POLYNOMIAL REPRESENTATION OF THE ZERO VELOCITY SURFACES IN THE SPATIAL ELLIPTIC RESTRICTED THREE-BODY PROBLEM

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1 POLYNOMIAL REPRESENTATION OF THE ZERO VELOCITY SURFACES IN THE SPATIAL ELLIPTIC RESTRICTED THREE-BODY PROBLEM Ferenc Szenkovits 1,2 Zoltán Makó 1,3 Iharka Csillik 4 1 Department of Applied Mathematics, Babeş-Bolyai University, M. Kogălniceanu 1, Cluj-Napoca, Romania fszenko@math.ubbcluj.ro 2 Department of Astronomy, Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary 3 Department of Mathematics and Informatics, Sapientia University, , Miercurea-Ciuc, Romania makozoltan@sapientia.siculorum.ro 4 Astronomical Institute of Romanian Academy, Cireşilor 19, Cluj-Napoca, Romania iharka@math.ubbcluj.ro 15. Sept Abstract Zero velocity surfaces are deduced in the restricted three-body problem by using the Jacobi-integral. These surfaces are the boundaries of the Hill-regions: regions where the motion of the third, massless particle around the two primaries is not possible. V. Szebehely generalized this result for the planar elliptic restricted three-body problem. In a recent paper Makó and Szenkovits (2004) presented a generalization of this result for the spatial elliptic restricted three-body problem, where the existence of an invariant relation was proved analogous to the Jacobi integral in the restricted problem. For small eccentricities, this invariant relation can be approximated with zero velocity surfaces, given by implicit equations, delimiting the pulsating Hill-regions. In this paper we present the polynomial representation of these zero velocity surfaces. Keywords: elliptic restricted three-body problem, zero velocity surfaces, Hill-regions. MSC2000: 70F07 1

2 1 INTRODUCTION 2 1 INTRODUCTION Many studies were dedicated to the classical gravitational three-body problem, involving different methods and theories. The development of modern computers and computational techniques gave the possibility to deal with these problems using more powerful methods. This approach lead to new results. Szebehely (1967), Marchal (1990) and many other researchers have dedicated extensive studies to this problem, pointing out various and interesting aspects. A particular case of the three-body problem is the restricted three-body problem. Here, the motion of a massless particle moving around two massive primaries is considered. If the motion of the primaries is circular, then the problem is known as the circular restricted three-body problem or simply yhe restricted three-body problem (RTBP). In the case when the two primaries revolve on elliptic orbits the problem is called elliptic restricted three-body problem (ERTBP). In the study of the RTBP the Jacobian integral plays an important role, since it makes possible certain general, qualitative statements regarding the motion without actually solving the equations of motion. It permits for example the establishment of certain forbidden regions from which the third body is excluded (see eg. Érdi, 2001). The application of this principle to celestial mechanics was first made by Hill (1878) showing that the Moon cannot depart from the Earth s neighbourhood arbitrarily far. These regions are called today Hill-regions. Szebehely demonstrated the existence of the pulsating Hill-regions in the planar case of the ERTBP (Szebehely, 1967). In a recent paper we generalized Szebehely s result to the spatial ERTBP (Makó and Szenkovits, 2004). Using these zones, we deduced necessary conditions of the gravitational capture of small bodies. In this paper we present the polynomial representation of the zero velocity surfaces in the ERTBP, by using spherical coordinates. 