Matter and motion in our neighbourhood

Adapted from IOAA 2013 (Volos), theory problem 1. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

In a uniform universe the density parameter of matter, ordinary and dark together, is ${\mathrm{\Omega }}_{\text{m}}={\rho }_{\text{m}}/{\rho }_{\text{c}}=0.32$. Take $H_0=67.8$ km/s/Mpc.

(1) Find the average density of matter in our neighbourhood. [16 points]

(2) Find the escape speed of a galaxy 100 Mpc away from the matter inside that distance, taking the matter density of (1). [14 points]

(3) Suppose that this galaxy moved on a circular orbit round the centre of our cluster of galaxies, with us at that centre. What would its angular speed on the sky be? [12 points]

(4) Two such galaxies start on the same line of sight but orbit at different radii. Could we ever see them separate? Answer yes or no. [8 points]

Answer(1) $2.76\times {10}^{-27}$ kg/m³. (2) 3835 km/s. (3) $8.8\times {10}^{-19}$ rad/s, the same at every distance. (4) No. In a uniform sphere every circular orbit has the same period, so the two galaxies stay lined up.