vlaiv

Indeed we do - but our vision is such that majority of brightness that we perceive as sharpness comes from green part of spectrum - specifically around 500nm. For that reason - ~500nm is often used as baseline for telescope performance assessment. or rather 500nm for night time use and 550nm for daytime use. I think that Baader Solar Continuum filter is very good filter for assessing visual telescope performance by means of camera analysis. Actually I think that your CCD is just sufficient to completely resolve image from Mak. If I'm not mistaken, at around 500nm - critical sampling rate for 3.75µm pixel size is F/15. Mak is F/15 or slower (due to slightly lower aperture than 180mm), so you should be able to sample image properly. There will be only slight pixel blur, but I think it is rather small in this context. However, I think there is difference between PSF obtained as differential from this edge softening and actual PSF. Edge gradient is result of 2D convolution yet we here treat it as 1D case. Let me show you what I mean. I generated airy pattern and made perfect edge in the image (just rectangle with value 1). Then I convolved image with Airy pattern and did following kernel on it: 0 0 0 1 -1 0 0 0 0 to extract the edge. Then I measured the edge (plot profile), and this is what I got: Notice that ripples are not at zero like linear cross section of Airy pattern is: This is because of 2d convolution so our values are not affected by values only to the left and right - but also those in "upper rows" and "lower rows" - which add to average and somewhat "smooth things out". Doing simple edge contrast won't produce proper PSF

That higher contrast might be due to resampling used or it might be genuine feature. There is another way to conduct experiment - we should just look at original images from greater distance. In my case - 1.5m produces similar effect as reduced images (since smaller images are 34% of original size). At over 2m away from screen - I can virtually detect no difference between original 4 images.

You did fitting of the curve to the data. What model did you use and why? MTF looks like gaussian, so my guess is that exponential of sorts was used for fitting. Did you use single wavelength of light? What you have here is averaged MTF for different wavelengths weighted according to QE response of your system. Remember - 400nm has almost twice the cut of frequency as 700nm (x1.75 precisely). MTF will look much more "hanging" if you combine different wavelengths. Try using raw data and doing row to row difference V(x,y) = V(x-1,y) - V(x,y) That is similar to differential filter (it will give you slope at constant intervals) - don't do any fitting but that should look roughly like profile of Airy disk. Stacking is good option to reduce noise.

I was hoping for few snaps thru the eyepiece as well So you spotted two main belts?

Here is interesting thing I concluded that @mikeDnight had issues with scale of Jupiter - once image gets too large for a given blur - it does not look as sharp, so I offered the same image only scaled down to be more like the size viewed at the eyepiece. Then I said - why don't I try to calculate actual Planet size if viewed at "standard" settings behind computer screen. So there are findings that are valid for "standard" settings - if your viewing conditions are different - your results might be different as well. I calculated that if I used magnification of x176 at the telescope - Jupiter at 47 arc seconds will have apparent diameter of 2.3°. If we are sitting half a meter - 50cm away from computer screen and we have computer screen that has 96dpi resolution, then 2.3° will be ~80px across for planetary disk. So standard settings for viewing following images are 96dpi display and 50cm viewing distance. Left is 4" Jupiter and right one is 12" Jupiter. Interestingly - not much difference between the two. It looks like one would need higher magnification in order to see the difference between what aperture can deliver. I can still detect some softness in left image versus right one - meaning that x176 is too much for 4" instrument - you don't need as much magnification to fully resolve what aperture can deliver - that goes to show that x50 per inch is just not needed to fully resolve everything that telescope can deliver.

Let me try again then, maybe this will better approximation to what you see in 4": What do you think, does it look more like image of Jupiter you see thru 4"?

I certainly don't mind seeing pictures here.

I really want to try this comparison in real life. Although image on computer screen can be quite telling - looking at the eyepiece is different experience. Closest approximation to seeing free view that I could think of is looking at a terrestrial object that has appropriate size of features and possibly contrast levels. Either printed image or maybe physical object like marble or even image on phone screen. I don't own excellent optics, but I do want to see difference between 4" Mak and 8" Newtonian first hand. I do have experience with what 8" can deliver - but there is always that residual - "how much was it down to seeing and how much down to quality of optics?". If I can get similar results in actual testing to these theoretical values - that will give me confidence that comparison between 0.8 Strehl and 0.98 Strehl scope is also relevant. Taking images during tests is also interesting proposition - that way we can document actual test.

