Scatter Calculator

[Ags]

I started a thread recently asking for information about the performance of the Skywatcher Skymax 180 Pro. It started quite a discussion about the relative contrast offered by refractors, SCTs, Maks and Newts.

I've tried to write a small program to calculate the actual level of light scatter for a given scope. The program takes as input the aperture, the size of the central obstruction if any, and number and width of vanes supporting the central obstruction. It also includes the transmission of the telescope.

The calculation is based on the idea that diffraction scattering is caused by the edges in the optical system, counterbalanced by the raw light gathering of the scope. Obviously as aperture goes up, the ratio of light gathering to edge scatter steadily increases, so larger scopes are sharper. Any further edges (like vanes and secondaries) increase the scatter. I think it doesn't matter where the edges are in the light column because the light is focussed to a point anyway.

As a final flourish the transmission of the telescope is used to reduce the light gathering of the scope, so lower transmission means more scatter proportionally to light gathered. (Obviously, transmission would also dim the diffraction scatter, but it would add new scattered light so I have ignored this in my calculation).

Anyway here are the results of my program as it stands today - I hope you can help me refine the calculations!

The results are given in two forms - a meaningless dimensionless ratio, and then as the equivalent 'perfect' telescope with 100% transmission and no central obstruction. Remember this is a calculation of contrast NOT light gathering.

120mm frac = 0.034722222222222224

120mm frac equates to an ideal 115.19999999999999mm telescope

204mm newt = 0.058886595383529014

204mm newt equates to an ideal 67.92717381516044mm telescope

102mm mak = 0.07159282582610962

102mm mak equates to an ideal 55.87152mm telescope

127mm mak = 0.05749974987608803

127mm mak equates to an ideal 69.56552mm telescope

C8 SCT = 0.03431571681105691

C8 SCT equates to an ideal 116.564664mm telescope

C9.25 SCT = 0.031957837768968704

C9.25 SCT equates to an ideal 125.16491349999997mm telescope

The obvious conclusion is that Newts have poor contrast due to their vanes, ad SCTs are equivalent to fracs 2/3 of their aperture. I think there is little difference between SCTs and Maks.

Acording to my formula, the use of super-thin vanes hardly makes a difference. According to my limited knowledge of optics (I trained as a chemist) this is correct - 0.5mm width is still enormous compared to the wavelength of visible light, so I say thin vanes improve light gathering but have no meaningful effect on reducing diffraction. Am I wrong?

Another thing my calculator does not consider is the effect of baffling and other features to reduce stray light.

[Ags]

I haven't calculated the Mak 180 Pro (which started all this off) as I don't know it's transmission and exact CO size.

[Ags]

I thought my brilliant scatter calculator would be met with more enthusiasm... ;-)

I refined the calculations a bit and added in some more popular telescopes to the default results. According to my calculations, the Skymax 180 Pro has less contrast than a C8 (as you might expect from a smaller telescope with a slightly larger central obstruction). Here are the results again, this time only showing how well real telescopes perform in terms of contrast versus 'ideal' telescopes with perfect glass and no central obstruction):

120mm frac equates to an ideal 115mm telescope

110mm newt equates to an ideal 35mm telescope

130mm newt equates to an ideal 42mm telescope

150mm newt equates to an ideal 49mm telescope

204mm newt equates to an ideal 68mm telescope

250mm newt equates to an ideal 84mm telescope

300mm newt equates to an ideal 101mm telescope

102mm mak equates to an ideal 56mm telescope

127mm mak equates to an ideal 70mm telescope

C8 SCT equates to an ideal 117mm telescope

C9.25 SCT equates to an ideal 135mm telescope

Skymax 180 equates to an ideal 102mm telescope

[John]

Well personally I'm a bit baffled by the results - don't know where to start really

The figures your formula has produced certainly don't seem to measure up to actual performance experiences as far as I can see - unless I'm missing something .....

[Ags]

That's beta software for you! ;-)

I think the formula might be a bit heavy in adjusting for vanes, I'm hoping someone with a better understanding of optics can improve on my simplistic edges/area formula.

