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I've seen a formula θ = 1.22 λ/D here and there, but as if it's used in two "meanings".
Sometimes the θ is taken to be the (half of) angular size of the artificial star that is just beyond resolution.
Sometimes it's the angle of the first minimum of the airy disk from the normal going through the middle of the aperture.

How does an artificial star form an airy disk? I'd think there were two angles: the angular size of the artificial star and the angle from the normal where the first minimum of the diffraction pattern emerges.

And in case of reflectors, which is the aperture? The tube or the mirror?

Senior Embedded SW Designer
Telescope: OrionOptics XV12, dobsonian, all manual.
LAT 61° 28' 10.9" N, Bortle 5

Another version of the formula θ(arc sec) = 4.5/D(inches) is probably easier to use. For your XV12 that is 0.375".

At 100 m distance, a pinhole needs to be < 0.2mm. Pretty easy to achieve with a small hole poked in aluminium foil. And then if you reflect this in a stainless steel polished ball bearing or other curved surface, you will have a very small point light source.

"34 South: The Hilltops Observatory"
Central West NSW
Bortle 1-2 skies, 148 E, 34 S

Amateur astronomer since 1978
Astronomical interests : astrophotography, visual observing, nightscape photography, solar eclipse chasing
asteroidal occultations, nightscape astrophotography workshops

web site : http://joe-cali.com/
SCOPES - ATM 18" Dob, Vixen VC200L, ATM 6"f7, ED80, M70
MOUNTS- EM-200, iEQ45, Push dobsonian with Nexus DSC, Various homemade EQ's
CAMERAS : Pentax K1, K5, K01 / VIDEO CAMS : TacosBD, Lihmsec

OK, the tube doesn't make the diffraction pattern on the mirror, but the mirror makes it on the image?
But θ = 1.22 λ/D still bothers me. It's the angle of first diffraction ring (minimum = airy disk edge), but it seems to be also used as the limit of the telescope resolution (target object angular size).

Or maybe the angle is so small anyway, that it can be considered a point source, and only the airy disk size matters, and the magnification can be neglected?
In that case you start from the airy disk, and work backwards. The maximum angular size of the artificial star (from primary mirror to the artificial star) IS the angular size of the airy disk in the image (as seen from the primary mirror)?
And the angle is considered so small that the magnification doesn't have any notable effect (zero times magnification is still zero)?

Senior Embedded SW Designer
Telescope: OrionOptics XV12, dobsonian, all manual.
LAT 61° 28' 10.9" N, Bortle 5

The star, real or artificial, radiates light that illuminates the entire lens or mirror which then focusses it down to a point. Yes the first Airy disc radius is the resolution limit for two point sources. Closer than that and the Airy discs overlap. The other reason for putting the artificial star a long way out is so that you can reach focus. A close point source will come to focus way out past the normal plane of focus. You may not reach it or need to use a lot of extension.

"34 South: The Hilltops Observatory"
Central West NSW
Bortle 1-2 skies, 148 E, 34 S

Amateur astronomer since 1978
Astronomical interests : astrophotography, visual observing, nightscape photography, solar eclipse chasing
asteroidal occultations, nightscape astrophotography workshops

web site : http://joe-cali.com/
SCOPES - ATM 18" Dob, Vixen VC200L, ATM 6"f7, ED80, M70
MOUNTS- EM-200, iEQ45, Push dobsonian with Nexus DSC, Various homemade EQ's
CAMERAS : Pentax K1, K5, K01 / VIDEO CAMS : TacosBD, Lihmsec

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.

... Henk. Telescopes: 6" Mak-Newt (Comet Hunter), ES ED127CF, ES ED80, Zhumell Z12, AT6RC, Venture RX-7, Celestron Skymaster 20x80, Mounts and tripod: Losmandy G11S with DIY OnStep, AVX, LXD55, Tiltall, Cameras: Fuji X-a1, Canon SX40, Xt, XSi, T6, ELPH 100HS, DIY: OnStep controller, Barndoor trackers for 10" Dob and camera, Afocal adapter, Foldable Dob base, Az/Alt Dob setting circles, Accessories: TV Paracorr 2, Baader MPCC Mk III, ES FF, SSAG, Plossls, Barlows, Telrad, Laser collimators (Seben LK1, Z12, Howie Glatter), Cheshire, 2 Orion RACIs 8x50, Software: KStars-Ekos, DSS, PHD2, Nebulosity, Photo Gallery, Gimp, CHDK

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.

It's hard to follow that. It mostly about spherical aberration even if it mentions collimation too.
I guess KE = knife edge, but what are s and t? Deviations of light paths due to mirror being paraboloid instead of spherical?

