turboscrew wrote: ↑Sun Dec 27, 2020 5:29 pm
SkyHiker wrote: ↑Sat Dec 26, 2020 6:56 am
The distance criterion has to do with how different the wave front of a star at infinity vs. a perfect pinhole at a finite distance looks at the focal plane. Nils Olof Carlin explains this at http://web.telia.com/~u41105032/misc/nulltest.htm
. This does not involve Dawes criterion at all. Nils does not consider the size of the pinhole, perhaps that is less critical than the distance but there should be math for it too.
I tried to figure out the link, but I couldn't.
For the Y-B distance, I got: (Y-B)^2 = B^2 + (1 - 2B/2R)Y^2 + ((1/2R)^2)(Y^4)
Carlin's equal was: (Y-B)^2 = B^2 + (Y^2 - 2B(1/2R)Y^2) + ((1/2B - 1/2R)^2)(Y^4)
I can't figure out the (1/2B - 1/2R) instead of (1/2R) in the last term.
And I haven't figured out the t and s.
Sorry for replying late. I wanted to wait until I did the math from scratch and I am slowly getting there. Nils Olof is no longer here so all we can say is that his article is short on details that he left out to keep it readable. He mentioned that there was some grueling algebra behind it, which I am finding out, well not too bad really but I'm getting there. What else is there to do with travel restrictions, vacation and rain outside!
Your Y-B squared distance is simply based on the Pythagorean sum of squares between the points Y and B, accounting for the offset Y^2/2R due to the curvature. This is not the distance that matters. What matters is the phase difference relative to the ideal circular wave front, which is quite different from that at the intersection of the Y axis. Nils Olof defines s and t as the errors relative to that circular wave front.
Here's what I think (but I am not certain about it). His B term for instance, is (B-Y^2/2R)+Y^2/2B. The term within parentheses is the distance between B and the vertical down from Y. The last term is a correction term for a circle of radius B with B at the center (just like you would get Y^2/2R at the center radius R). The t is defined relative to the circle with center B, likewise s is related to a analogous circle around A. They are defined implicitly and must be derived in the solution to the problem.
PS I should have mentioned that the distance is valid for wave fronts that center exactly around a point. The trouble is that this only happens with rays coming from infinity, not from a finite distance. That discrepancy is what is causing the problem, I believe.
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