Ring torque on janus and the melting of Enceladus

Voyager 2 images show parts of Enceladus' surface to be very smooth, lacking craters down to the resolution limit of 4 km. This absence of craters indicates geologically recent resurfacing, probably due to internal melting. However, calculations of current heating mechanisms, including radioactive decay and tidal heating due to Enceladus' resonance with Dione, yield heating rates too small to cause melting. The orbital mean motion of Janus (1980S1) is slightly less than twice that of Enceladus and, according to theoretical calculations, is currently decreasing as Janus' orbit evolves outward due to resonant torques from Saturn's rings. If Janus were ever locked into a stable 2:1 orbital commensurability with Enceladus, the resulting angular momentum transfer could have sufficiently enhanced the eccentricity of Enceladus' orbit for the ensuing tidal heating to have melted Enceladus' interior. The existence of a Laplace-like three-body resonance including Dione, although unlikely, would have increased heating. If Janus were indeed held in resonance with Enceladus until recently (10⁷–10⁸ years B.P.) when the lock was disrupted by an unspecified event (possibly a catastrophic collision which simultaneously created the coorbital pair, or by the influence of Dione) both the recent internal activity of Enceladus and the proximity of Janus to Saturn's rings may be explained. However, the predicted rapid time scale for ring evolution due to resonant torques from Saturn's inner moons remains a major problem.     

Dynamics of two satellites in the 2/1 Mean–Motion resonance: application to the case of Enceladus and Dione

The dynamics of a pair of satellites similar to Enceladus–Dione is investigated with a two-degrees-of-freedom model written in the domain of the planar general three-body problem. Using surfaces of section and spectral analysis methods, we study the phase space of the system in terms of several parameters, including the most recent data. A detailed study of the main possible regimes of motion is presented, and in particular we show that, besides the two separated resonances, the phase space is replete of secondary resonances.     

The formation of commensurabilities in satellite systems due to the action of tidal forces

The five types of resonance possible between a pair of satellites at a 21 commensurability are described. By a modification of the method usually used in the restricted three-body problem, phase-plane diagrams are constructed for these resonances for the more general case where both satellite masses are non-zero. These phase-plane diagrams are used to discuss the different types of motion possible at the five resonances.It is shown that tidal forces can drive a pair of satellites towards a commensurability, and at the 21 commensurability it is possible for the satellites to be captured into a libration at any of the five resonances, the probability of capture depending on the eccentricities, inclinations, and masses of the satellites. The tidal hypothesis provides a reasonable explanation of the origin of the commensurabilities between Mimas and Tethys, and between Enceladus and Dione, in the satellite system of Saturn.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Late-stage impacts and the orbital and thermal evolution of Tethys

An inferred ancient episode of heating and deformation on Tethys has been attributed to its passage through a 3:2 resonance with Dione (Chen, E.M.A., Nimmo, F. [2008]. Geophys. Res. Lett. 35, 19203). The satellites encounter, and are trapped into, the e-Dione resonance before reaching the e-Tethys resonance, limiting the degree to which Tethys is tidally heated. However, for an initial Dione eccentricity >0.016, Tethys’ eccentricity becomes large enough to generate the inferred heat flow via tidal dissipation. While capture into the e-Dione resonance is easy, breaking the resonance (to allow Tethys to evolve to its current state) is very difficult. The resonance is stable even for large initial Dione eccentricities, and is not broken by perturbations from nearby resonances (e.g. the Rhea–Dione 5:3 resonance). Our preferred explanation is that the Tethyan impactor which formed the younger Odysseus impact basin also broke the 3:2 resonance. Simultaneously satisfying the observed basin size and the requirement to break the resonance requires a large (≈250 km diameter) and slow (≈0.5 km/s) impactor, possibly a saturnian satellite in a nearby crossing orbit with Tethys. Late-stage final impacts of this kind are a common feature of satellite formation models (Canup, R.M., Ward, W.R. [2006]. Nature 441, 834–839).     

