Modelling the AM CVn and Double Detonation Supernova Progenitor Binary System CD-30$^{\circ}$11223

We present a detailed modelling study of CD-30$^{\circ}$11223 (CD-30), a hot subdwarf (sdB)-white dwarf (WD) binary identified as a double detonation supernova progenitor, using the open-source stellar evolution software MESA. We focus on implementing binary evolution models carefully tuned to match the observed characteristics of the system including $\log g$ and $T_{\rm eff}$. For the first time, we account for the structure of the hydrogen envelope throughout the modelling, and find that the inclusion of element diffusion is important for matching the observed radius and temperature. We investigate the two sdB mass solutions (0.47 and 0.54 $M_{\odot}$) previously proposed for this system, strongly favouring the 0.47 $M_{\odot}$ solution. The WD cooling age is compared against the sdB age using our models, which suggest an sdB likely older than the WD, contrary to the standard assumption for compact sdB-WD binaries. Subsequently, we propose a possible alternate formation channel for CD-30. We also perform binary evolution modelling of the system to study various aspects such as mass transfer, orbital period evolution and luminosity evolution. Our models confirm CD-30 as a double detonation supernova progenitor, expected to explode $\approx55$ Myr from now. The WD accretes a $\approx0.17$ $M_{\odot}$ thick helium shell that causes a detonation, leaving a 0.30 $M_{\odot}$ sdB ejected at $\approx$750 km/s. The final 15 Myr of the system are characterised by helium accretion which dominates the system luminosity, possibly resembling an AM CVn-type system.