Resolving Power of a Telescope and CCD

1. Image Size 4. Optimum Detail
2. Resolving Power 5. Field of View
3. Pixel Size  

If you are imaging an object where fine detail is important then you need to get the best 'resolving power' performance from your telescope and camera.

Image Size

The diameter of the objective lens or mirror is D and its focal length is L, giving a focal ratio F:

F = L/D

Light rays from two objects have an angular separation θ. They pass through an objective of diameter D and focal length L and form an image on the CCD at prime focus B. The separation, d, of the objects on the CCD  is:

 d = Lsinθ

The effective focal length L can be increased or decreased using a Barlow, Focal Reducer or other form of projection, to increase or decrease the size of the image.

Resolving Power (Dawes Limit)

Due to diffraction, light rays have a natural tendency to spread out. Thus fine details of your target object will become slightly blurred in the eyepiece of a telescope or as projected onto a CCD. The ability of any optical system to capture fine detail, (its resolving power), is limited. According to the Dawes equation, the maximum resolving power, R, of a telescope is:

R = 125/D arcseconds

Where D is the diameter of the telescope objective in millimetres.  This theoretical limit is for perfect optics and seeing conditions and is seldom achieved in practice.

For example a 8" Meade LX200 has D=200mm so its maximum theoretical resolving power is 125/200= 0.625 arcseconds. However, in practice, you will seldom achieve resolution better than 1 arcsecond due to atmospheric 'seeing' conditions - and even that is on a 'really good night'. Of course if you are on the top of a mountain with perfect seeing then you may achieve better results.

Pixel Size (Nyquist Limit)

The ‘signal’ present in a CCD at the end of an exposure consists of variations in the electric charges of the pixels (photosites). Fine details of the subject are represented by differences in charge from one pixel to the next. This means we need at least two pixels, (a bright one and a dark one), to show any detail. No detail can be captured that is smaller than two pixels. This rule is formalised as the 'Nyquist Limit' and is based on Shannon’s sampling theorem. The theorem says that the measurement (sampling) frequency must be at least twice the frequency of the signal to be measured. Digitising pixel values is in effect ‘sampling’ of the image. 

Optimum Image Detail

If we want to capture the optimum amount of detail (e.g. surface detail of a planet) then we can see from the above that the resolving power and pixel size need to be reasonably matched. If the distance between the details in the image is d, and the pixel size is p then:

d ≈ 2p

By combining the above equations we can arrive at:

Fideal = (2 x pixel-size) x ( number of arcseconds in a radian) / (resolving power)

Which works out at:

Fideal = 3300 x pixel-size in mm.

For my cameras the optimum (theoretical) F factor is:

Camera Ideal F-ratio
Toucam-Pro F/18
Starlight  Express MX5-C F/31

Note the ideal F factor depends on pixel size, regardless of telescope size. You will need to investigate the various options for image projection to see how best to achieve the desired F-ratio.

Of course, a high F-ratio means a small field of view, low density of light in the image and the challenge of accurate guiding for long exposures. A compromise may need to be reached between optimum detail and these other factors. In practice I user F/20 for planetary work with both cameras as this is easily achieved with my F/10 'scope and a Barlow. Attempts using two stacked Barlows (F/40) yield a large but faint and fuzzy image. Even F/20 is quite difficult to handle in terms of tracking and image contrast. The alternative is to use F/10, take a large number of images, and stack them using the "drizzle" process. This can bring out more detail than any one individual image can obtain.

Field of View

CCD chips are usually quite small, so you only capture a small portion of the field of view visible through an eyepiece.

If the CCD is w pixels wide and each pixel is size p mm then the width of the whole CCD matrix d is:

d = wp

If θ radians is the field of view for the camera then:

θ = wp/L

This can be further simplified to state that:

(FoV in arcminutes) = 3437 x (width of CCD in mm)/(focal length in mm)

It may be useful to prepare a table of FoV values for your cameras/telescopes/projection methods. This will help planning the best set-up for an observing session. This is the table for my equipment.

Scope Projection Camera Field Width Field Height
         
8" LX200 x2 Barlow F20 Toucam 2' 6" 1' 33"
8" LX200 Prime F10 Toucam 6' 20" 4' 43"
8" LX200 Reducer F6.3 Toucam 9' 34" 7' 9"
8" LX200 x2 Barlow F20 MX5-C 6' 20" 4' 40"
8" LX200 Prime F10 MX5-C 9' 29" 7' 1"
8" LX200 Reducer F6.3 MX5-C 11' 52" 8' 51"
4" Helios Prime F5 Toucam 25' 24" 19' 1"
4" Helios Prime F5 MX5-C 33' 40" 24' 0"

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Tony Evans 2004-2014

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