Dynamical friction for accelerated motion in a gaseous medium

Dynamical friction arises from the interaction of a perturber and the gravitational wake it excites in the ambient medium. This interaction is usually derived assuming that the perturber has a constant velocity. In realistic situations, motion is accelerated as for instance by dynamical friction itself. Here, we study the effect of acceleration on the dynamical friction force. We characterize the density enhancement associated with a constantly accelerating perturber with rectilinear motion in an infinite homogeneous gaseous medium and show that dynamical friction is not a local force and that its amplitude may depend on the perturber’s initial velocity. The force on an accelerating perturber is maximal between Mach 1 and Mach 2, where it is smaller than the corresponding uniform motion friction. In the limit where the perturber’s size is much smaller than the distance needed to change the Mach number by unity through acceleration, a subsonic perturber feels a force similar to uniform motion friction only if its past history does not include supersonic episodes. Once an accelerating perturber reaches large supersonic speeds, accelerated motion friction is marginally stronger than uniform motion friction. The force on a decelerating supersonic perturber is weaker than uniform motion friction as the velocity decreases to a few times the sound speed. Dynamical friction on a decelerating subsonic perturber with an initial Mach number larger than 2 is much larger than uniform motion friction and tends to a finite value as the velocity vanishes in contrast to uniform motion friction.

💡 Research Summary

The paper investigates how dynamical friction (DF) – the gravitational drag exerted by a gaseous medium on a moving perturber – is altered when the perturber undergoes constant linear acceleration rather than moving at a fixed speed. Traditional DF theory, epitomized by the Chandrasekhar and Ostriker formulas, assumes a steady velocity and therefore treats the drag as a local function of the instantaneous Mach number. The authors challenge this assumption by solving the linearized fluid equations (continuity, Euler, and Poisson) for a point mass moving with velocity v(t)=v₀+at in an infinite, homogeneous, isothermal gas of density ρ₀ and sound speed cₛ.

Using Green‑function techniques, they derive an explicit expression for the density perturbation (the wake) that depends on the entire past trajectory of the perturber. This non‑locality means that the drag at any instant is a functional of the history of the Mach number, the initial velocity v₀, and the constant acceleration a. A key length scale emerges, Lₐ = cₛ²/a, the distance over which the Mach number changes by unity due to acceleration. The analytical results are valid when the physical size of the perturber rₛ ≪ Lₐ, ensuring that the point‑mass approximation holds.

The study identifies four distinct regimes:

1. Accelerating sub‑sonic perturbers (M<1) – If the perturber has never been supersonic, the wake has not yet formed a Mach cone, and the drag closely matches the classical Ostriker sub‑sonic formula. The drag is essentially local and depends only on the instantaneous Mach number.

2. Accelerating trans‑sonic perturbers (1<M<2) – In this window the drag peaks, exceeding the steady‑state value. The wake’s leading edge is a nascent Mach cone that concentrates overdensity ahead of the perturber, producing a stronger gravitational pull.

3. Accelerating highly supersonic perturbers (M≫2) – Once a fully developed Mach cone exists, further acceleration only marginally enhances the drag relative to the steady‑state supersonic expression. The wake geometry is already set, so the additional kinetic energy does not dramatically reshape it.

4. Decelerating supersonic perturbers – As the Mach number declines, the drag drops sharply, becoming weaker than the steady‑state value once M falls below ~2. The wake lags behind, and the reduced cone angle lessens the gravitational pull.

A particularly striking result concerns decelerating sub‑sonic perturbers that started supersonically (initial M₀>2). The wake generated during the earlier supersonic phase persists, exerting a finite drag even as the velocity approaches zero. This contrasts with the classical sub‑sonic drag, which vanishes linearly with speed. Hence, the drag retains a “memory” of past supersonic episodes.

The authors discuss the astrophysical implications. In galactic nuclei, massive black holes that are accelerated by DF themselves will experience a drag that is not simply a function of their current speed; their past orbital history matters. In protoplanetary disks, dust grains or planetesimals that are accelerated by gas drag will feel a stronger or weaker DF depending on whether they have crossed the sonic barrier. Spacecraft re‑entry problems also benefit from a more accurate drag model that accounts for the non‑local wake.

Limitations of the work include the linear perturbation assumption (non‑linear shocks and turbulence are ignored) and the point‑mass approximation (finite size effects, tidal truncation, and self‑gravity of the wake are not treated). The authors suggest that future high‑resolution hydrodynamic simulations should test the analytic predictions in the non‑linear regime and explore the transition when rₛ becomes comparable to Lₐ.

In conclusion, the paper demonstrates that dynamical friction is intrinsically non‑local when the perturber accelerates or decelerates. The drag can be larger than the steady‑state value in the trans‑sonic regime, slightly enhanced for highly supersonic acceleration, and markedly reduced for deceleration, while a lingering supersonic wake can sustain a finite drag even at vanishing speed. These findings refine our understanding of momentum exchange between bodies and gaseous media across a wide range of astrophysical and engineering contexts.