How Do We Find Out Distances to the Stars?

Question posed by Carlos, and I'm pretty sure someone else asked me it at some point too...

For stars relatively close to us, astronomers use the effect of parallax to work out how far away they are. I've spoken about parallax in a couple of other posts, but for a simple description of the effect, have a go at this:

Parallax

Go outside, close one eye, and put a finger up in front of you at arm's length. Line your finger up with a distant building, try, electricity pylon or other such landmark. Now close your eye and open the other one. Your finger is no longer lined up with the landmark: as far as your eyes are concerned, your finger appears to have moved, whereas the object in the distance has stayed still.

By Booyabazooka at en.wikipedia

Now, in your mind's eye, swap the object in the distance for the general background of stars and space. Swap your finger for a closer star that we're trying to find a distance for, and imagine that the Sun is sat between your eyebrows. The Earth orbits the Sun, and once every six months occupies the space where either your left eye or right eye is.

Now, when the Earth is in the right-eye position, we take a photo of the star and its background. We then wait six months until Earth is now in the left-eye position and take another photo of the same star. Now we compare the two photos to see how the star appears^[1] to have moved over these six months, and measure the angle between the two apparent positions.

Imagine a triangle drawn using the star at one vertex, and the Earth and  Sun at the other two: we now have a triangle in which we know the length of one side as well as two of the angles (the angle at the star's vertex is the one we have measured, and the angle at the Sun's vertex is a right-angle). We want to find the length of the side between the Sun and the star^[2] , and we can do this using trigonometry^[3] .

Parallax is only useful for finding the distances of stars up to a few thousand light years away. Beyond this, the apparent change in a star's position is too small to be measured by today's equipment, and other methods have to be employed. The most popularly used one of these utilises a particular kind of star as a 'standard candle'.

Cepheid Variables

Cepheid variable stars are a class of star that have both a high luminosity (they're very bright) and a very strong relationship between their luminosity and pulsation period (i.e, the brightness of the star increases and decreases over time, and how fast this cycle happens is closely linked to the star's maximum brightness).

The distance to relatively nearby Cepheid variables has been determined using parallax, and astronomers have noted the relationship between their luminosity and pulsation period at known distances. This same relationship can be applied to Cepheid variables in distant galaxies by measuring their pulsation period and working out what their luminosity should be.

It's fairly obvious even on Earth that objects get dimmer as they get further away. In fact, this follows an inverse square law, which means that (for example) for two objects with the same luminosity, the one that is twice as far away appears 1/4 as bright; an object that is 10 times further away appears 1/100 as bright, and so on.

We can then use a Cepheid variable star's apparent brightness, by comparing it to its calculated luminosity, to tell us how far away it is. Finding the distance to a particular Cepheid variable naturally gives us a value for the distance to the galaxy or cluster within which it resides.

This distance-measuring technique is the primary one used in calculating the distances required to measure the expansion of the universe, and uncertainties in the calculated values for the Hubble constant are due in part to uncertainties in the use of Cepheid variables in distance-measuring.

Footnotes:

1. Remember, the star hasn't moved in the last six months. Well, not enough that we can detect it anyway. We have moved, and it's our changing viewpoint that makes it look like the star has moved. [back]
2. Given that the distance to the star is much, much larger than the distance between the Sun and the Earth, then the distance from Earth to the star can be considered to be the same as the distance from the Sun to the star. [back]
3. If you'd like a more detailed run-through of how to calculate a distance using trig, try asking at The Actual Maths! [back]