Skinny Triangle Simplification
There is one important way we can speed up our calculations using angular measure. It's an interesting trick all astronomers are familiar with. It applies when the angle of the object we are observing is very small (much less than 1 degree). This is always the case when dealing with celestial objects.
Skinny triangles have a side length which is almost the same as their heights. Compare this to wider triangles that have much longer sides as compared to the heights. Now here is the trick. If we are dealing with skinny triangles we can assume they are right triangles and use trigonometry to solve for length BC.
Below is an interactive illustration which shows the result of this assumption. Click and drag the points B & C to change the width of the triangle and notice the error drops as the triangle narrows:
This means we can assume that skinny triangles are right triangles and we can use our relationships from trigonometry! This is just a way of saving time, it doesn't involve any new mathematics. Let's review.
If some object has an angular size of one degree, what does that immediately tell us?
Tan(angular measure) = diameter/distance
Tan(1) = 0.017 = 1/57.3
distance = 57.3 * diameter
The distance to an object with an angular measure of 1 degree is 57.3 times its diameter. This leaves us with this very simple equation to use:
angular size in degrees = diameter/distance * 57.3
This can be rearranged in all sorts of ways to find what you are looking for. For example:
distance to planet = diameter / 57.3 * angular size degrees
When using seconds instead of degrees we just multiply 57.3 by 3600 (since there are 3600 seconds in a degree). This gives us 206265 which is a constant to convert radians to arcseconds. Here is the equation as we will see it most often:
δ = d / D * 206265
δ = angular measurement in arcseconds
d = diameter of object in kilometers
D = distance to object in kilometers
Let’s try it out in the next exercise!