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drollere
Pooh-Bah

Reged: 02/02/10

Loc: sebastopol, california
number of doubles per aperture
#5219146 - 05/12/12 06:34 PM

in the topic "How do I start?" one of my posts led to an off topic thread that i've moved here.

at issue was my claim that up to 40,000 double stars would be within resolvable reach of a 100mm (4") scope. a few people demurred with that estimate and asked how it was calculated.

first the answer: i calculated from the current separation the "rayleigh aperture" necessary to resolve it: D(mm) = 140/rho; this is REZ AP. (dawes would be 116/rho.) then i calculated the minimum aperture necessary to see the magnitude of the fainter star of the pair as =1.1*10^((m2-2.7)/5). (the 10% bump in the aperture is to get above the "dark sky" assumption in the formula for light grasp.) this is MAG AP. then i took the larger of the two apertures, and sorted the dataset on that value.

this ignores a third issue: the brightness necessary to stimulate foveal resolution in very close but matching pairs. thus, if you have a 12" telescope your resolution limit is 0.46 and your magnitude limit is 15.1 ... but you won't be able to separate a matched 15 mag. pair at 0.46"!

as remedy, i calculated a third aperture that would raise the fainter component to a visual brightness equal to a mag. 10 star viewed through a 12" telescope; this is TEN AP. (i find mag. 10 in my 12" is roughly where my foveal vision gives out.)

since each magnitude increase is a change of 2.5 times the luminance, the diameter of the mirror must increase by the square root of 2.5, or roughly 1.6, for each magnitude step below 10. thus, in order to observe an 11th mag. star at a brightness equal to a 10th mag. star in a 12" scope, you need a 19" aperture.

then i checked if the separation (rho) was greater than 10x the resolution limits defined by MAG AP or REZ AP (whichever aperture was larger); if it was, then the companion would be visible at the limit magnitude. if not, then i increased the aperture by adding the difference between MAG AP and either REZ PA or TEN AP multiplied by the resolution ratio (1/10 to 1).

(this may seem kludgy, but i don't believe the more "refined" nomograms or multivariate calculations are really any more accurate, without some measure of observer skill, instrument collimation, sky conditions, etc.)

here is the chart of number of resolvable pairs per millimeter of aperture in the revised calculation.

now it appears that a 4" (100mm) scope can resolve up to 13,000 double stars, and a 10" scope can resolve roughly 38,000 doubles. in fact a simple linear calculation of 130xD(mm) is a good rule of thumb.

i checked the doubles chosen by this method at different apertures, and the results seemed plausible and consistent.

someone in the previous topic mentioned that many doubles are separated by arcminutes rather than arcseconds. this led me to look at the *easiest* doubles identified by this method. here is a sampling of those with bayer/flamsteed designations:

ZET UMA (STF 1744 AC)
92 sig 1,2 TAU (STFA 11)
20,21 NU 1,2 CRB (STFA 29)
8 ALP VUL (STFA 42)
77,78 THE 1,2 TAU (STFA 10)
17 CVN (STFA 24)

anyway, this is one of the advantages of having WDS in spreadsheet format: you can really drill down on any question.

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GlennLeDrew
Postmaster

Reged: 06/18/08

Re: number of doubles per aperture [Re: drollere]
#5221633 - 05/14/12 10:41 AM

Bruce,
I always find such database analyses fascinating. They help to give the big picture, and provide illuminating perspectives. Thanks!

I note the few small 'hiccups', or steps in the line's plot on the graph. How do you interpret them?

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drollere
Pooh-Bah

Reged: 02/02/10

Loc: sebastopol, california
Re: number of doubles per aperture [Re: GlennLeDrew]
#5224142 - 05/15/12 08:36 PM

yes, i noticed those when i validated the data. they arise because the separation measurements in WDS are rounded to the tenth arcsecond, so it's not feasible to compute continuous values without interpolation. there will be a long run of perhaps 100 stars, all with a separation of 0.5, etc., and with very similar or identical secondary magnitudes. this becomes more noticeable higher in the curve because the magnitude of the stars is fainter so there are more of them.

