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Putting the "Rule of Thumb" to test

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#26 WRAK

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Posted 08 November 2012 - 03:14 AM

After weeks of clouds and fog finally no fog and clear sky patches between fast moving clouds with some additional veil clouds above - you have to take what you get. Moon still under horizon, naked eye magnitude limit bit less than +3mag.
Having prepared a small double star session in Peg I had first to wait for a cloud to clear, meanwhile I tried Delta Cyg again - with x140 a fuzzy blob instead of the usual needle point with only a fuzzy hint for the secondary. Reducing the aperture in steps down from 140mm to 65mm I got suddenly a much better image with a crisp star disk and I could convince myself to seeing still a hint of the companion.
Then to Peg and to 3.9" STF2958 +6.6/9.1mag - with 140mm and x70 hint of companion in the right position, with x140 clear split, but the companion seems fainter than +9.1mag. RoT (116/sep*delta-m) would give this one about 75mm for reflector and 2/3=50mm for refractor. I reduced the aperture down in steps down to 70mm with still faint glimpses of the companion, maybe a bit optimistic for these conditions.
I have now come to the conclusion that with all due respect to Bruce and Fred the discussed RoT is too simple to work.
Especially the calculation of difference of brightness as delta-m=m2-m1 does not make much sense as the magnitudes are on a logarithmic scale with the base 2.512 which means the difference in brightness is 2.512^m2-2.512^m1.
As I have with an iris diaphragm now a fine tool for finding limit apertures for doubles within the reach of my scope I will use each observation session to fill up a small database with aperture limits for all observed doubles and then doing some number crunching with the principle of least squares error sum. Meanwhile I can study the many valuable approaches of this topic from professional and amateur astronomers (Lord, Peterson, Haas, Arguelles, Funakoshi ...) to get some ideas for a senseful formula.
Wilfried

#27 fred1871

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Posted 08 November 2012 - 09:56 PM

Wilfried, your sky sounds like mine - "weeks of clouds"...
What you're finding with the diaphragm on your refractor is interesting, though I'm not sure it tells us as much as might appear. I think it's become clear that small apertures do better, comparatively, than medium apertures, and medium apertures do better, comparatively, than large apertures. A great deal of this effect is due to atmospheric ("seeing") factors.

A smaller factor, though it could have a slight but noticeable effect, is the longer f-ratio you get when stopping down your refractor, because it improves eyepiece performance with many eyepiece designs. The spot diagrams in Rutten and van Venrooij's book demonstrate this. I get a similar small improvement with my Petzval f/5.7 refractor by using a 2.5x Televue Powermate, which gives an effective ~f/14 for the eyepiece. Star images are neater with this than the same magnification achieved by eyepiece only. And, no, I'm not using Huyghenian eyepieces. A mix of orthoscopics, Naglers, Pentax XW, etc.

STF 2958 in Pegasus is a double I'm familiar with, and I have notes on it from several occasions in the last few years. With my 140mm refractor it's an easy one - clearly double at 62x, though I like it better at 114x. I don't have the impression that the secondary star is fainter than m9.1, at least not by a significant amount. So that impression might be an effect of your local light pollution levels. My local naked-eye limit is, on the better moonless nights, around mag 5, though dust or water vapour can reduce this sometimes (4-4.5).

Where did you get your iris diaphragm? I haven't noticed these being listed commercially.

Your list of people to read is a mixed collection. Peterson won't tell you much about telescope limits. He used one small telescope at quite low power (45x) and his diagram tells us more about the limits of vision with such an arrangement.

Sissy Haas's table in her book is based on a study of stars reported separated (in some sense) in the long list of observing notes in the Revue des Constellations. You'll need to look at the telescope sizes used to see how useful that is.

Chris Lord is a different matter. His is a detailed and comprehensive study, and he takes into account past work (Lewis, Treanor) as well as his own observing with a number of telescopes from 3-inch (76mm) to 10-inch (254mm). It's very useful.

Arguelles - well, I have difficulty with the "difficulty index", not least because it crams into a tiny part of the index anything that's difficult. My feeling is that Arguelles needs to re-analyse his data into a different form in order to be useful. Funakoshi I've looked at but haven't analysed yet.

