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fred1871
professor emeritus


Reged: 03/22/09

Loc: Australia
Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5546013 - 11/29/12 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.


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5549082 - 12/01/12 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.brayebrookobservatory.org/BrayObsWebSite/BOOKS/TELESCOPIC%20RESOLUTION.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


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5550658 - 12/02/12 04:48 PM

After a first reading of the Trenaor article on this topic (http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1946Obs....66..255T&db_key=AST&page_ind=0&plate_select=NO&data_type=GIF&type=SCREEN_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


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fred1871
professor emeritus


Reged: 03/22/09

Loc: Australia
Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5551225 - 12/02/12 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.

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.


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5551600 - 12/03/12 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


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fred1871
professor emeritus


Reged: 03/22/09

Loc: Australia
Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5552970 - 12/03/12 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.


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5553935 - 12/04/12 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


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5555148 - 12/05/12 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


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fred1871
professor emeritus


Reged: 03/22/09

Loc: Australia
Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5556558 - 12/05/12 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.


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5556876 - 12/06/12 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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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5557854 - 12/06/12 04:46 PM

Fred, the following is kind of mysterious for me: Treanor states "The intensity of the third maximum is to that of the Airy disk as 415:100,000. The magnitude difference is thus 2.5log10 I1/I2 = 5.95 approximately." but this value would according to my information be valid for the sixth maximum and in the third maximum we have 1.5% of the total energy of the Airy disk and therefore the magnitude difference is 4.56. Maybe a little error of Treanor with whatever impact on his results or do I miss something here?
Wilfried


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fred1871
professor emeritus


Reged: 03/22/09

Loc: Australia
Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5558309 - 12/06/12 09:22 PM

Wilfried, I think part of the difficulty with brightness levels is that there is a difference between the total amount of light in a diffraction ring, and the maximum brightness of the ring, because the rings have a gradation of brightness across their width in the same way that the disc has a gradation of brightness across it.

So the total energy of the first ring, for example, is 7% of the total image light of a star, with 84% in the disc. That might suggest a ratio of 12:1 between disc and ring, but it's not that simple.

Although the disc has a gradation of brightness from centre to edge, the light is more concentrated in the disc, where the ring, being much larger, has its light spread out more.

Lewis (on page 374 of his article in "The Observatory") gives a table of illumination levels at a series of distances from the centre of the disc. The maximum illumination of the first bright ring is given as .017 (relative to 1.000 for the disc centre). That gives a ratio just short of 60:1, or near enough, 4 magnitudes instead of ~2.6 (12:1).

So in the case of Delta Cygni, where the secondary star is mag 6.3, it will be noticeably brighter than the brightest part of the first ring, which is equivalent to about mag 6.9. As well, the separation (2.6 arcseconds) will put the secondary's star image not on the brightest part of the bright ring, but towards the inner edge of that bright ring, where illumination is much less.

Rayleigh for 60mm is 2.3", and the brightest part of the first ring is around 3.0-3.1" from the disc centre. Where the secondary star is located, at 2.6", the illumination of the bright ring is about half the maximum, so closer to mag 7.6. Yes, I'm ignoring illumination fall-off in the secondary star disc, which makes the apparent size of the disc smaller, because that's a factor to be looked at later - though I suspect it makes not much difference overall, at least to the visibility of the star, placed where it is in diffraction terms.

The above may go some way towards explaining why a double such as Delta Cygni can be seen as a double with a 60mm refractor, when the telescope has good optics and seeing is very good, so it ceases to be a factor.

I'd expect that a similar delta-m and separation would be less readily observed with stars that are less bright. The light gathering capacity of 60mm has no difficulty with 3rd and 6th magnitude stars. At magnitudes of say 6.3 and 9.7 we'd be facing lack of light as a problem with a fairly close pair. In between magnitudes? - perhaps someone with a 60mm or similar scope might like to create a list of pairs around 2.5" separation, with delta-m of 3 to 3.5, but with the primary star at mags 4, 5 or 6, and see what's possible. Is Delta Cygni near a "sweet spot" for small telescopes? And are there some other bright pairs that show this pattern?

