### Motivation

Whenever a massive object passes through a rarefied medium, it draws surrounding matter toward it. As a result, this material creates an overdense wake behind the object that exerts its own gravitational pull, retarding the original motion. Such dynamical friction arises whether the medium consists of non-interacting point particles, e.g., a stellar cluster, or a continuum fluid, e.g., an interstellar cloud. Material can also fall onto the mass, which imparts both mass and momentum to the particle. The combination of these contributions contribute to the overall gravitational friction force.

Despite the widespread occurrence of gaseous dynamical friction, there is still no generally accepted derivation of the force, even after 70 years of effort. In the case where the gas is collision-less, the analytic formulation is well known. However, with an isothermal gas, the flow in the vicinity of the gravitating mass is complex both temporally and spatially, as many simulations have shown. These two papers attempted to derive an analytical expression for this dynamical friction force.

### Subsonic accretion flow

Our method of derivation is to ignore the complex non-linear region close to the central mass, and instead focus on the far-field gas that is slightly perturbed from its original path. In the assumption of steady-state, the perturbations on the gas here is a result of both the gravitational force from the central mass and the gas pressure forces from the overdense wake. By measuring the net momentum flux into a sphere drawn around the central mass, we effectively measure the amount of momentum that is lost as a dynamical friction drag force.

These perturbation analysis techniques combine the equations of fluid dynamics and the assumption that vorticity is conserved in the flow. For the subsonic case, where the gas is moving slower than the speed of sound, there are no shocks which can develop turbulence and break these conservation assumptions.

### supersonic accretion flow

When the gas becomes supersonic, the presence of a Mach cone complicates the flow. In this case, we must apply conservation laws to monitor the gas flow across the Mach cone, which, in the limit of steady state, can exist even out to large distances. In short, there's a lot of messy math that has to be juggled around.

However, a result that immediately arises is that the mass accretion rate onto the central point particle plays a role in the overall dynamical friction force.

### Dynamical Friction Force

This study has pivoted on the close relationship between the dynamical friction force, i.e. the transfer of linear momentum from gas to a gravitating object, and the transfer of mass to that same object. This relationship is embodied in our central result, that the friction force is the mass accretion rate multiplied by the far-field velocity of the gas. This relation assumed a conservation of vorticity and a second-order perturbation analysis, which might not be satisfied in the supersonic case where the downstream wake can be dense and turbulent. Nonetheless, this marks one of the first attempts to successfully derive an analytic friction force in the case of a mass moving through a gas.