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# Reflector/Refractor equivalence formula

162 replies to this topic

### #51 maknewtnut

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Posted 02 April 2013 - 11:09 PM

The trouble with a rule of thumb is that people's thumbs vary greatly in length.....

Dave

Dave nailed it with his analogy. A vast majority of the arguments summarized by the thread title are pure bunk.

### #52 timps

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Posted 02 April 2013 - 11:22 PM

When they refer to the percentage of the central obstruction, Is that a percentage by area or diameter?
Celestron 14" for example: secondary mirror obstruction by area is 10.3% but by diameter is 32.1%.

### #53 jrcrilly

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Posted 02 April 2013 - 11:38 PM

When they refer to the percentage of the central obstruction, Is that a percentage by area or diameter?

Could be either (percentage by area is figured by taking the square root of the percentage by diameter; nobody actually measures the area). You can tell which has been stated by the figure. It will only make sense as one or the other (10%-15% range would be area while 30%-45% would be diameter). Which is chosen generally depends on the context. Light loss varies as the area while contrast varies as the diameter.

### #54 Jon Isaacs

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Posted 02 April 2013 - 11:43 PM

Eddgie,
I was recently given Suiter's 1994 edition by a friend. In it, the MTF chart is always normalized to the *fraction* of the maximum spatial frequency. This is appropriate when considering a telescope in the afocal configuration, where diffraction effects scale as the exit pupil.

My emphasis in this discussion is on the appearance of the image at the eyepiece and at given exit pupil. The better quality instrument, irrespective of aperture, will deliver the better quality view.

The eye sees an image, and knows not what is the aperture delivering it. All it 'knows' is whether the image is good or not so good as regards such things as diffraction and aberrations. If at some particular exit pupil one rates the view less afflicted by diffraction and aberrations for a smaller aperture, then that smaller aperture, in some respect at least, delivers by definition better contrast transfer.

Any two telescopes, no matter how much they may differ in aperture but which have identical MTF charts, will deliver identical contrast at the same exit pupil.

Glenn:

I am with Eddgie on this one... One needs to step back from what is "always done" and consider what one wants to know when attempting to analyze the difference between two scope that differ in aperture. This is the question as I see it:

For a given spacial frequency, which scope will provide the superior contrast. All we want to know is for a given detail on the surface of Jupiter, which scope will show it with more contrast. If you normalize by aperture, then the scale of the object must be normalized and that is not what we want, we are looking for absolute numbers for comparison.

In the case of the C-14 and the 6 inch, the same exit pupil occurs at 2.3 times the magnification, the details with the same contrast are much finer in the larger scope.

Jon

### #55 Kevin Barker

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Posted 03 April 2013 - 02:42 AM

Interesting stuff. here is my 2c worth....

A crude rule of thumb for the equivalent obstructed and unobstructed equivalent aperture is take away the obstruction from the aperture for an obstructed scope. This seems to hold for up to 10-11 inches. And for obstructions from 20-35 %.

However obstructed scopes need to be well cooled, well baffled, perfectly collimated and have a very good figure if they are to produce similar contrast as an unobstructed "apochromat" equal to their aperture minus the obstruction. With this reasoning a 10 inch planetary Newtonian with a 2.6 inch secondary should be able to hold it's own against a 7 inch apochromat.

The apochromat could well seem to be more appealing visually and have a higher contrast to brightness than the Newtonian.

I have an 8 inch f-6 Dobsonian which has a 2 inch secondary. It does indeed show similar views to a good five inch apochromat when it is well cooled. The missing inch may be because of the less than perfect control of scattered light.

I think once achromatic refractors start to obey the f rule of 3 times the aperture in inches they also behave close to an obstructed aperture minus the obstruction.

### #56 GlennLeDrew

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Posted 03 April 2013 - 04:22 AM

Jon,
It's axiomatic that at given spatial frequency a larger aperture delivers better contrast. This is the essence of the aperture race. And as I stated, I am in complete agreement.

My emphasis is on what the observer sees *at the scale of the retina's resolving power*, so to speak.

We have two otherwise identical telescopes. A 2" is examining a 1st magnitude star and a 20" is examining a 6th magnitude star. Each is working at, say, the same 1mm exit pupil. The view through each eyepiece will be indistinguishable.

We have two optically excellent telescopes, one a 2" refractor and the other a 20" Cat. As before, each looks to a 1st and 6th magnitude star, respectively. At the same exit pupil, the Cat's image suffers additional diffraction. An observer conducting a blind test would judge it inferior.

This is the crux of my argument. Discounting the obvious increase in resolving power (and light grasp) which aperture affords, what is the *quality of the image on the retina* at any given exit pupil diameter?

We all know that resolving power scales directly as the linear aperture. This is so fundamental that once learned can be relegated to the back of the mind. The MTF chart is not concerned whatsoever with the actual, absolute resolving power, being normalized as it is to the theoretical maximum. The MTF chart is all about representing the departure from perfection of a same-size circular aperture. Two widely disparate apertures delivering the same optical quality will have identical MTF charts.

