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Despite over forty years of active research, the nature of the white dwarfprogenitors of Type Ia supernovae remains unclear. However, in the last decade,various progenitor scenarios have highlighted the need for detonations to bethe primary mechanism by which these white dwarfs are consumed, but it isunclear how these detonations are triggered. In this paper we study howdetonations 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 detonationinitiation in terms of the comparison between the spontaneous wave speed andthe Chapman-Jouguet speed. We develop an analytic expression for thespontaneous wave speed and utilize it to determine a semi-analytic criterionfor the minimum size of a hotspot with a linear temperature gradient between apeak and base temperature for which detonations in burning carbon-oxygenmaterial can occur. Our results suggest that spontaneous detonations may easilyform under a diverse range of conditions, likely allowing a number ofprogenitor scenarios to initiate detonations that burn up the star.

A Semi-analytic Criterion for the Spontaneous Initiation of Carbon Detonations in White Dwarfs