Rayleigh Criterion: the Smallest Angle an Aperture Can Separate

θ = 1.22λ/D is the whole formula, and on its own it tells you nothing about your actual problem. This page gives you the limit, then compares it with the separation you are really looking at and says whether the two things merge, sit exactly on the limit, or come apart — and draws the two spots as you would see them.

Last updated 11 Nov 2026

Diffraction limit, and whether your pair is resolved

Aperture in millimetres, arcseconds out

WavelengthIn nanometres. 550 nm is the usual convention — the middle of the visible band and the peak of the eye’s daylight response — and it is what observatory specifications quote unless they say otherwise. The field reaches 100 000 nm so you can see what happens in the infrared: resolution is proportional to wavelength, so the same dish is four times blunter at 2.2 µm than at 550 nm, which is the whole reason infrared astronomy needs bigger mirrors.

Aperture diameterIn millimetres, measured across the clear aperture — the objective lens or the primary mirror, not the tube. A dark-adapted human pupil is about 7 mm and a daylight one about 3 mm. A common beginner’s telescope is 100 to 150 mm; the Hubble Space Telescope is 2400 mm. For a reflector with a secondary mirror in the way, use the full primary diameter: the obstruction changes how the light is distributed between the central spot and the rings, but not the position of the first dark ring, which is what this formula is about.

Separation of the two things you are looking atIn arcseconds. One arcsecond is 1/3600 of a degree — roughly a one-rupee coin seen from five kilometres. For scale: the Moon is about 1860 arcseconds across, Jupiter about 40, the double star Albireo about 35, and Mizar’s close pair about 14. This field is what turns the formula into an answer: the verdict and the picture below both compare it with the limit.

Distance to themIn metres. Used only to turn the angular limit into a physical size: at 1000 m, an angular limit of one arcsecond is about 4.8 mm. It does not affect the limit itself, which is a property of the aperture and the wavelength and of nothing else. Leave it at 1000 if you are working in angles.

The two diffraction discs at your separation, with the brightness profile across them underneath. Each disc is drawn one Rayleigh limit in radius, so when the centres are one radius apart you are looking at exactly the criterion.

1.3840arcsecExample

A 100 mm telescope at 550 nm, looking at a double star 2.0 arcseconds apart

One formula, one constant, and the comparison that matters

θ = 1.22λ/D (radians) · θ[arcsec] = θ[rad] × 206 265 · smallest detail = θL · verdict = (your separation)/θ

1.22: the first zero of the diffraction pattern of a circular aperture, in units of λ/D. A long slit of width a gives its first zero at λ/a with no 1.22 at all; the extra factor is the price of the aperture being round. It is 1.21967…, the first zero of a Bessel function, and it is not adjustable
θ: an angle in radians, always. Every error on this topic that is not a unit error is someone reading 1.22λ/D as degrees. Multiply by 206 265 for arcseconds, which is the unit telescope makers and double-star observers actually use. This page does that conversion for you and shows the radian value too
D: the diameter of the aperture, in the same units as λ before you divide. Resolution improves in direct proportion to D: a 200 mm telescope resolves twice as finely as a 100 mm one. It does not depend on focal length, on magnification or on the eyepiece, which is why ‘how much does it magnify’ is the wrong question to ask about a telescope
the ratio: your separation divided by the limit. Below about 0.85 the pair reads as one object; between 0.85 and 1.25 it is a single elongated blob with a shallow dip; above 1.25 it comes apart cleanly. Those boundaries are this page’s own way of describing a continuous change, not a published standard
Dawes’ limit: the observers’ alternative, 4.56 arcseconds per inch of aperture, or 115.8/D with D in millimetres. It is empirical, derived from what people could actually split, and it is smaller than Rayleigh’s — 1.16 against 1.38 arcseconds for a 100 mm aperture. Both are shown. The gap between them is the gap between a theorist’s definition and an observer’s experience

Worked example

A 100 mm telescope at 550 nm, looking at a double star 2.0 arcseconds apart
EVERYTHING IN METRES FIRST. λ = 550 nm = 5.50×10⁻⁷ m. D = 100 mm = 0.100 m. Both must be in the same unit or the ratio is meaningless, and this is where most of the lost marks are.
THE LIMIT IN RADIANS. θ = 1.22 × 5.50×10⁻⁷ / 0.100 = 6.71×10⁻⁶ rad. That is the answer the formula gives, and it is in radians whether or not anyone says so.
INTO ARCSECONDS. Multiply by 206 265 (the number of arcseconds in a radian): 6.71×10⁻⁶ × 206 265 = 1.3840 arcseconds. A useful shortcut worth remembering: at 550 nm, θ in arcseconds is simply 138.4/D with D in millimetres.
NOW COMPARE IT WITH THE REAL PAIR. The stars are 2.0 arcseconds apart and the limit is 1.3840, so the ratio is 2.0/1.3840 = 1.45. Comfortably above 1.25, so they resolve: two separate points with dark sky between them, given steady air.
WHAT WOULD NOT WORK. If the pair were 1.0 arcsecond apart the ratio would be 0.72 and no eyepiece in the world would split them with this telescope. The aperture you would need is 1.22λ/θ = 1.22 × 5.50×10⁻⁷ / (1.0/206 265) = 138 mm. Aperture, not magnification — that is the entire lesson of this page.
A SANITY CHECK ON YOUR OWN EYES. Put D = 3 mm, a daylight pupil: θ = 138.4/3 = 46 arcseconds, about three quarters of an arcminute. People test at roughly one arcminute, a little worse than the diffraction limit, which is the right way round — the retina and the optics of the eye add to diffraction rather than beating it.