Hmmm... suspect I need some more thinking time on this... meanwhile, I'll hope the above ideas about brightness levels in the diffraction image might be in the right direction as part of explaining the sometimes surprising performance of small telescopes.


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5559189 - 12/07/12 12:10 PM

Fred, do you have an online link to this article of Lewis?
Wilfried


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5559592 - 12/07/12 04:42 PM

Quote:

...Lewis (on page 374 of his article in "The Observatory") gives a table of illumination levels at a series of distances from the centre of the disc. The maximum illumination of the first bright ring is given as .017 (relative to 1.000 for the disc centre). That gives a ratio just short of 60:1, or near enough, 4 magnitudes instead of ~2.6 (12:1)...




Fred, applied this to my small data set - does not help much as it increases the average error about 25% compared with my conservative approach concerning the delta-m between diffraction rings.
Wilfried


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fred1871
professor emeritus


Reged: 03/22/09

Loc: Australia
Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5559830 - 12/07/12 07:04 PM

Wilfried, the Lewis article is in The Observatory, for 1914, vol 37, pp372-379. The mention of it in Argyle (Observing and Measuring Visual Double Stars has the wrong volume number - in both 1st and 2nd editions).

I'm not suggesting my attempted analysis will be immediately applicable to an algorithm with predictive capacity. Rather, I was trying to explain why I thought a "special case" could occur. Generally, I'd expect placement neatly between the diffraction rings to be the best model; edging away from that gets into low probabilities of visibility. Delta Cygni seems to me a pair that
"sometimes" is visible with 60mm - that telescope, that observer, that night.

No algorithm is going to be completely accurate because of variable factors - observer eyesight, observer experience, interaction of telescope and seeing, are a few. I was reminded of this two nights ago when observing some difficult uneven pairs - even changing the eyepiece had an effect on the visibility of dim close companions, and the most difficult flickered in and out of visibility.

So, I was dealing with a matter - small telescopes, not medium or large - that did not fit the type of model I'd proposed for medium telescopes. And my feeling is that there are windows where the small telescope does better, comparatively, than medium telescopes. That would require a different algorithm, or predictive equation, and one with boundaries placed on it to allow for the modest light-gathering of small scopes.

I thought that my closer analysis of the diffraction pattern might be informative in the issue of Delta Cygni. I haven't yet got as far as applying it to medium-size scopes or other examples of doubles. But I have had the experience, not often, of having a "small scope" experience with my 140mm refractor, where I could definitely separate a double I'd expect to be somewhat beyond that aperture based on the RoT. I've not had that experience with somewhat larger telescopes such as my 235mm SCT, or a C14 (35cm) that I had access to in the past. With them, the ragged edge of visibility was pretty much where it was expected on the best nights. So I'm inclined to think something similar to the RoT has good predictive capacity within certain limitations. I agree it needs refining. And I'm beginning to see why Lewis didn't attempt an overall model, but gave a series of patterns (wry comment).


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5560421 - 12/08/12 05:37 AM

Fred, this is going in small steps to a common ground I think:
- splitting doubles with small scopes is a separate game with own rules (I feel even no longer sure if Dawes is valid for small scopes as also his numbers are probably derived from observations with larger scopes)
- there is no one singular RoT but a set of rules like for example:
--- m1 < 6 and delta-m < 1 then Dawes
--- m1 < 6 and delta-m < 2.5 then Rayleigh
--- delta-m > 10 then TLM including NELM
--- inbetween maybe a rule derived from crude number crunching
- any serious RoT should not only provide one number for required separation or aperture but also the average error range
- this error range should allow deriving probabilities for splitting with a given aperture and the same time cover the vast number of side effects not taken into calculation


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5565412 - 12/11/12 05:16 AM

Applied part of this step by step approach to Lord's data set for 3" and 6" refractor, but it did not bring much improvement - and once again I got the impression that this data set is not consistent on the "limit".
If for example 3" STF51 +6/12mag is a limit observation for a 150mm refractor how can then said the same of 3" A2225 +7.5/12mag and therefore 25% less delta-m?
Wilfried


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fred1871
professor emeritus


Reged: 03/22/09

Loc: Australia
Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5566446 - 12/11/12 06:59 PM

Wilfried, I think it's clear that Chris Lord's observations set does not have enough examples that are at or near the limits of the telescopes. If all his data is used, the range (from not difficult to very difficult) means there's no clear pattern. I think only his most difficult with each particular telescope examples are useful for establishing limits.

