historical resolution "limits"
Posted 07 December 2013 - 07:12 PM
after a little inquiry i limited my selection to the catalogs by w.f. struve (STF), o. struve (STT), s. burnham (BU) and j. herschel (HJ), primarily because i could get reasonably reliable information on the telescope apertures these gentlemen were using within specific time periods.
i deleted all records that are not listed as binary systems in WDS, to eliminate detection made easier by multiple targets. i also deleted all observations with historical dates outside the working era of each astronomer (e.g., many "first" epochs are in the 20th century), or in periods where the astronomer was possibly using a mixture or an uncertain selection of instruments. of course a larger or more disciplined selection of catalogs would be possible with more research.
i calculated the lambda/D resolution limit of the aperture (R = 113.4/D), then divided this into the separation in arcseconds, and plotted this metric (rho/R) against the magnitude difference (MD) between the primary and secondary components -- a "treanor diagram". i inserted as a reference my easy to calculate "rule of thumb" boundary (RoT = R, if MD<1; otherwise RoT = R*MD).
the several observations that appear in the area rho/R < 1 and MD<1 are not obvious errors, since the same exceptions appear in treanor's original diagram. this is interesting to me because the separation criterion here is actual detection -- the astronomers were not attempting to resolve a system they already knew to be binary, but detecting a binary through cursory scrutiny. add to that, the lambda/D criterion i prefer is 98% of the dawes criterion and 82% of the rayleigh criterion.
the most acute discoveries were by burnham (curse you, burnham!) and o.struve -- STT is in fact a weird amalgam of subarcsecond, nearly matched pairs mixed in with wide and extremely unequal pairs, many of them likely optical. STF covers a more "predictable" spread of values. HJ does not even come up to my "rule of thumb", however he was using a 470mm metal aperture, so seeing and cooldown may have been issues. (i excluded observations in the period where he was also using the "7 foot" telescope for micrometer observations.)
in another thread i conjectured that calculating difficulty as the ratio between the magnitude difference and the magnitude range between the primary star and the magnitude limit of the aperture, or MD/(ML-M1), might be more useful:
this does not seem radically to change the picture of relative performance, although as i suggested in my conjecture the information about the system brightness is now evident. (because the vertical scale is a proportion on various M1 and LM, the RoT line is undefined.)
burnham is clearly reaching into fainter systems, because his MD values are a considerable proportion of the difference between the primary star brightness and the limit magnitude;
STF and STT now have very similar limits along the bottom of the distributions, because they were using the same instruments (dorpat and pulkovo);
HJ seems to be working with brighter systems partly because his large aperture has increased the limit magnitude. the average primary magnitude of burnham systems within the plot area is 7.8, for herschel it is 8.14; but their limit magnitudes are 13.5 and 16.0. so there may be merit in using an MD/(ML-M1) metric, at least to display the extraordinary performance that makes burnham perhaps the most visually acute of all double star astronomers. (using a clark refractor couldn't hurt, either.)
if anyone wants to play around with/refine the data, i've uploaded the excel spreadsheet here:
Posted 08 December 2013 - 03:06 AM
Looks like a useful approach to establishing a guide for visual limits on pairs of unequal magnitudes would be to use the best of the Burnham observations -- there's data there covering from two to four and a half differences in magnitude. Very few people will exceed his observations, which means a more challenging approach to the topic.
Posted 08 December 2013 - 04:17 AM
It'd be interesting to factor in individual local issues to all observers like seeing conditions per region, optical through-put of the unique instruments (light scatter). Some things we can't know with certainty but it'd be intersting to be able to factor in these effects. Such is histories veil though.
Burnhams results begs for a Super Thumb! Again though it'd be interesting to know his instrument particulars and environmentals. Its splitting hairs perhaps but I wonder about it.
This great work Bruce. Both.
