In a star cluster there are 100 stars of the same brightness. Find the total magnitude of the cluster if the apparent brightness of one star is 8m.

The equality of the magnitudes of the stars of the cluster means the equality of the illumination that they create on the surface of the Earth.
Let us denote this illumination, identical for each of the stars, by E.
Then the total illumination created by all N stars of the cluster will be
E (N) = NE, N = 100, m = 8m.
From the definition of the magnitude, we find the total magnitude of the cluster
m (N) = m – 2.5 log [(E (N) / E] = m – 2.5 log N = 8 – 2.5 • 2 = 8m-5m = 3m

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