Senior Embedded SW Designer
Telescope: OrionOptics XV12, dobsonian, all manual.
LAT 61° 28' 10.9" N, Bortle 5

Lady Fraktor wrote: ↑Sat Dec 26, 2020 7:19 pm
The formula I posted in the other thread is an adaption of Nils Olof Carlin maths for this.

What I wonder is:
Isn't R ~ 115.9/A the anglular size of airy disk?
And what has that to do with the angular size of the artificial star?

AH! the Rayleigh limit is the angle between two stars that can (barely) be separated, because the airy disks overlap half way. It gives a two peak curve.
But then, why doesn't focal length (magnification) affect the angular distance of the stars? Or do two stars with angular separation θ cause two airy disks with angular separation of also θ? Kind of sounds odd. I would have thought that magnification would made the separation of airy disks bigger.

My problem is not calculating that, but the thinking behind the math.

Senior Embedded SW Designer
Telescope: OrionOptics XV12, dobsonian, all manual.
LAT 61° 28' 10.9" N, Bortle 5

Lady Fraktor wrote: ↑Sat Dec 26, 2020 7:19 pm
The formula I posted in the other thread is an adaption of Nils Olof Carlin maths for this.

What I wonder is:
Isn't R ~ 115.9/A the anglular size of airy disk?
And what has that to do with the angular size of the artificial star?

AH! the Rayleigh limit is the angle between two stars that can (barely) be separated, because the airy disks overlap half way. It gives a two peak curve.
But then, why doesn't focal length (magnification) affect the angular distance of the stars? Or do two stars with angular separation θ cause two airy disks with angular separation of also θ? Kind of sounds odd. I would have thought that magnification would made the separation of airy disks bigger.

My problem is not calculating that, but the thinking behind the math.

The diameter controls the angular size of the airy disk and hence the Rayleigh limit. Varying the focal length only changes the linear image size of the airy disk but not the angular size.

"34 South: The Hilltops Observatory"
Central West NSW
Bortle 1-2 skies, 148 E, 34 S

Amateur astronomer since 1978
Astronomical interests : astrophotography, visual observing, nightscape photography, solar eclipse chasing
asteroidal occultations, nightscape astrophotography workshops

web site : http://joe-cali.com/
SCOPES - ATM 18" Dob, Vixen VC200L, ATM 6"f7, ED80, M70
MOUNTS- EM-200, iEQ45, Push dobsonian with Nexus DSC, Various homemade EQ's
CAMERAS : Pentax K1, K5, K01 / VIDEO CAMS : TacosBD, Lihmsec

Lady Fraktor wrote: ↑Sat Dec 26, 2020 7:19 pm
The formula I posted in the other thread is an adaption of Nils Olof Carlin maths for this.

What I wonder is:
Isn't R ~ 115.9/A the anglular size of airy disk?
And what has that to do with the angular size of the artificial star?

AH! the Rayleigh limit is the angle between two stars that can (barely) be separated, because the airy disks overlap half way. It gives a two peak curve.
But then, why doesn't focal length (magnification) affect the angular distance of the stars? Or do two stars with angular separation θ cause two airy disks with angular separation of also θ? Kind of sounds odd. I would have thought that magnification would made the separation of airy disks bigger.

My problem is not calculating that, but the thinking behind the math.

The diameter controls the angular size of the airy disk and hence the Rayleigh limit. Varying the focal length only changes the linear image size of the airy disk but not the angular size.

Now I got it. This picture helped me to finally get it.

off_axis.jpeg (11.04 KiB) Viewed 507 times

This is about spherical mirror, but it's the same thing with parabolic. And this, of course, also applies to convex mirror.
The false image just forms behind the mirror.

Senior Embedded SW Designer
Telescope: OrionOptics XV12, dobsonian, all manual.
LAT 61° 28' 10.9" N, Bortle 5

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.

Senior Embedded SW Designer
Telescope: OrionOptics XV12, dobsonian, all manual.
LAT 61° 28' 10.9" N, Bortle 5

OzEclipse wrote: ↑Sat Dec 26, 2020 1:56 am
The star, real or artificial, radiates light that illuminates the entire lens or mirror which then focusses it down to a point. Yes the first Airy disc radius is the resolution limit for two point sources. Closer than that and the Airy discs overlap. The other reason for putting the artificial star a long way out is so that you can reach focus. A close point source will come to focus way out past the normal plane of focus. You may not reach it or need to use a lot of extension.

There's more to it. Only from infinity will a light bundle parallel to the optical axis focus to a point, for a parabolic reflector. When the light source is not at infinity, there will be spherical aberration and light emitted by a point source will no longer focus to a point. How bad that is depends on F ratio, for instance. This will affect the Airy disk pattern as well.

