About

Angular resolution defines the smallest angular separation at which an optical system can distinguish two point sources. The fundamental physical limit arises from diffraction at the aperture boundary, quantified by the Rayleigh criterion: θ = 1.22 × λ ÷ D. Underestimating this limit leads to wasted magnification, false detail in astrophotography, and incorrect assumptions in microscopy or remote sensing. The calculator also outputs the Dawes empirical limit and the Sparrow criterion for comparison. Results are reported in arcseconds, arcminutes, degrees, milliradians, and radians.

Assumptions: circular, unobstructed aperture with uniform illumination. Central obstruction (e.g., Newtonian secondary mirrors) degrades contrast but does not significantly shift the Rayleigh angle. Atmospheric seeing typically limits ground-based telescopes to ≈ 1″ regardless of aperture. This tool approximates the diffraction-limited case only.

Formulas

The Rayleigh criterion gives the minimum angular separation at which the central maximum of one Airy pattern falls on the first minimum of the other:

θ_Rayleigh = 1.22 × λD

The Dawes limit is an empirical formula derived from observations of equal-brightness double stars. It uses aperture in millimeters and outputs arcseconds directly:

θ_Dawes = 116D mm arcsec

The Sparrow criterion defines the limit where the combined intensity pattern of two point sources shows no dip between them:

θ_Sparrow = 0.95 × λD

The Airy disk linear radius at the focal plane (requires focal length):

r_Airy = 1.22 × λ × fD

Where: θ = angular resolution (radians), λ = wavelength of light, D = aperture diameter, f = focal length, r_Airy = linear radius of first Airy disk minimum. The constant 1.22 is the first zero of J₁(x) ÷ x, where J₁ is the Bessel function of the first kind.

Reference Data

Instrument / Aperture	Aperture D	Rayleigh Limit (550 nm)	Dawes Limit	Typical Use
Human Eye	7 mm	19.8″	16.6″	Naked-eye observation
50 mm Binoculars	50 mm	2.77″	2.32″	Birdwatching, casual astronomy
70 mm Refractor	70 mm	1.98″	1.66″	Beginner telescope
100 mm Refractor	100 mm	1.38″	1.16″	Planetary observation
150 mm Newtonian	150 mm	0.92″	0.77″	Deep sky, double stars
200 mm SCT	200 mm	0.69″	0.58″	Astrophotography
250 mm Dobsonian	250 mm	0.55″	0.46″	Visual deep sky
300 mm Cassegrain	300 mm	0.46″	0.39″	Research-grade amateur
500 mm Ritchey-Chrétien	500 mm	0.28″	0.23″	Professional observatory
1 m Research Telescope	1000 mm	0.14″	0.12″	Professional research
Keck Telescope	10000 mm	0.014″	0.012″	Adaptive optics research
JWST	6500 mm	0.021″ (at 550 nm)	0.018″	Space infrared astronomy
Hubble Space Telescope	2400 mm	0.058″	0.048″	Space optical/UV
Camera 50 mm f/1.8	27.8 mm	4.98″	4.17″	Photography lens
Camera 200 mm f/2.8	71.4 mm	1.94″	1.63″	Telephoto lens
Radar Dish 3 m (10 GHz)	3000 mm	50.8′	-	Radar / microwave
Radio Dish 100 m (1.4 GHz)	100000 mm	15.7′	-	Radio astronomy (21 cm line)

Frequently Asked Questions

The factor 1.22 is the first zero of the function J₁(πx) ÷ (πx), derived from Fraunhofer diffraction through a circular aperture. Specifically, the first zero of the Bessel function J₁(x) occurs at x ≈ 3.8317, and 3.8317 ÷ π ≈ 1.2197. For a rectangular slit, no such factor applies and the limit is simply λ ÷ D.

Atmospheric turbulence limits ground-based resolution to approximately 0.5″ to 2″ at most sites, regardless of aperture. A 200 mm telescope has a diffraction limit of 0.69″, but seeing typically caps performance at 1″ to 3″. Adaptive optics systems can partially correct wavefront distortion and approach the diffraction limit. This calculator outputs the diffraction-limited theoretical value only.

The Dawes limit (116 ÷ D_mm) is empirically calibrated for visual observation of equal-brightness double stars under good seeing. It is wavelength-independent by design (it implicitly assumes green light around 550 nm). Use Rayleigh when working at specific wavelengths, in imaging systems, or outside the visible band. Dawes is approximately 16% tighter than Rayleigh at 550 nm.

A central obstruction (secondary mirror) does not significantly shift the position of the first Airy minimum, so the Rayleigh angle remains nearly unchanged. However, it transfers energy from the central disk into the diffraction rings, reducing contrast. For a typical 30% linear obstruction, the Strehl ratio drops by roughly 8%. The Sparrow limit and contrast-based resolution can degrade measurably, making an obstructed system perform worse than the Rayleigh number suggests.

For small angles, the minimum resolvable linear separation s at distance R is s = R × tan(θ) ≈ R × θ (with θ in radians). For example, a resolution of 1″ (4.848 × 10⁻⁶ rad) at 1 km resolves features as small as 4.85 mm.

Yes. The Rayleigh and Sparrow formulae apply to any circular aperture at any wavelength. Enter the wavelength in the appropriate unit (e.g., 21 cm for the hydrogen line). The Dawes limit is only valid for visible-light, visual double-star observation and should be disregarded for radio frequencies. Note that radio interferometry (e.g., VLBI) uses baselines as the effective aperture D, not the physical dish diameter.