On inertial forces (indirect terms) in problems with a central body

Gravitational systems in astrophysics often comprise a body – the primary – that far outweights the others, and which is taken as the centre of the reference frame. A fictitious acceleration, also known as the indirect term, must therefore be added to all other bodies in the system to compensate for the absence of motion of the primary. In this Research Note, we first stress that there is not one indirect term but as many indirect terms as there are bodies in the system that exert a gravitational pull on the primary. For instance, in the case of a protoplanetary disc with two planets, there are three indirect terms: one arising from the whole disc, and one per planet. We also highlight that the direct and indirect gravitational accelerations should be treated in a balanced way: the indirect term from one body should be applied to the other bodies in the system that feel its direct gravitational acceleration, and only to them. We point to situations where one of those terms is usually neglected however, which may lead to spurious results. These ideas are developed here for star-disc-planets interactions, for which we propose a recipe for the force to be applied onto a migrating planet, but they can easily be generalized to other astrophysical systems.