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Appendix A: Alternative scalings for the optical depth

Alternative scalings to those adopted in Sect. 3.3 can be considered. In our hypotheses, the total volume V is independent of the orbit semimajor axis a because the size of the region enclosing the relative minimum of the gravitational potential along the magnetic field lines scales with the radius of the star R_∗, given that the boundary conditions controlling the field geometry are fixed at the surface of the star. Therefore, we expect that in general the vertical extension of the slab depends on R_∗ rather than the distance a of the planet. In other words, the volume V should be proportional to although the coefficient of proportionality is not necessarily that adopted above, that is, 4π×2.5, which corresponds to the smallest vertical extension of the slab ~ 2 R_∗.

The orbit semimajor axis a controls the evaporation rate Ṁ and the timescale t_a for the filling of the potential well along the field lines. With the scaling adopted above, these two dependences cancel each other in the final expression for the mean density n, but other scalings for t_a are plausible and generally lead to a dependence of n on a.

The shortest possible timescale corresponds to the free-fall time in the case of negligible pressure gradients and Lorentz force, and neglecting the ram pressure of the stellar wind. In this case, , where M_∗ is the mass of the star, and the optical depth appearing in Eq. (1) becomes(A.1)Conversely, the longest timescale is that given by the diffusion process considered in Sect. 3.3. An intermediate possibility is that of a syphon flow along the magnetic field lines that directly carries the evaporated matter into the potential well. The typical velocity of that flow is of the order of the sound speed c_s (cf. the evaporation model in the presence of planetary and stellar magnetic fields by Adams 2011), thus t_a ~ a/c_s. With that scaling, we find(A.2)where c_s is constant in the case of a steady isothermal flow.

© ESO, 2014