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WRAK
scholastic sledgehammer


Reged: 02/18/12

Size of spurious disk
      #5645468 - 01/27/13 04:36 AM

We had this topic here several times and so far I had no reason to question the assumption that for a +6mag star the size of the spurious disk is about 50-60% of the size of the Airy disk (first minimum of the diffraction pattern). This means that the two spurious disks of an equal bright +6mag double star should slightly overlap at Rayleigh and certainly overlap at Dawes. But there have always been observation reports with "dark space" between the disks at Dawes or even less separation - and I have always taken these reports with caution because they seemed to disagree with established diffraction theory.
Now Chris Taylor writes in Argyle's book on "Observing ... Double Stars" on page 139 that he measured the size of the spurious disk of a +5-6mag star with a x825 magnification with 0.311" with an potential error of 0.037" and that this means only about 37% of the size of the Airy disk (with a reflector with a CO of 0,163 but he confirmed this result with his 4" refractor - although there should be a small difference in the relative size of the Airy Disk due to CO).
This would explain many dark space reports on close doubles and even explain many positive observations significantly below Dawes.
Wilfried


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Asbytec
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Reged: 08/08/07

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Re: Size of spurious disk new [Re: WRAK]
      #5645500 - 01/27/13 05:17 AM

Wilfried, it's a fascinating question. I have always used the 50% value at 6th magnitude and scaled it from there.

The dark space in a Dawes split is about 5% fall of in contrast between the maxima. It's not completely dark, but detectable to the human eye and appears dark. So, the spurious discs do overlap, it seems, at Dawes limit. And maybe even at Raleigh limit with 28% drop off in peak intensity. Of course, the latter does look "black."

Let me read up on the topic and hopefully some folks will chime in. I have always wanted to know how to calculate the spurious disc size, too, more accurately according to theory, aberrations present, and obstruction effects.

I believe this effect is further complicated with brighter stars because their spurious discs look uniformly bright. I am not sure at what point they may begin to appear no uniform as luminosity falls off. But, observing close 7th magnitude pairs in fairly good seeing did seem to show a contrast differential within each component's spurious disc.

"...the human eye response to light intensity is mainly logarithmic, hence better illustrated with logarithmic PSF. For instance, the intensity gap between central peak and second maxima in aberration-free aperture is 57 to 1, respectively; the eye, however, sees the peak as less than twice brighter (this applies when both are well within eye's detection threshold; as the fainter 1st bright ring nears detection threshold and falls bellow it, the perceived intensity differential dramatically increases)."

"For instance, the intensity gap between central peak and second maxima in aberration-free aperture is 57 to 1, respectively; the eye, however, sees the peak as less than twice brighter (this applies when both are well within eye's detection threshold; as the fainter 1st bright ring nears detection threshold and falls bellow it, the perceived intensity differential dramatically increases)."

"Moderately larger disc still should allow clear resolution, due to the intensity low forming between two star images, with the discs likely appearing less than perfectly round."


http://www.telescope-optics.net/telescope_resolution.htm

Here's one treatment of resolution. I am not sure it's exactly right. I read it and found I did not trust or understand it entirely. But it might offer some clues.
http://www.cloudynights.com/documents/resolution.pdf

Edited by Asbytec (01/27/13 05:42 AM)


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Cotts
Just Wondering
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Reged: 10/10/05

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Re: Size of spurious disk new [Re: WRAK]
      #5645639 - 01/27/13 09:03 AM

Wilf and Norme. I've been reading all the threads on this topic with fascination. You guys are really making an effort to nail down the science of 'splitting'. These threads should be required reading for all doublestar geeks (like me, for instance). At the eyepiece, for me however, things are far simpler.

All I ask myself is, "Can I see two stars?" Then I have succeeded in detecting duplicity. The degree of overlap can then be given a descriptor word like 'peanut', 'oblong' etc. Since the vast majority of amateur double star observers are not measuring the pairs (and those who do are using video and computer reduction methods) then the arbitrary 'dark sky split', Dawes' limit and Rayleigh's criterion are not necessary.

Another factor to consider is that the seeing is rarely good enough to closely examine the diffraction pattern at 800x with any telescope and if the seeing IS that good then I won't spend the night measuring the diameter of the spurious disc or the thickness of the rings or the various radii - I'll be viewing really close, difficult pairs.....