In principle yes - as long as we have wavefront of miscollimated scope as is - we can create simulation and add MTF to comparison. I think there is very small difference between 1/8 and 1/10 PV to be honest. Look at difference between 1/4PV and 1/5.6PV - I was expecting more, but it actual image - it's not that much at all. It is hard to see side by side - best way to notice differences is to have images blinking and they are minimal.

Here are results of 4" and 5" ideal clear aperture telescope vs 12" 1/5.6 PV with 20% CO and diffraction limited 12" 1/4 PV with 20% (I threw in diffraction limited scope in 12 for comparison) MTF diagram - black is 4", blue is 5", red is 12" 1/4PV and orange is 12" 1/5.6PV Top row: 4", 5", bottom row: 12" 1/4PV, 12" 1/5.6PV

Not really. If they are close in aperture size - like 4" and 6" - then optical quality and central obstruction plays a part. But once aperture is large enough and both scopes are diffraction limited - aperture wins in cases where there is no atmospheric influence of it is minimized. Atmosphere often reduces aperture to equivalent of 2-3" anyway in poor seeing and in such cases effect it has on smaller aperture is not as detrimental as on larger aperture. It is often said that smaller aperture "cuts thru seeing" better.

Isn't that MDF?

That is actually derived data. These three contain actual data needed to generate MTF graph (any of the three - they are just different representations) This is also important point: That data is produced for/with 546nm wavelength. Reflectors produce same wavefront regardless of wavelength used, but refractors do not. If we want to perfectly simulate refractor - we need wavefront per each wavelength (close ones are very similar - but say each 30nm is good sampling - about 10 different wavelengths used for 400-700nm range).

Well, actually raw RGB of camera space is significantly different than sRGB that is commonly used in display devices. All color conversion processing usually boils down to one matrix multiplication - that is already there to convert from raw RGB to sRGB and possibly gamma function (needed in case of sRGB). For example - this is response of RGB sensor: and this is virtual response of sRGB (note the negative values - this is because sRGB has narrower gamut than human vision and can display some colors that humans can see):

Yes, one of the problems that we face when doing these type of comparison is that refractors have different strehl based on wavelength. That is something that is very hard to simulate as far as rendering a target / simulated view of the planet. We both need to take defocus into account but also need to have target in much more wavelengths than standard 3 - red green and blue. Ideally we would do something like at least 10 different wavelengths and then combine results.

Since we are talking about pure telescope performance without influence of atmosphere - I would say either marbles or printed image of Jupiter stapled about 150 meters away . In fact, given that you'll be using 12" scope - a target at least 100 meters away to minimize spherical aberration from close focusing.

Ok, so it shell be 12" 20% obstructed newtonian with spherical term that is equivalent to Strehl of 0.9 versus perfect 4" or 5" APO? Which one shall we go for? Maybe both to see the difference? I'll do both MTF and simulated views of Jupiter in perfect seeing later - got to take one of my dogs to a vet now.

Just saying Strehl 0.9 - does not give me enough information to actually plot MTF. Infinite number of different wavefronts will produce Strehl 0.9. If we want to compare actual scope to other actual scope or perfect scope for that matter - we need either: wavefront diagram or Zernike polynomials. Say one went and did optics test of a telescope. Likely result will come in form of report that looks like this: Above diagram is what is needed to produce PSF and hence MTF of actual telescope. This is in fact for primary mirror - but in reality one would do such report for whole system. Alternatively, we could assume one particular type of aberration - say spherical and then find which term corresponds to Strehl of 0.9 (say 1/5.6th of a wave or something like that) - then I could generate such wavefront and based on that do comparison.

Everything that has been said so far applies to Strehl 1 optics - so absolutely 100% perfect wavefront optics. We are comparing exceptional specimens of any particular design. In reality - things are going to be only "worse". Quality of the telescope just "lowers" MTF in some way. It will not impact cut off frequency - that is solely dictated by aperture size. Everything else - like central obstruction or optical aberrations just makes MTF with lower values. Something like this: Actual telescope performance can be obtained by examining either wavefront or PSF. General relationship is: PSF is power spectrum of FT of wavefront aberrations MTF is frequency spectrum of FT of PSF In order to have MTF we need either wavefront or PSF. Seeing is exceptionally tough topic to tackle - since it is very random. Best that we can do is run some sort of simulations by having for example range of wavefront parameters - average aberration and standard deviation from that value. We also need how fast it is changing and then we need to integrate over period of time - usually 30ms or so - time that our brain / eye system perceives as single image. Established way to handle wavefront aberrations is via Zernike polynomials. That is orthonormal basis set for deformation of unitary circle. This means that any sort of wavefront aberration can be decomposed into Zernike polynomials - some of which we know under different name like - lower / higher order spherical, defocus, piston, tilt, coma, astigmatism and so on ... Seeing is approximated by mean value for each of these polynomials and standard deviation - so we can then use random process to "bend" wavefront that is otherwise produced by telescope and that will give us integral deformation of PSF which will in turn give us MTF

