  Despite over forty years of active research, the nature of the white dwarf
progenitors of Type Ia supernovae remains unclear. However, in the last decade,
various progenitor scenarios have highlighted the need for detonations to be
the primary mechanism by which these white dwarfs are consumed, but it is
unclear how these detonations are triggered. In this paper we study how
detonations are spontaneously initiated due to temperature inhomogeneities,
e.g., hotspots, in burning nuclear fuel in a simplified physical scenario.
Following the earlier work by Zel'Dovich, we describe the physics of detonation
initiation in terms of the comparison between the spontaneous wave speed and
the Chapman-Jouguet speed. We develop an analytic expression for the
spontaneous wave speed and utilize it to determine a semi-analytic criterion
for the minimum size of a hotspot with a linear temperature gradient between a
peak and base temperature for which detonations in burning carbon-oxygen
material can occur. Our results suggest that spontaneous detonations may easily
form under a diverse range of conditions, likely allowing a number of
progenitor scenarios to initiate detonations that burn up the star.
