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WRAK
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Reged: 02/18/12

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Re: Putting the "Rule of Thumb" to test new [Re: Astrojensen]
      #5583679 - 12/22/12 08:11 AM

Quote:

Quote:

Using a violet or blue violet filter on many a star would dim it substantially but the diffraction pattern never changes size.




That is actually wrong. They DO change size with wavelength. The spurious disk ALSO change size with changing brightness.


Clear skies!
Thomas, Denmark




Aperture and spectrum together with brightness of a light source give a specific diffraction pattern - looking at this given diffraction pattern with a filter does not change the diffraction pattern itself but influences only what you see of it.
Wilfried


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WRAK
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Reged: 02/18/12

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Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5583680 - 12/22/12 08:13 AM

Some extra spices in this discussion are always appreciated - but related to the topic of this thread ("RoT for required aperture for splitting unequal bright or fainter than +6mag double stars") this idea of using tainted filters to "improve" the diffraction pattern is of minor relevance. Even if it works it would only mean a small shift at the base of a RoT now suggested as Dawes limit - but this would anyway be covered by the unavoidable statistical error as such an approach can not be part of a standard setup.
Wilfried


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fred1871
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Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5584421 - 12/22/12 05:45 PM

Agreed, that using filters for changing resolution levels is a side issue - standardising the wavelength where the eye is most sensitive eliminates a variable, that can be re-introduced if one wishes, as a factor modifying the numbers.

Where I'll support Thomas is in his (briefly) pointing out there are two matters regarding how we see a diffraction pattern - first, the size does change with wavelength - see the standard equation which includes lambda (wavelength) in determining the angle subtended by the image.

The other factor is perception - a fainter image will have an apparently smaller spot size (Airy disc) than a brighter star, because the eye sees less of the full diameter of the disc. Here, of course, the spacing of the rings is unchanged, though they become dimmer too.


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WRAK
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Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5585110 - 12/23/12 05:12 AM

Quote:

...Where I'll support Thomas is in his (briefly) pointing out there are two matters regarding how we see a diffraction pattern - first, the size does change with wavelength - see the standard equation which includes lambda (wavelength) in determining the angle subtended by the image.

The other factor is perception - a fainter image will have an apparently smaller spot size (Airy disc) than a brighter star, because the eye sees less of the full diameter of the disc. Here, of course, the spacing of the rings is unchanged, though they become dimmer too.



Fully agree. Wavelength is a mixed bag given by the spectrum of a star, so what we see is a multitude of wavelenghts and for simplicity we calculate with an average value for yellow light. This assumed the diffraction pattern is a given as mentioned. If you look at this diffraction pattern through a filter you dim out some wavelenghts which gives only a relative enhancement of others, so this does not change the diffraction pattern but only your perception of it - if this is of any benefit is another question.
Same goes for the size of the spurious disk - fainter stars have smaller spurious disks. If you dim your image with a filter the spurious disk will get seemingly smaller - if this would be of any benefit for splitting doubles then it would be easier to split fainter doubles with same separation than brighter doubles. And this is certainly not the case as we all know the contrary.
Wilfried


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fred1871
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Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5585160 - 12/23/12 07:21 AM

To spell it out a little more - I had in mind using a relatively narrowband cutoff filter, so that light outside a certain wavelength range was largely eliminated. Even with a passband of around 100nm, the effect shows nicely in the Damian Peach images I mentioned. The blue image (350-450nm) shows separate star discs; the green image (450-550nm) shows an elongated single image. Same object, same telescope, etc.

Regarding fainter pairs - an issue here is the human eye losing resolution as light levels drop. Which is a major factor in why fainter pairs can't be resolved as well as brighter ones. However, I've noticed that the loss of resolution comes in at a fainter level than I'd expected based on Lewis - and some of the observers he used had the same experience. SW Burnham (again) did very well with dimmer close fairly even pairs using the smaller telescopes he observed with. NOt as exceptionally well with the larger ones.

