Telescope Resolution Calculator (Dawes & Rayleigh)

Background.

This calculator gives the two numbers double-star observers argue about — the Rayleigh criterion and the Dawes limit — side by side for any aperture, and then answers the question behind them: what aperture would I actually need to split this pair? It also reports the size of the Airy disc both as an angle on the sky and as a physical diameter in the focal plane, and the aperture at which the night's seeing has already taken over from the optics.

The two criteria are not two versions of the same calculation, and confusing them is the most common error on this topic. The Rayleigh criterion is diffraction physics. Light passing a circular aperture forms an Airy pattern whose first dark ring lies at an angle of 1.22 λ/D radians, and Rayleigh's convention declares two equally bright points resolved when the peak of one sits on that first zero of the other. It is a theoretical statement about an ideal, unobstructed aperture, and it involves the wavelength, so it changes if you observe in red rather than green.

The Dawes limit is not diffraction at all. William Rutter Dawes published it in 1867 as an empirical fit to his own visual splits of double stars — roughly equal pairs near sixth magnitude — and it comes out tighter than Rayleigh, at 4.56 arcseconds divided by the aperture in inches. It is tighter because a trained eye registers a notch or an elongation in a blended blob long before the two peaks are cleanly separated. It carries no wavelength, so it does not respond to a filter, and it does not generalise: a pair of very unequal brightness, or fine planetary detail, is a different problem that Dawes never measured.

One consequence of holding both criteria at once is worth stating because it is checkable. If you set the two expressions equal to one another the aperture cancels, and what is left is a wavelength — about 460 nanometres. That is to say Dawes' empirical rule behaves like a Rayleigh criterion in blue light rather than green, which is consistent with the way a dark-adapted eye's sensitivity shifts towards the blue at low light levels. Neither Rayleigh nor Dawes states that number; it falls out of holding both, and this page's test suite asserts it, because an error in either coefficient would move it.

Units and conventions are fixed. Aperture and focal length are in millimetres, wavelength in nanometres, every angle in arcseconds, and the Airy disc's physical diameter in micrometres. One radian is 648000/π arcseconds, computed rather than typed in. The inch used for the Dawes conversion is the international inch of exactly 25.4 millimetres, so 4.56 arcseconds per inch is 115.824 arcseconds per millimetre of aperture — most handbooks round that to 116, which is a fifth of a percent adrift.

The model assumes scalar diffraction from a perfect, unobstructed circular aperture with no atmosphere. All three assumptions are optimistic. A Newtonian's secondary mirror pushes energy out of the central disc into the rings, which slightly helps a bare split while hurting contrast on planets. Optical figure, collimation and tube currents all cost more than they should. And above everything sits the atmosphere: on a two-arcsecond night the seeing alone blurs anything above about a seventy-millimetre aperture down to that figure, which is why the calculator reports the crossover aperture rather than leaving you to assume that a bigger telescope always resolves finer. Treat three significant figures as the honest precision — Dawes' 4.56 is itself a two-decimal empirical fit, and seeing is a fluctuating statistic rather than a constant. Any input of zero or less is a genuine singularity and is rejected against the field that caused it.

What is telescope resolution calculator?

Resolution, or resolving power, is the smallest angle between two point sources at which a telescope still shows them as two rather than one. It is measured in arcseconds — one arcsecond is 1/3600 of a degree, roughly the angle a one-euro coin subtends at five kilometres.

Resolution is set by the aperture and by nothing else in the telescope. Focal length, magnification and eyepiece quality change how comfortably you can see the detail; they cannot create detail the aperture has not collected. This is why a 60 mm refractor advertised at 500× shows no more than a 60 mm refractor at 120×.

The Rayleigh criterion, θ = 1.22 λ/D, is the diffraction limit of an ideal circular aperture. The Dawes limit, 4.56″/D(inches), is an empirical result from visual double-star work and comes out about 16 % tighter. The Airy disc is the bright central blob of the diffraction pattern, whose diameter out to the first dark ring is exactly twice the Rayleigh angle.

How to use this calculator.

1. 01Enter the aperture in millimetres — the clear diameter of the objective lens or primary mirror.
2. 02Enter the focal length. It is used only to size the Airy disc in the focal plane, through the focal ratio.
3. 03Leave the wavelength at 550 nm for ordinary visual work, or enter your filter's centre wavelength.
4. 04Enter the separation of the pair you are trying to split, in arcseconds.
5. 05Enter the seeing you expect. Two arcseconds is an ordinary suburban night; one arcsecond is a good night at a good site.
6. 06Compare the two limits: Dawes is the optimistic answer for a roughly equal pair, Rayleigh the conservative one.
7. 07Read the verdict, and check the 'seeing takes over above' figure before blaming the telescope for a failed split.