I think a detailed study of Lewis's table, cherry-picking certain observers who habitually pushed the limits, is more useful than his aceraged results. The averages tend to be too easy to establish limits, unsurprising given the observing program choices of some of those listed, and the putting together of large with medium with fairly small telescopes.

Large telescopes do noticeably less well than moderate to small ones. Louis Bell, in The Telescope (1922), commented on this in regard to the results Lewis had tabulated for SW Burnham, in regard to the Dawes Limit - we're not even looking at uneven pairs here. Bell noted that Burnham did better than the Dawes Limit with 6-inch aperture, and 9.4-inch - "with none of the others did he reach it and in fact fell short of it by 15 to 60%" - that's with 12, 18.5, and 36-inch telescopes. "All observations being by the same notably skilled observer and representing discoveries of doubles, so that no aid could have been gained by familiarity...."

Treanor (1946) in addressing the issue of resolution, particularly of unequal pairs, ends up with a distinction between larger and smaller telescopes, and in graphing results has a split between under 15-inches and over 15-inches, using Lewis's data.

Treanor attempts to determine a "modified Rayleigh Limit", based essentially on diffraction theory, as per Rayleigh. He looks at the relative brightness level of the diffraction rings, translated to magnitudes, and requires a star to fall on an interspace in the rings based on its brightness relative to the rings, which become increasingly dim as we move outwards from the diffraction disk.

If we list his figures, in terms of the Dawes Limit (treated as a practical criterion), we get the following:
delta-m 1mag 1.2DL as a limit
2m 1.5DL
3m 1.8DL
4.4m 2.3DL
5.9m 3.24DL
Some of these figures are approximate, based on measuring Treanor's graph.

Treanor remarks that "this curve [the modified Rayleigh Limit curve] appears to form a tolerable limit to the observations, though as a criterion of resolution under astronomical observing conditions, it appears rather optimistic".

The RoT shows clearly the effect that various of us have noted - it needs modification at small delta-m levels.

So, to list - if we take the RoT in its refractor version, as 2/3Dm as a multiplier for DL, we get:
Dm 1 RoT 0.67 DL - clearly not valid
Dm 2 RoT 1.33 DL - close to Treanor's 1.5DL
Dm 3 RoT 2.0 DL - close to Treanor's 1.8DL
Dm 4.4 RoT 2.9DL - Treanor 2.3DL
Dm 5.9 RoT 4.0 DL - Treanor 3.24 DL

Treanor is more optimistic than RoT as the Dm increases - observing experience suggests that the Treanor levels with large Dm factors are very hard to achieve, fitting Treanor's own comments.

Treanor also mentions the work of Danjon and Couder in the 1930s (in Lunettes et Telescopes) regarding the "correction to be applied to obtain the effective from the theoretical resolving power for given apertures and states of atmospheric turbulence. Their tables show the increased importance of turbulence with large apertures, even in rather good observing conditions". Chris Lord has also tried to factor this in.

A further useful point - "Moreover, these authors point out that slight imperfections of spherical aberration, even within the lambda/4 tolerance limit, will greatly increase sensitivity to turbulence". I'd add that central obstructions have a similar effect in increasing sensitivity to turbulence, as well as moving more energy into the diffraction rings.

Much of the above is based on studies I did back in 1996-98and wrote up at the time, in looking at resolution limits for unequal doubles. Since then I've increasingly realised the complexity of the issues, but I still think Treanor's work very valuable in working toward better predictions of what's possible.

So - a few more things to think about. I'll stop there, for now, before I turn this into a monster length note. :grin:

#28 WRAK

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Posted 09 November 2012 - 04:26 AM

... Where did you get your iris diaphragm? I haven't noticed these being listed commercially...