Concentrating on the *absolute* differences in resolving power resulting naturally from differences in aperture is all well and good. But that this scales linearly with aperture is so easily appreciated it hardly bears more than the briefest thought.

What we *really* desire to know is this; how far does my telescope depart from perfection? The MTF chart tells us clearly, and this is of direct relevance to the quality of the view over the range of exit pupils useable.

For an afocal instrument (with eyepiece, used visually), the quality of the view *as perceived on the retina* is what ultimately matters. In this context, then, contrast transfer must be rated with respect to the perfect wavefront emerging through a given exit pupil.

The importance of the telescope and eye *as a system* is too easily overlooked. The exit pupil is the coupling interface which locates the entrance pupil (objective) at the eye's pupil. The eye cares not a whit what other optics lie in front of it. All that matters is the pupil diameter and the wavefront passing through it.

If that wavefront is not aberrated, the dimension of the Fresnel pattern on the retina scales precisely as the f/ratio of the light cone defined by the exit pupil (or iris, if the smaller.) This is irrespective of the telescope aperture. And if any particular aberration or aberrations is present (say, 1/2 wave if spherical aberration), no matter the aperture, for given exit pupil it will have identical apparent extent on the retina.

This is why the exit pupil is so important. It is the normalizer for image surface brightness, diffraction and extent of aberrations.

If we assume for the moment that both systems are otherwise essentially perfect, how can we can say that the C14 delivers better contrast than a 6" APO when we know from its MTF chart that the additional obstruction impacts contrast??? At given exit pupil (smaller ones, of course) we will see in the C14 the degradation of contrast compared to the slightly cleaner image in the 6".

Yes, I know. The much larger image in the C14 WELL MORE THAN COMPENSATES for its slightly poorer contrast transfer. But just because that greatly larger aperture, with its commensurately better resolving piwer, so handily bests a superior but smaller instrument in no way means it delivers better contrast transfer.

If it did, then contrast transfer is to a *very* great degree merely interchangeable with resolving power, scaling almost purely with aperture. That's superfluous, don't you think?

To me, contrast transfer, at least in the context of an afocal system, is kind of like the Strehl ratio; it's related only to what a perfect, same-size, (in this case unobstructed) aperture could deliver.

### #57 Mark Harry

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Posted 03 April 2013 - 05:31 AM

About the ganymede observation with 6+14" scopes---
***
Isn't the resolving power of a 6" around 1 arc-second?
I would think if so, an investigation why the features weren't seen is mandatory.
(Daves, and Marks remarks about rule of thumb- right on the mark!)
M.

### #58 Jon Isaacs

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Posted 03 April 2013 - 06:52 AM

Concentrating on the *absolute* differences in resolving power resulting naturally from differences in aperture is all well and good. But that this scales linearly with aperture is so easily appreciated it hardly bears more than the briefest thought.

Glenn:

It seems that all too often, the fact that "it's axiomatic that at given spatial frequency a larger aperture delivers better contrast" is simply forgotten. This is particularly true in this forum, I see it stated time and time again that refractors, regardless of aperture, have better contrast than other scopes...

This is the issue I believe Eddgie is addressing, this is the issue I constantly address: That contrast for a given spacial frequency is a function of aperture, that in terms of planetary viewing, a 10 inch Newtonian with a 20%CO will have better contrast than a 4 inch anything. To try to explain this, I use phrases like "all scopes are obstructed", "by far, the most important diffraction effect that affects contrast is the result of the most important obstruction, the Outer Obstruction, more commonly known as the Aperture."

When evaluating a telescope, I am an observer, I am looking to see what I can see, the telescope itself, it ought to be a black box. I don't really care if it's a 4 inch scope that is as perfect as a 4 inch scope can be or a 10 inch scope that has average optics, what I want to know is which one will provide me with the more detailed views of Jupiter, which one will provide me with best view of Stephan's Quintet... which one will provide me with the best view of the North American Nebula..

When one normalizes by aperture, it's pretty apparent that an apochromatic refractor, because of the simple, unobstructed, will provide better contrast. What is not so clear though is that a somewhat larger scope with a central obstruction can overcome that advantage and provide equal or superior contrast to the refractor. An "Equivalance Formula" should be able to provide a basis for this relationship.

The value of MTFs, contrast transfer, is that if they are not normalized, then they can provide a basis for analysis to determine some sort of equivalency relationship. It's pretty apparent that a 12.5 inch F/6 Newtonian is going to have a big advantage in spacial contrast and resolution over the most perfect 6 inch refractor ever built, but how about an 8 inch Newtonian, how does it fit?

And then when one includes the practical aspects, thermal management in all it's many aspects, optical quality, sensitivity to seeing.. it gets really complicated.

For me, to understand that the contrast scales with exit pupil is interesting and worth understanding but it is also academic. What counts happens at the eyepiece, when comparing two scopes, I am not using the same exit pupil in both scopes, I am using the optimal exit pupil in each scope... What do I see looking at Jupiter with my 16 inch Newtonian versus my 4 inch apo?

