| article: The Resolution of a Telescope - Dawes, Rayleigh and Sparrow |
Diffraction limited optics
A lot has been written about the ultimate resolution limit and performance of telescopes.
In general we can say that any aperture with a finite size will cause diffraction and hence it's resolution will be limited.
The finite aperture (front lens, main mirror) must cut a part of the incoming plane wave front.
This missing part is disturbing the otherwise perfect interference of the propagating waves in a certain way.
This is resulting in a modultaion of the wave front called the Point Spread Function (PSF).
Instead of the perfect image of a point this PSF (for an otherwise perfect telescope!) creates something like pictured below (when sliced in the center of this 3-dimensional shape):
PSF of unobstructed, perfect aperture [OSLO light]
The Airy Disk for a perfect and unobstructed telescope is encircling 84% of the energy from the point light source.
16% are spread to the rings with the first ring containing the biggest part.
A telescope with such a perfect Airy Disk is also said to have a Strehl Ratio of 100%.
Because of possible aberrations (i.e. Spherical aberration, Coma, Astigmatism, Chromatical aberration) and other things like wrong focus, central obstruction and spider vanes a real life's telescope has a Strehl Ratio of less than 100%.
Somehow arbitrarily a telescopes with a Strehl Ratio better than 80% is called to be diffraction limited. This is corresponding to a resulting wavefront error of λ/4.
Estimating the Resolution Limit: Airy, Rayleigh, Dawes and Sparrow
Obviously the resolution limit or resolving power is connected to that Airy Disk, because no detail imaged by the telescope can be smaller than this disk.
The radius of the Airy Disk can be estimated with:
q lin = 1.22 * ( f * λ ) / D (for linear resolving in cycles/mm)
q ang = 1.22 * λ / D (for angular resolving in arc sec)
For both cases the Airy Disk is getting smaller when the diameter of the aperture is getting bigger.
For the linear case the Airy Disk is getting smaller when f is getting smaller or in other words when the f/# is getting faster.
On the other hand the Airy Disk is shrinking with smaller wavelengths. Blue light is resolving better than red light.
Because the human eye beeing most sensitive to green light the wavlength of 550 nm is very often picked for the calculations.
Rayleigh's resolution limit
Lord Rayleigh now said that the ultimate resolution limit of a telescope is reached when the center of one Airy Disk is just one radius away of the other one.
2 PSF separated by one radius
Rayleigh's resolution limit [arc sec] = 140 / Aperture Diameter [mm] (for green light)
Dawes's resolution limit
Dawes found out by own observations that he could resolve a binary star with both stars having a magnitude of 6 slightly better than Lord Rayleigh claimed.
The Dawe's limit is hence an empirical one and can be written as:
Dawes's resolution limit [arc sec] = 116 / Aperture Diameter [mm] (for green light)
Sparrow's resolution limit
A more appropriate resolution limit has been proposed by C. Sparrow (
Hecht, 2002,
).
He is not very well known to the amateur astronomers however his proposal was veryfied by professional astronomers.
Sparrow claimed that when the combined signal by two PSFs becomes a flat top the signals can still be separated.
When we allow the combined signal to form not a flat top but a slightly curved shape the distance between the two PSFs is just 1/2 of the radius of the Airy Disk.
2 PSF separated by 1/2 radius
That should be the definit limit of resolution giving almost no contrast between the two PSFs.
Only slightly closer together we would see a pattern which cannot be distinguished from a single PSF.
Slightly optimistic Sparrow's resolution limit [arc sec] = 70 / Aperture Diameter [mm] (for green light)
Examples for various Telescopes:
| | Telescope | D [mm] | Rayleigh | opt. Sparrow | | APO FS-60C | 60 | 2.33 | 1.17 | | APO 120 mm | 120 | 1.17 | 0.58 | | 6 inch Newtonian | 150 | 0.93 | 0.47 | | 8 inch Newtonian | 200 | 0.70 | 0.35 | | 10 inch Newtonian | 250 | 0.56 | 0.28 | | 12 inch Newtonian | 300 | 0.47 | 0.23 | | 16 inch Newtonian | 400 | 0.35 | 0.18 |
From experienced high resolution astrophotographers like
Jean Dragesco (Dragesco, 1995) and experienced observers like
J.B. Sidgwick (Sidgwick, 1971)
we know that the Rayleigh and Dawes limits can be easily surpassed with excellent telescopes!
Given the fine quality of today's telescopes I am sure that most of the readers have already seen the Cassini Division of Saturn's ring system.
The author was able to glimpse it in his 60 mm APO refractor and clearly see it in his 150 mm Newtonian reflector.
But the Cassini Division is barely 0.5 seconds of arc when it is best placed towards the earth!
And we have seen that a 150 mm telescope should only resolve 0.9 to 0.8 seconds of arc while the 60 mm one is even worse having about 2 arc sec resolution limit.
The optimistic Sparrow gives 1.17 arc sec resolution for the 60 mm APO refractor in green light.
Now, when we assume that the blue-violett part of the light gives 30 % better resolution compared to green, we are quite close to the observational values.
Obviously it is fair to use the optimistic Sparrow limit as the resolution limit for a telescope manufactured close to perfection (Strehl close to 100%).
It should also be mentioned here that single features linke the Encke division in the ring system of Saturn are visible in 8 inch telescopes.
The Encke division is in the order of several 1/100 of arc seconds when at it's maximum (depending of the current angle between Saturn and Earth).
This does not mean that the Encke division is resolved, it's just visible.
If the Encke divison would actually be be formed by two small gaps you would not resolve that but see one single feature instead.
This phenomen can be regarded to be a sort of undersampling hence being imaged with the wrong contrast and the wrong extent.
So the image would not be useable for astrometry or photometry of the Encke division.
The discussed limits are only dealing with the highest spatial frequencies or finest details.
This may be reasonable for splitting close double stars, but planetary and galactical detail is usually at lower frequencies.
It must be emphasized that the discussed limits are not dealing with that.
For judging the performance of a telescope and a coupled CCD or digital camera in all spatial frequencies please see also:
The Performance of a Telescope - The Modulation Transfer Function MTF
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