Planet Migration in Self-Gravitating Discs

Planet Migration in Self-Gravitating Discs: Survival of Planets

Sahl Rowther & Farzana Meru

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Abstract

We carry out three-dimensional SPH simulations to study whether planets can survive in self-gravitating protoplanetary discs. The discs modelled here use a cooling prescription that mimics a real disc which is only gravitationally unstable in the outer regions. We do this by modelling the cooling using a simplified method such that the cooling time in the outer parts of the disc is shorter than in the inner regions, as expected in real discs. We find that both giant ( > M_Sat ) and low mass ( < M_Nep ) planets initially migrate inwards very rapidly, but are able to slow down in the inner gravitationally stable regions of the disc without needing to open up a gap. This is in contrast to previous studies where the cooling was modelled in a more simplified manner where regardless of mass, the planets were unable to slow down their inward migration. This shows the important effect the thermodynamics has on planet migration. In a broader context, these results show that planets that form in the early stages of the discs' evolution, when they are still quite massive and self-gravitating, can survive.

Cooling the disc

The balance between the heating and cooling rate determines the stability of self-gravitating discs. In this work we use a straight forward implementation to model the cooling based on the location in the disc. The cooling time is given by

\(t_{cool} = \beta \Omega^{-1}\).

Disc evolution

Past studies which used a constant \(\beta \) found that planets are unable to slow their rapid inward migration and quickly reach the inner boundary of the disc in a couple of orbits (Baruteau et al. 2011) . However, using this simple model to cool the disc results in the entire disc being gravitationally unstable (see Fig 1a). Realistic self-gravitating discs are expected to only be gravitationally unstable in the outer regions.

Using a variable \(\beta \) allows us to mimic a realistic self-gravitating disc (see Fig 1b). The longer cooling time in the inner disc results in a gravitationally stable region with very little density fluctuations. The aim of this work is to determine whether planets can slow down their inward migration in the smoother inner disc.

Figure 1: (a) Constant \(\beta \) -- Gravitationally unstable throughout the disc. (b) Variable \(\beta \) -- Only gravitationally unstable in the outer regions.

Results

Planet migration

To study the importance of thermodynamics on planet migration, we embed a single planet in the outer regions for both a constant and a variable \(\beta \) disc.

In both cases, the planet initially rapidly migrates inwards at a similar speed. However, once the planet reaches the inner regions of the disc, the migration track is quite different. Since the entire disc is gravitationally unstable, the planet simply continues migrating inwards with a constant \(\beta \). However, with a variable \(\beta \), the planet is able to slow down in the inner regions of the disc which is gravitationally stable.

Figure 2: A Jupiter mass planet in a constant \(\beta \) disc (a) only stops migrating after reaching the inner edge of the disc. Whereas, with a variable \(\beta \) (b), the planet's migration slows down earlier once it reaches the gravitationally stable inner disc.

Analysis of coorbital material and torques

The coorbital material around the planet plays a key role in its migration. Initially the planet begins its migration in the gravitationally unstable part of the disc, where the density fluctuates a lot. These fluctuations results in an underdense region forming in front of the planet which drives its inward migration in both models. Since the entire disc is gravitationally unstable with a constant \(\beta \), the corotation torque remains negative throughout and the planet is unable to slow down (Figure 3a). However with a variable \(\beta \), the planet is able to slow down once it reaches the gravitationally stable inner disc. This smoother region results in the structure around the planet becoming symmetric, thus decreasing the negative corotation torque and slows the planet down (Figure 3b).