Telescope contrast and resolution

[vlaiv]

19 minutes ago, alpal said:

It looks like a very good solution.
I think it is referring to splitting a double star?
The maximum ratio of energy between the diffraction rings is 5.25.

Ok, I'll show you why it is not a very good solution and you decide if you agree with that.

From a single number characterizing quality of the view of a telescope in terms of blurriness, I would expect at least these two things:

1. Given two scopes number performs consistently - meaning if number suggests that one scope should perform better than the other - that it really performs better then the other scope (in terms of resolution / contrast / blurriness).

2. That there is correspondence between quantity and described attribute. If number suggests that difference between two different scopes should be small - we should not see much difference, and if number suggests that difference should be large - majority of people agree that there is significant difference.

With above given definition of Contrast ratio - it fails point number one in following case:

Take first telescope to be 76mm Newtonian with 20% CO and take 8" Newtonian with 20% CO. Both will have Contrast ratio of 3.27 - which suggests that they should perform the same, but actual resolution / contrast / blurriness will be miles apart between these two scopes.

For point number two, let's compare 4" APO scope and 4" 20% CO Newtonian. One has Contrast ratio of 5.25 and other has contrast ratio of 3.27. That is 60% increase in number. What is the optical difference? - pretty much none.

Let's now take that same Newtonian with 20% CO and compare it with another Newtonian with 30% CO. Here we have Contrast Ratio of 3.27 and 2.03. That is increase of 61%. Here, you will be able to tell the difference in quality of the view and contrast.

For same difference in Contrast ratio between scopes with same aperture we have different perceived improvement in quality.

Does such number strike you as useful measure of something? If I tell you that I have two scopes, one has Contrast ratio of 4 and other has Contrast ratio of 3, and I don't tell you anything else - would you be able to tell which one is going to provide better view and by how much?

[alpal]

3 hours ago, vlaiv said:

Does such number strike you as useful measure of something? If I tell you that I have two scopes, one has Contrast ratio of 4 and other has Contrast ratio of 3, and I don't tell you anything else - would you be able to tell which one is going to provide better view and by how much?

On the contrary - the article gives a telling picture that I post here now:

 

Edited December 4, 2020 by alpal

[vlaiv]

11 hours ago, alpal said:

On the contrary - the article gives a telling picture that I post here now:

In which part is that disagreeing with what I've written above?

I also assessed that 30% CO will be visibly blurrier than 20% CO and we know that up to 20% CO there is no really difference to unobstructed that can be seen (12.5% in above image is in that range).

Take aberrator and make simulations of 4" APO vs 8" 25% CO Newtonian on same image and see the difference - which will be sharper to again confirm what I written above.

In fact - you don't need aberrator - you can do it above with the method I showed you in ImageJ.

 

[alpal]

11 hours ago, vlaiv said:

In which part is that disagreeing with what I've written above?

I also assessed that 30% CO will be visibly blurrier than 20% CO and we know that up to 20% CO there is no really difference to unobstructed that can be seen (12.5% in above image is in that range).

Take aberrator and make simulations of 4" APO vs 8" 25% CO Newtonian on same image and see the difference - which will be sharper to again confirm what I written above.

In fact - you don't need aberrator - you can do it above with the method I showed you in ImageJ.

 

Because you wrote

"Ok, I'll show you why it is not a very good solution"

I still haven't found the software or calculator that I remember seeing many times about 10 years ago.

[vlaiv]

10 hours ago, alpal said:

Because you wrote

"Ok, I'll show you why it is not a very good solution"

I still haven't found the software or calculator that I remember seeing many times about 10 years ago.

I did write that - and I explained why:

1. Because it does not work with scopes of different aperture: 5.25 contrast index of 4" scope means more blurred image than 3.17 contrast index of 8" scope and higher contrast index should mean sharper image.

2. Because you can't really tell how much more blurred image you'll be getting (if at all) by examining numbers - sometimes same number mean almost no increase in image blur and sometimes that same number means visible increase in image blur.

