Magnetospheric flows in X-ray pulsars II: Heating, cooling and ionization degree at sub-critical accretion

Magnetospheric accretion flows in X-ray pulsars shape their spectra, polarization, and variability. We model the thermal balance of the flow enveloping the neutron star magnetosphere in the sub-critical regime ($L \lesssim 10^{37}\,\mathrm{erg\,s^{-1}}$), where radiation forces do not control the dynamics and single Compton scatterings dominate. The energy budget includes Compton heating by surface X-rays, compressional (adiabatic) heating in the converging flow, and radiative cooling dominated by free-free emission and contributed also by cyclotron emission. We show that the interplay of these processes leads to efficient cooling of the flow in the inner magnetosphere. We compute the flow temperature profile as a function of luminosity and find that near the stellar surface the temperature can fall to a few tens of eV at $L < 10^{35}\,\mathrm{erg\,s^{-1}}$. Under such conditions, the accreting plasma, modelled here as pure hydrogen, is no longer fully ionized. In the strong magnetic fields typical for X-ray pulsars, such temperatures permit partial recombination of electrons and protons into neutral hydrogen. As a result, a significant fraction of the flow becomes weakly ionized, while external illumination ionizes this gas only partially within a geometrically thin layer immediately above the neutron star surface. This implies that magnetospheric accretion at low luminosities proceeds through a partially ionized medium, in contrast to the commonly assumed fully ionized flow.