Secular effects of tidal friction on the planet-satellite systems of the solar system

We separate the tidal evolution of a planet-satellite system with zero eccentricity in two phases:phase 1—from the formation of the system to satellite's corotation (satellite's corotation means that its spin angular velocity equals the orbital angular velocity);phase 2—after satellite's corotation.We study the planet-satellite system during phase 1 with Darwin's graphical method and obtain an upper limit to satellite'sQ which discloses whether or not it is corotating. Moreover we obtain some qualitative information about the future evolution of the corotating satellites.The present work does not give any new result for the Earth-Moon case and for the Neptune-Triton case.     

On the stability of circumbinary planetary systems

The dynamics of circumbinary planetary systems (the systems in which the planets orbit a central binary) with a small binary mass ratio discovered to date is considered. The domains of chaotic motion have been revealed in the “pericentric distance–eccentricity” plane of initial conditions for the planetary orbits through numerical experiments. Based on an analytical criterion for the chaoticity of planetary orbits in binary star systems, we have constructed theoretical curves that describe the global boundary of the chaotic zone around the central binary for each of the systems. In addition, based on Mardling’s theory describing the separate resonance “teeth” (corresponding to integer resonances between the orbital periods of a planet and the binary), we have constructed the local boundaries of chaos. Both theoretical models are shown to describe adequately the boundaries of chaos on the numerically constructed stability diagrams, suggesting that these theories are efficient in providing analytical criteria for the chaoticity of planetary orbits.     

Empirical evidence for tidal evolution in transiting planetary systems

Most transiting planets orbit very close to their parent star, causing strong tidal forces between the two bodies. Tidal interaction can modify the dynamics of the system through orbital alignment, circularization, synchronization and orbital decay by exchange of angular moment. Evidence for tidal circularization in close-in giant planet is well known. Here, we review the evidence for excess rotation of the parent stars due to the pull of tidal forces towards spin-orbit synchronization. We find suggestive empirical evidence for such a process in the present sample of transiting planetary systems. The corresponding angular momentum exchange would imply that some planets have spiralled towards their star by substantial amounts since the dissipation of the protoplanetary disc. We suggest that this could quantitatively account for the observed mass–period relation of close-in gas giants. We discuss how this scenario can be further tested and point out some consequences for theoretical studies of tidal interactions and for the detection and confirmation of transiting planets from radial velocity and photometric surveys.     

Secular love and tidal numbers of synchronously orbiting satellites

The secular Love and the secular tidal numbers have been computed for eight synchronously orbiting satellites in the solar system for which the triaxiality parameters and satellite-centric gravitational constant are available. Excepting Deimos the total mass of which should be first refined, the secular Love and tidal numbers are rather close to unity, as a rule. That is why, the centrifugal and tidal distortions can be assumed responsible for the actual figures of the synchronously orbiting satellites resulting from the primordial spheres, as well as, their static equilibrium nearly satisfied. The hypothesis of the origin of synchronously orbiting satellites by accretion in orbits is supported by the results obtained.     

Secular tidal changes in lunar orbit and Earth rotation

Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = ?0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.     

