Angular Size
Introduction
The main goal of this lab is to get you to understand how Astronomers measure the actual sizes of astronomical objects.
Since we can't take a meter stick to Jupiter in order to measure the actual size of its Great Red Spot, we have to use some basic trigonometry to help us measure these objects.
Don't worry! We actually end up using almost no trigonometry to determine the actual sizes - but we have to show you how we simplify it in order for you to understand what's going on.
Part 1: Angles and Geometry: a Review
Everything in this lab course will be about triangles - trigonometry is basically the study of triangles. Start by taking a look at the following triangle:
The first thing we need to do is label the sides:
Now, what's the nerdiest thing you can possibly think of to do with these sides?
I know!
Compare the lengths of the sides to each other! You can divide A and B, or B and C, or C and B, and so on - just do fractions a lot.
What you find if you compare the sides, is a very special relationship between the sides, and this is called the Pythagorean Theorem. This relates the squares of the sides as so:
A2 + B2 = C2
The side labeled 'C' has a special name, and it's called the Hypotenuse, and it's the longest side of a triangle.
With this, you can maybe see that if we can get information about the sides, we can calculate the length of another side without having to actually measure it!
But this is not so great for Astronomy, because at a minimum you need information about 2 of the sides - which is hard to do since we can't drag a tape-measure from here to Jupiter.
So for Astronomy, we have to use another relationship between the sides: Sine, Cosine, and Tangent! These are the nerdy ratios we just talked about, and it turns out those ratios between the sides are related to the angles between the sides of the triangle!
The shorthand mnemonic for these ratios is called SOH-CAH-TOA, which may bring back traumatic memories from high school math, but we will repair that relationship today.
Toa is short for the Tangent of the angle = Opposite side over Adjacent side.
Ok, let's take a look at the triangle but study one of the angles, call it ø (the greek letter phi).
So now let's apply our algorithm:
Toa: Tan(ø) = A / B
And if you want to calculate what the angle is, you have to unwrap the tangent by taking the inverse tangent:
ø = Tan-1(A/B)
Look for the inverse tangent function on your calculator! Here it is on the google calculator:
Brief aside: Degrees or Radians?
Here in the United States, we deal with an annoying hodgepodge of units. The internationally recognized system of units, called "SI" units is officially adopted in the United States, but in practice only scientists and engineers use them on a regular basis. Which is STUPID.
For example, instead of miles per hour, nearly every other country uses kilometers per hour.
Instead of Fahrenheit, nearly every other country uses Celsius.
And other countries use SI units because SI units are easier to understand and to work with and create fewer problems. Which brings us to SI units for angles.
In this lab, we have to measure angles - after all this is the Angular Size lab - and the two units that are commonly used are degrees and radians. Both of these measure the sizes of angles.
You may recall a full circle is 360 degrees, (360˚), half of a circle is 180˚, a quarter is 90˚ (also called a right angle), and half of 90˚ is a 45˚ angle, which is the angle you cut wood to make picture frames fit nicely at the corners.
The degree is very easy to define - just take a circle and cut it up into 360 equal sized pieces. Each one is 1 degree - see below, a circle divided up into 10 degree segments:
Now, the Radian (often shortened to rad because it's so cool) is defined a little bit differently. A circle has a radius, which is the distance from the center to the edge of the circle. If you take that radius, and count how many times it fits around the circle, it's actually 3 radii, and a little bit more. It turns out to be π radii! 3.14159...radii!
And each of these angles that subtends the length of a radius along the circumference is called a radian.
Take a look at the following animation showing how the radian is defined, you will want to watch it from the beginning and might have to wait for it to start over:
This is where (pi) π comes from! So there are then 2π radians in a complete circle, or 2π rad = 360˚.
This means that π rad = 180˚, and π/2 radians = 90˚, and π/4 radians = 45˚.
But don't be freaked out! Remember that π is just a number that is about 3.14.
So π/2 is just 3.14/2, which is about 1.57.
