Connecting the m-dots: accretion rates and thermonuclear burst recurrence times on neutron stars and white dwarfs

We present a compilation of observed recurrence times ($t_{\rm rec}$) and infer the corresponding local mass-accretion rates ($\dot m$) for type I X-ray bursts, milliHertz quasi-periodic oscillating sources and recurrent novae eruptions. We construct models of the $t_{\rm rec}-\dot m$ relation for accreting white dwarfs and neutron stars and find that both are roughly consistent with a global inverse linear relation, connecting for the first time thermonuclear runaways on neutron stars and white dwarfs. We find that theoretical models of pure He bursts are in agreement with the best $t_{\rm rec}$ measurements in ultra-compact X-ray binaries at low $\dot m$ (4U~$0614+09$ and 2S~0918-549). We suggest that the transient Z source XTE~J1701-462 is a slow rotator, based on its mHz QPO properties. Finally, we discuss the implications for thermonuclear ignition and point out that the difference in eruption/burst energy ($E_{b_{WD}}/E_{b_{NS}}=2\times 10^4$) is consistent with the difference in area between neutron stars and white dwarfs $\left((R_{WD}/R_{NS})^2=4\times 10^4\right)$. We conclude that ignitions of thermonuclear shell flashes on neutron stars and white dwarfs depend primarily on the specific mass accretion rate and do not depend on the nature of the underlying compact object.