Fred, thanks for your comprehensive answer, I will certainly have a look at the work of Lewis and Treanor.
Regarding the iris diaphragm: I know one commercial version from Baader (http://www.baader-pl..._iris_gross.gif) but this is sold only together with the expensive H-alpha sun filter.
So I just searched for "Iris Blenden" in the internet and found the offer from SAHM of interest (http://www.irisblend...is_sprengN.html). I handicrafted a kind of cap over the sliding baffle with the iris diapraghm (outer diameter 198mm and max. inner diameter 145mm) on top of a 10mm foam strip fitting the diameter of the baffle. Weight is about the same as of a heavy eyepiece and is easily balanced by retracting the baffle. Nice tool to convert with reasonable effort any double star observation into a limit observation.
There are even larger iris diaphragms available from Edmund optics (http://www.edmundopt...howall#products) usable for larger refractors.
Wilfried

#29 WRAK

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Posted 22 November 2012 - 03:16 AM

Persistent fog prohibits observation sessions so my list of limit aperture observations is growing rather slow - 20 objects so far. But this gives me time to check the details of other Rules of Thumb for further investigation.
The most prominent is certainly Lords RoT and while the mathematical content of his paper (http://www.brayebroo... RESOLUTION.pdf) is a bit intimidating the performance of his algorithm is less impressive. Compared with the so far discussed too simple approach of "Requ.App=Dawes/Sep*Delta-M" Lord's formula "S = 1.033 * 10 ^ [ 1/n * ( Abs(delta mag) - 0.1 ) ] * rho" is certainly far more advanced as it includes a 10^Delta-M component and therefore takes account of the nonlinear character of increasing Delta-M. But it disregards the increasing difficulty of splitting doubles of given separation but decreasing brightness - so you get for a 2" +4/7mag and for a 2" +7/10mag double the same result and this is certainly wrong. Lord gives a correllation coefficient of 0,9998 which is quite high but does not provide an indication of the average error of his formula. Applied to my small data set (performance factor n choosen as unrealistic 10 to get the smallest possible error of the formula with my data) I get in terms of required aperture for a given separation an average error of 40mm - and this seems quite huge with lagest deviations -97 resp. +37mm which means a bias towards requirement of too small apertures for fainter doubles as is to expect from the mentioned weakness of this formula. Quintessence: A reasonable good RoT will have to allow not only for Delta-M but also for increasing faintness of the double.
To be continued.
Wilfried

#30 fred1871

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Posted 22 November 2012 - 09:28 PM

Wilfried, I can only agree that Chris Lord's mathematical content is a bit intimidating. For that reason I use his nomogram as an easy way to read off resolution levels. My experience is that some of the time the result is backed up by doubles I've observed with various telescopes, but not always. Fainter pairs certainly don't follow the rule of brighter ones, especially when they're uneven as well.

I think Lewis was on the right track with his work, but his numbers don't always match up to observing experience because he didn't distinguish, as I remarked before, obervers pushing the limits versus those not doing so - and didn't separate out categories of telescope size, small, medium, large. They all went into the same categories of doubles.

I'm currently compiling lists of the more difficult pairs I've observed in the various categories, and with the telescopes with which I've done the greatest numbers of pairs - 20cm and 35cm SCTs, and 14cm, 15cm and 18cm refractors. I have less material on pairs I've seen with Newtonians (15cm and 25cm).

Many of the doubles I have notes on are not on the Haas list for the current project on unequal pairs. Some of them are, and I'll send my notes on those to Haas.

Meanwhile - I think anyone compiling observations will find their limits on fainter pairs are not as good as on brighter ones, which shows up in Lewis's data lists as well. With the 18cm refractor I had access to some years ago, close pairs around magnitude 8.0 to 8.5 were not much harder than those around magnitude 6, but those around magnitude 9.0 to 9.5 were noticeably harder. That was with a suburban sky with a naked-eye limit about 4.5 on moonless nights.

And in the case of doubles near the limit of visibility the observer's eyesight becomes a significant factor as well. We're not all equal either in sharpness of vision nor in ability to see faint stars. That remains true even if we only compare experienced observers, who have the practise (experience) to make the most of their eyesight.

#31 WRAK

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Posted 23 November 2012 - 03:54 AM

... I'm currently compiling lists of the more difficult pairs I've observed ...

