The angular resolution calculator above applies the Rayleigh criterion to a circular aperture and then does something most versions skip: it compares the answer against the atmospheric seeing you enter and tells you which of the two is actually limiting your image. That second step is not a refinement. For any ground-based telescope larger than a modest amateur instrument it is the whole story, and quoting the diffraction limit alone gives a figure the telescope will never deliver.
Arb Digital builds free tools that state their assumptions rather than burying them. This page covers what the 1.22 factor means, why the Dawes limit differs from the Rayleigh figure, how the same equation applies to camera lenses and radio dishes, and where the whole approach stops being the right model.
What This Angular Resolution Calculator Does
You give it an aperture diameter and a wavelength. It returns the smallest angular separation at which two point sources are still distinguishable, expressed in arcseconds in the headline and in radians in the grid. Arcseconds are the astronomer's unit; radians are what every downstream calculation actually wants.
The grid adds three more figures. The Dawes limit is an older empirical criterion for splitting double stars, which comes out slightly more optimistic than Rayleigh and is still quoted throughout amateur astronomy. The detail figure multiplies the angle by the target distance to give a physical size — the smallest feature you could resolve on a wall, a licence plate or a lunar crater. The final item combines the diffraction limit with the seeing you entered in quadrature, which is the realistic expectation rather than the theoretical ceiling.
The subtitle names which of the two dominates. That single word is often the most useful output on the page, because it tells you whether buying more aperture will improve your resolution or only your light grasp.
How to Use It
- Enter the clear aperture, not the tube diameter. For a telescope it is the diameter of the primary mirror or objective lens. For a camera lens it is focal length divided by f-number, so a 200 mm lens at f/2.8 has a 71 mm aperture.
- Pick a wavelength that matches the light you are collecting. Resolution is directly proportional to wavelength, so an infrared image at 2,200 nm from the same aperture is four times coarser than a visible one at 550 nm.
- Set a target distance if you want a physical size. The detail figure is simply the angle in radians multiplied by the distance, so it works equally for a wall across the street and for the Moon.
- Enter honest seeing. Two arcseconds is a fair suburban night. Half an arcsecond is a genuinely excellent mountain site. Zero disables the correction, which is right for a space telescope or an optical bench.
- Read which limit dominates before drawing conclusions. If seeing is the limiting factor, a larger aperture will not sharpen the image on a normal night.
The Formula: How Angular Resolution Is Calculated
Light passing through a circular aperture diffracts into an Airy pattern — a bright central disc surrounded by faint rings. The first dark ring of that pattern sits at an angle θ = 1.22λ ÷ D radians, where λ is the wavelength and D is the aperture diameter in the same units. The Rayleigh criterion says two point sources are just resolvable when the centre of one Airy disc falls on the first dark ring of the other, which makes that same angle the resolution limit. OpenStax University Physics Volume 3, section 4.5 on circular apertures and resolution, derives the 1.22 factor and states the criterion in exactly this form.
The 1.22 is not arbitrary. It comes from the first zero of the first-order Bessel function that describes diffraction through a circle, at 3.8317, divided by π. A slit rather than a circle gives 1.00 instead, which is why the same telescope resolves differently along a spider vane's diffraction spike than it does radially.
Work through the default. A 200 mm aperture at 550 nm gives θ = 1.22 × 550×10⁻⁹ ÷ 0.2 = 3.355×10⁻⁶ radians. Multiplying by 206,265 arcseconds per radian gives 0.692 arcseconds. At a target 1,000 m away that angle spans 3.4 mm, which is roughly the smallest feature a 200 mm telescope could separate on a distant wall in perfect conditions.
Why Atmospheric Seeing Usually Wins on the Ground
Every ground-based telescope looks through several kilometres of turbulent air. Pockets of slightly different temperature act as weak, constantly shifting lenses, so a point source arrives not as a clean Airy disc but as a blurred, boiling blob a few arcseconds across. That blur is called seeing, and it does not care how good your optics are.
The consequence is a hard threshold. A 150 mm telescope has a Rayleigh limit near 0.92 arcseconds, which is comparable to a good night's seeing, so on a good night the optics are the limit. A 400 mm telescope has a limit near 0.35 arcseconds — far finer than typical seeing of two arcseconds, so on almost every night the atmosphere is the limit and the extra aperture buys light rather than detail. This is the single most important practical fact about telescope resolution, and it is why professional observatories are built on high, dry mountains and why adaptive optics exists at all.
Above the atmosphere the picture changes completely. NASA's page on the Hubble Space Telescope's optics notes that its 2.4 m mirror distinguishes objects 0.05 arcseconds apart in visible light, which is close to the diffraction limit this calculator returns for that aperture. A ground telescope of the same size on an ordinary night does roughly thirty-five times worse.
Rayleigh, Dawes and the Criterion You Choose
The Rayleigh criterion is a convention, not a law of nature. It defines "resolved" as the first minimum of one pattern landing on the peak of the other, which produces a visible dip of about 26 % between the two peaks. That dip is detectable, but it is not the smallest one a trained eye or a good detector can find.