The formula.

The Airy pattern of a circular aperture has its first zero where the Bessel function J₁ first vanishes, at 3.831705970. Dividing by π gives the coefficient 1.2196699, which Rayleigh and every observing handbook since have rounded to 1.22. This page uses 1.22 as published; the rounding makes the answer 0.027 % larger than the exact Bessel value, which is negligible against the one-to-two-percent uncertainty in a manufacturer's stated aperture, but it is a rounding and it is recorded rather than hidden.

Work through the default configuration. A 200 mm aperture at 550 nm gives θ = 1.22 × 550 × 10⁻⁹ ⁄ 0.200 = 3.355 × 10⁻⁶ radians. Multiplying by 206264.806247 arcseconds per radian gives 0.6920184250 arcseconds — the familiar shorthand 138.4⁄D(mm). Dawes on the same aperture is 4.56 × 25.4 ⁄ 200 = 0.57912 arcseconds exactly, about 16 % tighter. The Airy disc is twice the Rayleigh angle, 1.3840368499 arcseconds, which at f/6 corresponds to 2.44 × 0.55 µm × 6 = 8.052 micrometres in the focal plane. Inverting for a one-arcsecond pair: 138.4036849918 mm on Rayleigh, or 115.824 mm on Dawes. And a two-arcsecond night blurs anything above 69.2018424959 mm of aperture down to that same two arcseconds.

ROUNDING STAGE. Nothing is rounded part-way through. All arithmetic runs at forty significant digits, and rounding happens once, at the return boundary, to ten decimal places. The figures quoted inside the verdict sentence are rounded to two or three decimals for readability, from the same unrounded values.

SIGNIFICANT FIGURES. Three at most, and often fewer. Dawes' 4.56 is a two-decimal empirical fit from visual observation; seeing is a statistic that changes minute to minute; and manufacturers' apertures are nominal.

APPROXIMATION REGIME AND WHERE IT BREAKS. Scalar Fraunhofer diffraction, unobstructed circular aperture, perfect figure, no atmosphere. A central obstruction redistributes light from the disc into the rings — marginally helping a bare split, measurably hurting planetary contrast. Unequal pairs are outside Dawes' calibration entirely: a magnitude-2 primary with a magnitude-9 companion may be unsplittable at ten times the Dawes separation. And the atmosphere normally dominates: the crossover aperture output exists precisely so that this limitation sits beside the number rather than in a footnote.

INVALID DOMAIN. Aperture, focal length, wavelength, target separation and seeing must all be strictly positive; each of them divides something. A zero aperture, a zero target separation or a zero seeing would each produce an infinity, so the calculator raises a labelled error against that field instead.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Tele Vue Optics, Inc., 'Telescope Formulas'. Manufacturer engineering note, verified by retrieval to state verbatim: Dawes Limit (Resolving Power) = '4.56 arc seconds / Objective diameter in inches', and Focal Ratio = 'Objective focal length / objective diameter'. This is the independent second authority used to check the Dawes coefficient. Open access. Retrieved 2026-07-29.
2. [2]Rayleigh, Lord (J. W. Strutt) (1879). 'Investigations in optics, with special reference to the spectroscope.' Philosophical Magazine, Series 5, 8(49), 261–274. Origin of the resolution criterion implemented here. Print / bibliographic reference — no authoritative free full text is offered rather than linking a scan of uncertain provenance.
3. [3]Dawes, W. R. (1867). 'Catalogue of Micrometrical Measurements of Double Stars.' Monthly Notices of the Royal Astronomical Society 27. Origin of the empirical 4.56″ separating-power rule. Secondary sources cite this 1867 catalogue at both p. 158 and p. 217; both are recorded here rather than one chosen silently. Print / bibliographic reference; the ADS record is JavaScript-rendered and the article scan refuses automated retrieval, so no URL is shipped rather than one that reads as dead.
4. [4]Born, M. & Wolf, E. (1999). Principles of Optics, 7th (expanded) edition, §8.5.2 (the Airy pattern; first zero of 2J₁(v)/v at v = 3.8317) and §8.6.2 (resolving power of image-forming systems), Cambridge University Press. Source of the exact 1.2196699 coefficient that 1.22 rounds. Print / bibliographic reference; the publisher's page refuses automated retrieval, so no URL is shipped.
5. [5]National Institute of Standards and Technology, Special Publication 811 (2008 ed.), Appendix B.8 'Factors for Units Listed Alphabetically': inch (in) → metre, 2.54 E-02, printed in boldface, which the table's legend defines as exact. Used for the 4.56″/inch → 115.824″/mm conversion. Open access. Retrieved 2026-07-29.