# Telescope and Sensor

On this page I want to investigate how telescope and camera sensor can be tuned to each other and what differences there are between deep sky images on the one hand and moon, sun and planet images on the other hand.

Notes

• Despite all the calculating, these are all just rough "guidelines", which are successfully violated again and again in practice...
• Page DSO Photography for Dummies - Telescope and Sensor presents this topic shorter, simpler and restricted to DSO photography.

### 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.

## Introduction

### Questions...

There are a large number of astronomy cameras from different manufacturers on the market. A distinguishing feature is the size of the cells of the camera sensor, also called pixel size. Hobby astronomers, who want to get into astrophotography or EAA (Electronically Augmented Astronomy) or even buy another astronomy camera, are thus faced with the question of what pixel size the sensor of such a camera should have in order to fit the focal length of their telescope or telescopes optimally ("optimal adaptation"). Conversely, for a given camera, i.e. with a given pixel size of the sensor, the question arises what focal length your telescope should have so that it fits the sensor optimally. This raises a number of questions: Why does pixel size matter? What does "optimal fit" mean in this context? And how do you find it? I would like to answer these questions in the following!

#### Digitization...

Unfortunately the answer to these questions is not easy and has to be somewhat "theoretical". First of all, we have to realize that using a digital camera on a telescope is a process in which an analog signal, the optical telescope image, is converted into a digital one, namely the image produced by the camera sensor. Ideally, this conversion, also called digitization, should be lossless, so that in the digital version even fine or, more so, the finest details of the original are preserved. For example, if you digitize music for a CD, the aim is to transfer all audible frequencies, i.e. all frequencies between 20 and 20,000 Hertz. But how do you achieve (as far as possible) loss-free digitization and how does this look like in concrete case of a telescope with a digital camera attached?

#### The Digitization of Spatial Signals (Images)

While when digitizing temporal signals the analog signal is measured (sampled) in rapid temporal succession, spatial signals are measured (sampled) "side by side", that is, spatially distributed and often temporally in parallel. In digital photography, where two spatial dimensions are to be captured, this "spatial juxtaposition" is realized by rectangular sensors, which are built up from a matrix of tiniest light-sensitive cells, called pixels. Here, too, the aim is to preserve the details, that is, to prevent objects and spatial structures that are as small as possible from disappearing. In the case of a telescope, these are the smallest stars that a telescope can show. The size of these "smallest stars" is determined by the resolving power (resolution) of the telescope, which depends on the aperture of the telescope. So these "smallest stars" have to be obtained when imaging with a digital camera!

#### And now to the Initial Question, the Question of the Pixel Size!

A camera connected to a telescope captures the optical image produced by the telescope with a sensor consisting of a rectangle of tiny sensor cells, the "pixels". And, as we know from digital photography, the number of pixels that a camera sensor has is important - and for a given sensor size, this number also determines the size of the pixels, which we usually care little about. This is, however, different in astrophotography; here the size of the pixels plays a role, and precisely in the question of how to achieve the best lossless digitalization possible. Our somewhat "imprecise" initial question, namely, what size the pixels should have in order to achieve an "optimal adaptation" of the telescope and camera sensor, can now be reworded as: What size should the pixels of the camera sensor have so that the optical signal can be digitized without losses so that that even the finest stars that the telescope can show are preserved.

#### The Answer, theoretical and general at first...

This question is first of all answered in general terms by the Nyquist theorem: It states that the "sampling rate" must be at least twice as high as the highest frequency to be transmitted. For CDs, therefore, 44 kHz is chosen in order to transmit 20 kHz safely. In the case of spatial signals (we speak of so-called "spatial frequencies", more difficult for laypersons to imagine...), the "receiving grid" of sensor cells must be at least twice as fine as the finest details of the original image, which should still be preserved.

#### And now practical!

For astronomy cameras, this means that the smallest imageable stars must fall on at least two pixels for them to be imaged "optimally" (if they fall on three pixels, the stars become even rounder...). The finest stars that a telescope can show correspond in size to its resolving power, so a pixel must be half the size or less than the resolving power of the telescope used. So we basically got the answer to the question asked at the beginning! What is still missing are formulas to calculate the optimal pixel size, because the resolving power is given in arcseconds and the pixel size in micrometers. I have found such and other formulas on the Internet and would like to present them in short form below. More detailed formulas and derivations as well as reasons for certain factors and values can be found on page Telescope and Sensor.

#### Even more practical: The air turbulence (Seeing)!

In astronomical practice, there is unfortunately still a complication! The air tends to be restless and turbulent, in English we speak of "seeing" (I will use this term in the following), and this enlarges the star images to some degree. In practice, this does not have an effect on short time exposures (moon, sun, planets), but it does have an effect on photos with longer exposure times, like deep sky photos. For these photos, the telescope resolution is therefore not important, but the larger seeing value (as a FWHM value), which in principle is a measure of the size of a "bloated star". This case can be treated with the formulas mentioned above by entering the desired FWHM value into the formulas instead of the resolution (see below).