Which might lead us back to the mag 10 limit, suggested by Couteau, that was discussed a little while ago. Modest apertures don't run into the Couteau barrier as readily because of their lesser light gathering.

Much more to discuss but I'll have to go quiet for a few days given the arrival of Christmas.


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WRAK
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Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5585735 - 12/23/12 02:40 PM

Have finally come to a usable formula for doubles with separations larger than 10" - required aperture should be depend only on the magnitude limit of the scope reduced by light pollution (are there today locations for amateurs without any?).
There is a lot of rather complicated calculations for TLMs (considering even the color of the eyes of your grandpa) on the market but a rather simple of general use goes TLM = 2,7+5*LOG10(D_mm). Remains the topic light pollution expressed in terms of naked eye limit magnitude NELM down to values below +3mag for locations in large cities. With the help of some limit observations I found with statistical analysis the rather unwieldy relation TLM(NELM) = 3.2*LOG10(D_mm)+6.7392231622074*(NELM/6.5)^0.533365942734249. This gives for doubles with sep>10 a required aperture = 10^((M2-6.7392231622074*(NELM/6.5)^0.533365942734249)/3.2) - for example 110mm for POU3851 14.2" +9.5/10.5mag.
This was the last item on my list for a RoT proposal based on my quite small data set on limit observations. Between the holidays I will make an Excel spreadsheet with this RoT proposal with a link for downloading it if anybody is interested in testing it.
Next step is certainly enlarging the data set of limit observations for further enhancement and higher statistical reliability.
Wilfried


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WRAK
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Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5603144 - 01/03/13 02:39 PM

Quote:

...I will make an Excel spreadsheet with this RoT proposal with a link for downloading ...




This was a bit premature - when checking the current model with a test data set covering all relevant data points for separation, delta-m, m1, m2 and light pollution for graceful behavior I found several glitches. These are partly due to the too small data base of real limit observations and partly due to not so clever assumptions concerning the model functions. One example is the relation of increasing m1 (increasing faintness) and required aperture depending on separation - it seems that after a steep increase counteracting the fast decline in required aperture by increasing separation there comes a plateau of indifference. First tries so solve this with logarithmic or trigonometric functions did not work out so I thin I will have to work out just another if-then condition.
So there will be a delay for posting a spreadsheet with a proposed model for testing.
Wilfried


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WRAK
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Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5608610 - 01/06/13 03:06 PM

I am still not satisfied with the interacting of the different components of my RoT model but I managed some improvements and eliminated the component for doubles with separation > 10 arcsec and the component for faint companions on the border of telescope magnitude limit for now.

Here the promised link for downloading the spreadsheet with the present state of the RoT model for small refractors: "http://www.sterngucker.eu/XLS/WRAKs RoT.xlsx" as a starter filled with doubles from Taurus sorted by required aperture.
You can use the spreadsheet by simply enter the data of doubles manually but it is certainly better to get a list of all double stars in your required field of view (for example with the help of a tool like AstroPlanner) and insert it with copy and paste. You can then sort the list by required aperture and select the part of interest to you for your observation planning.

Any comments on obvious or not so obvious errors or suggested improvements are welcome.

Besides trying to improve the quality of the structure of the model the main future step will certainly be the enlargment of the existing data set of limit observations to get a better statistical relevance for the parameter values of the model. Another vital point of interest would be to add a component regarding the CO for reflectors.
Clear skies for you and me.
Wilfried


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Asbytec
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Reged: 08/08/07

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Re: Putting the "Rule of Thumb" to test new [Re: fred1871]
      #5609959 - 01/07/13 11:03 AM

Wilfred, sorry for coming in late. I have not read the other post where the RoT was suggested, but what is the rational for the division of sep by delta-m? Most math is derived from logic, but I am not familiar with how that came about. It is intuitive, however.

