Measuring the Area of a Region of Sky
Date: 05/30/2003 at 11:31:28 From: Randy Subject: Area of a Circle in Degrees Dr. Math, I enjoy reading your answers, and feel you are offering a great service. Keep up the good work. I'm learning astronomy and have been trying understand the geometry of the sky. My question is this: When I look through my telescope, what percentage of the sky am I looking at? I know the area in my field of view in either degrees x degrees or arcmin x arcmin. Is it possible to calculate the area of the visible sky, in either square degrees or square arcminutes, if there is such a thing? If yes, then a simple area of view/total area will give me my answer. I can calculate the area when a radius is known, but what about when one is not known such as in degrees? For example the area of a half of a circle with a radius of 1 is 6.283.
Date: 05/30/2003 at 12:27:20 From: Doctor Peterson Subject: Re: Area of a Circle in Degrees Hi, Randy. I think this answer from the Dr. Math archives will help: Degrees in a Sphere? Steradians http://mathforum.org/library/drmath/view/55358.html You will see there that we can measure the "area" of a region of sky in steradians, which means the area of that region of a sphere with radius 1. It's easiest to work this out for a circular region that is cut out from the "sky" by a cone whose apex is at the center of the sphere. If the "radius" of your field of view (that is, half the angle of view) is theta, we can find the area of the "spherical cap" cut out by the cone on the sphere using the formula found in the Dr. Math Geometric Formulas FAQ: Sphere Formulas http://mathforum.org/dr.math/faq/formulas/faq.sphere.html#spherecap S = 2 pi r h The height h of the cap is 1-cos(theta), and r is taken as 1, so our "area" is 2 pi (1 - cos(theta)) steradians Since there are 4 pi steradians in a whole sphere (that is, the whole unit sphere has area 4 pi), this is (1 - cos(theta))/2 of the sky. For example, if you see two degrees across, making the "angular radius" of your field of view 1 degree, you are seeing (1 - cos(1 deg))/2 = 0.000076 of the sky, or about 1/13,000 of it. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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