Calculating sizes of lunar & planetary features

How to estimate the size of a feature in a lunar or planetary image e.g. in this capture of the Apollo 17?

Three parameters are required:

effective focal length used
pixel size of camera
distance to object captured

The sampling of one (square) pixel in arcseconds calculates to

(206.265 / (focal length in mm) )* (pixel size in microns)

In the case of the image above a C9 with a 2x barlow and no other extensions has been used, the effective focal lenth is 2350mm*2=4700mm. The pixel size of the CCD used is 5.6µm^2.

-> alpha = (206.265 / 4700)* 5.6µm = 0.246"

With simple trigonometry the size of one pixel projected onto the target object can be calculated:

a = (tan(alpha/2)*d)*2

For the image above the distance to the moon was 378767km (not considerung the curvature of the moon) so one pixel edge measures

a = ( tan( 0.246" / 2 ) * 378767km ) *2 = ( tan( 0.246° / (60*60*2) ) * 378767km ) * 2 = 0.45km

Of course this applies to a plane parallel to the CCD image plane only. The parallax is not considered since the object distance to pixel size ratio is very high.

Let's verify the above by an example. The crater in the middle at the top edge of the image above is Vitruvius. It measures ~67px*63px in the image: Considering the moon's spherical shape we assume Vetruvius as a circle of 67px diamter and calculate it's true size as 0.45km*67 = 30.15km
Size data vor Vetruvius varies between 30km and 31km so the estimated size looks pretty good

This may rise some related questions:

What is the minimum size of lunar features my scope can resolve ?

There are some well known criterions for minimum resolvable or detectable details. One is the Rayleigh criterion according to which two pointlight sources can be told apart at least when the central maximum of one airy disk is located at the first minimum of the other light source:

R[arcsec] = 138 / D[mm] or R[arcsec] = 5.45 / D[inch] with D = scope aperture

The first thing that comes to our mind for very close pointlight sources are doublestars. Keep in mind that the size of the airy disk varies with the light's wavelength - bigger for longer wavelengths, smaller for shorter. The above definition of the criterion is correct for yellow light at 550 nanometers.

A bit more optimistic than Rayleigh is the empiric Dawes Limit from William Dawes. He calls a double as split as soon as it's visible as an eight-like shape, his formular:

R[arcsec] = 116 / D[mm] or R[arcsec] = 4.56 / D[inch] with D = scope aperture

Let's use the Rayleigh Criterion for some calculations and assume the moon's distance to the observer is d=370.000km and we look at a small feature with size x located near the center of the moon's disk with a D=235mm scope:

According to Rayleigh R = 138/235mm = 0.59"
To get the min. feature size: tan(R/2) = (x/2)/d => x =tan(R/2)*2*d
With R[°]=0.59"/3600: x = 1.06km

In planetary highres-imaging we often distinguish between resolving features and detecting them. Think of a lunar craterlet with a low angle of sunlight. Resolving the craterlet means seeing at least a bright sunlit craterlet wall and a dark one in shadows. Detecting the craterlet means to have at least one pixel coloured differently than the surrounding, we can tell there's something but not the true nature of the object.

Luckily extended linear features, especially with high contrast, can be detected beyond the Rayleigh Criterion in the range of R/3.5 or even R/5. This means the scope above can detect linear features up to 0.12" width if all parameters like seeing, colimation, optical quality etc. support this. For lunar features like rilles in low sunlight this means a width of 0.21km is detectable.