To the Stars

Maths can be a dry subject to learn when you do not know of its many applications.This post will be about one of the many applications of trigonometry. Trigonometry is used extensively in the GPS system to determine your position. But I’d like to go a bit farther out in space and show how distances to stars are determined.

Let me first set up the example and then define some new measurement units before I actually develop the solution. This is because of the large distances and small angles used in astronomy.

A common method to calculate the distance of nearby stars is to use parallax. You can see parallax in action by stretching out your hand and extend your index finger, then alternately close one eye and see how your finger changes position compared to objects further away. This effect is rather pronounced with your finger because the distance between your eyes is not much smaller than the distance to your finger.

What astronomers do is to see the relative position of of a nearby star at different places compared to farther stars that do not change their position much over time. But any distance on earth is very small compared to the distance to a star so the effect is too small to measure. What to do?

Well as we orbit our star (the sun), we could make a measurement on one side of the orbit, say in June, then on the other side in December. This would be a much larger distance than anything we could do on earth and the resulting parallax effect would be measurable. So the set up of the problem looks like this:

So the two readings are needed to determine the angle šœƒ. Once šœƒ s known, we can use it to calculate d. From the previous post on trig functions, you can see that for this problem:

\[\tan\hspace{0.33em}\mathit{\theta}\hspace{0.33em}{=}\hspace{0.33em}\frac{r}{d}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{d}\hspace{0.33em}{=}\hspace{0.33em}\frac{r}{\tan\hspace{0.33em}\mathit{\theta}}\]

But there are some practical problems here. Even though we are using the largest possible distance while restricted to the earth’s surface to measure the parallax angle, this angle is still very very small. So astronomers use arc seconds (arcsec). This is 1/3600 of a degree. Also, as distances to stars are very large, the unit of distance between the earth and the sun used is 1 AU (astronomical unit). 1 AU equals 149.6 million kilometers. Using these units results in the distance expressed in parsecs (yes, parsecs is a distance measurement, not a time measurement as used in the Star Wars movies). However, we will just use AU’s for distance and degrees for the angle in the example.

So how far is a star with a measured parallax angle of 0.0002 degrees?

\[{d}\hspace{0.33em}{=}\hspace{0.33em}\frac{{1}\hspace{0.33em}{AU}}{\tan\hspace{0.33em}{(}{0}{.}{0002}{)}}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{0.00000349065}\hspace{0.33em}{=}\hspace{0.33em}{286479}{.}{6}\hspace{0.33em}{\mathrm{AU}}\]

You can see why astronomers use arcsec. Using the all knowing Google, converting 286479.6 AU gives 42,900,000,000,000 km or 4.53 light years or 1.39 parsecs.