A biologist inside a globular cluster

Adapted from IOAA 2012 (Rio de Janeiro), theory problem L1. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

An astronomer on Earth observes a globular cluster of angular diameter $\alpha$, in radians, at distance $D$. It holds $N$ stars, all of the same absolute magnitude $M_0$, spread evenly through a sphere. A biologist is at the centre of the cluster.

(a) What is the difference between the combined magnitude of all the stars as the biologist measures it and as the astronomer measures it?

(b) What diameter of telescope would the astronomer need to see the cluster with the same total brightness that the biologist sees with the naked eye? Take the pupil of the eye as 6 mm.

(c) What would the difference in part (a) be if the biologist saw only a field of angular diameter $\alpha$?

AnswerThe biologist sees the cluster $2.5{\log }_{10}(12/{\alpha }^2)$ magnitudes brighter than the astronomer does, a telescope of diameter $0.021/\alpha$ metres would make up the difference, and through a field of diameter $\alpha$ the biologist would see $-2.5{\log }_{10}(12{\sin }^2(\alpha /4)/{\alpha }^2)$, about 0.31 magnitudes fainter than the astronomer.