Posted 08 December 2013 - 04:17 AM
Posted 08 December 2013 - 02:45 PM
my plot has a different purpose. the criterion here is not whether a known binary system can be resolved with skillful effort, but whether an unknown system can be detected in a random field or unfamiliar selection of stars.
there is a great deal about the observing methods and conditions that i don't know, and about the equipment -- notably, transmission (as mentioned above) or the diffraction magnification (exit pupil) utilized. i find remarkable that burnham, to my knowledge, was known for potting around in random star fields, looking for shells on the beach as it were, which implies a leisurely attitude to the work. in contrast, wilhelm struve was methodically examining every star visible in his finder scope as a fundamental survey project and at the rate of 6 or 8 systems *per minute*, which suggests a more arduous and disciplined attitude. herschel had the distinction (?) of being the only double star astronomer to survey the entire sky (probably why, in all photos of him, he looks so damn tired), and his close systems with the 20 foot telescope are mostly within the easily noticed ("matching") 1 magnitude difference.
the fact that burnham's discoveries are located so far in the lower left of the plot either means that everything else was already discovered, or that he was intentionally looking for a different kind of visual cue, outside the bandwidth of previous astronomers.
there is clear indication that detection was occurring at separations below the resolution limit of the telescope. i use the lambda/D criterion, equivalent to the dawes criterion which treanor used in his 1946 diagram, and his shows similar performance. to me this suggests that o.struve and burnham were *searching* at exit pupils below 1.0, possibly below 0.5 where a "rod" or "egg" would be discernable.
these data were rather hastily edited and may not (especially in burnham's case) exclude discoveries made with larger instruments. (those shown for burnham are all before 1874, when i believe he first had working access to larger instruments than his 150mm clark.)
it's the distributions rather than individual data points that should be relied on. these distributions do illustrate some fundamental differences in the catalogs and, surprisingly to me, reinforce the impression that a simple "rule of thumb" is useful as a detection benchmark.
Posted 08 December 2013 - 03:42 PM
Bruce, interesting points, especially if the RoT quest would be scientific research requiring then very precise definitions of the test environment and observing procedures.
well, wilfried, a perceptual psychologist can explain to you why the conception of "conclusive results" requires stimulus, presentation and motivational control that volunteer data and naturalistic conditions cannot provide. the issue in perceptual research is never whether you can split two overlapping distributions but, given the difference in means and the variances around the means, whether the split is informative about perception or predictive of task performance in a new sample of observers. i eagerly await reports on those two points from any research effort, because those are the only criteria that matter...
This is in my opinion here not the case as the question which aperture is required to resolve a specific binary is only of interest for amateurs - this includes then all inadequacies given when amateurs are acting.
Then the so far missed step of testing the predictive power of published RoT approaches by the creators themselves ... Even if this would be scientific work such a step is not necessarily part of developing a theory if the published content is precise enough to allow falsification by experiments (empirical or logical) of others means fellow scientists.
To make a field study if a RoT is of good use is an effort usually not possible for amateur astronomers. I try to get as much feedback as possible in this forum for my own efforts but this is certainly far from a systematic field study.
But again - we are no scientists here. And the very topic has a fundamental weakness - even if we could consider all parameters involved it is impossible to develop a RoT predicting all situations perfectly as long as the base data on binaries regarding separation and magnitudes is not free of errors.
So we have to try to get a grip on a non perfect situation and do our best.
Posted 08 December 2013 - 09:16 PM
I've successfully resolved stars I might not have even recognized to be binary if not for the prior knowledge they indeed are. It would have taken a lot of patients, magnification, and skill along with a sound search method to determine some of these pairs existed at all. They might have been missed on any given night, but they could have been resolved.
