POLYNOMIAL REPRESENTATION OF THE
ZERO VELOCITY SURFACES IN THE
SPATIAL ELLIPTIC RESTRICTED
THREE-BODY PROBLEM
Ferenc Szenkovits1,2
Zoltán Makó1,3
Iharka Csillik4
1
Department of Applied Mathematics, Babeş-Bolyai University,
M. Kogălniceanu 1, 400084 Cluj-Napoca, Romania
email: [email protected]
2
Department of Astronomy, Eötvös Loránd University,
Pázmány Péter sétány 1/A, 1117 Budapest, Hungary
3
Department of Mathematics and Informatics, Sapientia University,
530104, Miercurea-Ciuc, Romania
email: [email protected]
4
Astronomical Institute of Romanian Academy,
Cireşilor 19, 400487 Cluj-Napoca, Romania
email: [email protected]
15. Sept. 2004
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
1
1
INTRODUCTION
2
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 m1 and m2 revolve on elliptical orbits under their mutual
gravitational attraction and the motion of a third, massless body is studied.
The orbit of m2 around m1 , in an inertial system, is
a 1 − e2
,
(1)
r=
1 + e cos f
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 m2 . The ξ˜η̃ 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
2
HILL-REGIONS IN THE ERTBP
3
ζ̃
6
r1
,
,
P1 (−µ, 0, 0)
,
s , r2
!!
!
,
,
!!
!
,
,
!!
!
s !
,
,
f
Π
,
,
,
,
,
s,
P2 (1 − µ, 0, 0)
ξ˜
˜ η̃, ζ̃
P3 ξ,
s
,
,
,
,
η̃
Figure 1: The spatial ERTBP
the
rotation, the system also pulsates, to keep the primaries in fixed positions
ξ˜1 = −µ, η̃1 = ζ̃1 = 0, ξ˜2 = 1 − µ, η̃2 = ζ̃2 = 0 .
In this system the equations of motion of the third massless particle are:
 00
∂ω
0
˜

 ξ − 2η̃ = ∂ ξ̃ ,
η̃ 00 + 2ξ˜0 = ∂ω
(2)
∂ η̃ ,


ζ̃ 00 = ∂ω
,
∂ ζ̃
where the derivatives are taken with respect to the true anomaly f , and
−1
ω = (1 + e cos f )
Ω,
with
Ω
=
1
2
ξ˜2 + η̃ 2 − eζ̃ 2 cos f +
+q
µ
2
(ξ̃−1+µ)
+η̃ 2 +ζ̃ 2
1−µ
q
2
(ξ̃+µ) +η̃2 +ζ̃ 2
+
(3)
+ 12 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 !2
2
Rf ζ̃ 2 sin h
dξ˜
dη̃
dζ̃
(4)
+
+
= 2ω − e 1+e
cos h dh −
df
df
df
0
−2e
Rf
0
Ω sin h
dh
(1+e cos h)2
− C.
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
−e
1 + e cos f
Zf
0
ζ̃ 2 sin h
dh − 2e
1 + e cos h
Zf
0
Ω sin h
2 dh
(1 + e cos h)
= C.
(5)
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 ) ,
where
Ω=
and
r1 =
1−µ
µ
1
1 ˜2
ξ + η̃ 2 − eζ̃ 2 cos f +
+
+ µ (1 − µ) ,
2
r1
r2
2
r
r
2
2
2
2
˜
ξ + µ + η̃ + ζ̃ , r2 =
ξ˜ − 1 + µ + η̃ 2 + ζ̃ 2 .
(6)
(7)
(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,
η̃ = r sin ϕ sin θ,

ζ̃ = r cos θ.
The implicit equation of ZVS in spherical coordinates is:
2 (1 − µ) 2µ
r2 sin2 θ − e cos2 θ cos f +
+
+ µ (1 − µ) = C (1 + e cos f ) , (9)
r1
r2
where
r1 =
q
p
2
r2 + 2rµ cos ϕ sin θ + µ2 , r2 = r2 + 2r (µ − 1) cos ϕ sin θ + (µ − 1) .
If we use:
A = sin2 θ − e cos2 θ cos f,
B = cos ϕ sin θ,
E = C (1 + e cos f ) − µ (1 − µ) ,
3
THE POLYNOMIAL REPRESENTATION OF ZVS
5
equation (9) has the following form:
2 (1 − µ)
Ar2 + p
r2 + 2µBr + µ2
2µ
+q
= E.