What different apertures can separate at 550 nm

Two things to notice. The first is how quickly the formula outruns reality: ordinary air blurs stars to about one arcsecond, so every aperture from 150 mm upwards is already limited by the atmosphere rather than by diffraction on a typical night. That is why a 200 mm telescope does not usually show twice the detail of a 100 mm one — it collects four times the light, which is a real and separate benefit, but the sharpness is capped by the sky. The second is the last row: Keck’s diffraction limit is better than Hubble’s because it is four times the diameter, and it reaches that limit only with adaptive optics bending a flexible mirror hundreds of times a second to undo the air. The Dawes column is consistently about 16 per cent smaller than the Rayleigh column, because it records what observers could actually do rather than where the first dark ring sits.
Aperture	Rayleigh θ	Dawes’ limit	Detail at 1 km	Typical use
3 mm (daylight pupil)	46.13″	38.60″	223 mm	Unaided eye in sunlight
7 mm (dark-adapted pupil)	19.77″	16.54″	95.8 mm	Unaided eye at night
50 mm (binoculars)	2.768″	2.316″	13.4 mm	Hand-held binoculars
100 mm	1.384″	1.158″	6.71 mm	Small refractor
200 mm	0.692″	0.579″	3.36 mm	Already beaten by the air most nights
2400 mm (Hubble)	0.0577″	0.0483″	0.280 mm	Above the atmosphere
10 000 mm (Keck)	0.01384″	0.01158″	0.0671 mm	Needs adaptive optics to reach this

Why a bigger lens sees more detail, and why magnification does not

Light going through a hole spreads out. That is diffraction, and it means a point of light — a star, a distant lamp — never arrives as a point. It arrives as a small bright disc surrounded by faint rings, called the Airy pattern after the Astronomer Royal who worked it out. The disc has a size, and that size is set by the aperture: the wider the hole, the smaller the disc. Two stars close together produce two overlapping discs, and once the discs overlap enough you cannot tell there are two.

Rayleigh’s criterion fixes where ‘enough’ is. He proposed that the pair is just resolved when the centre of one disc falls on the first dark ring of the other, which happens at an angular separation of θ = 1.22λ/D. At that separation the combined brightness profile has a dip of about 26 per cent between the two peaks — shallow, but real. Push them closer and the dip fills in; pull them apart and it deepens into black sky. The diagram above draws exactly that profile at your numbers.

The 1.22 is the only interesting part of the formula. A long slit of width a sends its first diffraction zero to λ/a, with no numerical factor at all. A round hole sends it to 1.22λ/D, and the 1.22 comes from the first zero of a Bessel function — the circular-aperture equivalent of the sine that appears for a slit. If you have met the grating condition you have already met the slit version; the 1.22 is what you add when the aperture is a telescope rather than a letterbox.

And here is the part that costs money. The limit depends on the aperture and the wavelength. It does not depend on the focal length, the eyepiece, or the magnification. A 60 mm telescope advertised as ‘525×’ resolves 2.3 arcseconds whatever eyepiece you screw into it; at 525× you are looking at a hugely enlarged blur. The useful range of magnification is set by the aperture too — see the magnification page for the 2×-per-millimetre ceiling that follows from exactly this argument. If you want to separate closer things, you need a bigger hole. There is no other way, and no eyepiece can supply it.

Where this goes wrong in exams. Three places. Forgetting that θ is in radians, and treating 1.22λ/D as though it were degrees — an error of a factor of 57. Mixing the units of λ and D, usually nanometres against millimetres, before dividing. And using the radius instead of the diameter, which doubles the answer. A good habit is the shortcut: at 550 nm, θ in arcseconds is 138.4/D with D in millimetres. If your answer is nowhere near that, something has gone wrong upstream.

One honest caveat. The criterion is a convention and the boundary is not sharp. A careful observer on a steady night can split a pair a little closer than Rayleigh’s limit, which is why Dawes’ empirical limit — about 16 per cent smaller — exists at all; and image processing on a clean, well-sampled picture can do better still. Equally, on a night of poor seeing a 300 mm telescope performs like a 100 mm one, because the atmosphere has blurred everything to about an arcsecond before the light ever reaches the mirror. The formula tells you what the aperture permits. The sky decides what you get.

