Bondi-Hoyle-Lyttleton accretion flow in a stratified layer

TL;DR

This work extends the classical Bondi–Hoyle–Lyttleton framework to a vertically Gaussian-stratified medium by modeling the downstream wake as a one-dimensional tail and deriving steady-state continuity and momentum equations. A shooting method extracts physically admissible solutions, revealing that the tail is densest and slowest when the layer thickness $H$ is near the gravitational radius scale $ξ_{0}$, and providing an empirical fit for the stagnation point location $x_{0}$. The study characterizes how the mass accretion rate $\dot{M}$ and the drag forces depend on $H/ξ_{0}$, showing that $\dot{M}$ is maximized for $H\llξ_{0}$ at fixed surface density and remains roughly constant for $H\gtrsim 2ξ_{0}$ at fixed midplane density; the 2D infinitely-thin limit yields a simple analytic solution. Comparisons with linear theory highlight differences in far-field wake predictions, and the results are in good agreement with existing 3D simulations for thick discs, while underscoring the limitations of the BHL assumptions in certain regimes. Overall, the work provides a scalable framework to assess accretion and dynamical-friction feedback for perturbers embedded in stratified discs.

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 $ξ_{0}\equiv GM/v_{0}^{2}$, where $v_{0}$ its relative velocity with respect to the medium. The location of the stagnation point is evaluated as a function of $H$ and $ξ_{0}$, and an empirical fitting formula is provided. While the distance to the stagnation point is maximized when $H\simeq ξ_{0}$, the mass accretion rate attains its maximum value for $H \ll ξ_{0}$ 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 $2ξ_{0}$. Additionally, we investigate how both the drag force resulting from mass accretion and the gravitational drag arising from its tail depend on $H/ξ_{0}$. 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.

Figures (8)

• Figure 1: Schematic of the flow (not to scale) in the BHL approach (top panel) and in the C11-C13 model (bottom panel). The solid lines indicate the streamlines and the red arrows represent the velocity in the tail. The dotted lines correspond to the streamline with impact parameter $2\xi_{0}$. The tail is slower in the BHL model. While the flow in the tail is assumed plane-parallel in both models, it presents shear in the C11-C13 model. Streamlines with impact parameter $\leq 2\xi_{0}$ are assumed to entry into the accretor, but they are not modelled in the C11- C13 model.
• Figure 2: $\tilde{x}_{0}$ versus $\tilde{H}$. The blue band includes those values of $\tilde{x}_{0}$ that satisfy conditions (I)-(VI). The dots indicate the central value in that interval. The red line represents the analytical fit given in Eq. (\ref{['eq:anlytical_x0_H']}).
• Figure 3: Dimensionless velocity $\tilde{v}$ (upper panel) and linear density $\mu$, normalized to $\sqrt{2\pi}\Sigma\xi_{0}$ in the middle panel, and to $2\pi\rho_{0}\xi_{0}^{2}$ in the lower panel, shown for different values of $\tilde{H}$ indicated on each curve. For a given $\tilde{H}$, the value of $\tilde{x}_{0}$ was chosen as the central point of the blue band in Figure \ref{['fig:x0_vs_Htilde']}.
• Figure 4: $\tilde{v}$ at $\tilde{x}=10$ (top panel) and at $\tilde{x}=0.19$ (bottom panel) as a function of $\tilde{H}$ in the solutions given in Figure \ref{['fig:v_and_mu']} (black circles), together with the predicted values using Equation (\ref{['eq:v_function']}) with the corresponding values of $\tilde{x}_{0}$ (red squares).
• Figure 5: Mass accretion rate as a function of $\tilde{H}$. In the top panel, the accretion rate is normalized to $\sqrt{2\pi} \Sigma v_{0} \xi_{0}$, whereas it is normalized to $\pi \rho_{0}v_{0}(2\xi_{0})^{2}$ in the bottom panel. The solid curves correspond to the BHL model whereas the dashed lines for C13 model.