So, a sample of the above RoT shows 42 Ori to require a separation of about 2.2" arc ( 0.76 * 2.9 = 2.2" arc for 150mm aperture) to be recognized as a double if one did not know beforehand. At this separation 42 Ori would definitely be much easier than it is currently (~1.2" arc) and a more pessimistic RoT would likely succeed. However, 42 Ori can be resolved in a 150mm aperture if one knows what to look for in favorable conditions. (Interestingly, it's a fairly continuous bright knot on the ring in 8/10 seeing and a speck can be seen with the use of a neutral density filter.)
I like where the above RoT is going, plug in variables for aperture and magnitude difference to determine the separation requires. Simple. The simpler the better. And you're correct, IMO, it's not just the equipment that resolves stars, it's in large part the observer that resolves them. So, a more optimistic or accurate RoT requires so many variables and does not include differences between observers. It becomes accurate in terms of equipment and conditions but it probably cannot account for variations between observers and requires some sort of empirical probability prediction.
It's a fascinating field of discovery. I just wish there were some easy one's left for us to discover.
Posted 09 December 2013 - 12:13 AM
A question - why did you choose "R = 113.4/D" ?? - that's slightly closer than the Dawes Limit, which is 4.56*25.4 = 115.8.
Burnham is of course a good choice for this study because of his ability to push limits. John Herschel is not. In his early double star period, JH was generally using small refractors (3.7-inch and 5-inch) and mostly recording wide pairs. Later, in his surveys of the Northern and Southern skies, he used the 20-foot (~18.5-inch) speculum metal reflector. His southern "sweeps" of discovery were made with an eyepiece giving 180x, so a field of perhaps 15'. This approach was borrowed from his Northern sweeps with that telescope. Doubles were found along with nebulae and clusters and recorded; some of them had rough measures made; more often if he measured doubles it was on re-visiting with an equatorial refractor and micrometer. The result is not very many close (under 1 second) pairs, and not many of high inequality if also close.
Regarding previous attempts at rules, RoTs, and limit possibilities - Lewis simply averages averages for the most part; Treanor compares Lewis's data with diffraction theory in terms of Rayleigh; Arguelles gets all his hard cases unhelpfully into one tiny corner; Lord tries for everything, but there are problems; Haas in her book simply analyses observing notes from Revue des Constellations; and Napier-Munn has a more scientific and analytic approach as befits someone who is a university professor accustomed to doing data analysis. So we have quite a variety of approaches. Treanor needs care in reading - he uses the Dawes Limit, but is specifically relating data points to the Rayleigh Criterion. Not take most of 'em seriously? - well, depends on what you're looking for. I still find Lewis and Treanor useful. And Napier-Munn's graphed results for 35cm of interest.
Detection below the resolution limit of the telescope used - yes, pretty common with Burnham; can occur with others. But with Burnham more often because he re-visits a lot of "I think it's double" from observing with the 6-inch, to confirm and measure with the 18.5-inch or 36-inch. Most observers didn't have that approach, of finding "possibles" with a small telescope then checking them with a much bigger one.
One observer who might be useful here is Couteau, who comments specifically on using the form of the diffraction image, below Dawes, to determine separations, regarding this as better than the micrometer in such cases. He might be found to have discovered doubles below the Dawes Limit for the 15-inch and 20-inch scopes he mostly used for discovery surveys.
I'd suggest a similar possibility with Aitken, who remarks on finding doubles with the Lick 12-inch, later measured with the 36-inch, if only one could find out which pairs these were.
So that's part of why Burnham pushes the boundaries better than most - small telescope finds a "suspect", bigger one later shows it double and allows measuring. And, yes, because Burnham comes after the Struve surveys, a lot of the easy stuff is already discovered, and measured, within the limits of the telescopes used. And, as you note, the sheer speed of work of Wilhelm (FGW) Struve in particular is striking, but could lead to missing some doubles.
Agree with your summary point, of course - the distributions are what are needed - for which lots of data points will have to be obtained. Wilfried has pointed out the usefulness of diaphragms to obtain limit observations more closely and with good frequency - that way more data points are obtainable. The limitation is that diaphragms are best with refractors (no CO); similarly, the use of off-axis masks with large reflectors. Otherwise, you're dealing with the extra variable of CO along with aperture.
