Accurate numerical potential and field in razor-thin axisymmetric discs

We demonstrate the high accuracy of the density splitting method to compute the gravitational potential and field in the plane of razor-thin, axially symmetric discs, as preliminarily outlined in Pierens & Hure (2004). Because residual kernels in Poisson integrals are not C^infinity-class functions, we use a dynamical space mapping in order to increase the efficiency of advanced quadrature schemes. In terms of accuracy, results are better by orders of magnitude than for the classical FFT-methods.Comment: 11 pages, 5 color figures, 2 table

Three-dimensional evolution of radiative circumbinary discs: the size and shape of the inner cavity

The evolution of circumbinary discs and planets is often studied using two-dimensional (2D) numerical simulations, although recent work suggests that 3D effects may significantly alter the structure of the inner cavity created by the binary. In this study, we present the results of 3D hydrodynamical simulations of circumbinary discs that orbit around analogues of the Kepler-16 and Kepler-34 systems, including the effect of stellar heating and radiative cooling on the thermal disc structure. We find that compared to their 2D counterparts, the structures of the cavities in 3D circumbinary disc models appear to reach a quasi-stationary state more rapidly, and in a subset of our runs the evidence for this is unambiguous. Furthermore, the sizes and eccentricities of the inner cavity are smaller in 3D compared to 2D. We attribute this difference to enhanced spiral wave dissipation in disc regions above the midplane, where the cooling time is of the order of the dynamical timescale, resulting in smaller inner cavity sizes in 3D disc models. Our results suggest that migrating planets should park closer to the central binary in 3D models of circumbinary discs, and point to the importance of including the 3D structure when simulating circumbinary discs and planets.Comment: Accepted in A&

On the formation and migration of giant planets in circumbinary discs

We present the results of hydrodynamic simulations of the formation and subsequent orbital evolution of giant planets embedded in a circumbinary disc. We assume that a 20 earth masses core has migrated to the edge of the inner cavity formed by the binary where it remains trapped by corotation torques. This core is then allowed to accrete gas from the disc, and we study its orbital evolution as it grows in mass. For each of the two accretion time scales we considered, we performed three simulations. In two of the three simulations, we stop the accretion onto the planet once its mass becomes characteristic of that of Saturn or Jupiter. In the remaining case, the planet can accrete disc material freely in such a way that its mass becomes higher than Jupiter's. The simulations show different outcomes depending on the final mass m_p of the giant. For m_p=1 M_S (where M_S is Saturn's mass), we find that the planet migrates inward through its interaction with the disc until its eccentricity becomes high enough to induce a torque reversal. The planet then migrates outward, and the system remains stable on long time scales. For m_p > 1 M_J (where M_J is Jupiter's mass) we observed two different outcomes. In each case the planet enters the 4:1 resonance with the binary, and resonant interaction drives up the eccentricity of the planet until it undergoes a close encounter with the secondary star. The result can either be ejection from the system or scattering out into the disc followed by a prolonged period of outward migration. These results suggest that circumbinary planets are more likely to be quite common in the Saturn-mass range. Jupiter-mass circumbinary planets are likely to be less common because of their less stable evolution.Comment: 12 pages, 12 figures. Accepted for publication in A&

Do we expect to find the Super-Earths close to the gas giants?

We have investigated the evolution of a pair of interacting planets embedded in a gaseous disc, considering the possibility of the resonant capture of a Super-Earth by a Jupiter mass gas giant. First, we have examined the situation where the Super-Earth is on the internal orbit and the gas giant on the external one. It has been found that the terrestrial planet is scattered from the disc or the gas giant captures the Super-Earth into an interior 3:2 or 4:3 mean-motion resonance. The stability of the latter configurations depends on the initial planet positions and on eccentricity evolution. The behaviour of the system is different if the Super-Earth is the external planet. We have found that instead of being captured in the mean-motion resonance, the terrestrial planet is trapped at the outer edge of the gap opened by the gas giant. This effect prevents the occurrence of the first order mean-motion commensurability. These results are particularly interesting in light of recent exoplanet discoveries and provide predictions of what will become observationally testable in the near future.Comment: 7 pages, to appear in the proceedings of the conference "Extra-solar Planets in Multi-body Systems: Theory and Observations"; eds. K. Gozdziewski, A. Niedzielski and J. Schneider, EAS Publication Serie

Hydrodynamical turbulence in eccentric circumbinary discs and its impact on the in situ formation of circumbinary planets