S = 116/D. Multiplying by D and dividing by S yields D = 116/S. Sure, resolution is determined by both separation and brightness between the two stars. The primary offers the bright diffraction background and the companion must be bright enough to show through. So, dividing S by delta M provides a way to scale D, when delta M is large S/delta M is small. This in turn increases D. Makes sense, intuitively.

But is the variable sep/delta M valid? I guess that's what you want to know. I guess it would help to understand the logic of why delta M was introduced in such a way. It is important variable, though, just asking why it was applied there.

One thing I suspect is missing is accounting for the bright diffraction field surrounding a star. That is a variable, too. As we have been discussing, a perfect unobstructed aperture puts 16% of the light into that small bright diffraction radius. It's possible a 9th magnitude star could be detected much closer to the primary.

However, an obstruction of .3D puts 32% of the star's light into the diffracted field, and it draws down the peak intensity of both the primary and the companion. In this instance, a brighter diffracted pattern and the companions lower peak intensity push that detection radius out much further. Or requires a lower delta M at a given radius.

It seems possible to calculate a RoT for one type of scope that would not apply to another. So, if there were a variable introduced to account for the bright diffraction field delta M has to contend with, then an appropriate aperture would be estimated. Of course, since a refractor is unobstructed, this variable would fall to zero and we'd be back to D = [116/(S/delta M) * 0.]

Actually, I am thinking that variable might actually be a constant of some value dependent on the relative diameter of the obstruction. It would simply scale aperture upward proportional to CO. Again, if unobstructed, it would have a value of zero.

Edit: I see where the value 2/3rd is applied for unobstructed scopes. Sounds reasonable.

So, lets say aperture is known because I have one scope, and I want to know the possible separations for observing doubles. I would rewrite the equation to S * delta M = 116/D, then divide by delta M to get S = 116/D * 1/delta M. But problems arise with delta M in the denominator, when it is zero for example. And S decreases with larger delta M, it's not the behavior I am looking for. Something in the logic (or the my math) does not appear to work: the left side does not always equal the right side, I think. The equality does not hold.


Anyway, just getting caught up and went off on a tangent. Still reading.

Edited by Asbytec (01/07/13 11:44 AM)


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Asbytec
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Re: Putting the "Rule of Thumb" to test new [Re: Asbytec]
      #5610067 - 01/07/13 11:58 AM

Quote:

I do not consider light pollution as serious influence - as stated elsewhere I even have the experience that the seeing her is most of the times better than in darker places with a stronger drop in temperature in the evening.



I would agree, when doing a rule of thumb simple is best. There are just too many variables, such as seeing, collimation, focus, cooling, light pollution, observer experience, etc. Those should be assumed either perfect, negligible, or constant. The variables that matter are separation, diffraction brightness, and companion magnitude (the latter two are factors of delta M.) If the the resolution RoT fails in light polluted skies, resolution should occur under dark skies being limited only by the laws of diffraction (governing ring intensity of the primary and peak PSF of the companion.) If seeing is bad, the RoT will fail, too. It does not mean it's not a valid RoT, it just needs appropriate conditions to work. That's why it's a RoT and not a precise mathematical equation.

Someone should say, a 10" SCT can split this pair. And indeed it can. However, like everything else, if conditions do not permit...then they do not permit. Wait for a better night for success that is STILL possible IAW the basic and simple RoT. Allow the observer to decide if a 9th mag pair is appropriate for his local conditions. Maybe, or maybe not. But the split IS doable - the important consideration - for someone else under better conditions.


