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:
If there is no effective overlap, the two sources are cleanly resolved.
If θ = 0, the 2 point light sources will coincide exactly and CANNOT be distinguished at all.
θ_R = (1.21966989 ...)*(λ/D) ≅ 1.220*(λ/D) ,where
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.
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.
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).
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.
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.
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.
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.
Credit/Permission: Spencer Bliven (AKA User:Quantum7),
2014 /
Public domain.
Image link: Wikimedia Commons.
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