Resolving Power of a Telescope and CCD
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.
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