First supernovae in dwarf protogalaxies

(Abridged) We explore numerically the chemical, thermal, and dynamical evolution of a shell formed by a high-energy supernova explosion ($10^{53}$ erg) in dwarf protogalaxies with total mass $10^7 M_\odot$ at a redshift $z=12$. We consider two initial configurations for the baryonic matter, one without rotation and the other having the ratio of rotational to gravitational energy $\beta=0.17$. The (non-rotating) dark matter halo is described by a quasi-isothermal sphere. We find that the dynamics of the shell is different in protogalaxies with and without rotation. For instance, the Rayleigh-Taylor instability in the shell develops faster in protogalaxies without rotation. The fraction of a blown-away baryonic mass is approximately twice as large in models with rotation ($\sim 20$) than in models without rotation. On the other hand, the chemical evolution of gas in protogalaxies with and without rotation is found to be similar. The relative number densities of molecular hydrogen and HD molecules in the cold gas ($T \le 10^3$ K) saturate at typical values of $10^{-3}$ and $10^{-7}$, respectively. The clumps formed in the fragmented shell move with velocities that are at least twice as large as the escape velocity. The mass of the clumps is $\sim 0.1-10 \msun$, which is lower than the Jeans mass. We conclude that the clumps are pressure supported. A supernova explosion with energy $10^{53}$ ergs destructs our model protogalaxy. The clumps formed in the fragmented shell are pressure supported. We conclude that protogalaxies with total mass $\sim 10^{7} M_\odot$ are unlikely to form stars due to high-energy supernova explosions of the first stars.