#### Why "optimal pixel size"? Types of sampling

The formulas for pixel size on the Internet typically refer to an "optimal pixel size", and I have used this term as well. In fact, the Nyquist theorem has only an upper limit on pixel size, and therefore the pixels might be as small as you like. So there must be practical reasons for the upper limit to be the optimum and therefore also the lower limit, although in certain cases, such as the image scale formulas listed below, you may want to aim for a range around the optimum.

To the upper limit first! If a star falls on less than two pixels, the digitized image becomes coarser than the original. In "technical jargon", this is referred to as "undersampling". The Nyquist theorem helps us to avoid this! Now to the lower limit! Basically, the larger the pixels of a sensor are, the more light-sensitive it is (and the pixels themselves as well). Small pixels therefore lead to a lower sensitivity, and therefore the pixels should be as large as possible to keep exposure times short. They are, as we learned above, when a star falls on exactly two pixels. The range around this optimum is also called "good sampling". However, smaller pixels are not only less sensitive to light, but in the case of astronomy, where we are dealing with weak signals, the smaller the pixels, the more the signals, i.e. stars, spread over increasingly more pixels. This further weakens an already weak signal. On the other hand, the more pixels an object is distributed over, the more details appear (provided that these details can be reproduced). Therefore, in applications where there is enough light available, such as in lunar, solar, and planetary photography, this approach, called "oversampling", is used in practice. For this purpose, formulas have been developed which calculate an optimal compromise between details and exposure time (see below).

### Outlook

In the following I present some simple formulas for the optimal adaptation of telescopes and sensors, for which there are often also "rules of thumb" that simplify the calculations. The formulas for pixel size and telescope focal length are a direct application of the approach just described. For the other formulas I, did not find any derivations, but they are also based on the principles described here.

Because of what is written about seeing, I distinguish in the following between deep sky photography (long exposures) and moon, sun and planetary photography (short exposures), even though the "basic formulas" have the same basis.

## Deep Sky Photos

In the following I present formulas that are used for deep sky photography; there are "rules of thumb" for them, which make things easier in practice and which I provide here (the exact formulas are presented in the appendix):

1. If you are looking for a suitable camera for deep-sky photography, you will use the formulas for pixel size and telescope focal length (formulas 1a-d deliver "theoretical" values) where you can also consider the influence of the seeing (formulas 2a/b).
2. If a camera is already at hand, you will want to determine the image scale for different telescopes in your own equipment (formula 4), where there is also the possibility to take the seeing into account (formulas 5a/b).
3. And finally, the recommended focal length range of a telescope can be determined for a sensor with the help of the image scale (with and without seeing influence; formulas 6a-c).

### (1) Pixel Size

#### Depending on the Resolution

For the optimum pixel size or telescope focal length, the following "rules of thumb" formulas have been developed in which the resolving power of the telescope after Rayleigh is indirectly a determining factor (for the derivation of the formulas and more exact formulas see the appendix):

• Pixel size [µm] = Focal ratio [mm] * 0.3355 (Formula 1a)
• Focal length [mm] = Pixel size [µm] * Aperture [mm] / 0.3355 (Formula 1b)

These formulas are typically not used for deep sky photos and presented here for reference only (they are used in a table further down).

#### Depending on the 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 or telescope focal length; here, are the corresponding "rules of thumb" (for the derivation of the formulas and the exact formulas see the appendix):

• Pixel size [µm] = Focal length [mm] * FWHM ["] / 412.5 (Formula 2a)
• Focal length [mm] = Pixel size [µm] / FWHM ["] * 412.5 (Formula 2b)

Example (TLAPO1027)

• Focal length 714 mm; seeing = 3" (according to H.J. Strauch, the average value for Central Europe) >> pixel size = 5,2 [µm].
>> This fits quite well for the Atik Infinity with 6.45 µm pixel size!
• Focal length 714 mm; Pixel size Atik Infinity = 6.45 [µm]; seeing = 3" (according to H.J. Strauch the average value for Central Europe) >> Telescope focal length = 887 [mm]
>> The focal length difference is not too big, the Atik Infinity fits quite well on the TLAPO1027!

#### Depending on the Size of the Airy Disk

The diameter of the Airy disk, which is the effective aperture diameter of an optical system, determines its resolving power. Two points can be separated reliably according to the Rayleigh criterion if the maxima of their images are separated by at least the radius of the Airy disk. The diameter also indicates the minimum size with which stars are imaged in the telescope.

