Least-Action Principle Applied to the Kepler Problem

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Least-action Principle Applied to the Kepler Problem

S. Olszewski Institute of Physical Chemistry of the Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland Reprint requests to Prof. S. O.; e-mail: [email protected]

Z. Naturforsch. 59a, 375 – 381 (2004); received March 1, 2004

The least-action principle is examined for the two-body Kepler problem. This examination allows one to couple the eccentricity parameter of the Kepler orbit with the size of the major semiaxis of that orbit. The obtained formula is applied to an estimate of eccentricities characteristic for a set of the planetary and satellitary tracks.

Key words: Eccentricity of the Kepler Orbits; Least-action Principle.

1. Introduction: Dependence of the Eccentricity e the motion of the system. In the next step, the same on the Major Semiaxis a0 of the Kepler Orbit action variables J1,J2,... occur useful in the quantiza- tion process which takes an easy form according to the The Kepler problem is one of the best known in clas- well-known Bohr-Sommerfeld rules [9, 10]. sical mechanics. Being based partly on observations, it But beyond of J1,J2,... an elementary action func- accounts of the motion of a planet, or satellite, along an tion S0 can be defined, equal to double the kinetic en- elliptical trajectory extended about a gravitational cen- ergy of a body integrated over a time interval passed ter. The period T of the motion and the major semiaxis by that body along its trajectory. The least-action prin- a0 are coupled by the well-known Kepler law ciple states that – for a conservative system – this ac- 2 = π2 3, tion function should be at minimum [8, 11]. In the next GMST 4 a0 (1) step, the derivative of S0 calculated with respect to the energy E of the body at some time t provides us with where G is the gravitational constant, M is the mass p S the time interval of the motion of the body, beginning of the Sun in the planet case, or the mass of the planet from some time t : in the satellite case. In (1) it is assumed – for the sake 0 of simplicity – that the mass MS remains at rest. Si- ∂S 0 = t −t . (2) multaneously, the eccentricity e defining the elliptical ∂E 0 trajectory of a planet, or satellite, is considered as in- p dependent of a0 or T [1 – 6]. The aim of the present paper is to point out that (2) can In spite of the fundamental importance of the Kepler be satisfied only on the condition that the eccentricity problem its mechanics seems to be discussed insuffi- e of the planetary, or satellitary, track depends on the ciently. Here we have in mind the least-action principle major semiaxis a0 of the Kepler orbit. applied to a conservative system. The principle makes The Cartesian coordinates of a body on the Kepler reference to the variational foundations of mechanics orbit are and is based on the idea of the action function. In prac- = ( − ), tice, the action variables J1,J2,..., considered for the x a0 cosu e (3) Kepler problem, are parameters conjugated to the cor- 2 1/2 responding angle variables of the moving body [7, 8]. y = a0(1 − e ) sinu, (3a) Both kinds of the action and angle variables serve to where the coordinate u depends on the time t by the transform the Hamiltonian into a function dependent equation [8] solely on J1,J2,... . This technique is useful because it enables one to calculate the frequencies of the pe- 2π t ≡ ωt = u − esinu. (3b) riodic motion without finding a complete solution of T

0932–0784 / 04 / 0600–0375 $ 06.00 c 2004 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen · http://znaturforsch.com 376 S. Olszewski · Least-action Principle Applied to the Kepler Problem

