Before I forget, earlier in this or another thread you asked me about the definition of "angular diameter" and I missed it at the time, only seeing it yesterday reviewing the thread. If you are not yet clear on that let me know - especially since it is critical to this discussion.
>> I may not have done it right I felt I could resolve the pair standing a maximum of 120 inches from the monitor at 50%.
I can only tell you the way I look at it, and I am not necessarily right.
At 100" I can drive a truck between the two stars. There is no argument. As I slowly walk back to about 135" I increasingly enter in an increasingly hot argument with myself, and fortunately there is no one else here at the moment to hear it .
At some point in the 110-135 area it is unclear to me if they are "touching" or cleanly split and the problem is that that tiny sliver of black is beyond the basic resolution of the two stars (a smaller subject in itself). Also, it is cleaner when I first look, and then my eyes start to water a bit. That is a big problem I have with observing sessions and it may be exacerbated by wearing contact lenses.
Your 120 number is equivalent to 157 on my 90 DPI monitor. I could not call that cleanly split no matter how hard I tried. You got me . The most I could possibly argue is about 140 or so.
In the real world, this is a moving subject . Even on the stillest of nights there is some jiggling and dancing around. And for that reason my instinct is to be on the conservative side, even with this very stationary subject.
Of course, everyone looks at this differently and that is why I previously said two people with identical visual acuity could come to honest but different conclusions. And for that reason I do take "out of the norm" observing reports with some grains of salt.
Interestingly, a real astronomical binary pair so close that they are not fully split at any magnification but "notched" or just elongated typically look like live (bacterial) Bacillus. Just like the videos where they wiggle around while going about their business. I think it is quite strange .
>> - can we broaden this same topic to include monitors or start a separate thread?
Pythagoras agrees with Apple's spec . Your 109 DPI number should be correct.
I just physically measure the horizontal width of my monitors and divide into the reported horizontal pixel dimension. I assume you did the same. Does that cover it?
Here is a revised formula to more directly account for differing monitor resolution:
>> - when I download the full-sized jpegs from the Nikon D800E samples .... I am gobsmacked...
I guess Nikon marketing did their job . It would be nice if they shot standard scenes that crossed camera bodies. I don't know how to answer your question. If I had the camera in hand I would shoot the same scene with it and my other favored bodies to compare.
I added two more objects to the image and re-posted it below. That image also includes my revised resolution computation using DPI.
The lowest object now is a single star. When doing real world binary observations it is not uncommon to have other stars in the field although they are rarely the same magnitude (brightness). But it does help to get a sanity check.
The middle object is more interesting. You can see, by zooming in, that the two airy disks are merged. Binary observers call this "notched" or "peanut-ed".
The middle object was my best attempt (limited to a 10 pixel star size) at illustrating a binary pair at the Rayleigh or Dawes limit of resolution. Both definitions are close to each other although Dawes number is a little tighter (by about 15%) and represents a "lower MTF" so to speak.
Dawes did the work first, sometime in the mid-19th century. His work was purely empirical, testing himself and other observers with various telescopes.
Rayleigh developed the physics and crunched the numbers, maybe 20-30 years later (not sure of the publication dates). The fact that they agree so closely, in retrospect, illustrates that reality reflects the theory, and also that around the 1850's or so Dawes had access to optics that were every bit as good as anything made today.
The Rayleigh definition is "Two point sources are regarded as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other."
Translated, the Rayleigh limit is simply the radius (half the diameter) of the entire central airy disk.
I personally have trouble applying the Rayleigh formula to double star observing for two reasons. First, while the extent of the Airy disk in theoretical and computer generated models is quite fuzzy, my perception, the real world, with a small scope, is that it is quite distinct. I suspect that what I perceive to be the full extent of the Airy disk is smaller than what we might think it is looking at computer generated plots (See Wikipedia.
The sharp delineation I perceive is reflected in my E Lyra model posted here. Right, wrong or indifferent .
Second, the recent periastron (closest approach) of the famous Gamma Virginis pair took them well outside the reach of almost all amateur scopes (way too tight to resolve) and then slowly worked their way out to separations resolved by increasingly smaller scopes.
I cleanly split G Virginis, at about 1.5" separation, with my smaller Questar, which has a theoretical notched Rayleigh limit of 1.3" and that should correlate to a minimum of 2.6". For that matter, I should not be able to cleanly split E Lyra, yet I can and I do it year in and year out. For that reason things can get confused when I talk about theory and practice at the same time .
I also watched a lot of Bacillus, going about their business, in the nasty March/April air, over many nights, while trying to do that split . As I recall, I used magnifications up to 600x or so... nasty stuff.
My confusion may not be applicable to photographic resolution in general but it may help to explain why some things I've said here could possibly be nit-picked by a good editor .
I do think it is extremely helpful to understand diffraction from the perspective of binary star observation and that is why I went into some detail here and prepared the model. This is the only time in nature that we humans can observe an Airy disk. The theory of diffraction and resolution was developed by astronomers, for astronomers, mainly just so they could understand what they were seeing and measuring.
My upper figure is E Lyra as I see it in my scope. If I prepared a set of aperture masks for that scope, simulating the aperture diaphragms in our lenses, each successively smaller mask would result in smaller f/ratios. And the diameter of the Airy disks would steadily increase, to the point they are touching, and then more and more heavily notched.
At some point it would look like the middle figure, where the central points of the Airy disks are separated by the radius of one disk. That is the point where "no further resolution is possible" despite increased magnification.
A similar concept occurs as we stop down our lenses, as it relates to the pixel density. However, the analogy becomes quite strained because, as you can see here, the Airy disk has structure. When our pixels hit the wall, the Airy disk is the size of a single pixel, which does not well represent a circle. The effect is quite clear, though, when photographing a line chart. The black and white bars become increasingly grayer to the point where they can not be distinguished at all.
Just thought I would start to connect that final dot. However, in terms of the COC we need to resolve our images at a certain size and distance, our eyes suffer the same problem (or worse aberrations) as we see here in the E Lyra model and there we do not have that problem of pixel binning.
While you are pondering the upper E Lyra model, compare to the single star at the bottom. Also compare to the middle model of two point sources at the very limits of diffraction. At what distance and resolution is the middle notched model no different than the single star?