How this instrument works
Light squeezed through a circular hole never converges to a point. It fans out into a bright central disc wrapped in faint rings — a pattern George Biddell Airy computed in 1835 and which now carries his name. Radius of that first dark ring is what fixes an instrument's power to keep two neighbours apart: 1.22 λ ⁄ D, where 1.22 is 3.8317 divided by π, and 3.8317 is where Bessel function J₁ first crosses zero.
Lord Rayleigh turned that geometry into a working rule in 1879 — two equally bright points count as resolved once one Airy peak sits on its neighbour's first dark ring. It is a convention about contrast, not a hard wall. William Dawes, splitting double stars by eye in 1867, found observers could do slightly better: 116 arcseconds over aperture in millimetres, against Rayleigh's 138. Sparrow's criterion is tighter again, and deconvolution routinely goes past all three.
Every symbol here assumes an unobstructed circular aperture, optics good to a quarter wave, and wavefronts arriving flat from far away. Break any assumption and real instruments fall short: a secondary mirror drains light from core into rings, polishing errors soften contrast, and atmospheric turbulence pins ground telescopes near one arcsecond however wide their glass — which is precisely why adaptive optics and orbiting observatories exist. Microscopy uses a close cousin, 0.61 λ ⁄ NA, since immersion fluid changes what an aperture collects.
- Set Wavelength to your observing band — 550 nm for visual work, 2 µm near-infrared, 1.3 mm for millimetre arrays.
- Enter Aperture diameter as clear glass or mirror width. Not focal length, not f-number — only diameter counts here.
- Read Angular resolution (radians) for optical maths, or Angular resolution (arcseconds) for anything pointed at sky.
- Compare against local seeing. Below roughly 150 mm at sea level, atmosphere rarely limits you; above that, it usually does.
Worked example — a 100 mm refractor on a double star
A 100 mm refractor, aimed at a tight double star in green light near 550 nm. Substitute straight in: θ = 1.22 × 5.5 × 10⁻⁷ ⁄ 0.1 = 6.71 × 10⁻⁶ radians. Multiply by 206265 arcseconds per radian and that lands at 1.38 arcseconds.
Dawes, working empirically on equal sixth-magnitude pairs, would quote 1.16 arcseconds for that same tube — tighter, because a trained eye tolerates less contrast than Rayleigh's rule allows. Either way, a pair separated by 2 arcseconds splits cleanly, while a 1-arcsecond pair stays one fuzzy dot until you fetch more aperture. Step up to 200 mm and that answer halves to 3.36 × 10⁻⁶ radians, or 0.69 arcseconds — steady air permitting.
Questions
Why 1.22 and not simply λ ⁄ D?
That factor comes straight from circular geometry. Diffraction through a round hole produces an Airy pattern whose first dark ring falls at 3.8317 radians of phase — first zero of Bessel function J₁ — and 3.8317 ⁄ π = 1.2197. A long narrow slit, having no circular symmetry, gives plain λ ⁄ D with no prefactor at all. Square and hexagonal apertures land elsewhere again, which is why segmented mirrors such as JWST produce six-pointed spikes rather than smooth haloes.
Should I work in radians or arcseconds?
Radians are SI and native to this formula, since an angle is a length ratio and carries no dimension. Astronomers quote arcseconds because useful figures then land near 1 rather than near 10⁻⁶. Conversion factor is 206264.806, which is 648000 ⁄ π. Both outputs appear side by side, so pick whichever your notes already use.
Does a longer focal length sharpen my images?
No. Only Aperture diameter enters this formula. Focal length sets image scale and magnification — how large a blur appears, never how small it is. Pushing magnification past roughly twice aperture in millimetres yields empty magnification: a bigger, dimmer, equally unresolved smudge. Photographers meet an identical wall as f-number climbs, where diffraction rather than lens quality caps detail beyond about f/11 on small sensors.
Why does my large telescope never hit this figure?
Air. Turbulence scrambles wavefronts on scales set by Fried's parameter, typically 10 to 20 cm at visible wavelengths, so anything wider than that is seeing-limited rather than diffraction-limited. Long exposures from good mountain sites settle around 0.5 to 1.5 arcseconds regardless of mirror size. Adaptive optics correct much of it in real time; putting glass above atmosphere avoids it entirely, which is how a 2.4 m Hubble mirror reaches roughly 0.05 arcseconds.
How does this compare with microscope resolution?
Same physics, different bookkeeping. A microscope objective collects a wide cone, so numerical aperture NA = n sin α replaces a bare diameter, and lateral resolution becomes 0.61 λ ⁄ NA — Abbe wrote λ ⁄ (2 NA) for much the same idea in 1873. Oil immersion raises n to about 1.5, pushing NA past 1.4 and green-light resolution near 240 nm. Both expressions share one heritage: 0.61 is simply half of 1.22.
Can anything beat this limit?
Yes, by changing what D means or by exploiting extra information. Interferometry replaces aperture with baseline: linked dishes spanning 10,000 km at 1.3 mm gave Event Horizon Telescope roughly 25 microarcseconds. Deconvolution recovers detail below Rayleigh's contrast threshold when signal-to-noise is generous. Fluorescence methods such as STED and PALM switch molecules on and off individually, earning a 2014 Nobel Prize precisely for stepping around a barrier once thought absolute.