If you want good measure of how system behaves - you have one, it is MTF diagram. It is specifically designed to work with contrast - function on that graph is contrast, 1.0 being full contrast and 0 being no contrast and all the values in between acting accordingly. It is based on physics, hence not arbitrary, it is absolute (as in not relative) and hence will show you differences between any two scopes, regardless of aperture.

It also includes other aberrations as well, so you'll be able to compare telescope with spherical aberration to one without, for example.

Single number simply can't give you that information.

 

[alpal]

Thanks Vlaiv,

there is a good article here:

https://www.astronomy-electronics-centre.com.au/article_resolvingpower.htm

QUOTE

Quote

Because of their constructional characteristics, apochromatic refractors generally produce visual and photographic images which are sharper, higher in contrast and  -  inch per inch of aperture  -  brighter than those produced by reflecting telescopes of the same and even larger apertures. Furthermore, as I have already said above, refractors give jet black sky backgrounds, thus making it easier to view and photograph fainter deep-sky objects, which are normally not seen through the ‘light-buckets’ because of their tendency to produce dark grey backgrounds instead of jet-black ones. From both the visual and photographic point of view  -  everything else being equal  -  reflecting telescopes (especially fast and very fast focal ratios ones)   cannot match the optical quality of their refractor counterparts. This is because, apart from other factors, their central obstructions have a negative impact on their contrast. And this is particularly the case when these fast focal ratio telescopes are used visually.
 
There is no escape from the fact that there clearly is a loss of contrast in Newtonian and Cassegrain telescope systems. But surprisingly, as far as resolution is concerned, contrast appears somewhat enhanced in obstructed systems! For a 25% obstruction, the loss of contrast is only 15%; while for a 50% obstruction, the contrast loss goes up to 55%. Needless to say, an obstruction of 75% (which would hardly ever be used) would destroy most of the contrast. Most fast reflecting astrographs have a large secondary obstruction, which makes them unsuitable for visual observation. Also,  although theoretically these instruments are said to be all right for astrophotography, I still have to see a deep-sky picture which matches the overall high quality of, for example, any of the Takahashi FS, TSA, FSQ, and TOA Series of apochromatic refractors! Here I am not saying that other brands of apochromats are no good  -  far from it. What I am saying, however, is that, so far, Takahashi is still number one. Differences in contrast, resolution, and colour correction between the Takahashi refractors and other good brands may be subtle, but they are visible!

 

[vlaiv]

1 hour ago, alpal said:

Thanks Vlaiv,

there is a good article here:

https://www.astronomy-electronics-centre.com.au/article_resolvingpower.htm

QUOTE

I'm sorry to say - that article is not very good.

 

[alpal]

1 hour ago, vlaiv said:

I'm sorry to say - that article is not very good.

 

Why?

Well yes - that article was a little bit biased as
the author is the Australian and New Zealand Takahashi dealer - Claudio Voarino:
https://www.astronomy-electronics-centre.com.au/

cheers
Allan

Edited December 6, 2020 by alpal

[vlaiv]

11 hours ago, alpal said:

Why?

Like you, yourself noticed - it is biased towards Takahashi refractors. I don't mind that, Taks are certainly fine scopes (I have not looked thru one myself, but I have heard/read about them so many times), however, text contains quite a bit of half truths and even some things that are just plain wrong.

Let's start from the quote you posted above.

14 hours ago, alpal said:

Because of their constructional characteristics, apochromatic refractors generally produce visual and photographic images which are sharper, higher in contrast and  -  inch per inch of aperture  -  brighter than those produced by reflecting telescopes of the same and even larger apertures.

This is "half-truth". Apos, if properly executed, indeed produce better high power visual images - inch per inch of aperture, but as we have seen, things are not as clear cut when we take in consideration larger aperture.

My 8" dob, although around 0.8 strehl, walks all over my 80mm APO that has strehl over 0.96 on planets. Small Apo simply can't deliver detail / resolution that 8" scope can.