Although the results are perhaps odd for Newtonians, I think it is giving mostly uncontroversial results - refractors are first, followed by Maks/SCTs and then Newtonians. In any case even though large Newts appear to have a low contrast by my formula, they would still have high light gathering and resolution, so they would vastly outperform the small Maks they seem comparable to in terms of contrast.

I suppose you would expect Maks to do better than SCTs, but with a larger central obstruction the maths points the other way. But the results are almost identical for these two types. Perhaps there is something additional I need to factor in for the greater focal length of Maks?

I'm not saying the calculation would be simple, but it is easy to add layers of complexity like this to a program.

[FraserClarke]

The classical way to do this is to fourier transform the pupil of the telescope (that's what you see when you look at an out-of-focus image) to get the point-spread-function. The fourier transform of a filled circle (i.e a refractor with no obscurations in the beam) is the classical airy pattern of rings. If you put a central obstruction and spiders in (newtonian, e.g), you'll get a different pattern with the classical diffraction spikes coming out.

The really tricky bit when it comes to scattering though is dealing with the surfaces. For example, a dusty mirror/schmidt plate is going to give a lot more scattering than a clean one. What sort of scattering you get depends on the details of the scattering surface and the incoming light. That is VERY hard to do and get right properly.

I'm rather confused by your 'ideal' telescope? How do you convert a 300m Newtonian into a 101mm 'ideal telescope'? Almost irrespective of the scattering, the larger aperture will have better resolving power (ignoring seeing) because it is larger. The contrast may not be so good of course, and it seems like that is what you're trying to quantify? But it's not clear how you get there...

Contrast is also very much a function of distance from the bright thing you're looking at. Do you include that in any way??

I don't mean to sound negative by the way! It's very interesting to quantify the performance of different telescopes -- I'm also just aware it is very hard and can be quite subjective to what you're actually trying to measure...

[Ags]

Hmmm. I was hoping to keep the maths straightforward (at the cost of some gross simplifications) - but I'll look into fourier transforms... I haven't done one in twenty years, I hope that part of my brain has not petrified.

The effect of surfaces is very hard to calculate as you say, and factors like cleanliness and baffling are not amenable to calculation. I tried to reduce all these factors to the transmission level of the telescope, but that may be a simplification too far.

The ideal telescope is simply the equivalent scatter produced according to my rough forumla by a telescope with no CO and 100% transmission. Obviously the 300mm newt beats the 101mm ideal telescope on resolving power and light gathering, so it wins hands down with all factors considered. But it would not beat a 300mm refractor - it wouldn't come close. I thought the 'ideal telescope' would be a simple way of visualising the results, but it looks like it causes more confusion!

[FraserClarke]

Hmmm. I was hoping to keep the maths straightforward (at the cost of some gross simplifications) - but I'll look into fourier transforms... I haven't done one in twenty years, I hope that part of my brain has not petrified.

Fortunately in the past 20 years people have written nice FFT libraries for most languages, so you don't have to remember At least that is my approach...

I guess you can make the assumption that if the light doesn't go into the perfect diffraction-limited image, it gets spread around evenly in the field of view. Probably not a bad assumption to start with.

Good luck! Will be interested to see what you come up with.

[michael.h.f.wilkinson]

The standard way to assess contrast transfer (as a function of spatial or angular resolution of details) is through the modulation transfer function (MTF). You cannot capture this in a single figure, because MTF is a curve. It can be estimated quite easily from the Fourier transform of the autocorrelation function of the aperture (or power spectrum of the aperture). This will tell you haw sine waves of different frequencies in the image are reduced in contrast.

Actually, I am quite baffled at the claim that a 12" Newt equates to a 10" "ideal apo". The old claim that you needed a reflector twice the size of the APO to get the same resolving power has been debunked time and time again. Of course, you are looking at contrast, but even there I do not believe the results.

[Ags]

I've dug up some formulae for diffraction that I will try to work into my program, along the lines of the Fourier transform suggested. I think my approximation holds good for refractors and to some extent for catadiatropics, but it falls apart for newtonians.

[michael.h.f.wilkinson]

Even for catadioptrics I think it needs improvement