Trigonometric parallax

Hello!

Now before I start, I know, I know, people won't be randomly be finding the distance to a star they found over the course of six months, but for people who actually take their jobs seriously, they literally find the distances to a star for a living, and because of that, knowing the technique they use is important. Astronomy is weird sometimes.

All jokes aside, finding the distances between the Earth to a star is quite important in astronomy and astrophysics, as it helps astronomers find relationships between a star's luminosity and its color (which is usually based on temperature). That could help us to find the distance to literally everything in the observable universe.

Now, even though this may seem unrelated, I am going to make you manipulate your own eyes in a very creative way (sentence dramatized for effect). Put your finger in front of your eyes and stare at it. Now, I force you to close one eye and eyeball the location of the finger in relation to the background. Now close your eyes and open your opposite eye. Does the relative location seem different? This simple concept is essential for finding the distance to a star, believe it or not.

Diagram created in desmos.com/geometry

This diagram shows how the distance to a star is calculated.

In this diagram which is totally to scale which is not to scale, we can see that we can find the measure of the angle (which we'll call \(\theta\)) with the vertex at the center of the star based on how far the star moves in the six months Earth orbits around the Sun. We can also find out the other two angles as the triangle created is an isosceles triangle and the measure of the other two angles can be determined by \(\frac{180-\theta}{2}\). The distance from the Earth to the Sun is 1 AU (astronomical unit), so one of the magnitude of the sides of the triangle is 2 AU.

We can now use some basic trigonometry (if you have been to high school) to solve for the distance \(d\):

\(\tan(\frac{\theta}{2})=\frac{1 AU}{d}\)

\(d=\frac{1 AU}{\tan{\frac{\theta}{2}}}\)

For people who do not know, \(\tan(\theta)\) refers to the ratio of the side opposite to an angle \(\theta\) to the side adjacent to the angle, which is not the hypotenuse (Only for right angled triangles)

And that is the distance! Now you know what "real" astronomers actually do daily as their jobs to get paid^{(citation needed)}.

Thanks for reading!

Note: I have attempted to include sarcasm in the blog! Tell me if you like the usage of sarcasm or not!