A Two-Dimensional Hydrostatically Equilibrium Atmosphere of a Neutron Star with Given Differential Rotation

An analytic solution has been found in the Roche approximation for the axially symmetric structure of a hydrostatically equilibrium atmosphere of a neutron star produced by collapse. A hydrodynamic (quasione-dimensional) model for the collapse of a rotating iron core in a massive star gives rise to a heterogeneous rotating protoneutron star with an extended atmosphere composed of matter from the outer part of the iron core with differential rotation (Imshennik and Nadyozhin, 1992). The equation of state of a completely degenerate iron gas with an arbitrary degree of relativity is taken for the atmospheric matter. We construct a family of toroidal model atmospheres with total masses $M \approx 0.1 \div 0.2 M_{\odot}$ and total angular momenta $J \approx (1 \div 5.5) \cdot 10^{49} erg \cdot s$, which are acceptable for the outer part of the collapsed iron core, in accordance with the hydrodynamic model, as a function of constant parameters $\omega_{0} and r_{0}$ of the specified differential rotation law $\Omega = \omega_{0}\exp{\Big[-\frac{(r\sin{\Theta})^{2}}{r_{0}^{2}}\Big]}$ in spherical coordinates. The assumed rotation law is also qualitatively consistent with the hydrodynamic model for the collapse of an iron core.