observed data. As we already stated, the model is not suit- mj , the ratio between the mean radius of the planet r0 and able for the case of Venus because of its eccentricity: we the ring radius Rj , the ring radius r0, the revolution period

6 Trev. By evaluating such factors for each planet we can ex- the basic assumptions of the ring model produces suscep- plain the dissimilarities in precession frequencies attained tibility to the relativistic effect only for one planet, with a by the ring model. correction of eight percentç we can conclude that the major We stressed the disadvantage in the central force ap- contribution to the precession motion has to be ascribed to proximation of not correctly performing the relativistic ef- the classical effect. fect on the apsidal precession of Mercury: as a matter of facts, the relative error between the theorical value and the observed data exhibits the same order for Mercury and for 6 Addendum: Coefficients nu- the rest of the planets, with the exception of Venus. Such a limit clearly appears if one releases the assumptions of merical values Bj,n and Dj,n complanar and circular orbits. The relativistic correction can be summed up to the classic term in order to obtain We list below the numerical values of the coefficients Bj,n e the correct precession frequency for Mercury. Dj,n, calculated using Matlab, for each planet of the Solar The observed values fit adequately with the theorical system. The units of the coefficients in the SI system arem m4 values obtained with the ring model. Furthermore, leaving [Bj,n] = [Dj,n] = s2 . Mercury

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 Bj,n

(1) 24 24 23 23 23 22 B2,n 4.32 · 10 2.32 · 10 9.69 · 10 3.64 · 10 1.29 · 10 4.40 · 10 (1) 24 23 23 22 21 20 B3,n 2.01 · 10 5.64 · 10 1.23 · 10 2.42 · 10 4.49 · 10 8.01 · 10 (1) 22 21 20 19 18 17 B4,n 6.10 · 10 7.38 · 10 6.95 · 10 5.88 · 10 4.70 · 10 3.61 · 10 (1) 24 22 20 18 16 14 B5,n 4.53 · 10 4.70 · 10 3.79 · 10 2.76 · 10 1.89 · 10 1.24 · 10 (1) 23 20 18 15 12 10 B6,n 2.19 · 10 6.74 · 10 1.61 · 10 3.48 · 10 7.07 · 10 1.38 · 10 (1) 21 18 15 11 8 5 B7,n 4.13 · 10 3.15 · 10 1.87 · 10 9.96 · 10 5.01 · 10 2.43 · 10 (1) 21 17 13 10 6 2 B8,n 1.27 · 10 3.93 · 10 9.50 · 10 2.07 · 10 4.24 · 10 8.36 · 10

Venus

n = 1 n = 2 n = 3 n = 4 n = 5

(2) 25 25 25 25 24 B3,n 2.45 · 10 2.40 · 10 1.83 · 10 1.26 · 10 8.14 · 10 (2) 23 23 23 22 21 B4,n 7.44 · 10 3.14 · 10 1.03 · 10 3.06 · 10 8.52 · 10 (2) 25 24 22 21 19 B5,n 5.52 · 10 2.00 · 10 5.64 · 10 1.43 · 10 3.42 · 10 (2) 24 22 20 18 16 B6,n 2.67 · 10 2.87 · 10 2.40 · 10 1.81 · 10 1.28 · 10 (2) 22 20 17 14 11 B7,n 5.03 · 10 1.34 · 10 2.78 · 10 5.17 · 10 9.09 · 10 (2) 22 19 16 13 9 B8,n 1.54 · 10 1.67 · 10 1.41 · 10 1.07 · 10 7.68 · 10

n = 6 n = 7 n = 8 n = 9

(2) 24 24 24 24 B3,n 5.07 · 10 3.08 · 10 1.83 · 10 1.08 · 10 (2) 21 20 20 19 B4,n 2.29 · 10 5.99 · 10 1.54 · 10 3.89 · 10 (2) 17 16 14 12 B5,n 7.88 · 10 1.77 · 10 3.89 · 10 8.43 · 10 (2) 13 11 9 7 B6,n 8.76 · 10 5.83 · 10 3.80 · 10 2.45 · 10 (2) 9 6 3 B7,n 1.54 · 10 2.54 · 10 4.10 · 10 6.53 (2) 6 3 −3 B8,n 5.30 · 10 3.56 · 10 2.34 1.52 · 10

Earth

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

(3) 23 23 22 21 21 20 D1,n 7.40 · 10 2.08 · 10 4.54 · 10 8.93 · 10 1.66 · 10 2.95 · 10 (3) 25 25 25 25 25 24 D2,n 3.81 · 10 3.74 · 10 2.85 · 10 1.96 · 10 1.27 · 10 7.90 · 10 n = 7 n = 8 n = 9 n = 10 n = 11 n = 12

(3) 19 18 18 17 16 15 D1,n 5.14 · 10 8.76 · 10 1.47 · 10 2.44 · 10 4.02 · 10 6.55 · 10 (3) 24 24 24 23 23 23 D2,n 4.80 · 10 2.86 · 10 1.68 · 10 9.72 · 10 5.58 · 10 3.18 · 10

