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

## TL;DR

This paper develops a semi-analytic criterion for the spontaneous initiation of carbon detonations in white dwarfs, based on temperature inhomogeneities, aiding understanding of Type Ia supernova progenitors.

## Contribution

It introduces an analytic expression for spontaneous wave speed and a criterion for hotspot size necessary to trigger detonations in white dwarf material.

## Key findings

- Detonations can spontaneously initiate under diverse conditions.
- Hotspots with specific temperature gradients can trigger detonations.
- The criterion helps evaluate different supernova progenitor scenarios.

## Abstract

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.