2D MHD simulations of bow shock formation for WASP-189b under varying stellar wind conditions. Six combinations of stellar wind speed (υ_sw = 1−3υ_esc) and plasma density (ρ_atm or ρ_sw), with a stellar magnetic field of 79.05 G, are shown. Top: magnetic field strength (contours) and wind streamlines (yellow arrows); black circle marks the magnetopause stand-off distance (R_mp = 1.09–1.58 R_p). Bottom: fast-mode wavefronts and shock cones (brown), showing bow shock formation only for M_F > 1. Panels a and b demonstrate that high υ_sw combined with high ρ is required for a distinct bow shock.