An asteroid that falls into a star

Adapted from IOAA 2024 (Rio de Janeiro), theory problem 3. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

An asteroid of mass $m$ is seen at a distance $d$ from a star of mass $M$, moving at speed $v=\sqrt{GM/d}$. The distance $d$ is much larger than the radius of the star. Give each answer in terms of $M$, $d$ and constants.

(a) If the asteroid is moving straight towards the star, how long will it take to hit it? [8 points]

(b) If it is instead moving straight away from the star, how long will it take to hit it? [2 points]

Answer(a) $t=(\pi /2-1)\sqrt{d^3/GM}$. (b) $t=(3\pi /2+1)\sqrt{d^3/GM}$.