Here we assumed that the beginning of the motion is at ∂S0 = ∂ ∂ the perihelion of the Kepler orbit [8]. Obviously, t 0 = S0 = a0 = t2 at u 0; the frequency of the motion about the gravi- ∂Ep ∂Ep tation center has been put equal to ω. ∂a0 For a moment let us assume that e is constant in a ∂ω way similar to a . Because of (3), (3a) and (3b) we ω + 2 ( + ) 0 2a0 a0 ∂ u2 esinu2 (8) have = a0 GMS 2 2 2a0 2 + 2 = 2 2 +( − 2) 2 du x˙ y˙ a0 sin u 1 e cos u 3ω( + ) dt a0 u2 esinu2 1 = = (u2 + esinu2). 1 GM ω = a2(1 − e2 cos2 u)ω2 (4) S 0 (1 − e2 cosu)2 In the calculation of (8) we took into account + = 2ω2 1 ecosu. a0 1 − ecosu GMSmp Ep = − , (9) 2a0 The integral of the expression (4), extended over an which is the energy of the planet or satellite, as well as interval of t beginning with t = t = 0 and ending at 1 the relations (1) and (3b). Since from the third Kepler some t = t ,gives 2 law given in (1) we have

t2 t2 ∂T 3 T 1 + ecosu = , (10) (x˙2 + y˙2)dt = a2ω2 dt ∂a 2 a 0 1 − ecosu 0 0 0 0 u 2ω 2 the derivative of the expression a0 entering (7) calcu- = 2ω ( + ) (5) a0 1 ecosu du lated with respect to a0 gives 0 = ∂(a2ω) ∂ω = 2ω( + )u u2 , 0 = 2a ω + a2 a0 u esinu = ∂a 0 0 ∂a u 0 0 0 1 3 T = 2a ω + 2πa2 − (11) 0 0 T 2 2 a since u(t1)=u(0)=0 and 0 1 = a0ω. du ω 2 = ; (6) dt 1 − ecosu Our task is now to compare the result of (8) with t2 = t2(u2) calculated from (3b) for an arbitrary fraction of see (3b). The integral giving the elementary action S 0, the period T of rotation of the celestial body about the equal to the integral of the double kinetic energy ex- gravitational center. This is done for different values of tended along the time interval (0,t2), is t2 = λT, (12)

t2 where λ is a constant: 0 < λ ≤ 1. The mentioned com- S = m (x˙2 + y˙2)dt 0 p parison is presented in Table 1, where – for the sake 0 (7) of convenience – the variable u = u has been taken u = u 2 = m a2ω(u + esinu) 2 , as a parameter. It is evident that solely at λ = 1 and p 0 u = 0 2 λ = 1 there exists an agreement between t2 = t2(u2) taken from (3b) and t2 calculated from (8); the case where mp is the mass of a planet or satellite. λ = 0 has been omitted as being a trivial one. The fundamental equation (2), valid for any planet A discrepancy obtained between t2 in (3b) and t2 or satellite trajectory, becomes therefore given by Eq. (8), illustrated by the second and the third S. Olszewski · Least-action Principle Applied to the Kepler Problem 377

Table 1. A comparison of the time interval λT of the motion In the next step, because of the relation of a celestial body performed along the Kepler orbit with the time interval calculated from the elementary action function; Ep see (3b), (8) and (12). The orbit eccentricity e is assumed ∂ m GM 2π2a 1 equal to a constant number. T is the circulation period of the t p = t S = t 0 = a ω2t , (16) body about the gravitational center. The beginning point of 2 ∂ 2 2 2 2 0 2 a0 2a0 T 2 the motion is assumed in the perihelion when the position of the center of force is assumed in the Sun. the expression (15), substituted into the ﬁrst part of (8), u2 λ obtained from t2 λ obtained from t2 gives see (3b) for a reference calculated calculated in (8) between t and u according to (3b) ∂ ∂ 2 2 √ √ 1 1 2 e 2 y 1 1 2 1 2 0 = a ωe + a ωy + a ω + a ω π − π e + π e 0 0 0 0 4 8 4 8 4 2 2 ∂a0 ∂a0 1 π 1 − 1 1 + 1 (17) 2 4 2π e 4 2π e ∂ √ √ = 1 ω( + )+ 2ω ( + ), 3 π 3 − 2 e 3 + 2 e a0 e y a0 e y 4 8 4π 8 4π 2 ∂a0 π 1 1 2 √ 2 √ or 5 π 5 + 2 5 − 2 4 8 4π e 8 4π e 3 3 1 3 1 ∂ π + π e − π e 2 4 √2 4 √2 e + y = −2a0 (e + y), (17a) 7 π 7 + 2 7 − 2 ∂a0 4 8 4π e 8 4π e π 2 11 where

column of Table 1, implies that e entering S 0 should y = esin(ωt2 + e ). (18) depend on a0: The solution of (17a) is e = e(a0). (13) 1/2( + ) = ln a0 e y const (19)