I think we both know the answer.

Jon

### #59 Eddgie

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Posted 03 April 2013 - 08:41 AM

Because while the resloving power is about 1 arc second, the detials will have lost most of their contrast at this size.

That is the whole point of MTF.

If you start out with a detail that has 100% contrast (Black on white) that is only one arc second across, it will have lost almost all of that contrast in the 6" scope!

That is what MTF is about.

All telescopes loose contrast.

By the time you get to detail this small, it will have so little contrast left that it is impossible to see unless it started with 100% contrast to begin with.

Lets take the example of the curve of Reggio Galilee. This is a Mare like feature on Ganymede. It describes a curve that is ony about 1 arc second in width and one side is flattened.

But the contrast of this area is very poor.. Perhaps ony 20% against the surrounding landscape (dark gray against light gray).

Becaucause this curve starts with low contrast, in the scope that can ony resolve a one arc second feture, 98% of teh contrast would be lost at the focal plane. A feature that started with 20% of the contrast at the target now ony has about .5% contrast at the focal plane. It has become invisible to the human eye, which requires between 2.5% contrast (brightly illluminted target) and perhaps 5% contrast.

In the C14 that has twice the spatical response, this feature that started with 20% contrast still at about .4 of the scopes maxiumum spatial frequency.

This means that a 1 arc second detail in this scope will still possess over 50% of the contrast that it started with.

A one arc second feature on Ganymede in this scope (regardless of the magnification used) will still show with 10% contrast at the focal plane.

As you can see, the feature has lost too much contrast to be visible in the 6" scope at all. If it is less than one arc second in size, it completely merges with the background.

But in the C14, it is still visible with 10% contrast. Difficult, but well within the eye's ability to resolve.

Contrast transfer has nothing to do with the observer or the eyepiece or the magnification.

This is the image at the focal plane being formed by the optics.

Resolition figures published with telescopes give you the resolving power for Double Stars.

A double star is a case where the contrast starts with 100% and reprsents the best possible case for the telescope because you are looking at a black seperation between two bright Airy Disks. That black "line" that appears between the two Airy disks represents the size of a feature where the contrast is almost completly gone. We only see it becuase the the contrast starts with 100%.

So, all of this other talk about magnification and image scale for different size telscope when viewed visually miss the point.

The point is that at the focal plane, different instruments transfer contrast (get detail from the target that starts with a given contrast and form the image on the focal plane with a lower contrast than on the target) at different rates.

By the time the 6" APO shows the image, unless it starts with 100% contrast (like the line between a double star) a one arc second detail will have lost so much contrast as to be inpossible to see.

The 14" aperutre will only have lost about 50% of the contrast detail for a target this size. If the detail started with 20% contrast, 10% will still remain.

That is how MTF charts work.

They tell you how much contrast a line pair of a given width will lose when it is rendered at the focal plane.

This can be used to infer how much contrast for a similar sized detail will be preserved.

Io in the 6" APO is almost featureless even on the very best of nights.

In the C14, it takes great patiance and excellent seeing, but there are occasions where I can resolve detail.

And I should be able to. THe C14 preserves far more contrast for these small sized details than the 6" APO does.

In fact, the C14 preserves more contrast for any size detail than the 6" APO does.

And that is what the MTF chart show you. They show that both instruments loose contrast on detail, but the sloping line shows you how much contrast is lost for a given size detail

The C14 has contrast transfer that is sperior across the entire range of detail size.

All planetary detail is shown with better contrast in the C14 than in the 6" APO.

Even on nights when seeing is less than great, large scale detail still stands out better in the C14 than the 6" APO. While small scale detail may be blurred out, the observer still enjoys the better contrast of the larger instrument.

### #60 Eddgie

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Posted 03 April 2013 - 09:48 AM

The section continues, exploring various aspects of the central obstruction, equivalence formula, etc.
It appears that (from that section, and his formula on page 163), that all other things being equal (including the crucial element of seeing...) that the unobstructed equivalent of a 14" with a 32% would be 9.52", according to Suiter. On that score, the 34% CO equivalent of a 6" unobstructed aperture would be a tad more than 9 inches?

To harken back to the OPs question, the formula is really only general, and works better for visual observing than for imaging. For imaging, it is not at all accurate because the obstruction doesn't matter for the finest detail the scope can resolve. Even a large obstruction doesn't loose much additional contrast past about .7 on the MTF plot meaning that for the finest detail, the size of the obstuction doesn't matter.

Here is a plot that shows what is going on.

This is a plot for a perfect 9.25" 36% obstructed instrument

The red line would show how much contrast would be lost for even a perfect 9.25" apeture.

The dashed red line shows how much contrast would be lost for the obstructed instrument.

The blue line shows how much contrast would be lost for the perfect 6" instrument.

Now the forumla for S' Max would indicate that the 9.25" scope will have lost almost all contrast for a line frequencey of 2 line pair per arc second on the target.

