Evolution of the First Supernovae in Protogalaxies: Dynamics of Mixing
  of Heavy Elements
  The paper considers the evolution of the supernova envelopes produced by
Population III stars with masses of $M_*\sim 25-200 M_\odot$ located in
non-rotating protogalaxies with masses of $M\sim 10^7 M_\odot$ at redshifts
$z=12$, with dark-matter density profiles in the form of modified isothermal
spheres. The supernova explosion occurs in the ionization zone formed by a
single parent star. The properties of the distribution of heavy elements
(metals) produced by the parent star are investigated, as well as the
efficiency with which they are mixed with the primordial gas in the supernova
envelope. In supernovae with high energies ($E\simgt 5\times 10^{52}$ erg), an
appreciable fraction of the gas can be ejected from the protogalaxy, but nearly
all the heavy elements remain in the protogalaxy. In explosions with lower
energies ($E\simlt 3\times 10^{52}$ erg), essentially no gas and heavy elements
are lost from the protogalaxy: during the first one to threemillion years, the
gas and heavy elements are actively carried from the central region of the
protogalaxy ($r\sim 0.1 r_{vir}$, where $r_{vir}$ is the virial radius of the
protogalaxy), but an appreciable fraction of the mass of metals subsequently
returns when the hot cavity cools and the envelope collapses. Supernovae with
high energies ($E\simgt 5\times 10^{52}$ erg) are characterized by a very low
efficiency of mixing of metals; their heavy elements are located in the small
volume occupied by the disrupted envelope (in a volume comparable with that of
the entire envelope), with most of the metals remaining inside the hot,
rarified cavity of the envelope. (abridged)
