Hydrodynamical simulations with strong indirect terms in FARGO-like codes

Numerical aspects of the non-inertial frame and artificial viscosity

¹ Institut für Astronomie und Astrophysik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
² Zentrum für Astronomie (ZAH), Institut für Theoretische Astrophysik (ITA), Universität Heidelberg, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany

^★ Corresponding author; lucas.jordan@uni-tuebingen.de

Received: 15 April 2024
Accepted: 26 November 2024

Abstract

Context. Binary star systems allow us to study the planet formation process under extreme conditions. In the early stages, these systems contain a circumbinary disk and a disk around each star. To model the interactions between these disks in the frame of one of the stars, strong fictitious forces must be included in the simulations. The original FARGO and the FARGO3D codes fail to correctly simulate such systems if the indirect term becomes too strong.

Aims. We present a different way to compute the indirect term that, together with a tensor artificial viscosity prescription, allows the FARGO code to simulate the circumbinary disks in a non-inertial frame of reference. In this way, the FARGO code can be used to study interactions between circumstellar and circumbinary disks.

Methods. We first evaluated the accuracy of the standard implementation and our proposed indirect term prescription using a simple N-body test case. We then analytically estimated the effect of the default artificial viscosity used in the FARGO code in the limit of large distances to the N-body system. Finally, we evaluated the effects of the different prescriptions by performing hydrodynamical simulations in a non-inertial frame of reference.

Results. By updating the indirect term prescription and the artificial viscosity, we were able to successfully simulate a circumbinary disk in a frame that is centered on the less massive star. We find that updating the indirect term becomes relevant when the indirect term becomes stronger than the direct gravitational forces, which occurs for mass ratios of q ≳ 5%. The default artificial viscosity used in the FARGO code inherently produces artificial pressure in a non-inertial frame of reference even in the absence of shocks. This leads to artificial mass ejection from the Hill sphere, starting at brown dwarf masses (q ≳ 1%). These problems can be mitigated by using a tensor artificial viscosity formulation. For high mass ratios, q ≳ 1%, it is also becomes important to initialize the disk in the center-of-mass frame. We expect our proposed changes to be relevant for other grid-based hydrodynamic codes where strong indirect terms occur, or for codes that use artificial viscosity.