This is of course for visual. For photographic high power applications - obstructed scopes even have slight advantage over unobstructed. We can go over that in detail, but here is simple explanation:

In planetary photography, we perform a step called frequency restoration (or rather, it is commonly called - sharpening, but proper term would be frequency restoration) - we restore below curve to be 1 or close to 1 on whole interval. Lower it is in the graph, and further to the right - harder it is to bring it back. Obstructed telescopes, have this curve lower than unobstructed in left part of the graph and at "higher" values - which is easier to restore and have edge on right side of the graph.

One should not mix visual and photographic when it comes to sharpness and level of detail, as for example, this was taken with 4" telescope that costs ~ $200 (and has central obstruction):

No 4" telescope in the world, Takahashi or not, will give you that level of detail at the eyepiece. Or to put it in more "scientific" terms - eye can't correct above curve, while computers can with appropriate mathematics involved.

15 hours ago, alpal said:

Furthermore, as I have already said above, refractors give jet black sky backgrounds, thus making it easier to view and photograph fainter deep-sky objects, which are normally not seen through the ‘light-buckets’ because of their tendency to produce dark grey backgrounds instead of jet-black ones. From both the visual and photographic point of view  -  everything else being equal  -  reflecting telescopes (especially fast and very fast focal ratios ones)   cannot match the optical quality of their refractor counterparts. This is because, apart from other factors, their central obstructions have a negative impact on their contrast. And this is particularly the case when these fast focal ratio telescopes are used visually.

For dark backgrounds and quality of the view, I suggest you read this:

https://www.scopereviews.com/best.html

To get the idea of what Newtonian can deliver in comparison to premium APO refractor.

When photographing deep sky objects - we are really not concerned about contrast, or quality of telescope. It is all about SNR. Quality of optics, comes 4th after - seeing, mount quality and aperture.

15 hours ago, alpal said:

There is no escape from the fact that there clearly is a loss of contrast in Newtonian and Cassegrain telescope systems. But surprisingly, as far as resolution is concerned, contrast appears somewhat enhanced in obstructed systems! For a 25% obstruction, the loss of contrast is only 15%; while for a 50% obstruction, the contrast loss goes up to 55%. Needless to say, an obstruction of 75% (which would hardly ever be used) would destroy most of the contrast. Most fast reflecting astrographs have a large secondary obstruction, which makes them unsuitable for visual observation. Also,  although theoretically these instruments are said to be all right for astrophotography, I still have to see a deep-sky picture which matches the overall high quality of, for example, any of the Takahashi FS, TSA, FSQ, and TOA Series of apochromatic refractors! Here I am not saying that other brands of apochromats are no good  -  far from it. What I am saying, however, is that, so far, Takahashi is still number one. Differences in contrast, resolution, and colour correction between the Takahashi refractors and other good brands may be subtle, but they are visible!

Yes, sure. We all know that top planetary photographers all use high quality Takahashi 4"-5" refractors, right?

So I started reading the article to see what is else written, and I found this:

Quote

Unfortunately, the nature and purpose of both the angular and linear resolving power are often misunderstood. For example, many amateur astronomers still labour under the misconception that, everything else being equal, larger aperture telescopes always yield higher resolving power than smaller aperture ones. As we will learn in this article, this is not generally the case, and the selection of deep-sky images used to illustrate this essay will prove that you don’t have to buy a ‘light bucket’ in order to obtain spectacular pictures of the night sky.

and this:

Quote

We will also investigate Dawes’ Limit Law and its restrictions, the Rayleigh Limit, the Contrast Transfer, as well as the very best way to determine the angular resolving power of a telescope system by using the Strehl Ratio benchmark.

and then I stopped reading further ...

[alpal]

Thanks Vlaiv,

thanks for trying to explain these issues.
I'm not sure why telescopes with such large
secondary obstructions still seem to work so well.

 

[Ags]

Don't forget the Hubble Space Telescope is an obstructed instrument. It does pretty good on deep space!