Fred, the approach of setting "difficult pairs" equal to "limit observations" is as I see it the problem of most studies done so far in this area. I am quite often surprised how much room is still for decreasing the aperture for "difficult pairs".
One exception is Peterson - he included also non splits in his analysis and could this way better define the "frontiere" between splits and non splits. He had also a different approach concerning the relevant factor with the magnitude of the secondary. Next I will check his formula with my so far small data base.
Wilfried

#32 fred1871

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Posted 23 November 2012 - 04:30 AM

I have a list of "failed to see" objects as well. But often enough when I go back to these, same telescope on a better night, I can see them. Not always, of course. Some stay permanently beyond reach for a given telescope with my eyes and even the best sky conditions.

Peterson is of minor relevance in these matters. His study was ultimately a work on what he could and could not see with a low magnification that did not allow his telescope to reach its limits either for separation or faintness. It does demonstrate that as stars get fainter, eventually they need to be further away from the primary to be seen. But I think we knew that. Perhaps the significant point of his plot is that the separation limit is steady until a certain level of faintness is reached, and after that pairs need to be wider.

However, because of the low magnification, even for a 3-inch telescope, this is less informative than it might have been.

#33 WRAK

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Posted 23 November 2012 - 02:04 PM

I have checked Petersons algorithm (min.sep = (10^(5/8(m2-TML+2.4)))*Seeingfactor with TLM=9.1+(5*LOG10(D_inches)) and Seeingfactor is the greater of 1) Dawes limit or 2) the size of the seeing disk)) which is therefore based on the difference between the magnitude of the companion and the telescopes magnitude limit - the closer the harder to resolve, not much of a surprise.
Applying to real data results in a mixed bag with a lot of heavy deviations from observations and a far worse average error than Lords approach.
Never the less I think that the magnitude of the companion is a relevant factor for a satisfying RoT formula.
Your mentioned go/no go cases seem perfect limit observations and I would be grateful if you could share some of these observations with me - if possible with an (estimated) naked eye magnitude limit (seems also be a factor of great influence) and focal length (who knows?).
Wilfried

#34 fred1871

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Posted 23 November 2012 - 06:55 PM

If Peterson had observed with a higher magnification his limit of resolution would have been much closer than 3 arcseconds. Even a 3-inch refractor can easily beat that - so the "wall" effect in his diagram represents the limit of his visual acuity with that telescope - an apparent separation of ~125" (45x at 3" separation).

In the distant past I had a 3-inch refractor and could separate pairs below 2", and see "figure-8" shapes on even pairs around 1.5", roughly the Dawes Limit. And I used higher magnifications to do that. Likewise, higher power showed fainter stars than magnitude 11, even from a suburban site with a mag 4-4.5 limit.

Looking at Peterson's diagram again, one interesting feature is the lack of information on the magnitude of primary stars - we have only secondaries' magnitudes. This will make a difference with the closer pairs, and with fainter secondary stars. If Peterson can see some mag 8.7-8.8 stars at 3.0"-3.5" from the primary, were the primary stars magnitude 5 or 6 or 7 or 8? So that when he sees companions around magnitude 9 at 10" or 20" separation, is this because the primary stars are much brighter with those examples? - one might expect this to be the case.

James Mullaney in his book on Double and Multiple Stars recommends that "observers desiring to create a Peterson diagram for their telescope use higher powers than that employed in the original study" because the "resolving magnification" is "at least 25x per inch of aperture". That would be a minimum of 75x for a 3-inch telescope. When I had a 3-inch, I commonly used 133x (9mm Ortho, 1200mm focal length) for the closer doubles, and 200x (6mm Ortho) for the most difficult. That's 45x per inch and 67x per inch respectively. If I were to do a repeat of Peterson's work with a 3-inch, I'd probably use 133x as my standard eyepiece - allows the Dawes Limit pairs to be detected, and goes nearly as faint as the aperture makes possible. I'd also keep a record of the primary stars magnitudes. And, yes, it would become much less simple than Peterson's work suggested.

The above comments are not meant as an entire dismissal of Peterson's work. It has its usefulness - though not much in the direction it was often interpreted, because it's too much a study of what Peterson could see (the limit of Peterson's eye under a badly chosen magnification limit) rather than what he could have seen by pushing the scope to the limits of what it could show him. The latter is what Dawes and Lewis and Treanor and Lord and Arguelles etc - and ourselves - have been investigating. There will always remain the variable which is the observer - even under near-perfect seeing and with near-perfect telescopes. Factoring that in to the equation - the capable and experienced but not exceptional observer - will be a bit fuzzy.

