Possible evolution channels shown on the critical mass ratio vs. the initial stellar radius diagram (lower panel) and the Hertzsprung–Russell diagram (upper panel) for binary systems containing a 3.2 M_⊙ donor star. The lower panel shows how the critical initial mass ratios change with the donor star’s radius. The radius is increasing with evolution and the colored regions correspond to the onset of mass transfer at that radius for the given mass ratio. Three different critical mass ratios, q_ad, q_th, and ﹩{q}_{{{\rm{L}}}_{2}}﹩, are used to identify the boundaries among the possible evolutionary channels, dynamical-timescale mass transfer, thermal-timescale mass transfer, and overfilling of the outer Lagrangian surface L₂. In the lower panel, different filling colors indicate what kind of mass-transfer channel the donor star may undergo for the initial mass ratio in that range. The thick red dashed line and thick solid blue line show the more effective constraint leading to a common-envelope phase. For clarity, we omit the stellar models at evolutionary phases in which the donor star’s radius is smaller than in a preceding phase, such as during core helium burning, or just beyond the terminal main sequence. The red squares in the upper panel indicate the possible evolutionary channels of the MS and HG donors. These correspond to the thick red dashed line on the left of the lower panel. The blue triangles show the possible evolutionary channels of the RGB and early AGB donors. These correspond to the thick solid blue line in the middle of the lower panel. The red stars in the upper panel reveal the possible evolutionary channels of the late AGB donors, and they correspond to the thick red dashed line on the right of the lower panel.