Formation of Extremely Low-mass White Dwarfs Binaries

Motivated by the discovery of a handful of pulsating, extremely low mass white dwarfs (ELM WDs, mass $M \lesssim 0.17\, M_\odot$) which likely have WD companions, this paper discusses binary formation models for these systems. Formation of an ELM WD by unstable mass transfer (MT) or a common envelope (CE) event is unlikely. Stable Roche-lobe overflow with conservative MT produces only $M \gtrsim 0.2\, M_\odot$. This paper discusses the formation of ELM WD using angular momentum losses by magnetic braking. The initially more massive star becomes the companion WD through a CE event. The initially less massive star becomes the ELM WD as an evolved donor in a Cataclysmic Variable binary. Evolutionary models are constructed using the Modules for Experiments in Stellar Astrophysics (MESA), with ELM WD progenitors in the range $M_d=1.0-1.5\, M_\odot$ and WD companions in the range $M_{\rm a} \simeq 0.4-0.9\, M_\odot$. A prescription to reduce magnetic braking for thin surface convection zones is included. Upon the thinning of the evolved donor's envelope, the donor star shrinks out of contact and MT ceases, revealing the ELM WD. Systems with small masses have previously been suggested as possible AM CVN's. Systems with large masses, up to the limit $M \simeq 0.18\, M_\odot$ at which shell flashes occur on the WD cooling track, will tend to expand out to orbital periods $P_{\rm orb} \gtrsim 15\, {\rm hr}$. In between this range, ELM WD may become pulsators both as pre-WD and on the WD cooling track. Brickhill's criterion for convective mode driving is used to estimate the location of the blue edge of the g-mode instability strip.