Computing the Apparent Angular Diameter of the Moon As Viewed From Any Given Distance Measured From the Lunar Surface
When a planet is viewed from a very great distance, we can generally treat its angular diameter as the diameter of a flat circular disc of the same diameter without significant error. However, this approximation only works for great distances, like a million miles or kilometers or more. When viewing a sphere from much closer, such as the moon from 150000 miles, then the actual curvarure of the spherical surface significantly effects the apparent size and a circular approximation is insufficient. The following general formulas all take into account the spherical curvature of the lunar surface based on the mean value of the lunar radius as given above. Assuming the mean lunar radius ( R ) and distance (d ) from the surface, in the same units, the angular diameter () of the moon may be found by:
The radius and distance can be expressed in any units as long as both use the same units. NOTE: ArcCos(x) refers to the arc (or inverse) circular cosine function. Given the value (x ) of the cosine, this function returns the corresponding angle (a ). When writing a computer program, in many programming languages, the inverse trigonometric functions generally return the corresponding angles in radians. When this is the case, then the returned radians can be easily converted into the more convenient degrees by a simple conversion factor.
degrees = radians * 180 / = radians * 180 / 3.1415926535897932
The distance ( d ) refers to the minimum radar distance between the eye and the spherical lunar surface below.
For ( R, d ) expressed in miles (R = 1079.4 mi):
For example, if we were d=76546 miles from the lunar surface, assuming the mean lunar radius (1079.4 mi ), its apparent angular diameter would work out to:
= 0.027811381010052 rad = 1.5934747543063°
For ( R, d ) expressed in kilometers (R = 1737.1 km):
© Jay Tanner  2018
