2022

A journey between galaxies

Round: Theory
Points: 10
Level: Silver
Taught in: Section 13.6

Adapted from IOAA 2022 (Kutaisi), theory problem 4. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

We set out for a planet that is $d_{0} = 10$ Mpc away when we leave. The Universe keeps expanding by the Hubble–Lemaître law, and the Hubble constant $H = 70$ km/s/Mpc does not change.

(a) Write the distance between Earth and the planet at a time $t$ after we leave. [2 points]

(b) What is the smallest constant speed $v_{0}$ of the rocket, relative to the space it passes through, that will reach the planet? Can a rocket at $v_{0} = 1000$ km/s reach it? If so, how long does the journey take? [8 points]

I solved it without the solution

Answer(a) $d = d_{0} e^{H t}$ . (b) The rocket needs more than 700 km/s, so yes, at 1000 km/s it arrives after about 16.8 billion years. The official solution misprints the smallest speed as 70 km/s and the time as $5.3 \times 10^{16}$ s, though its 17 billion years is right.

This problem also appears among the olympiad problems of Section 13.6, Distances in an expanding universe, which teaches the ideas it uses. Equation and figure numbers refer to the course.