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fred1871
sage

Reged: 03/22/09

Loc: Australia
Re: number of doubles per aperture [Re: GlennLeDrew]
#5224230 - 05/15/12 09:32 PM

bruce,
I can only agree with Glenn that it's a fascinating (and useful) exercise.

The new methodology appears to have given a much more realistic estimate of how many doubles are accessible to a particular size telescope. One thing it does demonstrate is how many more doubles are within reach of mid-size scopes (10-inch for example) than smallish (such as 4-inch).

Well done. It obviously isn't possible to give absolute numbers because of observer differences, to name only only one of several variables.

But it certainly makes obvious that for the amateur observer, there are far more doubles than, say, galaxies available to observe. To see (even as an averted vision hint) 13,000 galaxies will take much more than a 4-inch telescope - 16-inch? 20-inch?

A point to note - you mention that "i find mag. 10 in my 12" is roughly where my foveal vision gives out"... interesting, as Paul Couteau, an authority on doubles, mentions mag 10 as a rough limit for accurate measures (visually) regardless of the aperture of the telescope, and he had extensive experience with apertures up to a 30-inch refractor. So, maybe there are rapidly diminishing returns, beyond 10th magnitude in terms of seeing pairs and not only measuring them.

I observed a lot of doubles, including faint ones, with a C14 some years ago, and found the dimmer pairs, below 10th mag, more difficult than I'd expected. However moderately bright - say mag 8.5 and 9.0 pairs at 0.5" - were cleanly splittable on the steadiest nights at around 600x.

Back on the main topic - thanks for the investigation. A usefully revealing analysis. It shows there's more than a lifetime's doubles within reach of a 10-inch, especially if you go back from time to time to re-observe changing binaries.

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drollere
Pooh-Bah

Reged: 02/02/10

Loc: sebastopol, california
Re: number of doubles per aperture [Re: fred1871]
#5225127 - 05/16/12 01:19 PM

Quote:

Well done. It obviously isn't possible to give absolute numbers because of observer differences, to name only only one of several variables.

yes, i have a good understanding of the psychometric tradition and its methods, and that makes me skeptical of the precision implied by estimates of limiting magnitude or "resolvability" applied to individual observers or instruments.

the *shape* of the curve is diagnostic of the *domain* of visual tests, and since magnitude limits and resolution limits are both roughly log scales, it seems reasonable that the curve is roughly linear across a large range.

Quote:

A point to note - you mention that "i find mag. 10 in my 12" is roughly where my foveal vision gives out"... interesting, as Paul Couteau, an authority on doubles, mentions mag 10 as a rough limit for accurate measures (visually) regardless of the aperture of the telescope

i don't recall seeing this in couteau's book, do you have the page number? i suspect his cutoff may result from the increase in light grasp offset by the increased sensitivity to seeing.

thanks for the kudos. i agree, there are more doubles than one can exhaust in normal observing.

a key point, which is not often emphasized, is that doubles are a self similar observational domain. this is probably most obvious with matched binaries. you can pile on more aperture, but the new pairs that you can resolve are identical to the ones you could see before. in that sense doubles are "aperture free". whatever aperture you have is going to show you pretty much what any other aperture will show you. this is quite unlike the case with nebulae and galaxies, where more aperture changes the view fundamentally.

true, there are astonishing or breathtaking doubles that require a minimum aperture to be enjoyed. but those are also self similar in an astonishing and breathtaking way. the diversity of multiples is constant across magnitudes.

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JimP
Pooh-Bah

Reged: 04/22/03

Loc: South Carolina
Re: number of doubles per aperture [Re: drollere]
#5225231 - 05/16/12 02:08 PM

Well, yes and no. You may be absolutely correct from a technical/mathematical pov but if I want to split Antares or the B component of Gamma Andromeda, or Sirius then I want to split Those particular stars and not another that may superficially look like them. Not trying to start a "debate" just my opinion. I agree that there are plenty of doubles to keep one busy for a lifetime but, for me, if I were going to concentrate on doubles there would be some minimum aperture I would require.

best,

Jim

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blb
Post Laureate

Reged: 11/25/05

Loc: Piedmont NC
Re: number of doubles per aperture [Re: JimP]
#5225283 - 05/16/12 02:43 PM

Quote:

... if I were going to concentrate on doubles there would be some minimum aperture I would require.