2
r2 + 2 (µ − 1) Br + (µ − 1)
(10)
If E ≥ 0, id est
µ (1 − µ)
,
C
then we can transform the equation (10) in polynomial form.
If
E ≥ Ar2 ,
1 + e cos f ≥
(11)
(12)
then we may raise to the second power both sides of the equation
2 (1 − µ)
p
r2
+ 2µBr +
µ2
2µ
+q
r2
= E − Ar2 .
2
+ 2 (µ − 1) Br + (µ − 1)
and we get:
4 (1 − µ) µ
=
2
(r2 + 2µBr + µ2 ) r2 + 2 (µ − 1) Br + (µ − 1)
r
2
E − Ar2 −
2
4µ2
4 (1 − µ)
−
2.
r2 + 2µBr + µ2
r2 + 2 (µ − 1) Br + (µ − 1)
If the condition
2
E − Ar2 ≥
2
4 (1 − µ)
4µ2
+
2
r2 + 2µBr + µ2
r2 + 2 (µ − 1) Br + (µ − 1)
is also verified, then the implicit equation of the ZVS is
2
16 (1 − µ) µ2
=
2
(r2 + 2µBr + µ2 ) r2 + 2 (µ − 1) Br + (µ − 1)
#2
"
2
2
4
(1
−
µ)
4µ
(13)
E 2 + A2 r4 − 2EAr2 − 2
−
2
r + 2µBr + µ2
r2 + 2 (µ − 1) Br + (µ − 1)
Equation (13) is equivalent to the algebraic equation
f (r) = 0,
with
f (Z) =
16
X
k=0
where
ak+1 Z k ,
(14)
(15)
3
THE POLYNOMIAL REPRESENTATION OF ZVS
6
a1 = A4 ,
a2 = 8A4 µB − 4A4 B,
a3 = −4A3 E − 24A4 µB 2 + 4A4 B 2 − 4A4 µ + 24A4 µ2 B 2 + 2A4 + 4A4 µ2 ,
a4 = 20A4 µB + 16A3 BE − 32A3 EµB − 48A4 µ2 B 3 + 16A4 B 3 µ − 36A4 Bµ2
+32A4 µ3 B 3 + 24A4 µ3 B − 4A4 B,
a5 = 64A4 µ2 B 2 − 16A3 B 2 E + 16A3 µE + 10A4 µ2 + 16A4 µ4 B 4
+16A4 µ2 B 4 − 16A4 µB 2 − 32A4 µ3 B 4 + 6A4 µ4 − 96A3 µ2 B 2 E − 12A4 µ3
−16A3 µ2 E + 48A4 µ4 B 2 − 4A4 µ + 96A3 EµB 2 − 96A4 µ3 B 2 + A4
−8A3 E + 6E 2 A2 ,
a6 = −80A4 µ4 B 3 − 64A3 B 3 Eµ + 32A4 µ5 B 3 + 16A3 BE + 56A4 µ3 B
−60A4 µ4 B + 64A4 µ3 B 3 + 48E 2 A2 µB + 192A3 µ2 B 3 E
−24E 2 A2 B − 128A3 µ3 B 3 E + 4A4 µB − 80A3 EµB − 16A4 µ2 B 3
+24A4 µ5 B + 144A3 Eµ2 B − 24A4 Bµ2 − 96A3 µ3 EB,
a7 = −24E 2 A2 µ + 24E 2 A2 µ2 − 144E 2 A2 µB 2 + 4A4 µ6 − 12A4 µ5 +
14A4 µ4 − 4E 3 A − 4A3 E − 16A2 µ2 − 8A2 + 16A2 µ + 12E 2 A2
−8A4 µ3 + 2A4 µ2 − 64A3 µ2 B 4 E − 256A3 µ2 B 2 E − 192A3 µ4 EB 2
+16A3 µE + 24A4 µ6 B 2 − 72A4 µ5 B 2 + 48A3 µ3 E + 76A4 µ4 B 2
−40A3 µ2 E − 32A4 µ3 B 2 + 64A3 EµB 2 − 64A3 µ4 B 4 E + 128A3 µ3 B 4 E
