Structure and instability of the ionization fronts around moving black holes

In this paper, we focus on understanding the physical processes that lead to stable or unstable ionization fronts (I-fronts) observed in simulations of moving black holes (BHs). The front instability may trigger bursts of gas accretion, rendering the BH significantly more luminous than at steady state. We perform a series of idealized three-dimensional radiation hydrodynamics simulations resolving the I-fronts around BHs of mass MBH and velocity $v$∞ accreting from a medium of density nH. The I-front, with radius RI, transitions from D-Type to R-Type as the BH velocity becomes larger than a critical value $v_\mathrm{R}\sim 40\, \mathrm{km\,s}^{-1}$. The D-Type front is preceded by a bow-shock of thickness ΔRI that decreases as $v$∞ approaches $v$R. We find that both D-Type and R-Type fronts can be unstable given the following two conditions: (i) for D-Type fronts the shell thickness must be ΔRI/RI < 0.05 (i.e. $v_\infty \gtrsim 20\, \mathrm{km\,s}^{-1}$), while no similar restriction holds for R-Type fronts; (ii) the temperature jump across the I-front must be TII/TI > 3. This second condition is satisfied if $T_\mathrm{I}\lt 5000\, \mathrm{K}$ or if $n_\mathrm{H}\, M_\mathrm{BH} \gtrsim 10^6\, M_\odot \, \mathrm{cm^{-3}}$. Due to X-ray pre-heating typically $T_\mathrm{I} \sim 10^4\, \mathrm{K}$, unless the D-Type shell is optically thick to X-rays, which also happens when $n_\mathrm{H}\, M_\mathrm{BH}$ is greater than a metallicity-dependent critical value. We thus conclude that I-fronts around BHs are unstable only for relatively massive BHs moving trough very dense molecular clouds. We briefly discuss the observational consequences of the X-ray luminosity bursts likely associated with this instability.