Orbital evolution of Saturn’s new outer satellites and their classification

Based on data for twelve recently discovered outer satellites of Saturn, we investigate their orbital evolution on long time scales. For our analysis, we use the previously obtained general solution of Hill’s double-averaged problem, which was refined for libration orbits, and numerical integration of the averaged system of equations in elements with allowance for Saturn’s orbital evolution. The following basic quantitative parameters of evolving orbits are determined: extreme eccentricities and inclinations, as well as circulation periods of the pericenter arguments and of the longitudes of the ascending nodes. For four new satellite orbits, we have revealed the libration pattern of variations in pericenter arguments and determined the ranges and periods of their variations. Based on characteristic features of the orbits of Saturn’s new satellites, we propose their natural classification.     

Orbital evolution of new distant Neptunian satellites and ω-librators in the satellite systems of Saturn and Jupiter

We used data on the recently discovered three outer Neptunian satellites to analyze the long-period evolution of their orbits. We estimated the ranges of eccentricities and inclinations as well as approximate circulation periods of the pericenter arguments and the longitudes of the ascending nodes. The results were mainly obtained by using two different versions of the averaged Hill problem. Plane sections of the phase space of satellite orbital elements are given. We discuss the peculiarity of the evolution of several satellite orbits related to the librational variation of the pericenter argument ω. The ω-librators of Saturn’s system were found to qualitatively differ from the libration orbit in the system of Jupiter.     

Orbital evolution of Uranus’s new outer satellites

Data on three recently discovered satellites of Uranus are used to determine basic evolutional parameters of their orbits: the extreme eccentricities and inclinations, as well as the circulation periods of the pericenter arguments and of the longitudes of the ascending nodes. The evolution is mainly investigated by analytically solving Hill’s double-averaged problem for the Uranus-Sun-satellite system, in which Uranus’s orbital eccentricity e _U and inclination i _U to the ecliptic are assumed to be zero. For the real model of Uranus’s evolving orbit with e _U≠0 and i _U≠0, we refine the evolutional parameters of the satellite orbits by numerically integrating the averaged system. Having analyzed the configuration and dynamics of the orbits of Uranus’s five outer satellites, we have revealed the possibility of their mutual crossings and obtained approximate temporal estimates.     

Chaotic motions of prometheus and pandora

Recent HST images of the saturnian satellites Prometheus and Pandora show that their longitudes deviate from predictions of ephemerides based on Voyager images. Currently Prometheus is lagging and Pandora leading these predictions by somewhat more than 20°. We show that these discrepancies are fully accounted for by gravitational interactions between the two satellites. These peak every 24.8 days at conjunctions and excite chaotic perturbations. The Lyapunov exponent for the Prometheus-Pandora system is of order 0.3 year⁻¹ for satellite masses based on a nominal density of 0.63 g cm⁻³. Interactions are strongest when the orbits come closest together. This happens at intervals of 6.2 years when their apses are antialigned. In this context, we note the sudden changes of opposite signs in the mean motions of Prometheus and Pandora at the end of 2000 occurred around the time their apsidal lines were antialigned.     

Orbital evolution of Jupiter’s new outer satellites

Data on recently discovered outer satellites of Jupiter are used to analyze their long-period orbital evolution. We determine the extreme eccentricities and inclinations, as well as the circulation periods of the pericenter arguments and of the longitudes of the ascending nodes. The satellite orbital elements are plotted against time. The methods of analysis are identical to those that we used to analyze the orbital evolution of Uranus’s and Saturn’s new outer satellites.     

Dynamical history of coplanar two-satellite systems

One of the possible early states of the Earth-Moon system was a system of several large satellites around the Earth. The dynamical evolution of coplanar three-body systems is studied; a planet (Earth) and two massive satellites (proto-moons) with geocentric orbits of slightly different radii. Such configurations may arise in multiple satellite systems receding from a planet due to tidal friction. The numerical integration of the equations of motion shows that initially circular Keplerian orbits are soon transformed into disturbed elliptic orbits which are intersecting. The life-time of such a coplanar system between two probable physical collisions of satellites is roughly from one day to one year for satellite systems with radii less than 20R⊕, and may reach 100 yr for three-dimensional systems. This time-scale is short in comparison with the duration of the removal of satellites due to tides raised on the planet, which is estimated as 10⁶–10⁸ yr for the same orbital dimensions. Therefore, the life-time of a system of several proto-moons is mainly determined by their tidal interactions with the Earth. For conditions which we have considered, the most probable result of the evolution was coalescence of satellites as the consequence of the collisions.     