We just write π/2 for brevity and accuracy.
So in this way, a 90˚ angle is the same as a 1.57 radian angle!
1.57 radians = 90˚
Now what you have to be careful of, is that your calculator is set to radians if you want radians, and degrees if you want degrees!
In degrees, you can take the tangent of (45 degrees), but in radians, you would take the tangent of (π/4 radians), or (0.785 radians) if you don't like the pi symbol.
Small Angle Formula
Now as promised, I will show you how astronomers can get out of having to do any trigonometry! And that is using the small angle formula!
To begin, let's start by creating a "unit circle," or a circle that has a radius of 1.
Now let's put a second radius on there, make it red, to create an angle, and call the angle ø.
Next, let's imagine what taking the tangent of that angle would get us: Tan(ø) = Opposite / Adjacent.
The Adjacent side is just the black line radius of length 1, but the opposite side is a whole new triangle, green, blue and black labeled below.
Since the adjacent side is equal to one, the opposite side of the triangle (blue) is equal to the tangent of the angle.
And now take the Sin(ø) = Opposite / Hypotenuse
Since the hypotenuse is equal to one, the length of the pink line is just equal to the sine of the angle.
Ok, now let's make the angle smaller, see if you can notice something about what happens as the angle gets small: sin(ø) starts to become the same length as tan(ø)! The blue line and the pink line get closer to the same length! As do the red lines and the green lines.
And let's go to an even smaller angle!
So hopefully you can see how as the angle gets smaller, the two triangles created by sine and tangent become equal, and this leads to the following result:
For Small Angles:
sin(ø) = tan(ø) = ø
So
tan(ø) = ø = opp / adj
We will exclusively use the 2nd formula, tan(ø) = ø, because that relates the opposite and the adjacent sides of a triangle, which is great for astronomical objects as we will see. If we can measure the distance to an object, then we can calculate its size!
This is because the distance to an object is the adjacent side of a triangle, and its actual size is the opposite side of the triangle! The angle is its angular size, which is its apparent size, and something we can measure.
Thus the final small angle formula that we need for angles measured in radians becomes:
ø = opp/adj
Now you should be able to do part 1!
Part 2: Applying the Small Angle Formula
So what's the point of all this? Well the goal is to be able to use some simple geometry to determine the true sizes of astronomical objects without going there.
The main idea here, is that the further away an object is, the smaller it appears. We call this 'apparent size' the angular size of an object - because you're not actually measuring the object's true size, you're measuring the angle it takes up in your vision.
Behold! You all know that great seats to see a Taylor Swift concert are close the stage, because you can see her better - she has a larger angular size. She's not actually taller or bigger - you're simply closer so she appears bigger.
Imagine taking a picture of Taylor in the nose bleed section, far away from the stage, vs taking a picture close to her.
Note that in the following image - her actual height 'H' does not change! Only the angle and your distance. You get closer, she appears bigger, you move further away, she looks smaller!
Taylor's height, and distance, and angular size are related by the small angle formula:
angular size = ø = Opp / Adjacent = Taylor's Actual Height / Your Distance From Her
This is true for anything at all - every object appears smaller when it gets further away from you, and that relationship is rigid.
So your goal in part 2 is to understand this relationship by calculating the true width of the andromeda galaxy, and comparing that with how large it 'appears', its 'apparent size', or its 'angular size'.
In this part, you will apply the small angle formula to the Andromeda Galaxy.
The ultimate goal in this part is to calculate the actual size of the Andromeda Galaxy, without going to it and measuring it in person - All you can do from Earth is measure the angular size of Andromeda.
This is exactly what astronomers do!
Since astronomical objects are so far away, we can use the small angle formula because they're so small.
If we can simply measure their angular size, we can multiply that by the distance to get the true size!
Remember, 'ø' is pronounced 'phi' (either fee or fye), and means 'angular size'.
ø = Height/Distance
So
Height = ø*Distance
So what you have to do in this part is to convert the angular size of Andromeda into light years!