Convergence of the Critical Cooling Rate for Protoplanetary Disk Fragmentation Achieved: The Key Role of Numerical Dissipation of Angular Momentum

Abstract

We carry out simulations of gravitationally unstable disks using smoothed particle hydrodynamics(SPH) and the novel Lagrangian meshless finite mass (MFM) scheme in the GIZMO code (Hopkins 2015). Our aim is to understand the cause of the non-convergence of the cooling boundary for fragmentation reported in the literature. We run SPH simulations with two different artificial viscosity implementations, and compare them with MFM, which does not employ any artificial viscosity. With MFM we demonstrate convergence of the critical cooling time scale for fragmentation at β crit ≈ 3.. Non-convergence persists in SPH codes, although it is significantly mitigated with schemes having reduced artificial viscosity such as inviscid SPH (ISPH) (Cullen & Dehnen 2010). We show how the non-convergence problem is caused by artificial fragmentation triggered by excessive dissipation of angular momentum in domains with large velocity derivatives. With increased resolution such domains become more prominent. Vorticity lags behind density due to numerical viscous dissipation in these regions, promoting collapse with longer cooling times. Such effect is shown to be dominant over the competing tendency of artificial viscosity to diminish with increasing resolution. When the initial conditions are first relaxed for several orbits, the flow is more regular, with lower shear and vorticity in non-axisymmetric regions, aiding convergence. Yet MFM is the only method that converges exactly. Our findings are of general interest as numerical dissipation via artificial viscosity or advection errors can also occur in grid-based codes. Indeed for the FARGO code values of β crit significantly higher than our converged estimate have been reported in the literature. Finally, we discuss implications for giant planet formation via disk instability.

Towards a population synthesis model of self-gravitating disc fragmentation and tidal downsizing II: the effect of fragment–fragment interactions

“…The previous five years of work in this area, which include particle-based and grid-based simulations suggest the critical α is unlikely to greatly exceed 0.1. Most recently, meshless hydrodynamic simulations have demonstrated convergence at α ≈ 0.13, confirming that numerical dissipation is the cause (Deng et al 2017).…”

Formation of the First Stars and Black Holes

“…Namely, its surface density must be low enough to maintain the Toomre Q above unity, and the cooling time has to be longer than the local orbital time (Gammie, 2001). For the latter condition, to so-called Gammie criterion), simulations using improved numerical resolution have demonstrated convergence as a function of resolution after a decade of discordant results (Deng et al, 2017). If the cooling time drops below the local orbital time the Toomre Q will decrease and eventually drop below unity, entering the fragmentation regime.…”

On the diversity and statistical properties of protostellar discs

“…This resolution is sufficient to resolve the local Jeans mass throughout the calculation, which is necessary to model fragmentation of collapsing molecular clouds correctly (Bate & Burkert 1997;Truelove et al 1997;Whitworth 1998;Boss et al 2000;Hubber, Goodwin & Whitworth 2006). More recently, there has been much discussion in the literature about the resolution necessary to resolve fragmentation in isolated gravitationally unstable discs (Nelson 2006;Meru & Bate 2011, 2012Hopkins & Christiansen 2013;Rice et al 2014;Young & Clarke 2015Lin & Kratter 2016;Takahashi, Tsukamoto & Inutsuka 2016;Baehr, Klahr & Kratter 2017;Deng, Mayer & Meru 2017;Klee et al 2017). As yet, there is no consensus as to the resolution that is necessary and sufficient to capture fragmentation of such discs.…”