2. Equation for e(a0) or 1/2 + (ω + ) = . ( ) This dependence can be examined when the Fourier a0 e esin t2 e const 19a expansion for u is applied [6, 12, 13]: It is evident from (19a) that e depends both on a 0 ∞ 1 and t2. For example an explicit differentiation of (19a) u = ωt + 2 ∑ J (ne)sin(nωt). (14) n with respect to a0 gives successively the equations n=1 n − / / ∂e Here J is the nth-order Bessel function of the ﬁrst 1 2 + 1 2 = n ea0 2a0 ∂ 0 (20) kind. Because e is a small number, the expansion (14) a0 – as well as the power expansions for J – can be lim- π n for ωt = and ωt = 3 π, ited to either one or two ﬁrst terms. By substituting the 2 2 2 2 = √ √ expansion (14) for u u2 into the derivative of S0,we 1 − / 1 / ∂e ea 1 2 + e +a1 2 + e = ( ) obtain 0 2 0 2 ∂ 0 20a 2 2 a0 S ∂ 0 π 7 for ωt2 = and ωt2 = π, and mp ∼ 1 4 4 = a ω[ωt +e +esin(ωt +e )] (15) ∂ 0 2 2 √ √ a0 2 1 − / 1 / ∂e ea 1 2 − e +a1 2 − e = ( ) 0 2 0 2 ∂ 0 20b ∂ ∂ ∂ 2 2 a0 + 2ω e + e (ω + )+ (ω + ) e , a0 sin t2 e ecos t2 e ∂a0 ∂a0 ∂a0 ω = 3 π ω = 5 π 1 for t2 4 and t2 4 . If the terms 2 e and e en- where tering the brackets in (20a) and (20b) are omitted, these equations are reduced to the same differential equa- e ≈ esin(ωt2). (15a) tion for e(a0), which is given in (20). For ωt2 = π and 378 S. Olszewski · Least-action Principle Applied to the Kepler Problem

ωt2 = 2π Eq. (20) is fulﬁlled identically. A solution and perform the variational calculation with the aid of of (20) is τ instead of t. For the case of the Kepler problem a natural substitution for τ is 1/2 = a0 e const (21) u = u(t), (22a) which is a result proportional to an average of (19a) T where u is the eccentric anomaly introduced in Sect. 2 calculated over 2 . Evidently, the average of (19a) calculated over the whole time period T is equal to zero. and coupled with t by the formula (3b). In effect, for Equation (21) implies that the eccentricities e1 and e2 the problem of a planar motion we obtain the require- of two non-interacting bodies 1 and 2 circulating about ment [15, 16] a common gravitational center are referred to the major semiaxes a and a of the bodies trajectories by dx 2 dy 2 01 02 δ Ep −V mp + du = 0, (23) the equation du du

/ where E is the total energy (9) and e a1 2 p 1 = 02 . (21a) 1/2 e2 a GMSmp 01 V = − (9a) (x2 + y2)1/2 The results of (21) and (21a) are used in Section 4. is the potential energy of the moving body. Because 3. Least-action Principle for the Kepler Problem, of (3) and (3a) we have Examined in the Framework of the Euler-Jacobi 2 + 2 = 2( − )2, Formulation x y a0 1 ecosu (24) and in view of (3) and (3a) Equation (20) can be obtained also from a direct ap- dx 2 dy 2 plication of the so-called Euler-Jacobi formulation of + = a2[sin2 u +(1 − e2)cos2 u] u u 0 the least-action principle [14 – 16]. This formulation d d (24a) had as its main idea, to transform the time t into some = 2( − 2 2 ), a0 1 e cos u new variable

