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 c1 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 “Kozai-resonance” in special celestial-mechanics 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 “Lidov-Kozai-resonance” 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 “Lidov-Kozai-mechanism”. 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 three-body 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 m3 order solution, which is more accurate, than the solution of the double-averaged Hill problem (order m2). 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 three-body 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 double-averaged 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 numerical-analytical 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 numerical-analytical 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 giant-planets, 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://cfa-www.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 – ieclip (t),3 - w(t), deg, 4 - W(t), deg,
strict numerical calculation; ·········· numerical-analytical 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 numerical-analytical 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 ascending-node 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 Lidov-Kozai-mechanism. 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 w-librators discovered among the planetary satellites by solving the double-averaged 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 Lidov-Kozai-resonance 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 (c2 < 0). However, the computations made by more accurate numerical and numerical-analytical 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 c2 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 c2 » 0, but c1 is close to its bifurcation value of 0.6. Under such conditions, the (w, e) plane develops a pair of singular points. Integral surfaces с2 (w, е) deform along с1 = 0.95; 0.6; 0.05; 0 (Fig. 5, from above down). First, a sufficiently smooth surface develops singularities (с1 = 0.6) and then does depressions and “discharges” (с1 = 0) along which the phase flow «discharges» onto the boundary of the domain (е = 1). When с1 » 0.6 and с2 »0 the evolutionary equations become extremely sensitive to any perturbations of the double averaged Hill problem and, hence, their solution may undergo qualitative changes.
с1 = 0.95
с1 = 0.6
с1 = 0.05
с1 = 0
Fig. 5. Integral surfaces с2 (w, е) (left column) and curves c2 = const
(right column) for different values с1 = 0.95; 0.6; 0.05; 0.
The Fig. 6 shows the separatrices с2 = 0 in the plane of the initial parameters (i0, w0). The circles and triangles, respectively indicate the w-librators 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 (i0, w0);
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 (c2 > 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 с1 = 0.6 and the above orbits in the plane of initial parameter values (i0, e0).
Fig. 8. Real and supposed w - librators in the plane (i0, e0); sign D - w circulates for 4 satellite orbits.
When с1 » 0.6, it is the unaccounted about six-year 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)-l1(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 10000-year long time interval.
The phase pattern of «classical» w - librators, which are in the state of the Lidov-Kozai resonance, resembles a rather narrow ring-shaped 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.
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 № 07-02-92169-NTsNI_a and 07-02-91229-YaF as well as the grant of scientific school № NSh-1123.2008.1.
C.Beaugé, D.Nesvorny, Astron. J. 133, 2537 (2007).
V.V.Beletsky, Essays on the Motion of Celestial Bodies. Translated from
the Russian by Andrey Iakob. Birkhäuser Verlag. Basel. Boston. Berlin.
372 p. (2001).
A.A.Christou, Icarus, 174, 215 (2005).
M.Cuk, J. A.Burns, Astron. J. 128, 2518 (2004).
N.V.Emelyanov, Astron. & Astrophys., 435, 1173-1179 (2005).
N.V.Emelyanov, A.A.Kanter, Solar System Research, 39, 2, 112-123 (2005).
D.Jewitt, N.Haghighipour, Annu. Rev. Astron. Astrophys, 45, 261-295 (2007).
H.Kinoshita, H.Nakai, Cel. Mech. & Dyn. Astron., 75, 125-147 (1999).
H.Kinoshita, H.Nakai, Cel. Mech. & Dyn. Astron., 98, 67-74 (2007).
Y. Kozai, Astron. J. 67, pp. 591-598 (1962).
M.L. Lidov, Iskusstvennye Sputniki Zemli, 8, pp. 5-45, (1961, in Russian).
M.L. Lidov, In Dynamics of Satellites (M. Roy Ed.), pp. 168-179,
Springer-Verlag, Berlin. Göttingen. Heldelberg (1963a),
Symposium Paris, May 28-30, 1962.
M.L. Lidov, Planet. Space Sci., 9, pp. 719-759 (1962).
Translated by H.F. Cleaves from Iskusstvennye Sputniki Zemli,
No. 8, p.5, 1961.
M.L. Lidov, AIAA Journal, 1, 8, pp. 1985-2002 (1963b).
Translated from Iskusstvennye Sputniki Zemli,
(Artificial Earth Satellites) (Academy of Sciences Press,
Moscow, 1961), No. 8, pp.5-45, 1961.
Translated by Jean Findlay, Green Bank, West Va.
Reviewed by Yoshide Kozai, Smithsonian Astrophysical Observatory,
M.L.Lidov, Trudy Inst. Teor. Astron., XVII, 54-61, 1978 (in Russian).
N.D.Moiseev, Trudy Gos. Astron. Inst. im. P.K.Sternberga, XV,
pp. 100 – 117, 1945 (in Russian).
D.Nesvorny, Jose L.A.Alvarellos, Luke Dones, Harold F.Levison,
Astron. J. 126, 398 (2003).
J. Kovalevsky, Acad. Sci. 258, 18 (1964).
J. Kovalevsky, IAU Symp. No.25 (Ed. G.I. Contopoulos), London,
Academ. Press, p. 326 (1966).
A.A.Orlov, Bull. Inst. Teor. Astron., X, 5(118), p. 360 (1965a, in Russian).
A.A.Orlov, Proc. of XV Intern. Congress on Astronautics
Paris, Gautier-Villars, Warshawa, PWN-Polish Sci. Publ., 1 (1965b).
V.S.Ural’skaya, Solar Syst. Res. 37, 337 (2003).
M.A.Vashkov’yak, Astronomy Letters, 25, 476-481 (1999).
M.A.Vashkov’yak, Astronomy Letters, 29, 695-703 (2003).
M.A.Vashkov’yak, Astronomy Letters, 31, 64-72 (2005).
M.A.Vashkov’yak, N.M.Teslenko, Astronomy Letters, 31, 140-146 (2005).
M.A.Vashkov’yak, N.M.Teslenko, Astronomy Letters, 33, 780-787 (2007).