摘要：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 packageFositeat 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 timescaletcrit ∼ 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.
