$Varying separation of Airy diffraction patterns$

Caption: Two overlapping Airy diffraction patterns (in some optical device with a circular aperture) for two identical monochromatic point light sources at optical infinity separated in angle θ. Going downward in the image, θ goes 2θ_R, θ_R, and (1/2)*θ_R, where θ_R is the Rayleigh criterion.

Features:

An example of multiple point light sources is the situation of viewing multiple stars though an optical telescope where the primary mirror or primary lens acts as the circular aperture and the rest of the optics is a complication we need NOT discuss here.
Note that if θ is very large, the 2 Airy diffraction patterns will not overlap at all effectively.
If there is no effective overlap, the two sources are cleanly resolved.
As θ between the 2 point light sources is reduced, the patterns eventually overlap and resolution into 2 Airy diffraction patterns is reduced. This means the resolution of the 2 point light sources is reduced.
If θ = 0, the 2 point light sources will coincide exactly and CANNOT be distinguished at all.
The generally recognized fiducial limit of angular resolution is called the Rayleigh criterion given by the formula
```
 θ_R = (1.21966989 ...)*(λ/D) 
     ≅ 1.220*(λ/D)  , 
```
where
1. θ_R is in radians: 1 radian = 57.2957795 ... degrees ≅ 57.3° ≅ 60°.
2. λ is the wavelength of the monochromatic light.
3. D is the diameter of the circular aperture.
4. λ and D must be in the same units in the standard version of the formula shown above.
5. The irrational number coefficient 1.21966989 ... = (3.83170597 ...)/π, where 3.83170597 ... is the 1st zero (AKA root) of Bessel function J_1(x) (i.e., the 1st order Bessel function of the first kind): thus J_1(3.83170597 ...)=0 (see Wikipedia: Angular resolution: Rayleigh criterion). The zeroth zero (AKA root) of J_1(x) is x = 0: i.e., J_1(0)=0.
If θ = θ_R, the center of one Airy diffraction pattern lies on the 1st minimum of the other. This is the case in the 2nd panel of the image.
If the 2 point light sources have θ < θ_R, they are unresolved by the Rayleigh criterion and in many cases are practically unresolved.
However, you can actually resolve 2 point light sources at somewhat smaller angles than the Rayleigh criterion if you are being very precise and see the separate Airy diffraction patterns.
So the Rayleigh criterion is NOT an absolute limit of angular resolution merely a fiducial one and often a practical one.
The 3rd panel in the image shows that case of θ = (1/2)*θ_R is distinguishable from complete overlap, and so does actually resolve the 2 point light sources.
However, if your measuring device was poor θ = (1/2)*θ_R may be effectively an unresolvable separation.
The Rayleigh criterion can be written in terms of fiducial values:
```
 θ_R = (1.21966989 ...)*(λ/D) radians ≅ 1.220*(λ/D) radians   standard form

     = (25.16'')*(λ_μm/D_cm) = (9.905'')*(λ_μm/D_in)          fiducial-value form

     = (4.952'')*[(λ_μm/(0.5 μm))/D_in]                       fiducial-value form 

     ≅ (5'')*[(λ_μm/(0.5 μm))/D_in]                           approximate fiducial-value form, 
```
where θ_R('') is in arcseconds ('') (1° = 3600''), λ_μm is wavelength in microns (μm), D_cm is aperture diameter in centimeters, and D_in is aperture diameter in inches (1 in = 2.54 cm exactly in modern definition).
For visible light (fiducial range 0.4--0.7 μm), one can often just use the last version of the above formula for crude calculations.
Actually, a slightly different angular resolution criterion is often used for visual astronomy. This criterion is Dawes limit (see Wikipedia: Dawes limit and Wikipedia: Angular resolution: Rayleigh criterion).
The Dawes limit was determined by an empirical study of what angular resolution humans could obtain when observing close binary systems (see Wikipedia: Angular resolution: Rayleigh criterion).
The Dawes limit is
```
             θ_DL('') = (4.56'')/D_in  . 
```
There is NO explicit wavelength dependence since the formula was obtained for effective human eye wavelength-averagved psychophysical sensitivity to stars.
Actually, people often just conflate the Rayleigh criterion and the Dawes limit because they have such similar formulae, but they are really NOT the same thing.
If one equates θ_DL('') and θ_R('') and solves for λ_μm, one obtains a Dawes wavelength λ_μm_DL = 0.4604 μm.
The Dawes wavelength can only be considered a characteristic or average wavelength for psychophysical sensitivity for resolving stars assuming that the Dawes limit is, in fact, approximately the Rayleigh criterion.
The λ_μm_DL = 0.4604 μm is in the spectral color blue (≅ 0.450--0.495 μm).
But starlight is a mixture of spectral colors and, in fact, the mixture never looks blue to yours truly on the sky.
So if the Dawes limit is approximately Rayleigh criterion, it is NOT because starlight is pure blue.
If Dawes limit is approximately Rayleigh criterion, we would predict the Dawes wavelength to be close to the maximum of psychophysical luminosity function for scotopic vision (i.e., human eye vision under poorly lit conditions which is where visual astronomy is done). This maximum is at ∼ 0.498 μm.
Well, the two values are NOT so far apart, but NOT so close that we can say for sure that we've proven Dawes limit is approximately the Rayleigh criterion.
Note the maximum of psychophysical luminosity function for photopic vision (i.e., human eye vision under well lit conditions) occurs at ∼ 0.555 μm---which is NOT very close to the Dawes wavelength.

Credit/Permission: Spencer Bliven (AKA User:Quantum7), 2014 / Public domain.
Image link: Wikimedia Commons.
Local file: local link: optics_rayleigh_criterion.html.
File: Optics file: optics_rayleigh_criterion.html.