Flux-tube cross-section area change, density and velocity profile for three examples of streamlines. Panel a shows the course of the streamlines (red, green, and blue) with the corresponding shock frame of references (normal and tangent vectors ft and î, respectively) used for the Rankine–Hugoniot relations. Panel b shows the flux-tube cross-section area change along the streamlines. Panels c and d show the 1-D curves of the density and velocity profiles along the three streamlines (red, green, and blue) from the bow shock to X_MSM = −4R_M, respectively. Again, the average solar wind parameters during the entire MESSENGER service between 2011 and 2015 are used with an Alvén Mach number of M_A = 5.8, ram pressure of P_ram = 10.7 nPa, and plasma beta of β = 0.72. The gray shaded area indicates the region where the flux tube becomes extremely compressed around the magnetopause (A/A_BS ≤ 0.75), and Eq. (21) does not yield a solution.