2 Hill-Regions in the ERTBP In the elliptic restricted three-body problem (ERTBP) the two massive primaries, with masses m 1 and m 2 revolve on elliptical orbits under their mutual gravitational attraction and the motion of a third, massless body is studied. The orbit of m 2 around m 1, in an inertial system, is r = a ( 1 e 2) 1 + e cos f, (1) where r is the mutual distance, a and e are the semimajor axis and the eccentricity of the elliptical orbit, and f is the true anomaly. There are several systems of reference that can be used to describe the elliptic restricted three-body problem. In our study a nonuniformly rotating and pulsating coordinate system is used. In this system of reference (Figure 1) the origin is in the center of mass of the two massive primaries (Sun and Earth for example), and the ξ axis is directed towards m 2. The ξ η coordinateplane rotates with variable angular velocity, in such a way, that the two massive primaries are always on the ξ axis, and the period of the rotation is 2π. Besides

3 2 HILL-REGIONS IN THE ERTBP 3 ζ P 1 ( µ, 0, 0) f Π P 2 (1 µ, 0, 0) ξ r 2 r 1 ) P 3 ( ξ, η, ζ η Figure 1: The spatial ERTBP the rotation, the system also pulsates, to keep the primaries in fixed positions ( ξ1 = µ, η 1 = ζ 1 = 0, ξ 2 = 1 µ, η 2 = ζ 2 = 0). In this system the equations of motion of the third massless particle are: ξ 2 η = ω ξ, η + 2 ξ = ω η, (2) ζ = ω ζ, where the derivatives are taken with respect to the true anomaly f, and ω = (1 + e cos f) 1 Ω, with ( ξ2 ) Ω = η 2 e ζ 2 1 µ cos f + + (3) 2 ( ξ+µ) + η2 + ζ 2 + µ ( ξ 1+µ) + η2 + ζ 2 2 µ (1 µ). Performing the same operations, which in the RTBP leads to the Jacobiintegral (see Szebehely, 1967), in the case of the spatial ERTBP we obtain an invariant relation of the form: ( ) 2 d ξ + df ( ) ( 2 d η + df d ζ df ) 2 f = 2ω e f 2e 0 0 ζ 2 sin h 1+e cos hdh (4) Ω sin h (1+e cos h) 2 dh C.

4 3 THE POLYNOMIAL REPRESENTATION OF ZVS 4 This is the generalization of Szebehely s invariant relation (Szebehely, 1967, p. 595) for the spatial ERTBP. The zero velocity surfaces (ZVS) in the ERTBP are: 2Ω f 1 + e cos f e 0 ζ 2 sin h f dh 2e 1 + e cos h 0 Ω sin h 2 dh = C. (5) (1 + e cos h) These surfaces delimite the Hill-regions, in which the motion of the third particle is not possible. For small values of the eccentricity e, if the motion of the third, masslese body is bounded and the collisions are excluded, then the integral terms in (5) may be neglected, and the approximate equation of Hill-surfaces is: 2Ω = C (1 + e cos f), (6) where and Ω = 1 ( ξ2 ) + η 2 e ζ 2 cos f + 1 µ + µ + 1 µ (1 µ), (7) 2 r 1 r 2 2 r 1 = ( ξ + µ ) 2 + η2 + ζ 2, r 2 = ( ξ 1 + µ ) 2 + η2 + ζ 2. (8) Graphic representation of the Hill-regions gives many ideas to determine the properties of these important regions. Unfortunately the Hill-surfaces are given by the implicit equation (6) and therefore the MATLAB surface display functions cannot be applied to display this surface. 3 The polynomial representation of ZVS Equation (6) of the zero velocity surfaces can be transformed in a polynomial form by using spherical coordinates: ξ = r cos ϕ sin θ, η = r sin ϕ sin θ, ζ = r cos θ. The implicit equation of ZVS in spherical coordinates is: r 2 ( sin 2 θ e cos 2 θ cos f ) + where r 1 = r 2 + 2rµ cos ϕ sin θ + µ 2, r 2 = If we use: 2 (1 µ) r 1 + 2µ r 2 + µ (1 µ) = C (1 + e cos f), (9) A = sin 2 θ e cos 2 θ cos f, B = cos ϕ sin θ, E = C (1 + e cos f) µ (1 µ), r 2 + 2r (µ 1) cos ϕ sin θ + (µ 1) 2.