The diameter D (length, angular size) of the Airy disk is calculated according to the following "rules of thumb" (for the exact formulas see the appendix):

• Length:
• D [µm] = 2.44 * 0.55 * Focal ratio
• D [µm] = 1.344 * Focal ratio (Formula 3a)
• Angle:
• D ["] = 276.73 / Aperture [mm] (Formula 3b)

Often only the rounded value "277" is used. In angular measure, the Airy disk is twice as large as the Rayleigh resolving power (on which it is based), because the resolving power refers to the radius, while the Airy disk is typically used with the diameter.

When observing DSO, the airy disk may be larger than the current seeing values, measured as FWHM values (in seconds). In such a case, the larger value, i.e. the size of the airy disk, has to be used. For a comparison with the FWHM value, the size of the airy disk in seconds is needed, for determining the pixel size, its size in µm. The latter has to be halved for arriving at the sensor's pixel size, because the airy disk size refers to two pixels.

Example (Vaonis Vespera)

• A focal ratio of F/4 and a wavelength of 0.55 µm (550 nm) lead to a diameter of 5.37 µm >> the "ideal" sensor pixel size is 2.68 µm.
• An aperture of 50 mm and a wavelength of 0.00055 mm (550 nm) lead to a diameter of 5.54" >> is above a FWHM of 5".

### (2) Image Scale

The image scale (in arcseconds per pixel; for the derivation of the formulas see the appendix) is used as a measure of the quality of the fit of telescope and sensor if a sensor is already given. Depending on the value of the image scale, a distinction is made between "oversampling", "undersampling" and "good sampling"*. "Good sampling" corresponds to an optimum fit, for which there are guiding values for the image scale that differ for deep sky photos and for moon, sun, and planet photos. For the latter, "oversampling" (smaller than the "ideal" values) is also often used. Undersampling (larger than the "ideal" values) should be avoided in any case.
*) See the Baader Planetarium glossary, article Der Begriff sampling, over,- under- und good sampling (www.sbig.de/universitaet/glossar-htm/sampling.htm) with sample images for these sampling variants.

#### For "Good Sampling

The image scale (in arcseconds per pixel) is calculated according to:

• Image scale ["/pixel] = 206.265 * Pixel size [µm] / Focal length [mm] (Formula 4; rule of thumb)

Often only the rounded value of "206" is being used.

This value is used to judge the quality of the fit of a camera sensor/telescope combination. For deep-sky photos, the rule of thumb for "good sampling" is to aim for an image scale of about 1 to 2 seconds per pixel (other specifications that I found are: 1.25, 1.5, 1.5-2, 1-2.5 and even 0.7-3)*. Values for the image scale above 2 are called "undersampling", values below 1 are called "oversampling".

*) Reasons for these guiding values are usually not given, but obviously they are based on typical values for the seeing (in Central Europe). More about this below!

Example (TLAPO1027)

• Focal length 714 mm, aperture 102 mm, aperture ratio 1/7; pixel size Atik Infinity 6.45 [µm] >> image scale = 1.86 ["/pixel]
>> That is still acceptable for deep sky photos...

#### Depending on the Seeing

According to H.J. Strauch, one simply halves the seeing value (FWHM) in practice and uses this as the desired image scale value. This is, In principle, the application of the Nyquist theorem, which states that the sampling rate should be twice the frequency of the sampled analog signal. Thus, the image scale calculated according to formula 4 is not checked according to whether it lies between the "ideal" values 1 and 2, but rather whether it is close to the image scale value determined by the FWHM value. More on this below!

To determine the pixel size of a sensor at a given telescope focal length, the formula for the image scale has to be transformed; the same applies to the telescope focal length at a given pixel size:

• Pixel size [µm] = Focal length [mm] * (FWHM ["] / 2) / 206.265 (Formula 5a; rule of thumb)
• Focal length [mm] = 206.265 * Pixel size [µm] / (FWHM ["] / 2) (Formula 5b; rule of thumb)

Example (TLAPO1027)

• According to the "rule of halving", a local seeing of 4" on average means that an image scale of 2 should be aimed for.
This results in a pixel size of 6.9 [µm] for the TLAPO1027 with 714 mm focal length; the Atik Infinity with 6.45 [µm] pixel size would fit.
The Atik Infinity with 6.45 [µm] pixel size would result in a focal length of 665.2 mm, which is close to the focal length of the TLAPO1027 with a focal length of 714 mm.

#### Astronomy.tool "Tweak"

Astronomy.tools writes about the sampling rate: "There is some debate around using this for modern CCD sensors because they use square pixels, and we want to image round stars. Using typical seeing at 4" FWHM, Nyquist's formula would suggest each pixel has 2" resolution which would mean a star could fall on just one pixel, or it might illuminate a 2 x 2 array, so be captured as a square." In order to achieve "round" stars, the authors of the Website propose to sample with the 3-fold frequency of the analog signal - but they do this only partially.

