Spatial Offsets in Bow Shocks

Spatial offsets of high radial velocity material in paraboloid bow shocks. Top: A paraboloid viewed from the side is shown, where the thin arrows indicate the velocity of the material immediately behind the bow shock. For a viewing angle phi, material along the projected apex of the bow moves tangentially to the line of sight, so the maximum negative radial velocity appears at some point interior to the bow. Bottom: The paraboloid appears as a fatter parabola when projected onto the plane of the sky. The intersection of the line x' = z' with the parabola defines a characteristic size h of the bow. For arbitrary viewing angles and arbitrary paraboloids, as long as the cooling distance behind the shock is small compared with h, the location of largest radial velocity V_max (the maximum redshift for a redshifted bow or the maximum blueshift for a blueshifted bow) is always offset from the apex of the projection of the paraboloid on the plane of the sky by an amount d = 0.25 h. This result is very useful for testing the applicability of a bow shock model when velocity-resolved images are available.