5 3 THE POLYNOMIAL REPRESENTATION OF ZVS 5 equation (9) has the following form: Ar (1 µ) r2 + 2µBr + µ 2 + 2µ r (µ 1) Br + (µ 1) 2 = E. (10) If E 0, id est µ (1 µ) 1 + e cos f, (11) C then we can transform the equation (10) in polynomial form. If E Ar 2, (12) then we may raise to the second power both sides of the equation 2 (1 µ) r2 + 2µBr + µ + 2µ = E Ar r (µ 1) Br + (µ 1) 2 and we get: 4 (1 µ) µ (r 2 + 2µBr + µ 2 ) (r (µ 1) Br + (µ 1) 2) = If the condition [ E Ar 2 ] 2 4 (1 µ) 2 r 2 + 2µBr + µ 2 4µ 2 r (µ 1) Br + (µ 1) 2. [ E Ar 2 ] 2 4 (1 µ) 2 r 2 + 2µBr + µ 2 + 4µ 2 r (µ 1) Br + (µ 1) 2 is also verified, then the implicit equation of the ZVS is 16 (1 µ) 2 µ 2 (r 2 + 2µBr + µ 2 ) (r (µ 1) Br + (µ 1) 2) = [ ] E 2 + A 2 r 4 2EAr 2 4 (1 µ) 2 2 r 2 + 2µBr + µ 2 4µ 2 r (µ 1) Br + (µ 1) 2 (13) Equation (13) is equivalent to the algebraic equation with where f (Z) = f (r) = 0, (14) 16 k=0 a k+1 Z k, (15)

6 3 THE POLYNOMIAL REPRESENTATION OF ZVS 6 a 1 = A 4, a 2 = 8A 4 µb 4A 4 B, a 3 = 4A 3 E 24A 4 µb 2 + 4A 4 B 2 4A 4 µ + 24A 4 µ 2 B 2 + 2A 4 + 4A 4 µ 2, a 4 = 20A 4 µb + 16A 3 BE 32A 3 EµB 48A 4 µ 2 B A 4 B 3 µ 36A 4 Bµ 2 +32A 4 µ 3 B A 4 µ 3 B 4A 4 B, a 5 = 64A 4 µ 2 B 2 16A 3 B 2 E + 16A 3 µe + 10A 4 µ A 4 µ 4 B 4 +16A 4 µ 2 B 4 16A 4 µb 2 32A 4 µ 3 B 4 + 6A 4 µ 4 96A 3 µ 2 B 2 E 12A 4 µ 3 16A 3 µ 2 E + 48A 4 µ 4 B 2 4A 4 µ + 96A 3 EµB 2 96A 4 µ 3 B 2 + A 4 8A 3 E + 6E 2 A 2, a 6 = 80A 4 µ 4 B 3 64A 3 B 3 Eµ + 32A 4 µ 5 B A 3 BE + 56A 4 µ 3 B 60A 4 µ 4 B + 64A 4 µ 3 B E 2 A 2 µb + 192A 3 µ 2 B 3 E 24E 2 A 2 B 128A 3 µ 3 B 3 E + 4A 4 µb 80A 3 EµB 16A 4 µ 2 B 3 +24A 4 µ 5 B + 144A 3 Eµ 2 B 24A 4 Bµ 2 96A 3 µ 3 EB, a 7 = 24E 2 A 2 µ + 24E 2 A 2 µ 2 144E 2 A 2 µb 2 + 4A 4 µ 6 12A 4 µ A 4 µ 4 4E 3 A 4A 3 E 16A 2 µ 2 8A A 2 µ + 12E 2 A 2 8A 4 µ 3 + 2A 4 µ 2 64A 3 µ 2 B 4 E 256A 3 µ 2 B 2 E 192A 3 µ 4 EB 2 +16A 3 µe + 24A 4 µ 6 B 2 72A 4 µ 5 B A 3 µ 3 E + 76A 4 µ 4 B 2 40A 3 µ 2 E 32A 4 µ 3 B A 3 EµB 2 64A 3 µ 4 B 4 E + 128A 3 µ 3 B 4 E +144E 2 µ 2 B 2 A 2 + 4A 4 µ 2 B A 3 µ 3 EB 2 24A 3 µ 4 E + 24E 2 B 2 A 2 a 8 = 96A 2 µ 3 B + 144A 2 µ 2 B 112A 2 µb 24E 2 A 2 B 216E 2 µ 2 A 2 B +120E 2 A 2 µb 32E 3 AµB + 16E 3 AB + 32A 2 B + 240A 3 µ 4 EB +64A 3 µ 2 B 3 