# Image scale

Image scale of the telescope / Nyquist-Shannon-Sampling-Theorem

The image scale gives a statement about how much sky section (in arc seconds) can be represented on one pixel.
The Nyquist-Shannon-Sampling-Theorem comes from signal and control engineering and roughly states that an analog frequency f (in this case the object signal) must be digitally sampled with a frequency greater than 2*f in order to be able to reconstruct the output signal exactly.

The section of the sky theoretically mapped on one pixel must therefore be scanned with two pixels instead of one. In other words, the resolving power must be twice as good.

The magnification of the telescope-camera system results from the following relationship:

The following results from trigonometry :

β – angle of the image scale
P – pixel size in [mm]
f – telescope focal length in [mm]

The following relationship applies due to the definition of radians:

The size of the angle (in radians) is equal to the ratio of the arc length 's' to the circle radius 'f'.

Since the pixel size 'P' (which, due to its very small value, corresponds approximately to the arc length 's') and therefore the angle 'β' are very small, the small angle approximation can be applied. This states that small angles of the trigonometric tan functions correspond approximately to the angle in radians, which can be illustrated by combining the two formulas above:

This means that the tan function can be removed, which simplifies the formula:

β – image scale in [rad]
P – pixel size in [mm]
f – telescope focal length in [mm]

To obtain a value in arc seconds and to be able to enter the pixel size in µm, the formula must be converted accordingly:

β – image scale in ["]
P – pixel size in [µm]
f – telescope focal length in [mm]

In Germany, a FWHM of 2" - 4" prevails. So that the diffraction discs "inflated" by seeing can always be sampled with two pixels, the resolving power β must therefore lie between 1" - 2". In astrophotography, the smaller value is often sampled with three pixels, so that a value range of 0.67" - 2" is applied for this FWHM.

A suitable combination of telescope and camera can be determined via the calculation tool 'Deep Sky' in the menu point 'Tools'.