How can astronomers use variable to find distance?

Answer 1

Cepheid variables have a predictable relationship between the period of variability and their absolute brightness. Astronomers can calculate how far away they are by comparing the apparent brightness (as measured in the sky) to the absolute brightness (as calculated from the period). In the 1920's, Edwin Hubble was able to measure Cepheid variables in the Andromeda galaxy, and establish that it really was another galaxy like the Milky Way.

Answer 2

Initially, astronomers made observations by other methods that allowed them to work out the relationship in the first place:

1. Use parallax to work out the distance to a number of variable stars
2. Use the apparent magnitude to work out the absolute magnitude (ie how bright the star really is).
3. Realise that for a certain class of variable stars, there is a direct relationship between the period of variability and the absolute magnitude

Now, we can observe a Cepheid variable and, based on its period we can work out its absolute magnitude, and from its apparent magnitude, we can work out how far away it is.