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| + | [[Area of regions on the lunar sphere]] |
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| − | Usenet: Space FAQ 04/13 - Calculations
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| − | http://www.faqs.org/faqs/space/math/
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| − | ==Area of regions on the lunar sphere==
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| − | To calculate area on the lunar sphere bounded by pair of latitudes and a pair of longitudes:
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| − | The general calculation for the area on the Moon (in square kilometers) bounded by a pair of longitudes and a pair of latitudes is as follows:
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| − | {2 x Pi x r ^2 [{1 - sine ( Latitude-1)} - {1 - sine ( Latitude-2) }]} x [(Longitude-1 - Longitude-2)/360 degrees] x 3,020,644
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| − | Here is the proof:
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| − | First, calculate the area of the strip between two parallels of latitude.
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| − | That is the same as the difference in the area of two spherical caps about the same apex.
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| − | Those areas are most easily calculated as solid angles with units of steradians.
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| − | The steradian is defined as "the solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere having an area r^2."
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| − | The area of a spherical cap (A = 2 x pi x r x h), where h = distance between the apex of the cap and the center of the base plane of the cap.
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| − | To calculate h:
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| − | h = r - [r * sine (Latitude)]
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| − | So:
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| − | Where Latitude-2 > Latitude-1 (i.e. A1 > A2):
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| − | Then, for Latitude-1: A1 = 2 x pi x r x [r-{ r x sine (Latitude-1)}]
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| − | = 2 x pi x r x r [1-sine(Latitude-1)]
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| − | = 2 x pi x r ^2 x [1-sine(Latitude-1)]
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| − | And for Latitude-2: A2 = 2 x pi x r ^2 x [1-sine(Latitude-2)]
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| − | So area of strip between two latitudes is
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| − | A(strip) = A1 - A2 = [2 x pi x r ^2 x [1-sine(Latitude-1)]]-[2 x pi x r ^2 x [1-sine(Latitude-2)]]
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| − | = 2 x Pi x r ^2 [{1-sine(Latitude-1)} - {1-sine(Latitude-2) }]
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| − | That gives the area in steradians.
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| − | The area of a sphere is 4 * pi steradians = 4 pi r2
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| − | The mean radius of the Moon = 1738 km
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| − | To convert from steradians to square kilometers, multiply by r2 = 3,020,644
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| − | Surface area of the entire Moon = 37,958,532 Square kilometers
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| − | Once we have calculated the area of the strip, then it is a simply matter to pro-rate for the longitude, as follows:
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| − | The area of the curved rectangle subtended by the pairs of latitude and longitude can be expressed as follows:
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| − | Area of the strip multiplied by the decimal fraction of the longitudinal circumference.
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| − | The decimal fraction of the longitudinal circumference (in degrees) is calculated as follows:
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| − | (Longitude-1 - Longitude-2)/360 degrees
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| − | Hence the general calculation for the area on the Moon (in square kilometers) bounded by a pair of longitudes and a pair of latitudes is as follows:
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| − | {2x Pi x r ^2 x [{1-sine(Latitude-1)}-{1-sine(Latitude-2)}]}x[(Longitude-1-Longitude-2)/360 degrees]x 3,020,644
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| | [[Category:Physics]] | | [[Category:Physics]] |
| − | [[Category:Selonology]] | + | [[Category:Selenology]] |
