Wilhelm Kley

Planet-Disk Interaction and Orbital Evolution

As planets form and grow within gaseous protoplanetary disks, the mutual gravitational interaction between the disk and planet leads to the exchange of angular momentum and migration of the planet. We review current understanding of disk-planet interactions, focusing in particular on physical processes that determine the speed and direction of migration. We describe the evolution of low-mass planets embedded in protoplanetary disks and examine the influence of Lindblad and corotation torques as a function of the disk properties. The role of the disk in causing the evolution of eccentricities and inclinations is also discussed. We describe the rapid migration of intermediate-mass planets that may occur as a runaway process and examine the transition to gap formation and slower migration driven by the viscous evolution of the disk for massive planets. The roles and influence of disk self-gravity and magnetohydrodynamic turbulence are discussed in detail, as a function of the planet mass, as is the evolution...

A torque formula for non-isothermal type I planetary migration – I. Unsaturated horseshoe drag

We study the torque on low-mass planets embedded in protoplanetary discs in the two-dimensional approximation, incorporating non-isothermal eects. We couple linear estimates of the Lindblad (or wave) torque to a simple, but non-linear, model of adiabatic corotation torques (or horseshoe drag), resulting in a simple formula that governs Type I migration in non-isothermal discs. This formula should apply in optically thick regions of the disc, where viscous and thermal diusion act to keep the horseshoe drag unsaturated. We check this formula against numerical hydrodynamical simulations, using three independent numerical methods, and nd good agreement.

A comparative study of disc–planet interaction

We perform numerical simulations of a disc-planet system using various grid-based and smoothed particle hydrodynamics (SPH) codes. The tests are run for a simple setup where Jupiter and Neptune mass planets on a circular orbit open a gap in a protoplanetary disc during a few hundred orbital periods. We compare the surface density contours, potential vorticity and smoothed radial profiles at several times. The disc mass and gravitational torque time evolution are analysed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5 per cent for the Eulerian simulations. The SPH results predict the correct shape of the gap although have less resolution in the low-density regions and weaker planetary wakes. The disc masses after 200 orbital periods agree within 10 per cent. The spread is larger in the tidal torques acting on the planet which agree within a factor of 2 at the end of the simulation. In the Neptune case, the dispersion in the torques is greater than for Jupiter, possibly owing to the contribution from the not completely cleared region close to the planet.

A torque formula for non-isothermal Type I planetary migration – II. Effects of diffusion

We study the effects of diffusion on the non-linear corotation torque, or horseshoe drag, in the two-dimensional limit, focusing on low-mass planets for which the width of the horseshoe region is much smaller than the scaleheight of the disc. In the absence of diffusion, the non-linear corotation torque saturates, leaving only the Lindblad torque. Diffusion of heat and momentum can act to sustain the corotation torque. In the limit of very strong diffusion, the linear corotation torque is recovered. For the case of thermal diffusion, this limit corresponds to having a locally isothermal equation of state. We present some simple models that are able to capture the dependence of the torque on diffusive processes to within 20 per cent of the numerical simulations.

Nested-grid calculations of disk-planet interaction

We study the evolution of embedded protoplanets in a protostellar disk using very high resolution nested- grid computations. This method allows us to perform global simulations of planets orbiting in disks and, at the same time, to resolve in detail the dynamics of the flow inside the Roche lobe of the planet. The primary interest of this work lies in the analysis of the gravitational torque balance acting on the planet. For this purpose we study planets of dierent masses, ranging from one Earth-mass up to one Jupiter-mass, assuming typical parameters of the protostellar disk. The high resolution supplied by the nested-grid technique permits an evaluation of the torques, resulting from short and very short range disk-planet interactions, more reliable than the one previously estimated with the aid of numerical methods. Likewise, the mass flow onto the planet is computed in a more accurate fashion. The obtained migration time scales are in the range from few times 10 4 years, for intermediate mass planets, to 10 6 years, for very low and high mass planets. These are longer than earlier assessments due to the action of circumplanetary material. Typical growth time scales depend strongly on the planetary mass. Below 64 Earth-masses, we nd this time scale to increase as the 2=3-power of the planets mass; otherwise it rises as the 4=3-power. In the case of Jupiter-size planets, the growth time scale is several times ten thousand years.

