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The Saturn, Janus and Epimetheus Dynamics As a Gravitational Three-Body Problem in the Plane

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The Saturn, Janus and Epimetheus Dynamics As a Gravitational Three-Body Problem in the Plane INVESTIGACION´ REVISTA MEXICANA DE FISICA´ 55 (2) 97–105 ABRIL 2009 The Saturn, Janus and Epimetheus dynamics as a gravitational three-body problem in the plane A. Bengochea and E. Pina˜ Department of Physics, Universidad Autonoma´ Metropolitana - Iztapalapa, P.O. Box 55 534, Mexico, D.F., 09340 Mexico, e-mail: [email protected], [email protected] Recibido el 7 de abril de 2008; aceptado el 13 de marzo de 2009 Using a coordinate system given by the principal axis of inertia, as determined by an angle, and also two distances related to the principal moments of inertia and an auxiliary angle as coordinates, we consider the Three Body Problem, interacting through gravitational forces in a plane. The dynamics of the triple Saturn-Janus-Epimetheus has been considered in these coordinates as an adiabatic perturbation of the classical equilateral triangle Lagrange solution and of the collinear Euler solution. The co-orbital motion remembering the Saturn-Janus- Epimetheus behavior is then developed theoretically based on numerical and experimental evidence. Keywords: Three-body problem; Saturn; Janus; Epimetheus. Se usa un sistema de coordenadas dado por los ejes principales de inercia, determinados por un angulo,´ y ademas,´ dos distancias relacionadas a los momentos principales de inercia y un angulo´ auxiliar. Consideramos al problema de tres cuerpos, que interacciona a traves´ de fuerzas gravitacionales en un plano. La dinamica´ de la terna Saturno-Jano-Epimeteo se ha descrito en estas coordenadas como una perturbacion´ adiabatica´ de las soluciones clasicas´ triangular equilatera´ de Lagrange y la solucion´ colinear de Euler. El movimiento co-orbital semejante al comportamiento de Saturno-Jano-Epimeteo se desarrolla teoricamente´ basados en evidencia numerica´ y experimental. Descriptores: Problema de tres cuerpos; Saturno; Jano; Epimeteo. PACS: 45.50.Pk; 95.10.Ce 1. Introduction Yoder et al. [8, 9], derived a simple analytic approxima- tion to the motion of all the Trojans and applied it to the Janus Voyager 1 flight confirmed the existence of two minor satel- and Epimetheus motion. Llibre and Olle´ [10] proved the ex- lites of Saturn: Janus and Epimetheus, providing a first es- istence of stable horseshoe orbits for the value of the mass timate of their masses as well as their orbital elements. A parameter of Saturn and Janus in the context of the circular detailed history of this discovery was given by Aksnes [1]. restricted three body problem. These two moons became familiar after the Voyager 1 Waldvogel & Spirig [11–13], studied the problem of Sat- travel in space. They received a strong attention from differ- urn’s coorbital satellites by means of a singular perturbation ent researchers. approach, that is, the motion is initially described by an outer Some of them consider their orbits from a numerical point and an inner approximation valid when the satellites are far of view Janus and Epimetheus roughly share the same mean apart or close together, respectively; the complete description orbit of a radius of 151440 km from Saturn and a frequency of the motion requires the matching of both approximations. near 518 deg day¡1 [2], they are called coorbital. One of these moons is faster than the other by the small amount of Corresponding to the Hill’s problem, Petit & velocity produced by a difference of radius of 50 km. Each Henon [14, 15] dealt with satellite encounters in the frame- 4 years [3] the two moons become 15000 km apart, but their work of Hill’s problem from the analytical and numerical mutual gravitational interaction prevents collision, and they points of view. switch orbits, the one in the interior orbit becoming exterior Cors and Hall [16] approach this problem analytically and viceversa, the exterior becoming interior at the moment by introducing small parameters into the usual general three of close approach called the encounter. Colombo [4], esti- body problem in the plane, truncating high order terms and mates the amplitude of the librations of the moons in a ro- deriving dynamic information from the resulting equations. tating frame. The Cassini imaging observations at Saturn in A new perspective is undertaken in this paper with the use 2006 resulted in new data [5] and analysis to fix the parame- of new coordinates but regarding the Three-Body Problem ters for the dynamics of the Saturn’s moons and numerically without the simplifications of the restricted models. The use calculate their orbits. of new coordinates was recommended by the extreme simpli- The main theoretical approach has been made in the con- fication that the integrable Euler and Lagrange