Planet Migration In A Polytropic DiskChia-Ying Chiang1,4, Chi Yuan1, Chien-Chang Yen1,2, and Hui Zhang1,3

1Institute of Astronomy and Astrophysics, Academia Sinica2Department of Mathematics, Fu Jen Catholic University

3Department of Astronomy, NanJing university 4Institute of Astronomy, National Tsing Hua University

Abstract Due to discovery of "hot Jupiters", which could hardly be understood by the present-day planet formation theory, astrophysicists turned to investigate the migration process of the planet. Most of the studies are limited to disks of isothermal gas without consideration of self-gravity of the disk, on the polar coordinates. Instead, we treat the protoplanetary accretion disc of polytropic gas without self-gravity. Simulations are performed on Cartesian coordinates by using the Antares code, a high-order Godunov code, we have developed. Since we use the Cartesian coordinates, we don't need to cut a hole in the center of the disk. On the other hand, we must specify a softening length for the potential of the central star, to avoid the singularity. In the numerical simulations, we study the orbit of the planet as it migrates. We compare our results with those of isothermal disks. We find for soften length equals to 30 grids, corresponding to 0.3 AU, the planet in a polytropic gas disk takes less time to migrate to the inner part of the disk. We find negative torque exerted on the planet by the disk material in the Roche lobe of the planet dominates, the planet loses its angular momentum to gas surrounding it and goes inward. However, when resolution has been upgraded, simulation results show that the planet may even go out eventually after an initial inward migration.

Density evolution of isothermal and polytropic disks

Planet experienced torque

Planet migration orbit

Gap formationIn isothermal disk, begined with uniform density, the planet experiences Type I migration first, then Type III migration, and finally Type II, the gap forms. Because of initial sharp density profile in the polytropic disk, gap does not form but be filled by material diffused from the center part of the disk. As resolution has been upgraded, at first planet experiences similar migration process, but a gap starts to form without gas filling in, and the planet stop rapid migration inward.

Comparisons of migration orbitFigures on the right-hand side show comparisons of migration orbit in different cases. Planet migration in a polytropic disk is much faster than in an isothermal disk. Nevertheless, results of the same case in different resolution are not in the same manner. As resolution is higher, the planet would even go outward.

ConclusionsFrom figures of torque, we believe that swift planet migration is associated with net negative torque, which is dominated by torque induced by planet Roche lobe. In addition, gap formation would influence planet migration rate.

References1. R.P.Nelson, J.C.B.Papaloizou, F.Masset W.Kley, “The migration and growth of protoplanets in protostellar discs”, MNRAS, 2000