Which orbits can one kick reach?

Adapted from IOAA 2011 (Katowice), theory problem 7. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

A satellite moves on a circular orbit round Earth with momentum $\mathbf{p}$. An explosive charge gives it an extra impulse $\mathrm{\Delta }\mathbf{p}$ equal in size to $\mathbf{p}$. Let $\alpha$ be the angle between $\mathbf{p}$ and $\mathrm{\Delta }\mathbf{p}$, and $\beta$ the angle between the satellite's position vector from Earth's centre and $\mathrm{\Delta }\mathbf{p}$. For each orbit below, say whether it can be reached, and if so give values of $\alpha$ and $\beta$ that reach it. For example, $\alpha =180°$ and $\beta =90°$ leave the satellite at rest, so that it falls straight down.

(a) A hyperbola with its perigee at the point of the explosion.

(b) A parabola with its perigee at the point of the explosion.

(c) An ellipse with its perigee at the point of the explosion.

(d) A circle.

(e) An ellipse with its apogee at the point of the explosion.

AnswerAll five are possible, with $\beta =90°$ and (a) $\alpha$ less than 90°, (b) $\alpha =90°$, (c) $\alpha$ between 90° and 120°, (d) $\alpha =120°$, (e) $\alpha$ between 120° and 180°. For any $\alpha$ other than 0° and 180° the kick has a part out of the plane of the old orbit, so the new orbit is tilted.