+144E 2 µ2 B 2 A2 + 4A4 µ2 B 2 + 384A3 µ3 EB 2 − 24A3 µ4 E + 24E 2 B 2 A2
a8 = −96A2 µ3 B + 144A2 µ2 B − 112A2 µB − 24E 2 A2 B − 216E 2 µ2 A2 B
+120E 2 A2 µB − 32E 3 AµB + 16E 3 AB + 32A2 B + 240A3 µ4 EB
+64A3 µ2 B 3 E − 96A3 µ5 EB − 20A4 µ4 B + 96A3 Eµ2 B − 16A3 EµB
+96E 2 B 3 A2 µ + 8A4 µ7 B − 28A4 µ6 B + 36A4 µ5 B + 144E 2 µ3 A2 B
−128A3 µ5 B 3 E + 320A3 µ4 B 3 E − 256A3 µ3 B 3 E + 192E 2 µ3 B 3 A2
−288E 2 µ2 B 3 A2 − 224A3 µ3 EB + 4A4 µ3 B,
a9 = 96A2 µ3 + E 4 − 72E 2 A2 µ3 + 32EAµ2 − 32EAµ − 384A2 µ2 B 2
+192A2 µB 2 − 24E 2 A2 µ + 36E 2 µ4 A2 + 60E 2 A2 µ2 + A4 µ8 + 96E 3 AµB 2
+288E 2 µ4 A2 B 2 + 16E 3 Aµ − 96E 2 A2 µB 2 − 576E 2 µ3 A2 B 2 + 6A4 µ6
−4A4 µ5 + A4 µ4 − 8E 3 A − 112A2 µ2 − 16A2 − 4A4 µ7 + 64A2 µ − 48A2 µ4
+6E 2 A2 − 32B 2 A2 + 16EA − 16A3 µ2 B 2 E − 16A3 µ6 E + 288A3 µ5 EB 2
−304A3 µ4 EB 2 − 96A3 µ6 EB 2 − 16E 3 B 2 A + 32A3 µ3 E − 8A3 µ2 E
+384E 2 µ2 B 2 A2 − 96E 3 µ2 B 2 A + 96E 2 µ4 B 4 A2 − 192E 2 µ3 B 4 A2
+96E 2 µ2 B 4 A2 + 128A3 µ3 EB 2 + 48A3 µ5 E − 56A3 µ4 E − 192µ4 A2 B 2
+384A2 µ3 B 2 − 16E 3 µ2 A,
3
THE POLYNOMIAL REPRESENTATION OF ZVS
7
a10 = −4E 4 B + 224EAµB + 192EAµ3 B − 288EAµ2 B − 640A2 µ3 B
+480A2 µ2 B − 192A2 µB − 64EAB − 192µ5 BA2 + 480µ4 BA2
−144E 2 µ2 A2 B + 24E 2 A2 µB + 144E 3 Aµ2 B − 80E 3 AµB + 16E 3 AB
+8E 4 µB + 144E 2 µ5 A2 B + 32A2 B + 80A3 µ4 EB − 32A3 µ7 EB
+112A3 µ6 EB − 144A3 µ5 EB − 64E 3 B 3 Aµ − 128µ5 B 3 A2 + 320µ4 B 3 A2
−96E 3 µ3 AB − 360E 2 µ4 A2 B − 64B 3 A2 µ + 336E 2 µ3 A2 B + 192E 3 µ2 B 3 A
−128E 3 µ3 B 3 A + 384E 2 µ3 B 3 A2 − 96E 2 µ2 B 3 A2 − 480E 2 µ4 B 3 A2
+192E 2 µ5 B 3 A2 − 16A3 µ3 EB + 256µ2 B 3 A2 − 384µ3 B 3 A2 ,
a11 = 224A2 µ3 − 8E 2 + 16E 2 µ − 16E 2 µ2 + 2E 4 − 48E 2 A2 µ3 + 144A2 µ5
+768EAµ2 B 2 − 384EAµB 2 + 96EAµ4 − 192EAµ3 + 224EAµ2
−128EAµ − 352A2 µ2 B 2 + 64A2 µB 2 + 84E 2 µ4 A2 + 24E 2 µ6 A2
−72E 2 µ5 A2 + 12E 2 A2 µ2 + 384µ4 EAB 2 + 64E 3 AµB 2 − 432E 2 µ5 A2 B 2