The S' Max for the 6" apeture (if you have been following along) is 1.32 arc seconds per line pair on the target.

First, no one should be able to argue that the angular resolution is better for a 9" scope than a 6" scope. There are numerous formulas out there that show that, and contrast transfer is after all, a function of angular resolution. It describes how far away from the geometric center of the Airy Disk light energy will fall. Everyone knows that the Airy Disk is smaller as apeture grows (all other things being equal).

Back to the chart. The 1 on the X axis indicates that the 9.25" instrument can resolve 2 line pair per arc second on the target.

The 6" instrument cannot reasolve lines this small at all. It can only reslove line paris that are 1.32 arc seconds indicated by the fact that the S' max is only 65% of the S' Max of the 9.25" scope.

Now, follow the graph over to where you see the little green line between the obstructed and perfect aperture lines.

This represents line pairs that are about .33 of the S' Max of the 9.25" Scope, or lines that are about .66 lines per arc second (in other words, line pairs about 1.5 arc seconds wide at the target).

What the MTF chart shows for line pairs (or detail about 1.5 arc second in size at the traget) is this.

If these lines started as a 1.5 arc second wide line pair with 100% contrast on the target as black and white alternating sine waves, in the perfect 9.25" scope at .33 S' Max, they will have lost about 39% of their contrast.

They will appear as very dark gray lines alternating with very light gray lines, but the contrast is high ehough that they will still appear to the eye as almost black and white.

Continue down to where the green line is between the two scopes in question. Note that the perfect 6" aperture will show those same lines with only about 46% contrast. Now, rather than looking more black and white, they are starting to look more medium dark gray and medium light gray. They are starting to blend together because the light from the Airy Disks coming from the points along the white line are spreading into the area of the black lines, causing the contrast to lower.

Now, follow the green line down to the obstructed apeture.

In this case, the contrast as fallen to 43% maybe.

Now this is not a lot of contrast difference at all. Even the very very best observers will struggle to see a difference this small, though on a brightly illuminated target, people can often judge contrast differences as small as about 2.5%.

This graph shows though that a 9.25" 36% obstruted instrument is not quite as good as a perfect 6" apeture for details that are bigger than about 1.4 or 1.5 arc seconds in size on the target.

And the 6" aperture retains that contrast advantage for all lareger size detail though as the detail gets larger, the advantage dwindles. It peaks for line pairs about 1.3 to 1.5 arc seconds wide though.

But past about .5 of the S' Max of the bigger scope notice that it quickly gets back to even footing with the perfect apeture.

This represents contrast for the smallest possible detail the scope can resolve.

And this should make perfect sense if you use a double star as an example.

The space between the double stars can represent a feature on the surface of a planet.

Suppose you had a feature that was the same length as the spacing between a pair of double stars that was 1.5 arc seconds from geometric center to geometric center.

In a small aperture, the big extension of the Airy Disks would cause them to meet very near the center of this line.

If you used a bigger scope, the Airy Disk would be much smaller, and as a consequence, you would see more of the dimension of that line that was not covered by the Airy Disk of a star at either end.

And that is what you get when you viewin a planet. Each point of every feature emits ligth that speads out the distancee of the Airy Pattern based on the size, quality and obstruction of the telscope that forms the image.

If the telescope that forms the image conentrasts that energy in a smaller circle, or keeps more of the energy in the Airy Disk itself rather than in rings far away, it may preserve more enegy from the point that formed it.

MTF describes how this energy is distrubuted around the geometric point that formed it.

So, a 9.25" 36% obstructed aperture, when used visually, will have conrast transfer that is fairly close to a 6" perfect apeture.

This is a crucial qualification though. Again, this is only if the obstucted scope is perfect.

Finding 6" APOs with perfect optics is not difficult. No leanding manufacturer is going to sell you an expensive APO that is less than near perfect.

Near perfect C9s though are not the norm. And when you add the typeical inperfections, it can cause the MTF line to further sag.

And it doesn't take much for the line to sag enough so that the contrast falls to a 10% difference.

And when it does, the difference starts to show at the eyepeice.

If you know the amount of error though, you can model it.

### #61 Eddgie

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Posted 03 April 2013 - 09:59 AM

Now it gets interesting.

Here, I have modeled the same scope as appears above.. A 9.25" 36% obstructed apeture.

This one though, is less than perfect. I have modeled in a bit of spherical abberation and a bit of astigmatism, both of which are not at all uncommon in mass produced scopes.

By compraison, just about any 5" APO from a specialty provider you can buy these days will have optics that consistently border on perfect.

Notice now that the sag has incresed a bit. While it does not seem like much, suddenly, at the important visual frequenceis, the 9.25" 36% obstructed aperure is not transferring contrast any better than perhaps a perfect 5" instrument.

And this is perhaps why accounts when comparisons are made differ.

The "Rule of tumb" formula is only really good for visual observing. For imaging, obstruction is not usualy an issue.

