Orbital Evolution of the Outer Satellites of Giant Planets.
Methods of Analysis and Results

Fig. 1. Lidov
Drops the Moon on the Earth.
Friendly Caricature of
Professor I.V.Novozhilov (MSU)
3°. Note that the property of almost orthogonal
satellite orbits, established by Lidov, was inconsistent with quiet existence of the
close Uranian satellites known at that time. Their almost circular and almost
equatorial orbits are inclined by about 98° to the plane of the ecliptic (or the
plane of the Uranocentric orbit of the Sun). However, Lidov’s analysis showed
that solar perturbations play no crucial part in the orbital evolution of close
Uranian satellites. The double averaged Hill problem with the allowance for the flattening of the
central planet as considered by Lidov is nonintegrable (integral c_{1 }does not exist in the
general case with no axial symmetry). However, one of its integrable cases
describes the evolution of equatorial satellite orbits when the equatorial
plane of the planet is orthogonal to the plane of motion of the perturbing
body. This case approximately takes place in the system of inner Uranian
satellites. It allows the existence of almost circular orbits whose evolution
is determined mostly by flattening of the planet (Lidov, 1963a). Hence the flattening
of Uranus prevents its close satellites from falling onto the planet, and allows the
conflict to be resolved in
terms of a real physical model.
Returning to the double averaged Hill problem (without the
allowance for flattening of the planet) let us point out that the stationary
singular point and librational variation of the pericenter are usually referred
to as the “Kozairesonance” in special celestialmechanics publications after
the Japanese researcher Y.Kozai, who studied the asteroid variant of the
problem one year after Lidov had made his analysis. The work that Lidov published in 1961, became known abroad only after
1962, when it was presented at the international meeting in Paris and was
published in English in two leading journals on space research (Lidov, 1962,
1963b). Given that the asteroid case of the problem maintains all the
qualitative features found by Lidov, it would be fair to use the term
“LidovKozairesonance” as suggested by Professor A.I.Neishtadt. For the same
reason, it would be fair for the possible substantial variation of the
eccentricity and the inclination angle of the satellite orbit is referred to as
the “LidovKozaimechanism”. These terms have been used in recent years by some
of the authors who analyzed the orbital evolution of a number of outer
satellites of giant planets.
They were the discoveries
of outer satellites that made it necessary not only to perform a qualitative
analysis of the double averaged Hill problem, but also to derive the time
dependences of the real orbital elements of such satellites for arbitrary
initial conditions. The
general solution of the evolutionary problem was constructed in papers
(Kinoshita, H.Nakai, 1999, 2007) and in (Vashkov’yak, 1999).
J. Kovalevsky and especially A. A.
Orlov further analyzed the satellite variant of the averaged problem. The
equations, that describe the satellite orbital evolution in the restricted
threebody problem, were derived using the Zeipel method up to third order
infinitesimal in the small parameter m
 the ratio of the mean motions of the perturbing body and the satellite. Thus
the above authors (Kovalevsky, 1964, 1966; Orlov, 1965a, 1965b) constructed an m^{3} order
solution, which is more accurate, than the solution of the doubleaveraged Hill
problem (order m^{2}). The results of Orlov were
essentially used by us in developing the numerical analytical method for the
analysis of the orbital evolution of distant satellites.
4°. Not only the model of double averaged restricted
circular threebody problem, but also the corresponding singly averaged model
may prove to be of use for the analysis of the orbital evolution of some
distant Jovian satellites. N. D. Moiseev introduced such an averaged problem
(or scheme) in 1945, and it later came to be named after him. In this problem
(Moiseev, 1945) only the fastest variable  the mean longitude of the satellite
 has been excluded from the perturbing function. At the same time, the
evolutionary equations contain the periodic functions of time that are due to
the motion of the perturbing body during one satellite revolution. The
doubleaveraged model described above does not contain these functions.
However, the effect of perturbations with the intermediate period equal to half
the orbital period of the planet may be great.
A combined numericalanalytical
method found its application in many problems of celestial mechanics along with
analytical and numerical methods. The version of this method that M. L. Lidov
proposed in 1978 for Hamiltonian systems includes the following three main
components (Lidov, 1978):
The
first
component is an analytical construction
of a sequence of canonical changes of variables to eliminate rapidly
oscillating functions of time from the Hamiltonian. These transformations
establish a relation between the osculating and average orbital elements. The second component is a
derivation of a new Hamiltonian written in terms of average elements (this
procedure is also performed analytically). The
third component is an
effective numerical integration of differential equations written in terms of
average elements with a long time step of the order of several orbital periods
of the satellite.
