The evolution of a gas shell , swept by the supernova remnant of a massive first generation star , is studied with H _ { 2 } and HD chemistry taken into account and with use of a semi-analytical approximation to the dynamics .
When a first-generation star , formed in a parent pregalactic cloud , explodes as a supernova with explosion energy in the range of 10 ^ { 51 } { erg } -10 ^ { 52 } erg at redshifts of z = 10 - 50 , H _ { 2 } and HD molecules are formed in the swept gas shell at fractional abundances of \sim 10 ^ { -3 } and \sim 10 ^ { -5 } , respectively , and effectively cool the gas shell to temperatures of 32 { K } -154 K. If the supernova remnant can sweep to gather the ambient gas of mass 6 \times 10 ^ { 4 } \thinspace M _ { \odot } -8 \times 10 ^ { 5 } M _ { \odot } , the gas shell comes to be dominated by its self-gravity , and hence , is expected to fragment .
The amount of swept gas necessary for fragmentation increases with the explosion energy and decreases with the interstellar gas density ( or redshift ) of the host cloud , which provides a lower boundary to the mass of the host cloud in which star formation is triggered by the first-generation supernova .
Also , the condition for fragmentation is very sensitive to the thermal state of interstellar gas .
Our result shows that for a reasonable range of temperatures ( 200 { K } \sim 1000 K ) of interstellar gas , the formation of second-generation stars can be triggered by a single supernova or hypernova with explosion energy in the above range , in a primordial cloud of total ( the dark and baryonic ) mass as low as \hbox { a few } \times 10 ^ { 6 } \thinspace M _ { \odot } .
For higher temperature in the interstellar gas , however , the condition for the fragmentation in the swept gas shell demands a larger supernova explosion energy .

We also follow the subsequent contraction of the fragment pieces assuming their geometry ( sphere and cylinder ) , and demonstrate that the Jeans masses in the fragments decrease to well below a solar mass by the time the fragments become optically thick to the H _ { 2 } and HD lines .
The fragments are then expected to break up into dense cores whose masses are comparable to the Jeans masses and collapse to form low mass stars that can survive to date .
If the material in the gas shell is mixed well with the ejecta of the supernova , the shell and low-mass stars thus formed are likely to have metals of abundance [ { Fe } / { H } ] \simeq - 3 on average .
This metallicity is consistent with those of the extremely metal-poor stars found in the Galactic halo .
Stars with such low metallicities of [ { Fe } / { H } ] < -5 as HE0107-5240 , recently discovered in the Galactic halo , are difficult to form by this mechanism , and must be produced in different situations .