A further factor is that unobstructed apertures will therefore be relatively small - refractors up to around 18cm (7-inch); or off-axis with larger reflectors, up to about 18-20cm unobstructed. Also the bigger reflectors that have small-CO ratios, and bigger apertures would be useful - David Gray's 16-inch DK is a fine example of a scope that can compare with the mid-size refractors of 15-18-inch. But not many of us have the use of instruments like that.
So we can readily get data for telescopes up to about 7-inches; but extending that data to cover 8-inch to 14-inch is going to be most often restricted to scopes of larger CO (SCT), or dealing with the vagaries of Newtonians (I've had a few!), most of them not long f-ratio versions like Pete's.
Wilfried's experiments with CO of varying size on his refractor suggests a promising approach, that could lead to another version of the RoT, for large-ish obstructions - the SCT-RoT.
To extend the RoT in whatever form is I think going to need telescopes in the of mid-size for amateur use, say 8-14-inch.
Side note - Sissy Haas's letter to Sky & Telescope a little while back suggests she's taking a particular interest in smaller telescope results for her project. Small scope results are not directly applicable to larger telescopes - even the redoubtable Burnham does better, relative to aperture, with 6- and 9.4-inch scopes, than with 18.5- and 36-inch. Common experience suggests a similar divide between telescopes around 3-4-inches compared to 8-10-inches.
Posted 09 December 2013 - 09:50 AM
Posted 09 December 2013 - 12:27 PM
(the small pluses follow treanor in calculating the magnitude ratio between the airy rings and the peak airy disk, and locating the values at the rez ratio position of the interspaces. for the first value, located at the rayleigh interspace, i've used 0.5 or FWHM as the "ring" brightness.)
what is starting to become plain is an area of complexity in the zone between 1 to 2 times the resolution ratio (rho/R) and magnitude differences down to about 2.5. this is almost exclusively the domain of otto, sherburne and paul. if rez ratio < 1 is the limitation of optics, and the "RoT" mag diff < rez ratio is a limitation of vision, then this area seems to be the transition zone of observer skill and/or acuity.
the many subresolution values for mueller are rounding errors: these are systems with a listed separation of 0.2" in an instrument with a lambda/D of 0.23. we can't tell if the actual separations at discovery were closer to 0.24 or 0.15. perhaps the comprehensive WDS has better values.
for amusement, here is the diagram extended out to 100 times the resolution ratio rho/R. the banding in herschel's discoveries occurs because he was estimating separation visually in 5 arcsecond intervals. jonckheere is clearly working a middle range of mostly faint systems between 5 and 20 times his resolution limit; wider systems would be more likely optical.
for each observer, the year range of the data, the aperture (mm), and average magnitude of the primary star for systems in the 0-10 plot:
w. struve (1829-37) 240 8.45
herschel (1831-37) 470 8.14
o. struve (1827-32) 240; (1843-54) 380 7.57
jonckheere (1906-11) 325 9.73
burnham (1870-74) 152 7.73
innes (1896-1902) 450; (1925-27) 670 8.79
mueller (1969-72) 500 9.59
there's obvious rounding in the apertures and this will affect the resolution limit, but i think not as much as the rounding of separation values in WDS.
i used my "short" WDS as the origin file, which is truncated at a primary magnitude of 10.5. i don't see any edging in the plot that would imply censoring, but i may import the data from whole WDS to make sure.
i mentioned in another topic that my "airy disk" threshold is around 60% of my telescope limit magnitude, and my "averted vision" threshold is about 75% of the LM. here's the plot of these ratios for the secondary magnitudes in the file, against the calculated LM of the apertures:
despite the scatter due to errors in aperture spec, LM calculation, and magnitude measurement, it's clear that a mean of about 60% obtains below about ~4 rez ratio, but by ~30 rez ratio the minimum has declined to around 75%. not only in resolution but also in light grasp, these guys were obviously working at the limits of their capabilities and instruments.
