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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
On the development of M. L.Lidov's techniques on
the evolution of satellite orbits
M. A. Vashkov'yak
Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences,
Moscow, Russia
In this paper basic results of the investigations carried out by M. L. Lidov
on the evolution of satellite orbits are presented. They have received further
theoretical and applied development.
1. As early as in 1961 Lidov initiated new approach in celestial mechanics
aimed to investigate the orbital evolution of artificial and natural satellites of
the planets. His methods allowed to calculate sufficiently exactly the orbital pa­
rameters of artificial Earth satellites and to get qualitative conclusions about the
behavior of satellite orbits of different classes on the basis of simplified integrable
problem [4,5], i.e. the so called double--averaged Hill's problem. This problem de­
scribes the evolution of the satellite orbit under the effect of secular perturbations
of the distant attracting point in the first approximation. Two integrals of this
problem c 0
and c 1
were known since long ago [8]. The third integral c 2
found by
Lidov allows to reduce the problem to the investigation of the behavior of integral
curves in the plane (!; e) with fixed c 1
and different values c 2
. The qualitative
analysis of integral curves has resulted in interesting celestial mechanics conclu­
sions as follows: 1) the satellite orbits with constant e and ! exist in the averaged
problem. This special point with the libration change of ! in its neighborhood
is called often ``Kozai--resonance'' by the name of Kozai who investigated a more
general case of the problem one year later [1]. Since the qualitative peculiarities
revealed by Lidov remain valid in this case the name ``Lidov--Kozai--resonance''
as proposed by A. I. Neishtadt would be justified; 2) in particular case c 1 = 0
when the satellite orbital plane is orthogonal to the orbital plane of the perturb­
ing body the evolution leads to transforming the orbit into the straight line. For
small values of c 1 the orbital evolution for any initial values of ! involves a strong
increase in e. As the semi--major axis a = c 0
is constant the distance of the peri­
center becomes equal to the planetary radius and satellite falls onto its surface.
This peculiarity of evolving orbits might be called ``Lidov--Kozai--mechanism''.
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It's interesting to note that the above peculiarity of almost orthogonal satellite
orbits conflicted with the real existence of the Uranus' satellites. Near circular and
equatorial orbits of these satellites are inclined to the ecliptic plane by 98 ffi . The
analysis of real physical model taking into account the Uranus' oblateness resolved
this contradiction and admitted the existence of circular orthogonal orbits [6].
Moreover, it leaded to the formulation of a new celestial mechanics problem
(double--averaged Hill's problem taking into account the oblateness of central
planet). For analyzing this problem two parameters are essential: fl --- which
characterizes the ratio of perturbing accelerations due to the planet's oblateness
and external body; ffl --- the inclination of equatorial planet's plane to the orbital
plane of the perturbing body. In general case this problem doesn't allow any
more three first integrals in involution. Nevertheless, this problem has a number
of integrable cases and particular solutions revealed in [7,3]. Of the most interest
is the coplanar case (ffl = 0) investigated earlier by numerical way [2].
2. In what follows the researches carried out in the recent years and repre­
senting the development of Lidov's works are briely described.
2.1. In our investigations of the double--averaged Hill's problem taking into
account the oblateness of the central planet there was established the invariance
of evolutionary system with respect to a certain transformation of the elements
and time. The families of the periodic solutions have been revealed and actually
constructed (mainly numerically). These solutions correspond to the so--called pe­
riodically evolving satellite orbits characterized by the change of all four elements
i, e,
!,\Omega with the same period. The analysis of the stability of the stationary
solutions has been carried out. In their small vicinity the symmetric and asym­
metric periodic solutions have been constructed. The numerical prolongation of
periodic solutions to the regions far from the stationary points permit to reveal
their closed families. The regions of the stability of discovered periodic solutions
have been determined by numerical way (see [9] and references therein).
2.2. The development of Lidov's techniques is also of much importance for
the theory of motion of natural satellites. It concerns primarily the orbits of new
satellites of the giant planets discovered from the end of 1997 to the beginning
of 2000. They move in the regions, where the influence of the solar perturbation
considerably exceeds the influence of the central planet's oblateness (fl Ü 1). For
the analysis of their evolution in the first approximation it is possible to use the
double--averaged Hill's problem in taking fl = 0. Qualitative investigation of this
problem initiated by Lidov has been completed by us in 1999 with construction
of the general solution depending on four arbitrary constants, i.e. initial values of
the elements. For calculation
of\Omega an analytical expression in the form of elliptic
integrals of the first and third kinds was received. Our subsequent papers have
dealt with the qualitative peculiarities of the evolution of new satellite orbits. The
simplest classification of these orbits was proposed together with the determina­
tion of the approximate characteristics of the evolution including the extreme
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values of e and i as well as the periods of the circulation of !
and\Omega\Gamma In the
framework of the employed double--averaged Hill's problem for four new Saturn's
satellites the libration of ! was detected. However, it should be mentioned that
their integral curves are located near to the separatrix. Therefore some libration
orbits can pass to the group of circulator if the elements of the satellite orbits
or the model of the evolution are defined more precisely. The above peculiarity
is extremely rare even among ensemble of thousand of asteroid orbits. Therefore,
the fact that among twelve orbits of recently discovered Saturn's satellites four
orbits turn out to be potential librators is strange enough.
Another phenomenon in satellite systems of Jupiter, Saturn and Uranus is
the distribution of the semi--major axes of the satellite orbits. In Jupiter's system
the external satellites are divided evidently into two groups, direct and indirect
ones. In Saturn's system there is the range of the semi--major axes where there
exist both direct and indirect orbits. In the Uranus' system all discovered by to­
day external satellites have indirect motion. Moreover, in all three systems there
are the regions of the semi--major axes free from satellite orbits. The revelation
of possible mechanism of ``avoiding'' above regions by the satellites presents a
very interesting problem. In the Uranus' system such ``emptiness'' which sepa­
rates internal and external satellites stretches approximately from 0.6 million km
(Oberon's orbit) to 7 million km (Caliban's orbit), where the perturbing influence
of the Sun is comparable with that of the oblateness of the central planet. A qual­
itative analysis of one of integrable cases of the double--averaged Hill's problem
taking into account planet's oblateness performed by Lidov [7] allows to presume
(as a hypothesis) the celestial mechanics explanation of the absence of the equa­
torial Uranus' satellites in the region a ? 1.3 million km. If there were equatorial
satellites in the above region, then their orbits should have begun to intersect
with internal satellite orbits. In this case the probability of close approaches and
collisions with internal satellites increases, as they were to go to more distant
orbits or fall down onto the internal satellites filling up their masses essentially.
Indirect confirmation of this fact is remarkable massivity of the internal Uranus'
satellites as compared with external ones. More details may be found in [10] and
references therein.
This study is supported by the Council of Grants of the President of Russia
and the State support of Leading Scientific Schools (grant No. 00--15--96036).
References
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