DSO Photography for Dummies - Telescope and Sensor

In preparation

On this page I want to investigate how telescope and camera sensor can be tuned to each other for deep sky images "the easy way".

For the Hurried...

The quality of the fit of a camera sensor with a given pixel size to a given telescope focal length can be judged on the basis of its image scale (rule of thumb):

Image scale ["/pixel] = 206.265 * Pixel size [µm] / Focal length [mm] =>> 200 * Pixel size [µm] / Focal length [mm]

The image scale should lie between the guiding values of 1 and 2 (often, a value of 1.5 is mentioned).

If you want to consider the seeing, halve the FWHM value ["] for the local seeing and use this value or these values as a guide:

e.g. FWHM = 3" => 1,5 or you pick a range, e.g. FWHM = 2"-4" => 1-2

The rules of thumb that are presented and derived on this page can be found in Appendix: Collection of Rules of Thumb.

In preparation

Final Words

In preparation

Appendix: Collection of Rules of Thumb

The rules of thumb presented here use a light wavelength of 550 nm.

Resolving Power

Based on the Nyquist Frequency		After Dawes		After Rayleigh
Resolving power [rad] = 0.00055 / Aperture [mm] Resolving power ["] = 113.45 / Aperture [mm]		= 0.00056 / Aperture [mm] = 115.7 / Aperture [mm]		= 0.00067 / Aperture [mm] = 138.4 / Aperture [mm]

Pixel Size and Telescope Focal Length (Based on Resolving Power)

Based on the Nyquist Frequency		After Dawes		After Rayleigh
Pixel size [µm] = Focal ratio * 0.275 Focal length [mm] = Pixel size [µm] * Aperture [mm] / 0.275		= Focal ratio * 0.2805 = Pixel size [µm] * Aperture [mm] / 0.2805		= Focal ratio * 0.3355 = Pixel size [µm] * Aperture [mm] / 0.3355

Pixel Size and Telescope Focal Length (Based on Seeing)

Pixel size [µm] = Telescope focal length [mm] * FWHM ["] / 412.5
Telescope focal length [mm] = Pixel size [µm] / (FWHM ["] * 412.5

Airy Disk

D [µm] = 2.44 * 0.55 * Focal ratio = 1.344 * Focal ratio
D ["] = 276.73 / Aperture [mm]

Image Scale

Image scale ["/pixel] = 206.265 * Pixel size [µm] / Focal length [mm]

Appendix: Derivations of the Formulas

Optimum Pixel Size, Optimum Telescope Focal Length

The derivation follows:

Sternen-Surfer - Pixelgröße und Brennweite (Hans-Jürgen Strauch): sternen-surfer.jimdo.com/tipps/pixelgröße-und-brennweite

Often the question is asked what pixel size the sensor of a camera should have for a given telescope focal length. Conversely, for a given camera, i.e. pixel size, the question arises what focal length suitable telescopes should have. The following consideration may be helpful for this: Two objects can only be separated on the sensor if there is another pixel between them. The distance between these objects on the chip is therefore twice the pixel size:

The following can be taken from the diagram:

tan (angle) = 2 x Pixel size / Focal length or
Pixel size = tan (angle) * Focal length / 2 or
Focal length = 2 * Pixel size / tan (angle)

Note: A symmetrical approach with one pixel above and below the centerline is also conceivable. Then the angle would be "pixel size / focal length" and the Tangens would have to be taken twice (two right-angled triangles). In practice, the angle is so small that there are no numerical differences between the two variants. Even the linearization of the Tangens did not produce any differences in the numerical values for me.

Resolving Power

For the size of the angle, the resolving power of the telescope according to Rayleigh can be used under certain conditions, which is given by:

Resolving power [rad] = 1.22 * λ [mm] / Aperture [mm]
Resolving power [rad] = 1.22 * 0,00055 / Aperture [mm] = 0.000671 / Aperture [mm] = 0.00067 / Aperture [mm] (rule of thumb based on a wavelength λ = 550 nm given in mm)

Some authors use the Nyquist frequency instead (factor 1 => 0,00055) or the nearly identical Dawes criterion (factor 1,02 => 0,000561); for the respective rules of thumb see further below!