+456E 2 µ4 A2 B 2 + 144E 2 µ6 A2 B 2 + 16E 3 Aµ + 48E 3 µ3 A − 768EAµ3 B 2
−24E 4 µB 2 − 192E 2 µ3 A2 B 2 − 4E 3 A − 136A2 µ2 − 8A2 − 48µ6 A2
+4E 4 µ2 + 4E 4 B 2 + 48A2 µ − 4E 4 µ − 232A2 µ4 + 32EA − 24A3 µ6 E
+64B 2 EA − 4A3 µ8 E + 16A3 µ7 E − 192µ6 B 2 A2 + 576µ5 B 2 A2
+384E 3 µ3 AB 2 − 192E 3 µ4 AB 2 − 24E 3 µ4 A + 24E 2 µ2 B 2 A2
−256E 3 µ2 B 2 A − 64E 3 µ2 B 4 A − 64E 3 µ4 B 4 A + 128E 3 µ3 B 4 A + 24E 4 µ2 B 2
+16A3 µ5 E − 4A3 µ4 E − 864µ4 A2 B 2 + 768A2 µ3 B 2 − 40E 3 µ2 A,
a12 = −4E 4 B + 384EAµB + 1280EAµ3 B − 960EAµ2 B + 144E 2 µ2 B
−400A2 µ3 B + 128A2 µ2 B − 16A2 µB − 64EAB − 112E 2 µB − 96E 2 µ3 B
−592µ5 BA2 + 640µ4 BA2 − 36E 4 µ2 B − 96µ7 BA2 + 336µ6 BA2
−512µ2 B 3 EA + 384µ5 EAB + 768µ3 B 3 EA + 96E 3 Aµ2 B − 16E 3 AµB
+48E 2 µ7 A2 B − 168E 2 µ6 A2 B + 20E 4 µB − 960EAµ4 B + 216E 2 µ5 A2 B
+24E 4 µ3 B + 32E 2 B + 256µ5 B 3 EA − 640µ4 B 3 EA + 128B 3 EAµ
+16E 4 B 3 µ − 224E 3 µ3 AB − 120E 2 µ4 A2 B + 240E 3 µ4 AB
−96E 3 µ5 AB + 24E 2 µ3 A2 B + 64E 3 µ2 B 3 A + 320E 3 µ4 B 3 A
−256E 3 µ3 B 3 A − 128E 3 µ5 B 3 A + 32E 4 µ3 B 3 − 48E 4 µ2 B 3 ,
a13 = 16 + 48A2 µ3 − 16E 2 − 64µ + 64E 2 µ − 12E 4 µ3 + 112µ2
−48E 2 µ4 + 96E 2 µ3 − 112E 2 µ2 + E 4 + 160A2 µ5 + 48µ4 − 96µ3
+6E 4 µ4 + 704EAµ2 B 2 − 128EAµB 2 + 464EAµ4 − 448EAµ3
+272EAµ2 − 96EAµ − 384E 2 µ2 B 2 + 192E 2 µB 2 + 6E 2 µ4 A2
+6E 2 µ8 A2 + 36E 2 µ6 A2 − 24E 2 µ5 A2 − 24E 2 µ7 A2 + 1728µ4 EAB 2
+96µ6 EA + 32E 3 µ3 A + 48E 4 µ4 B 2 − 1536EAµ3 B 2 − 16A2 µ8
−288EAµ5 − 16E 4 µB 2 + 64A2 µ7 − 8A2 µ2 − 128µ6 A2 + 10E 4 µ2
−4E 4 µ − 120A2 µ4 − 32B 2 E 2 + 16EA + 384µ6 B 2 EA − 1152µ5 B 2 EA
+128E 3 µ3 AB 2 − 304E 3 µ4 AB 2 + 48E 3 µ5 A − 56E 3 µ4 A − 96E 3 µ6 AB 2
3
THE POLYNOMIAL REPRESENTATION OF ZVS
8
+288E 3 µ5 AB 2 − 16E 3 µ2 B 2 A − 16E 3 µ6 A + 16E 4 µ2 B 4 + 64E 4 µ2 B 2
+16E 4 µ4 B 4 − 32E 4 µ3 B 4 − 192µ4 E 2 B 2 + 384E 2 µ3 B 2
−8E 3 µ2 A − 96E 4 µ3 B 2 ,
a14 = −64B − 672µ2 B + 32EAµB + 800EAµ3 B − 256EAµ2 B − 192µ5 BE 2