#35 fred1871

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Posted 23 November 2012 - 11:05 PM

A few thoughts on limiting observations. With regard to pairs that are pretty equal in brightness, I can report that a binary such as Gamma Centaurus, that has currently closed beyond reach (~0.15"), was a fairly easy pair in the late 1990s at 1.1", but by 2009 at 0.4" was beyond my 140mm refractor. The same telescope has succeeded in elongating several pairs in the 0.50"-0.55" range. That appears to be its limit for showing definite elongation. Splitting requires more separation - Dawes and Rayleigh fit well.

On unequal pairs, 10 Arietis, mentioned in the thread on 72 Pegasi, did show as a pair with the 140mm - mags 5.82 and 7.87, a delta-m of 2.05m, at 1.3" - on a night of "good+" seeing it was split at 230x. That's one of the tighter uneven pairs I've separated. But I haven't tried it with an aperture mask to see where the "not visible" point occurs.

Likewise, Theta Gruis, which is on the Haas list of test pairs, I saw as a neat split some years ago at 180x with an 18cm refractor; last month with the 140mm refractor I could see it as a lesser star adjoining the primary at 250x, and a neat split at 400x - it was not far from overhead that night, and seeing was very good. Magnitudes are 4.45 and 6.80, delta-m 2.35, at 1.5". Again, I've not attempted it with less than 140mm.

Also in Grus, RST 5560 (part of DUN 248) was split with the 18cm refractor at 330x - mags for the close Rossiter pair are 6.15 and 8.93, so delta-m ~2.8, at a separation of 1.3". This one I'll try again with my 140mm refractor, when I get a sufficiently steady night. Same separation as 10 Ari, but larger delta-m. It might prove too much for the 140mm. The RoT for refractors suggests it'd need to be a little wider - 1.55" - to see as a double.

However I definitely did better than the refractor RoT some years back with the 18cm - Upsilon Gruis, mags 5.70 and 8.24 - delta-m 2.54 - separation only 0.9" - at 330x the companion could just be made out as a brightening near the first diffraction ring. This was on a clear moonless night of above average seeing.

The 18cm telescope was an f/9 apo from AP (Roland Christen), focal length 1600mm, normally used at f/9; I had access, not ownership; regrettably beyond my budget. The 140mm is a lowly Petzval achromat from Vixen, that I commonly use with a 2.5x Powermate (f/14) on the tougher pairs. Not sure how I'd rate the focal length of that arrangement. With the Powermate it's effectively 2000mm, instead of basic 800mm.

#36 WRAK

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Posted 24 November 2012 - 03:26 AM

... I commonly use with a 2.5x Powermate (f/14) on the tougher pairs. Not sure how I'd rate the focal length of that arrangement. With the Powermate it's effectively 2000mm, instead of basic 800mm.

Fred thats interesting - higher focal ratio is assumed to be combined with larger depth of focus and I always wondered if this effect is also to get with a Barlow/Powermate. Will have to try this whenever I have the next opportunity.
Focal ratio could very well play also a role for splitting thight unequal pairs - but I have not considered yet the complexity of a then required recursive function, seems better to make this not too complicated.
A RoT with reasonable small error range could go as follows: Dawes as base + Lord for delta-m + Peterson for m2 (idea, not algorithm) + function for m1 + function for NEML.
Wilfried

#37 fred1871

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Posted 24 November 2012 - 08:05 AM

Footnote to my previous note - I've now found another observation of RST 5560, again with the 18cm scope, this time just split at 180x - seeing was better on this night. So I'm now more optimistic that the 140mm might show it split as well (at higher power).

I don't think the Powermate increases the depth of focus - focus is quite critical, and I'd say similarly critical, at similar powers, whether with eyepiece only or using the Powermate with a longer focal length eyepiece.

Interesting suggestion on a more complex RoT. I'll have a look at it.