[vlaiv]

28 minutes ago, alpal said:

Thanks Vlaiv,

thanks for trying to explain these issues.
I'm not sure why telescopes with such large
secondary obstructions still seem to work so well.

 

Not sure where you got the idea that obstructed telescopes ought to perform so poorly?

In fact I think I know. If someone says that you'll get Contrast Loss of 55% for 50% CO - you instantly think that it will have half the performance of unobstructed scope. That is simply not the case.

Yes, there will be contrast loss in visual in comparison to unobstructed telescope of the same size, however it is not remotely as drastic as it may seem. In fact 20-25% central obstruction gives almost the same views as unobstructed aperture. With 20% CO - most of people would not be able to tell the difference and only experienced planetary observers would notice in side by side comparison to unobstructed telescope.

It is also important to understand that resolution is governed by aperture and contrast and resolution are almost synonymous terms. In fact resolution of a telescope is related to frequency at which contrast falls to 0, and blur that we get is gradual per wavelength contrast loss. This means that larger aperture will give you better contrast (or rather resolution) than smaller aperture.

All that information is contained in MTF, and that is the best way to compare theoretical max performance of two optical systems (just make sure both MTF graphs are plotted for same values - they often come "normalized" to max frequency, but max frequency depends on aperture).

[alpal]

4 hours ago, Ags said:

Don't forget the Hubble Space Telescope is an obstructed instrument. It does pretty good on deep space!

Hubble's secondary mirrors has a very small obstruction which is easy to do at   f 24.

https://en.wikipedia.org/wiki/Hubble_Space_Telescope

https://www.nasa.gov/mission_pages/hubble/story/index.html

Primary Mirror Diameter: 94.5 inches (2.4 m)
secondary Mirror Diameter: 12 inches (0.3 m)

Obstruction percentage 0.3/2.4 =  12.5 %

[alpal]

4 hours ago, vlaiv said:

Not sure where you got the idea that obstructed telescopes ought to perform so poorly?

In fact I think I know. If someone says that you'll get Contrast Loss of 55% for 50% CO - you instantly think that it will have half the performance of unobstructed scope. That is simply not the case.

Yes, there will be contrast loss in visual in comparison to unobstructed telescope of the same size, however it is not remotely as drastic as it may seem. In fact 20-25% central obstruction gives almost the same views as unobstructed aperture. With 20% CO - most of people would not be able to tell the difference and only experienced planetary observers would notice in side by side comparison to unobstructed telescope.

It is also important to understand that resolution is governed by aperture and contrast and resolution are almost synonymous terms. In fact resolution of a telescope is related to frequency at which contrast falls to 0, and blur that we get is gradual per wavelength contrast loss. This means that larger aperture will give you better contrast (or rather resolution) than smaller aperture.

All that information is contained in MTF, and that is the best way to compare theoretical max performance of two optical systems (just make sure both MTF graphs are plotted for same values - they often come "normalized" to max frequency, but max frequency depends on aperture).

Yes -  20 to 25% obstruction is almost negligible.
Many of the large astrographs have a 50% obstruction but still seem to
produce nice images.

[andrew s]

Enjoyable  thread. I have generally gone to telescope-ootics.net as my main source of information on this area.

I was taken by the comment that there is an inconstancy in the analysis which put the cut of frequency in the same place for unobstructed and obstructed apertures. 

As is pointed out the obstructed central peak is narrower then the unobstructed one and so can't have the same cut-off as the must have different transforms.

I did try to find any other references to this but found none. Placing the cut-off in the same place seems to be a convention. 

The discussion is here Discussion

Regards Andrew 

[vlaiv]

1 hour ago, andrew s said:

Enjoyable  thread. I have generally gone to telescope-ootics.net as my main source of information on this area.

I was taken by the comment that there is an inconstancy in the analysis which put the cut of frequency in the same place for unobstructed and obstructed apertures. 

As is pointed out the obstructed central peak is narrower then the unobstructed one and so can't have the same cut-off as the must have different transforms.

I did try to find any other references to this but found none. Placing the cut-off in the same place seems to be a convention. 

