The Toroidal Iron Atmosphere of a Protoneutron Star: Numerical Solution
  A numerical method presented by Imshennik et al. (2002) is used to solve the two dimensional axisymmetric hydrodynamic problem on the formation of a toroidal atmosphere during the collapse of an iron stellar core and outer stellar layers. An evolutionary model from Boyes et al. (1999) with a total mass of $25M_{\odot}$ is used as the initial data for the distribution of thermodynamic quantities in the outer shells of a high-mass star. We analyze in detail the results of three calculations in which the difference mesh and the location of the inner boundary of the computational region are varied. In the initial data, we roughly specify an angular velocity distribution that is actually justified by the final result - the formation of a hydrostatic equilibrium toroidal atmosphere with reasonable total mass, $M^{tot} = (0.117 \div 0.122)M_{\odot}$, and total angular momentum, $J^{tot} = (0.445 \div 0.472) x 10^{50} erg \cdot s$, for the two main calculations. We compare the numerical solution with our previous analytical solution in the form of toroidal atmospheres (Imshennik and Manukovskii 2000). This comparison indicates that they are identical if we take into account the more general and complex equation of state with a nonzero temperature and self-gravitation effects in the atmosphere. Our numerical calculations, first, prove the stability of toroidal atmospheres on characteristic hydrodynamic time scales and, second, show the possibility of sporadic fragmentation of these atmospheres even after a hydrodynamic equilibrium is established. The calculations were carried out under the assumption of equatorial symmetry of the problem and up to relatively long time scales $(\approx 10s)$. 
V. S. Imshennik, K. V. Manukovskii and M. S. Popov (ITEP Moscow)