Closing the gap to convergence of gravitoturbulence in local simulations^⋆

¹ Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstr. 15, 24118 Kiel, Germany
e-mail: jklee@astrophysik.uni-kiel.de
² Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany
³ Steward Observatory, The University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

Received: 11 June 2019
Accepted: 19 September 2019

Abstract

Aims. Our goal is to find a converged cooling limit for fragmentation in self-gravitating disks. This is especially interesting for the formation of planets, brown dwarfs, or stars, and the growth of black holes. While investigating the limit, we want to give a clear criterion for the state of convergence.

Methods. We ran two-dimensional shearingsheet simulations with the hydrodynamic package Fosite at high resolutions. Thereby, resolution and limiters were altered. Subsequently, we investigated the spectra of important physical quantities at the length scales where fragmentation occurs. In order to avoid prompt fragmentation at high resolutions, we started these simulations with a fully-developed gravitoturbulent state obtained at a lower resolution.

Results. We show nearly converged results for fragmentation with a critical-cooling timescale t_crit ∼ 10 Ω⁻¹. We can backtrace this claim by investigating the spectra of relevant physical variables at length scales around and below the pressure scale height. We argue that well-behaved results cannot be expected if counteracting quantities vary too much on these critical-length scales, either by change of resolution or numerical method. A comparison of fragmentation behaviour with the related spectra reveals that simulations behave similar, if the spectra are converged to the length scales where self-gravity leads to instabilities. Observable deviations in the results obtained with different numerical setups are confined to scales below these critical length scales.