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WRAK
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Reged: 02/18/12

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Re: Putting the "Rule of Thumb" to test new [Re: Asbytec]
      #5610167 - 01/07/13 12:44 PM

Norme, there have been so many intermediate steps in this process that I have to some degree lost the overview - but I cannot remember a suggestion for division of sep by delta-m. If you mean the very first step in our discussion D=116/(sep/delta-m) you should read this as D=116*delta-m/sep - sorry for the confusing notation.
In case you have not yet downloaded the spreadsheet with the current suggested RoT version (based on statistical analysis of a small data set of observations with a refractor with an iris diaphragm to be able to reduce the aperture to the limit of splitting) and therefore not read the introduction I repeat here shortly:
Rule of Thumb Model for estimating the required aperture for splitting unequal and faint doubles for small 1-6" refractors per 2013-01-05:
rA = required Aperture in mm = rA1+rA2+rA3+rA4+rA5 with a split probability of 50% and an average error of 10%
rA1 = Dawes limit = 116/sep with sep = separation in arcseconds
rA2 = f(dM) = function of delta magnitudes = dm'*7,6220905762799*(m2-m1)/sep^0,713969668906051 with dm' = switch for delta-m < 1, m1 = magnitude primary and m2 = magnitude secondary
rA3 = f(m1) = function of m1 = m1'*1,83163170523935*(m1-3,02833460276676)/(m1''*sep^-0,415228540946423+(1-m1'')*(14-m1)^-0,415228540946423) with m1' = switch for m1 <= 6 and m1'' = switch for sep <= 14-m1
rA4 = f(m2) = function of m2 = m2'*19,6060894270314*(m2-9,18074376678362) with m2' = switch for m2 <= 9
rA5 = f(NELM) = function of NELM = 1,81466984138841*(6,5-NELM)^0,901484005717405/6,5 with NELM = average naked eye magnitude limit for a given location.
The structure of the model was gained in a multitude of steps with some theoretical reasoning, formulating accordingly functions and running statistical analysis. After eliminating all approaches with bad results in terms of correlation and average error this is what remained, but there is still a lot of work to do and some parts of the model seem a bit counter intuitive and not according to any optical theory.
But we are looking here for a Rule of Thumb usable for selecting doubles for session planning and not a new optical theory.

One of the next steps could be an extension for reflectors with CO. My current basic idea for this is
- modifying the Dawes limit for reflectors regarding the somewhat smaller Airy disk depending on the CO size - means a little smaller than for refractor
- applying CO to the other components rA2-rA5 with a factor in relation to the CO percentage - means somewhat larger than for refractor
giving in total a small advantage for the reflector for close and equal bright doubles and an increasing advantage for refractor for doubles with increasing unequal brightness.

The problem in verifying this approach is the lack of limit observations with reflectors as the use of aperture masks is somewhat questionable (the percentage of CO would increase unreasonable) and all observations with fixed apertures are at "limit" only by chance - only doubles with a mixed bag of positive and negative results under similar conditions could be considered as "limit" for a given aperture/CO combination.
Wilfried


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Asbytec
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Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5610203 - 01/07/13 01:02 PM

Thanks, Wilfred, I know some very good minds are working on it. I am simply struggling to keep up.

Okay, I see much more went into the equation, and thank you for the correction.


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WRAK
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Reged: 02/18/12

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Re: Putting the "Rule of Thumb" to test new [Re: Asbytec]
      #5617561 - 01/11/13 04:30 PM

The evidence that Rayleigh and Dawes criterions have to be adapted for a smaller Airy disk radius for scopes with CO is growing even when empirical evidence is still a bit shaky. But also the in the mentioned paper of Lord published single observations show for a 150mm MAK significantly better performance near the Dawes limit than with a 6" refractor This means that for a potential CO component for the RoT Model two steps are necessary:
- Modify the Dawes limit base 116/sep accordingly to the smaller Airy disk radius due to CO (I have meanwhile found a numerical function for calculating this radius from the CO value)
- Apply a CO dependent factor on the other components regarding delta-m, m1>6 and so on. First shot would be an exponential function like multiplying these components with 1+CO^2 but with missing empirical data this is so far more a shot from the hip than an educated guess.
Wilfried


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WRAK
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Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5618801 - 01/12/13 10:57 AM

