Helium Accumulation and Thermonuclear Instabilities on Accreting White Dwarfs: From Recurring Helium Novae to Type Ia Supernovae

We investigate helium accumulation on carbon-oxygen (CO) white dwarfs (WDs), exploring a broad parameter space of initial WD masses ($0.65$--$1.0M_{\odot}$) and helium accretion rates ($10^{-10}$--$10^{-4}M_{\odot}\text{yr}^{-1}$). Our simulations, which were allowed to run for up to the order of a Gyr, reveal distinct regimes determined by the given accretion rate: at higher rates ($\gtrsim10^{-5}M_\odot\rm yr^{-1}$), the mass is repelled by radiation pressure without accretion; intermediate rates ($\sim10^{-8}$--$10^{-5}M_{\odot}\text{yr}^{-1}$) produce periodically recurring helium nova eruptions, enabling gradual WD mass growth; and lower rates ($\lesssim 10^{-8}M_{\odot}\text{yr}^{-1}$) facilitate prolonged, uninterrupted helium accumulation, eventually triggering a thermonuclear runaway (TNR) which for some cases is at sub-Chandrasekhar masses, indicative of a type Ia supernova (SNe) ignition, i.e. providing a potential single-degenerate channel for sub-Chandra SNe. Our models indicate that the WD mass and the helium accumulation rate critically determine the ignition mass and TNR energetics. We identify compositional and thermal signatures characteristic of each regime, highlighting observational diagnostics relevant to helium-rich transients. We discuss these theoretical results in the context of the observed helium nova V445 Puppis, emphasizing helium accretion's pivotal role in shaping diverse thermonuclear phenomena.