  The first stars to form in the universe may have been dark stars, powered by
dark matter annihilation instead of nuclear fusion. The initial amount of dark
matter gathered by the star gravitationally can sustain it only for a limited
period of time. It has been suggested that capture of additional dark matter
from the environment can prolong the dark star phase even to the present day.
Here we show that this capture process is ineffective to prolong the life of
the first generation of dark stars. We construct a Monte-Carlo simulation that
follows each Weakly Interacting Massive Particle (WIMP) in the dark matter halo
as its orbit responds to the formation and evolution of the dark star, as it
scatters off the star's nuclei, and as it annihilates inside the star. A rapid
depletion of the WIMPs on orbits that cross the star causes the demise of the
first generation of dark stars. We suggest that a second generation of dark
stars may in principle survive much longer through capture. We comment on the
effect of relaxing our assumptions.
