Solar impact by a retrograde launch ($u_{\min }^2=v_{e,E}^2+v_E^2(1-\sqrt{2R_{\odot }/(a+R_{\odot })}{)}^2$)

In the circular-Earth-orbit, unpowered patched conic approximation, the least Earth-relative excess speed for direct solar-surface contact lowers the heliocentric tangential speed to that of an ellipse with apocentre $a$ and pericentre $R_{\odot }$. The vis-viva equation gives $v_a=v_E\sqrt{2R_{\odot }/(a+R_{\odot })}$, so the retrograde excess magnitude is $v_E-v_a$ and the terrestrial surface launch speed is the title expression. Cancelling all heliocentric angular momentum instead produces radial fall to a point Sun and the larger estimate $\sqrt{v_{e,E}^2+v_E^2}$. Rotation, planetary gravity assists and atmospheric drag are outside this direct-flight model.