MadSci Network: Astronomy
Query:

Re: What fraction of the night sky do we see with a telescope?

Date: Tue Sep 13 23:22:37 2005
Posted By: Neil Saunders, Research fellow
Area of science: Astronomy
ID: 1121379861.As
Message:

hi Atinuke,

This is a great question. We can answer it using a bit of mathematics called trigonometry.

When you look through a cardboard tube, you see a circle at the end of the tube containing a section of the sky. What we want to know is the area of that circle compared with the area of the sky. Now, the area of a circle is found using this formula:

area = Π * r2
Π is the greek symbol pi (about 3.14) and r is the radius of the circle.

What units should we use to measure this area? We could measure the radius of the tube using a ruler and get an answer in centimetres. But that would be wrong, because the sky is much larger and much further away than our tube. If you imagine holding a soccer ball close to you and looking at the moon - the soccer ball would appear larger, but you know that the moon is a larger object than the soccer ball - the moon just looks smaller because it is further away. So we don't use units like centimetres or units - we use units of angular measurement called degrees.

Imagine that you are looking at your cardboard tube from the side. You'd see something like this:

 this       |
  is        |
  the       |-------------------------------------------  your eye is here
width of the|      this is the length of the tube (L)
the tube (D)|
If you drew a line from your eye to the "top" and the "bottom"of the other end of the tube in this diagram, you would see that you had 2 triangles, the same size. This is a bit hard to draw on this webpage, but here's an example with a much shorter tube:
 |\
 | \
 |  \
 |   \
 |__ a\ 
 |   a/ your eye is here
 |   /
 |  /
 | /
 |/
That letter a is the angle between the horizontal line from your eye to the end of the tube and the lines that you drew from your eye to the top and bottom at the end of the tube. It is also the radius of the end of the tube, as seen from your eye, in degrees.

So - we know the length (L) and the width (D) of our tube. There is a special number, called the tangent of the angle a, which is just the tube radius divided by the tube length. I would guess that a typical toilet paper tube is about 15 cm long and 5 cm in diameter, so the tangent of a is (5/2) / 15 = 0.167. Then we get the angle a using something called the inverse tangent and that comes to about 9.5 degrees. Finally, we put that into our formula for area of a circle and it gives us about 280 square degrees. So when you look through your tube, that's the area of sky that you can see.

Let's think about what the sky is. The earth is a sphere and to us, the sky also looks like a sphere surrounding us with the earth at the centre. In fact, astronomers call the sky the celestial sphere. So in fact, that small part of sky that you see through your tube is slightly curved. We pretended that it was flat in the first picture that we drew to make our calculations easier and you don't have to worry about that too much.

It turns out that you can calculate the area of the whole sky in square degrees too and the answer is about 41 254. If you divide that by the 280 square degrees that we see through our tube, the answer is about 146 - which is very close to the 1/144 that you stated in your question.

I hope that this helped you understand your question and is not too mathematical for you. Here are 2 good websites to help you some more:
How many stars can we see?
Why do astronomers measure size in degrees?

Neil


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