Plate Scale

The plate scale is a constant for an optical system derivable from the focal length of the optical system.

Consider a calculation of the angular resolution of a detector of pixel scale (p) attached to an idealized telescope with focal length (f). Take both in units of millimeters (mm).

Recall that the pixel scale is simply the size of an individual pixel in the detector. For this article, I assume a square pixel, but real-world pixels may be rectangular.

You can conceive of a right-triangle where the short leg is the pixel scale and the long leg is the focal length. In nearly all real-world cases, the focal length will be much larger than the pixel scale.

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In radians, the angular resolution would be arctan(p/f) [radians].

Converting this to arcsec('') would require multiplying by a conversion factor (180/π * 3600).

So the angular resolution would be  arctan(p/f)*(180/π * 3600) [''].

In the limit of f >> p, we can employ the small angle approximation for tangent (sin x/cos x), namely that:

arctan(p/f)≈p/f 

So the angular resolution can be written as (p/f)*(180/π * 3600) [''] .

For a system (such as ours) with swappable detectors, each with different pixel scales (p), it is useful to factor-out the following expression:

(1/f)*(180/π * 3600) [''/mm]

This expression is the plate scale. It is measured in arcseconds/mm or whatever angular/length units you prefer or is most useful.

To use the plate scale in determining the angular resolution of an optical system, just scale the plate scale by the pixel scale of the detector.