On the pressure of collisionless particle fluids. The case of solids settling in disks

Aims. Collections of dust, grains, and planetesimals are often treated as a pressureless fluid. We study the validity of neglecting the pressure of such a fluid by computing it exactly for the case of particles settling in a disk. Methods. We solve a modified collisionless Boltzmann equation for the particles and compute the corresponding moments of the phase space distribution: density, momentum, and pressure. Results. We find that whenever the Stokes number, defined as the ratio of the gas drag timescale to the orbital timescale, is more than 1/2, the particle fluid cannot be considered as pressureless. While we show it only in the simple case of particles settling in a laminar disk, this property is likely to remain true for most flows, including turbulent flows.

💡 Research Summary

The paper revisits a widely used approximation in protoplanetary‑disk studies: treating the solid component (dust, pebbles, planetesimals) as a pressure‑less fluid. The authors ask whether this simplification is justified when the particles are subject to gas drag and settle toward the mid‑plane. To answer the question they solve a modified collisionless Boltzmann (or Vlasov) equation that includes a linear drag term representing the aerodynamic coupling with the gas. The problem is reduced to one dimension (the vertical direction) and the gas is assumed to be laminar and static, so the only dynamical variable for the particles is the vertical coordinate (z) and the vertical velocity (v).

The key nondimensional parameter is the Stokes number
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