τ = τ(t) (22) so the expression submitted to variation in (23) becomes: 1/2 + GMSmp 2( − 2 2 ) 1/2 Ep ma0 1 e cos u (1 − ecosu)a0 / 2 2 2 2 1 2 GMSm a (1 − e cos u) = E m a2(1 − e2 cos2 u)+ p 0 p p 0 (1 − ecosu)a 0 (25) / = 2 + 2 + 2 − 2 2 2 1 2 Epmpa0 GMSmpa0 GMSmpa0ecosu Epmpa0e cos u 1/2 1/2 1 / 1 = GM m2a 1 + 2ecosu + e2 cos2 u 1 2 = GM m2a (1 + ecosu). 2 S p 0 2 p p 0

/ In the last but one transformation in (25) we took into 1 1 2 = δ GM m2a du + e cosudu = 0. account the formula (9). Due to (25), the variational 2 S p 0 problem of (23) becomes A full gyration of a body about a gravitational center provides us with the interval / 1 1 2 δ GM m2a (1+ecosu)du (26) 2 S p 0 0 ≤ u ≤ 2π. (27) S. Olszewski · Least-action Principle Applied to the Kepler Problem 379

In this case the e-dependent component of the integral Table 2. Theoretical eccentricities etheor of the planet tra- entering (26) vanishes. Consequently, by allowing for jectories calculated on the basis of (21a) compared with the obs variation of a0, we obtain from (26) and (27) that observed eccentricities e ; see [17]. The observed eccentricity of the Jupiter trajectory has been taken as the basic pa- / theor 1 1 2 rameter in order to define e of the orbits of the remaining δ GM m2a 2π planets. The major semiaxis a ≡ x of the planet trajectories 2 S p 0 0 (28) are also listed with the x value of Jupiter put equal to 1. 1/2 1 2 −1/2 Planet xetheor eobs Error of etheor = GMSm πa δa0 = 0 2 p 0 see [17] see (32) Mercury 0.07440 0.176 0.2056 0.17 or Venus 0.13903 0.129 0.0068 17.97 Earth 0.19220 0.109 0.0167 5.53 δa0 = 0. (28a) Mars 0.29286 0.089 0.0933 0.05 Jupiter 1.00000 – 0.0480 – In view of (28a) the term equal to the first component Saturn 1.83338 0.035 0.0560 0.60 of (26) can be considered as a constant. We assume that Uranus 3.68864 0.025 0.0460 0.84 solely the second component of the expression (26) Neptune 5.77782 0.020 0.0100 1.00 Pluto 7.59759 0.017 0.2480 13.60 may provide us with a coupling between e and a 0.By / 1 2 1 2 neglecting the constant factor of 2 GMSmp we ob- Table 3. Orbits of the satellites of Jupiter and their eccentrici- tain from the second integral in (26) between the limits ties. The observed eccentricity (eobs) of Elara has been taken theor u0 and u1: as a known parameter in the calculation of e of the remaining satellite orbits; (21a). The major semiaxis x = a0 of δ( 1/2 )( − )= . the orbit of Elara is taken as a unit distance. a0 e sinu1 sinu0 0 (29) Satellite xetheor eobs Error of etheor This gives us a dependence between e and a0 because see [17] see (32) the sin-difference in (29) can be considered as a multi- Callisto 0.160 0.517 0.010 50.70 plier which does not influence the variational process. Leda 0.946 0.213 0.146 0.46 Himalia 0.977 0.209 0.158 0.32 In effect, from (29) we have Lysithea 0.977 0.207 0.130 0.59 Elara 1.000 – 0.207 – 1 − / / 1 2δ + 1 2δ = , ( ) Ananke 1.763 0.156 0.170 0.11 a0 a0e a0 e 0 29a 2 Carme 1.904 0.150 0.210 0.40 Pasiphae 1.985 0.147 0.380 1.58 1 δ which – when divided by 2 a0 – gives the equation Sinope 2.019 0.146 0.280 0.92 equivalent to (20). The same result can be obtained by applying the variational procedure to the original ex- and taking into account that pression of the kinetic energy [8, 11, 13]: Jupiter = . , ( ) e 0 048 31a dx 2 dy 2 δ m + dt = 0, (30) theor p dt dt the eccentricities e of the remainder eight planets of the solar system can be calculated. The results for theor obs called the Euler-Maupertuis formulation of the least- e , together with the observed data e for e and the = action principle. planet distances a0 x, are listed in Table 2. The error given in Table 2, and subsequent Tables, is defined by 4. Calculation of