Extremely low-mass white dwarfs (ELM WDs) are helium WDs with a mass less than $sim$$0.3rm;M_odot$. Most ELM WDs are found in double degenerates (DDs) in the ELM Survey led by Brown and Kilic. These systems are supposed to be significant gravitational-wave sources in the mHz frequency. In this paper, we firstly analyzed the observational characteristics of ELM WDs and found that there are two distinct groups in the ELM WD mass and orbital period plane, indicating two different formation scenarios of such objects, i.e. a stable Roche lobe overflow channel (RL channel) and common envelope ejection channel (CE channel). We then systematically investigated the formation of ELM WDs in DDs by a combination of detailed binary evolution calculation and binary population synthesis. Our study shows that the majority of ELM WDs with mass less than $0.22rm;M_odot$ are formed from the RL channel. The most common progenitor mass in this way is in the range of $1.15-1.45rm;M_odot$ and the resulting ELM WDs have a peak around $0.18rm;M_odot$ when selection effects are taken into account, consistent with observations. The ELM WDs with a mass larger than $0.22rm;M_odot$ are more likely to be from the CE channel and have a peak of ELM WD mass around $0.25rm;M_odot$ which needs to be confirmed by future observations. By assuming a constant star formation rate of 2$rm;M_odot yr^{-1}$ for a Milky Way-like galaxy, the birth rate and local density are $5times10^{-4}rm;yr^{-1}$ and $1500rm;kpc^{-3}$, respectively, for DDs with an ELM WD mass less than $0.25rm;M_odot$.
Download