We use what is called the Distance Ladder which has its roots in the measure of the distance from the Earth to the Sun, and by using the properties of various types of stars as 'standard candles' gets us out to several 100 million parsecs.
For nearby galaxies where you can see individual luminous stars called Cepheid Variables we know from studies of these stars in the Milky Way that their light variation periods are related very accurately to their luminosities; the number of ergs per second of light they produce. Knowing just how bright they appear to us in the sky, we can figure out very accurately how far away they are, and how far away their host galaxies are.
For galaxies so distant that individual stars cannot be seen, astronomers such as Alan Sandage have been using supernova. There are several different classes of supernova, so you first have to establish which class the particular supernova is in by its 'light curve' that is the history of it's brightness change over a period up to 1 year after eruption. Supernova can be seen out to a billion parsecs, and if you believe that supernova of a certain class have a typical, maximum luminosity, you can again determine how far away they are from their apparent brightness in the sky. Currently, Alan Sandage using supernova and Wendy Freedman using Cepheids find a serious difference in the far distance ladder, one favoring a smaller universe and a faster expansion rate than the other. Only time will tell what the basis for the discrepancy is.
Copyright 1997 Dr. Sten Odenwald
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