First of all, the authors assign FWHM value ranges to the different seeing conditions, and by dividing these values by 3 or 2 they arrive at "recommended" value ranges for the image scale (which they call "pixel size"...) They divide the FWHM value at the lower limit not by 2, but by 3, which leads to the following table, in which I also included the "standard procedure" of "halving":

 Seeing Image Scale Seeing Conditions FWHM- Value Astronomy.tools H.J. Strauch* Remarks From To From To From To Exceptional good seeing 0.5" 1" 0.17 0.5 0.25 0.5 Good seeing 1" 2" 0.33 1 0.5 1 OK seeing 2" 4" 0.67 2 1 2 Mean value = 3" for Central Europe => 1.5 (H.J. Strauch) Poor seeing 4" 5" 1.33 2.5 2 2.5 Very poor seeing 5" 6" 1.67 3 2.5 3

*) According to the "rule of halving" (from H.J. Strauch), if you use the seeing ranges defined by Astronomy.tools

Using an online calculator on the Astronomy.tools Website, you can calculate the image scale for your configuration (it calculates according to the rule of thumb given above) and relate it to the values of the local seeing. So you do not check whether this value lies between 1 and 2 (or whatever is given...), but whether it lies within the limits given by the local seeing conditions.

Example

• The case of "OK Seeing " (local seeing between 2" and 4") leads to an image scale between 0.67 and 2 (or according to the "halving rule" of 1 to 2), which should therefore be aimed at.
This results in a pixel size for the TLAPO1027 with 714 mm focal length between 2.3/3.46 [µm] and 6.9 [µm]; the Atik Infinity with 6.45 [µm] pixel size would just about fit.

#### Where Do the Recommendations for the Value of the Image Scale Come from?

As already mentioned, Internet sources usually do not provide any justification for the "ideal" image scale values given. My suspicion that they are based on typical values for seeing in Central Europe seems to be confirmed by the table above.

The often mentioned value range of 1-2 for the scale of reproduction corresponds to "OK Seeing", the also often mentioned value of 1.5 corresponds to the "average seeing" of 3", which H.J. Strauch states for Central Europe. Other values or value ranges seem to be merely "variations" of this. In this respect, it is probably best to calculate the image scale for one's own or intended configuration and the expected seeing and compare it with the table above. Whether one then follows the interpretation of Astronomy.tools or that of H.J. Strauch and others is up to the individual...

### (3) Recommended Focal Length Range

With the help of the rule of thumb that the image scale should be between 1 and 2, one can also determine the focal length range recommended for a sensor and thus check if one's own telescopes are in a suitable focal length range. For the sake of simplicity, I use here the rule of thumb for the image scale, which I reform accordingly:

• Focal length [mm] = 206.265 * pixel size [µm] / Image scale ["/pixel] (Formula 6a; rule of thumb)

To determine the focal length range, I now insert the values "2" and "1" into the formula one after the other:

• Focal length [mm] = 206.265 * Pixel size [µm] / 2 to 206.265 * pixel size [µm] (Formula 6b/c; rule of thumb)

If you want to include seeing (see Astronomy.tools), just enter the corresponding values for the image scale (upper and lower limit, e.g. 0.67 and 2 for "OK Seeing") into the formula.

Example

• TLAPO1027: focal length 714 mm; PS 72/432: focal length 432 mm; Skymax-127: focal length 1500 mm; C8: focal length 2032 mm; C8R: focal length: 1280 mm; pixel size Atik Infinity 6.45 [µm]
Telescope focal length [mm] = 206.265 * 6.45 / 2 to 206.265 * 6.45 = 665.2 to 1330.4
>> Thus the TLAPO and the C8 with f/6.3 reducer fit into the recommended focal length range. With a 0.5-fold reducer, the C8 and also the Skymax-127 should fit.
• With "OK Seeing ", for local seeing between 2" and 4", an image scale between 0.67 and 2 (or according to the "halving rule" from 1 to 2) should be aimed for.
The Atik Infinity with 6.45 [µm] pixel size would result in a focal length between 665.2 mm and 1330.4/1986 mm, which includes the TLAPO1027's focal length of 714 mm.
Probably a camera with smaller pixels (e.g. ASI 224 with 3.75 [µm]) would be better suited to this telescope. Here the focal length range would be between 387 mm and 773/1154 mm.

## Moon, Sun, and Planet Photos

In the following I present formulas for moon, sun and planet photography, for which there are often "rules of thumb":

• If you are looking for a suitable camera for deep sky photography, you will use the formulas for pixel size and telescope focal length (formulas 1a/b). If you already have a camera at hand, you will want to determine the image scale for different telescopes in your own telescope park (formula 4)
• Furthermore, I present formulas for the case that oversampling is to be used, i.e. many details are to be shown (formulas 7a/b, 8).