E 96A 3 µ 5 EB 20A 4 µ 4 B + 96A 3 Eµ 2 B 16A 3 EµB +96E 2 B 3 A 2 µ + 8A 4 µ 7 B 28A 4 µ 6 B + 36A 4 µ 5 B + 144E 2 µ 3 A 2 B 128A 3 µ 5 B 3 E + 320A 3 µ 4 B 3 E 256A 3 µ 3 B 3 E + 192E 2 µ 3 B 3 A 2 288E 2 µ 2 B 3 A 2 224A 3 µ 3 EB + 4A 4 µ 3 B, a 9 = 96A 2 µ 3 + E 4 72E 2 A 2 µ EAµ 2 32EAµ 384A 2 µ 2 B A 2 µb 2 24E 2 A 2 µ + 36E 2 µ 4 A E 2 A 2 µ 2 + A 4 µ E 3 AµB E 2 µ 4 A 2 B E 3 Aµ 96E 2 A 2 µb 2 576E 2 µ 3 A 2 B 2 + 6A 4 µ 6 4A 4 µ 5 + A 4 µ 4 8E 3 A 112A 2 µ 2 16A 2 4A 4 µ A 2 µ 48A 2 µ 4 +6E 2 A 2 32B 2 A EA 16A 3 µ 2 B 2 E 16A 3 µ 6 E + 288A 3 µ 5 EB 2 304A 3 µ 4 EB 2 96A 3 µ 6 EB 2 16E 3 B 2 A + 32A 3 µ 3 E 8A 3 µ 2 E +384E 2 µ 2 B 2 A 2 96E 3 µ 2 B 2 A + 96E 2 µ 4 B 4 A 2 192E 2 µ 3 B 4 A 2 +96E 2 µ 2 B 4 A A 3 µ 3 EB A 3 µ 5 E 56A 3 µ 4 E 192µ 4 A 2 B A 2 µ 3 B 2 16E 3 µ 2 A,

7 3 THE POLYNOMIAL REPRESENTATION OF ZVS 7 a 10 = 4E 4 B + 224EAµB + 192EAµ 3 B 288EAµ 2 B 640A 2 µ 3 B +480A 2 µ 2 B 192A 2 µb 64EAB 192µ 5 BA µ 4 BA 2 144E 2 µ 2 A 2 B + 24E 2 A 2 µb + 144E 3 Aµ 2 B 80E 3 AµB + 16E 3 AB +8E 4 µb + 144E 2 µ 5 A 2 B + 32A 2 B + 80A 3 µ 4 EB 32A 3 µ 7 EB +112A 3 µ 6 EB 144A 3 µ 5 EB 64E 3 B 3 Aµ 128µ 5 B 3 A µ 4 B 3 A 2 96E 3 µ 3 AB 360E 2 µ 4 A 2 B 64B 3 A 2 µ + 336E 2 µ 3 A 2 B + 192E 3 µ 2 B 3 A 128E 3 µ 3 B 3 A + 384E 2 µ 3 B 3 A 2 96E 2 µ 2 B 3 A 2 480E 2 µ 4 B 3 A E 2 µ 5 B 3 A 2 16A 3 µ 3 EB + 256µ 2 B 3 A 2 384µ 3 B 3 A 2, a 11 = 224A 2 µ 3 8E E 2 µ 16E 2 µ 2 + 2E 4 48E 2 A 2 µ A 2 µ EAµ 2 B 2 384EAµB EAµ 4 192EAµ EAµ 2 128EAµ 352A 2 µ 2 B A 2 µb E 2 µ 4 A E 2 µ 6 A 2 72E 2 µ 5 A E 2 A 2 µ µ 4 EAB E 3 AµB 2 432E 2 µ 5 A 2 B E 2 µ 4 A 2 B E 2 µ 6 A 2 B E 3 Aµ + 48E 3 µ 3 A 768EAµ 3 B 2 24E 4 µb 2 192E 2 µ 3 A 2 B 2 4E 3 A 136A 2 µ 2 8A 2 48µ 6 A 2 +4E 4 µ 2 + 4E 4 B A 2 µ 4E 4 µ 232A 2 µ EA 24A 3 µ 6 E +64B 2 EA 4A 3 µ 8 E + 16A 3 µ 7 E 192µ 6 B 2 A µ 5 B 2 A E 3 µ 3 AB 2 192E 3 µ 4 AB 2 24E 3 µ 4 A + 24E 2 µ 2 B 2 A 2 256E 3 µ 2 B 2 A 64E 3 µ 2 B 4 A 64E 3 µ 4 B 4 A + 128E 3 µ 3 B 4 A + 24E 4 µ 2 B 2 +16A 3 µ 5 E 4A 3 µ 4 E 864µ 4 A 2 B A 2 µ 3 B 2 40E 3 µ 2 A, a 12 = 4E 4 B + 384EAµB EAµ 3 B 960EAµ 2 B + 144E 2 µ 2 B 400A 2 µ 3 B + 128A 2 µ 2 B 16A 2 µb 64EAB 112E 2 µb 96E 2 µ 3 B 592µ 5 BA µ 4 BA 2 36E 4 µ 2 B 96µ 7 BA µ 6 