7 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

(3) 24 24 24 23 23 23 B4,n 2.72 · 10 2.20 · 10 1.38 · 10 7.80 · 10 4.16 · 10 2.14 · 10 (3) 26 25 23 22 21 19 B5,n 2.02 · 10 1.40 · 10 7.53 · 10 3.65 · 10 1.67 · 10 7.35 · 10 (3) 24 23 21 19 17 15 B6,n 9.76 · 10 2.01 · 10 3.21 · 10 4.61 · 10 6.26 · 10 8.17 · 10 23 20 18 16 14 11 B7,n(3) 1.84 · 10 9.36 · 10 3.71 · 10 1.32 · 10 4.44 · 10 1.44 · 10 (3) 22 20 17 14 21 8 B8,n 5.64 · 10 1.17 · 10 1.89 · 10 2.74 · 10 3.75 · 10 4.94 · 10

Mars

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

(4) 27 26 25 24 23 22 B5,n 1.09 · 10 1.75 · 10 2.19 · 10 2.46 · 10 2.61 · 10 2.67 · 10 (4) 25 24 22 21 19 18 B6,n 5.26 · 10 2.51 · 10 9.30 · 10 3.11 · 10 9.77 · 10 2.96 · 10 (4) 23 22 20 17 15 13 B7,n 9.90 · 10 1.17 · 10 1.08 · 10 8.89 · 10 6.93 · 10 5.21 · 10 (4) 23 21 18 16 13 11 B8,n 3.04 · 10 1.46 · 10 5.47 · 10 1.84 · 10 5.86 · 10 1.79 · 10

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

(4) 23 22 21 20 19 18 D1,n 4.86 · 10 5.88 · 10 5.54 · 10 4.69 · 10 3.75 · 10 2.88 · 10 (4) 25 25 24 24 23 22 D2,n 2.50 · 10 1.06 · 10 3.48 · 10 1.03 · 10 2.87 · 10 7.70 · 10 (4) 25 25 25 25 24 24 D3,n 5.87 · 10 4.74 · 10 2.98 · 10 1.69 · 10 8.99 · 10 4.61 · 10 n = 7 n = 8 n = 9 n = 10 n = 11 n = 12

(4) 17 16 15 13 12 11 D1,n 2.16 · 10 1.59 · 10 1.15 · 10 8.21 · 10 5.82 · 10 4.09 · 10 (4) 22 21 21 20 19 19 D2,n 2.02 · 10 5.17 · 10 1.31 · 10 3.27 · 10 8.08 · 10 1.98 · 10 (4) 24 24 23 23 23 22 D3,n 2.31 · 10 1.13 · 10 5.47 · 10 2.61 · 10 1.24 · 10 5.80 · 10

Jupiter

n = 1 n = 2 n = 3 n = 4 n = 5

(5) 27 27 27 26 26 B6,n 7.15 · 10 3.98 · 10 1.72 · 10 6.70 · 10 2.46 · 10 (5) 26 25 24 23 22 B7,n 1.35 · 10 1.86 · 10 1.99 · 10 1.92 · 10 1.74 · 10 (5) 25 24 23 21 20 B8,n 4.13 · 10 2.32 · 10 1.01 · 10 3.98 · 10 1.47 · 10 n = 6 n = 7 n = 8 n = 9 n = 10

(5) 25 25 25 24 24 B6,n 8.69 · 10 2.99 · 10 1.01 · 10 3.36 · 10 1.11 · 10 (5) 21 20 19 17 16 B7,n 1.53 · 10 1.30 · 10 1.09 · 10 8.99 · 10 7.32 · 10 (5) 18 17 15 14 12 B8,n 5.26 · 10 1.83 · 10 6.23 · 10 2.09 · 10 6.94 · 10

n = 1 n = 2 n = 3

(5) 23 21 19 D1,n 1.42 · 10 1.48 · 10 1.19 · 10 (5) 24 23 21 D2,n 7.33 · 10 2.65 · 10 7.48 · 10 (5) 25 24 22 D3,n 1.72 · 10 1.19 · 10 6.42 · 10 (5) 24 23 22 D4,n 4.29 · 10 6.89 · 10 8.61 · 10

Saturn

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

(6) 27 26 26 25 25 24 B7,n 1.53 · 10 7.11 · 10 2.57 · 10 8.35 · 10 2.56 · 10 7.56 · 10 (6) 26 25 25 24 23 22 B8,n 4.69 · 10 8.88 · 10 1.31 · 10 1.73 · 10 2.16 · 10 2.60 · 10