All defects or design quailites (secondary obstruction, and optical defects) add to the sag of the MTF line.

When the scope is unobstructd, even a little quality error does not affect the contrast enought to be a concern.

But when the obstrucion is large, the quaity becomes much more critical.

Any meaningful amount of spehrical abberation or astigmatism can quickly lower the MTF performance so that it is reduce to contrast transfer no better than an apeture half its size.

So now, you have one person that says thier C9.25 is better than a 5" APO and one that says theirs is not as good.

Sample to sample quality varations could easlily account for that.

The purple line shows that with a little sperhcial abberation and a little astigmatism, the 9.25" apeture is now performing with less contrast transfer at the lower (visually important) frequencies than a perfect 5" instrument!!!!

And that is the beauty of MTF.

### #62 Sean Puett

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Posted 03 April 2013 - 10:20 AM

While it is true that larger apertures suffer more from seeing, I would say that there is when using both scopes side by side, there was rarely a night that I did not see more detail in my C14 than in my 6" APO.

Even on the best nights, I have not seen detail using the 6" APO that I routinely see in the C14.[/color]

And when seeing is so poor that I can't see more than in the 6" APO, then it just is not worth doing, because even the 6" APO will suffer on such nights.

Yours really must be a fantastic C14!
And I envy you your obviously paramount seeing conditions.

Chris

I live in an area with bad seeing as the norm. I have set up both my 4" refractor and 12" reflector together numerous times. When seeing is bad, both scopes are always affected. Occasionally, low mag images in the refractor are slightly better. Most of the time this is not the case. Seeing affects magnification limits, from my experience, and has little to do with aperture or telescope design. The only reason refractors sometimes seem to perform better in bad seeing is that they can operate at a much lower minimum magnification.

These contrast debates seem to ignore the fact that you need great contrast with a small scope to see what is easily seen in a larger scope. My 4" refractor shows the belts on Jupiter as a light pinkish to brownish tone that is low contrast. My 12" reflector shows them as dark reddish brown and it shows that there are more than 2 of them. Detail can be seen in the bands as well and again, it is easily seen. The downside is that it is very bright and you don't want to be fully dark adapted when looking at Jupiter.
I do get the contrast side of the debate in that on the moon, in my refractor, I can see more detail in the bright area than with my reflector. Maybe because I am completely blinded by looking at the moon with a 12" scope, maybe not. But, it is a more pleasing view in my refractor.

I really enjoy refractors. That being said, if my wife threatened to divorce me unless I cut down to one telescope, I would choose my 12"newt. It shows more detail on everything. There are things that do not fit in the fov though. I really hope she never says that to me...

### #63 GlennLeDrew

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Posted 03 April 2013 - 12:14 PM

In the context of finding the equivalence between an unobstructed and obstructed telescope, one is naturally concentrating on the image representation in the range out to about 5 times the resolution limit (0.2 on the MTF's ordinate). It is in this region, after all, where the most significant differences lie. And so the comparative representation of image contrast is of paramount importance.

What inspired my more generalized approach was the greatly unfair juxtaposition of a C14 and a 6" refractor. Unless that big Cat has horrid optics, it naturally will present better contrast over the full range of resolution provided by the much smaller refractor.

I digressed from the specific requirements of the thread and presented a more 'holistic' way to appreciate the images as presented by more widely disparate apertures. That is, to consider contrast transfer of a system as normalized to the exit pupil.

In a fashion, this approach still has validity here. Each telescope has the same exit pupil limit as regards the realization of its theoretical resolving power. That is, if for observer X some telescope realizes its resolution limit at an exit pupil of 1.1mm, so will all others. If aberrations are bad this limit might be reached at a somewhat larger exit pupil. But at given exit pupil, the image more afflicted by diffraction and aberrations belongs to the poorer telescope as regards optical quality. But then, this is so obvious, ain't it?

Getting back on track... A poorer but larger telescope often bests a smaller, "perfect" scope, by virtue of its greater resolving power. But then, this is so obvious, ain't it?

The \$64,000 question: Can a general equivalence formula be derived? If we restrict to the limit of several times the resolution limit, and consider only a limited variance on quality, I shouldn't see why not. That is, if we discard the poorer specimens from consideration, and consider only the small scale regime of the image, we should be able to predict with some confidence.

As long as it's always borne in mind that the problem is restricted to contrast transfer as regards the rendering of small details. In terms of image brightness, the larger obstructed aperture (nominally) is the better...

...which may have some small contribution to make, partially mitigating against its reduction in contrast.

Outside the small scale regime of several times the maximal resolving power, the larger obstructed scope is the better, for the contrast reduction becomes of little importance, with the brighter image handily making up for this.

### #64 KaStern

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Posted 03 April 2013 - 03:11 PM

Hi,

what type of refractor and what type of reflector

I can tell you that my 8"f/6 newt was able to give equally good views
of Jupiter and M13 when I compared it to an 7"f/6 TMB
during the german ITT astrofest. Secondary sice is 39mm.