I've had a similar problem in going through my observing notes. I have observing records for thousands of double stars, hundreds of them difficult with the telescope I used in each case. The problem is determining which examples are the borderline cases, those where I was working near the limit of the telescope. All this would be much simpler if I'd had a diaphragm on the scopes I used, so I could stop down the aperture on the difficult pairs on the best nights, to see how much aperture reduction was possible without losing sight of the secondary star.

Even so, I'm gradually getting a list of pairs that do look close to the limit of the scope used. When that's finished, I'll see what pattern I can find from my own observing. Because I've been using mid-size amateur telescopes - 14cm to 35cm aperture - it fits Treanor's 15-inches or less, while avoiding the "small scope syndrome", most obvious around 60-80-100mm aperture refractors.

I'm also starting to go through the observations of others, similar to Lewis's data collection, but including some observers he didn't look at, notably some in the southern hemisphere, and some post-1914 northern observers (as his study was published in 1914).


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WRAK
scholastic sledgehammer


Reged: 02/18/12

Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5567468 - 12/12/12 12:21 PM

Applied the step by step approach now to my own data set (including 2 of your observations and some of the closest to limit from Lewis for 3" and 6" scopes) and feel stuck now - the best I could do so far is an average error in aperture of 20mm and this seems not this good as this the same value I got with more or less crude number crunching with a program for statistical analysis with hypothetical functions without much regard of optical theories.
I think I will take a recreational break with some reading on this topic and wait for opportunities for observation sessions to get some more data on limit observations in the range of 50-140mm aperture. Maybe some enlightment will come with time.
Wilfried


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fred1871
professor emeritus


Reged: 03/22/09

Loc: Australia
Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5568074 - 12/12/12 06:42 PM

Wilfried, I'm inclined to think that +/- 20mm isn't too bad related to a 140mm aperture - though I'd like to see it closer. Down at the 50mm aperture level that variation tells us nothing useful. It's proportional, as I'm sure you know.

I've had the experience of observing a double star that another experienced observer could not see, same time, same telescope, same eyepiece. The two of us looking alternately. To me the companion was fairly obvious; to him it was invisible, then after "look in this position" it was "maybe I can see it". So we have the observer factor as well. Some observers will need a bigger telescope, or higher power, or more practice on doubles.

I haven't mentioned another published paper on resolving doubles that I think you (and others) would find of interest. It was published in the (free, online) JDSO (Journal of Double Star Observations) - the particular item was in volume 4 no 4, Fall 2008 - by Tim Napier-Munn, "A mathematical model to predict the resolution of double stars by amateurs and their telescopes" .

It details a study based on observations by quite a few observers with telescopes from 80mm to 508mm on a variety of doubles to see what the limits were. And it deals with probability factors for predicting "splittability".

One thing I'd mention here is a graph that plots resolution against delta-m by aperture - for 80mm, 203mm, 356mm. There were 4x80mm, 1x203mm, 1x356mm scopes. The 80mm scopes did less well than I'd expect; the 203mm also less well; but the 356mm about what I'd expect from the RoT.

The paper discusses background - diffraction, Rayleigh, Peterson, etc etc - but the useful thing is the study based on new observations by 15 observers with 25 different telescopes. "315 valid observations were made". That's a reasonably significant total, despite the aperture spread being very large. There's no full data list, and the author does say that he rejected some observations for a variety of reasons, just as the USNO orbit catalog plots show some observations are well out of line with the pattern. Haas will have a similar experience - my impression is that one particular observer is already offering false positives on some doubles.

Last night I finally got a clear and dark and STEADY sky - able to take 400x with clear discs and neat diffraction rings - so I was able to try some difficult pairs again, and succeeded with some of them. Notes to follow after I've checked current WDS data etc.


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