Abstract.We study the time evolution of two protoplanets still embedded in a protoplanetary disk. The results of two different numerical approaches are presented and compared. In the first approach, the motion of the disk material is computed with viscous hydrodynamical simulations, and the planetary motion is determined by N-body calculations including exactly the gravitational forces exerted by the disk material. In the second approach, only the N-body integration is performed but with additional dissipative forces included such as to mimic the effect of the disk torques acting on the disk. This type of modeling is much faster than the full hydrodynamical simulations, and gives comparative results provided that parameters are adjusted properly. Resonant capture of the planets is seen in both approaches, where the order of the resonance depends on the properties of the disk and the planets. Resonant capture leads to a rise in the eccentricity and to an alignment of the spatial orientation of orbits. The numerical results are compared with the observed planetary systems in mean motion resonance (GJ 876, HD 82943, and 55 Cnc). We find that the forcing together of two planets by their parent disk produces resonant configurations similar to those observed, but that eccentricity damping greater than that obtained in our hydrodynamic simulations is required to match the GJ 876 observations.

The stationary hydrodynamic equations for transonic viscous accretion discs in Kerr geometry are derived. The consistent formulation is given for the viscous angular momentum transport and the boundary conditions on the horizon of a central black hole. An expression for the thickness of the disc is obtained from the vertical Euler equation for general accretion flows with vanishing vertical velocity. Different solution topologies are identified, characterized by a sonic transition close to or far from the marginally stable orbit. A numerical method is presented that allows to integrate the structure equations of transonic accretion flows. Global polytropic solutions for the disc structure are calculated, covering each topology and a wide range of physical conditions. These solutions generally possess a sub-Keplerian angular momentum distribution and have maximum temperatures in the range 10 11 − 10 12 K. Accretion discs around rotating black holes are hotter and deposit less angular momentum on the central object than accretion discs around Schwarzschild black holes.

We consider time-dependent models for rotating accretion flows onto black holes, where a transition takes place from an outer cooling-dominated disc to a radiatively inefficient flow in the inner region. In order to allow for a transition of this type we solve the energy equation, both, for the gas and for the radiation field, including a radiative cooling flux and a turbulent convective heat flux directed along the negative entropy gradient. The transition region is found to be highly variable, and a corresponding variation expected for the associated total luminosity. In particular, rapid oscillations of the transition radius are present on a timescale comparable with the local Keplerian rotation time. These oscillations are accompanied by a quasi-periodic modulation of the mass flux at the outer edge of the advection-dominated accretion flow (ADAF). We speculate about the relevance for the high-frequency QPO phenomenon.Comment: 6 pages, 3 figures. Revised version, added system of equations. Accepted for publication in MNRA

The equations governing dissipative relativistic hydrodynamics are formulated within the 3+1 approach for arbitrary spacetimes. Dissipation is accounted for by applying the theory of extended causal thermodynamics (Israel-Stewart theory). This description eliminates the causality violating infinite signal speeds present in the conventional Navier-Stokes equation. As an example we treat the astrophysically relevant case of stationary and axisymmetric spacetimes, including the Kerr metric. The equations take a simpler form whenever the inertia due to the dissipative contributions can be neglected.Comment: 24 pages, LateX, uses mn.sty and AMS fonts, no figures, accepted for publication in the MNRAS, also available via http://www.lsw.uni-heidelberg.de/~jpeitz/Personal.htm

Non-ideal relativistic hydrodynamics is formulated with respect to ducial observers (FIDOs). Three types of theories for dissipative relativistic uids are considered, which have di erent causality properties, and di erent complexity of the equations that determine the dissipative uxes. The 3+1 equations for these uid theories are given in terms of the three-space quantities which correspond to those familiar from non-relativistic physics.