D Disk-Planet Interactions and Non-Circular Orbits
2004-03-01

We present the results of various 3D simulations of protoplanetary accretion disks with an embedded protoplanet. The protoplanet causes distortions in the disk. In the case of a 1 Jupiter mass protoplanet the protoplanet will have opened a gap in the disk and it will send large spiral waves through the disk. These non-axisymmetric features have a severe impact on the development of the protoplanet, most importantly on the accretion rate and the evolution of the orbit. To study these effects we performed numerical simulations using a finite difference scheme. This has been done by many previous authors, but we included a new degree of freedom into the simulations. In contrast to previous 3D simulations that have studied planetary migration on a fixed circular orbits, we do not assume the orbit of the protoplanet is circular or even in the plane of the disk. Instead we investigate the effect of eccentric and inclined orbits on the disk, and the effect of these distortions on the evolution of the planetary orbit. Introduction About 120 extrasolar planets have been discovered in the last 10 years. Their masses range from about the mass of Saturn up to 10 Jupiter masses, and because of the way they are discovered, their orbital periods are no longer than a few years. One of the most striking differences with our own solar system is not their tight orbits, which may be due to an observational bias, but that most of the planets are on eccentric orbits, as can be seen in Figure 1. Figure 1: The distribution of eccentricities of extrasolar planets as a function of semimajor axis There are several theories on how these eccentric orbits arise. Gravitational perturbance from passing stars or companion stars may cause eccentricity (Laughlin and Adams, 1998), however this is unlikely to explain the large quantity of eccentric orbits. A possible cause could be planet-planet interaction, but given the lack of planets on circular orbits this could hardly explain all the planets. Another theory forms Jupiter mass planets not by accretion but by gravitational collapse of a disk instability (Boss, 2000). Here we study the effect a 3D disk has on the eccentricity and inclination of embedded planets. Physical Model We solve the Navier-Stokes equations using a threedimensional finite difference scheme based on the NIRVANA code by Udo Ziegler with subsequent adaptations by Kley and d’Angelo. We use an isothermal equation-ofstate, and the planetary orbit is calculated using a order Runge-Kutta solver. The simulations start with a disk with constant surface density and a gap profile near the planet as calculated by Artymowicz (1994). The vertical profile is a Gaussian with scale height . The inner and outer boundary are managed using a non-reflecting closed boundary. This is set up by choosing a ramp function (with value 1 at the border and 0 at the inner edge of the damping region) that slowly changes the density in the region to its original value. This setup avoids mass loss at the open boundary and on the other hand it damps reflecting waves that will in turn distort the picture near the planet that arise from using closed boundary conditions. Simulation details The calculations were performed on a 3-dimensional grid in spherical coordinates grid with 108 cells in the radial direction , 24 cells in the vertical direction . The simulations were run for several hundred orbits. Our standard model starts with a planet on a slightly eccentric and inclined orbit ,+ .0/1 2 3 $ . The model includes a constant viscosity of 45 6!# 87 9 in dimensionless units, which is equivalent to an : -viscosity of :; ) = !# 7 ? . The evolution of the orbital elements over a period of 400 orbits can be seen in Figure 2. 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 0 50 100 150 200 250 300 350 400 a @ Time (orbits) Semimajor axis 0 0.01 0.02 0.03 0.04 0.05 0.06 0 50 100 150 200 250 300 350 400 e A Time (orbits) Eccentricity