The discussion is here Discussion

Regards Andrew 

Yes, we did mention that on one occasion, but I have no idea why would narrower central peak shift cut off frequency? After all, MTF is FT of Airy pattern and integration is complex operation - it is not immediately obvious one way or another.

Except for doing analytical integration, best way to check if there is shift in frequency due to narrowing of the peak is to do numerical analysis.

This is unobstructed vs 40% CO aperture. As far as I can tell - they have exact same cut off frequency. Here is same measurement "zoomed in" on cut off frequency:

Now that I have once again read above discussion, I think I know where error in reasoning in it comes from. Let's examine what is being said:

Quote

While this formalism should accurately describe contrast transfer between zero and cutoff frequency for annular aperture, it is not appropriate for determining the cutoff frequency. The reason is that annular pupil produces diffraction pattern reduced in size with respect to that of clear aperture of the same diameter - in effect, a pattern nearly identical to one produced by a larger clear aperture with spherical aberration, in which the level of aberration does not cause change in cutoff frequency. In other words, it is physically impossible that a near-identical impulse response (PSF) produces two different MTF cutoff frequencies.

I strongly object this argument (in bold - by me for emphasis) that something that is "nearly" identical - which clearly means different (however small difference - it is not asymptotically small / vanishing) should produce identical results.

Author even shows - how "nearly identical" these are:

Left side of this image shows PSF of 32% CO and 10% larger aperture with 1/4 PV spherical. These are clearly not identical and I see no reason why their FT should fall to zero at exact same point.

[vlaiv]

7 hours ago, alpal said:

Yes -  20 to 25% obstruction is almost negligible.
Many of the large astrographs have a 50% obstruction but still seem to
produce nice images.

This is because DSO / long exposure astrophotography is in completely different regime to high power planetary.

It is dominated by blur caused by atmosphere / seeing blur and tracking / guiding imperfection blur - both of which integrated over significant period of time - couple of minutes. This produces blur in the image that is order of magnitude stronger than that produced by optics (for 6-8" aperture, with smaller apertures this difference obviously reduces).

Sampling rate for planetary imaging is often 0.1-0.2"/px, while that of long exposure astrophotography is 1-2"/px and higher.

Differences between obstructed and unobstructed telescopes on these scales simply can't be seen, and mirrored telescopes have some significant advantages in DSO imaging over refractors - hence are much more often used as scientific instruments.

[andrew s]

@vlaiv I think the issue is the cutoff frequency is defined by convention as 1/(lamda F#) and not on the actual shape of the PSFs in the two case.

The comment you object to is not intended as you read it in my view. What he is saying is that the PSF of an unobstructed aperture diameter D is significantly more different to an obstructed aperture D than the obstructed aperture PSF is to the PSF of an unobstructed  aperture 10% larger than D with 1/4 wave P-V spherical aberration. 

Thus the  MTF of the obstructed aperture should be closer to that of the aberrated larger aperture than the smaller unaberrated one. I agree with this view. 

With your logic 

5 hours ago, vlaiv said:

These are clearly not identical and I see no reason why their FT should fall to zero at exact same point.

why should the cut of be the same for the more different PSF of the obstructed  and unobstructed apertures with the same diameter?

Regards Andrew 

Edited December 7, 2020 by andrew s

[vlaiv]

51 minutes ago, andrew s said:

 I think the issue is the cutoff frequency is defined by convention as 1/(lamda F#) and not on the actual shape of the PSFs in the two case.

I think you are mistaken there. If you have PSF - which happens to be defined as a function of distance from optical axis (in mm or other spatial units), then cut off frequency is mathematical concept - not physics one.

Given certain PSF, arguing that cut off frequency is at certain point rather than other - makes about as much sense as arguing if 7 x 5 equals 35 or not.

53 minutes ago, andrew s said:

What he is saying is that the PSF of an unobstructed aperture diameter D is significantly more different to an obstructed aperture D than the obstructed aperture PSF is to the PSF of an unobstructed  aperture 10% larger than D with 1/4 wave P-V spherical aberration. 

