A parabola from Earth to Mars

Adapted from IOAA 2010 (Beijing), theory problem 16. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

A spacecraft leaves Earth and is quickly brought to the speed, along the direction of Earth's motion round the Sun, that puts it on a parabola with the Sun at its focus. The parabola touches Earth's orbit. Take the orbits of Earth and Mars as circles in the same plane, of radius 1 au and 1.5 au, and count only the Sun's gravity for most of the flight.

(a) What angle $\psi$ does the spacecraft's path make with the orbit of Mars where it crosses it, ignoring the gravity of Mars? [10 points]

(b) Suppose Mars is very close to the crossing point when the spacecraft gets there. As seen from Mars, before the spacecraft is much affected by Mars's gravity, how fast is the spacecraft approaching, and from what direction relative to the Sun? [20 points]

Answer(a) $\psi =35.26°$. (b) The spacecraft approaches at 20.21 km/s. Seen from Mars it comes from the side of the Sun, 10.7° from the direction of the Sun, on the side from which Mars is moving away.