Pushing an asteroid's moon

Adapted from IOAA 2023 (Katowice), theory problem 12. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The DART spacecraft struck Dimorphos, a small moon of the asteroid Didymos, to test how an impact changes its orbit. Before the impact Dimorphos moved on a circular orbit round Didymos with a period of 11.92 h. The masses of Dimorphos and Didymos are $m=4.3\times {10}^9$ kg and $M=5.6\times {10}^{11}$ kg. At impact the spacecraft had mass $m_{\text{s}}=580$ kg and moved at $v_{\text{s}}=6.1$ km/s relative to Dimorphos. Ignore all other bodies.

(a) Find the expected change of the period, in minutes, if the collision was head-on, central and perfectly inelastic, so that the spacecraft stayed in Dimorphos. [20 points]

(b) The period was in fact observed to change by $\mathrm{\Delta }{P}_0=-33$ minutes, because the impact threw debris off Dimorphos, which carried momentum away. Find the momentum of the debris as a fraction of the momentum of Dimorphos before the impact. Take the mass of the debris to be much smaller than Dimorphos's. [15 points]

(c) Find the change of Dimorphos's speed, in mm/s, including the effect of the debris. [5 points]

Answer(a) The period should fall by about 10.0 minutes. (b) The debris carried away a momentum of 0.0107, about 1.1 per cent, of Dimorphos's momentum, more than twice the spacecraft's own. (c) Dimorphos slowed by 2.71 mm/s, 3.3 times as much as the spacecraft alone would have slowed it.