I know you did not claim that - I was sort of building up "case" in order to start discussion on some long held beliefs. One of those - smaller aperture apo will have better contrast than larger telescope with central obstruction. It might well be that 4" APO indeed has better contrast than say 6" Newtonian - but I don't think that is due to central obstruction. It could be due to much more effective baffling and mirror light scatter and such - but I don't think it is due to central obstruction. I wanted next to compare clear aperture to 25% CO in the fashion that I proposed. I have a feeling that difference will hover at or just above 10%. In fact - let's do it now. Here are graphs of clear aperture and 25% CO

While this is valid way to compare two telescopes: And here we see MTF plot of 125mm APO with perfect figure and 172mm Mak with 58mm CO and again perfect figure (I did not calibrate graph - it is still in pixels since I used FFT to generate these curves). We can use concept of JND to really compare two telescopes. Just noticeable difference is idea that we only perceive difference in some stimuli if it is above percentage of that actual stimulus value. In another words - we will see difference in intensity of the light if difference in intensity is about 10% - or 0.1. For stronger light we need stronger change before we see any difference. I'm going to propose plotting above graph slightly differently - let's take ratio of MTF curve between two scopes APO / MAK and see what it looks like If above line goes above 1.1 - there will be noticeable difference where APO wins (more than 10% of brightness on that frequency). When graph falls below ~ 0.91 (1/1.1 - or 10% or more brightness advantage for mak on given frequency) - Mak wins. From this graph we can see that APO never provides advantage that we can actually see - no frequency component will be more than 10% brighter in APO than in Mak - while Mak will show frequencies above certain frequency to be more than 10% brighter than APO.

How come that we see the stars then? Stars are much smaller features than 0.93 arc seconds - they are in micro arc second range and smaller than that.

If you feel it will contribute, I'll certainly put an effort to demonstrate it. I wanted first to address JND (just noticeable difference) and MTF attenuation part as well as to give a bit more understanding on what high and low frequencies contribute to image composition.

I guess to some extent there is already something similar in dark line on white background vs single star above?

To expand a bit further on topic and to present some "real life" examples, here are a few simulations that will help understand - sharpness, contrast and detail and resolution: This is set of features on planet X. Largest of these features is about 5.5 arc seconds in size. Others are progressively 15/16th, 14/16th ... 1/16th of the size of original - so the smallest one is ~0.345" in "size". Note that features consists of "smaller parts" - that are roughly 1/3rd to 1/5th of original size (length and width of spike). We observe planet X with 8" of clear aperture telescope in extraordinary seeing conditions, what do we see? Such telescope has airy disk diameter of ~1.26" and highest frequency component of ~1.94 cycles per arc seconds or equivalent wavelength of ~0.5156" (x2.44 times smaller than airy disk diameter, or x1.22 times smaller than airy disk radius). This is all for 500nm wavelength. We see that first few features exhibit what we would call - sharpness loss primarily. Second row also, but last feature in second row starts to show contrast loss as well. In third row dominant thing that we notice is contrast loss - but we still can recognize that thing is pentagonal in its structure. In last row - we completely loose resolution, contrast is severely impacted and last feature is almost invisible - we can sort of tell that maybe something is there. We can now start to get the idea - when feature size is about x5 that of cutoff wavelength (2-2.5 size of airy disk) - we start to see contrast loss. When feature is about the size of airy disk - or roughly x2.5 size of cutoff frequency - we start to loose all resolving capability - we cannot longer tell anything about the shape of the feature. How does smaller telescope compare to this? Here is perfect 4" clear aperture telescope aimed at the same target (take above stats and multiply by two - airy disk size will be 2.52", cutoff wavelength at about 1.03" and so on). Here we jump straight into contrast loss part - even first row is experiencing that. That sort of makes sense now since largest feature size is 5.5" and airy disk diameter is now 2.52" - so again roughly x2-x2.5. Third row is no longer resolved and fourth shows "disappearing" feature - where first two can be said to have something there - even third - but last is gone completely. Mind you - with contrast and disappearing features - base contrast plays major role - we can still see features that are beyond resolving power if they are very high contrast to start with - but their size - can't be smaller than airy disk (it can in one direction - for example rile or gap in rings can be long - but width is limited to roughly airy disk diameter).