The weak friction approximation and tidal evolution in close binary systems

The aim of this paper is to study the dynamical problem of tidal friction in a binary system consisting of deformable components, with the restriction that the angle of lag or advance of the tidal distortion with respect to the direction of the disturbing companion is small. The fractional distortion of the bodies due to rotation and tidal interaction is also treated as a first-order small quantity, and terms up to the fourth harmonic in the tidal potential are retained. In this linear approximation, the time-dependent tidal potential can be Fourier decomposed into a spectrum of simple harmonic terms, each of which is responsible for raising a partial wave in the body; each such partial wave can then be treated independently of the others. This is the method first employed by Darwin.In Section 2, it is assumed that the phase lag in the response of the body (due to dissipation of kinetic energy of deformation) is proportional to the forcing frequency, which is justified for small amplitude oscillations of a viscous fluid or visco-elastic body. A simple expression is then obtained for the potential function for the distortion in terms of the disturbing potential and the structure of the body.In Section 3, the distortion potential function is employed in deriving the componentsR, S andW of the disturbing force which are then substituted in the Gaussian form of the equations for variation of the elements. In Section 4, the Eulerian equations for motion of deformable bodies are derived, using the so-called mean axes of the body as the rotating axes of reference. In Section 5, it is shown that the dynamical effects of rotational distortion occur on a much shorter time scale than those arising from tidal friction, which allows one to consider the two phenomena as acting independently of one another. The collected set of Gaussian (orbital) and Eulerian (body) equations is re-written in terms of dimensionless variables for the tidal friction case, and the stability of the system is examined on the basis of these equations.In Section 6, the tidal friction equations are integrated numerically for the close binary system AG Persei and for the Earth-Moon system. In the former, the integrations were started from a highly elliptical orbit and the system was found to relax into a circular orbit, with synchronous rotation perpendicular to the orbit. In the latter, the integrations were performed backwards in time from the present day, and it was found that the lunar orbit rapidly becomes highly elliptical at the time of closest approach, thus indicating a probable capture of the Moon by the Earth. This result is in agreement with that obtained by other investigators; however, it is shown that the detailed behaviour of the system at the time of capture, in particular the inclination of the lunar orbit to the ecliptic, depends critically on the chosen rate of dissipation in the Moon's interior. A simple argument is presented which allows an estimation for the mean viscosity of a fluid body from the known age of the system: for the components of AG per, the result is 2×10¹¹ g cm^–1 s^–1, indicating that the stars must have possessed turbulent convective outer regions during some part of their tidal evolution, while for the Earth, the result, is 1.4×10¹² g cm^–1 s^–1. It is shown that the angle of tidal lag in nonsynchronous close binary systems in general is expected to be extremely small, and not observationally detectable.     

On the evolution of giant protoplanets forming in circumbinary discs

We present the results of hydrodynamic simulations of Jovian mass protoplanets that form in circumbinary discs. The simulations follow the orbital evolution of the binary plus protoplanet system acting under their mutual gravitational forces, and forces exerted by the viscous circumbinary disc. The evolution involves the clearing of the inner circumbinary disc initially, so that the binary plus protoplanet system orbits within a low density cavity. Continued interaction between disc and protoplanet causes inward migration of the planet towards the inner binary. Subsequent evolution can take three distinct paths: (i) the protoplanet enters the 4 : 1 mean motion resonance with the binary, but is gravitationally scattered through a close encounter with the secondary star; (ii) the protoplanet enters the 4 : 1 mean motion resonance, the resonance breaks, and the planet remains in a stable orbit just outside the resonance; (iii) when the binary has initial eccentricity   e _bin≥ 0.2  , the disc becomes eccentric, leading to a stalling of the planet migration, and the formation of a stable circumbinary planet.
These results have implications for a number of issues in the study of extrasolar planets. The ejection of protoplanets in close binary systems provides a source of 'free-floating planets', which have been discovered recently. The formation of a large, tidally truncated cavity may provide an observational signature of circumbinary planets during formation. The existence of protoplanets orbiting stably just outside a mean motion resonance (4 : 1) in the simulations indicate that such sites may harbour planets in binary star systems, and these could potentially be observed. Finally, the formation of stable circumbinary planets in eccentric binary systems indicates that circumbinary planets may not be uncommon.     

Angular momentum and tidal evolution

Five satellites of Neptune orbit under the synchronous zone. In this sense the Neptune's system is similar to that of Uranus (nine satellites) and differs from Jupiter (two) and Saturn (zero). The basic parameters describing the angular momentum within the Neptune's system and of its tidal evolution are estimated. The main character of the tidal dynamics is due to the retrograde Triton. The total tidal decrease in the spin angular momentum of Neptune is compared with those of Uranus, Jupiter and Saturn.     

Final tidal evolution of orbit-orbit resonances

In this paper we describe a model for the tidal evolution of an orbit-orbit resonance between two satellites of the same planet. We let the system evolve till infinity or until the resonance is destroyed. We find that there are asymptotic values for the eccentricities and inclinations. We list the possible final stages that a resonance can achieve, we give a few examples, and finally we discuss the limitations of the model and its possible applications to real systems.     