I agree and that would be my 4-inch TV102 refractor or one of my 5-inch mak-cass scopes. There are simply more doubles than I will be able to see with that size of telescope. There are actually more than enough doubles (6000+) for even the classic 60mm refractor.

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fred1871
sage

Reged: 03/22/09

Loc: Australia
Re: number of doubles per aperture [Re: drollere]
#5227797 - 05/18/12 05:36 AM

Regarding my comment borrowed from Couteau, and the mag 10 limit for accurate measures etc - he's pretty specific, and it's found on pages 180-181 of the 1981 MIT Press English language edition: it's in a discussion of "the penetrating power of an instrument", and he speaks of useful restrictions:

"limit consideration to pairs containing equal components, each of which is not fainter than magnitude 10. Experience shows that, whatever the aperture, magnitude 10 is a barrier. In a large instrument the images lose their sharpness and break up. Light is lost in the diffraction rings, and the eye does not receive very much more illumination."

On another matter, regarding the comments by JimP and Buddy - I can only agree about wanting a certain *minimum* aperture - and, like Buddy, I'd place it around the 4-inch refractor 5-inch MCT level. Less than that tends to not provide enough light for my taste, and leaves out plenty of particular pairs I'd like to see as doubles (JimP's point). Around the 6-inch to 8-inch size light grasp becomes decent, resolution can go pretty high, and the scope isn't yet too much bothered by seeing. I see it as a sweet spot. But I won't knock back bigger telescopes if they're optically good.

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drollere
Pooh-Bah

Reged: 02/02/10

Loc: sebastopol, california
Re: number of doubles per aperture [Re: JimP]
#5233104 - 05/21/12 07:41 PM

Quote:

Well, yes and no. You may be absolutely correct from a technical/mathematical pov but if I want to split Antares or the B component of Gamma Andromeda, or Sirius then I want to split Those particular stars and not another that may superficially look like them. Not trying to start a "debate" just my opinion. I agree that there are plenty of doubles to keep one busy for a lifetime but, for me, if I were going to concentrate on doubles there would be some minimum aperture I would require.

no debate, as you're not taking my words in the spirit they were offered (as you point out).

i am making a statement similar to "all granitic mountains have essentially the same fractal surface structure", and you're making the statement "K2 is not the same as mt. whitney."

i agree about minimum aperture, and for me it is a ten inch mirror. in many situations, that would also be my maximum aperture.

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drollere
Pooh-Bah

Reged: 02/02/10

Loc: sebastopol, california
Re: number of doubles per aperture [Re: fred1871]
#5233213 - 05/21/12 09:03 PM

Quote:

it's found on pages 180-181 of the 1981 MIT Press edition

thanks, frank! i have that edition but unfortunately my copy was misbound, so that the signatures are in jumbled order. sorting that out and flipping back and forth in the book deterred me from reading that chapter.

Quote:

"Experience shows that, whatever the aperture, magnitude 10 is a barrier. In a large instrument the images lose their sharpness and break up. Light is lost in the diffraction rings, and the eye does not receive very much more illumination."

this is a peculiar statement, an assertion inserted in the middle of mathematical reasoning about binary periods and distances.

the claim that "images in large instruments lose their sharpness and break up" seems to refer to seeing, or perhaps excessive magnification given couteau's habitual reference to large focal ratio refractors. but would observing from outer space answer this objection?

apparently not, because "light is lost in the diffraction rings". that seems odd, since it implies that the star and the diffraction ring are of similar brightness. but if two stars of equal magnitude and 1 rayleigh separation (as couteau requires in this passage) are easily separable at magnitude 6, then why not at magnitude 12? the ratios of brightnesses among all the diffraction components will be the same; each star will be about 40 times brighter than the first diffraction ring of its companion.

the point is that the justification or causal origin of the limit couteau asserts is not made clear, which means it can't be modeled or calculated in relation to actual pairs.

my reasoning was that aperture should deliver matched pairs of any given magnitude at a brightness *visually equivalent* to magnitude 10 stars in a 12" telescope, because that minimum illuminance seems "from experience" adequate to stimulate the foveal vision necessary to separate pairs near the *visual* resolution threshold. opposed to that is the claim that increasing aperture would *not* provide "very much more illumination", which seems illogical.

perhaps the issue is the context of "information time", meaning indirectly that the pairs must be bright enough to be measurable with a filar micrometer in order to determine their orbits: in other words, you can use larger apertures to resolve the stars visually but then they are too fuzzy to measure accurately. or maybe something is muddled in the translation.