With λ = 550 nm = 0,00055 mm for (average value for the light spectrum perceived by the human eye) and converting the resolving power into more common angular measures ("), we get (see "Sidepath" beow for the numbers):

Resolving power ["] = 1.22 * 206264.81 * 0.000550 / Aperture [mm] (Rayleigh) // 1.02 * 206264.81 * 0.000550 / Aperture [mm] (Dawes) // 1 * 206264.81 * 0.000550 / Aperture [mm] (Nyquist)
Resolving power ["] = 138.4 / Aperture [mm] (rule of thumb; Rayleigh) // 115.7 / Aperture [mm] (rule of thumb; Dawes) // 113.45 / Aperture [mm] (rule of thumb; Nyquist)

Inserting the formula for the resolving power (only shown for the Rayleigh criterion) and converting the pixel size to µm results in:

tan (1.22 * λ [mm] / Aperture [mm) = 2 * Pixel size [mm] / Focal length [mm]
tan (1.22 * λ [mm] / Aperture [mm) = 2 * Pixel size [µm] / (1000 * Focal length [mm])
tan (1.22 * λ [mm] / Aperture [mm) = Pixel size [µm] / (500 * Focal length [mm])

Resolving the formula to pixel size in µm, while all other measures are in mm, and inserting a wavelength of 550 nm in mm leads to:

Pixel size [µm] = Focal length [mm] * tan (1.22 * λ [mm] / Aperture [mm]) * 500
Pixel size [µm] = Focal length [mm] * tan (0.000671 / Aperture [mm]) * 500

Linearizing leads in first approximation to the following rule of thumb:

Pixel size [µm] = Focal length [mm] * 0.000671 * 500 / Aperture [mm]
Pixel size [µm] = Focal length [mm] / Aperture [mm] * 0.3355
Pixel size [µm] = Focal ratio * 0.3355 (rule of thumb)

With a given pixel size in µm, the focal length of the telescope im mm amounts to:

Focal length [mm] = Pixel size [µm] / (tan (1.22 * λ [mm] / Aperture [mm]) * 500)

Linearizing leads in first approximation to the following rule of thumb (550 nm in mm):

Focal length [mm] = Pixel size [µm] * Aperture [mm] / (500 * 1.22 * λ [mm])
Focal length [mm] = Pixel size [µm] * Aperture [mm] / (500 * 1.22 * 0.00055 [mm])
Focal length [mm] = Pixel size [µm] * Aperture [mm] / 0.3355 (rule of thumb, Rayleigh criterion)

Note: Instead of starting from the formuölar for the focal length, we might have also have started from the linearized formula for the pixel size, in order to arrive at the rule of thumb for the telescope focal length.

All these steps brbrought us to rules of thumb for pixel size and telescope focal length, which are rarely used, however, because instead of the resolving power of the telescope, values for local seeing are being used:

Pixel size [µm] = Focal ratio * 0.3355
Focal length [mm] = Pixel size [µm] * Aperture [mm] / 0.3355

For the Dawes criterion and the Nyquist frequency the following rules of thumb follow:

Focal length [mm] = Pixel size [µm] * Aperture [mm] / (500 * 1,02 * λ [mm])
Focal length [mm] = Pixel size [µm] * Aperture [mm] / (500 * 1,02 * 0,00055 [mm])
Focal length [mm] = Pixel size [µm] * Aperture [mm] / 0,2805 ((rule of thumb, Dawes criterion)

Focal length [mm] = Pixel size [µm] * Aperture [mm] / (500 * λ [mm])
Focal length [mm] = Pixel size [µm] * Aperture [mm] / (500 * 0,00055 [mm])
Focal length [mm] = Pixel size [µm] * Aperture [mm] / 0,275 ((rule of thumb, Nyquist frequency)

Mit all den obigen Schritten sind wir bei Faustformeln für die Pixelgröße und Teleskopbrennweite angelangt, die jedoch selten eingesetzt werden, weil statt des Auflösungsvermögens des Teleskops eher Werte für das lokale Seeing verwendet werden:

Pixelgröße [µm] = Öffnungsverhältnis * 0,3355 // 0,2805 // 0,275 (Rayleigh//Dawes//Nyquist frequency)
Brennweite [mm] = Pixelgröße [µm] * Öffnung [mm] / 0,3355 // 0,2805 // 0,275 (Rayleigh//Dawes//Nyquist frequency)

Sidepath: For the conversion of radians into angles, or more precisely, into degrees, minutes, and seconds, the following applies:

1 rad = 360 / (2 * pi) = 57.29°
1 rad = 60 * 360 / (2 * pi) = 3,437.75'
1 rad = 3600 * 360 / (2 * pi) = 206,264. 81" (or 1" = 0.000004848 rad)

Often the wavelength is entered in µm, while mm is used in the calculations. For this purpose, everything has to be be divided by 1000, which leads to the constant 206.265, which is often seen in rules of thumb or formulas.

