Angular Resolution Calculator
Enter the aperture diameter and light wavelength to find the diffraction-limited angular resolution of your optical instrument. Choose between the Rayleigh, Dawes, and Sparrow criteria, switch aperture units between millimetres and inches, and get results in arcseconds, arcminutes, and degrees. The calculator also solves in reverse: enter a desired resolution to find the minimum aperture you need. A linear resolution at a chosen distance is included for real-world applications like planetary observation or surveillance.
Formula
Worked example
A 200 mm telescope at 550 nm under the Rayleigh criterion: theta = 1.22 x (550e-9 m) / 0.200 m = 3.355e-6 rad = 0.692 arcsec. At a distance of 1000 m, the linear resolution is 1000 x 3.355e-6 = 3.355 mm.
What is angular resolution?
Angular resolution is the smallest angle between two point sources that an optical system can distinguish as separate objects. Below this limit, two stars, craters, or printed dots merge into a single blurred blob regardless of magnification. It is determined by diffraction, the bending of light waves at the edges of the aperture, and sets an absolute physical ceiling on image sharpness for any lens, mirror, or pupil of a given diameter.
The Rayleigh, Dawes, and Sparrow criteria explained
The Rayleigh criterion (factor 1.22) is the scientific standard: two point sources are just resolved when the central maximum of one Airy pattern coincides with the first minimum of the other. This is where the combined intensity dips to roughly 73.5% between the two peaks. The Dawes limit (factor 0.82) was derived empirically for visual double-star work with high-contrast stellar targets; it gives a slightly tighter limit because the eye can detect the saddle point in the combined profile before it satisfies the Rayleigh condition. The Sparrow criterion (factor 1.00) marks the point at which the combined profile shows no dip at all: the two peaks are just merged but the combined intensity plateau is still narrower than one fully unresolved source. For comparing instruments on a common footing, the Rayleigh criterion is the conventional choice.
How aperture and wavelength determine resolution
The formula theta = k x lambda / D shows that resolution scales directly with wavelength and inversely with aperture diameter. Doubling the aperture halves the minimum resolvable angle. Using a shorter wavelength (blue light at 450 nm vs red at 700 nm) shrinks the diffraction limit by 36%. In practice, this means that a 200 mm telescope at 550 nm can theoretically resolve features separated by about 0.69 arcsec, or roughly 3.4 mm on an object 1 km away. Ground-based telescopes are usually limited by atmospheric turbulence (seeing) to 1-3 arcsec unless adaptive optics are used to correct the wavefront in real time.
Reverse-solving: what aperture do I need?
For engineering or purchasing decisions, it is often more useful to start with a required resolution and work backwards to the aperture. Rearranging the formula gives D = k x lambda / theta. To split double stars separated by 0.5 arcsec at 550 nm using the Rayleigh criterion, you need at least D = 1.22 x 550e-9 / (0.5/206265) = 276 mm of aperture. This is why professional solar and planetary imagers routinely use apertures of 250-400 mm, and why space telescopes need metres of aperture to image the fine structure of distant galaxies.
Practical applications beyond astronomy
Angular resolution is relevant wherever two features must be distinguished through a diffraction-limited aperture. In microscopy, the Abbe diffraction limit uses the numerical aperture (NA) instead of the physical diameter, but the underlying physics is the same. Camera lenses reach their diffraction limit at small f-numbers where the aperture is large, and at small apertures (high f-numbers) diffraction softens the image even if the lens itself is optically perfect. In surveillance and remote sensing, the linear resolution at a known distance (s = r x theta) determines the smallest feature resolvable from a satellite or aircraft. The calculator provides this figure directly when you enter a distance.
Angular resolution benchmarks for common optical instruments
| Instrument | Aperture | Wavelength | Resolution (arcsec) | Resolution class |
|---|---|---|---|---|
| Human eye (bright) | 4 mm | 550 nm | 34.4 | Moderate |
| Human eye (dark-adapted, 7 mm) | 7 mm | 550 nm | 19.7 | Moderate |
| 60 mm refractor | 60 mm | 550 nm | 2.29 | Good |
| 100 mm refractor | 100 mm | 550 nm | 1.38 | Good |
| 200 mm SCT | 200 mm | 550 nm | 0.69 | Very good |
| 400 mm Dobsonian | 400 mm | 550 nm | 0.34 | Very good |
| Hubble Space Telescope | 2400 mm | 500 nm | 0.05 | Excellent |
| Very Large Telescope (VLT, adaptive) | 8200 mm | 550 nm | 0.016 | Excellent |
All values use the Rayleigh criterion (factor 1.22) at 550 nm wavelength. Atmosphere limits ground-based telescopes to ~1-3 arcsec without adaptive optics.
Frequently asked questions
What is the Rayleigh criterion and why is 1.22 used?
The Rayleigh criterion defines the minimum angular separation at which two point sources are just resolved: the central maximum of one Airy diffraction pattern falls on the first dark ring of the other. The factor 1.22 comes from the first zero of the Bessel function J1(x), specifically from 1.22 = 3.832/pi, which is where the circular aperture Airy pattern first goes to zero. For a slit aperture the factor is simply 1.00, but circular optics produce Airy patterns and the 1.22 factor applies.
How is the Dawes limit different from the Rayleigh criterion?
The Dawes limit uses a factor of 0.82 instead of 1.22, making it about 33% stricter. The 19th-century astronomer William Rutter Dawes derived it empirically from visual observation of double stars, where the eye can detect a slight elongation or saddle in the combined star image before the Rayleigh condition is met. It works best with high-contrast targets (bright stars on a dark sky) and experienced observers. For camera or CCD imaging, the Rayleigh criterion is the more appropriate standard.
Why does my telescope not achieve the theoretical diffraction limit?
Several factors limit real telescopes short of the theoretical floor. Atmospheric seeing introduces random wavefront distortions that smear images to typically 1-5 arcsec at sea level and 0.5-1.5 arcsec at good observatory sites. Optical aberrations, mirror or lens figure errors, thermal gradients in the tube, and vibration all degrade resolution further. The diffraction limit is the best possible performance for a perfect optic in vacuum; it is a ceiling, not a guarantee.
What wavelength should I use for the calculation?
For a general estimate use 550 nm (peak of human visual sensitivity and green light). For specific colours: blue is ~450 nm, red is ~650 nm, near-infrared is ~850 nm. If you are calculating resolution for a camera filter, use the centre wavelength of that filter. Shorter wavelengths give better resolution, which is why modern optical lithography uses deep ultraviolet light around 193 nm to print nanometre-scale transistors.
Can I use this calculator for a camera lens or microscope?
For camera lenses: use the physical aperture diameter (focal length divided by f-number) as the aperture input, and the wavelength of light you are imaging. This gives the diffraction-limited resolution at that aperture; stopping down to a smaller aperture (higher f-number) increases diffraction and degrades resolution. For microscopes: use the Abbe formula d = 0.61 lambda / NA instead, where NA is the numerical aperture printed on the objective lens. The angular resolution formula on this page applies to single-element apertures in air.
What is the Airy disk?
When a point source of light is imaged through a circular aperture, diffraction spreads the light into a bright central disk surrounded by fainter rings. This central disk is called the Airy disk, named after the 19th-century astronomer George Biddell Airy. Its angular radius (to the first dark ring) is 1.22 lambda / D, which is identical to the Rayleigh criterion. The calculator shows this value in arcseconds alongside the resolution under your chosen criterion, so you can compare.