2008

The distance to the Virgo cluster

Round: Data analysis
Points: 100
Level: Bronze
Taught in: Section D7

Adapted from IOAA 2008 (Bandung), data-analysis problem 1. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

The Virgo cluster, the nearest large cluster of galaxies, spreads over about 10° of the sky. The table gives its distance $d_{i}$ as found by eight different methods, each with its uncertainty $s_{i}$ .

Method	Distance (Mpc)
Cepheids	14.9 ± 1.2
Novae	21.1 ± 3.9
Planetary nebulae	15.2 ± 1.1
Globular clusters	18.8 ± 3.8
Surface brightness fluctuations	15.9 ± 0.9
Tully–Fisher relation	15.8 ± 1.5
Faber–Jackson relation	16.8 ± 2.4
Type Ia supernovae	19.4 ± 5.0

(a) Find the weighted mean of the distances, with weights $1 / s_{i}^{2}$ . This was worth 30 marks. (b) Find its uncertainty. This was worth 10 marks. (c) The galaxies of the cluster recede from us at 1136 km/s on average. Estimate the Hubble constant and its uncertainty. This was worth 35 marks. (d) Find the Hubble time $1 / H_{0}$ and its uncertainty. This was worth 25 marks. Use 1 Mpc = $3.086 \times 10^{19}$ km.

I solved it without the solution

AnswerThe distance to the Virgo cluster is 15.8 ± 0.5 Mpc, the Hubble constant is 72 ± 2 km/s per Mpc and the Hubble time is 13.6 ± 0.5 Gyr. The official solution finds 15.81 Mpc, because its sum of $d_{i} / s_{i}^{2}$ comes out 0.17 too large, and the same Hubble constant and Hubble time.

This problem also appears among the olympiad problems of Section D7, Statistics of a sample, which teaches the ideas it uses. Equation and figure numbers refer to the course.