Evolutionary characteristics of the orbits of outer Saturnian,Uranian, and Neptunian satellites

We present the results of our systematic study of the long-period orbital evolution of all of the outer Saturnian, Uranian, and Neptunian satellites known to date. The plots of the orbital elements against time give a clear idea of the pattern of the orbital evolution of each satellite. The tabular data allow us to estimate the basic parameters of the evolving orbits, including the ranges of variation in the semimajor axes, eccentricities, and ecliptical inclinations as well as the variation periods and mean motions of the arguments of pericenters and the longitudes of the nodes. We compare the results obtained by numerically integrating the rigorous equations of the perturbed motion of the satellites with the analytical and numerical-analytical results. The satellite orbits with a librational pattern of variation in the arguments of pericenters are set apart.     

Regions of possible motions for new Jovian satellites

We present the results of our simulation and study of the regions of possible motions for 46 newly discovered Jovian satellites. We show that the orbits of some satellites (such as S/2003 J02, S/2003 J03, S/2003 J04, S/2003 J10, S/2003 J12, and S/2003 J23) presently cannot yet be determined with an acceptable accuracy for planning observations, because the amount of observational information is insufficient.     

Constructive analytical solution of the evolution hill problem

A new analytical solution of the system of differential equations describing secular perturbations and long-period solar perturbations of mean orbits of outer satellites of giant planets was obtained. As distinct from other solutions, the solution constructed using von Zeipel’s method approximately takes into account, in the secular part of the perturbing function, the totality of fourth order with respect to the small parameter m of the ratio of the mean motions of the primary planet and the satellite. This enables us to describe more accurately the evolution of satellite orbits with large apocentric distances, which in the course of evolution may exceed the halved radius of the Hill sphere of the planet with respect to the Sun. Among these are the orbits of the two outermost Neptunian satellites N10 (Psamathe) and N13 (Neso). For these satellites, the parameter m amounts to 0.152 and 0.165, respectively. Different from a purely analytical solution, the proposed solution requires preliminary calculations for each satellite. More precisely, in doing so, we need to construct some simple functions to approximate more complex ones. This is why we use the phrase “constructive analytical.” To illustrate the solution, we compare it with the results of the numerical integration of the strict motion equations of the satellites N10 and N13 over time intervals 5–15 thousand years.     

Gravitational scattering within the Himalia group of jovian prograde irregular satellites

We test the hypothesis that gravitational scattering within the Himalia prograde irregular satellite group is responsible for the large velocity dispersion identified by Nesvorný et al. (2003, Astron. J. 126, 398-429). We carry out numerical simulations of the dynamical evolution of this group under several scenarios for Himalia's mass for ¹⁰⁸ yr. We find that modification of the satellite orbits is significant, 10-15% of their semimajor axes differences but <5% of the eccentricities. Additionally, the inclination of the satellite Lysithea may have been modified by a secular resonance with Himalia. We scale the observed orbital element diffusion to the age of the Solar System using a power law approximation. The projected changes in a and e can reduce the observable ejection speeds of Elara and Lysithea with respect to Himalia under the 100 m s⁻¹ mark, rendering them consistent with hydrocode simulation results (Michel et al., 2002, Icarus 160, 10-23). The dispersion magnitude required to migrate Elara to its present orbit implies that Himalia's size has been underestimated by Cassini (Porco et al., 2003, Science 299, 1541-1547) and/or that its density is significantly higher than that of extensively fractured or “rubble-pile” bodies. In addition, Lysithea could have been created as recently as the last 500 Myr of the Solar System's history. Simple calculations indicate that such diffusion would also be evident in groups of satellites genetically related to Phoebe at Saturn, Caliban and Sycorax at Uranus and Nereid at Neptune.     

Construction d'orbites periodiques de satellites dans un systeme d'axes tournants,I

Résumé On développe une méthode de construction d'orbites périoldiques dans un système d'axes tournants, pour un satellite gravitant autour d'un sphéroide. Les orbites sont quasi circulaires,i est l'inclinaison sur le plan équatorial de la planète. Pour les petites inclinaisons, la solution est donnée jusqu'aux termes enJ ₂ ² etJ ₄.Ce modèle peut être appliqué aux satellites de Saturne. Des valeurs observées des longitudes des noeuds ascendants de Mimas et Téthys, on donne une estimation des valeurs deJ ₂ etJ ₄ du potentiel de Saturne. La valeur deJ ₂ est très sensible aux valeurs adoptées pour le rayon équatorial de la planète.

