Isopoly models with super-radial expansion described by Eq. (9). The solutions embedding subsonic and supersonic regimes below 2.5 r_⊙ are displayed in black solid line and red solid lines, respectively. The left panels show isopoly solutions for the same initial parameters (T_p0 = 1.63 MK,  T_e0 = 0.71 MK,  r_iso|p = 13.6 r_⊙,  r_iso|e = 10.3 r_⊙,  γ_p = 1.52,  γ_e = 1.23), with varying expansion factor parameters (f_m, r_e, σ_e), and the right panels show the associated f(r) profiles; Panel a: Different values of f_m (maximum expansion factor obtained for large r); Panel b: Different values of r_e (radius at which the super-expansion almost stops); Panel c: Different values of σ_e (broadness of the expansion region). The f(r) profiles associated with panels a–c are displayed in panels d–f respectively. For all the f(r) parameters not displayed in the panel are set with (f_m = 20, r_e = 1.9 r_⊙, σ_e = 0.08 r_⊙). While PFSS is not used here, we still mark the region located below the source surface with the gray area as a guide for comparison.