8 n = 1 n = 2 n = 3 n = 4 n = 5

(6) 22 20 17 15 12 D1,n 7.75 · 10 2.39 · 10 5.71 · 10 1.23 · 10 2.50 · 10 (6) 24 22 20 18 16 D2,n 3.99 · 10 4.29 · 10 3.59 · 10 2.70 · 10 1.91 · 10 (6) 24 23 21 19 17 D3,n 9.36 · 10 1.92 · 10 3.07 · 10 4.42 · 10 6.00 · 10 (6) 24 23 21 20 18 D4,n 2.33 · 10 1.11 · 10 4.13 · 10 1.38 · 10 4.34 · 10 (6) 28 28 28 27 27 D5,n 8.05 · 10 4.48 · 10 1.94 · 10 7.54 · 10 2.77 · 10 n = 6 n = 7 n = 8 n = 9 n = 10

(6) 9 6 4 −2 D1,n 4.89 · 10 9.32 · 10 1.74 · 10 3.21 · 10 5.84 · 10 (6) 14 11 9 7 5 D2,n 1.31 · 10 8.70 · 10 5.68 · 10 3.65 · 10 2.32 · 10 (6) 15 13 12 10 8 D3,n 7.83 · 10 9.97 · 10 1.24 · 10 1.53 · 10 1.86 · 10 (6) 17 15 14 12 10 D4,n 1.31 · 10 3.88 · 10 1.12 · 10 3.21 · 10 9.04 · 10 (6) 26 26 26 25 25 D5,n 9.79 · 10 3.37 · 10 1.14 · 10 3.79 · 10 1.25 · 10

Uranus

n = 1 n = 2 n = 3 n = 4 n = 5

(7) 27 27 27 27 26 B8,n 7.65 · 10 5.84 · 10 3.47 · 10 1.86 · 10 9.36 · 10 n = 6 n = 7 n = 8 n = 9 n = 10

(7) 26 26 25 25 25 B8,n 4.54 · 10 2.15 · 10 9.96 · 10 4.55 · 10 2.06 · 10

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

(7) 22 19 16 12 9 6 D1,n 3.86 · 10 2.94 · 10 1.74 · 10 9.31 · 10 4.69 · 10 2.27 · 10 (7) 24 21 19 16 13 10 D2,n 1.99 · 10 5.29 · 10 1.10 · 10 2.04 · 10 3.59 · 10 6.08 · 10 (7) 24 22 19 17 15 12 D3,n 4.66 · 10 2.37 · 10 9.39 · 10 3.35 · 10 1.12 · 10 3.64 · 10 (7) 24 22 20 18 15 13 D4,n 1.16 · 10 1.37 · 10 1.26 · 10 1.04 · 10 8.13 · 10 6.11 · 10 (7) 28 27 26 25 24 23 D5,n 4.01 · 10 5.52 · 10 5.92 · 10 5.71 · 10 5.19 · 10 4.55 · 10 (7) 28 28 27 27 26 26 D6,n 4.05 · 10 1.88 · 10 6.79 · 10 2.21 · 10 6.77 · 10 2.00 · 10

Neptune

n = 1 n = 2 n = 3 n = 4 n = 5

(8) 22 18 15 11 7 D1,n 2.46 · 10 7.65 · 10 1.85 · 10 4.02 · 10 8.24 · 10 (8) 24 21 18 14 11 D2,n 1.27 · 10 1.38 · 10 1.16 · 10 8.81 · 10 6.31 · 10 (8) 24 21 18 16 13 D3,n 2.97 · 10 6.17 · 10 9.95 · 10 1.44 · 10 1.98 · 10 (8) 23 21 19 16 14 D4,n 7.41 · 10 3.57 · 10 1.34 · 10 4.50 · 10 1.43 · 10 (8) 28 27 25 24 22 D5,n 2.56 · 10 1.44 · 10 6.27 · 10 2.46 · 10 9.13 · 10 (8) 28 27 26 25 25 D6,n 2.58 · 10 4.89 · 10 7.19 · 10 9.53 · 10 1.19 · 10 (8) 28 28 27 27 27 D7,n 1.59 · 10 1.22 · 10 7.22 · 10 3.86 · 10 1.95 · 10

n = 6 n = 7 n = 8 n = 9

(8) 4 −4 −7 D1,n 1.63 · 10 3.13 5.91 · 10 1.10 · 10 (8) 8 5 2 −1 D2,n 4.35 · 10 2.92 · 10 1.93 · 10 1.25 · 10 (8) 10 7 4 D3,n 2.61 · 10 3.35 · 10 4.21 · 10 5.23 · 10 (8) 11 9 6 4 D4,n 4.38 · 10 1.30 · 10 3.81 · 10 1.10 · 10 (8) 21 20 18 17 D5,n 3.26 · 10 1.13 · 10 3.86 · 10 1.30 · 10 (8) 24 23 22 21 D6,n 1.43 · 10 1.68 · 10 1.93 · 10 2.18 · 10 (8) 26 26 26 25 D7,n 9.46 · 10 4.47 · 10 2.07 · 10 9.48 · 10 References 702 2013

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