Posted 09 December 2013 - 01:27 PM
to accent the positive i'll repeat what i feel my "rule of thumb" is good for. it is only good as a benchmark for what should be obvious. since its value is either R or deltaM*R, it's easy to calculate mentally. it's the first answer i look for if i can't resolve a close binary. and all it can say to me is, "there is probably something wrong here." if a binary is inside the RoT limit (faint, close), then i conclude i have to man up and get clever. if a binary is outside the RoT, then i check for dew, lens cap, and the possibility that i'm looking at the wrong star or have bad data to rely on (as in norme's discussion of 42 ORI, and the "bad data" of a 2.9 separation). in my observing routines that's all an RoT is good for, and it doesn't need two decimal places to do the job.
i always appreciate your scholarship, fred. my preference for lambda/D is first that it is effectively the width between diffraction rings and therefore truly the smallest diffraction limited interval in an image (it's also the FWHM radius of the airy disk, k = 1.03, another common unit in the structure of the artifact); second, that all "resolution limits" are just lambda/D multiplied by some factor "k" (e.g., dawes k = 1.02, rayleigh k = 1.22, etc., etc.), which only obscures the optics. there is a fuller discussion HERE. note for example that rayleigh's own estimate, once he actually did some research on the problem, was k = 1.09.
i explicitly chose herschel objects published in the interval when his catalogs were titled "...discovered with the 20 foot telescope." as i said, i limited burnham to the period in which i believe he was only using his clark. your larger point that there is uncertainty in linking a system to an aperture by a date is entirely correct.
i regretted omitting couteau but there were several scopes he was using in paris and nice, and of course aitken at lick. perhaps you can suggest reliably specific time intervals and apertures?
interestingly, napier-munn looked at CO as a "predictive" variable and found no use in it. personally i take that as a demonstration that the effect size is
CO < observer skill + effort
and skill in these matters is mostly patience and active looking. and as achievement research has shown, "the probability of successful performance is a function of task difficulty, skill, effort, and luck." (i can't tell if seeing is an aspect of difficulty or of luck.)
i have no use for sissy's "rule" endeavor or for her paint store color descriptions, but i find her immensely admirable for several reasons -- her passion, her humility, her enthusiasm to spread the gospel, her lapidary drawings, and her beautiful careful work in making a book.
Posted 09 December 2013 - 08:46 PM
Regarding Couteau, when I've looked at some of his original published papers, I've found he identifies which telescope was used. Yes, the papers are in French, but the tables of measures are clear without knowing the language, and his introductory remarks are readable with only a basic knowledge of the language.
So, looking at one such paper - published in Astr.Astrophys.Suppl., 1, 105-114 (1970), he gives measures for 100 new doubles, made with the 50cm refractor ('lunette') at Nice Observatory. As he says in the introductory notes, these results are from examining stars in the AGK2, starting in 1967, with the then-new 50cm (near enough, 20-inch). He gives basic statistics
on the number of pairs relative to stars examined, and notes 5000 stars per year are examined. There's also a program for the BD stars with the 76cm refractor, but in each case he indicates the separate Declination region examined.
Looking at the pages of results, stars are mostly identified by BD number, but are in the Declination zone, as expected, for the 50cm, not the 76cm. As would be expected, many of these are too wide to be of interest as "limit" observations. However, some are close - COU 256 for example has a measure of 0.27", mags 9.5/9.7, over 3 nights. COU 289, measure 0.17", mags estimated 9.6/9.6. This latter is closer than Dawes - DL for 50cm is 0.23". Pretty good for a not-bright double. Likewise COU 295 - 0.16", m 9.3/9.3, and COU 297 0.18", m 9.6/9.6.