Dave


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Cotts
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Re: Size of spurious disk new [Re: Cotts]
      #5645652 - 01/27/13 09:12 AM

P.S. The idea of quantifying double star resolution into a neat set of equations has been around since the days of FGW Struve and has not been successfully nailed down yet. Considering the number of parameters involved It might well be impossible:

Aperture
% central obstruction
Magnitude of star A
Delta mag A vs. B
Color of A, B

Lord tried and Sissy Haas is having a go at the Delta Mag thing but it is like catching a soap bubble with a pair of vice-grips.

Dave

Dave


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Asbytec
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Re: Size of spurious disk new [Re: Cotts]
      #5645690 - 01/27/13 09:45 AM

Dave, thank you. Yes, tropical seeing is /usually/ very good in a 6" aperture. It's been less than stellar, though, lately. Tonight I had 8 to 9/10 with a nearly full moon in the sky. Seeing the discs at any magnification was no problem, until magnitude neared 10th or more.

The proximity of the moon made dim difficult pairs even more so, or so it seemed. I did manage to split A 2705 at 9th and 10th magnitude and 1" arc sep but just could not go fainter. The spurious discs held nicely, but dimmer stars like A 2804 (nearly 10th mag and fainter companion) were begging for averted vision.

But, I agree, that's what I ask myself, "can I see two stars?" Not a rolling hint of brightness on one side of the ring like c Ori, but two discs or specks. A 2705 met that criteria, finally, as a tiny disc and a fainter speck.

Such nights are great for observing Jupiter, too, but the RoT project has my full attention.

You're correct, the RoT is very complicated and a difficult undertaking. But, I think if there is something to be gained by understanding the size of the spurious disc (either actually, theoretically, or visually), then some progress might be made.

Edited by Asbytec (01/27/13 09:51 AM)


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Size of spurious disk new [Re: Asbytec]
      #5646125 - 01/27/13 01:56 PM

Same book (Argyle), other observer with another scope (Jerry Spevak with 70mm refractor): STF2807 1.9" +8.7/8.8mag with touching spurious disks. Translating this into the smallest possible spurious disk fur such an observation I came up with ~45% of the Airy disk - but this is for a +8.7mag double and not in the +5-6mag range. So this observation seems to support rather the 50-60% hypothesis.
Taylor himself reports touching spurious disks for Beta Del with then (1996) 0.36". Regarding the effect of the rather small CO of 16,3% this would result in about 42% of the Airy Disk but this time for a brighter than +5mag double. This calculation does support his 37% claim to some degree but not fully.
So these numbers from Taylor and Spevak do not correspond very well.
I will be on the lookout for observation reports of equal bright doubles with touching spurious disks because these should allow to calculate the size of the spurious disk depending on the brightness of the components.
Dave - gratulation to your flexibility of changing your session plan according to seeing quality. I have the habit of sticking to my plan if possible at all. But as I use an iris diaphraghm I can translate any session plan into a limit observation session by reducing the aperture as far as possible. Wish I had better weather.
Your remark concerning past and current futile efforts for developping a useful RoT so far is certainly realistic - but thats just a challenge, not a reason to quit.
Wilfried


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EdZ
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Re: Size of spurious disk new [Re: Asbytec]
      #5646214 - 01/27/13 02:57 PM

Quote:

Here's one treatment of resolution. I am not sure it's exactly right. I read it and found I did not trust or understand it entirely. But it might offer some clues.
http://www.cloudynights.com/documents/resolution.pdf




I would like to point out one obvious error in my paper on resolution. About page 3, I say "a very bright star puts so much of the light into the spurious disk, the central disk can be 85% of the diaameter of the airy diosk." that is incorrect. While the central disk does contain 85% of hte light, that does NOT necessarily make it 85% of the diameter of the Airy disk.

That is the only error I know of to point out in that paper.

I recorded documentation in this forum many years ago on observations to test the size of the spurious disk. I'm sure now they are in the archives. There are several examples, but this one stands out. An observation of 16 Vul with a 150mm Ref.

