Diffraction sets a hard floor on how finely any lens or telescope can see. A point of light is never a point — through a round aperture of diameter D it spreads into an Airy disk. Two stars are just resolved when the peak of one disk falls on the first dark ring of the other: θ = 1.22 λ/D. Down the center, light flows in, the engine diffracts, the resolution limit comes out. The blue team builds and defends it; the red team tries to break it.
source Rayleigh (J. W. Strutt), “Investigations in optics, with special reference to the spectroscope,” Phil. Mag. S.5, 8, 261–274 (1879), doi:10.1080/14786447908639684 AMBER paywalled, no stable open scan. Rendered, not quoted.
A circular aperture diffracts a point source into the Airy pattern, intensity (2J₁(u)/u)² with u = πD sinθ/λ. Its first dark ring is the first zero of the Bessel function J₁, at u = 3.8317.
Divide by π: 3.8317/π = 1.2197. That is where the 1.22 comes from — not a fudge, a computed constant. Two point sources are just resolved when one peak sits on the other’s first ring:
θmin = 1.22 λ / D
For the current settings, computed live:
| quantity | value |
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The circular limit inherits directly from the single slit: a slit of width a spreads light to its first minimum at λ/a. Wrap that slit into a circle and the 2-D integral over the aperture replaces the factor 1 with 1.22 — the first zero of J₁ instead of the first zero of sinc.
Rayleigh (1879) turned that spreading into a rule for seeing: the same diffraction that blurs one slit sets the sharpest angle any telescope can ever resolve. Each sphere is the next one’s premise — the slit’s spreading is this sphere’s floor.
The blue team’s live check: recompute the Airy first zero, confirm θ falls with D and rises with λ, and verify the circle-vs-slit ratio is exactly 1.22. If red tampers, this badge is where it shows.
Three numbers feed the engine: the wavelength λ of the light, the diameter D of the round aperture, and the true angular separation of two point sources on the sky.
Bigger aperture → smaller Airy disk → finer resolution. Redder light → wider disk → coarser resolution. The question the panel answers: are the two sources far enough apart to be told apart?
Blue & cyan curves are the two Airy disks; the gold curve is their sum. When the sum still dips ≥26% between the peaks, the pair is resolved (the Rayleigh dip is ≈0.735).
Every number is computed on the spot from θ=1.22λ/D and the Airy intensity — never looked up.
What the machine proves: the minimum resolvable angle θmin = 1.22 λ/D, falling as D grows and rising with λ; the just-resolved condition (peak on first ring); and the diffraction-limited focal spot ≈ 1.22 λ f/D. The 1.22 is the first zero of J₁ over π — checked live below.
The blue team’s witness (left) confirms the factor live; the red team (right) tries to overstate it.
And on the ground it rarely matters: atmospheric seeing (~1″) usually dominates a big telescope’s diffraction limit unless adaptive optics or interferometry (VLBI, aperture synthesis) is used. The floor is real; whether you ever hit it is another question.
“1.22λ/D is a hard wall you can never beat.” Cut. It is a contrast convention; PSF fitting and deconvolution beat it for bright, isolated, known sources.
“θ = λ/D for a telescope.” Cut. That is the slit value. A round aperture carries the 1.22 factor from the first zero of J₁ — drop it and you overstate the resolution.
“1.22 is arbitrary.” Kept, corrected. 1.22 = 3.8317/π, the first zero of J₁ over π — a computed constant, in the engine.
The red team’s move: drop the 1.22 and use θ=λ/D (the slit value), overstating a circular aperture’s resolution. The blue team’s witness (window 7) is watching.
Drop the factor and the circle-vs-slit ratio collapses from 1.22 to 1.00 — the witness recomputes, disagrees with the Airy first zero, and turns red. Nothing is faked; the attack is real and it is caught.