https://lunarpedia.org/api.php?action=feedcontributions&user=80.38.37.26&feedformat=atomLunarpedia - User contributions [en]2024-03-29T08:01:44ZUser contributionsMediaWiki 1.34.2https://lunarpedia.org/index.php?title=Area_of_Regions_on_the_Lunar_Sphere&diff=10870Area of Regions on the Lunar Sphere2007-10-09T08:47:09Z<p>80.38.37.26: /* Area of regions on the lunar sphere */</p>
<hr />
<div>==Area of regions on the lunar sphere==<br />
<br />
To calculate area on the lunar sphere bounded by pair of latitudes and a pair of longitudes:<br />
<br />
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:<br />
<br />
<math>{2\pi r^2 [{1 - sin(Latitude_1)} - {1 - sin(Latitude_2)}]} [(Longitude_1 - Longitude_2)/360] \,\!</math> <br />
<br />
where <math>Longitude_1 \,\!</math> and <math>Longitude_2 \,\!</math> are in degrees<br />
<br />
Here is the proof:<br />
<br />
First, calculate the area of the strip between two parallels of latitude.<br />
<br />
That is the same as the difference in the area of two spherical caps about the same apex.<br />
<br />
Those areas are most easily calculated as solid angles with units of steradians.<br />
<br />
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>."<br />
<br />
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.<br />
<br />
To calculate h:<br />
<br />
<math>h = r - [r sin(Latitude)] \,\!</math><br />
<br />
So:<br />
<br />
Where <math>Latitude_2 > Latitude_1 \,\!</math> (i.e. <math>A1 > A2 \,\!</math>):<br />
<br />
Then, for <math>Latitude_1 \,\!</math>: <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]<br />
= 2\pi r x r [1-sin(Latitude_1)]<br />
= 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math><br />
<br />
And for <math>Latitude_2 \,\!</math>: <math>A2 = 2\pi r ^2 [1-sin(Latitude_2)] \,\!</math><br />
<br />
So area of strip between two latitudes is<br />
<br />
<math>A(strip) = A1 - A2 = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]<br />
= 2 \pi r^2 [{1-sin(Latitude_1)} - {1-sin(Latitude_2) }]<br />
\,\!</math><br />
That gives the area in steradians.<br />
<br />
The area of a sphere is <math>4\pi*steradians = 4\pi r^2 \,\!</math><br />
<br />
The mean radius of the Moon = 1738 km<BR><br />
<br />
Surface area of the entire Moon = 37,958,532 Square kilometers<BR><br />
<BR><br />
Once we have calculated the area of the strip, then it is a simple matter to pro-rate for the longitude, as follows:<br />
<br />
The area of the curved rectangle subtended by the pairs of latitude and longitude can be expressed as follows:<br />
<br />
Area of the strip multiplied by the decimal fraction of the longitudinal circumference.<br />
<br />
The decimal fraction of the longitudinal circumference (in degrees) is calculated as follows:<br />
<br />
<math>(Longitude_1 - Longitude_2)/360 \,\!</math> degrees<br />
<br />
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:<br />
<br />
<math>{2\pi r^2 [{1-sin(Latitude_1)}-{1-sin(Latitude_2)}]}x[(Longitude_1-Longitude_2)/360 degrees] \,\!</math><br />
<br />
==External Links==<br />
* Usenet: Space FAQ 04/13 - Calculations<br />
http://www.faqs.org/faqs/space/math/<br />
<br />
<br />
[[Category:Physics]]<br />
[[Category:Selenology]]</div>80.38.37.26https://lunarpedia.org/index.php?title=Area_of_Regions_on_the_Lunar_Sphere&diff=10869Area of Regions on the Lunar Sphere2007-10-09T08:45:33Z<p>80.38.37.26: /* Area of regions on the lunar sphere */</p>
<hr />
<div>==Area of regions on the lunar sphere==<br />
<br />
To calculate area on the lunar sphere bounded by pair of latitudes and a pair of longitudes:<br />
<br />
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:<br />
<br />
<math>{2\pi r^2 [{1 - sin(Latitude_1)} - {1 - sin(Latitude_2)}]} [(Longitude_1 - Longitude_2)/360] \,\!</math> <br />
<br />
where <math>Longitude_1 \,\!</math> and <math>Longitude_2 \,\!</math> are in degrees<br />
<br />
Here is the proof:<br />
<br />
First, calculate the area of the strip between two parallels of latitude.<br />
<br />
That is the same as the difference in the area of two spherical caps about the same apex.<br />
<br />
Those areas are most easily calculated as solid angles with units of steradians.<br />
<br />
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>."<br />
<br />
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.<br />
<br />
To calculate h:<br />
<br />
<math>h = r - [r sin(Latitude)] \,\!</math><br />
<br />
So:<br />
<br />
Where <math>Latitude_2 > Latitude_1 \,\!</math> (i.e. <math>A1 > A2 \,\!</math>):<br />
<br />
Then, for <math>Latitude_1</math>: <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]<br />
