Distance Ladder Maths

Measuring distances is very difficult to do in space, because we can not go to far objects and physically measure them. Rather than finding actual distances, astronomers find relative distances and then calculate the actual distance. With the distance ladder, close stars become our tape measure, and with a bit of calibration, we scale our tape measure up to galaxies.

Main Sequence Fitting

We first need Parallax. After about 150 light years, parallax is useless. Instead, we use main sequence fitting.

Main sequence fitting uses similar stars in different places to calculate distance, and uses an adapted version of the Stefan-Boltzmann law:

main sequene fitting1 (1)

If we take a close star object of known distance (calculated from parallax) and measure its luminosity, we could find the distance to a similar star in a far away galaxy .This is because two stars similar in temperature, metallicity and size will have similar luminosity.

We can set up set up two equations for the close and far star that gets rid of their luminosity and uses only intensity (much easier found) and distance.

main sequene fitting2

Above, I cancelled out the constants and L, leaving only the distances and intensities. This is how we “calibrate” our methods; finding a distance with one technique, then using it to adjust the next one. Hence, we use the name distance ladder.

If we analyse each star more we can correct the errors and get a distance with 95-97% accuracy.

Standard Candles

Standard candles are simply objects suitable for main sequence fitting, as they have universally regular luminosities. Cepheid variables and type 1a Supernovae are our top objects.

There’s lots of Cepheids in the close by Large Magellanic cloud, so all we need is to find a distant galaxy that has them. Once a galaxy has been found, we can use the intensity-distance equation and find a distance!

How do astronomers use distance ladder?

If you plot some data of nearby Cepheid variables against one in a distant galaxy, you’d get a graph that looks similar to this. In the real world, data would rarely have such a nice correlation.

cepheid

The blue ones are the close variable stars, and the purple is the one in the far away galaxy.

You can read off the graph the intensities of the purple and appropriate blue star (needs the same pulse period) and sub it into the intensity/luminosity formula from before!

In further galaxies, type 1a supernovae can also be used, but there’s less of them out there.

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