Patched conic approximation

The patched conic approximation models different portions of a spacecraft trajectory by separate two-body Kepler orbits, matching position and velocity where one attracting body gives way to another. For a surface departure from a nonrotating planet, conservation of energy gives $u^2=v_e^2+w^2$, where $u$ is surface launch speed, $v_e$ is the planet's surface escape speed, and $w$ is the planet-relative hyperbolic excess speed. Adding that excess velocity vector to the planet's orbital velocity supplies the heliocentric initial condition. It neglects the simultaneous full gravitational action of both bodies near the matching region.

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.