Tidal evolution of Mimas, Enceladus, and Dione

The tidal evolution through several resonances involving Mimas, Enceladus, and/or Dione is studied numerically with an averaged resonance model. We find that, in the Enceladus-Dione 2:1 e-Enceladus type resonance, Enceladus evolves chaotically in the future for some values of k₂/Q. Past evolution of the system is marked by temporary capture into the Enceladus-Dione 4:2 ee^′-mixed resonance. We find that the free libration of the Enceladus-Dione 2:1 e-Enceladus resonance angle of 1.5° can be explained by a recent passage of the system through a secondary resonance. In simulations with passage through the secondary resonance, the system enters the current Enceladus-Dione resonance close to tidal equilibrium and thus the equilibrium value of tidal heating of 1.1(18,000/QS) GW applies. We find that the current anomalously large eccentricity of Mimas can be explained by passage through several past resonances. In all cases, escape from the resonance occurs by unstable growth of the libration angle, sometimes with the help of a secondary resonance. Explanation of the current eccentricity of Mimas by evolution through these resonances implies that the Q of Saturn is below 100,000. Though the eccentricity of Enceladus can be excited to moderate values by capture in the Mimas-Enceladus 3:2 e-Enceladus resonance, the libration amplitude damps and the system does not escape. Thus past occupancy of this resonance and consequent tidal heating of Enceladus is excluded. The construction of a coherent history places constraints on the allowed values of k₂/Q for the satellites.     

Motion of dust in mean motion resonances with planets

Effect of stellar electromagnetic radiation on the motion of spherical dust particle in mean motion orbital resonances with a planet is investigated. Planar circular restricted three-body problem with the Poynting–Robertson (P–R) effect yields monotonic secular evolution of eccentricity when the particle is trapped in the resonance. Planar elliptic restricted three-body problem with the P–R effect enables nonmonotonous secular evolution of eccentricity and the evolution of eccentricity is qualitatively consistent with the published results for the complicated case of interaction of electromagnetic radiation with nonspherical dust grain. Thus, it is sufficient to allow either nonzero eccentricity of the planet or nonsphericity of the grain and the orbital evolutions in the resonances are qualitatively equal for the two cases. This holds both for exterior and interior mean motion orbital resonances. Evolutions of argument of perihelion in the planar circular and elliptical restricted three-body problems are shown. Numerical integrations show that an analytic expression for the secular time derivative of the particle’s argument of perihelion does not exist, if only dependence on semimajor axis, eccentricity and argument of perihelion is admitted. Connection between the shift of perihelion and oscillations in secular eccentricity is presented for the planar elliptic restricted three-body problem with the P–R effect. Period of the oscillations corresponds to the period of one revolution of perihelion. Change of optical properties of the spherical grain with the heliocentric distance is also considered. The change of the optical properties: (i) does not have any significant influence on the secular evolution of eccentricity, (ii) causes that the shift of perihelion is mainly in the same direction/orientation as the particle motion around the Sun. The statements hold both for circular and noncircular planetary orbits.     

New results on the possible chaotic motion of Enceladus

New results are shown for the possible long term evolution of the orbit of Enceladus perturbed by Dione in the 2:1 resonance in the planar elliptic restricted problem. The numerical integrations of the secular variation equations (Ferraz-Mello and Dvorak, 1986) yield different types of chaotic behaviour. We also studied the onset of chaos for this dynamical system. Additionally, some illustrative pictures of the whole phase trajectory show the complex structure of phase space.     

Dynamical evolution of the Prometheus–Pandora system

The system of Saturn's inner satellites is saturated with many resonances. Its structure should be strongly affected by tidal forces driving the satellites through several orbit–orbit resonances. The evolution of these satellites is investigated using analytic and numerical methods. We show that the pair of satellites Prometheus and Pandora has a particularly short lifetime (<20 Myr) if the orbits of the satellites converge without capture into a resonance. The capture of Pandora into a resonance with Prometheus increases the lifetime of the couple by a few tens of Myr. However, resonances of the system are not well separated, and capture results in a chaotic motion. Secondary resonances also disrupt the resonant configurations. In all cases, the converging orbits of these two satellites result in a close encounter. The implications for the origin of Saturn's rings are discussed.     

A second fundamental model for resonance

We analyse a simple one degree of freedom Hamiltonian system depending upon a parameter \(H = - 3(\delta + 1)R + R^2 - 2\sqrt {2R} \cos r\) . This model is much closer to resonance problems arising in Celestial Mechanics than the pendulum. We deduce from it the conditions of capture into resonance or escape from resonance for systems drifting slowly. We apply this analysis to the Enceladus-Dione resonance.     