### (1) Good Sampling

#### Pixel Size, Telescope Focal Length

As written above, when photographing these objects with exposure times of fractions of a second, the turbulence in the atmosphere is practically "frozen". This makes it possible to calculate with the theoretical resolution of the telescope; here only the rules of thumb:

 Based on the Nyquist Frequency After Dawes After Rayleigh Pixel size [µm] = Focal ratio * 0.275 Focal lengt [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 [µm] (Formula 1a); Focal length [mm] (Formula 1b)

Example (TLAPO1027, Rayleigh/Dawes/Nyquist)

• Focal length 714 mm, F/7, resolution 1.15" >> pixel size = 2,35 / 1,96 / 1,9 [µm]
>> This does not fit well for the Atik Infinity with 6.45 µm pixel size! It would have to be operated with binning.
• Resolving power 1.15, aperture 102 mm; pixel size Atik Infinity 6.45 [µm] >> telescope focal length = 1960.95 [mm]
>> This requires a 3-fold tele extender

#### Image Scale

From the following formulas, the image scale can be determined, provided the telescope focal length and sensor (pixel size) are given:

• Image scale ["/pixel] = 206.265 * Pixel size [µm] / Focal length [mm] (Formula 4; rule of thumb)

Example (TLAPO1027)

• Focal length 714 mm; pixel size Atik Infinity 6.45 [µm] >> image scale = 1, 86 ["/pixel] (rule of thumb)

I have not been able to find any other standard values for the image scale of these objects (moon, sun, planets), although certain sources write that they exist...

### (2) Oversampling, Optimum Aperture Ratio, Optimum Focal Length

For moon, sun and planetary photos taken with webcams or CCD/CMOS cameras, it may be useful to "oversample" the images in order to capture more details. In doing so, the light is distributed over more pixels than required by the Nyquist criterion to achieve the image resolution, because the loss of sensitivity is not a major factor (if the seeing allows for showing the details). However, an arbitrary increase of the focal length is not reasonable. Instead, a compromise between focal length and image brightness (and thus the exposure time) is aimed at. For this purpose, the optimum aperture ratio "fo" is calculated according to a formula given by Stefan Seip (see the appendix) or according to the following rules of thumb:

• fo (SW) = pixel size [μm] * 3.57 (Formula 7a; rule of thumb)
• fo (color) = pixel size [μm] * 5.00 (Formula 7b; rule of thumb) (according to a posting by Gerd Düring the B&W value of 3,57 is also valid for color)

The easiest way to determine the optimum focal length is:

• Optimum focal length = fo* Aperture (Formula 8; Rule of thumb)

Examples

(1) Atik Infinity Colour camera, pixel width 6.45 μm. For this, the formula with factor 5 results in an optimum aperture of 32.25 (i.e. 32) and thus an optimum aperture ratio of about f/32 (1:32).

Application to my telescopes:

• TLAPO1027: focal length 714 mm, aperture 102 mm, f/7 ratio.
Optimum focal length = 32 x 102 mm = 3264 mm. The telescope focal length should be extended from 714 to 3264 mm (by the factor 4.57 = 5).
• PS 72/432: Focal length 432 mm, aperture 72 mm, focal ratio 1/6.
Optimum focal length = 32 x 72 mm = 2304 mm. The telescope focal length should be extended from 432 to 2304 millimeters (by a factor of 5.33 = 5).
• C8: Focal length 2032 mm, aperture 203 mm (203.2), focal ratio 1/10.
Optimum focal length = 32 x 203.2 mm = 6502 mm. The telescope focal length should be extended from 2032 to 6502 mm (by a factor of 3.2 = 3).

(2) Camera ASI 224 MV Color, pixel width 3.75 μm. For this, the formula with factor 5 results in an optimum aperture of 18.75 (i.e. roughly 16) and thus an optimum aperture ratio of about f/16 (1:16).

Application to my telescopes:

• TLAPO1027: focal length 714 mm, aperture 102 mm, f/16 (1:16).
Optimum focal length = 18.75 x 102 mm = 1912.5 mm. The telescope focal length should be extended from 714 to 1900 mm (by a factor of 2.68, i.e. about 2.5 or 3).
• PS 72/432: Focal length 432 mm, aperture 72 mm, focal ratio 1/6.
Optimum focal length = 18.75 x 72 mm = 1350 mm. The telescope focal length should be extended from 432 to 1350 mm (by a factor of 3.125 = 3).
• C8: Focal length 2032 mm, aperture 203 mm (203.2), aperture ratio 1/10.
Optimum focal length = 18.75 x 203.2 mm = 3810 mm. The telescope focal length should be extended from 2032 to 3810 mm (by a factor of 1.875 = 2).

## Applications

In the following, I present tables with calculation results based on the above formulas for my and some other telescopes and for camera sensors that are relevant for me. At the end of this section, I try to check the suitability of three sensor sizes for my telescopes using a reduced table.

