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Cepheids are stars that vary in luminosity. The image below shows a cepheid in M100 as it varies in brightness over a period of days, (the cepheid is in the center of each frame).
The incredible importance of cepheids for measuring cosmological distances was discovered in 1912 by Henrietta Leavitt. She discovered that absolute magnitude of cepheids can be determined directly from the period of the variation in their brightness.
This relationship is important because it means that cepheids can be used as a "standard candle". All we have to do is measure the period of their luminosity changes, and we know their absolute magnitude, and then, once again, we can use the following equation to determine their distance from us:
Cepheids provide us the next step in the distance ladder because they are very bright (about 1000 times more luminous than the Sun), and they are found in distant galaxies; hence, they allow us to make that leap from distances within the Milky Way to the distances to remote galaxies.