The observation reports of Fred concerning BU 1040 and BU 91 made me aware of a serious flaw in my current RoT model regarding the influence of light pollution - it is simply far to simple to add a few mm in required aperture for a bad light pollution because this is certainly an exponential component with decreasing faintness especially of the secondary. This is curious insofar as I have already made a concept for a component for very faint companions based on a TML function including NEML. The formula for the req. aperture for resolution of a star with magnitude m is 10^((m+LN((6.5/neml)^2.512)-2.7)/5) giving for example 138mm for +13,4mag and perfect NEML of +6.5mag as good approximation of the theoretical TML of a 140mm scope and approximately the same required aperture for +11.4mag with a rather bad NEML of +2.9mag. Even if these formula might be a bit too optimistic for the TML with bad light pollution it can be used to calculate the required extra aperture for light pollution giving for example an add on of 30mm from 54mm to 84mm for a +11.4mag star with a NEML of +4.5mag. So for faint companions it seems to make sense to first calculate the NEML dependent TML required aperture and add then the additional factors for delta-m, m1>6 etc.
Will think about it.
Wilfried


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Asbytec
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Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5618883 - 01/12/13 11:37 AM

Wilfred, I am becoming aware of the complexity of developing a rule of thumb. I applaud your efforts and wish you success.

The complexity and difficulty seems to come from not only the large number of variables, from NELM to observer experience to seeing, but also the vast range of those variables, from 6/10 to 9/10 seeing to the entire Bortle scale. Accounting for all of these seems to turn a rule of thumb into a sophisticated calculation.

I would think most observers have a feel for their NELM and seeing conditions, and maybe about any other condition that affects them. So, I would think a RoT might be just that, under reasonable conditions one might expect to split this star limited only by diffraction. If the companion is a 13th magnitude star in 7/10 seeing, the observer can estimate his own chances of splitting a double (that can be split) with his aperture under his conditions.

It may be possible to refine a RoT with each successive layer of complexity, but maybe it would be best to have one that is "diffraction lmited" (if you don't already), then scale and test each layer of complexity (NELM and seeing, etc.) individually.

Each time a variable is added to your RoT, the level of complexity rises. Maybe exponentially. I would think it's best to use the KISS rule for a rule of thumb. Then, once established, add a layer of complexity. Working a puzzle with fewer pieces is easier.

Just some thoughts, Wilfried, in support of your project. Not a complaint against it. I offer them for you to (re)consider (I am sure you have) because I want you to succeed.


Edited by Asbytec (01/12/13 11:41 AM)


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WRAK
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Reged: 02/18/12

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Re: Putting the "Rule of Thumb" to test new [Re: Asbytec]
      #5619485 - 01/12/13 05:24 PM

Norme, thanks. I understand your reasoning for keeping it simple and in the beginning of this task I started with quite a simple approach. But meanwhile I found that it is not this feasable to add components step by step as they are interacting. Main problem seems to me is not the resulting complexity but the much too small data set so I am happy for each observation report on doubles of interest giving me an opportunity to eliminate errors in my current approach.
Wilfried


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Asbytec
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Re: Putting the "Rule of Thumb" to test new [Re: WRAK]
      #5620045 - 01/13/13 12:00 AM Attachment (17 downloads)

Wilfried, I suspected there might be some interaction that required simultaneous solving. I applaud your work.

One thing that keeps bugging me is phase interference. I can't help but wonder if there is not another level of complexity where out of phase waves from the primary interfere with the companion at some angular distances.

"Since, according to Van Cittert-Zernike Theorem, light arriving from stars is coherent in amateur-size telescopes, as long as it is near monochromatic, it is an interesting question how much this coherence factor, combined with the coherence-lowering polychromatic spectrum and OPD differential between two close stars influences their actual resolution limit in the field."

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

I would think, a very simple rule of thumb might just say a companion must be both above the visible threshold (aperture limiting mag and NELM) and at least 5% brighter than the surrounding ring structure.