... Henk. Telescopes: 6" Mak-Newt (Comet Hunter), ES ED127CF, ES ED80, Zhumell Z12, AT6RC, Venture RX-7, Celestron Skymaster 20x80, Mounts and tripod: Losmandy G11S with DIY OnStep, AVX, LXD55, Tiltall, Cameras: Fuji X-a1, Canon SX40, Xt, XSi, T6, ELPH 100HS, DIY: OnStep controller, Barndoor trackers for 10" Dob and camera, Afocal adapter, Foldable Dob base, Az/Alt Dob setting circles, Accessories: TV Paracorr 2, Baader MPCC Mk III, ES FF, SSAG, Plossls, Barlows, Telrad, Laser collimators (Seben LK1, Z12, Howie Glatter), Cheshire, 2 Orion RACIs 8x50, Software: KStars-Ekos, DSS, PHD2, Nebulosity, Photo Gallery, Gimp, CHDK

OzEclipse wrote: ↑Sat Dec 26, 2020 1:56 am
The star, real or artificial, radiates light that illuminates the entire lens or mirror which then focusses it down to a point. Yes the first Airy disc radius is the resolution limit for two point sources. Closer than that and the Airy discs overlap. The other reason for putting the artificial star a long way out is so that you can reach focus. A close point source will come to focus way out past the normal plane of focus. You may not reach it or need to use a lot of extension.

There's more to it. Only from infinity will a light bundle parallel to the optical axis focus to a point, for a parabolic reflector. When the light source is not at infinity, there will be spherical aberration and light emitted by a point source will no longer focus to a point. How bad that is depends on F ratio, for instance. This will affect the Airy disk pattern as well.

And it also depends on the distance of the star image from the optical axis?

Senior Embedded SW Designer
Telescope: OrionOptics XV12, dobsonian, all manual.
LAT 61° 28' 10.9" N, Bortle 5

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.

... Henk. Telescopes: 6" Mak-Newt (Comet Hunter), ES ED127CF, ES ED80, Zhumell Z12, AT6RC, Venture RX-7, Celestron Skymaster 20x80, Mounts and tripod: Losmandy G11S with DIY OnStep, AVX, LXD55, Tiltall, Cameras: Fuji X-a1, Canon SX40, Xt, XSi, T6, ELPH 100HS, DIY: OnStep controller, Barndoor trackers for 10" Dob and camera, Afocal adapter, Foldable Dob base, Az/Alt Dob setting circles, Accessories: TV Paracorr 2, Baader MPCC Mk III, ES FF, SSAG, Plossls, Barlows, Telrad, Laser collimators (Seben LK1, Z12, Howie Glatter), Cheshire, 2 Orion RACIs 8x50, Software: KStars-Ekos, DSS, PHD2, Nebulosity, Photo Gallery, Gimp, CHDK

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.

I read that twice, but I guess I'll read it 3rd time tomorrow with the Carlin's page open aside.

Senior Embedded SW Designer
Telescope: OrionOptics XV12, dobsonian, all manual.
LAT 61° 28' 10.9" N, Bortle 5

SkyHiker wrote: ↑Mon Dec 28, 2020 9:29 pm
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 ... for instance ... (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...

7x50 Helios Apollo ✶ 8x42 Bresser Everest ✶ 73mm f/5.9 WO APO ✶ 4" f/5 TeleVue Genesis ✶ 6" f/10 Celestron 6SE ✶ 0.63x reducer ✶ 1.8, 2, 2.5 and 3x Barlows ✶ eyepieces from 4.5 to 34mm

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).

Y^2/2R is the curve of the mirror - a parabola whose focal point is F (= R/2 here).
{ For parabola y = a(x-x0)^2+y0, vertex is (x0, y0) and focal point is (X0, y0+1/(4a)) }
And Y^2/2B is.. what? At least Carlin says that:

Consider a point on the mirror, at a radius Y from the optical axis. The distances from this point to the KE and source, respectively, are taken to be:

(A + (Y2/2)(1/A - 1/R) + s) and (B + (Y2/2)(1/B - 1/R) + t);

I just can't see how the Y^2/2B fits into the "distance from this point to source".

I wish Carlin would have been a bit more verbose...

Senior Embedded SW Designer
Telescope: OrionOptics XV12, dobsonian, all manual.
LAT 61° 28' 10.9" N, Bortle 5

I attached my calculations about how rays emanating from a point source that is placed at a finite distance from a parabolic mirror, are reflected. It is just geometry based and exact in the sense that no ideal lens formulas are used that assume some approximation.

While I obtained closed formulas for the points of intersections with the optical axis and corresponding path lengths to those points, it does not really say anything about Airy discs because I have no clue how interference of rays in close proximity is described. I found P-V values numerically based on finding the points where the deviation from the optical axis is minimal, for a number of use cases.

All this is probably worth nothing but it resulted in some pretty pictures, I attached one here.

I also attached the Scilab script to execute the math, for who wants to make more pretty pictures.