I observed 16 Vul. mags 5.8-6.2/ separation 0.8" I used a 150mm f/8 refractor at magnifications from about 300x to 540x. At powers between 300 and perhaps 420x or 450x, it appeared to be completely separated. These lower powered observations would seem at first to support how some observers describe such observations as completely split. However as I approached the highest powers from 450x to 540x I was able to see more detail, that could not be seen at any lower powers. I clearly observed the spurious disks to have appox 10% overlap at powers over 500x. This observation supports that the spurious disk is approx 50%-60% of the Airy disk.

Other examples of observations very well documented in this forum are 52 Ori with a C5 (see the post "Limits of the C5") and the AB components of 16 CnC with a 150mm refractor, neither of which would support the spurious disks being less than 50% of the Airy disk.

All in all, over the last 10 years, I've documented dozens of observations in this forum, numerous in the mag 8 equal double category. (Actually, this forum has a treasure trove of documented observations.) I would say mag 8 equal double observations may get down to the 45% size of Airy disk, but the research of the actual observations recorded in here, that often give a percent overlap, would give a much more acurrate prediction.

In the "Limits of the C5" observations, I also explored the affect of blue stars on the size off the Airy and spurious disk, looking for a way to beat the 1.1" Rayleigh limit of the C5, IIRC to no avail.

edz


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Asbytec
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Re: Size of spurious disk new [Re: EdZ]
      #5647111 - 01/27/13 10:42 PM

Ed, thank you for clarifying. I believe that was the point I (personally, and maybe only me) got confused, and in the ensuing discussion about percentages of light in the Airy disc.

Is there a mathematical treatment for determining the diameter of the disc (without taking the human eye into account), maybe slicing the PSF at the visual threshold and calculating it's diameter?

Another active thread mentioned you comments in this thread, off to read it now...
http://www.cloudynights.com/ubbthreads/showflat.php/Cat/0/Number/3149322/Main...


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fred1871
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Re: Size of spurious disk new [Re: Asbytec]
      #5647130 - 01/27/13 10:54 PM

Lewis, following Airy (much earlier), gives some of the mathematics in his 1914 paper - along with levels of relative illumination in the diffraction image, etc. Of course, this is for optics without central obstruction.

Norme, the problem of where to slice - that is, what is perceptible to the eye - is I think more of an issue than the nature of the image. EdZ has indicated one aspect of that with his comments on what is seen at varying levels of magnification. In other words - how do you decide/establish "the visual threshold" to know where to slice?


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Size of spurious disk new [Re: EdZ]
      #5647364 - 01/28/13 02:40 AM

Quote:

... I would like to point out one obvious error in my paper on resolution "... the central disk can be 85% of the diaameter of the airy diosk." that is incorrect...
That is the only error I know of to point out in that paper.
...



Ed, thanks for pointing this out.
May be you could add some remarks to the topic "effect of CO" because your statement "If both scopes were the same size aperture, regardless of f#, the angular size of the Ariy disk would be the same" is certainly contradicting established diffraction theory stating the contrary that the size of the Airy disk is very well influenced by (the size of) CO.
Wilfried


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EdZ
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Reged: 02/15/02

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Re: Size of spurious disk new [Re: WRAK]
      #5647442 - 01/28/13 06:01 AM

Well, you raised two completely different issues in that you just mentioned aperture and CO. I addressed the affect of CO in that paper. I treat aperture and CO in separate sections and talk about one effect at a time, so my comments regarding each are not contradictory. Likewise I treat wavelength and magnitude each in another section and treat them separately.

One of the most difficult issues when dealing with a topic that has perhaps as many as 10 different influences that affect the outcome is to not mix up influences in the discussion. The only way to do that is discuss one influence at a time, as I did in the passage where you mentioned my statement regarding aperture. By itseelf it is completely correct and not contradictory at all. By discussing one influence at a time, the reader is not left asking, OK which of those issues caused that outcome?

Let me also point out, the CO robs light from the central peak and throws that light into the sunsequent rings, making the rings much brighter in a scope with a CO. By doing so, it reduces the size of the spurious (visual central) disk. It has very little , if any, affect on the size of the Airy disk.

edz


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EdZ
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Re: Size of spurious disk new [Re: EdZ]
      #5647488 - 01/28/13 07:04 AM

Quote:


I have always wanted to know how to calculate the spurious (visible) disc size,

Is there a mathematical treatment for determining the diameter of the (visible) disc




I do recall reading a paper years ago in which a table was presented with fractions for the visible disk. I don't recall if those were determined mathematically or from observations..

