Formation of Extremely Low-mass White Dwarf Binaries

Motivated by the discovery of several pulsating, extremely low-mass white dwarfs (ELM WDs, mass M less than or similar to 0.18 M-circle dot) that likely have WD companions, this paper discusses binary formation models for these systems. ELM WDs are formed using angular momentum losses by magnetic braking. Evolutionary models are constructed using the Modules for Experiments in Stellar Astrophysics (MESA), with ELM WD progenitors in the range 1.0 <= (Md)/M-circle dot less than or similar to 1.5 and WD companions in the range 0.4 less than or similar to M-a/Me-circle dot less than or similar to 0.9. A prescription to reduce magnetic braking for thin surface convection zones is included. Upon the thinning of the evolved donor envelope, the donor star shrinks out of contact and mass transfer (MT) ceases, revealing the ELM WD. Systems with low masses have previously been suggested as possible AM CVNs. Systems with high masses, up to the limit M similar or equal to 0.18 M-circle dot at which shell flashes occur on the WD cooling track, tend to expand out to orbital periods P-orb greater than or similar to 15 hr. In between this range, ELM WDs may become pulsators both as pre-WDs 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. In the appendix, we show that the formation of an ELM WD by unstable MT or a common-envelope event is unlikely. Stable Roche-lobe overflow with conservative MT produces only M greater than or similar to 0.2 Me-circle dot.