In this paper, there are not as many of the significantly uneven pairs that are close. But there are some. COU 269 is listed at 0.61", mags 9.7/12.5. Not one for Wilfried's 140mm; and I'll not attempt it with my 235mm SCT either. And COU 296 is for those who like faint combined with large delta-m - mags 9.5 and 14.2 at 1.24". Then there's COU 304 - 0.50", mags 9.3/12.0.
Going through Couteau's publications would not be a quick job, but it might yield a reasonable number of pairs useful for evaluating resolvability of dim pairs, and some both dim and notably unequal. There will be fewer of the bright uneven pairs because Couteau is recent. Even so, he has discovered some that are bright, unequal and close - Theta CrB is one; another is Tau Ari, which is in this particular paper (COU 259).
Regarding Aitken, I don't know how to distinguish his 12-inch from his 36-inch pairs, as he doesn't say (Couteau notes this too); but as he said that some pairs down to 0.2" were found with the 12-inch, presumably anything closer than that is 36-inch.
Although Napier-Munn found CO not useful as a "predictive" variable, Wilfried's recent experiments with CO on his refractor suggest that it does have an effect - in particular, that CO above 0.3 can make invisible some companions that were seen with no CO and with small CO, such as 0.15, with the same aperture.
Burnham - when he produced the catalog of his own discoveries, in discussing each double he remarks on which telescope was used in discovering it; and sometimes indicates that it was first suspected with the 6-inch, confirmed later with a bigger scope. That might help evaluating where to plot data points - a suspect isn't the same as a definite elongation, or a split. And there were a few false positives among these suspects.
Posted 10 December 2013 - 12:48 AM
Quote: innes (1896-1902) 450; (1925-27) 670 8.79
Innes started discovering doubles in Australia, in 1894-5, with a 6-inch refractor and a 16-inch reflector. He then moved to the Cape Observatory during 1896, where he had use of the old 7-inch refractor. In 1898 the McClean telescopes, a 24-inch photographic refractor and 18-inch visual refractor on the same mounting, came into use at the Cape; and as the 24-inch had faults, it was returned to Grubb for fixing, which initially allowed Innes a lot of time with the 18-inch for double stars. His period on the 18-inch is restricted from early 1901 when the 24-inch returns; his last use of it appears to be March 1903.
Innes left the Cape Observatory to become Director of a new observatory at Johannesburg, variously named over time; he gets a 9-inch refractor for it, which allows him to recommence observing (discovering) doubles from 1907, until the big refractor - 26.5-inch - becomes available in 1925.
If you wish to bypass going through all of Innes's early papers, it's safe to assume discoveries from 1907-1924 are with the 9-inch.
Another Southern observer, who used only one telescope, is RA Rossiter at the Lamont-Hussey Observatory at Bloemfontein, South Africa. He used the (new) 27-inch refractor there from 1928 until the early 1950s. In 1937 he published a paper (written 1933) in the Memoirs of the Royal Astronomical Society, vol 65, p27 et seq. There he provides a summary of the first 2350 doubles he discovered (his eventual total was 5,534, the record for anyone, according to Rob. Argyle).
Rossiter interestingly gives parameters for the doubles listed - he adopted an approach similar to Aitken, requiring doubles to be within certain separation limits to be recorded. So, in general, he required the limit for mag 9.0 to be within 5.0"; for mag 10.0, within 3.2"; etc. The formula was log d = 2.5 - 0.2m (d in arcseconds, m is combined visual mag). He relaxed this a little to limits obtained by using 2.625 instead of 2.5 to cover borderline cases; this allows 6.7" at mag 9.0, 4.2" at mag 10.0, etc. How many doubles were noticed and not recorded? - because "too wide" - we can only speculate.
There are some notably difficult pairs in Rossiter's list - RST 625 BC is mags 13.5/13.5 at 1.0"; RST 631 is mags 10.1/12.4 at 0.5"; RST 634 is mags 9.1/13.6 at 0.9"; RST 650 is mags 8.6/14.3 at 1.2"; and for a faint'n'close even pair, RST 824 mags 11.3/11.3 at 0.3". Plenty more where those came from.
