On the dynamics of resonant super-Earths in disks with turbulence driven by stochastic forcing

¹ Université de Bordeaux, Observatoire Aquitain des Sciences de l’Univers, BP 89, 33271 Floirac Cedex, France
² Laboratoire d’Astrophysique de Bordeaux, BP 89, 33271 Floirac Cedex, France
e-mail: arnaud.pierens@obs.u-bordeaux1.fr
³ Department of Astronomy and Astrophysics, University of Santa Cruz, CA 95064, USA
⁴ DAMTP, University of Cambridge, Wilberforce Road, Cambridge , CB30WA UK

Received: 31 January 2011
Accepted: 21 March 2011

Abstract

Context. A number of sytems of multiple super-Earths have recently been discovered. Although the observed period ratios are generally far from strict commensurability, the radio pulsar PSRB1257 + 12 exhibits two near equal-mass planets of  ~4 M_⊕ close to being in a 3:2 mean motion resonance (MMR).

Aims. We investigate the evolution of a system of two super-Earths with masses  ≤ 4 M_⊕ embedded in a turbulent protoplanetary disk. The aim is to examine whether resonant trapping can occur and be maintained in the presence of turbulence and how this depends on the amplitude of the stochastic density fluctuations in the disk.

Methods. We performed 2D numerical simulations using a grid-based hydrodynamical code in which turbulence is modelled as stochastic forcing. We assumed that the outermost planet is initially located just outside the position of the 3:2 mean motion resonance with the inner one, and we studied the dependence of the resonance stability on the amplitude of the stochastic forcing.

Results. For systems of two equal-mass planets we find that in disk models with an effective viscous stress parameter α ~ 10^-3, damping effects due to type I migration can counteract the effects of diffusion of the resonant angles, in such a way that the 3:2 resonance can possibly remain stable over the disk lifetime. For systems of super-Earths with mass ratio q = m_i/m_o ≤ 1/2, where m_i(m_o) is the mass of the innermost (outermost) planet, the 3:2 resonance is broken in turbulent disks with effective viscous stresses 2 × 10^-4 ≲ α ≲ 1 × 10^-3, but the planets become locked in stronger p + 1:p resonances, with p increasing as the value for α increases. For α ≳ 2 × 10^-3, the evolution can eventually involve temporary capture in a 8:7 commensurability but no stable MMR is formed.

Conclusions. Our results suggest that, for values of the viscous stress parameter typical to those generated by MHD turbulence, MMRs between two super-Earths are likely to be disrupted by stochastic density fluctuations. For lower levels of turbulence, however, as is the case in presence of a dead-zone, resonant trapping can be maintained in systems with moderate values of the planet mass ratio.