Abstract We compute the density and velocity profiles along the tail induced by a body of mass M, embedded in the midplane of a vertically-stratified media with scaleheight H, adopting a one-dimensional model as in the Bondi-Hoyle-Lyttleton problem. In analogy to what occurs in the case of a homogeneous medium, there exist a family of solutions that satisfy the boundary conditions. A shooting method is employed to isolate those solutions that fulfill a specific set of physical and mathematical constraints. The tail is found to be both densest and slowest when the scaleheight H is equal to the gravitational radius ₀ GM/v₀^2, where v0 its relative velocity with respect to the medium. The location of the stagnation point is evaluated as a function of H and ξ0, and an empirical fitting formula is provided. While the distance to the stagnation point is maximized when H ≃ ξ0, the mass accretion rate attains its maximum value for H ≪ ξ0 at fixed surface density. When instead the midplane density is held constant and H is varied, the accretion rate hardly changes once H exceeds about 2ξ0. Additionally, we investigate how both the drag force resulting from mass accretion and the gravitational drag arising from its tail depend on H/ξ0. We highlight how the effect of varying the degree of mixing in the tail influences the resulting drag force. Finally, for the particular case of an infinitely thin layer, we provide a simple analytical solution, which may serve as a useful pedagogical reference.
F J Sánchez-Salcedo (2026) studied this question.