+480E 2 µ2 B − 192E 2 µB − 640E 2 µ3 B − 24E 4 µ2 B + 1184µ5 EAB
+24E 4 µ5 B − 60E 4 µ4 B + 4E 4 µB − 1280EAµ4 B + 56E 4 µ3 B − 480µ4 B
+192µ5 B + 320µB + 32E 2 B + 768µ3 B + 192µ7 BEA − 672µ6 BEA
−16E 3 µ3 AB + 80E 3 µ4 AB − 64B 3 E 2 µ − 128µ5 B 3 E 2 + 320µ4 B 3 E 2
−144E 3 µ5 AB − 32E 3 µ7 AB + 112E 3 µ6 AB + 32E 4 µ5 B 3 − 80E 4 µ4 B 3
+64E 4 µ3 B 3 − 16E 4 µ2 B 3 + 480µ4 BE 2 + 256µ2 B 3 E 2 − 384µ3 B 3 E 2 ,
a15 = 32 − 8E 2 − 192µ + 64B 2 + 48E 2 µ − 8E 4 µ3 + 496µ2 − 232E 2 µ4
+224E 2 µ3 − 136E 2 µ2 − 1344µ3 B 2 − 576µ5 B 2 + 192µ6 B 2
+592µ4 − 704µ3 + 96µ6 − 288µ5 + 144E 2 µ5 − 48µ6 E 2 + 4E 4 µ6
−12E 4 µ5 + 14E 4 µ4 + 240EAµ4 − 96EAµ3 + 16EAµ2 − 352E 2 µ2 B 2
+64E 2 µB 2 + 256µ6 EA + 76E 4 µ4 B 2 + 24E 4 µ6 B 2 − 72E 4 µ5 B 2
−320EAµ5 − 384µB 2 + 2E 4 µ2 + 576µ5 B 2 E 2 + 16E 3 µ5 A
−192µ6 B 2 E 2 − 4E 3 µ4 A + 32µ8 EA − 128µ7 EA − 4E 3 µ8 A
+16E 3 µ7 A − 24E 3 µ6 A + 4E 4 µ2 B 2 + 960µ2 B 2 − 864µ4 E 2 B 2
+768E 2 µ3 B 2 + 1152µ4 B 2 − 32E 4 µ3 B 2 ,
a16 = −672µ6 B + 36E 4 µ5 B − 64B + 336E 2 µ6 B − 1344µ2 B
−28E 4 µ6 B − 2400µ4 B − 96E 2 µ7 B + 448µB + 2272µ3 B
+4E 4 µ3 B + 640µ4 BE 2 + 8E 4 µ7 B − 20E 4 µ4 B − 16E 2 µB
−592µ5 BE 2 − 400E 2 µ3 B + 128E 2 µ2 B + 1632µ5 B + 192µ7 B,
a17 = −896µ3 + 16 − 128µ + 448µ2 + 1136µ4 + 48E 2 µ3 − 8E 2 µ2
−120E 2 µ4 − 16E 2 µ8 + 64E 2 µ7 + 544µ6 − 960µ5 + 48µ8 − 192µ7
+160E 2 µ5 − 128µ6 E 2 + E 4 µ8 − 4E 4 µ7 + 6E 4 µ6 − 4E 4 µ5 + E 4 µ4 .
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:
n
˜ η̃, ζ̃ ∈ R3 / ξ˜ = r cos ϕ sin θ, η̃ = cos ϕ sin θ,
ZV S (f, e, C, µ) =
ξ,
ζ̃ = 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 ≥ Ar2 ,
2
E − Ar2 ≥
2
4 (1 − µ)
4µ2
+
2
2
2
r + 2µBr + µ
r2 + 2 (µ − 1) Br + (µ − 1)
)
.
4
4
CONCLUSIONS
9
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, 2004.
[4] Marchal, C.: The three-body problem, Elsevier, Studies in Astronautics,
1990.
[5] Szebehely, V.: Theory of orbits, Academic Press, New-York, 1967.