Eccentricities for the Planet and the relation Satellite Trajectories. Discussion e> error = − 1, (32) e< Equation (21) or (21a) can be readily applied to the calculation of e of the Kepler tracks. From the nine where e> is the larger and e< is the smaller quantity planets arranged according to their increasing distance of the pair of eobs and etheor obtained for a given planet from the Sun, the planet Jupiter is a central one; this is (or satellite); see Tables 2 – 5. also the heaviest planet of the system [17]. Putting The error (32) is below unity for Mars, Saturn and Uranus, which have their orbits closest to that of Jupiter ≡ Jupiter = a0 x 1 (31) Jupiter. More distant planet orbits, which are Earth and 380 S. Olszewski · Least-action Principle Applied to the Kepler Problem Table 4. Orbits of the satellites of Uranus and their eccentric- in their number, are placed on not much distant orbits obs ities. The observed eccentricity (e ) of Umbriel has been and have not very much different masses. The assump- theor taken as a known parameter in the calculation of e of the tion of a coplanar position of the orbits having a com- remaining satellite orbits of Uranus; (21a). The major semi- mon gravitational center can be another source of the axis x = a0 of the orbit of Umbriel is taken as a unit distance. error. theor obs theor Satellite xe e Error of e A much better situation is attained for the satellites see [17] see (32) Miranda 0.488 0.0050 0.0170 2.40 of Jupiter, for which the well-established observed ec- Ariel 0.718 0.0041 0.0028 0.46 centricities beginning with Callisto and the next eight Umbriel 1.000 – 0.0035 – satellites – taken successively with an increasing dis- Titania 1.640 0.0027 0.0024 0.13 tance from the Jupiter center – are considered; see Ta- Oberon 2.193 0.0024 0.0007 2.43 ble 3. The observed eccentricity of Elara, having a central position of its orbit distance from the Jupiter cen- theor Table 5. Theoretical eccentricities e of the orbits taken ter, has been taken as a starting point to calculate the as a basis in the calculations presented in Table 2, 3, and 4 (Jupiter, Elara and Umbriel). The etheor = e are calculated orbit eccentricities of the remainder of the Jupiter satel- i lites. A very large error of etheor is found solely for Cal- from (33) with the aid of the observed data (ei−1,ei+1 for theor the eccentricities and a0,i−1,a0,i+1 for the major semiaxes) listo, but from the seven remaining e , six of them of the orbits neighboring to i and the major semiaxis a0,i of (those for Leda, Himalia, Lysithea, Ananke, Carme and the orbit i. Sinope) have their error below 1, and the error of one of Considered Bodies on the etheor eobs Error of etheor etheor (that of the orbit of Pasiphae) is not much above celestial neighboring see [17] calculated that number. body orbits from (32) (i)(i − 1,i + 1) A similar comparison of the eccentricities is done Jupiter Mars, Saturn 0.0630 0.0480 0.31 for the five satellites of Uranus (Miranda, Ariel, Um- Elara Lysithea, Ananke 0.1780 0.2070 0.16 briel, Titania, Oberon) for which the observed eccen- Umbriel Ariel, Titania 0.0024 0.0035 0.46 tricity of Umbriel has been chosen to establish the remainder of the satellite eccentricity data; Table 4. The Neptune, have their error equal to 5.5 and 1.0, respec- satellites which have their orbits closest to that of Um- tively. The largest error – much above 10 – is obtained briel give the error of etheor much less than 1.0 (Ariel for Venus and Pluto. Rather accidentally, the error of and Titania), whereas the same error for the exter- theor e for Mercury is much below 1.0. These data show nal satellite orbits, those of Miranda and Oberon, ex- that for numerous planets