Rossiter comes to the Southern Hemisphere after various surveys that have found a lot of the easier doubles; unsurprising then that so many of his doubles are close and faint. However the examples I've listed above are a small selection of some of the more extreme pairs, from a quick look through this initial listing.
Posted 10 December 2013 - 11:58 AM
Posted 10 December 2013 - 12:23 PM
Posted 10 December 2013 - 01:01 PM
Wow! To be so knowledgeable and well writen (except for your lack of the proper use of capital letters), only to end it on such a sour note. I would be ashamed. Who are you? A retired profesional astronomer? or just another amateur?
i have no use for sissy's "rule" endeavor or for her paint store color descriptions,...
Posted 11 December 2013 - 04:29 PM
i'd be ashamed too, buddy. however i ended on this note:Wow! To be so knowledgeable and well writen (except for your lack of the proper use of capital letters), only to end it on such a sour note. I would be ashamed.
i have no use for sissy's "rule" endeavor or for her paint store color descriptions,...
... but i find her immensely admirable for several reasons -- her passion, her humility, her enthusiasm to spread the gospel, her lapidary drawings, and her beautiful careful work in making a book.
Posted 11 December 2013 - 04:57 PM
Bruce, there's a problem for you with one of your listed observers - RTA Innes. I'm currently working on a detailed study of him, and your listing of telescopes doesn't fit.
i'm amused to learn that precisely the period i excluded is the one you know is trustworthy. (inclusion was based on comments in couteau 1979, p. 14-24 passim.) thanks for the help, i'll emend the data posted online per your guidance.
Rossiter interestingly gives parameters for the doubles listed - he adopted an approach similar to Aitken, requiring doubles to be within certain separation limits to be recorded. So, in general, he required the limit for mag 9.0 to be within 5.0"; for mag 10.0, within 3.2"; etc. The formula was log d = 2.5 - 0.2m (d in arcseconds, m is combined visual mag).
i will include a selection of rossiter (1928-33, 686mm) per your information. btw, can you provide any guidance with espin? i'm interested to include "faint star" catalogs compiled at smaller apertures.
btw, the calculation you quote is the "aitken rule", usually cited from aitken's "binary stars" (p.35) but, so far as i can find, actually proposed by E.C. pickering as a "reply" to aitken's 1911 mutterings about too much data:
it's not generally appreciated, but the rule actually projects a constant orbital radius (defined by the constant), using the primary star magnitude as the basic metric for both observational distance and orbital radius (hence the log separation, which gets us back into an inverse square metric on brightness). unfortunately, the orbit becomes larger as the star gets brighter, out of proportion to the increase in stellar mass with brightness. (try some examples and you'll see how.)
the rule is sometimes disparaged as a bad astrophysical theory, when it's clear from aitken's use of it that he was trying to throw out pairs that were likely optical, or at least so wide as to be fixed and therefore of no scientific interest.
Posted 11 December 2013 - 05:07 PM
Bruce, appreciate your effort to keep a RoT as simple as possible snd I found it an excellent starting point for my own research in this area but I had to move on as it failed in the field test I made with a meanwhile large number of limit observations.
a yeoman once approached the king of england and on bended knee did inquire:
Sire, ye have ordained and declared that a foot shall be the length of twelve thumbs in all the land! But I am mightily perplexed, sire, for I do find my foot is but a ten of thumbs, aye, even when ill washed and shod of wood!
the king, who had eaten well and felt himself in good humor at this impudence, raised his ermine trimmed robe and extended his silk hosed and velvet slippered appendage, and declared to the court:
then, with a gracious sweep of his ermine trimmed sleeve, he upraised his thumb as 'twere a rare carbuncle of cathay, and said with most cheerful mien:
Posted 11 December 2013 - 07:49 PM
yule cheer to all. see you next year.
