Sonic-point ratio of Prad/Pgas. Different contributions to the ratio of Prad/Pgas at the sonic point, as imposed by the outer wind, are indicated on a logarithmic scale. The left panel shows the same sequence of wind models as in Fig. 3, i.e., models with a fixed terminal wind velocity of ν∞ = 1800 km s-1 as a function of the adopted mass-loss rate Ṁ. The right panel shows the same model sequence as in Fig. 5, i.e., models for which Ṁ is derived consistently with the critical condition (Eq. (3)) as a function of ν∞. We plot the ratio of Prad/Pgas that follows from our numerical models (solid blue line), from the approximate relation Eq. (25) (black dashed line), and Eq. (31) (red circles). The grey dotted and dash-dotted lines indicate the (weighted) sonic-point optical depths τs and respectively. The red solid line indicates the ratio τs/η, and the red dashed line the factor . The black solid line indicates the offset factor in Eq. (31).
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