= Solar impact by a retrograde launch
{title2=$u_{\min}^2=v_{e,E}^2+v_E^2(1-\sqrt{2R_\odot/(a+R_\odot)})^2$}
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.
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