Evolution of eccentricity and inclination of hot protoplanets embedded in radiative discs

Abstract

We study the evolution of the eccentricity and inclination of protoplanetary embryos and low-mass protoplanets (from a fraction of an Earth mass to a few Earth masses) embedded in a protoplanetary disc, by means of three dimensional hydrodynamics calculations with radiative transfer in the diffusion limit. When the protoplanets radiate in the surrounding disc the energy released by the accretion of solids, their eccentricity and inclination experience a growth toward values which depend on the luminosity to mass ratio of the planet, which are comparable to the disc's aspect ratio and which are reached over timescales of a few thousand years. This growth is triggered by the appearance of a hot, under-dense region in the vicinity of the planet. The growth rate of the eccentricity is typically three times larger than that of the inclination. In long term calculations, we find that the excitation of eccentricity and the excitation of inclination are not independent. In the particular case in which a planet has initially a very small eccentricity and inclination, the eccentricity largely overruns the inclination. When the eccentricity reaches its asymptotic value, the growth of inclination is quenched, yielding an eccentric orbit with a very low inclination.As a side result, we find that the eccentricity and inclination of non-luminous planets are damped more vigorously in radiative discs than in isothermal discs.

Impact of thermal effects on the evolution of eccentricity and inclination of low-mass planets

“…This implies that the eccentricity and inclination damping on a non-luminous embryo is completely dominated by thermal effects, which are nearly one order of magnitude more important than those arising from wave launching. This strong damping is in agreement with the findings of Eklund & Masset (2017) who found that the eccentricity and inclination of a non-luminous low-mass planet embedded in a radiative disc was indeed much stronger than that expected from Tanaka & Ward's formulae. When > 1, i.e.…”

“…• The numerical experiments of Eklund & Masset (2017), which were performed prior to the existence of this work and were therefore not aimed at checking analytical predictions with a high accuracy, do show that planets are subjected to effects similar to those described here (i.e. excitation of eccentricity and inclination at large luminosity, as opposed to a strong damping of these quantities when they are nonluminous).…”

“…They called this effect the hot trail effect. The disc's response to the heat release indeed adopts a trailing, cometary shape for eccentricities well below the disc's aspect ratio (Eklund & Masset 2017). Unlike the response carried by density waves, the effect of heat release can therefore be captured by a simple calculation of dynamical friction even in the subsonic regime (Masset & Velasco Romero 2017).…”

“…Little is known, however, on the impact of thermal diffusion on the gravitational interaction between an eccentric or inclined non-luminous planet and a gaseous disc. Eklund & Masset (2017) performed numerical experiments in isothermal and radiative discs, and found the eccentricity and inclination damping to be much stronger in the latter than in the former, but this effect, which was not the primary focus of that work, was not systematically quantified, and may have been misrepresented by the low resolution with which it was captured.…”

Thermal torque effects on the migration of growing low-mass planets

“…There are a number of shortcomings in our analysis. While we take into account the effects of the planet's luminosity on the time evolution of its semi-major axis, we neglect its impact on the eccentricity and inclination, which is potentially quite strong (Eklund & Masset 2017;Chrenko et al 2017). Sizeable eccentricities and inclinations may significantly alter the migration paths of planetary embryos, and may also affect the accretion rate on the embryos (Chrenko et al 2017) and the rate of collisions between embryos.…”