Complete determination of the orbital parameters of a system with N+1 bodies using a simple Fourier analysis of the data

Here we show how to determine the orbital parameters of a system composed of a star and N companions (that can be planets, brown-dwarfs or other stars), using a simple Fourier analysis of the radial velocity data of the star. This method supposes that all objects in the system follow keplerian orbits around the star and gives better results for a large number of observational points. The orbital parameters may present some errors, but they are an excellent starting point for the traditional minimization methods such as the Levenberg-Marquardt algorithms.Comment: 4 page

Tidal Evolution of Exoplanets

Tidal effects arise from differential and inelastic deformation of a planet by a perturbing body. The continuous action of tides modify the rotation of the planet together with its orbit until an equilibrium situation is reached. It is often believed that synchronous motion is the most probable outcome of the tidal evolution process, since synchronous rotation is observed for the majority of the satellites in the Solar System. However, in the 19th century, Schiaparelli also assumed synchronous motion for the rotations of Mercury and Venus, and was later shown to be wrong. Rather, for planets in eccentric orbits synchronous rotation is very unlikely. The rotation period and axial tilt of exoplanets is still unknown, but a large number of planets have been detected close to the parent star and should have evolved to a final equilibrium situation. Therefore, based on the Solar System well studied cases, we can make some predictions for exoplanets. Here we describe in detail the main tidal effects that modify the secular evolution of the spin and the orbit of a planet. We then apply our knowledge acquired from Solar System situations to exoplanet cases. In particular, we will focus on two classes of planets, "Hot-Jupiters" (fluid) and "Super-Earths" (rocky with atmosphere).Comment: 30 pages, 19 figures. Chapter in Exoplanets, ed. S. Seager, to be published by University of Arizona Pres

Spin-orbit coupling and chaotic rotation for coorbital bodies in quasi-circular orbits

Coorbital bodies are observed around the Sun sharing their orbits with the planets, but also in some pairs of satellites around Saturn. The existence of coorbital planets around other stars has also been proposed. For close-in planets and satellites, the rotation slowly evolves due to dissipative tidal effects until some kind of equilibrium is reached. When the orbits are nearly circular, the rotation period is believed to always end synchronous with the orbital period. Here we demonstrate that for coorbital bodies in quasi-circular orbits, stable non-synchronous rotation is possible for a wide range of mass ratios and body shapes. We show the existence of an entirely new family of spin-orbit resonances at the frequencies $n\pm k\nu/2$, where $n$ is the orbital mean motion, $\nu$ the orbital libration frequency, and $k$ an integer. In addition, when the natural rotational libration frequency due to the axial asymmetry, $\sigma$, has the same magnitude as $\nu$, the rotation becomes chaotic. Saturn coorbital satellites are synchronous since $\nu\ll\sigma$, but coorbital exoplanets may present non-synchronous or chaotic rotation. Our results prove that the spin dynamics of a body cannot be dissociated from its orbital environment. We further anticipate that a similar mechanism may affect the rotation of bodies in any mean-motion resonance.Comment: 6 pages. Astrophysical Journal (2013) 6p

Transit light curve and inner structure of close-in planets

Planets orbiting very close to their host stars have been found, some of them on the verge of tidal disruption. The ellipsoidal shape of these planets can significantly differ from a sphere, which modifies the transit light curves. Here we present an easy method for taking the effect of the tidal bulge into account in the transit photometric observations. We show that the differences in the light curve are greater than previously thought. When detectable, these differences provide us an estimation of the fluid Love number, which is invaluable information on the internal structure of close-in planets. We also derive a simple analytical expression to correct the bulk density of these bodies, that can be 20% smaller than current estimates obtained assuming a spherical radius.Comment: 6 pages, 3 figure

Deformation and tidal evolution of close-in planets and satellites using a Maxwell viscoelastic rheology

In this paper we present a new approach to tidal theory. Assuming a Maxwell viscoelastic rheology, we compute the instantaneous deformation of celestial bodies using a differential equation for the gravity field coefficients. This method allows large eccentricities and it is not limited to quasi-periodic perturbations. It can take into account an extended class of perturbations, including chaotic motions and transient events. We apply our model to some already detected eccentric hot Jupiters and super-Earths in planar configurations. We show that when the relaxation time of the deformation is larger than the orbital period, spin-orbit equilibria arise naturally at half-integers of the mean motion, even for gaseous planets. In the case of super-Earths, these equilibria can be maintained for very low values of eccentricity. Our method can also be used to study planets with complex internal structures and other rheologies.Comment: 16 pages, 13 figures, 2 table

On the equilibrium figure of close-in planets and satellites

Many exoplanets have been observed close to their parent stars with orbital periods of a few days. As for the major satellites of the Jovian planets, the figure of these planets is expected to be strongly shaped by tidal forces. However, contrarily to Solar System satellites, exoplanets may present high values for the obliquity and eccentricity due to planetary perturbations, and may also be captured in spin-orbit resonances different from the synchronous one. Here we give a general formulation of the equilibrium figure of those bodies, that makes no particular assumption on the spin and/or orbital configurations. The gravity field coefficients computed here are well suited for describing the figure evolution of a body whose spin and orbit undergo substantial variations in time.Comment: 5 pages, 2 figure

Secular evolution of a satellite by tidal effect. Application to Triton

Some of the satellites in the Solar System, including the Moon, appear to have been captured from heliocentric orbits at some point in their past, and then have evolved to the present configurations. The exact process of how this trapping occurred is unknown, but the dissociation of a planetesimal binary in the gravitational field of the planet, gas drag, or a massive collision seem to be the best candidates. However, all these mechanisms leave the satellites in elliptical orbits that need to be damped to the present almost circular ones. Here we give a complete description of the secular tidal evolution of a satellite just after entering a bounding state with the planet. In particular, we take into account the spin evolution of the satellite, which has often been assumed synchronous in previous studies. We apply our model to Triton and successfully explain some geophysical properties of this satellite, as well as the main dynamical features observed for the Neptunian system.Comment: 4 pages, 1 figur

Dynamics of co-orbital exoplanets in a first order resonance chain with tidal dissipation

Co-orbital planets (in a $1:1$ mean motion resonance) can be formed within a Laplace resonance chain. Here, we develop a secular model to study the dynamics of the resonance chain $p:p:p+1$, where the co-orbital pair is in a first-order mean motion resonance with the outermost third planet. Our model takes into account tidal dissipation through the use of a Hamiltonian version of the constant time-lag model, which extends the Hamiltonian formalism of the point-mass case. We show the existence of several families of equilibria, and how these equilibria extend to the complete system. In one family, which we call the main branch, a secular resonance between the libration frequency of the co-orbitals and the precession frequency of the pericentres has unexpected dynamical consequences when tidal dissipation is added. We report the existence of two distinct mechanisms that make co-orbital planets much more stable within the $p:p:p+1$ resonance chain rather than outside it. The first one is due to negative real parts of the eigenvalues of the linearised system with tides, in the region of the secular resonance mentioned above. The second one comes from non-linear contributions of the vector field and it is due to eccentricity damping. These two stabilising mechanisms increase the chances of a still-to-come detection of exoplanets in the co-orbital configuration