The long term stability of coorbital moons of the satellites of Saturn: I. Conservative case

The aim of this work is to understand the absence of objects along the orbits of Mimas and Enceladus in contrast to their presence at the orbits of neighbouring Tethys and Dione from the point of view of dynamical stability. Large scale numerical simulations of 360 test particles within the coorbital regions of these four saturnian satellites were carried out for 4×¹⁰⁵ yr or 1.6×¹⁰⁸ revolutions of the innermost moon Mimas. The tidal forcing of the satellites' orbits was not taken into account in these simulations. We have quantitatively reproduced the Mimas-Tethys 4:2 and Enceladus-Dione 2:1 mean motion resonances in the system and devised a scheme by which the parameter space of the coorbital resonance is sampled uniformly by our test particles. We observe that 6 out of the 36 integrated horseshoe particles of Enceladus escaped the coorbital region. All 54 tadpole particles remained stable. The main cause of instability for Enceladus coorbitals appears to be the overlap between the coorbital resonance and the 2:1 mean motion resonance between the particle and Dione. This leads particles with starting semimajor axes near the horseshoe-tadpole separatrix to be ejected from the resonance, as proposed by Morais [Morais, M.H.M., 2000. The effect of secular perturbations and mean motion resonances on trojan dynamics. Ph.D. thesis, Univ. of London], over timescales of ∼8×¹⁰⁷ revolutions of Enceladus. For Mimas we observe a larger number of coorbital escapes overall, both of tadpole (7/54) and horseshoe (29/36) librators. An analysis of the observed dynamical evolution suggests a two-stage process at work: The semimajor axis of particles with starting conditions near the horseshoe-tadpole separatrix undergoes a slow random walk over timescales of ¹⁰⁵ yr through a mechanism similar to that at Enceladus but involving the 4:2 inclination resonance with Tethys. These particles are eventually injected into a region of short-term (?¹⁰⁴ yr) instability just inside the nominal boundary of stable, symmetric horseshoe motion. The presence of the 4:2 eccentricity triplet at that location is the most likely culprit for the instability. In both the cases of Mimas and Enceladus small-amplitude tadpoles remain stable until the end of the integration. The existence of fast escapers at Mimas provides a dynamical avenue for the short-term survival of impact ejecta in horseshoe orbits within Mimas' coorbital region.     

On the thermal history of Saturn’s satellites Titan and Enceladus

The thermal histories of two geologically active satellites of Saturn—Titan and Enceladus—are discussed. During the Cassini mission, it was found that there are both nitrogen-containing compounds—NH₃ and N₂-and CO₂ and CH₄ in the water plumes of Enceladus; at that, ammonia is the prevailing form. This may testify that during evolution, the material of the satellite was warmed up to T ∼ 500–600 K, when NH₃ (the form of nitrogen capable of being accreted) could only be partly converted into N₂. Contrary to Enceladus, the temperature inside Titan probably reached values higher than 800 K or even higher than 1000 K, since the process of the chemical dissociation of ammonia was completely finished on this satellite and its atmosphere contains only molecular nitrogen. While the internal heating of Titan up to high temperatures can be explained by its large mass, the heating source for Enceladus’ interior is far from evident. Such traditional heating sources as the energy of gravitational differentiation and the radiogenic heating due to shortliving ²⁶Al and ⁶⁰Fe could not be effective. The first one is because of the small size of Enceladus (RE ≈ 250 km), and the inefficiency of the second one is caused by the fact that the satellite was formed not earlier than 8–10 Myr after the formation of calcium and aluminum-enriched inclusions in carbonaceous chondrites (CAI), i.e., after ²⁶Al had completely decayed. In the present paper, we propose other heating mechanisms-the heat of long-living radioactive elements and tidal heat, which could provide the observed chemical composition of the water plumes of Enceladus rather than only the differentiation of its protomatter into the ironstone core and the ice mantle.     

Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth

We explore the evolution of the angular velocity of an elastic Earth model, within the Hamiltonian formalism. The evolution of the rotation state of the Earth is caused by the tidal deformation exerted by the Moon and the Sun. It can be demonstrated that the tidal perturbation to spin depends not only upon the instantaneous orientation of the Earth, but also upon its instantaneous angular velocity. Parameterizing the orientation of the Earth figure axis with the three Euler angles, and introducing the canonical momenta conjugated to these, one can then show that the tidal perturbation depends both upon the angles and the momenta. This circumstance complicates the integration of the rotational motion. Specifically, when the integration is carried out in terms of the canonical Andoyer variables (which are the rotational analogues to the orbital Delaunay variables), one should keep in mind the following subtlety: under the said kind of perturbations, the functional dependence of the angular velocity upon the Andoyer elements differs from the unperturbed dependence (Efroimsky in Proceedings of Journées 2004: Systèmes de référence spatio-temporels. l’Observatoire de Paris, pp 74–81, 2005; Efroimsky and Escapa in Celest. Mech. Dyn. Astron. 98:251–283, 2007). This happens because, under angular velocity dependent perturbations, the requirement for the Andoyer elements to be canonical comes into a contradiction with the requirement for these elements to be osculating, a situation that parallels a similar antinomy in orbital dynamics. Under the said perturbations, the expression for the angular velocity acquires an additional contribution, the so called convective term. Hence, the time variation induced on the angular velocity by the tidal deformation contains two parts. The first one comes from the direct terms, caused by the action of the elastic perturbation on the torque-free expressions of the angular velocity. The second one arises from the convective terms. We compute the variations of the angular velocity through the approach developed in Getino and Ferrándiz (Celest. Mech. Dyn. Astron. 61:117–180, 1995), but considering the contribution of the convective terms. Specifically, we derive analytical formulas that determine the elastic perturbations of the directional angles of the angular velocity with respect to a non-rotating reference system, and also of its Cartesian components relative to the Tisserand reference system of the Earth. The perturbation of the directional angles of the angular velocity turns out to be different from the evolution law found in Kubo (Celest. Mech. Dyn. Astron. 105:261–274, 2009), where it was stated that the evolution of the angular velocity vector mimics that of the figure axis. We investigate comprehensively the source of this discrepancy, concluding that the difference between our results and those obtained in Ibid. stems from an oversimplification made by Kubo when computing the direct terms. Namely, in his computations Kubo disregarded the motion of the tide raising bodies with respect to a non-rotating reference system when compared with the Earth rotational motion. We demonstrate that, from a numerical perspective, the convective part provides the principal contribution to the variation of the directional angles and of length of day. In the case of the x and y components in the Tisserand system, the convective contribution is of the same order of magnitude as the direct one. Finally, we show that the approximation employed in Kubo (Ibid.) leads to significant numerical differences at the level of a hundred micro-arcsecond.     

On constructing the general Earth’s rotation theory

In the present paper the equations of the orbital motion of the major planets and the Moon and the equations of the three–axial rigid Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and lunar orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and lunar evolution). Hence, the theory of the Earth’s rotation can be presented by means of the series in powers of the evolutionary variables with quasi-periodic coefficients with respect to the planetary–lunar mean longitudes. This form of the Earth’s rotation problem is compatible with the general planetary theory involving the separation of the short–period and long–period variables and avoiding the appearance of the non–physical secular terms.     

Inhomogeneities of the stellar winds of hot supergiants

The causes of variability of line profiles in the spectra of O supergiants are analyzed. It is suggested that the main cause of the variability is the motion in the atmosphere of dense clumps of matter (inhomogeneities or clouds) along the Une of sight between star and observer. The profiles of C IV and Si IV UV resonance lines in the spectra of bright OB supergiants are calculated for spherically symmetric atmospheres and for atmospheres with inhomogeneity along the line of sight. The dependence of the line profiles on the distance of the inhomogeneity from the center of the star is investigated. It is shown that the formation and evolution with time of discrete absorption components (DACs) in the profiles of C IV and Si IV UV resonance lines can be explained within the framework of the proposed model of variability of line profiles. The parameters of the inhomogeneities moving in the atmosphere to produce DACs are estimated. Translated from Astrofizika, Vol. 41, No. 3, pp. 423–441, July–September, 1998.     