An f/8 achromat falls short due to it`s colour aberration
that lowers contrast transfer.

But there are some unobstructed reflectors out there an the best of these
can rival an equally apertured apochromat.

And there are some reflector types wich are very much compromised.

In addition one can find very differing optical qualities in real world telescopes.

And not at least there are very many miscollimated reflectors
amd some miscollimated refractors too.

So in the end you can only do the following:
Get a scope of sufficiently good quality, collimate it,
let it cool down and observe the objects the scope is suited for.

Cheers, Karsten

### #65 jhayes_tucson

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Posted 04 April 2013 - 12:47 AM

Now it gets interesting.

Here, I have modeled the same scope as appears above.. A 9.25" 36% obstructed apeture.

This one though, is less than perfect. I have modeled in a bit of spherical abberation and a bit of astigmatism, both of which are not at all uncommon in mass produced scopes.

By compraison, just about any 5" APO from a specialty provider you can buy these days will have optics that consistently border on perfect.

Notice now that the sag has incresed a bit. While it does not seem like much, suddenly, at the important visual frequenceis, the 9.25" 36% obstructed aperure is not transferring contrast any better than perhaps a perfect 5" instrument.

And this is perhaps why accounts when comparisons are made differ.

The "Rule of tumb" formula is only really good for visual observing. For imaging, obstruction is not usualy an issue.

All defects or design quailites (secondary obstruction, and optical defects) add to the sag of the MTF line.

When the scope is unobstructd, even a little quality error does not affect the contrast enought to be a concern.

But when the obstrucion is large, the quaity becomes much more critical.

Any meaningful amount of spehrical abberation or astigmatism can quickly lower the MTF performance so that it is reduce to contrast transfer no better than an apeture half its size.

So now, you have one person that says thier C9.25 is better than a 5" APO and one that says theirs is not as good.

Sample to sample quality varations could easlily account for that.

The purple line shows that with a little sperhcial abberation and a little astigmatism, the 9.25" apeture is now performing with less contrast transfer at the lower (visually important) frequencies than a perfect 5" instrument!!!!

And that is the beauty of MTF.

One of the things that impresses me about this group is how much many of you guys know about optics; however, some of this discussion has veered completely into the weeds so I want to clarify a few points:

1) ALL optical systems have wavefront errors. Over many years, I've tested hundreds of components and systems with state-of-the-art PSI and dynamic interferometers and it is extremely rare to see any component with PV errors less than about 1/25 wave (and even that is a difficult level to measure on an absolute accuracy scale--but that is another subject.) Any contention that refractors have fewer errors than reflectors is a generalization that just ain't so. There are high quality reflecting, refracting, and catadioptric systems. It all depends on the design, manufacturing tolerances, and alignment of the system. Keep in mind that on-axis performance isn't all that counts either. Few (if any) of these systems are shift invariant and the size of the aplanatic patch is generally small, which means that spatial frequency response will vary significantly with field angle.

2) A theoretically perfect reflector with a 32.5% obscuration meets the Rayleigh (as well as the Marechal) criteria for diffraction limited performance in both Strehl and MTF. In the real world, it is possible to balance the degradation in performance due to wavefront errors against the obscuration ratio such that the performance remains diffraction limited but that will require a smaller obscuration.

3) You guys are getting way too hung up on individual frequency response components and forgetting that actual image detail is composed of the Fourier sum of the transmitted frequency components (assuming a zero PTF.) A larger aperture always allows a wider range of frequency components to exist in the details of an extended image. That produces sharper edge response making small details easier to see. Furthermore, an obscured aperture actually has slightly better high frequency response (meaning higher contrast at higher spatial frequencies) than an unobscured aperture. Small losses in contrast in the middle frequencies generally have little effect on the perceived "sharpness" of an extended object. It is the high frequency response that drives the "clarity" of small details in an extended object. Obviously, if the mid-frequency transmission falls "too-much", image quality can begin to suffer, which is why we use MTF as a tool to evaluate system performance in the first place.

Johnson's law correlates the cycles per dimension required for observing features on a target (with incoherent illumination.)

Detect = 1 cyc/dimension
Orient = 1.4 cyc/dimension
Recognize = 4.0 cyc/dimension
Identify = 6-8 cyc/dimension
Recognize (50% accuracy) = 7.5 cyc/dimension
Recognize (90% accuracy) = 12 cyc/dimension

The more cycles/dimension that you can produce is what allows better imaging and that always requires bigger aperture.

BTW, if you want to get hung up on the shape of the MTF curve, you might as well start building telescopes with square apertures. A square aperture has a (non-diagonal) linear decrease in MTF. Get over it, keep the MTF curve within the diffraction limit, and you’ll be fine.