### Calculations for My and Other Telescopes and Some Sensor Sizes

I calculated the following table using an Excel spreadsheet based on the formulas presented here.

#### Optimum Pixel Size

The optimum pixel size is calculated either by using the Rayleigh resolution or the seeing according to the halving rule (in some cases, the size of the airy disk may override these values, because it is larger).

 Telescope Resolution ["] Optimum Pixel Size [µm] Examples Focal Length [mm] Aperture [mm] f After Rayleigh After Dawes After Nyquist Via Resolution Via Seeing (FWHM) BS Rayl. Daw. Nyq. 2" 3" 4" 5" " Vespera 200 50 4 2.77 2.31 2.27 1.34 1.12 1.10 0.97 1.45 1.94 2.42 5.53 Stellina 400 80 5 1.73 1.45 1.42 1.68 1.40 1.38 1.94 2.91 3.88 4.85 3.46 APO 80/480 480 80 6 1.73 1.45 1.42 2.01 1.68 1.65 2.33 3.49 4.65 5.82 3.46 Heritage 100P 400 100 4 1.38 1.16 1.13 1.34 1.12 1.10 1.94 2.91 3.88 4.85 2.77 TLAPO1027 714 102 7 1.36 1.13 1.11 2.35 1.96 1.93 3.46 5.19 6.92 8.65 2.17 PS 72/432 432 72 6 1.92 1.61 1.58 2.01 1.68 1.65 2.09 3.14 4.19 5.24 3.84 eVscope, Newton 114/450 450 114 4 1.21 1.01 1.00 1.32 1.11 1.09 2.18 3.27 4.36 5.45 2.43 Newton 114/500 500 114 4.4 1.21 1.01 1.00 1.47 1.23 1.21 2.42 3.64 4.85 6.06 2.43 Heritage P130 650 130 5 1.06 0.89 0.87 1.68 1.40 1.38 3.15 4.73 6.30 7.88 2.13 6" Newton, Explorer 150PDS 750 150 5 0.92 0.77 0.76 1.68 1.40 1.38 3.64 5.45 7.27 9.09 1.84 6" Newton 900 150 6 0.92 0.77 0.76 2.01 1.68 1.65 4.36 6.54 8.73 10.91 1.84 6" Newton 1200 150 6 0.92 0.77 0.76 2.68 2.24 2.24 5.82 8.73 11.64 14.54 1.84 8" Newton, GSD 680 1200 200 6 0.69 0.58 0.57 2.01 1.68 1.65 5.82 8.73 11.64 14.54 1.38 Skymax-102 1300 102 12.7 1.36 1.13 1.11 4.28 3.58 3.40 6.30 9.45 12.61 15.76 2.71 Skymax-127 1500 127 11.8 1.09 0.91 0.89 3.96 3.31 3.25 7.27 10.91 14.54 18.18 2.18 Skymax-127R 750 127 5.9 1.09 0.91 0.89 1.98 1.66 1.62 3.64 5.45 7.27 9.09 2.18 Celestron C5 1250 125 10 1.11 0.93 0.91 3.36 2.81 2.75 6.06 9.09 12.12 15.15 2.21 Celestron C5R 787.5 125 6.3 1.11 0.93 0.91 2.12 1.77 1.73 3.82 5.73 7.64 9.55 2.21 Celestron C8 2032 203 10 0.68 0.57 0.56 3.36 2.81 2.75 9.85 14.78 19.70 24.63 1.36 Celestron C8R 1280 203 6.3 0.68 0.57 0.56 2.12 1.77 1.73 6.21 9.31 12.41 15.51 1.36 Celestron C8R2 1016 203 5 0.68 0.57 0.56 1.68 1.40 1.38 4.93 7.39 9.85 12.31 1.36 Celestron C14 3500 350 10 0.40 0.33 0.32 3.35 2.81 2.75 16.97 25.45 33.94 42.42 0.79

#### Image Scale

The rule of thumb was used for the image scale, because the exact formula delivers the same numerical values. The more exact value of "206.265" was used instead of "206". The optimum focal length is used for moon, sun, and planetary photos; the optimum aperture is pixel size [µm] * 5.