BA 2 512µ 2 B 3 EA + 384µ 5 EAB + 768µ 3 B 3 EA + 96E 3 Aµ 2 B 16E 3 AµB +48E 2 µ 7 A 2 B 168E 2 µ 6 A 2 B + 20E 4 µb 960EAµ 4 B + 216E 2 µ 5 A 2 B +24E 4 µ 3 B + 32E 2 B + 256µ 5 B 3 EA 640µ 4 B 3 EA + 128B 3 EAµ +16E 4 B 3 µ 224E 3 µ 3 AB 120E 2 µ 4 A 2 B + 240E 3 µ 4 AB 96E 3 µ 5 AB + 24E 2 µ 3 A 2 B + 64E 3 µ 2 B 3 A + 320E 3 µ 4 B 3 A 256E 3 µ 3 B 3 A 128E 3 µ 5 B 3 A + 32E 4 µ 3 B 3 48E 4 µ 2 B 3, a 13 = A 2 µ 3 16E 2 64µ + 64E 2 µ 12E 4 µ µ 2 48E 2 µ E 2 µ 3 112E 2 µ 2 + E A 2 µ µ 4 96µ 3 +6E 4 µ EAµ 2 B 2 128EAµB EAµ 4 448EAµ EAµ 2 96EAµ 384E 2 µ 2 B E 2 µb 2 + 6E 2 µ 4 A 2 +6E 2 µ 8 A E 2 µ 6 A 2 24E 2 µ 5 A 2 24E 2 µ 7 A µ 4 EAB 2 +96µ 6 EA + 32E 3 µ 3 A + 48E 4 µ 4 B EAµ 3 B 2 16A 2 µ 8 288EAµ 5 16E 4 µb A 2 µ 7 8A 2 µ 2 128µ 6 A E 4 µ 2 4E 4 µ 120A 2 µ 4 32B 2 E EA + 384µ 6 B 2 EA 1152µ 5 B 2 EA +128E 3 µ 3 AB 2 304E 3 µ 4 AB E 3 µ 5 A 56E 3 µ 4 A 96E 3 µ 6 AB 2

8 3 THE POLYNOMIAL REPRESENTATION OF ZVS E 3 µ 5 AB 2 16E 3 µ 2 B 2 A 16E 3 µ 6 A + 16E 4 µ 2 B E 4 µ 2 B 2 +16E 4 µ 4 B 4 32E 4 µ 3 B 4 192µ 4 E 2 B E 2 µ 3 B 2 8E 3 µ 2 A 96E 4 µ 3 B 2, a 14 = 64B 672µ 2 B + 32EAµB + 800EAµ 3 B 256EAµ 2 B 192µ 5 BE E 2 µ 2 B 192E 2 µb 640E 2 µ 3 B 24E 4 µ 2 B µ 5 EAB +24E 4 µ 5 B 60E 4 µ 4 B + 4E 4 µb 1280EAµ 4 B + 56E 4 µ 3 B 480µ 4 B +192µ 5 B + 320µB + 32E 2 B + 768µ 3 B + 192µ 7 BEA 672µ 6 BEA 16E 3 µ 3 AB + 80E 3 µ 4 AB 64B 3 E 2 µ 128µ 5 B 3 E µ 4 B 3 E 2 144E 3 µ 5 AB 32E 3 µ 7 AB + 112E 3 µ 6 AB + 32E 4 µ 5 B 3 80E 4 µ 4 B 3 +64E 4 µ 3 B 3 16E 4 µ 2 B µ 4 BE µ 2 B 3 E 2 384µ 3 B 3 E 2, a 15 = 32 8E 2 192µ + 64B E 2 µ 8E 4 µ µ 2 232E 2 µ E 2 µ 3 136E 2 µ µ 3 B 2 576µ 5 B µ 6 B µ 4 704µ µ 6 288µ E 2 µ 5 48µ 6 E 2 + 4E 4 µ 6 12E 4 µ E 4 µ EAµ 4 96EAµ EAµ 2 352E 2 µ 2 B 2 +64E 2 µb µ 6 EA + 76E 4 µ 4 B E 4 µ 6 B 2 72E 4 µ 5 B 2 320EAµ 5 384µB 2 + 2E 4 µ µ 5 B 2 E E 3 µ 5 A 192µ 6 B 2 E 2 4E 3 µ 4 A + 32µ 8 EA 128µ 7 EA 4E 3 µ 8 A +16E 3 µ 7 A 24E 3 µ 6 A + 4E 4 µ 2 B µ 2 B 2 864µ 4 E 2 B E 2 µ 3 B µ 4 B 2 32E 4 µ 3 B 2, a 16 = 672µ 6 B + 36E 4 µ 5 B 64B + 336E 2 µ 6 B 1344µ 2 B 28E 4 µ 6 B 2400µ 4 B 96E 2 µ 7 B + 448µB µ 3 B +4E 4 µ 3 B + 640µ 4 BE 2 + 8E 4 µ 7 B 20E 4 µ 4 B 16E 2 µb 592µ 5 BE 2 400E 2 µ 3 B + 128E 2 µ 2 B µ 5 B + 192µ 7 