  Our understanding of population III star formation is still in its infancy.
They are formed in dark matter minihalos of $\rm 10^5-10^6 M_{\odot}$ at
$z=20-30$. Recent high resolution cosmological simulations show that a
protostellar disk forms as a consequence of gravitational collapse and
fragments into multiple clumps. However, it is not entirely clear if these
clumps will be able to survive to form multiple stars as simulations are unable
to follow the disk evolution for longer times. In this study, we employ a
simple analytical model to derive the properties of marginally stable
steady-state disks. Our results show that the stability of the disk depends on
the critical value of the viscous parameter $\alpha$. For $\alpha_{crit} = 1$,
the disk is stable for an accretion rate of $\rm \leq 10^{-3} M_{\odot}/yr$ and
becomes unstable at radii about $\rm \geq 100 AU$ in the presence of an
accretion rate of $\rm 10^{-2} M_{\odot}/yr$. For $0.06 < \alpha_{crit} < 1$,
the disk can be unstable for both accretion rates. The comparison of the
migration and the Kelvin-Helmholtz time scales shows that clumps are expected
to migrate inward before reaching the main sequence. Furthermore, in the
presence of a massive central star the clumps within the central 1 AU will be
tidally disrupted. We also find that UV feedback from the central star is
unable to disrupt the disk, and that photo-evaporation becomes important only
once the accretion rate has dropped to $\rm 2 \times 10^{-4} M_{\odot}/yr$. As
a result, the central star may reach a mass of 100 $\rm M_{\odot}$ or even
higher.