4) I not sure that I understand what it means to "normalize the MTF to an exit pupil." MTF is commonly normalized to spatial frequency with the maximum at 1/lambda*F...where F= F/#. That means that all F/5 telescopes will have contrast fall to zero at 0.4 cyc/micron at a wavelength of 0.5 microns--regardless of aperture. That .4 cyc/micron projects onto the sky from the focal plane (not the exit pupil) at different angles which depends on the focal length of the system. For a 6", F/5 telescope that translates to 1.48 cyc/arc-sec and for a 12", F/5 telescope, 2.96 cyc/arc-sec.

5) Finally, while it is completely valid to analyze the MTF performance of the telescope objective, you guys haven’t included the eyepiece in the discussion. Remember, MTF can only be cascaded for incoherently connected components. An eyepiece is coherently coupled to the objective and must be considered a part of the system to correctly analyze full system performance. That means that you can completely destroy the performance of even a perfect objective with a poor eyepiece.

John

### #66 GlennLeDrew

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Posted 04 April 2013 - 02:08 AM

John,
That was an excellent contribution to the discussion.

Regarding your point #4. The phrase "normalize the MTF to the exit pupil" is a clumsy way of mine to try to get across the following point. Any equally good telescope (as a system, with eyepiece of course) working at a given exit pupil will present to the eye an identical degree/scale of visible diffraction on an image point. If the Fresnel pattern subtends some angular scale in a small scope at some particular exit pupil, it will be the same for a larger scope at the same exit pupil.

In other words, the resolving power as a fraction of the maximum is a function of the exit pupil.

Again, a clumsy way to place a basic principle of the afocal system into some perspective, so as to provide something of a more complete picture.

### #67 Sasa

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Posted 04 April 2013 - 04:14 AM

I take the example of the Celestron Omni XLT 150, which has a 150mm objective with a 46.5mm CO. Now according to Jarad's formula, that should make the Omni XLT equivalent to a 103.5mm refractor for planetary performance. But in my proposed reworked formula, the "Refractor Equivalence" of the Omni XLT should be a 126.75mm scope. Now, people may say this is wishful thinking, and I cannot say with any certainty that it isn't, but it would be nice to see how an Omni XLT 150 compares with 4" ED, 110mm ED, and 120mm ED (as well as the equivalent achromatic) scopes. I own neither the 110 nor 120mm ED scopes, nor any achromat, though there's a STRONG temptation for me to buy the XLT and compare it to my 102mm ED scope. According to Jarad's formula, it should still surpass it, but really, just barely, and the planetary performance should really be about the same. Stay tuned, but anyone else wishing to evaluate my recalculation of that old Reflector ~ Refractor Equivalence formula is more than welcome to educate me and the rest of the CN Brotherhood.

I had Orion Optics N150/750 with about 33% central obstruction with lambda/8 optics and Hilux coating. Quite often it was outperformed on planets by my 80mm apochromat (Lomo 80/480 triplet). There is no comparison with my ED100. This 4" refractor showed me much more on Mars and Jupiter than 150mm Newton ever did (but in this case it was not side-by-side comparison as was in case with 80mm lens).

I think the obvious difference has nothing to do with theoretical expectations. In my case, I'm storing telescopes at home at room temperature. And I was often travelling for dark skies with former 150mm Newton. I think that thermalization and constant need of colimation were the main reasons why even the small 80mm refractor was outperforming much bigger brother on planets. This is the reason why I sold 150mm Newton at the end and bough ED100. I still consider it, after few years of using ED100, as one of the best moves that I made. On the other hands, I have no doubts that a wisely used 150mm Newton with properly designed OTA would outperform 100mm refractor on planets (not necessary at f/5, it would probably require something longer, like f/8).

### #68 Jon Isaacs

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Posted 04 April 2013 - 06:32 AM

5) Finally, while it is completely valid to analyze the MTF performance of the telescope objective, you guys haven’t included the eyepiece in the discussion. Remember, MTF can only be cascaded for incoherently connected components. An eyepiece is coherently coupled to the objective and must be considered a part of the system to correctly analyze full system performance. That means that you can completely destroy the performance of even a perfect objective with a poor eyepiece.

Indeed.. but these days, there are very few "poor eyepieces" and very good eyepieces are not expensive... The differences between eyepieces is subtle, the difference between apertures is not.

Jon

### #69 GlennLeDrew

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Posted 04 April 2013 - 08:47 AM

The difference between observers' eyes is greater than found across the bulk of available modern (and not so modern) eyepieces. On axis, at any rate.

### #70 olivdeso

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Posted 04 April 2013 - 10:59 AM

So, a 9.25" 36% obstructed aperture, when used visually, will have conrast transfer that is fairly close to a 6" perfect apeture.

This is a crucial qualification though. Again, this is only if the obstucted scope is perfect.

Finding 6" APOs with perfect optics is not difficult. No leanding manufacturer is going to sell you an expensive APO that is less than near perfect.

Near perfect C9s though are not the norm. And when you add the typeical inperfections, it can cause the MTF line to further sag.

And it doesn't take much for the line to sag enough so that the contrast falls to a 10% difference.

And when it does, the difference starts to show at the eyepeice.