Frequently asked questions

What is the Rayleigh criterion?

Two point sources are said to be just resolved when the centre of one’s diffraction pattern falls on the first dark ring of the other’s. For a circular aperture that happens at an angular separation θ = 1.22λ/D radians, where λ is the wavelength and D the aperture diameter. At that separation the brightness profile across the pair shows a dip of about 26 per cent between the peaks. It is a convention chosen because it matches what a careful observer can actually see, not a hard physical boundary.

Why 1.22?

Because the aperture is circular. The diffraction pattern of a round hole is an Airy pattern, and its first dark ring sits at 1.22λ/D rather than at the λ/a you get from a long slit. The number is the first zero of a Bessel function, 1.21967 to five figures. If a question gives you a slit rather than a circular aperture, drop the 1.22 — putting it in is as wrong as leaving it out of the circular case.

Does more magnification let me see finer detail?

No. The limit 1.22λ/D contains the aperture and the wavelength and nothing else — no focal length, no eyepiece, no magnification. Once the detail is below the diffraction limit, magnifying the image enlarges the blur and nothing more. A useful amount of magnification is needed to let your eye USE the detail the aperture has already collected, which is why about 1 to 2× per millimetre of aperture is the standard working range, but beyond that you gain nothing at all.

What is the resolution of the human eye?

With a 3 mm daylight pupil, the diffraction limit at 550 nm is 1.22 × 550×10⁻⁹/0.003 = 2.24×10⁻⁴ rad, which is about 46 arcseconds or three quarters of an arcminute. In practice people test at roughly one arcminute — a little worse than the diffraction limit, because the spacing of the retina’s receptors and the imperfections of the eye’s own optics add to the blur. One arcminute corresponds to about 0.3 mm at arm’s length, which is roughly the finest print you can read.

Why do large telescopes still look blurry from the ground?

Because the atmosphere, not the aperture, is the limiting factor. Turbulence smears a star into a blob of about one arcsecond even on a good night, so any telescope larger than roughly 150 mm is working below its own diffraction limit most of the time. The extra aperture still collects more light and shows fainter objects, which is a real gain. Recovering the sharpness needs either going above the air, as Hubble does, or adaptive optics, which measures the distortion and bends a flexible mirror to cancel it hundreds of times a second.

What is the difference between Rayleigh’s limit and Dawes’ limit?

Rayleigh’s is theoretical: the separation at which one Airy disc’s centre lands on the other’s first dark ring, 1.22λ/D. Dawes’ is empirical: in 1867 William Rutter Dawes reported that an observer could split equal double stars down to about 4.56 arcseconds per inch of aperture, which is 115.8/D with D in millimetres. Dawes’ figure is about 16 per cent smaller, because practised observers detect an elongation before the dip is as deep as Rayleigh’s definition requires. Both are quoted on this page; neither is a hard wall.

Related calculators

References

OpenStax, College Physics for AP® Courses, section 27.6, Limits of Resolution: The Rayleigh Criterion (read 11 November 2026). Gives the criterion as θ = 1.22λ/D with θ in radians, states that the 1.22 comes from the first minimum of the diffraction pattern of a circular aperture, and defines D as the diameter of the aperture, lens or mirror. Its Example 27.5 works the Hubble Space Telescope’s 2.40 m primary at 550 nm and obtains a minimum resolvable angle of about 2.80×10⁻⁷ rad, which is the 0.0577 arcseconds in the table on this page.
NCERT, Physics Textbook for Class XII, Part II, Chapter 10, Wave Optics (chapter text read 11 November 2026). Cited for what it does not contain: the rationalised chapter carries no resolving-power formula and no 1.22λ/D. Its only statement on the subject is the qualitative remark in section 10.6 that the finite resolution of the eye and of instruments such as telescopes and microscopes is due to diffraction, and a Points to Ponder note that the limit is set by the wavelength. A CBSE candidate should check the current syllabus before assuming this formula is examinable. The copy read was a third-party mirror of the official PDF, not ncert.nic.in itself.
University of Michigan Lowbrow Astronomers, Telescope Formulas (D. Scobel, 2003; read 11 November 2026). An observers’ club page hosted by the university, cited for the practical side: it states resolving power as nominally proportional to aperture diameter, and notes that slightly imperfect optics can actually help when splitting close double stars. It gives no numerical coefficient for the resolution limit.
Dawes’ limit could not be sourced to its original publication for this page. The figure quoted here — 4.56 arcseconds per inch of aperture, which is 115.8/D with D in millimetres — is the value used throughout the observing literature and is attributed to W. R. Dawes in 1867, but no primary document was read while writing this page. Treat it as a well-established observers’ convention rather than a measured constant.