It's not so easy to figure out the distance to far away objects in the sky - see an older post on the cosmic distance ladder. However, it's possible for even beginning astronomy students to climb the First Three Rungs of the Cosmological Distance Ladder. The top of the ladder isn't so easy though. The 2011 Nobel Prize in Physics was awarded for "for the discovery of the accelerating expansion of the Universe through observations of distant supernovae". A key element of that discovery was the use of Type Ia supernova to determine distances. Type Ia supernova are supposed to the explosion of a white dwarf star, conveniently it seems to be possible to determine the absolute luminosity of those explosions, which indicates the actual distance.
The preprint Type Ia Supernovae and the discovery of the Cosmic Acceleration is a nice explanation of the history of this discovery by Alejandro Clocchiatti a member of one of the groups which made the discovery - the High-z Supernova Search Team.
Kepler's supernova
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