I know that there is greater difference between obstructed and unobstructed PSF then there is between 1/4 wave P-V and obstructed PSF, but it really does not matter since all three are transformed into Fourier domain and we can't judge if their transforms will be similar or different and to what extent - unless we do actual calculation (which btw shows that same apertures have same cut off frequency and larger has higher cut off frequency).

Fourier transform is integral, a sum, right? Here is an example how things with sums can be misleading.

7 + 3 + 6 + 6 = 22

8 + 1 + 8 + 5 = 22

8 + 1.01 + 8 + 5 = 22.01

First four numbers (unobstructed) is very different than second set of numbers (obstructed) which is in turn extremely similar to third set of numbers (10% larger with 1/4 wave PV), yet first and second set sum to exactly the same number, while third set does not.

1 hour ago, andrew s said:

why should the cut of be the same for the more different PSF of the obstructed  and unobstructed apertures with the same diameter?

Because calculations say so . This is purely in domain of mathematics, which says that if something acts as convolution kernel (PSF) then that is the same as doing multiplication in Fourier domain - hence Fourier transform of convolution kernel is filter.

Fourier transforms of Airy patterns for obstructed and unobstructed aperture of given size have same cut off frequency, but significantly different shape.

Fourier transforms of Airy patterns of obstructed aperture and 10% larger aperture with 1/4 wave PV spherical - are much more similar in shape but have different cut off frequency. In fact - we have seen that for certain aperture, regardless of obstruction or not, cut off frequency is constant - that is the property of corresponding Airy patterns (PSFs) and their Fourier transforms.

What can be argued and is much more complicated to derive is physics thing - which no one called into question here. That is - why does light at circular aperture, obstructed or not - gives Airy pattern at focus plane.

I've seen once derivation, and it's not hard, but it assumes simple wave mechanics (not QED). It's a bit like Feynman's explanation for reflection of light where each photon has internal clock (phase of wave at that point) and they propagate in all directions and bounce in all directions and are summed (as vectors) to form a wave in destination point and then magnitude is used as probability of finding photon there (intensity of PSF).

All that summing actually ends up being integral over aperture, and wave thing is

e to the power of i * pi * theta and finally square is used to get probability / intensity and we end up with Airy pattern being power spectrum of Fourier transform of aperture.

It is not convention to have MTF be auto correlation of aperture, but rather consequence of the fact that

Aperture --> Airy pattern (power spectrum)

Airy pattern --> MTF (frequency spectrum)

And then we have this:

Here we have that power spectrum is the same as function times its own conjugate.

[Ags]

All this maths reminds me of where I started this thread - how to calculate diffraction effects with simple maths and elementary physics (Huygens-Fresnel Principle). I am sure all this exotic maths is very beautiful, but I do think running a simple computer model where you sum the wavelets is very transparent and easily understood.
 

I think trying to put a number to contrast is just a bit of fun, like devising a way to score renaissance paintings from one to ten. My 6 inch newt had mathematically better contrast than my 4 inch mak, but to my eyes at least the newt stars looked hairy and horrible. Compared to the little mak, the newt was a dreadful instrument for viewing double stars, regardless of its theoretically better contrast and resolution.

[andrew s]

@vlaiv Airy first derived an approximate solution (GB Airy, trans. Camb. Phil. Soc., % (1835), 283.) but a modern derivation using Fraunhofer diffraction of classical waves is given in Principle of Optics by Born & Wolf. I can scan the pages if your interested.  There is little point in going for a QED solution when the classical field approximations are perfectly adequate.

2 hours ago, vlaiv said:

I

Fourier transforms of Airy patterns of obstructed aperture and 10% larger aperture with 1/4 wave PV spherical - are much more similar in shape but have different cut off frequency. In fact - we have seen that for certain aperture, regardless of obstruction or not, cut off frequency is constant - that is the property of corresponding Airy patterns (PSFs) and their Fourier transforms.