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Construction of periodic orbits of satellites in a moving system of axes, I

We give an algorithm for the construction of periodic orbits in a rotating frame for the cases of satellites moving around an oblate planet.The orbits are near to the circular case; the asymptotic developments of the periodic solutions are completely calculated for the termsJ ₂ andJ ₄ of the potential. The solutions for small inclinations are given up toJ ₂ ² .The families of solutions depend on three parameters: the semi-major axis, the inclination of the generating orbit and the initial position on this orbit.These solutions can be applied to the motion of the Saturnian satellites. From the observed longitudes of the ascending nodes of Mimas and Tethys, we estimate the valuesJ ₂ andJ ₄ of the Saturnian potential, the value ofJ ₂ very strongly depends on the adopted value of the planet's equatorial diameter.

     

Jupiter's obliquity and a long-lived circumplanetary disk

It has been claimed [Canup, R.M., Ward, W.R., 2002. Astron. J. 124, 3404-3423; Ward, W.R., 2003. In: AGU, Fall Meeting 2003] that a long-lived minimum mass circumplanetary gas disk is inconsistent with Jupiter's low obliquity. Here we find that while Jupiter's obliquity may constrain its characteristics it does not rule out a long-lived massive (compared to the mass of the Galilean satellites) disk. This is because the argument assumes a Solar System much like that of the present day with the one exception of a circumjovian disk which is then allowed to dissipate on a long timescale (¹⁰⁶-¹⁰⁷ yr). Given that the sequence of events in Solar System history that fit known constraints is non-unique, we choose for the sake of clarity of exposition the orbital architecture framework of Tsiganis et al. [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461], in which Jupiter and Saturn were once in compact, nearly coplanar orbits, and show that in this case Jupiter's low obliquity is consistent with the SEMM (solids-enhanced minimum mass) satellite formation model of Mosqueira and Estrada [Mosqueira, I., Estrada, P.R., 2003a. Icarus 163, 198-231; Mosqueira, I., Estrada, P.R., 2003b. Icarus 163, 232-255]. We suggest that a low inclination starting condition may apply, but stress that our SEMM satellite formation model could be compatible with Jupiter's obliquity even for mutually inclined giant planets.     

Refined model for the evolution of distant satellite orbits

We consider a model that describes the evolution of distant satellite orbits and that refines the solution of the doubly averaged Hill problem. Generally speaking, such a refinement was performed previously by J. Kovalevsky and A.A. Orlov in terms of Zeipel’s method by constructing a solution of the third order with respect to the small parameter m, the ratio of the mean motions of the planet and the satellite. The analytical solution suggested here differs from the solutions obtained by these authors and is closest in form to the general solution of the doubly averaged problem (∼m ²). We have performed a qualitative analysis of the evolutionary equations and conditions for the intersection of satellite orbits with the surface of a spherical planet with a finite radius. Using the suggested solution, we have obtained improved analytical time dependences of the elements of evolving orbits for a number of distant satellites of giant planets compared to the solution of the doubly averaged Hill problem and, thus, achieved their better agreement with the results of our numerical integration of the rigorous equations of perturbed motion for satellites.     

Mass distribution of Perseid meteoroids

An analysis of the Perseid meteoroid mass distribution is given. It is shown that particle mass distributions are qualitatively the same along the entire orbit of the stream. The extra minima in the cross sections of the stream at the ascending and descending branches of the curve of the parameter S indicate a jetlike nature of the stream. The variations of the nodal longitudes of maximum stream activity versus the minimum observed mass of meteoroids are found along the entire orbit of the stream. The positions of maximum activity for particles with minimum detectable masses larger than 1 and 10^?3 g are shifted by 1.4 degrees in solar longitude, with larger longitudes for smaller particles.     

Secular perturbations of asteroids in secular resonance

When the precessional rate of the orbital plane of an asteroid is nearly equal to that of Jupiter, the orbital inclination of the asteroid changes quite largely due to this near equality of their precessional rates, which is called a secular resonance. In the vicinity of the exact resonance the difference of their longitudes of nodes librates with quite a long period of order of 1×10⁶ yr. In this paper we treat this secular resonance by a method of semianalytical secular perturbations with use of numerical averaging for both non-resonant and resonant asteroids and show that the results by the semi-analytical treatment agrees qualitatively with those obtained by direct numerical integrations of asteroid's orbits.     

