Super-critical column accretion on to strongly magnetized neutron stars in ULX pulsars

We carry out axisymmetric two-dimensional radiation hydrodynamical simulations of super-critical accretion columns on to strongly magnetized neutron stars. The effect of the strong magnetic field is taken into account by inhibiting the fluid motion across the radial magnetic field of the neutron stars. It is found that the high-density matter falls on to the neutron star along the sidewall of the column. Within the column, two high-density inflow regions shaped like a hollow cone are found for the case of extremely high mass accretion rates, ${\dot{M}}/(L_{\rm Edd}/c{\,}^2) \sim 5\times 10^{2}$, where $\dot{M}$, $L_{\rm Edd}$, and $c$ are the mass accretion rate on to the neutron star, the Eddington luminosity, and the speed of light, respectively. The less dense matter in the gap between the high density inflow regions is blown away via the radiative force. The resultant structure of the inflow looks like a triple hollow cone. Matter falls on to the neutron star only through the sidewall for the case of moderately high mass accretion rates, ${\dot{M}}/(L_{\rm Edd}/c{\,}^2)\, {\sim 3 \times 10^1}$. A low-density outflow fills the interior of the column. In this case, the inflow structure looks like a single hollow cone. Although the copious photons are generated in the inflow regions via a shock, the photons escape from the sidewall of the column and the radiation force does not prevent inflow. The resulting luminosity of the sidewall exceeds $\sim\! 30$ times the Eddington luminosity for neutron stars, which is consistent with the observed luminosity of ultra-luminous X-ray pulsars.