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Models of low-mass helium white dwarfs including gravitational settling, thermal and chemical diffusion, and rotational mixing

A large number of extremely low-mass helium white dwarfs (ELM WDs) have been discovered in recent years. The majority of them are found in close binary systems suggesting they are formed either through a common-envelope phase or via stable mass transfer in a low-mass X-ray binary (LMXB) or a cataclysmic variable (CV) system. Here, we investigate the formation of these objects through the LMXB channel with emphasis on the proto-WD evolution in environments with different metallicities. We study, for the first time, the combined effects of rotational mixing and element diffusion (e.g. gravitational settling, thermal and chemical diffusion) on the evolution of proto-WDs and on the cooling properties of the resulting WDs. We present state-of-the-art binary stellar evolution models computed with MESA for metallicities between Z=0.0002 and Z=0.02, producing WDs with masses between 0.16-0.45 M$_{\odot}$. Our results confirm that element diffusion plays a significant role in the evolution of proto-WDs that experience hydrogen shell flashes. The occurrence of these flashes produces a clear dichotomy in the cooling timescales of ELM WDs, which has important consequences e.g. for the age determination of binary millisecond pulsars. Rotational mixing is found to counteract the effect of gravitational settling in the surface layers of young, bloated ELM proto-WDs and therefore plays a key role in determining their surface chemical abundances. We predict that these proto-WDs have helium-rich envelopes through a significant part of their lifetime, a crucial ingredient for understanding the newly observed ELM proto-WD pulsators. The hydrogen envelope at detachment, although small compared to the total mass of the WD, contains enough angular momentum such that the spin frequency of the resulting WD on the cooling track is well above the orbital frequency.