IIRC that paper was found at
Publications of the Brayebrook Observatory
a double star resource leading to numerous technical papers

and this Brayebrook technical paper in particular
Telescopic Resolution of Unequal Binaries by C.J.R. Lord

The resources thread pinned at the top of this forum included links which at one time were active tto that website. Unfortunately they are no longer active.

edz


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Size of spurious disk new [Re: EdZ]
      #5647601 - 01/28/13 08:52 AM Attachment (14 downloads)

Ed, sorry for switching topic from "size of spurious disk in relation to size of Airy disk" (we agree here so far at least to some degree) to "size of Airy disk depending on CO" without further notice - but it seems we disagree here totally.
There are at least two clear references that the radius of the Airy disk (diameter of the first minimum) changes with the size of CO. First one is the paper of Lord "A report upon the analysis of the telescopic resolution of double stars of unequal brightness" with a table on page 6 (see uploaded image) indicating that the size of the Airy disk changes significantly with the size of CO.
Second reference is to be found at http://www.telescope-optics.net/obstruction.htm in table 8 again with the same content.
Else there exist several references in the literature about double stars that the size of the Airy disk gets smaller with increased CO without giving exact numbers.
In another thread (http://www.cloudynights.com/ubbthreads/showflat.php/Cat/0/Number/5611524/page/1/view/collapsed/sb/5/o/all/fpart/1) I was already wondering why this aspect is kind of ignored but this discussion came to no conclusive result.
Wilfried


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Asbytec
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Re: Size of spurious disk new [Re: EdZ]
      #5647629 - 01/28/13 09:14 AM

Ed, I am reading, again, your treatment of resolution. On page 15, you begin discussing the effects of obstruction. As an example, you use a 6" aperture and a star with 83.8% light in the Airy disc - some of which is not visible. So, a dim star 60% as bright would be scaled down resulting in a visible disc of a smaller radius. The math seems pretty straight forward, 83.8% maximum intensity within the Airy disc * 60% ~ 50% the Airy diameter. For example, a 6" scope with an Airy diameter of 1.82" arc, the visible disc would have a diameter of 1.82" arc * 50% or 0.91" arc, which is the Raleigh limit in this case. A dimmer star, say 40%, would have a visible disc of 1.82" arc * 40% ~ 0.73" arc.

This is the convention I used when looking at an estimated visible disc size from bright stars down to the telescope's theoretical limiting magnitude (assuming 5% needed for the faintest stars to be seen above the pitch blackness of space.) Basically I just scaled their apparent, visual radii from 60% for mag 6 down to 5% for mag 14 (approx TLM.) I cannot remember what confused me during that exercise, other than maybe the slope of the PSF or the radius at any point not being that easy to work with.

So far, I have not gotten to the idea the Airy disc in an obstructed scope, the actual diffracted Airy disc, is smaller by a factor of 1 - co^2, where 1 is the aperture normalized to 1 and co is the obstruction by percentage of the aperture. In a 30% obstructed 150mm scope, the Airy disc becomes 138/150 * (1 - 0.3^2) = 0.84" arc, not 0.91" arc as Raleigh determined for a clear aperture. Added diffraction affects the outcome of the PSF.I think this explains my own success in excellent seeing of detecting a dark (not black) Dawes space on very tight nearly equal doubles, 7 tau at 0.72" arc +/- and around 7th magnitude being an example.

The maximum spacial frequency increases by that same factor, D/(1 - co^2) resulting from a smaller Airy disc, and peak intensity (normalized to 1) by (1 - co^2)^2. So, using the math above, the peak intensity for an obstructed scope might need to be adjusted down from 83.8% maximum peak intensity.

http://www.telescope-optics.net/obstruction.htm


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EdZ
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Re: Size of spurious disk new [Re: WRAK]
      #5647654 - 01/28/13 09:29 AM

thank you for finding Lord's table.

take note that although the location of the minimum of the Airy disk changes, the size of the visible disk, while it does change, changes less.

It's worth noting, as represented in this table, the position of the minimum of the Airy disk does change, but since it is a minimum, this position cannot be seen. Note how very little the position of the first ring changes and how little the size of the visible disk changes.

What has the most effect on the resovability of the double, the position of the minimum or the size of the central visual disk?