Synchronous spin-orbital resonance locking of large planetary satellites

A numerical investigation of the chaotic rotation of large planetary satellites before their synchronous spin-orbital resonance locking with regard to tidal friction is carried out. The rotational dynamics of seven large satellites greater than 1000 km in diameter and with known inertial parameters (Io, Europa, Ganymede, Callisto (J1–J4), Tethys (S3), Iapetus (S8), and Ariel (U1)) in the epoch of synchronous resonance locking is modeled. All of these satellites have a small dynamic asymmetry. The planar case is considered in which the satellite’s axis of rotation is orthogonal to the plane of orbit. The satellites possessing an initial rapid rotation pass through various resonant states during the tidal evolution. Here, the probability of their locking into these states exists. The numerical experiments presented in this paper have shown that, with a rather high arbitrariness in the choice of initial states, the satellites during the course of the tidal evolution of their rotational motion have passed without interruption through the regions of the 5: 2, 2: 1, and 3: 2 resonances in the phase space and are locked into the 1: 1 resonance. The estimate for the tidal deceleration time is obtained both theoretically and on the numerical experimental basis.     

热海王星系统HD 106315的近$2:1$ 平运动共振捕获与轨道演化

通过结合理论分析和数值模拟方法,可以对热海王星系统HD 106315轨道迁移中的近2:1平运动共振捕获机制以及潮汐作用下的演化过程进行研究.在轨道迁移阶段,初始轨道半长径、初始偏心率以及行星c的偏心率衰减系数K会对系统轨道构型产生影响.数值模拟结果显示当初始轨道半长径分别为ab～0.4 au、ac～0.8 au,偏心率eb和ec均小于0.03时, HD 106315b和HD 106315c在中央恒星的引力作用以及原行星盘粘滞作用下向内迁移, 65000 yr左右两颗行星均可迁移至当前观测位置附近并形成近2:1平运动共振捕获.此外,中央恒星的潮汐效应也可能会对行星系统共振构型产生影响,理论分析表明当行星潮汐耗散系数Q=100时,潮汐效应造成的轨道半长径衰减使系统轨道周期比发生的变化可能是系统脱离共振构型的原因.数值模拟结果显示, HD 106315系统内两颗行星Q103时,来自中央恒星的潮汐效应并不会使行星系统产生明显的偏心率和轨道半长径衰减,不足以使HD 106315行星系统在剩余寿命内脱离2:1平运动共振轨道构型.     

Effects of resonances on the stability of retrograde satellites

The stability evolution of family f of the planar circular restricted three-body problem in the Earth–Moon case is explored numerically using the Poincaré surface of section. It is shown that third order resonances are the main cause of the reduction of the stability region of retrograde satellites. Several branches of family f are also computed and these are seen by the configuration of their family characteristics to roughly determine the stability region. Previous results on smaller mass ratios of primaries are thus extended to the Earth–Moon system.     

Tidal interactions in multi-planet systems

We study systems of close orbiting planets evolving under the influence of tidal circularization. It is supposed that a commensurability forms through the action of disk induced migration and orbital circularization. After the system enters an inner cavity or the disk disperses the evolution continues under the influence of tides due to the central star which induce orbital circularization. We derive approximate analytic models that describe the evolution away from a general first order resonance that results from tidal circularization in a two planet system and which can be shown to be a direct consequence of the conservation of energy and angular momentum. We consider the situation when the system is initially very close to resonance and also when the system is between resonances. We also perform numerical simulations which confirm these models and then apply them to two and four planet systems chosen to have parameters related to the GJ 581 and HD 10180 systems. We also estimate the tidal dissipation rates through effective quality factors that could result in evolution to observed period ratios within the lifetimes of the systems. Thus the survival of, or degree of departure from, close commensurabilities in observed systems may be indicative of the effectiveness of tidal disipation, a feature which in turn may be related to the internal structure of the planets involved.     

Dynamics of planets in retrograde mean motion resonance

In a previous paper (Gayon and Bois 2008a), we have shown the general efficiency of retrograde resonances for stabilizing compact planetary systems. Such retrograde resonances can be found when two-planets of a three-body planetary system are both in mean motion resonance and revolve in opposite directions. For a particular two-planet system, we have also obtained a new orbital fit involving such a counter-revolving configuration and consistent with the observational data. In the present paper, we analytically investigate the three-body problem in this particular case of retrograde resonances. We therefore define a new set of canonical variables allowing to express correctly the resonance angles and obtain the Hamiltonian of a system harboring planets revolving in opposite directions. The acquiring of an analytical “rail” may notably contribute to a deeper understanding of our numerical investigation and provides the major structures related to the stability properties. A comparison between our analytical and numerical results is also carried out.