Radiating Bondi Flows I: Dimensionless Framework and Constant Opacity Solutions

Abstract

In this paper, we extend the foundational work of Bondi (1952) to include the effects of radiative feedback in gas-pressure-dominated environments. We construct steady-state spherically symmetric accretion solutions including radiative heating and cooling. Under the simplifying assumption of a constant opacity, the solutions are controlled by four dimensionless parameters: the adiabatic index $γ$, optical depth through the Bondi radius $τ_B$, dimensionless luminosity at infinity $\tilde{L}_\infty$, and a characteristic dimensionless cooling time $β$. We present numerical solutions across the dimensionless parameter space $(τ_B, \tilde{L}_\infty, β)\in [10^{-3}, 10^3]$. Contrary to radiation-pressure-dominated environments, radiative feedback primarily operates to suppress accretion -- particularly at high $τ_B$, $\tilde{L}_\infty$, and/or $β$. We also present analytic descriptions confirming the suppressive nature of this feedback and give the scalings for the accretion rate $\dot{M}\sim \tilde{L}_\infty^{-5/4}$ at large $\tilde{L}_\infty$, $\dot{M}\sim τ_B^{-10/11}β^{-5/11}$ at large $τ_B$, and $\dot{M}\sim (\tilde{L}_\inftyτ_B)^{-5/8}$ for large $\tilde{L}_\inftyτ_B$. We discuss the potential role of convection in these steady-state solutions, and the particular relevance to problems of planet formation where radiative heating is significant, but the system remains in the gas-pressure-dominated regime.

Figures (11)

• Figure 1: Adiabatic Bondi solutions for $\gamma=1$ (circle), $\gamma=7/5$ (triangle), $\gamma=5/3$ (square). The solid lines show Mach number profiles with corresponding density profiles as dashed lines. Markers are placed at the location of the sonic point with the $\gamma=5/3$ sonic point located at $r=0$. Dot-dashed lines plot the standard expected asymptotic scalings for reference.
• Figure 2: Example of families of curves and iterative procedure used to determine the sonic point (and therefore the accretion rate) for a model with $(\gamma, \tau_B,\tilde{L}_\infty, \beta) = (7/5,1,10,1)$. Each curve is an integration colored according to the choice of mass accretion rate. Cool-hued curves belong to the family of solutions which undershoot the sonic point, while warm hues are solutions classified as overshooting. Faint dashed lines show the steps of our iterative search to determine the sonic point, with a solid black line marking the converged transonic solution. An adiabatic solution would have $\dot{M}=10\pi\rho_\infty c_{s,\infty}r_B^2$, or a value of $5/2$ in the presented colorbar. It is seen that such an accretion rate would overshoot the sonic point, and that radiative feedback skews the transonic solution inwards to a comparably lower accretion rate for this set of dimensionless parameters.
• Figure 3: Slices at fixed $\beta\in(10^{-3},1, 10^3)$ in the surveyed parameter space, mapping the steady-state accretion rates relative to the adiabatic Bondi rate. Each black dot corresponds to a computed steady-state solution on our model grid with the surrounding square colored according to the value of $f_{\rm acc}$. Cells for which no transonic solution could be found are colored gray. Solid white contours are also drawn for each decade of $f_{\rm acc}$. For reference, dashed lines for the conditions $\tilde{L}_\infty = \tau_B\beta$ and $f_{\rm acc}=(\tilde{L}_\infty\tau_B)^{-5/8}$ with $f_{\rm acc} = (10^{-3}, 10^{-2}, 10^{-1}, 1)$ are included (see Section \ref{['sec:anal']} for the origin of these scalings).
• Figure 4: A volumetric rendering over the fiducial parameter space of our numerical solutions, showing the models which exhibit significant radiative disequilibrium $(E_r\neq aT^4)$. A blue cube is drawn at the location of any model which has $\max(D_{\rm eq}) > 1$. Note the z-axis is inverted for visualization purposes, so large values of $\beta$, favoring disequilibrium, are lower in a vertical sense.
• Figure 5: Panels (a-e): Radiating steady-state solutions for the models with $\tau_B=10^{-3}$, $\beta=10^{-3}$. Each panel plots the radial profile of a dependent variable of the problem $(\tilde{\rho}, \mathcal{M}, s, \tilde{L}, \tilde{E}_r)$ as solid lines with each curve corresponding to a different choice of luminosity $\tilde{L}_\infty$. Panel (e) contains additional dotted lines for the temperature solutions $\tilde{T}^4$. Panel (f): The mass accretion rate normalized to the adiabatic rate $f_{\rm acc}$ as a function of model luminosity $\tilde{L}_\infty$.