Fast-mode magnetosonic Mach number under WASP-189b-like stellar wind conditions. The colour map shows M_F − 1 as a function of log₁₀(ρ) and υ_sw, for a stellar magnetic field of 79.05 G. White contours mark the M_F = 1 boundary. Vertical magenta lines indicate estimated planetary upper atmosphere and stellar wind densities; horizontal dashed lines denote wind speeds of 1−3υ_esc. Regions where M_F > 1 are conducive to bow shock formation.