B, a 17 = 896µ µ + 448µ µ E 2 µ 3 8E 2 µ 2 120E 2 µ 4 16E 2 µ E 2 µ µ 6 960µ µ 8 192µ E 2 µ 5 128µ 6 E 2 + E 4 µ 8 4E 4 µ 7 + 6E 4 µ 6 4E 4 µ 5 + E 4 µ 4. The ZVS depends on the parameters f, e, C and µ. If 1 + e cos f µ(1 µ) C then ZVS is the following set: ZV S (f, e, C, µ) = {( ξ, η, ζ) R 3 / ξ = r cos ϕ sin θ, η = cos ϕ sin θ, ζ = C (1 + e cos f) µ (1 µ), where r is the real root of the polinom f (Z), θ [0, π], ϕ [0, 2π) and E Ar 2, [ E Ar 2 ] } 2 4 (1 µ) 2 r 2 + 2µBr + µ 2 + 4µ 2 r (µ 1) Br + (µ 1) 2.

9 4 CONCLUSIONS 9 4 Conclusions In the spatial elliptic restricted three-body problem the existence of an approximate equation of the zero velocity surfaces is presented, for small eccentricities of the primaries orbits and when the orbit of the third body is bounded and collisions with primaries are exclused. This implicit equation may be transformed to an algebraic equation by using spherical coordinates. For any given values of the angles θ and ϕ the polar distance r from the center of mass can be determined by using equation (14). This form of the equation of the ZVS is useful fot the graphical representation of the ZVS and it also makes possible the study of the topological type of these surfaces, for different values of the eccentricity e and true anomaly f. Acknowledgements The warm hospitality and help of the staff of the Department of Astronomy of the Eötvös Loránd University in Budapest where the firs author was accepted as visiting researcher is gratefully acknowledged. We thank Prof. Bálint Érdi for the very useful discussions. This work was supported by grants from the Hungarian Academy of Sciences through János Bolyai grant and the Research Programs Institute of Foundation Sapientia. References [1] Érdi, B.: Dynamics of the Solar system, Eötvös University Press, Budapest, 2001 (in Hungarian). [2] Hill, G.W.:Am. J. Math. 1878, 1, 129. [3] Makó, Z. and Szenkovits, F: Capture in the circular and elliptic restricted three-body problem. Celestial Mechanics and Dynamical Astronomy, Vol. 90, No. 1 2, [4] Marchal, C.: The three-body problem, Elsevier, Studies in Astronautics, [5] Szebehely, V.: Theory of orbits, Academic Press, New-York, 1967.