Black widow evolution: magnetic braking by an ablated wind

Black widows are close binary systems in which a millisecond pulsar is orbited by a companion a few per cent the mass of the sun. It has been suggested that the pulsar's rotationally powered $\gamma$-ray luminosity gradually evaporates the companion, eventually leaving behind an isolated millisecond pulsar. The evaporation efficiency is determined by the temperature $T_{\rm ch}\propto F^{2/3}$ to which the outflow is heated by the flux $F$ on a dynamical time-scale. Evaporation is most efficient for companions that fill their Roche lobes. In this case, the outflow is dominated by a cap around the L1 point with an angle $\theta_g\sim (T_{\rm ch}/T_g)^{1/2}$, and the evaporation time is $t_{\rm evap}=0.46(T_{\rm ch}/T_g)^{-2}\textrm{ Gyr}$, where $T_g>T_{\rm ch}$ is the companion's virial temperature. We apply our model to the observed black widow population, which has increased substantially over the last decade, considering each system's orbital period, companion mass, and pulsar spin-down power. While the original (Fruchter et al. 1988) black widow evaporates its companion on a few Gyr time-scale, direct evaporation on its own is too weak to explain the overall population. We propose instead that the evaporative wind couples to the companion's magnetic field, removes angular momentum from the binary, and maintains stable Roche-lobe overflow. While a stronger wind carries more mass, it also reduces the Alfv\'en radius, making this indirect magnetic braking mechanism less dependent on the flux $t_{\rm mag}\propto t_{\rm evap}^{1/3}$. This reduces the scatter in evolution times of observed systems, thus better explaining the combined black widow and isolated millisecond pulsar populations.