= 2\pi r x r [1-sin(Latitude_1)]<br />
= 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math><br />
<br />
And for <math>Latitude_2</math>: <math>A2 = 2\pi r ^2 [1-sin(Latitude_2)] \,\!</math><br />
<br />
So area of strip between two latitudes is<br />
<br />
<math>A(strip) = A1 - A2 = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]<br />
= 2 \pi r^2 [{1-sin(Latitude_1)} - {1-sin(Latitude_2) }]<br />
\,\!</math><br />
That gives the area in steradians.<br />
<br />
The area of a sphere is <math>4\pi*steradians = 4\pi r^2 \,\!</math><br />
<br />
The mean radius of the Moon = 1738 km<BR><br />
<br />
Surface area of the entire Moon = 37,958,532 Square kilometers<BR><br />
<BR><br />
Once we have calculated the area of the strip, then it is a simple matter to pro-rate for the longitude, as follows:<br />
<br />
The area of the curved rectangle subtended by the pairs of latitude and longitude can be expressed as follows:<br />
<br />
Area of the strip multiplied by the decimal fraction of the longitudinal circumference.<br />
<br />
The decimal fraction of the longitudinal circumference (in degrees) is calculated as follows:<br />
<br />
<math>(Longitude_1 - Longitude_2)/360 \,\!</math> degrees<br />
<br />
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:<br />
<br />
<math>{2\pi r^2 [{1-sin(Latitude_1)}-{1-sin(Latitude_2)}]}x[(Longitude_1-Longitude_2)/360 degrees] \,\!</math><br />
<br />
==External Links==<br />
* Usenet: Space FAQ 04/13 - Calculations<br />
http://www.faqs.org/faqs/space/math/<br />
<br />
<br />
[[Category:Physics]]<br />
[[Category:Selenology]]</div>80.38.37.26https://lunarpedia.org/index.php?title=Area_of_Regions_on_the_Lunar_Sphere&diff=10868Area of Regions on the Lunar Sphere2007-10-09T08:44:46Z<p>80.38.37.26: /* Area of regions on the lunar sphere */</p>
<hr />
<div>==Area of regions on the lunar sphere==<br />
<br />
To calculate area on the lunar sphere bounded by pair of latitudes and a pair of longitudes:<br />
<br />
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:<br />
<br />
<math>{2\pi r^2 [{1 - sin(Latitude_1)} - {1 - sin(Latitude_2)}]} [(Longitude_1 - Longitude_2)/360] \,\!</math> <br />
<br />
where <math>Longitude_1 \,\!</math> and <math>Longitude_2 \,\!</math> are in degrees<br />
<br />
Here is the proof:<br />
<br />
First, calculate the area of the strip between two parallels of latitude.<br />
<br />
That is the same as the difference in the area of two spherical caps about the same apex.<br />
<br />
Those areas are most easily calculated as solid angles with units of steradians.<br />
<br />
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>."<br />
<br />
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.<br />
<br />
To calculate h:<br />
<br />
<math>h = r - [r sin(Latitude)] \,\!</math><br />
<br />
So:<br />
<br />
Where <math>Latitude_2 > Latitude_1 \,\!</math> (i.e. <math>A1 > A2 \,\!</math>):<br />
<br />
Then, for <math>Latitude_1</math>: <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]<br />
= 2\pi r x r [1-sin(Latitude_1)]<br />
= 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math><br />
<br />
And for <math>Latitude_2</math>: <math>A2 = 2\pi r ^2 [1-sin(Latitude_2)]</math><br />
<br />
So area of strip between two latitudes is<br />
<br />
<math>A(strip) = A1 - A2 = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]<br />
= 2 \pi r^2 [{1-sin(Latitude_1)} - {1-sin(Latitude_2) }]<br />
</math><br />
That gives the area in steradians.<br />
<br />
The area of a sphere is <math>4\pi*steradians = 4\pi r^2</math><br />
<br />
The mean radius of the Moon = 1738 km<BR><br />
<br />
Surface area of the entire Moon = 37,958,532 Square kilometers<BR><br />
<BR><br />
Once we have calculated the area of the strip, then it is a simple matter to pro-rate for the longitude, as follows:<br />
<br />
The area of the curved rectangle subtended by the pairs of latitude and longitude can be expressed as follows:<br />
<br />
Area of the strip multiplied by the decimal fraction of the longitudinal circumference.<br />
<br />
The decimal fraction of the longitudinal circumference (in degrees) is calculated as follows:<br />
<br />
<math>(Longitude_1 - Longitude_2)/360</math> degrees<br />
<br />
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:<br />
<br />
<math>{2\pi r^2 [{1-sin(Latitude_1)}-{1-sin(Latitude_2)}]}x[(Longitude_1-Longitude_2)/360 degrees]</math><br />
<br />
==External Links==<br />
* Usenet: Space FAQ 04/13 - Calculations<br />
http://www.faqs.org/faqs/space/math/<br />
<br />
<br />
[[Category:Physics]]<br />
[[Category:Selenology]]</div>80.38.37.26