We show that the explosion of the first supernovae can trigger low-mass star formation via gravitational fragmentation of the supernova-driven gas shell .
If the shell mass does not exceed the host galaxy gas mass , all explosions with energies E _ { SN } \geq 10 ^ { 51 } erg can lead to shell fragmentation .
However , the minimum ambient density required to induce such fragmentation is much larger , n _ { 0 } > 300 \mbox { cm } ^ { -3 } , for Type II supernovae than for pair-instability ones , which can induce star formation even at lower ambient densities .
The typical mass of the unstable fragments is \sim 10 ^ { 4 - 7 } M _ { \odot } ; their density is in the range 110 - 6 \times 10 ^ { 7 } \mbox { cm } ^ { -3 } .
Fragments have a metallicity strictly lower than 10 ^ { -2.6 } Z _ { \odot } and large values of the gravitational-to-pressure force ratio f \simeq 8 .
Based on these findings , we conclude that the second generation of stars produced by such self-propagating star formation is predominantly constituted by low-mass , long-living , extremely metal-poor ( or even metal-free , if mixing is suppressed ) stars .
We discuss the implications of such results for Pop III star formation scenarios and for the most iron-poor halo star HE0107-5240 .