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A numerical method presented by Imshennik et al. (2002) is used to solve the twodimensional 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 ⊙ is used as the initial data for the distribution of thermodynamic quantities in the outer shells of a high-mass star. Our computational region includes the outer part of the iron core (without its central part with a mass of 1M ⊙ that forms the embryo of a protoneutron star at the preceding stage of the collapse) and the silicon and carbon-oxygen shells with a total mass of (1.8-2.5)M ⊙ . 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 = (0.117-0.122)M ⊙ , and total angular momentum, J tot = (0.445-0.472) × 10 50 erg 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 (∼10 s). c 2003 MAIK "Nauka/Interperiodica".

Abstract-We analyze the behavior of the outer envelope in a massive star during and after the collapse of its iron core into a protoneutron star (PNS) in terms of the equations of one-dimensional spherically symmetric ideal hydrodynamics. The profiles obtained in the studies of the evolution of massive stars up to the final stages of their existence, immediately before a supernova explosion (Boyes et al. 1999), are used as the initial data for the distribution of thermodynamic quantities in the envelope. We use a complex equation of state for matter with allowances made for arbitrary electron degeneracy and relativity, the appearance of electron-positron pairs, the presence of radiation, and the possibility of iron nuclei dissociating into free nucleons and helium nuclei. We performed calculations with the help of a numerical scheme based on Godunov's method. These calculations allowed us to ascertain whether the emersion of the outer envelope in a massive star is possible through the following two mechanisms: first, the decrease in the gravitational mass of the central PNS through neutrino-signal emission and, second, the effect of hot nucleon bubbles, which are most likely formed in the PNS corona, on the envelope emersion. We show that the second mechanism is highly efficient in the range of acceptable masses of the nucleon bubbles (≤0.01M ⊙ ) simulated in our hydrodynamic calculations in a rough, spherically symmetric approximation.

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