The angular resolution calculator handles every common diffraction-limit scenario in one place: the classical Rayleigh criterion for telescopes and cameras, the empirical Dawes limit for visual double-star observing, the Abbe diffraction limit for optical microscopes, pixel scale conversion for astrophotography sensors, linear resolution on a target at a given distance, and the inverse problem of finding the minimum aperture needed for a specified resolution.
Everything runs in your browser — no numbers are sent to a server.
How it works
Angular resolution is set by diffraction, the unavoidable spreading of light around aperture edges. The key formulas are:
Rayleigh criterion (telescopes, cameras, binoculars):
theta = 1.22 × lambda / D
where lambda is the wavelength of light and D is the aperture diameter — both in the
same units. The result is in radians; the calculator converts it automatically to
arcseconds, arcminutes, degrees, and microradians.
Dawes limit (visual telescopes — empirical):
theta_arcsec = 116 / D_mm
A practical rule derived by William Rutter Dawes from observing close double stars. It gives a slightly optimistic figure compared with Rayleigh because it exploits the eye’s ability to detect intensity gradients rather than requiring a fully dark gap between two peaks.
Abbe diffraction limit (microscopes):
d = lambda / (2 × NA)
where NA (numerical aperture) = n × sin(theta_half), n being the refractive index of
the medium between lens and sample. Oil-immersion objectives reach NA ≈ 1.4, pushing
d below 200 nm for visible light.
Pixel scale (astrophotography):
s = (pixel_size_µm / focal_length_mm) × 206.265 [arcsec/px]
The constant 206.265 converts from radians to arcseconds (= 1/tan(1 arcsec) ≈ 1 rad in arcseconds). The Nyquist limit says you need at least 2 pixels per resolution element, so an adequate image scale is s < 0.5 × FWHM of the seeing or diffraction disk.
Linear resolution on a target uses the small-angle approximation:
s = theta_rad × distance
Useful for surveillance optics, aerial reconnaissance, or knowing how large a crater the Hubble Space Telescope could resolve on the Moon.
Worked example
A 100 mm refractor observing at 550 nm (green light):
- Rayleigh limit: theta = 1.22 × 550 × 10⁻⁹ / 0.1 = 6.71 µrad = 1.38 arcsec
- Dawes limit: 116 / 100 = 1.16 arcsec
Paired with a camera whose pixels are 4.63 µm and a focal length of 690 mm:
- Pixel scale: (4.63 / 690) × 206.265 = 1.38 arcsec/px
That is almost perfect Nyquist sampling of the diffraction disk — one pixel per Rayleigh resolution element, meaning you are neither wasting resolution nor over-sampling.
To resolve detail 1 arcsecond wide you need: D = 1.22 × 550 × 10⁻⁹ / 4.848 × 10⁻⁶ = 138 mm aperture minimum.
| Aperture (mm) | Rayleigh @ 550 nm | Dawes limit |
|---|---|---|
| 60 | 2.30 arcsec | 1.93 arcsec |
| 100 | 1.38 arcsec | 1.16 arcsec |
| 200 | 0.69 arcsec | 0.58 arcsec |
| 500 | 0.28 arcsec | 0.23 arcsec |
Formula note
The 1.22 prefactor in the Rayleigh criterion is the first zero of the Bessel function J1(x), divided by pi — it arises from treating the aperture as a circular disk and computing the first dark ring of the Airy diffraction pattern. For a slit aperture (spectrographs, diffraction gratings) the prefactor is 1.0 instead of 1.22. The calculator uses the circular-aperture value throughout.