cases attained text of the restricted problems in the plane. Dermott and in these coordinates; and the relation of these two integrable Murray [6, 7], gave the description of the coorbital motion cases with the actual dynamics of Janus and Epimetheus rec- of Janus and Epimetheus based on a combination of numeri- ognized by most of the cited authors. Our main purpose is to cal integration and perturbation theory, and extrapolate some find in this paper new analytical approximations for the mo- results to the case when the third mass is not negligible. tion of Janus and Epimetheus in these variables, at least not 98 A. BENGOCHEA AND E. PINA˜ close to their encounter where we study the motion doing nu- which fixes the two directions of a and b in the plane orthog- merical integration as many of the previous authors working onal to m, and we assume the normalizations in this problem. a M aT = b M bT = ¹ ; (8) 2. The Pina-Jim˜ enez´ coordinates that define the vectors a and b with no dimensions. To study the Janus-Epimetheus dynamics we consider the These two vectors are easily computed from the previous general Three Body Problem, interacting through gravita- orthogonality conditions. One has tional forces in a plane. µ ¶ 1 1 1 The study of this dynamics is made in the coordinate sys- a = ¹y ; ; : (9) a m ¡ x m ¡ x m ¡ x tem of Pina˜ and Jimenez´ [17–19] that is further simplified by 1 a 2 a 3 a the previous hypothesis of plane two-dimensional motion. and The masses of the three bodies m1, m2 and m3 are dif- µ ¶ ferent, and ordered by the inequalities m1 À m2 > m3. 1 1 1 b = ¹yb ; ; ; (10) Since the motion takes place in a plane, it is sufficient to m1 ¡ xb m2 ¡ xb m3 ¡ xb take into account only one rotation angle à in order to trans- form from the inertial referential, to the frame of principal where ya and yb are normalization factors, and xa and xb are inertia axes, instead of three Euler angles. the roots of the quadratic equation In addition to this angle three other coordinates were in- x2 x troduced, named σ, R1, R2, where σ is an angle, and R1 and ¡ 2 ® + 3 = 0 ; (11) ¹2 ¹ R2 are two distances closely related to the two independent inertia moments where 2 µ ¶ I1 = ¹R ; (1) 1 1 1 1 ® = ¹ + + : (12) m m m and 1 2 3 2 The two solutions to that quadratic equation are I2 = ¹R ; (2) 2 p 2 and ¹ is the mass xa = ¹(® + ® ¡ 3) (13) r m1 m2 m3 and ¹ = ; (3) m1 + m2 + m3 p 2 xb = ¹(® ¡ ® ¡ 3) : (14) which will be used frequently in this paper. The cartesian inertial coordinates, with the center of grav- These quantities are defined in this form only for different ity at rest, written in terms of the new coordinates are masses. In that case we have the inequalities µ ¶ µ ¶ xj R2 cos σ cos à + R1 sin σ sin à rj = = aj m1 > xa > m2 > xb > m3 ; (15) yj R2 cos σ sin à ¡ R1 sin σ cos à µ ¶ R2 sin σ cos à ¡ R1 cos σ sin à that imply + bj ; (4) R2 sin σ sin à + R1 cos σ cos à a1 > 0; a2 < 0; a3 < 0; b1 > 0; b2 > 0; b3 < 0 : (16) where the aj and bj are constants forming two constant, lin- early independent vectors a and b, in the mass space, orthog- We define the angles σ1, σ2, σ3 by means of onal to the vector m = (m1; m2; m3): aj = kj cos σj ; bj = kj sin σj (17) a1m1 + a2m2 + a3m3 = b1m1 + b2m2 + b3m3 = 0 : (5) where k are positive constants. The angles σ obey the in- We introduce the following notation for the matrix j j equalities 0 1 m1 0 0 M = @ 0 m2 0 A : (6) 0 < σ1 < ¼=2 < σ2 < ¼ < σ3 < 3¼=2 : (18) 0 0 m3 To express the potential energy (G is the gravitational con- In order to complete the definition of vectors a and b we stant) assume Gm2m3 Gm3m1 Gm1m2 T 2 V = ¡ ¡ ¡ ; (19) b M a = 0 ; b M aT = 0 ; (7) p q r Rev. Mex. F´ıs. 55 (2) (2009) 97–105 THE SATURN, JANUS AND EPIMETHEUS DYNAMICS AS A GRAVITATIONAL THREE-BODY PROBLEM IN THE PLANE 99 we need the relationship among the distances between parti- We note à is a cyclic variable that is not present in the cles p, q, r, and the new coordinates. kinetic energy nor in the potential energy. Therefore 0 1 0 1 2 2 2 ³ ´ p R1 + R2 @K 2 _ @ 2 A @ 2 2 A Pà = = ¹R à ¡ σ_ sin(2θ) (25) q = Be (R1 ¡ R2) cos 2σ ; (20) _ 2 2 2 @à r (R1 ¡ R2) sin 2σ where Be is the constant matrix, depending only on the is a constant parameter. masses, The equations of motion for the other three coordinates 0 2 2 2 2 1 are 2 a1+b1 2 a1¡b1 2 m1 2 m1 2 m1 a1 b1 B C d B C ¹R_ ¡ ¹R[θ_2 + Ã_ 2 +σ _ 2 ¡ 2σ _ Ã_ sin(2θ)] 1 B a2+b2 a2¡b2 C Be= B m2 2 2 m2 2 2 m2 a b C : (21) dt ¹2 B 2 2 2 2 2 2 2 C @ A V ¤(σ; θ) 2 2 2 2 + 2 = 0 ; (26) 2 a3+b3 2 a3¡b3 2 R m3 2 m3 2 m3 a3 b3 d 1 @V ¤ ¹R2θ_ + 2¹R2σ_ Ã_ cos(2θ) ¡ = 0 ; (27) It has been convenient to introduce polar coordinates for dt R @θ R and R namely 1 2 d h i 1 @V ¤ ¹R2σ_ ¡ ¹R2Ã_ sin(2θ) ¡ = 0 ; (28) R1 = R cos θ ; R2 = R sin θ : (22) dt R @σ Writing these quantities in terms of the coordinates and where V ¤(σ; θ) is the function defined by V = ¡V ¤=R.
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