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 $M_mathrm{BH}$ and velocity $v_infty$ accreting from a medium of density $n_mathrm{H}$. The I-front, with radius $R_mathrm{I}$, transitions from D-type to R-type as the BH velocity becomes larger than a critical value $v_mathrm{R}sim 40,mathrm{km/s}$. The D-type front is preceded by a bow-shock of thickness $Delta R_mathrm{I}$ that decreases as $v_infty$ approaches $v_mathrm{R}$. 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 $Delta R_mathrm{I}/R_mathrm{I}<0.05$ (i.e., $v_infty gtrsim 20,mathrm{km/s}$.), while no similar restriction holds for R-type fronts; ii) the temperature jump across the I-front must be $T_mathrm{II}/T_mathrm{I}>3$. This second condition is satisfied if $T_mathrm{I}<5000,mathrm{K}$ 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.
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