Angular momentum and tidal evolution of Saturn's system

It has been demonstrated that dynamically the Saturnian system is analogous to the Jovian system; however, it is not an analogue of the Solar system as a whole. The departures in the figure parameters of the tri-axial Saturnian satellites orbiting in 1 : 1 resonance, from equilibrium figure parameters are not large in general, and the tidal and centrifugal distorting forces can be supposed to be responsible for the actual figures. The estimates for different dynamical parameters of the system support the hypothesis that the tri-axial satellites in 1 : 1 resonance were formed from the same protoplanetary nebula that gave rise to Saturn.     

Secular frequencies of 3-D exoplanetary systems

Using a 12th order expansion of the perturbative potential in powers of the eccentricities and the inclinations, we study the secular effects of two non-coplanar planets which are not in mean–motion resonance. By means of Lie transformations (which introduce an action–angle formulation of the Hamiltonian), we find the four fundamental frequencies of the 3-D secular three-body problem and compute the long-term time evolutions of the Keplerian elements. To find the relations between these elements, the main combinations of the fundamental frequencies common to these evolutions are identified by frequency analysis. This study is performed for two different reference frames: a general one and the Laplace plane. We underline the known limitations of the linear Laplace–Lagrange theory and point out the great sensitivity of the 3-D secular three-body problem to its initial values. This analytical approach is applied to the exoplanetary system Andromedae in order to search whether the eccentricities evolutions and the apsidal configuration (libration of ) observed in the coplanar case are maintained for increasing initial values of the mutual inclination of the two orbital planes. Anne-Sophie Libert is FNRS Research Fellow.     

Secular apsidal configuration of non-resonant exoplanetary systems

Using a high-order (order 12) expansion of the perturbative potential in powers of eccentricities [Libert, A.-S., Henrard, J., 2005. Celest. Mech. Dynam. Astron. 93, 187-200], we study the secular effects of two coplanar planets which are not in mean motion resonances. The main results concern eccentricity variations, oscillation amplitude of the angular difference of the apsidal lines (Δ?) and frequency of such an oscillation. We show that this analytical approach describes correctly the behaviour of most of the exosystems and underlines the known limitations of the linear Laplace-Lagrange theory. Apsidal configuration of υ Andromedae, HD 168443, HD 169830, HD 38529, HD 74156 and HD 12661 are examined. We also point out the great sensitivity of the υ Andromedae system to the initial values (e₁(0), e₂(0) or Δ?(0)).     

Effect of tidal evolution in determining the ages of eclipsing-variable early main sequence close binary systems

New Claret evolutionary model-tracks, constructed for the first time for studying close binary systems (CBS) including tidal evolution constants, are used to determine the age of 112 eclipsing-variable stars in the Svechnikov-Perevozkina catalog by the method of isochrones. There is some interest in comparing the calculated ages with previous estimates obtained for these same close binary systems using evolutionary modeltracks for individual stars taking their mass loss into account. A correlation of the ages of the principal and secondary components is noted, which is most marked for massive close binaries with principal components having masses M₁ ≥ 3 M_⊙. A rejuvenating effect is found to occur for the systems studied here as calculated on the new tracks; it is most distinct for low-mass close binaries with a total mass M₁ + M₂ ≤ 3.5 M_⊙ and is predicted theoretically in terms of magnetic braking. The calculated broadband grid of isochrones, from zero-age main-sequence (ZAMS) to the age of the galaxy, can be used for estimating the ages of close binaries from other catalogs. Ages are given for the 112 eclipsing-variable close binaries with detached components lying within the main sequence. __________ Translated from Astrofizika, Vol. 50, No. 2, pp. 299–312 (May 2007).     

Secular evolution of the orbits of hypothetical satellites of Uranus

The problem of the secular perturbations of the orbit of a test satellite with a negligible mass caused by the joint influence of the oblateness of the central planet and the attraction by its most massive (or main) satellites and the Sun is considered. In contrast to the previous studies of this problem, an analytical expression for the full averaged perturbing function has been derived for an arbitrary orbital inclination of the test satellite. A numerical method has been used to solve the evolution system at arbitrary values of the constant parameters and initial elements. The behavior of some set of orbits in the region of an approximately equal influence of the perturbing factors under consideration has been studied for the satellite system of Uranus on time scales of the order of tens of thousands of years. The key role of the Lidov–Kozai effect for a qualitative explanation of the absence of small bodies in nearly circular equatorial orbits with semimajor axes exceeding ~1.8 million km has been revealed.