 Telescope Image Scale ["/Pixel] Optimum Focal Length [mm] (Color) Examples Focal Length [mm] Aperture [mm] f Pixel Size [µm] 2.4 2.9 3.75 4.63 6.45 2.4 2.9 3.75 6.45 Vespera 200 50 4 2.48 2.99 3.87 4.78 6.65 --- --- --- --- Stellina 400 80 5 1.24 1.50 1.93 2.39 3.33 960.00 1160.00 1500.00 2580.00 APO 80/480 480 80 6 1.03 125 1.61 1.99 2.77 960.00 1160.00 1500.00 2580.00 Heritage 100P 400 100 4 1.24 150 1.93 2.39 3.33 1200.00 1450.00 1875.00 3225.00 TLAPO1027 714 102 7 0.69 084 1.08 1.34 1.86 1224.00 1479.00 1912.50 3289.50 PS 72/432 432 72 6 1.15 138 1.79 2.21 3.08 864.00 1044.00 1350.00 2322.00 eVscope, Newton 114/450 450 114 4 1.10 133 1.72 2.12 2.96 1368.00 1653.00 2137.50 3676.50 Newton 114/500 500 114 4.4 0.99 120 1.55 1.91 2.66 1368.00 1653.00 2137.50 3676.50 Heritage P130 650 130 5 0.76 0.92 1.19 1.47 2.05 1560.00 1885.00 2437.50 4192.50 6" Newton, Explorer 150PDS 750 150 5 0.66 080 1.03 1.27 1.77 1800.00 2175.00 2812.50 4837.50 6" Newton 900 150 6 0.55 0.66 0.86 1.06 1.48 1800.00 2175.00 2812.50 4837.50 6" Newton 1200 150 6 0.41 050 0.64 0.80 1.11 1800.00 2175.00 2812.50 4837.50 8" Newton, GSD 680 1200 200 6 0.41 050 0.64 0.80 1.11 2400.00 2900.00 3750.00 6450.00 Skymax-102 1300 102 12.7 0.38 046 0.59 0.73 1.02 1224.00 1479.00 1912.50 3289.50 Skymax-127 1500 127 11.8 0.33 040 0.52 0.64 0.89 1524.00 1841.50 2381.25 4095.75 Skymax-127R 750 127 5.9 0.66 0.80 1.03 1.27 1.77 1524.00 1841.50 2381.25 4095.75 Celestron C5 1250 125 10 0.40 0.48 0.62 0.76 1.06 2436.00 2943.50 3806.25 6546.75 Celestron C5R 787.5 125 6.3 0.63 0.76 0.98 1.21 1.69 2436.00 2943.50 3806.25 6546.75 Celestron C8 2032 203 10 0.24 029 0.38 0.47 0.65 2436.00 2943.50 3806.25 6546.75 Celestron C8R 1280 203 6.3 0.39 047 0.60 0.75 1.04 2436.00 2943.50 3806.25 6546.75 Celestron C8R2 1016 203 5 0.49 059 0.76 0.94 1.31 2436.00 2943.50 3806.25 6546.75 Celestron C14 3500 350 10 0.14 017 0.22 0.27 0.38 4200.00 5075.00 6562.50 11287.50

### Deep Sky: Image Scale Depending on the Seeing

An image scale taken from the above table can now be compared either with the "good sampling" value range of 1-2 (or the value 1.5) or with the following image scale values, which are provided for different seeing conditions:

 Seeing Image Scale Seeing Conditions FWHM- Value Astronomy.tools H.J. Strauch* Remarks From To From To From To Exceptional good seeing 0.5" 1" 0.17 0.5 0.25 0.5 Good seeing 1" 2" 0.33 1 0.5 1 OK seeing 2" 4" 0.67 2 1 2 Mean value = 3" for Central Europe => 1.5 (H.J. Strauch) Poor seeing 4" 5" 1.33 2.5 2 2.5 Very poor seeing 5" 6" 1.67 3 2.5 3

*) Following the "rule of halving " (after H.J. Strauch), if you use the seeing ranges defined by Astronomy.tools.

### Focal Length Ranges for Different Sensor Sizes

The focal lengths were calculated for an image scale of 2 and 1.

 Camera/Telescope Pixel Size Focal Length (2) Focal Length (1) Range Opt. Pixel Site for Native Tubes Fits... µm mm mm mm µm Vespera 2.9 299.1 598.2 300-600 1.34 PS72/432, APO 80/480 Stellina 2.4 247.5 495.0 250-500 1.68 PS72/432, APO 80/480 eVscope, ASI120, ASI224 3.75 386.7 773.5 400-800 1.32 TLAPO1027, SM127R, PS72/432, APO 80/480, C5R ASI294 4.63 477.5 955.0 450-1000 depending on the focal length of the tube used TLAPO1027, SM127R, C5R Atik Infinity 6.45 665.2 1330.4 650-1300 depending on the focal length of the tube used TLAPO1027, C8R, C8R2, SM127R, C5, C5R

Instead, the focal lengths can also be calculated for the border values of the respective seeing conditions.

