## How far away is that star?

On a clear night, far away from the city lights, one can look up and enjoy the beauty of the starry sky. This display must have enticed people for as long as people existed and I’m sure the question has often come up: how far away are those stars?

Well, there is an interesting tale of discovery related to the progression of measuring sticks that give the ability to determine the distances to astronomical objects. Part of this tale is how Edwin Hubble discovered that the universe is expanding.

The realization that we live in an expanding universe complicates the answer to the question of how far away astronomical objects are. Apart from the fact that the distances change, there is also the issue of what distance we observe at a given point in time. If I use the apparent brightness of a star with a known absolute brightness, then one may think (at least I would have) that the implied distance is between us (the earth) and the location of the star at the time the light was emitted. This is not the case.

The above diagram tries to explain what happens. The black dots represent a star or galaxy (the source of the light) at different locations in an expanding universe. The blue dot is the earth which is kept it at a fixed location in the expanding universe. The red circles represent the expanding sphere of light after being emitted by the source at some point in the past. Assuming that the universe expands uniformly, we see the source would always remain at the center of the expanding sphere. Moreover, since the observed apparent brightness is given by the total emitted power divided by the total surface area of the sphere, the associated distance is the distance from the earth to the current location of the source. This is called the proper distance to the source.

Amazing, we are able to know the distance to an object at its current location even if we cannot see that object now. Who knew?