Eccentric gaseous discs are unstable to a parametric instability involving the resonant interaction between inertial-gravity waves and the eccentric mode in the disc. We present 3D global hydrodynamical simulations of inviscid circumbinary discs that form an inner cavity and become eccentric through interaction with the central binary. The parametric instability grows and generates turbulence that transports angular momentum with stress parameter $\alpha \sim 5 \times 10^{-3}$ at distances $\lesssim 7 \;a_{bin}$, where $a_{bin}$ is the binary semi-major axis. Vertical turbulent diffusion occurs at a rate corresponding to $\alpha_{diff}\sim 1-2\times 10^{-3}$. We examine the impact of turbulent diffusion on the vertical settling of pebbles, and on the rate of pebble accretion by embedded planets. In steady state, dust particles with Stokes numbers ${\it St} \lesssim 0.1$ form a layer of finite thickness $H_d \gtrsim 0.1 H$, where $H$ is the gas scale height. Pebble accretion efficiency is then reduced by a factor $r_{acc}/H_d$, where $r_{acc}$ is the accretion radius, compared to the rate in a laminar disc. For accreting core masses with $m_p \lesssim 0.1\; M_\oplus$, pebble accretion for particles with ${\it St} \gtrsim 0.5$ is also reduced because of velocity kicks induced by the turbulence. These effects combine to make the time needed by a Ceres-mass object to grow to the pebble isolation mass, when significant gas accretion can occur, longer than typical disc lifetimes. Hence, the origins of circumbinary planets orbiting close to their central binary systems, as discovered by the Kepler mission, are difficult to explain using an in situ model that invokes a combination of the streaming instability and pebble accretion.Comment: Accepted in MNRA

Planet Formation in Binary Stars: The case of Gamma Cephei

Over 30 planetary systems have been discovered to reside in binary stars. For small separations gravitational perturbation of the secondary star has a strong influence on the planet formation process. It truncates the protoplanetary disk, may shortens its lifetime, and stirs up the embedded planetesimals. Due to its small semi-major axis (18.5 AU) and large eccentricity (e=0.35) the binary $\gamma$ Cephei represents a particularly challenging example. In the present study we model the orbital evolution and growth of embedded protoplanetary cores of about 30 earth masses in the putative protoplanetary disk surrounding the primary star in the $\gamma$ Cep system. We assume coplanarity of the disk, binary and planet and perform two-dimensional hydrodynamic simulations of embedded cores in a protoplanetary disk. The presence of the eccentric secondary star perturbs the disk periodically and generates strong spiral arms at periapse which propagate toward the disk centre. The disk also becomes slightly eccentric (with e_d = 0.1-0.15), and displays a slow retrograde precession in the inertial frame. For all initial separations (2.5 to 3.5 AU) we find inward migration of the cores. For initial semi-major axes (a_p \gsim 2.7), we find a strong increase in the planetary eccentricity despite the presence of inward migration. Only cores which are initially far from the disk outer edge have a bounded orbital eccentricity which converges, roughly to the value of the planet observed in the $\gamma$ Cep system. We have shown that under the condition protoplanetary cores can form at around 2.5 AU, it is possible to evolve and grow such a core to form a planet with final outcome similar to that observed.Comment: 12 pages, 17 figures, accepted by Astronomy & Astrophysic

How does disk gravity really influence type-I migration ?

We report an analytical expression for the locations of Lindblad resonances induced by a perturbing protoplanet, including the effect of disk gravity. Inner, outer and differential torques are found to be enhanced compared to situations where a keplerian velocity field for the dynamics of both the disk and the planet is assumed. Inward migration is strongly accelerated when the disk gravity is only accounted for in the planet orbital motion. The addition of disk self-gravity slows down the planet drift but not enough to stop it.Comment: 4 pages, accepted for publication in A&A Letter

Self-gravity at the scale of the polar cell

We present the exact calculus of the gravitational potential and acceleration along the symmetry axis of a plane, homogeneous, polar cell as a function of mean radius a, radial extension e, and opening angle f. Accurate approximations are derived in the limit of high numerical resolution at the geometrical mean of the inner and outer radii (a key-position in current FFT-based Poisson solvers). Our results are the full extension of the approximate formula given in the textbook of Binney & Tremaine to all resolutions. We also clarify definitely the question about the existence (or not) of self-forces in polar cells. We find that there is always a self-force at radius except if the shape factor a.f/e reaches ~ 3.531, asymptotically. Such cells are therefore well suited to build a polar mesh for high resolution simulations of self-gravitating media in two dimensions. A by-product of this study is a newly discovered indefinite integral involving complete elliptic integral of the first kind over modulus.Comment: 4 pages, 4 figures, A&A accepte

Generation of potential/surface density pairs in flat disks Power law distributions

We report a simple method to generate potential/surface density pairs in flat axially symmetric finite size disks. Potential/surface density pairs consist of a ``homogeneous'' pair (a closed form expression) corresponding to a uniform disk, and a ``residual'' pair. This residual component is converted into an infinite series of integrals over the radial extent of the disk. For a certain class of surface density distributions (like power laws of the radius), this series is fully analytical. The extraction of the homogeneous pair is equivalent to a convergence acceleration technique, in a matematical sense. In the case of power law distributions, the convergence rate of the residual series is shown to be cubic inside the source. As a consequence, very accurate potential values are obtained by low order truncation of the series. At zero order, relative errors on potential values do not exceed a few percent typically, and scale with the order N of truncation as 1/N**3. This method is superior to the classical multipole expansion whose very slow convergence is often critical for most practical applications.Comment: Accepted for publication in Astronomy & Astrophysics 7 pages, 8 figures, F90-code available at http://www.obs.u-bordeaux1.fr/radio/JMHure/intro2applawd.htm

Modelling circumbinary protoplanetary disks I. Fluid simulations of the Kepler-16 and 34 systems

S.L and Z.M.L are supported by the STFC. P.J.C is grateful to NERC Grant NE/K004778/1. S.J.P. is supported by a Royal Society University Research Fellowship