Seeing

For DSO images, the influence of seeing is usually taken into account when fitting a camera sensor to a telescope. Instead of the resolution, the local seeing is used in the form of an FWHM value (in arcseconds) in the formula for the pixel size:

Pixel size [µm] = Focal length [mm] * tan (FWHM ["] / 3600 * PI/ 180) * 500
Pixel size [µm] = Focal length [mm] * tan (FWHM ["] / 206264.81) * 500

Linearizing (first approximation) leads to:

Pixel size [µm] = Focal length [mm] * FWHM ["] / 206264.81 * 500
Pixel size [µm] = Focal length [mm] * FWHM ["] / 412.5 (rule of thumb)

For telescope focal length, we get::

Focal length [mm] = Pixel size [µm] / (tan (FWHM ["] / 3600 * PI / 180) * 500)
Focal length [mm] = Pixel size [µm] / (tan (FWHM ["] / 206264.81) * 500)

Linearizing (first approximation) leads to:

Focal length [mm] = Pixel size [µm] / (FWHM ["] / 206264.81 * 500)
Focal length [mm] = Pixel size [µm] / (FWHM ["] * 412.5 (rule of thumb)

Airy Disk

The airy disk determines the minimal size, at which stars appear in a telescope. Its diameter D (length, angular size) is calculated according to the following formulas:

Length:
- D [µm] = 2.43932 * λ [µm] * Focal ratio
- D [µm] = 1.3441626 * Focal ratio (wavelength = 550 nm)
- Rules of thumb:
  - D [µm] = 2.44 * 0.55 * Focal ratio
  - D [µm] = 1.344 * Focal ratio(wavelength = 550 nm)
Angle measure:
- D ["] = 360 * 3600 / PI * arcsin (1.21966 * λ [mm] / (Aperture [mm]))
  (λ = wavelength in mm, e.g. 550 nm = 0,55 µm = 0,00055 mm)
- D ["] = 360 * 3600 / PI * 1.21966 * λ [mm] / (Aperture [mm])
  (linear approximation; λ = wavelength in mm, e.g. 550 nm = 0,55 µm = 0,00055 mm)
- Rules of thumb:
  - D ["] = 276.73 / (Aperture [mm]) ≈ 277 / (Aperture [mm]) (λ = 550 nm)

(The numerical values correspond to the Rayleigh criterion.)

Image Scale

Here the diagram above is valid as well, but only for one pixel (the triangle is made of one pixel and the focal length):

Image scale ["/pixel] = atan (Pixel size [µm] / (1000 * Focal length [mm])) * 3600 * 180 / PI

Linearisation (approximation) and rule of thumb:

Image scale ["/pixel] = Pixel size [µm] / (1000 * Focal length [mm])) * 3600 * 180 / PI
Image scale ["/pixel] = 206.265 * Pixel size [µm] / Focal length [mm] (rule of thumb)

Optimum Aperture Ratio, Optimum Focal Length

The optimum aperture ratio follows from a formula given by Stefan Seip:

fo (BW) = Pixel size [μm] / (0.51 x λ [nm]) = Pixel size [μm] / (0.51 * 550) * 1000
fo (BW) = Pixel size [μm] * 3.57 (rule of thumb)
fo (Color) = Pixel size [μm] / (0.51 x λ [nm]) * √2 = Pixel size [μm] / (0.51 * 550 [nm]) * 1000 * √2
fo (Color) = Pixel size [μm] * 5.00 (rule of thumb)

With: fo = Aperture for a diffraction-limited image, pixel size = edge length of a pixel (μm), λ = wavelength of light (550 nm for "white" light) The factor √2 takes into account the influence of the Bayer filter in color sensors.

The optimum focal length can then be determined in two ways:

Optimum focal length = fo* Aperture
Optimum focal length = Focal length * fo * Focal ratio = Focal length * fo / (f value of the telescope)

Typically, with this approach, the telescope focal length must be extended using suitable Barlow lenses or focal extenders.