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A radiation-hydrodynamic model of accretion columns for ultra-luminous X-ray pulsars

Prompted by the recent discovery of pulsed emission from an ultra-luminous X-ray source, M82 X-2 ("ULX-pulsar"), we perform a two-dimensional radiation-hydrodynamic simulation of a super-critical accretion flow onto a neutron star through a narrow accretion column. We set an accretion column with a cone shape filled with tenuous gas with density of $10^{-4} {\rm g}~ {\rm cm}^{-3}$ above a neutron star and solve the two dimensional gas motion and radiative transfer within the column. The side boundaries are set such that radiation can freely escape, while gas cannot. Since the initial gas layer is not in a hydrostatic balance, the column gas falls onto the neutron-star surface, thereby a shock being generated. As a result, the accretion column is composed of two regions: an upper, nearly free-fall region and a lower settling region, as was noted by Basko \& Sunyaev (1976). The average accretion rate is very high; ${\dot M}\sim 10^{2-3} L_{\rm E}/c^2$ (with $L_{\rm E}$ being the Eddington luminosity), and so radiation energy dominates over gas internal energy entirely within the column. Despite the high accretion rate, the radiation flux in the laboratory frame is kept barely below $L_{\rm E}/(4πr^2)$ at a distance $r$ in the settling region so that matter can slowly accrete. This adjustment is made possible, since large amount of photons produced via dissipation of kinetic energy of matter can escape through the side boundaries. The total luminosity can greatly exceed $L_{\rm E}$ by several orders of magnitude, whereas the apparent luminosity observed from the top of the column is much less. Due to such highly anisotropic radiation fields, observed flux should exhibit periodic variations with the rotation period, provided that the rotation and magnetic axes are misaligned.