#38 WRAK

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Posted 25 November 2012 - 05:50 AM

Fred, regarding 10 Ari and Theta Gru - could you please estimate your naked eye magnitude limit at these observations?
Wilfried

#39 fred1871

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Posted 26 November 2012 - 01:25 AM

Wilfried, I doubt the naked eye limit was a factor in those observations, given the fairly bright stars. I don't have exact numbers for each observing night, usually only a note on moonlight if there, and haze or cloud if present.

These were clear sky observations without moonlight, so the naked eye limit was likely 4.0-4.5 for the location of the 18cm refractor, and ~4.5-5.0 for the 140mm.

#40 WRAK

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Posted 29 November 2012 - 12:51 PM

Check of the concept of Funakoshi did not bring new insights. 2002 he suggested an algorithm based on a multitude of "delta-m^n" components leading to unreasonable differences to real values. 2009 he suggested an improvement in form of an additional component based on a modification of the Dawes limit "(1-k)*116" with k as "threshold of Airy disk's light intensity (0<=k<=1)" but also this new approach produces unreasonable differences to real values.
Similar components in the algorithm as in Lord's (means Dawes and delta-m factors) but even worse results.
Back to the start.
Wilfried

#41 fred1871

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Posted 29 November 2012 - 06:53 PM

Wilfried, I agree with you about Funakoshi. He re-runs some material from previous work on uneven pairs, but I didn't find anything new that was also likely to be useful. His algorithms, as you say, appear to give rather erratic results.

Not entirely back to the start.

I think Treanor with his work based on diffraction theory is useful, in terms of best possibilities (very good seeing, very good optics, very good eyesight being assumed).

Also the RoT seems to be fairly accurate within certain limits of telescope and delta-m. It also fits quite well with some of the observations that appear close to the limit, from my own observing, and from looking through the best results (most difficult pairs seen) in a selection of the data Lewis compiled, and with telescopes in the 6-12-inch range (15-30cm).

As I've said earlier, I think small scopes (6-10cm) do better for their size, and big ones (40cm+ and especially 50cm+) do less well for their size, due to atmospheric seeing effects.

But I do agree, there's no easy answer in sight, and a fair way to go on looking for a general purpose algorithm, rather than a within-bounds-RoT.

#42 WRAK

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Posted 01 December 2012 - 04:53 PM

For lack of own observations due to ongoing cloudy and foggy nights I thought it clever to add some of the obervations Lord lists in his paper (http://www.brayebroo... RESOLUTION.pdf) to my data and selected those closest to Dawes to ensure as good as possible "limit" observations.
Then came the moment of doing some calculations with the different RoT approaches - no one delivered any good results for the added pairs, not even Lord's own algorithm.
I then applied Lord's algorithm to all of his for 3" and 6" refractors listed observations only to find that the average error in required aperture is 39mm - and for example "to split this double with 66% probability you need an aperture between 81 and 159mm" is certainly not a useful result.
So I returned to Lord's paper to check the details and found some sobering facts:
- I found at least one bogus observation (J781 - if observed with 3" as listed then it should have been evident, that this double is about 2mag fainter as advertised - WDS is meanwhile corrected)
- I found some observations clearly to be not on the limit as there were for same scopes other doubles with same separation but with higher delta-m listed
- There are no observations below 75mm (3") - but many unequal doubles have their limit in this range
- There are no observations between 75 and 150mm (6") - but most of the for an amateur interesting unequal doubles have their limits in this range
- All observations are with given apertures, so any listed "limit" observation has a random character and an error range covering the gap to the next smaller resp. larger scope.
So in total the used data shows statistically serious flaws in my opinion especially in the range of small telescopes.
Then Lord grouped the observations, calculated average observation ratios based on Dawes limit and developed his algorithm on base of these averages - this procedure does not seem statistically correct to me.
Especially the nonexistence of observations in the range of 60-140mm (with the exception of 3") makes the approach of Lord completely dubious at least if applied for this range and most of us do exactly this.
There is no Chris Lord bashing intended here but this is quite disappointing – but maybe I made wrong conclusions, so any corrections would be welcome.
Wilfried
PS to Fred: Thanks for the hint with Treanor, will have a look at his work

#43 WRAK

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Posted 02 December 2012 - 04:48 PM