Posted 11 December 2013 - 08:39 PM
His usefulness will be, as you say, in detecting faint companion stars - Espin avoided working on the close pairs for which in his day a refractor was thought better suited.
The obituary note on him in the Monthly Notices of the RAS comments: ..."his program consisted in the examination of B.D. stars, and every year he published in the Monthly Notices lists of the pairs he found ... His last list ... contained the reference number 2575 - the number of his own discoveries". Espin's work before 1900 was mostly on red stars and spectroscopy, not doubles. So, if you use the discoverers listing from the WDS site, it will give you under Espin a list of his papers and when and where published; and the ADSABS site will enable reading these, one by one. After that, cross-correlate for modern photometry (Tycho-2, etc). A lot of work there.
Given the size of Espin's telescopes used for doubles, 17- and 24-inch apertures, I'm not sure how useful he will be for "..."faint star" catalogs compiled at smaller apertures".
His reflectors, as was usual by the late 19th century (he acquired the 17-inch in 1885) were silver-on-glass mirror scopes, and in Espin's case both were by the notable maker George Calver.
A side note on history - in reading older materials which mention reflectors, it's necessary not to be misled by the use of the term "spec" in referring to them - the term does not mean they're old-style speculum metal mirrors. For many decades after silver-on-glass had taken over "spec" continued to be used as an easy abbreviation for a reflector; likewise the term "OG" was often used in that period for a refractor. Some observers continued these usages well into the 20th century.
Re Couteau on Innes - there are a few detail errors in Couteau's notes on the history of observers. Not many, but it's well to check. Innes was the most significant I've found.
And I overlooked a comment Couteau made about Aitken's work - he remarked that the Lick 12-inch was used for surveying Declinations +60 to the North Pole. The identification problem applies to the rest of the survey work, where both telescopes, 12-inch and 36-inch, were used, without noting which.
As Couteau says, this had the consequence that some areas of the Northern sky were surveyed only with the 12-inch; so the later work at Nice observatory, much of it with the 20-inch, found many new pairs because of the larger telescope as well as finding pairs that might in the earlier period have been too close to detect, and had widened later.
Posted 13 December 2013 - 10:44 AM
I have done two calculations.
One with your original formula giving a correlation coefficient of 0.6122 and a standard deviation of 11.39 arcseconds indicating a huge average error and a very limited predictive power. A quick look in single values shows a rather good behavior for small delta_m's (no surprise because then we are near or equivalent to Dawes) and cases with similar values for separation and delta_m. Wide separations, large delta_m and faint components are the great weaknesses of this formula.
For direct comparison with my proposed aperture approach I did then a second calculation with a transformation of your formula giving pA=113,4*delta_m/sep with delta_m=1 for difference in magnitude < 1.
Result is then a correlation coefficient of 0.71 with a standard deviation of 68.12mm means somewhat better correlation but still questionable predicitve power and again with huge errors.
In both cases I assume that even this rather modest correlation between RoT and observations is based mostly on the Dawes equivalence for a good part of the data set and without this the result would be a complete desaster.
You may still insist that your formula is of use for you but you might now understand why it does not work for me.
Posted 18 February 2014 - 08:34 PM
a footnote ... or endnote ... since you are a scholar, fred, i wanted to alert you to a source for the resolution/mag.diff chart (AKA, "treanor plot") that is even earlier than treanor:
Bruce, nice to see someone taking up a project I commented on a year or two ago, then didn't attempt - in part, because it meant re-doing Lewis, this time with modern photometry to improve the results for unequal pairs.
G.P. Kuiper, "Problems of Double Star Astronomy," part 1, p.28
he uses a log/log format, which is an interesting interpretation of separation and detectability (it's true, differences in separation mean less at wider separations). and his interest is sampling bias rather than detection limits per se.