Telescopes and plate scale

Some definitions:

$f =$ focal length, mm
$N =$ f/number: ratio of focal length to aperture or beam diameter
$D =$ physical diameters, mm
$s =$ scale at a focal plane, arcsec/mm
$\theta_{field} =$ angular field of view

To simplify the optical analysis, I will consider optical elements that are focused at infinity. We will deal with mirrors and lenses that take incident collimated beams - parallel ray bundles, such as from a star at nearly-infinite distance - and turn them into images at a finite distance, or vice versa, turning images into collimated beams. In the case of collimated beams, a mirror or lens turns the off-axis angle $\theta$ of an incident beam into an off-axis displacement $r$ of the image in the focal plane, and the amount of the displacement is governed by the lens focal length $f$: $r_{offaxis} = \theta_{offaxis} f$, where $\theta$ is in radians.

Figure 1: Telescope designs with locations of the secondary focus. The difference between Cassegrain and Gregorian is in the location and curvature of the secondary mirror, and in the field curvature of the focal plane. The Nasmyth focus is a variant of either, using a flat tertiary to change the physical location of the focus. Nasmyth foci are useful in alt-azimuth mounted telescopes.

Large telescopes are usually derived from a Cassegrain or Gregorian design; newer designs are almost all alt-azimuth mounting and frequently use a flat tertiary to provide a Nasmyth focus. Cassegrain telescopes have a convex secondary mirror, while Gregorians have a concave secondary which is located past the primary focus. Cassegrain designs are more common because the overall structure is shorter and the enclosure can be smaller. However, the Gregorian has a focal plane which is concave away from the telescope, toward the instrument, while Cassegrains are the opposite. This sense of curvature can make it easier to design wide field imagers for a Gregorian.¹

The primary mirror in modern large telescopes is quite fast, with f-number $\sim 1-3$, but the secondary mirror slows the system down, with f/5 to f/15 being a common range. In Figure 1, note that the angle of convergence at the secondary focus is narrower than it is at the prime focus. If one regards the beam as a cone of light, the f-number is simply the ratio of height of the cone to the base.

The plate scale at the secondary focal plane of a telescope and the physical diameter of the field of view depend on the focal length of the telescope $f_{tel}$ (the factor 206265'' converts from radians to arcseconds):

$$s_{tel} = 206265'' / f_{tel},$$

$$f_{tel} = D_{tel} N_{tel},$$

$$D_{field} = \theta_{field} / s_{tel}.$$

The scale at the secondary focal plane is usually large enough that it is not a good match for modern detectors. For example, at the f/11 focus of the 6.5-m Magellan, the plate scale is 2.9''/mm. A CCD detector with 15 $\mu$m pixels would have 0.043''/pixel, which grossly oversamples a reasonable atmospheric seeing ($\sim 0.6''$).