All stars, late in their lifetime, change from being main sequence stars to becoming swollen red giants. Most stars change from the swollen red giant phase to pulsating variable stars before they finally die. These are Cepheid Variables, which expand and contract, glowing brightly and fading with a period – between 1 and 70 days - which depends on their luminosity.
This means we can work out how far they are. We can measure their period quite easily, and knowing the relationship between period and luminosity, illustrated in the following graph, we can estimate their luminosity
then use
where
is the distance to the star and
is the intensity oof the light from the star received on Earth to find the distance d to the star.
Because cepheid variables are red giants – very luminous – they can be used to estimate quite large distances.