Quote:

I can only agree about wanting a certain *minimum* aperture - and, like Buddy, I'd place it around the 4-inch refractor 5-inch MCT level. Less than that tends to not provide enough light for my taste, and leaves out plenty of particular pairs I'd like to see as doubles (JimP's point).

perhaps, but isn't this really a discussion about *what* one is looking to achieve or experience with double star astronomy, displaced into a discussion of apertures?

i would choose a larger aperture, primarily so that i can do planetary observing as well, even in poor seeing; but to enjoy star color and rich fields a 3" counterweight mounted binocular and a recliner chair would be preferable for me.

"particular pairs" implies a checklist or challenge list criterion, which is historically determined ... if 10" refractors were common and inexpensive since the 18th century, today's portfolio of "challenge" pairs would be different ...

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GlennLeDrew
Postmaster

Reged: 06/18/08

Re: number of doubles per aperture [Re: drollere]
#5233626 - 05/22/12 06:30 AM

If atmospheric seeing was not a limitation, in principle there would be no barrier to employable aperture and the concomitant faintness of pairs observable. Compared to a 10", a 100" would get you 5 mag. fainter and 10X smaller separations resolvable. And this could (should?) nonetheless be achievable in moments of sufficiently good seeing.

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fred1871
sage

Reged: 03/22/09

Loc: Australia
Re: number of doubles per aperture [Re: drollere]
#5233683 - 05/22/12 08:00 AM

bruce, I read Couteau as referring to the effect of seeing as a limitation of large apertures - bigger telescopes are more greatly affected than smaller ones. They're generally further from the potential they'd have in (never-happening) perfect seeing - as in space.

An example of the failure to achieve potential is obvious in the case of the discoveries of SW Burnham - I think it was Thomas Lewis's study long ago that showed how Burnham achieved better than might be expected with smaller telescopes - such as his 6-inch refractor - and relatively decreasing improvements with bigger telescopes, such as the 18.5-inch Dearborn and 36-inch Lick telescopes.

In other words, the gains of aperture are not linear. Bigger apertures give more illumination - but it's not always useful illumination. I think you ran into that issue, judging from your earlier comments.

Regarding my other comments - I was merely making the point that the sheer number of doubles is only one of a number of factors in double star observing - yes, there are plenty, a lifetime's worth, even with a 3-inch refractor. I've owned a 3-inch refractor, and enjoyed it, but I don't see it as something I'd want to be limited to.

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drollere
Pooh-Bah

Reged: 02/02/10

Loc: sebastopol, california
Re: number of doubles per aperture [Re: fred1871]
#5235976 - 05/23/12 03:59 PM

yes, it's plausible that seeing is the sole complicating factor. i've always found seeing expressed as a resolution or lucky imaging limit (e.g., fried's r naught) ... this is the first time i have seen it presented as a fixed magnitude ceiling (although the effects of seeing on telescope limiting magnitude are well known).

the problem statement implies that increases in magnitude with aperture vs. increases in seeing distortion with aperture are two curves with different slopes that normally cross at around the mag. 10 point.

but to glenn's point, the magnitude curve would be fixed while the seeing curve would shift, depending on conditions, so that mag. 10 would not be the fixed limit that couteau implies.

i interpret his discussion as shaped by his intent to calculate the number of binaries whose orbits are solvable because they are measurable, given the mechanical attributes of a filar micrometer, the minimum possible visual width of the filaments, etc., etc. so the purely "lucky visual" aspects of the problem, as glenn implies, make the problem more complicated.

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