The dynamics of tadpole and horseshoe orbits: II. The coorbital satellites of saturn

The recently discovered coorbital satellites of Saturn, 1980S1 and 1980S3, are shown to be librating in horseshoe orbits. By considering the effects of tangential forces on the semimajor axes of the satellite orbits, we derive an accurate relation between the sum of the satellite masses and (a) their minimum angular separation, (b) the variation of their angular separation with time and (c) the libration period. Observations of (b) and (c) are the most practical methods of determining the satellite masses. The orbits of the coorbital satellites of Dione and Tethys are discussed. We demonstrate the possibility of calculating a new value for the mass of Dione and we show that one of the coorbital satellites of Tethys could be moving in a horseshoe orbit even though another satellite is librating in a tadpole orbit about the leading Lagrangian equilibrium point L₄. The origin of coorbital satellites and the stability of their orbits are discussed.     

Gas-drag-assisted capture of Himalia's family

To elucidate the capture of Jupiter's outer moons, we reverse-evolve satellites from their present orbits to their original heliocentric paths in the presence of Jupiter's primordial circumplanetary disk (Lubow et al., 1999, Astrophys. J. 526, 1001-1012; Canup and Ward, 2003, Astron. J. 124, 3404-3423). Our orbital histories use a symplectic integrator that allows dissipation. We assume that the present satellites Himalia, Elara, Lysithea, Leda, and S/2000 J11 are collisional fragments of a single parent. Our simulations show that this “prograde-cluster progenitor” (PCP) could be derived from objects with heliocentric orbits like those of the Hilda asteroid group. We show analytically that this capture is energetically possible. We also compare the spectroscopic characteristics of the prograde cluster members (Grav et al., 2003, Icarus, submitted for publication) with those of the Hildas, and conclude that the surface color of the prograde-cluster progenitor is consistent with an origin within the Hilda group. Accordingly, gas drag in the primordial jovian nebula is found to offer a plausible explanation for the origin of the prograde cluster. A similar capture mechanism is proposed for Saturn's Phoebe.     

Studies of multiple stellar systems – III. Modulation of orbital elements in the triple-lined system HD 109648

The triple-lined spectroscopic triple system HD 109648 has one of the shortest periods known for the outer orbit in a late-type triple, 120.5 d, and the ratio between the periods of the outer and the inner orbits is small, 22:1. With such extreme values, this system should show orbital element variations over a time-scale of about a decade. We have monitored the radial velocities of HD 109648 with the CfA Digital Speedometers for 8 yr, and have found evidence for modulation of some orbital elements. While we see no definite evidence for modulation of the inner binary eccentricity, we clearly observe variations in the inner and the outer longitudes of periastron, and in the radial velocity amplitudes of the three components. The observational results, combined with numerical simulations, allow us to put constraints on the orientation of the orbits.     

Orbites periodiques de satellites

In a previous paper (Stellmacher, 1981, hereafter mentioned as Paper I), we have given an algorithm for the construction of periodic orbits in a rotating frame, for satellites around an oblate planet. In the present paper, we apply this theory to the Mimas-Tethys case; we obtain the following results:

1. Without resonance, it is possible to find a rotating system in which the solution is a periodic one. The angular velocity of this rotating frame is calculated as function of the masses of the two satellites.
2. Including the resonant terms and assuming an exact commensurability of the implied frequencies, we demonstrate that the condition for periodic solutions in the rotating system as defined in (a) is: the initial position of the satellites at conjunction lies on an axis defined by (Ω₁+Ω₂)/2 or (Ω₁+Ω₂)/2 + π/2;Ω₁ and Ω₂ are the longitudes of the ascending nodes of the satellite's orbits. The solution still is a periodic one, thus all the conjunction occur in either axis.
3. In the Mimas Tethys case there is only approximately commensurability between these frequencies. The two satellites are considered as oscillators whose amplitudes and phases are functions of time. The equation of the libration can be established; we find the usual form, but for each satellite the generating solution is a periodic solution (as defined in Paper I), but not a Keplerian one. It follows a determination of the masses which slightly differs from that given by Kozai (1957), when the same values of the observed quantities are used for calculations.
4. The equation of the libration is: $$\ddot z + n_1^2 h^2 \sin z + n_1 q\dot z\sin z = 0$$

     

Peculiarities of Uranus’s satellite system

The absence of Uranus’s equatorial satellites in the region of approximately equal influence of its oblateness and solar perturbations is explained in terms of an improved physical model. This model is more complete than the previously studied case of an integrable averaged problem. The model improvement stems from the fact that the inclination of Uranus’s equator to the ecliptic differs by 90° and that the orbital evolution of Uranus due to secular planetary perturbations is taken into account. The lifetime of Uranus’s hypothetical satellites in orbits with semimajor axes 1.3–7 million km can be estimated by numerically integrating the evolution equations to be ～10⁴ yr. This is the time scale on which the evolution of the orbits leads to their intersection with the orbits of inner satellites.