I would like to mention, this very topic was discussed extensively in hundreds of posts in this forum quite a few years ago. I do not recall specifics, but I seem to recall differences of opinion with the data represented here.

edz


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Size of spurious disk new [Re: EdZ]
      #5647790 - 01/28/13 10:50 AM Attachment (13 downloads)

Quote:

...What has the most effect on the resovability of the double, the position of the minimum or the size of the central visual disk?...




Regardless of what seems plausible or not I tend to stick with the numbers like Norme - the Rayleigh criterion is defined with the first minimum of the diffraction pattern, the first minimum gets smaller with increasing CO, so CO influences the limits for splitting of equal doubles. This is also supportet with observation reports with better performance of reflectors for splitting equal bright doubles in the +6mag range.
Wilfried


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Asbytec
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Re: Size of spurious disk new [Re: EdZ]
      #5647912 - 01/28/13 11:44 AM

The Airy disc is also slightly brighter than it would be if the radius stayed the same with an obstruction added. So, the obstruction dims it (obstruction and diffraction) and a slight decrease in diameter brightens it. This change in luminosity should affect the spurious disc size. For example, as you know, an unobstructed scope puts 84% of the light at peak intensity and a 30% obstructed scope 68%.

So, if your calculating for a 150mm scope, Raleigh limit is 138/150 = 0.92. The Airy diameter is 1.84" arc. Then, according to the math above, a dim star near 6th magnitude puts 50% of the light in the visible disc so it's diameter would be 1.84 * .5 = 0.92" arc spurious disc diameter (the Raleigh limit.)

For an 30% obstructed scope, the same "Raleigh limit" figure is 0.84" arc * 2 = 1.68" arc Airy disc diameter. However, it is also much dimmer to begin with due to diffraction and obscuration. A perfect 30% obstructed aperture produces a peak intensity of 68% (total diffraction and obscuration) and not 84% (diffraction only.) Normalized to 1, that is a Strehl-like result of 1 and 0.83, respectively.

So, following the same logic as in the clear aperture for a 6th magnitude star, it seems the Airy disc would be 1.68" arc * 0.5 = 0.83" arc radius (due to total diffraction) and dimmer by (1 - co^2 = 0.91%), or 0.84" arc * 91% = 0.76" arc compared to 0.92" arc unobstructed. An obstructed aperture Airy disc is, in a perfect optic, only 83% as bright as in a clear aperture. The rest is in the rings, respectively, 16% clear aperture and 32% with 0.3D obstruction.

Alternatively, total peak intensity (hence visual diameter) due to both diffraction and obscuration - normalized to 1 - is (1 - co^2)^2, or about 83% of the peak intensity of a clear aperture where 0.92" arc * 83% = 0.76" arc spurious disc diameter for a 150mm and 0.3D obstructed aperture. Provided the math of multiplying the Airy diameter by a percentage of visible disc holds, ya?

This seems to have some effect for very close doubles, Dawes and Sparrow limits are pushed back, where obstructed apertures have an advantage at spacial frequencies near D/(1 - co^2) just as the MTF says it will. (Though the MTF is slightly misleading because each scope's spacial frequency is normalized to 1, so the curve meets at 1.0 spacial frequency instead of D/[1 - co^2].) Of course, without taking into account the eye function, which is a wild card variable, and assuming seeing permits.

Edited by Asbytec (01/28/13 12:11 PM)


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EdZ
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Re: Size of spurious disk new [Re: Asbytec]
      #5648133 - 01/28/13 01:08 PM

Quote:

Regardless of what seems plausible or not I tend to stick with the numbers like Norme - the Rayleigh criterion is defined with the first minimum of the diffraction pattern, the first minimum gets smaller with increasing CO, so CO influences the limits for splitting of equal doubles. This is also supportet with observation reports with better performance of reflectors for splitting equal bright doubles in the +6mag range.
Wilfried




My observations with 5", 6" and 8" SCTs do not support the increase in resolvability that you indicate here. Equally as important when compared to observations with my unobstructed scopes they do not show performance moved any considerable distance along the scale to indicate they are that much closer.