Disk eccentricity and embedded planets

Aims. We investigate the response of an accretion disk to the presence of a perturbing protoplanet embedded in the disk through time dependent hydrodynamical simulations. Methods. The disk is treated as a two-dimensional viscous fluid and the planet is kept on a fixed orbit. We run a set of simulations varying the planet mass, and the viscosity and temperature of the disk. All runs are followed until they reach a quasi-equilibrium state. Results. We find that for planetary masses above a certain minimum mass, already 3 M Jup for a viscosity of v = 10 -5 , the disk makes a transition from a nearly circular state into an eccentric state. Increasing the planetary mass leads to a saturation of disk eccentricity with a maximum value of around 0.25. The transition to the eccentric state is driven by the excitation of an m = 2 spiral wave at the outer 1:3 Lindblad resonance. The effect occurs only if the planetary mass is large enough to clear a sufficiently wide and deep gap to reduce the damping effect of the outer 1:2 Lindblad resonance. An increase in viscosity or temperature in the disk, which both tend to close the gap, have an adverse influence on the disk eccentricity. Conclusions. In the eccentric state the mass accretion rate onto the planet is greatly enhanced, an effect that may ease the formation of massive planets beyond about 5 M Jup that are otherwise difficult to reach.

Migration of protoplanets in radiative discs

Context. In isothermal discs, the migration of protoplanets is directed inwards. For small planetary masses, the standard type I migration rates are so high that this can result in an unrealistic loss of planets into the stars. Aims. We investigate the planet-disc interaction in non-isothermal discs and analyse the magnitude and direction of migration for an extended range of planet masses. Methods. We performed detailed two-dimensional numerical simulations of embedded planets including heating/cooling effects as well as radiative diffusion for realistic opacities. Results. In radiative discs, small planets with M planet < 50 M Earth migrate outwards at a rate comparable to the absolute magnitude of standard type I migration. For larger masses, the migration is inwards and approaches the isothermal, type II migration rate. Conclusions. Our findings are particularly important for the first growth phase of planets and ease the problem of too rapid inward type I migration.

Two planets orbiting the recently formed post-common envelope binary NN Serpentis

Planets orbiting post-common envelope binaries provide fundamental information on planet formation and evolution. We searched for such planets in NN Ser ab, an eclipsing short-period binary that shows long-term eclipse time variations. Using published, reanalysed, and new mid-eclipse times of NN Ser ab obtained between 1988 and 2010, we find excellent agreement with the light-travel-time effect produced by two additional bodies superposed on the linear ephemeris of the binary. Our multi-parameter fits accompanied by N-body simulations yield a best fit for the objects NN Ser (ab)c and d locked in the 2:1 mean motion resonance, with orbital periods P-c similar or equal to 15.5 yrs and P-d similar or equal to 7.7 yrs, masses M-c sin i(c) similar or equal to 6.9 M-Jup and M-d sin i(d) similar or equal to 2.2 M-Jup, and eccentricities e(c) similar or equal to 0 and e(d) similar or equal to 0.20. A secondary chi(2) minimum corresponds to an alternative solution with a period ratio of 5:2. We estimate that the progenitor binary consisted of an A star with similar or equal to 2 M-circle dot and the present M dwarf secondary at an orbital separation of similar to 1.5 AU. The survival of two planets through the common-envelope phase that created the present white dwarf requires fine tuning between the gravitational force and the drag force experienced by them in the expanding envelope. The alternative is a second-generation origin in a circumbinary disk created at the end of this phase. In that case, the planets would be extremely young with ages not exceeding the cooling age of the white dwarf of 10(6) yrs.

Circumbinary disk evolution

We study the evolution of circumbinary disks surrounding classical T Tau stars. High resolution numerical simulations are employed to model a system consisting of a central eccentric binary star within an accretion disk. The disk is assumed to be infinitesimally thin, however a detailed energy balance including viscous heating and radiative cooling is applied. A novel numerical approach using a parallelized Dual-Grid technique on two different coordinate systems has been implemented. Physical parameters of the setup are chosen to model the close system of DQ Tau and AK Sco, as well as the wider systems of GG Tau and UY Aur. Our main findings are for the tight binaries a substantial flow of material through the disk gap which is accreted onto the central stars in a phase dependent process. We are able to constrain the parameters of the systems by matching both accretion rates and derived spectral energy distributions to observational data where available.

Three-dimensional Simulations of a Planet Embedded in a Protoplanetary Disk

The dynamical influence of Jupiter-sized planets still embedded in protostellar disks is studied by means of numerical simulations. The three-dimensional structure of the disk is fully taken into account. Assuming a disk mass of 3.5 × 10-3 M☉ within 2 and 13 AU, typical values for the vertical thickness H/r = 0.05 and viscosity of the disk α ≈ 4 × 10-3, we find for the mass accretion rate onto the planet a value of 6 × 10-5 MJ yr-1 and for the migration timescale a value of 105 yr. These results are in excellent agreement with previously obtained results from two-dimensional calculations of infinitesimally thin disks. We argue that this agreement is to be expected if the vertical height of the disk is similar or smaller than the size of the Roche lobe of the embedded planet. Three-dimensional models having a lower planetary mass or different H/r values display small deviations from two-dimensional results.