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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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| − | <math>{2\pi r^2 [(1 - sin(Latitude_1)) - (1 - sin(Latitude_2))]} [(Longitude_1 - Longitude_2)/360] \,\!</math>
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| − | where <math>Longitude_1 \,\!</math> and <math>Longitude_2 \,\!</math> are in degrees
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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 <math>r^2 \,\!</math>."
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| − | The area of a spherical cap (<math>A = 2\pi r h \,\!</math>), 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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| − | <math>h = r - [r sin(Latitude)] \,\!</math>
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| − | So:
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| − | Where <math>Latitude_2 > Latitude_1 \,\!</math> (i.e. <math>A1 > A2 \,\!</math>):
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| − | Then, for <math>Latitude_1 \,\!</math>: <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]
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| − | = 2\pi r x r [1-sin(Latitude_1)]
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| − | = 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math>
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| − | And for <math>Latitude_2 \,\!</math>: <math>A2 = 2\pi r ^2 [1-sin(Latitude_2)] \,\!</math>
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| − | So area of strip between two latitudes is
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| − | <math>A(strip) = A1 - A2 = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]
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| − | = 2 \pi r^2 [(1-sin(Latitude_1)) - (1-sin(Latitude_2) )]
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| − | \,\!</math>
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| − | That gives the area in steradians.
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| − | The area of a sphere is <math>4\pi*steradians = 4\pi r^2 \,\!</math>
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| − | The mean radius of the Moon = 1738 km<BR>
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| − | Surface area of the entire Moon = 37,958,532 Square kilometers<BR>
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| − | <BR>
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| − | Once we have calculated the area of the strip, then it is a simple 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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| − | <math>(Longitude_1 - Longitude_2)/360 \,\!</math> 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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| − | <math>{2\pi r^2 [(1-sin(Latitude_1))-(1-sin(Latitude_2))]}x[(Longitude_1-Longitude_2)/360 degrees] \,\!</math>
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| | ==External Links== | | ==External Links== |