After a first reading of the Trenaor article on this topic (http://articles.adsa...GIF&classic=YES) I find his approach based on the theory of the diffraction pattern and based on Rayleighs criterium certainly of interest - this need a follow through.
The data base for his final conclusions is identical to Lord's (Lewis) and lacks therefore valid observations in the small telescope range of 60 to 150mm (with exception of 75mm) so these have to considered with care.
Wilfried

#44 fred1871

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Posted 02 December 2012 - 10:58 PM

Wilfried, your analysis of Lord's work is very interesting. And the point that various examples of stars split are not near the limit of the telescope used does indicate a problem with setting boundary conditions for resolution. This I suspect will be a problem for the Haas Project as well (one of several).

What is not surprising is that Lord's resulting algorithm has difficulties when used for small telescopes. These appear at times to re-write the rules for splitting doubles, and a good example, apart from some observations mentioned in this forum in the past, is Jerry Spevak's work as written up in Bob Argyle's book on "Observing and Measuring Visual Double Stars".

Spevak, with a 70mm refractor, detected doubles ("elongated")down to around 0.7"-0.8" - the Dawes limit for that aperture is 1.65". Curiously, he reported pairs near the Dawes limit not as "touching", which one might expect, but "notched" - the one example of "touching" is at 1.9", near the Rayleigh limit.

But there are so many reports from experienced observers with good telescopes, detecting duplicity in pairs that might be thought beyond the aperture, that there's a need to take notice - clearly, small telescopes can perform better for their size than larger ones. Quantifying that will be an interesting job. :grin:

With regard to Treanor: I don't think he has ultimate answers, but by using Lewis's large database and analysing it in a different way, a way that I think is far more productive of reasonable results and numbers that come closer to the reality of what's possible, he provides what I see as some "limiting conditions" for medium and large telescopes. That's in terms of the Rayleigh approach to doubles via diffraction theory. If it turns out that some doubles can be detected without being a close fit to the simle version of diffraction theory - so that a secondary star might be seen as an dimmer extension to the primary, without falling neatly into the dark interspace between the diffraction disc and the first bright ring - then that would be an advance on our knowledge, and indicate observing possibilities beyond what we'd expected.

And perhaps this condition might be more readily met with small telescopes where seeing is less problematic, and the amount of light gathered is less.

That's obviously a preliminary thought on my part - to be followed up, like so much else.

Meanwhile, I'd suggest continuing with Treanor - he has some useful ideas in that study. And you might get some new insights that could extend his study.

#45 WRAK

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Posted 03 December 2012 - 07:29 AM

If I understand Treanor right then he assumes that unequal pairs can be split at the minimums of the diffraction pattern with the lowest possible brightness of the companion equal to the brightness of the next maximum.
This would mean:
- pairs with a delta-m up to 2.86 can be split at Rayleighs limit (=radius of the airy disk)
- pairs with 2.86 < delta-m > 3.88 can be split at the second minimum
- pairs with 3.88 < delta-m > 4.56 can be split at the third minimum
- pairs with 4.56 < delta-m > 5 can be split at the fourth minimum.
The minimum aperture for these splits would be the scopes with the required size of the diffraction pattern - assuming that average wave length can be used for all pairs would make this a feasible approach for a RoT.
One curious side effect of this concept would be that with a larger aperture than required as mentioned above you will not be able to resolve a binary as the secondary will then sit in the next maximum an can therefore not be seen as long as the aperture is not big enough that the secondary is again outside of this maximum. This has certainly to be checked - but this would also be an explanation, why there are so many 60mm observations of doubles which seem to be difficult to split with somewhat larger apertures.
Wilfried

#46 fred1871

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Posted 03 December 2012 - 10:13 PM

Treanor does not seem to accept that pairs with delta-m up to 2.86 can be split at Rayleigh limit, and his graph indicates that - even though one might read his discussion as allowing such a high delta-m at Rayleigh.

The nearest we get is the graphed result of Howe's bright double stars, where a delta-m of 2.4 is almost at the Rayleigh limit. However the graph suggests that at 2.86 magnitudes the limit is about 1.5 x Rayleigh.

