# Angular Resolution

Angular resolution is the minimum separation between distinguishable features in an image. Objects smaller than the angular resolution are indiscernible in the picture. The theoretical maximum angular resolution is determined by the diameter of the aperture of the optical instrument.

Flat-Earthers like to demand images of the lunar landers taken with a telescope as proof that the Apollo Moon landings did occur. In reality, no optical telescope on Earth is large enough to resolve the landers.

Dawes’ limit is a formula to express the maximum resolving power of a telescope: R = 11.6/D, where R is the resolving power of the instrument in arcseconds, and D is the aperture diameter, in centimeters.

The following are several objects in space, their angular sizes, and the required minimum size of the aperture of a telescope to obtain a sharp 500-pixel image of each object, calculated by the Dawes’ limit.

• The Moon. Angular size: 0.52°. Minimum required aperture: 1.55 cm.
• The ISS. Size: 109 m. Distance: 410 km. Angular size: 0.015°. Minimum required aperture: 53 cm.
• Jupiter. Size: 139820 km. Distance: 628743036 km. Angular size: 0.0127°. Minimum required aperture: 63 cm.
• A geostationary satellite. Size: 10 m. Distance: 35786 km. Angular size: 0.000016°. Minimum required aperture: 500 m.
• Apollo lander, descent stage. Size: 9.4 m. Distance: 384400 km. Angular size: 0.0000014°. Minimum required aperture: 5.75 km.
• M87* black hole. Size: 0.00238156 light-years. Distance: 53490000 light-years. Angular size: 0.00000000255°. Minimum required aperture: 3158 km.

## The Case of Nikon P1000

The so-called “flat Earth camera” Nikon P1000 has a 4.3-539mm 1:2.8–8 lens. At its longest zoom, it has an aperture diameter of 539 mm / 8 = 67 mm = 6.7 cm. By applying the Dawes’ limit, we can determine its maximum resolving power is R = 11.6/6.7 cm = 1.73 arcseconds = 0.00048°. If we want to capture a sharp 500 px image of an object on the surface of the Moon, then the object needs to have an angular size of 0.00048° × 500 / 2 = 0.12° (the factor of 2 is the Nyquist sampling rate). If the object has an angular size of 0.12°, then at the Moon’s distance, it has a size of 384400 km × tan(0.12°) = 805 km.

Therefore, for the P1000 to be able to capture a clear 500-pixels image of an object on the lunar surface, the thing needs to be 805 km wide, or about half the lunar radius.  The Nikon P1000 is not a magical device flat-Earthers purport them to be. At best, it is comparable to the lowest end class of astronomy telescopes. It will not be able to resolve Apollo lunar landers on the lunar surface, just like any Earth-based telescopes, even the best of them. The fact it cannot see the landers is not proof the landers are not there.