If you know the amount of error though, you can model it.

having compared the TEC160ED to a C9 and a C11, I can confirm this.

I would add, that the C9 should be perfectly colimated and at thermal equilibriuum. I mean collimated on the Airy pattern, which already requires a excelent seeing for these diameter.
So on the field, the TEC160ED will often perform at its maximum while the C9 will require much more care and good seeing condition to deliver similar visual performances.

### #71 timmbottoni

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Posted 04 April 2013 - 12:37 PM

Fascinating topic!

I keep seeing the comparison of mirrored optics that are perfect used in the mtf and formula comparisons. I would love to see an objective way to compare not perfect mirrored optics whether newtonian or sct which is much more realistic, to a high quality apo refractor.

And I like the approach I see by some who are taking into consideration all of the factors including the human eye, seeing, cool down, etc. Somewhere I remember reading that our eyes can resolve the most detail with a 2mm exit pupil which should also be a factor

Timm

### #72 timmbottoni

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Posted 04 April 2013 - 12:43 PM

Maybe I missed it but is anyone considering the difference in the amount of light lost due to the two reflective surfaces required for reflectors?

Wouldn't this significantly reduce light gathering whereas a refractor loses much less light throughput?

I have no idea how much or how this is calculated by the way

Timm

### #73 Jon Isaacs

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Posted 04 April 2013 - 12:58 PM

Fascinating topic!

I keep seeing the comparison of mirrored optics that are perfect used in the mtf and formula comparisons. I would love to see an objective way to compare not perfect mirrored optics whether newtonian or sct which is much more realistic, to a high quality apo refractor.

And I like the approach I see by some who are taking into consideration all of the factors including the human eye, seeing, cool down, etc. Somewhere I remember reading that our eyes can resolve the most detail with a 2mm exit pupil which should also be a factor

Timm

I think the same assumption is being made about the Newtonian optics as are being made for the refractor optics. High quality apo optics are about 1/8th-1/10th wave. This is possible with a Newtonian...

As far as maximum resolution of the eye occurring at a 2mm exit pupil, I think that is a number mentioned in terms of seeing faint DSOs. Consider resolving a double star. The 2.3 arc-second pairs of the double-double are easily resolved in an 80mm but not at a 2mm exit pupil. A 2mm exit pupil corresponds to 40x, in my experience, the cleanest splits of these pairs are over 100x in an 80mm and if I am pushing the limits of an 80mm, a 0.5mm exit pupil or smaller is desirable.

The eye works better with more light...

Seeing, that's important but variable... some folks live where seeing is rarely excellent, then a small, fast cooling scope might be good. Some folks live where the seeing is good and often excellent. Then a bigger scope that can take advantage of the seeing is useful.

Last night, I got a nice clean split of 52 Orionis, "wide enough to drive a car through"... Skytools 3 tells me it's 1.0 arc-seconds... that didn't happen with a 4 inch or a 6 inch...

Jon

### #74 olivdeso

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Posted 04 April 2013 - 01:14 PM

There is another factor which may count: the vitreous floaters.

They are much more annoying at x2D on a refractor than at x1D on a reflector

### #75 jhayes_tucson

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Posted 04 April 2013 - 02:48 PM

Maybe I missed it but is anyone considering the difference in the amount of light lost due to the two reflective surfaces required for reflectors?

Wouldn't this significantly reduce light gathering whereas a refractor loses much less light throughput?

I have no idea how much or how this is calculated by the way

Timm

Throughput due to transmission issues is generally not a big issue--it depends on the quality of A/R coatings for the refractor and the reflective coatings for the reflector. Modern, multi-layer reflective coatings can easily achieve 96-99% reflectivity values. To find the total throughput simply multiply the reflectivity of each mirror together. So, for a two-mirror system, the throughput will be in the range of 92-98%. The biggest light loss in the reflector is due to the shadow of the secondary. Just remember that throughput is not what determines peak brightness in the Airy pattern. That is determined by the Strehl ratio. For circular apertures with zero wavefront errors it is easy to show (using a little Fourier optics) that the Strehl relative to an unobscured aperture is given by the square of the throughput (due to the obscuration.) So for an obscuration ratio of 32.5%, the throughput will be just shy of 90% and the relative Strehl will be 0.80 (all relative to a refractor.)

For a refractor system, the quality of the A/R coatings and the absorptivity of the glass are what matters. For small systems (<10"), the absorptivity of the glass is generally negligible so we assume the transmission to be 1.0. Broadband A/R coatings range in efficiency from about 2% to 0.25% (very difficult even at one wavelength!). If we assume that the broadband transmission at each surface is about 99% (probably a bit high), then we can compute the total transmission. For an air-spaced, double element achromat or apochromat, there are four surfaces and the total transmisision will be about 95%. Three element apochromatic objectives may be oiled together or cemented so they will likely have a little higher transmission value; but, that’s harder to compute on the back of an envelope. In either case, if the coatings are good, the light loss due to the coatings is roughly equivalent so it’s not worth worrying about.

John

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