Our proposed numericalanalytical
method for the computation of the orbital evolution of distant satellites is
also based on the main constructive idea of Lidov, the Zeipel method, and the
series of Orlov’s papers (Vashkov’yak, 2005).
5°. The first new outer satellites among the large family of such objects
discovered were the Uranian satellites U16 (Caliban) and U17 (Sycorax) found in
1997. The ephemerides of these and of other outer satellites of giantplanets,
discovered later, are calculated in the papers (Emelyanov, 2005; Emelyanov,
Kanter, 2005). These ephemerides are updating constantly in accordance with new
observations and are available on the servers (http://lnfm1.sai.msu.su/neb/nss/index.htm)
and http://www.imcce.fr/sat.
Our following presentation of the computed orbital evolution of these
and other satellites is based on the data from the special Internet server set
up by N.V.Emel'yanov (SAI MSU). One can download from this server, in
particular the orbital elements refined with using all the observations available
for the times of their discovery. We use these elements as the initial values
in computations employing various methods. These initial orbital elements
usually differ from the preliminary elements posted on the Minor Planet
Electronic Circular Internet site
(http://cfawww.harvard.edu/mpec/RecentMPECs.html). We computed the time
dependencies of the orbital elements of the outer satellites and, in
particular, those of Caliban and Sycorax, using the three methods described
above over the time intervals ranging from one to several periods of precession
of nodes. In the Fig. 2, 3 and in the other figures the solid lines show the results obtained by numerically integrating
the exact equations of motion; the dots
 correspond to the results obtained using the numerical analytical method,
and the dashed lines correspond
to the analytical solution of the double averaged Hill problem.
Fig. 2 U16 (Caliban);
Abscissa axis – time t
(years),
ordinate axis: 1 – e(t), 2
– i_{eclip }(t),3  w(t), deg, 4  W(t), deg,
strict numerical calculation; ··········
numericalanalytical method;
analytical method (double averaged Hill problem).
Fig.
3. U17 (Sycorax)
The orbit of satellite U23 (Margaret) (Fig. 4) differs from those of
Caliban and Sycorax by the librational variation of the argument of the
pericenter and by its prograde motion.
Fig. 4. U23
(Margaret)
The data for the outer Uranian satellites lead us to conclude that the
results obtained using numerical method agree well with those obtained using
numericalanalytical method. The time dependencies of the orbital elements
computed using analytical method differ from the corresponding time
dependencies computed using numerical methods. The differences are due mostly
to inaccurately computed periods of the variation of the pericenter arguments
and of the ascendingnode longitudes. However, in the cases where their results
agree qualitatively, the two methods yield sufficiently similar extreme
parameters of evolution.
The orbital evolution of two
Neptunian satellites with circulatory N9 (Halimede) and librational N11
(Sao) variations of the pericenter arguments is characterized by the
substantial amplitudes of oscillation of the eccentricities and inclinations in
accordance with the LidovKozaimechanism. The orbital elements of two Jovian
satellites – J18 and J46 and two Saturnian satellites – S9 and S24 show
similar temporal variations.
The orbits of a number of outer satellites proved to exhibit librational
variation of the arguments of the pericenter. This is a rather rarely
encountered feature even among the ensemble of several thousand asteroid orbits.
Y.Kozai was the first to discover an actual w  librational orbit of the asteroid Cincinnati. The
first wlibrators discovered among the planetary satellites by
solving the doubleaveraged Hill problem were the outer Saturnian satellites
S22 (Ijiraq) and S24 (Kiviuq). A number of similar orbits were later found in
the satellite systems of Jupiter  Carpo, Uranus  Margaret, and Neptune Sao
and Neso. All these satellites are demonstrative examples of natural
realization of the LidovKozairesonance in the satellite systems of giant
planets. The phenomenon of this resonance was also confirmed via numerical
integration of strict equations of motion of the satellites mentioned above.
6°. Note that in some cases the simple evolutionary
model initially yielded paradoxical
results. The analysis of the orbits of the outer Saturnian satellites S20
(Paaliaq) and S29 (Siarnaq) performed in framework of the double averaged Hill
problem implied the librational behavior of the variation of the pericenter
arguments w (c_{2} < 0). However, the computations made by more
accurate numerical and numericalanalytical methods showed that the actual
variations of w for these orbits are
circulatory. This qualitative discrepancy is due to the fact that the phase
point in the (w, e) plane is close to the separatrix of the integrable double
averaged Hill problem, where constant c_{2}
is very close to zero. The
allowance for minor perturbations revealed the actually circulatory behavior of
the evolution of w for the orbits mentioned above.