 

This is the root of the issue.  The issue is the Airy patterns are not the same. Increasing the aperture of a perfect circular aperture just scales the Airy disk and MTF. However, adding and obstruction does not do this it is in fact the difference of two "Airy disks" A positive contribution from the" unobstructed" aperture and a negative one from the obstruction.

So my question is why does the Fourier transform of an Airy disk have a cut of at all?  In Fraunhofer diffraction the PSF is proportional to the square of a Bessel function of the first kind.  I see no logical or physical reason for a cut off other than for practical, pragmatic reasons i.'e a convention.

I agree you need to do the calculations by that means doing the Fourier transforms of the PSF directly. It seems to me the equivalent methods assume the cutoff for them to be valid.

Regards Andrew

Edited December 7, 2020 by andrew s

[vlaiv]

56 minutes ago, andrew s said:

So my question is why does the Fourier transform of an Airy disk have a cut of at all?  In Fraunhofer diffraction the PSF is proportional to the square of a Bessel function of the first kind.  I see no logical or physical reason for a cut off other than for practical, pragmatic reasons i.'e a convention.

I agree you need to do the calculations by that means doing the Fourier transforms of the PSF directly. It seems to me the equivalent methods assume the cutoff for them to be valid.

I still don't think it is convention of any kind nor do we need to assume that in order to make integral finite / valid.

True proof of that really needs mathematical backing, and I'll try to do that (maybe not by solving whole integral thing, but if it is cut off - maybe we can show sum to be zero for some frequency higher than cut off), but in the meantime, let's look at some notable examples that really have finite FTs and acknowledge that FFT does produce effective cutoff (still not analytical proof - but it should behave the same regardless of the fact it is numerical method).

I would mention just plain box filter as example - because it is very similar to aperture (one can imagine box filter in polar 2d coordinates as being aperture).

Box filter has Sinc function as its Fourier transform. Note similarity between Sinc function and Airy pattern, or rather Sinc squared and Airy pattern:

Above is Sinc squared.

Now we want Fourier transform of Sinc squared. Look at that, it appears that Sinc squared and Triangle functions are Fourier pairs:

We can see that Triangle function has frequency cut off as well - not by convention but by actual value - zero for x values higher than certain cutoff point.

http://www.thefouriertransform.com/pairs/fourier.php

 

 

 

Edited December 7, 2020 by vlaiv

[vlaiv]

Just wanted to add following resource - still not proof, but confirmation none the less:

https://www.johndcook.com/blog/2015/12/16/sinc-and-jinc-integrals/

Jinc is defined similarly to sinc - J1/x which is core of Airy pattern (Jinc squared).

[alpal]

23 hours ago, vlaiv said:

This is because DSO / long exposure astrophotography is in completely different regime to high power planetary.

It is dominated by blur caused by atmosphere / seeing blur and tracking / guiding imperfection blur - both of which integrated over significant period of time - couple of minutes. This produces blur in the image that is order of magnitude stronger than that produced by optics (for 6-8" aperture, with smaller apertures this difference obviously reduces).

Sampling rate for planetary imaging is often 0.1-0.2"/px, while that of long exposure astrophotography is 1-2"/px and higher.

Differences between obstructed and unobstructed telescopes on these scales simply can't be seen, and mirrored telescopes have some significant advantages in DSO imaging over refractors - hence are much more often used as scientific instruments.

I've been think about this for a few days.
I am looking for a simple explanation.

For instance if we go here and calculate the size of the airy disc for old my 8"  f6 Newt.
http://www.wilmslowastro.com/software/formulae.htm#Airy

we get 1.26 arc seconds.

Using this calculator
https://astronomy.tools/calculators/telescope_capabilities

the Dawes limit is 0.57 arc seconds and the Rayleigh limit is 0.68 arc seconds.

Considering that the best seeing I've had is 2.9 arc seconds FWHM
then an obstruction of say 20 to 30% would be negligible in effect.
That would explain why large 50% obstructions can still provide
very good pictures on long exposures -

the Airy disc is far less in size than the seeing.