### Application to My Telescopes

In the following, I reduced the above table to my telescopes and try to apply the "rules". At the columns for the optimum pixel size, I highlight the 3" column (and slightly the 2" and 4" columns); at the image scale columns, I backlight suitable cells in color and highlight the sensor of my Atik Infinity camera. The result looks like this:

 Telescope Resolution ["] Optimum Pixel Size [µm] Image Scale Optimum Focal Length (Color) My Telescopes Foc. Len. [mm] Aperture [mm] f After Rayleigh Via Resol. Via Seeing (FWHM) BS For Pixel Size [µm] 2" 3" 4" 5" " 2.4 2.9 3.75 4.63 6.45 2.4 2.9 3.75 6.45 Vespera 200 50 4 2.77 1.34 0.97 1.45 1.94 2.42 5.53 2.48 2.99 3.87 4.78 6.65 --- --- --- --- TLAPO1027 714 102 7 1.36 2.35 3.46 5.19 6.92 8.65 2.17 0.69 0.84 1.08 1.34 1.86 1224.00 1479.00 1912.50 3289.50 PS 72/432 432 72 6 1.92 2.01 2.09 3.14 4.19 5.24 3.84 1.15 1.38 1.79 2.21 3.08 864.00 1044.00 1350.00 2322.00 eVscope 450 114 4 1.21 1.32 2.18 3.27 4.36 5.45 2.43 1.10 1.33 1.72 2.12 2.96 1368.00 1653.00 2137.50 3676.50 Celestron C5 1250 125 10 1.11 3.36 6.06 9.09 12.12 15.15 2.21 0.40 0.48 0.62 0.76 1.06 2436.00 2943.50 3806.25 6546.75 Celestron C5R 787.5 125 6.3 1.11 2.12 3.82 5.73 7.64 9.55 2.21 0.63 0.76 0.98 1.21 1.69 1500.00 1812.50 2343.75 4031.25 Celestron C8 2032 203 10 0.68 3.36 9.85 14.78 19.70 24.63 1.36 0.24 0.29 0.38 0.47 0.65 1500.00 1812.50 2343.75 4031.25 Celestron C8R 1280 203 6.3 0.68 2.12 6.21 9.31 12.41 15.51 1.36 0.39 0.47 0.60 0.75 1.04 2436.00 2943.50 3806.25 6546.75 Celestron C8R2 1016 203 5 0.68 1.68 4.93 7.39 9.85 12.31 1.36 0.49 0.59 0.76 0.94 1.31 2436.00 2943.50 3806.25 6546.75

The cells in the image scale columns are highlighted in cyan if they are in the range between 1 and 2 (halving rule); cells with values between 0.66 and 1 are highlighted in blue ("Tweak" according to Astronomy.tools). Text colors are meant to highlight relevant data.

I conclude from this:

• TLAPO1027: The Atik Infinity (6.45 µm) fits quite well this refractor at "OK Seeing"; the sensor of the ASI224 seems also to fit quite well (3.75 µm), especially under better seeing conditions.
• PS 72/432: Here, the sensor of the ASI224 (3.75 µm) fits perfectly; the 2.4 µm sensor would also be acceptable.
• eVscope: The Sony sensor used in the eVscope seems to be optimally chosen for this telescope.
• C5: The Atik Infinity would just about fit.
• C5R: The Atik Infinity would fit when using a 0.63 x reducer.
• C8: Here the Atik Infinity would still go straight, especially with good seeing. I do not know whether the camera will come into focus on the C8, it did so on a star friend's C8.
• C8R: With the f/6.3-Reducer/Corrector the Atik Infinity fits on the C8 and in my experience, it comes into focus.
• C8R2: With the f/6.5 Reducer/Corrector the Atik Infinity fits even better on the C8 I do not know whether the camera will come into focus with it. Probably the reducer vignettes.

The combination of both reducers also fits the C8, as I found out.

In short: on the TLAPO1027 the ASI224 (better with good seeing) and the Atik Infinity, on the PS 72/432 the ASI 224, on the Skymax-127 and C8 the Atik Infinity, but actually only with focal length reducers (on the 127R the ASI224 would also work).

I do not want to comment on the data for the optimum focal length here, except that 3x and 5x focal length extenders seem to be appropriate (which I own...).

## Final Words

It took me some time to recognize a certain "system" in the many formulas on the Internet. I hope that I present everything reasonably correct and understandable, so that other hobby astronomers can apply the formulas to their own equipment.

## Appendix: Collection of Rules of Thumb

The rules of thumb presented here are derived further below. They are based on 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 lengt [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]

### Optimum Aperture Ratio, Optimum Focal Length

• fo (BW) = Pixel size [μm] * 3.57
• fo (Color) = Pixel size [μm] * 5.00 (according to a posting by Gerd Düring the B&W value of 3.57 is also valid for color)
• Optimum focal length = fo* Aperture
• Optimum focal length = Focal length * fo * Focal ratio = Focal length * fo / (f value of the telescope)

## Appendix: Derivations of the Formulas

### Optimum Pixel Size, Optimum Telescope Focal Length

The derivation follows:

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

• 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)

• 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.

 gerd (at) waloszek (dot) de About me made by on a mac!
 21.03.2021