Star formation in relic HII regions of the first stars is investigated using magneto-hydrodynamical simulations with a nested grid method that covers \sim 10 orders of magnitude in spatial scale and \sim 20 orders of magnitude in density contrast .
Due to larger fraction of H _ { 2 } and HD molecules , its prestellar thermal evolution is considerably different from that in the first star formation .
Reflecting the difference , two hydrostatic cores appear in a nested manner : a protostar is enclosed by a transient hydrostatic core , which appears during the prestellar collapse .
If the initial natal core rotates fast at a rate with rotational to gravitational energy ratio \beta _ { 0 } \gtrsim 0.01 - 0.1 , the transient hydrostatic core fragments to \sim 10 \thinspace M _ { \odot } sub-cores at density \sim 10 ^ { 9 } { cm } ^ { -3 } .
With smaller rotation energy , fragmentation occurs at higher density while a single protostar forms without fragmentation if rotation is extremely slow with \beta _ { 0 } \lesssim 10 ^ { -6 } -10 ^ { -5 } .
If magnetic field is present , these threshold values of \beta _ { 0 } is boosted owing to angular momentum transport by the magnetic breaking .
Magnetic field also drives the protostellar outflows .
With strong magnetic field , two distinct outflows are observed : The slower one emanates from the transient hydrostatic core , while the faster one from the protostar .
These flows may affect the final stellar mass by ejecting some of masses in the initial core , and also may play some role in driving and maintenance of interstellar turbulence in young galaxies .