material that is isotropically re-emitted, and from the donor star in the form of a fast isotropic wind), are: $$h_\mathrm{a} = \frac{m_\mathrm{d}}{m_\mathrm{a}} \frac{J}{m_\mathrm{d}+m_\mathrm{a}},    \hskip1cm              h_\mathrm{d} = \frac{m_\mathrm{a}}{m_\mathrm{d}} \frac{J}{m_\mathrm{d}+m_\mathrm{a}},$$ and the total loss of angular momentum from the binary system can be written as: $$\label{amloss}
  \frac{\dot J}{J} = \frac{\dot m_\mathrm{d} }{m_\mathrm{d}+m_\mathrm{a}} \left [\frac{m_\mathrm{d}}{m_\mathrm{a}} (1 - \delta) (1-\beta)  +  \frac{m_\mathrm{a}}{m_\mathrm{d}}\delta  \right ].$$

Combing eq.(\[oam\_der\]) and eq.(\[amloss\]), we find that the orbital evolution for our calculations of evolutionary sequences during RLOF is: $$\label{orbit}
\frac{\dot a}{a} = \frac{\dot m_\mathrm{d} }{m_\mathrm{d}} \,
\frac{(1-\delta)(2m_\mathrm{d}^2 - 2m_\mathrm{a}^2 +\beta m_\mathrm{d}m_\mathrm{a}) - m_\mathrm{d} m_\mathrm{a}}{m_\mathrm{a} (m_\mathrm{d}+m_\mathrm{a})}.$$ Note that in our calculations we only choose to fix the constant conservation factor $\beta$ for each individual binary evolution, but not $\delta$, which is always a function of the current evolutionary state and MT rate. During the initial MT phase, when MT often operates on a timescale close to thermal, and the MT rate is high, $\delta$ is effectively 0. The inclusion of $\delta$ becomes more important during the late stage of the MT, when the MT operates on the nuclear timescale (for more details, see §5).

Analysis of the $\gamma$-formalism {#sec:three}
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[@Nelemans00] analyzed the formation of DWD by attempting to reconstruct 