As a result, the simple evolutionary model was found to be in need of
refinement for the domains of orbital parameters e, i and w, lying within the close neighborhood of the
separatrix. The same is also true for the orbit of the Saturnian satellite
S/2007 S1 and Neptunian satellite N10 (Psamathe). By contrast, the analysis of
the orbital evolution of the Neptunian satellite N13 (Neso) made in framework
of the double averaged Hill problem yields circulatory behavior of w, whereas more accurate methods show librational
variation of this parameter.
The evolutionary model had also to be refined in another case where
orbital parameters e, i and w are
such that not only c_{2 }» 0, but c_{1}
is close to its bifurcation value of 0.6. Under such conditions, the (w, e) plane develops a pair of singular points.
Integral surfaces с_{2} (w, е) deform along с_{1}
= 0.95; 0.6; 0.05; 0 (Fig. 5, from above down). First, a sufficiently smooth
surface develops singularities (с_{1}
= 0.6) and then does depressions and “discharges” (с_{1} = 0) along which the phase flow «discharges» onto the
boundary of the domain (е = 1). When с_{1} » 0.6 and с_{2}
»0 the evolutionary equations become extremely sensitive to any perturbations of the double averaged
Hill problem and, hence, their solution may undergo qualitative changes.
с_{1} = 0.95
с_{1} = 0.6
с_{1} = 0.05
с_{1} = 0
Fig. 5. Integral
surfaces с_{2} (w, е) (left column) and curves c_{2} =
const
(right column) for different values с_{1} = 0.95; 0.6; 0.05; 0.
The Fig. 6 shows the separatrices с_{2}
= 0 in the plane of the initial parameters (i_{0}, w_{0}). The circles and triangles,
respectively indicate the wlibrators and orbits that actually proved to be circulatory. The
asterisks indicate the peculiar orbits of three Jovian satellites J49 (Kore),
S/2003 J3, and J34 (Euporie).
Fig. 6. Real and supposed w  librators in the plane (i_{0}, w_{0});
sign D  w circulates
for 4 satellite
orbits.
If computed in framework of the simple model, the evolution of w for the
orbits of these satellites would have circulatory behavior (c_{2} > 0). However, the
computations made by the method of numerical integration showed (Fig. 7) that
the pericenter arguments w of these
orbits vary within limited intervals about the equilibrium value of 90°.
Fig. 7. Time dependences of the
elements of the evolving orbit for the Jovian satellite J34 (Euporie) as the
result of the numerical integration of the rigorous equations of motion.
This paradox is due to strong periodic solar perturbations that are neglected in the double averaged
problem. The Fig. 8 shows the curves с_{1}
= 0.6 and the above orbits in the plane of initial parameter values (i_{0}, e_{0}).
Fig. 8. Real and supposed w 
librators in the plane (i_{0}, e_{0});
sign D  w circulates
for 4 satellite
orbits.
When с_{1 }» 0.6, it is the unaccounted about sixyear
periodic perturbations (with half of the Jovian orbital period) that introduce
qualitative changes in the behavior of the evolution of w for the above orbits. Taking into account such
perturbations in the singly averaged model allowed qualitatively
to explain the librational behavior of the corresponding arguments of the
pericenters. It is possible to construct analytically special surface
f [e,
h = W(t)l_{1}(t), w = ± 90°] = Cdw/dt. (4)
The image of this surface is indicative of the
reversal of the sign of derivative of w with respect to t when w = ± 90°. Furthermore, for satellite J34 (Euporie) we also quantitatively
confirmed the domain of variation of w to span from 64° to 116°. For this satellite Fig. 9 shows the phase points on
the (w, e) plane, whose positions were computed by numerically
integrating the singly averaged Hill problem over a 10000year long time
interval.
The phase pattern of «classical» w  librators, which are in the
state of the LidovKozai resonance, resembles a rather narrow ringshaped
domain. By contrast, the phase points of the orbit of J34 (Euporie) fill
virtually the entire domain of libration. The orbits of the Jovian satellites
Kore and S/2003 J3 exhibit similar patterns. Note that the ranges of
eccentricity variations are wider than those obtained from the strict numerical
solution. Therefore the averaged evolutionary model for the orbits of these
three Jovian satellites is in need of further refinement.
Acknowledgements.
In conclusion, we consider it necessary to emphasize that this work on
the analysis of the orbital evolution of distant satellites is based on the
achievements of many celestial mechanists from different countries. I would
like to mark with gratitude the special role in my work on this subject of my
teachers  the Russian researchers Professors N.D.Moiseev, A.A.Orlov, and M.L.
Lidov.
This work is supported by the Russian Foundation for Basic Research
(RFBR) within the framework of two international grants № 070292169NTsNI_a
and 070291229YaF as well as the grant of scientific school №
NSh1123.2008.1.
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