The relative importance of metals and dust grains in the formation of the first low-mass stars has been a subject of debate .
The recently discovered Galactic halo star SDSS J102915+172927 ( Caffau et al .
2011 ) has a mass less than 0.8 ~ { } M _ { \odot } and a metallicity of Z = 4.5 \times 10 ^ { -5 } Z _ { \odot } .
We investigate the origin and properties of this star by reconstructing the physical conditions in its birth cloud .
We show that the observed elemental abundance trend of SDSS J102915+172927 can be well fitted by the yields of core-collapse supernovae with metal-free progenitors of 20 ~ { } M _ { \odot } and 35 ~ { } M _ { \odot } .
Using these selected supernova explosion models , we compute the corresponding dust yields and the resulting dust depletion factor taking into account the partial destruction by the supernova reverse shock .
We then follow the collapse and fragmentation of a star forming cloud enriched by the products of these SN explosions at the observed metallicity of SDSS J102915+172927 .
We find that [ 0.05 - 0.1 ] ~ { } M _ { \odot } mass fragments , which then lead to the formation of low-mass stars , can occur provided that the mass fraction of dust grains in the birth cloud exceeds 0.01 of the total mass of metals and dust .
This , in turn , requires that at least 0.4 ~ { } M _ { \odot } of dust condense in the first supernovae , allowing for moderate destruction by the reverse shock .
If dust formation in the first supernovae is less efficient or strong dust destruction does occur , the thermal evolution of the SDSS J102915+172927 birth cloud is dominated by molecular cooling , and only \geq 8 ~ { } M _ { \odot } fragments can form .
We conclude that the observed properties of SDSS J102915+172927 support the suggestion that dust must have condensed in the ejecta of the first supernovae and played a fundamental role in the formation of the first low-mass stars .