the dependence of e on a 0 ceeds 2.0. should be in fact much more complicated than that rep- An estimate of eccentricities of the orbits taken as resented by (21). The complication can be attributed the basis of calculations done in Table 2, 3, and 4 to the many-body interactions between planets. A de- (Jupiter, Elara and Umbriel, respectively) can be at- e a pendence of on 0, different than that given in (21), tained by considering the observed data for e and a 0 should apply especially to a group of relatively light of the two neighboring orbits of these bodies (Mars and relatively close planet orbits, like those of Mer- and Saturn for Jupiter, Lysithea and Ananke for Elara, cury, Venus, and Earth. Ariel and Titania for Umbriel); see Table 5. The final A comparison of the observed data with the theo- result for etheor is calculated as the arithmetical mean retical values obtained for e seems to be more diffi- of etheor obtained from any member of the mentioned cult for the satellite orbits than for the planetary ones orbit pair. Therefore, with the aid of (21a), we have because numerous eccentricity data for the satellite tracks are either variable, or uncertain, and the influ- 1 1/2 1/2 e = e − a + e + a , (33) ence of the many-body effects exerted on the solu- i 1/2 i 1 0,i−1 i 1 0,i+1 2a , tion of the two-body Kepler problem is here much in- 0 i tensified. Many planets – especially Jupiter, Saturn, where the index i labels the examined orbit, and Uranus and Neptune – have a large number of satellites whose a0 are approximately equal, and the masses a0,i−1 < a0,i < a0,i+1. (33a) of these satellites are often very similar [17]. For example, the largest error of etheor is found in the case The error of all etheor presented in Table 5 is much be- of the satellites of Saturn. These satellites are large low 1.0. S. Olszewski · Least-action Principle Applied to the Kepler Problem 381 5. Summary principle becomes equivalent to the Lagrange principle [18]. The least-action principle is applied to the motion In Sect. 1 we demonstrated that the Sommerfeld- of a celestial body along the Kepler orbit. This prin- Landau formula for the derivative of the action func- ciple – called also the Euler-Maupertuis principle – tion with respect to energy cannot be satisfied unless is only a special case of more general Lagrange or the eccentricity of the orbit is a function of the ma- Hamilton principles valid upon the following restric- jor orbital semiaxis. The corresponding equation, cou- tions [18]: (i) the Lagrangian function should not be pling the eccentricities and major semiaxes of the or- explicitly time dependent, so the energy of the sys- bits having a common gravitational center, is derived tem is a conserved quantity both on the actual and the in Section 2. The same equation can be derived by a varied paths; (ii) for the varied paths the changes of direct application of the least-action principle in both the body position at the terminal positions should van- the Euler-Jacobi and conventional (Maupertuis) varia- ish, which – for a closed path – makes this require- tional formulations; Section 3. The results of the equa- ment applicable to the beginning-end point of the path; tion which couples e and a0 are discussed in compari- (iii) the potential function entering the Lagrangian son with the observed data in Section 4. No influence should be velocity-independent. With the above re- of the mass ratio of the rotating body to that of the cen- strictions, which can be assumed to be satisfied for a tral body on the equation defining the orbit eccentricity body moving on the Kepler orbit, the Euler-Maupertuis is considered.

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