Something like below where the red are the PSF of the companion at varying magnitudes and separation, the yellow line is either a linear approximation or a curve defining the diffraction intensity, and the green line would be the RoT at varying NELM (white dashed lines.) But, alas, it's not that simple, eh?


Edited by Asbytec (01/13/13 04:01 AM)


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WRAK
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Reged: 02/18/12

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Re: Putting the "Rule of Thumb" to test new [Re: Asbytec]
      #5622244 - 01/14/13 08:25 AM

Norme, this approach reminds me somewhat on the concept of Treanor that a faint companion can be seen at the minimum next to the diffration ring of the same brightness. Basically this concept would be of high interest because it would be fully based on optical theory but it has some difficulties. I tried some weeks ago to translate this concept into numbers but found it difficult to calculate magnitudes for diffraction rings from the energy distribution in the diffraction pattern - how do you compare the light energy in a ring to the light energy in a spurious disk? Calculating with the percentage numbers I came up for refractors with a delta-m of 2.86 for the first, 3.88 for the second and 4,56 for the third ring but when I translated this numbers into required apertures for given doubles the results were not convincing. On the other side Lord comes up in his already often mentioned paper with a delta-m of 4.39 for the first ring - and here things get a bit out of hand: While it may be reasonable to see a 2.86mag fainter companion at the Rayleigh criterion this seems a bit unreasonable for a delta-m of 4.39mag.
May be you can find a good source for magnitude values of diffractions rings comparable to magnitudes of spurious disks then we could consider this concept once more as potential base for a RoT.
Wilfried


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Asbytec
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Re: Putting the "Rule of Thumb" to test [Re: WRAK]
      #5622538 - 01/14/13 11:51 AM

Yes, I was struck by Treanor's work and a little proud to have the same line of reasoning. I would yield to you on how to calculate the ring brightness in terms of delta magnitude. It might involve flux and an area terms, and that might be a bit hard to do for a range of magnitudes and scopes. So, again, I would yield to you or simply use Lord's figures.

Even then, it still might not work because I suspect there is more than brightness happening in the rings. There is also constructive and destructive interference at varying optical path differences. I just do not have a good feel accounting for these phase differences, nor know whether or not they are significant enough to matter.

I have been eager to observe some of the remaining Tau list and begin some of the Orion list, but the weather is just not clearing as expected. And my time is short, in two weeks I begin traveling through June of this year. Our observing season is over by then.

Anyway, I am only interested in the subject, pondering it, and discussing concepts. I am not trying to make life hard for you. Thinking about the RoT and observing a few stars, I do realize how daunting the task is. Never the less, it's a fascinating area of discovery.


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WRAK
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Re: Putting the "Rule of Thumb" to test new [Re: Asbytec]
      #5629482 - 01/18/13 08:40 AM

The need for more limit observations in my data set is great, not only for my own but also from other observers to cover different observing situations.
In the first moment it seems easy to go through existing literature and pick out observations but in detail it gets difficult because when should an observation considered to be at least near the limit (as observations with fixed apertures are "exactly at" the limit only by chance):
- the easy ones have to be eliminated or else the statitical analysis will show a bias towards easy splits. Existing criterions or RoTs are of some help here
- the overperforming ones have to be considered with care and existing criterions or RoTs are not always helpful here. And in this cases we have always the individual interpretation what is a successful resolution and what not - is an occasional flicker enough for the identification of a faint companion or is an elongation without a clear indication of the position of the secondary a positive result? Absurdly overperforming observations have even when doubtless "being true" certainly to be eliminated to avoid a statistical bias towards the overly difficult challenges
- the reported advertised data of pairs is especially for historic observations in many cases obsolete. The magnitudes are relativly easy to correct because they should have been the same 50 or more years ago. Separations are another thing because meanwhile these may be different now depending on orbits, so you have to take what has been reported then
- ...
Wilfried


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