Angular Size

 

Lots of people have difficulty understanding the concept of angular size, but it is very important for part four of this homework. It is a simple concept:
The further away something is, the smaller it appears to be. Even though the physical size of the object does not change as its distance does, it's apparent size or angular size does. That is all angular size is, a way to describe how large something appears to be in self-consistent units. One object may be smaller than another but if it the smaller object closer it can appear to be the same size. These two objects would be said to have the same angular size.

Example 1:There are 360 degrees in a circle no matter how big or small. In other words, a protractor always has 360 degree markings on it around it's edge, no matter how big or small the protractor is. If the protractor is very large then those markings might be one foot apart along the edge. If it is very small, like the kind you might put into your pocket or knapsack, then those marks will be only 1/32 of an inch apart, yet as seen from the center of any protractor no matter how large or small, the one degree markings always have the same angular separation from one another, namely one degree!

Example 2: As you know, the "size" an object appears to be does not change its actual size. A 6 foot tall person standing two hundred yards away is still 6 feet tall, even though they appear smaller than a dime held at arm's length. If you had an imaginary protractor with your eye at the center and the edge of it at the dime, you would see that dime has a certain size in degrees. Now if you made your imaginary protractor stretch all the way out to the 6 foot tall person you would see that person is smaller in degrees on the protractor than the dime is. Now if the 6 foot tall person walks up and tries to take the dime away from you, the imaginary protractor stretching to the person will now be the same size as the one that goes out to the dime; the person clearly now has a much larger angular size than the dime. Their angular size has changed with distance.

It turns out that for things with a small angular size, their distance equals the actual size of the object divided by it's angular size. If the angular size is half as much, it is twice as far away. Now we can measure angles in different units. The military uses something called a gradient. Most people are familiar with something called degrees1, but most science types use something called radians2.

Now in question four, you are asked to measure the angular size (in radians) of the ring around SN 1987A. You are given the angular size of the piece of sky of the picture of the ring and you have to estimate the angular size of the ring itself which is smaller than that of the whole picture. This is like being shown a dollar bill and being told it is 15 centimeters long and having to estimate how many centimeters long the picture of George Washington is, but your ruler only has inches on it.

 

Go back to the homework

1 We get degrees from ancient Babylon! The Babylonians attributed mystical importance to the numbers 2 and 3 and especially their product 6. They imagined the path of the sun against the stars through the year as a great circle (which it is) and assumed that 6's must be involved. They noticed that the sun took about 6x6x10 days to complete it's journey around this circle during the year, so they assumed it must be exactly 360 days and they were mis-measuring the sun's position or miscounting the days. So they divided up this great circle in the sky into 360 parts, and that is where degrees come from. Good grief, and I thought astronomy was bad for going all they back to Hipparchus's ancient magnitude system. Well everyone using degrees is using a system even older and more arbitrary!


2I like to use radians because they are easier to work with if you are trying to find the distance to something. If 1 degree=1.1 gradient units=.01754 radians, obviously you will get a different answer for distance if you divide actual size by each. Which one is right? Radians are. The others need to be converted to radians before the angular size/distance relationship will work. Radians also have a more intuitive definition. Try drawing a radian.

  1. Draw a big circle
  2. Take a length equal to the radius of the circle and lay it along one edge
  3. Draw an angle from the ends of the curved radius segment to the center of the circle.
  4. This makes an angle at the center of the circle. This angle is exactly one radian in size.

One radian=one radius worth of angle! 1 circle= 360 degrees = 2 * Pi radians, so 1 radian = 360/(2 * Pi) degrees which is about 57 degrees.