Your graphic would indicate a 5" 40% obstructed scope (my C5) can resolve nearly the same double as a 6" unobstructed scope (my CR150). Spend some time and perform 10 observations with each scope near and just below the difraction limits. You will find that just is not the case. The closest pair I've ever completely resolved with the C5 5" 40% obs was 1.1", with the CR150 0%obs 0.9"

In fact my C6 40% obstructed does not appreciably exceed my CR150.

edz


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Cotts
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Re: Size of spurious disk new [Re: WRAK]
      #5648358 - 01/28/13 02:31 PM

Quote:

Quote:

...What has the most effect on the resovability of the double, the position of the minimum or the size of the central visual disk?...




Regardless of what seems plausible or not I tend to stick with the numbers like Norme - the Rayleigh criterion is defined with the first minimum of the diffraction pattern, the first minimum gets smaller with increasing CO, so CO influences the limits for splitting of equal doubles. This is also supportet with observation reports with better performance of reflectors for splitting equal bright doubles in the +6mag range.
Wilfried




Wilfried. Your graph shows a very small difference in disc size from unobstructed (1.22 units - I assume arcsecs? ) to about 1.16 units in a 30% obstructed scope. About 5%. It is highly unlikely someone would use a 40% obstructed scope to do double star work so the entire range with useful double star scopes of 35% CO and less (interpolating a bit) is from 1.22 to 1.15 units which pretty much gets you into the noise theoretically and absolutely into the noise when actually observing. To simplify, I think the second decimal place here is not significant. And so, for practical purposes, at the eyepiece, the difference between these radii in an unobstructed scope and in a 35% obstructed scope is non-existant.

Dave

PS. The data in the WDS which we all use to determine separations only go to 2 significant digits in the separation ranges we are discussing here. Consider a WDS separation given as 1.2" This is a rounded figure and could actually be from 1.15" to 1.24", a range significantly greater than the variances in disc diameter vs. CO in your graph. Noisy data, indeed.

DC

Edited by Cotts (01/28/13 02:41 PM)


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EdZ
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Re: Size of spurious disk new [Re: Cotts]
      #5648545 - 01/28/13 03:52 PM

Also, as i noted above, what is really shown in Lord's table is that the position of the minimum is moving within the dark space. frankly this is an almost inconsequential effect. To say that the position of the 1st minimum is not at 1.22 lambda, but is at 1.058 lambda, when the edge of the central spurious disk has not moved inward to the same degree and the position of the 1st ring has moved even less, essentailly means a dark minimum has moved on a dark pallette.

Take note that there is very little movement in the position of the first ring and the (what he calls) edge of the disk center.

This edge of disk center is another point that needs discussion. He seems to have chosen an arbitrary point, that at which the light is 50% or one half of the peak intensity. Take note that results in all of his disks ranging in size from 42% to 45% of the Airy disk, somewhat small from what seems to be known of central visible disks.

For what magnitude stars is that? Are we to assume the disk edge cannot be seen beyond a point where the light intensity drop below 50% of peak intensity? Or is this simply an arbitrary point? I think it is just an easily identified arbitrary point used forr consistency.

Furtermore his values indicate the visible disk increasing in percentage size of the Airy disk as the central obstruction increases, but this has far more to do with his indicated movemeent of the 1st minimum than anything else. He does indeed indicate the central disk getting smaller as CO increases, as we would expect.

Take note also that even if we assume the 50% intensity is the low limit of visibility, his central disk sizes drop only 8% from a 0% obstructed to 40% obstructed scope. Since he has arbitrarily chosen the 50% intensity diameter, we will have to assume that would hold true also at the limit of visiblity at the disk edge, regardless at what intensity we can no longer see light or how large the central disk is. That is what is going to control your ability to completely resolve the double. As I said before, it's a very small difference.

We conveniently determine the limit of resolution of our scopes by assuming the size of the visible central disk as one half the Airy disk. We even sometimes adjust the size of the central visible disk larger or smaller according to what we know of intense or dim magnitude. Empirical data shows this visible disk can sometimes be found to be larger than one half the Airy disk. It is this sizing of the central visible disk that determines our ability to resolve.

However, here in this table Lord accurately provides us with the sizes of the central disks at a constant point. So we do not use Lord's positions of the 1st minimum to determine one half of that as the central disk size as resolvabilty when he has provided us with the size of the visible disk.

This table gives no indication of a 20% gain in resolvability .

edz


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