Remember also that all of Lewis's data points are from refractor observations, so the brightness of diffraction rings will be less than with obstructed telescopes.

Looking at Howe's original publication of doubles measures, with the Cincinnati 11-inch refractor, I'm left wondering if Lewis kept Howe's original magnitude estimates which were rough and in 0.5 mag steps, or whether he sought improved photometry for the study. A change in delta-m figures would move Howe's data point - likely to be lower, and therefore not as remarkable. As it is, that's the standout data point on the graph, so it needs investigation. Accepting Howe's measures, if we get modern photometry for the Howe pairs that fit the close and bright criteria, do we end up with the same data point?

Your thought on why smaller telescopes might do better is interesting.

It is the case that sometimes using a different aperture telescope can move a companion star from an interspace to a bright ring, hence making it less visible. Whether that's the whole story I doubt - seeing definitely impacts less on small telescopes, especially refractors.

Your telescope diaphragm could be used to study some pairs where the dimmer secondary star might be on the first diffraction ring at one aperture but not at another. The first ring, being much brighter than the 2nd, would be the more interesting to experiment with.

#47 WRAK

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Posted 04 December 2012 - 12:55 PM

A first very quick tests with the delta-m parameters as discussed above resulted in a similar to slightly better average error than with Lord. This seems encouraging - with a bit of tuning this could may be much better. What is missing and has to be added is the consideration of magnitudes of the primary fainter than +6mag.
If weather gets better I will certainly study the maximum faintness I can see within the first diffraction ring.
Wilfried

#48 WRAK

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Posted 05 December 2012 - 05:52 AM

Quote Treanor "As the result... it appears that... resolution occurs if the faint companion falls on a minimum of the brighter star, such that the adjacent maxima are not brighter than the faint star itself."
Some observations like for example Delta Cyg do not confirm this - with a delta-m of 3.4 resolution should occur at the second minimim with an aperture of about 97mm but there are reports of much smaller apertures down to 60mm which means a companion fainter than the first diffraction ring can be seen at or within it even if this sounds not very plausible.
Wilfried

#49 fred1871

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Posted 05 December 2012 - 10:21 PM

Wilfried, I haven't run the numbers to see what magnitude the first ring is equivalent to, but your noting that 97mm aperture puts the companion of Delta Cygni into the second minimum does suggest that calculating the aperture for putting it into the first interspace might be informative. I'll try it and see what the numbers look like.

Meanwhile, a star less bright than the first ring could presumably be seen if its magnitude allows it to form a noticeable brightening at one point on the ring; there should be an additive effect of light despite the interference effect from the star image being on the diffraction ring of the primary. Obviously some stars are too dim to be noticeable when they coincide with the diffraction ring. In the case of Delta Cyg the secondary star is bright so it could be seen with some small telescopes as a ring spot brightening.

The whole subject of uneven pairs with small telescopes has been very little looked at, presumably because, even in the 1800s, double star astronomers typically used medium or fairly large telescopes for serious observing. That meant most of the time from about 5-6-inch (13-15cm) aperture as a "small" telescope. 7-inch to 15-inch was more usual. Remember the point from my brief write-up on Robert Jonckheere - in the early 1900s his father provided him with an 8.7-inch (22cm) refractor which he decided was too small (!!) for his double star ambitions. Hence the 14-inch (~35cm) refractor he set up near Lille.

So I suspect there's a fairly unexplored area relating to small telescopes and double stars. I no longer have a small long-focus (traditional) refractor to experiment with this - my f/16 75mm telescope was sold long ago, and I'm not convinced that my 80mm f/5 is up to the job of assessing small scope performance.

But I do know that the old 75mm was surprisingly good on doubles, and gave very neat images with little scatter despite being an achromat. At that small aperture, chromatic aberration is not much of an issue at f/16.

#50 WRAK

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Posted 06 December 2012 - 04:21 AM

The first minimum would be ident with Rayleigh means 138/2.6 giving 53mm - may be a 50mm bino could do this job, but I doubt it. This would mean a 3.4 delta-m could be resolved at Rayleigh.
But I agree that a companion fainter as the first ring could be seen as a brighter spot on the ring.
Wilfried






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