Difference between revisions of "Area of Regions on the Lunar Sphere"

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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:
 
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:
  
<math>{2\pi r^2 [{1 - sin(Latitude_1)} - {1 - sin(Latitude_2)}]} [(Longitude_1 - Longitude_2)/360]</math>  
+
<math>{2\pi r^2 [(1 - sin(Latitude_1)) - (1 - sin(Latitude_2))]} [(Longitude_1 - Longitude_2)/360] \,\!</math>  
  
where <math>Longitude_1</math> and <math>Longitude_2</math> are in degrees
+
where <math>Longitude_1 \,\!</math> and <math>Longitude_2 \,\!</math> are in degrees
  
 
Here is the proof:
 
Here is the proof:
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Those areas are most easily calculated as solid angles with units of steradians.
 
Those areas are most easily calculated as solid angles with units of steradians.
  
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>."
+
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>."
  
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.
+
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.
  
 
To calculate h:
 
To calculate h:
  
<math>h =  r - [r sin(Latitude)]</math>
+
<math>h =  r - [r sin(Latitude)] \,\!</math>
  
 
So:
 
So:
  
Where <math>Latitude_2 > Latitude_1</math> (i.e. <math>A1 > A2</math>):
+
Where <math>Latitude_2 > Latitude_1 \,\!</math> (i.e. <math>A1 > A2 \,\!</math>):
  
Then, for <math>Latitude_1</math>:  <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]
+
Then, for <math>Latitude_1 \,\!</math>:  <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]
 
                       = 2\pi r x  r [1-sin(Latitude_1)]
 
                       = 2\pi r x  r [1-sin(Latitude_1)]
                       = 2\pi r ^2 [1-sin(Latitude_1)]</math>
+
                       = 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math>
  
And for <math>Latitude_2</math>:    <math>A2  = 2\pi r ^2 [1-sin(Latitude_2)]</math>
+
And for <math>Latitude_2 \,\!</math>:    <math>A2  = 2\pi r ^2 [1-sin(Latitude_2)] \,\!</math>
  
 
So area of strip between two latitudes is
 
So area of strip between two latitudes is
  
 
<math>A(strip) =  A1 - A2  = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]
 
<math>A(strip) =  A1 - A2  = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]
  = 2 \pi r^2  [{1-sin(Latitude_1)} - {1-sin(Latitude_2) }]
+
  = 2 \pi r^2  [(1-sin(Latitude_1)) - (1-sin(Latitude_2) )]
</math>
+
\,\!</math>
 
That gives the area in steradians.
 
That gives the area in steradians.
  
The area of a sphere is  <math>4\pi*steradians =  4\pi r^2</math>
+
The area of a sphere is  <math>4\pi*steradians =  4\pi r^2 \,\!</math>
  
 
The mean radius of the Moon = 1738 km<BR>
 
The mean radius of the Moon = 1738 km<BR>
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The decimal fraction of the longitudinal circumference (in degrees) is calculated as follows:
 
The decimal fraction of the longitudinal circumference (in degrees) is calculated as follows:
  
<math>(Longitude_1 - Longitude_2)/360</math> degrees
+
<math>(Longitude_1 - Longitude_2)/360 \,\!</math> degrees
  
 
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:
 
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:
  
<math>{2\pi r^2 [{1-sin(Latitude_1)}-{1-sin(Latitude_2)}]}x[(Longitude_1-Longitude_2)/360 degrees]</math>
+
<math>{2\pi r^2 [(1-sin(Latitude_1))-(1-sin(Latitude_2))]}x[(Longitude_1-Longitude_2)/360 degrees] \,\!</math>
  
 
==External Links==
 
==External Links==

Latest revision as of 14:47, 10 April 2019

Area of regions on the lunar sphere

To calculate area on the lunar sphere bounded by pair of latitudes and a pair of longitudes:

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:

<math>{2\pi r^2 [(1 - sin(Latitude_1)) - (1 - sin(Latitude_2))]} [(Longitude_1 - Longitude_2)/360] \,\!</math>

where <math>Longitude_1 \,\!</math> and <math>Longitude_2 \,\!</math> are in degrees

Here is the proof:

First, calculate the area of the strip between two parallels of latitude.

That is the same as the difference in the area of two spherical caps about the same apex.

Those areas are most easily calculated as solid angles with units of steradians.

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>."

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.

To calculate h:

<math>h = r - [r sin(Latitude)] \,\!</math>

So:

Where <math>Latitude_2 > Latitude_1 \,\!</math> (i.e. <math>A1 > A2 \,\!</math>):

Then, for <math>Latitude_1 \,\!</math>: <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]

                      = 2\pi r x  r [1-sin(Latitude_1)]
                      = 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math>

And for <math>Latitude_2 \,\!</math>: <math>A2 = 2\pi r ^2 [1-sin(Latitude_2)] \,\!</math>

So area of strip between two latitudes is

<math>A(strip) = A1 - A2 = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]

= 2 \pi r^2  [(1-sin(Latitude_1)) - (1-sin(Latitude_2) )]
\,\!</math>

That gives the area in steradians.

The area of a sphere is <math>4\pi*steradians = 4\pi r^2 \,\!</math>

The mean radius of the Moon = 1738 km

Surface area of the entire Moon = 37,958,532 Square kilometers

Once we have calculated the area of the strip, then it is a simple matter to pro-rate for the longitude, as follows:

The area of the curved rectangle subtended by the pairs of latitude and longitude can be expressed as follows:

Area of the strip multiplied by the decimal fraction of the longitudinal circumference.

The decimal fraction of the longitudinal circumference (in degrees) is calculated as follows:

<math>(Longitude_1 - Longitude_2)/360 \,\!</math> degrees

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:

<math>{2\pi r^2 [(1-sin(Latitude_1))-(1-sin(Latitude_2))]}x[(Longitude_1-Longitude_2)/360 degrees] \,\!</math>

External Links

  • Usenet: Space FAQ 04/13 - Calculations

http://www.faqs.org/faqs/space/math/