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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Geometrical analysis of solutions of restricted
three--body problem
V. I. Prokhorenko
Space Research Institute, Moscow, Russia
This investigation concerns the problem of the choice of the long living ellip­
tical orbits with high eccentricity. The aim is to construct a suitable approach for
this choice taking into consideration the evolution of the orbital parameters under
the perturbing influence of the external bodies. For the qualitative analysis of the
orbital evolution we use the solution of the restricted circular double--averaged
three--body problem for ff = a=a k !! 1 obtained by M.L. Lidov [1] in 1961. This
solution includes three first integrals:
c 0 = a; c 1 = '' cos 2 i; c 2 = (1 \Gamma '')(2=5 \Gamma sin 2 i sin 2 !)
and the quadrature t \Gamma t 0 = \Gamma2úa 3=2 (¯A) \Gamma1 L(c 1 ; c 2 ; '' 0 ; ''; ! 0 ), where
A = (15=2)ú(M k =M)(a=a k ) 3 '' k
\Gamma3=2 ;
L(c 1 ; c 2 ; '' 0 ; ''; ! 0 ) =
Z ''
'' 0
[(1 \Gamma '')'' 1=2 sin 2 i sin 2!] \Gamma1 d'' ;
a is the semi major axis; '' = 1 \Gamma e 2 , e is the eccentricity; i, ! are the inclination
and the argument of pericenter measured relative to the disturbing body orbit; M ,
M k are the masses of the central and disturbing bodies; ¯ is the product of the M
and the gravity constant f ; a k , '' k are the parameters of the orbit of the disturbing
body. The constants c 0 , c 1 , c 2 are determined as functions of the initial values of
the orbital elements: c 0 = a 0 , c 1 = '' 0 cos 2 i 0 , c 2 = (1 \Gamma '' 0 )(2=5 \Gamma sin 2 i 0 sin 2 ! 0 ).
The spherical coordinate system Or`–, where r = '', co--latitude ` = i, and
– = ! has been proposed [2] for the geometrical analysis. The origin of the frame
Or`– coincides with the central body S. The integral c 1 = const corresponds in
this frame to the revolution surface, while c 2 = const corresponds to a closed
line in this surface. The integral curves go around the separate point of the cen­
ter type corresponding to c 2 = c 2max (c 1 ) in the case c 2 ? 0. The parameter !
varies monotonously anti--clockwise with the period equal to 2ú. Such kind of
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evolution is marked as a rotation. The changing of parameters '' and i is vice
versa of oscillation character. In the case c 2 ! 0 the curves go around one of
the separate points corresponding to c 2 = c 2min (c 1 ), and ! = 90 ffi or ! = 270 ffi .
The evolution of the parameter ! is in this case of the libration type. The do­
mains of the different types of the ! evolution are separated by the separatrices
c 2 = 0. We use the similarity theory to present the non--dimensional period T \Lambda of
the evolution of the parameters '' and i as a product of the independent factors:
T \Lambda = \Gamma(4=15)(a \Gamma3=2
\Lambda =LD )jL c (c 1 ; c 2 )j. Here LD = ¯ k \Lambda a \Gamma3
k \Lambda '' \Gamma3=2
k ¯ \Gamma1=2
\Lambda is the non--
dimensional parameter of disturbances similarity dependent on the dynamical
characteristics of the central and disturbing bodies only; L c (c 1 ; c 2 ) is the double
quadrature L calculated for the limits ('' max , '' min ). The non­dimensional param­
eters, marked by the dot *, are expressed in terms of some character parameters:
dimension l, time Ü , and mass m. Let us use for L c (c 1 ; c 2 ) the name ''configura­
tion parameter of the orbit''. The investigation of the behavior of this function
of two variables helps to find some of latent regularities. Of most interest is the
existence of the quasi--symmetric solutions in the regions of the different types
of the ! evolution (libration or rotation). We mean the solutions with the equal
absolute values of the configuration parameters, corresponding to c 2 of the same
absolute values but of the opposite signs. We shall take into consideration the
finite dimension of the central body to estimate the ballistic lifetime. The region
of the satellite existence corresponds to the parts of the integral curves located
outside the spherical surface '' = ''\Lambda, corresponding to the impact of the satellite
with the central body. In accordance to [1] ''\Lambda = (2a \Lambda \Gamma 1)=a 2
\Lambda , where a \Lambda = a=R,
and R is the radius of the central body. The non--dimensional majorizing func­
tion of the ballistic lifetime is TB \Lambda = \Gamma(4=15)(a \Gamma3=2
\Lambda =LD )jL b (c 1 ; c 2 ; a \Lambda )j, where
L b (c 1 ; c 2 ; a \Lambda ) is the double quadrature L calculated for the limits ('' max , ''\Lambda) in
case '' max ? ''\Lambda – '' min . It should be mentioned that jL b (c 1 ; c 2 ; a \Lambda )j ! jL c (c 1 ; c 2 )j
and jL b j has just the same mirror quasisymmetry relative to c 2 = 0 as the function
jL c j has. The main result of this investigation is the creation of the geometrical
tool for the suitable choice of the orbits taking into account the ballistic lifetime
and the character of the orbital parameters evolution. The efficiency of this tool
was checked by the retrospective analysis of the orbits of the AES PROGNOZ
series, (a = 16:7RE ) launched in 1972--1995.
References
1. Lidov M. L. Orbit Evolution of Artificial Satellites of Planets under the
Action of Gravitational Perturbations of External Bodies. Iskusstv. Sputn.
Zemli, 1961, No. 8, 5--45 (in Russian).
2. Prokhorenko V. I. A Geometric Study of Solutions to Restricted Circular
Double--Averaged Three--Body Problem. Kosm. Issled., 2001, 39, 583--593
(in Russian).
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