Telescope resolution limits: Dawes, Rayleigh and why seeing usually wins

Dawes’ limit: an observer’s rule from 1867

The Rev. W. R. Dawes measured double stars for decades. In the introduction to his 1867 catalogue he reported a constant he had settled on about 35 years earlier by comparing telescopes of very different apertures: a 1-inch aperture just separates two 6th-magnitude stars whose centers are 4.56″ apart, with the atmosphere “moderately favourable”. The separating power of any aperture is that constant divided by the aperture in inches. Dawes printed a table from 1 to 30 inches; 8 inches gives 0.57″. He also noted that star brightness changes the result less than one might expect, because brighter pairs bear higher magnification.

In metric units the constant becomes 4.56 × 25.4 = 115.8 (computed). Tele Vue’s formula table lists the same inch form.

Dawes limit (″) = 4.56 ÷ aperture (inches) = 115.8 ÷ aperture (mm)

Rayleigh’s criterion: a diffraction definition

A telescope images a star as an Airy pattern: a central disc surrounded by faint rings. Sheffield’s telescope optics notes state Rayleigh’s criterion: two point sources count as just resolved when the peak of one star’s Airy pattern sits on the first dark ring of the other’s, leaving a 20 % dip in brightness between the two peaks. The angle to that first minimum is 1.22 λ/D radians. Multiplying by 206,265 arcseconds per radian and taking λ = 550 nm gives the metric form (computed):

Rayleigh limit (″) = 1.22 × λ ÷ D × 206,265 ≈ 138.4 ÷ D (mm) at 550 nm

The course’s worked example uses a 0.2 m amateur telescope at 550 nm and gets 0.69″; 138.4 ÷ 200 = 0.692″ agrees.

Why Dawes is tighter than Rayleigh

The two rules share the 1/D form and differ only in the constant. Setting 115.8 ÷ D equal to 1.22 λ ÷ D × 206,265 gives λ ≈ 460 nm (computed). In other words, Dawes’ visual limit equals the Rayleigh criterion evaluated in blue light; at 550 nm Dawes is about 16 % tighter. They measure different things. Rayleigh’s is a convention tied to a specific dip in brightness. Dawes’ is what one experienced observer could split, for equal 6th-magnitude pairs. Neither is a hard physical wall, and the image scale calculator shows both next to your pixel scale.

Values for apertures in our data

Computed from published apertures:

Telescope	Aperture	Dawes	Rayleigh (550 nm)
William Optics Cat 51	51 mm	2.27″	2.71″
Askar FRA400	72 mm	1.61″	1.92″
Sky-Watcher Esprit 100ED	100 mm	1.16″	1.38″
Askar 130PHQ	130 mm	0.89″	1.06″
Celestron C8 / EdgeHD 8	203.2 mm	0.57″	0.68″
Celestron RASA 11	279 mm	0.42″	0.50″
Celestron C14	355.5 mm	0.33″	0.39″

Both formulas describe a clear circular aperture. Several of these telescopes have a central obstruction, which the Sheffield derivation does not cover, so treat the obstructed rows as the textbook value only.

Seeing usually sets the limit in long exposures

The atmosphere spreads each star into a seeing disk. The AAVSO photometry guide calls it significantly larger than the Airy disk and puts typical seeing at 3–4″ in most locations. Diffraction Limited gives 2.5–3.5″ for non-alpine sites, with 2″ seen only on exceptional nights. Professional sites are in another class: ESO reports a median of 0.72″ at Paranal for 2016–2023.

Turning the Dawes formula around shows the aperture at which the telescope stops being the limit (computed):

Seeing FWHM	Aperture where Dawes limit equals the seeing
3″	39 mm
2.5″	46 mm
0.72″ (Paranal median)	161 mm

The smallest telescope in our data has a 50 mm aperture. Under 2.5″ seeing, every one of them has a Dawes limit finer than the seeing disk, so for long-exposure deep-sky images more aperture brings more light, not finer stars. The exception is work that captures moments of steadier air. Dawes’ own limit is visual, and for planetary imaging a ZWO staff member describes exposures of 20–30 ms used to freeze the seeing.

What this means for image scale

Projected onto the sensor, the Airy disc has a size that depends only on the focal ratio N: its first-minimum radius is 1.22 λ N (derived from 1.22 λ/D times the focal length). At 550 nm (computed):

• EdgeHD 8 at f/10: 6.71 µm, about 1.8 times a 3.76 µm pixel.
• Cat 51 at f/4.9: 3.29 µm, slightly smaller than a 3.76 µm pixel.
• RASA 8 at f/2: 1.34 µm, smaller than the smallest pixel in our camera data (1.45 µm, IMX715).

For deep-sky work, size the pixel to the seeing rather than to the aperture. With 3.76 µm pixels the EdgeHD 8 gives 0.38″ per pixel, finer than its own 0.57″ Dawes limit and about 6.5 pixels across a 2.5″ star; the Cat 51 gives 3.10″, coarser than both its Dawes limit and the seeing. The sampling calculator and the sampling and seeing guide take it from there, and the eyepiece calculator covers the visual side for the same apertures. The EdgeHD and SCT apertures are on the Celestron SCT page.