The Dawes limit, 116 divided by the aperture in millimetres, gives arcseconds and is empirical: it is what William Dawes found observers could actually achieve on equal-brightness double stars in the nineteenth century. It comes out roughly 16 % tighter than Rayleigh at visible wavelengths, which is why it is quoted more often in amateur telescope specifications. Neither number is wrong; they answer slightly different questions.
Modern image processing can go tighter still. Deconvolution and lucky imaging routinely recover detail below the Rayleigh limit when the signal-to-noise ratio is high and the point spread function is known, and fluorescence microscopy techniques have gone an order of magnitude past it by exploiting properties the classical analysis does not model. The Rayleigh figure is the right default for planning, not an inviolable wall.
The Same Equation Outside Astronomy
Nothing in the derivation is specific to telescopes. The same 1.22λ/D governs camera lenses, microscopes, radar and the human eye, and the results are worth knowing.
Your eye has a pupil around 3 mm in daylight, giving a diffraction limit near 46 arcseconds at 550 nm. Real visual acuity is closer to 60 arcseconds, limited by the spacing of cone cells rather than by diffraction — a rare case where the sensor rather than the optics sets the limit. A camera lens stopped down to f/16 has a diffraction blur circle large enough to soften a high-resolution sensor visibly, which is the reason stopping down past a point makes images worse rather than better; the camera field of view calculator covers the framing side of the same optics. Radio astronomy suffers most of all, because wavelengths are measured in centimetres rather than nanometres — a 25 m dish at 21 cm has a resolution of about half a degree, worse than the naked eye, which is exactly why radio interferometry was invented.
Where This Sits Next to Our Other Optics Tools
Adjacent pages on this site do different jobs. The wavelength calculator converts between wavelength, frequency and wave speed, which is how you get the input this page needs from a frequency. The diffraction grating calculator handles the interference maxima of a ruled grating, a different geometry with a different equation. The photon energy calculator turns the same wavelength into an energy in joules or electronvolts. For distances to the objects you are resolving, the astronomical distance converter handles astronomical units, light-years and parsecs, and the length converter covers terrestrial units. Everything is indexed on the free online tools hub.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Quoting the diffraction limit for a ground telescope — on most nights atmospheric seeing is several times worse and sets the real limit. Enter a seeing figure and read the combined result.
- Using the tube or filter diameter as the aperture — the equation needs the clear aperture of the objective. For a camera lens that is focal length divided by f-number.
- Mixing units between wavelength and diameter — this tool converts both to metres internally, but if you work the formula by hand, nanometres against millimetres is a factor of a million waiting to happen.
- Confusing angular resolution with magnification — magnification enlarges the blur along with the detail. Beyond a certain point it reveals nothing new and only makes the image dimmer.
- Applying the visible-light figure to infrared work — resolution scales directly with wavelength, so a near-infrared image from the same aperture is several times coarser.
Related Free Tools From Arb Digital
Get the wavelength input from a frequency with the wavelength calculator, or turn it into an energy with the photon energy calculator. For grating rather than aperture diffraction, use the diffraction grating calculator. The camera field of view calculator covers what fits in the frame rather than what can be separated within it. Distances come from the astronomical distance converter for space and the length converter for everything closer. The full set is on the free online tools hub.
Frequently Asked Questions
It is the convention that two point sources are just resolvable when the centre of one diffraction pattern falls on the first dark ring of the other. For a circular aperture that separation is 1.22 times the wavelength divided by the aperture diameter, in radians. It produces roughly a 26 per cent dip in brightness between the two peaks.
Because the aperture is circular. The first zero of the diffraction pattern from a circle occurs at 3.8317 in units where a slit would give pi, and 3.8317 divided by pi is 1.22. A rectangular slit of the same width uses a factor of 1 instead.
For anything above roughly 150 millimetres of aperture, on most nights, yes. Typical seeing of two arcseconds is worse than the diffraction limit of a 150 millimetre telescope, so larger instruments gather more light but do not resolve finer detail unless the site is exceptional or adaptive optics is used.
Rayleigh is a theoretical criterion from diffraction physics. Dawes is an empirical result from nineteenth-century observations of equal-brightness double stars, given by 116 divided by the aperture in millimetres, in arcseconds. Dawes comes out about 16 per cent tighter at visible wavelengths, and both are quoted in telescope specifications.
Multiply by 206,265, which is the number of arcseconds in a radian. It comes from 180 divided by pi to get degrees, then multiplied by 3,600. This calculator reports both units so you do not have to.
Worse. Resolution is directly proportional to wavelength, so doubling the wavelength doubles the smallest resolvable angle. This is why radio telescopes need enormous apertures or interferometry to achieve resolutions that a small optical telescope reaches easily.
Yes, under the right conditions. Deconvolution and lucky imaging recover detail below the Rayleigh figure when the signal is strong and the point spread function is known, and several fluorescence microscopy methods exploit properties the classical analysis does not model. The Rayleigh limit is the right default for planning rather than an absolute wall.
This tool is provided for educational and study use. It applies the idealised Rayleigh criterion for a perfect unobstructed circular aperture, and does not model central obstruction, optical aberrations, detector sampling or adaptive optics, so treat its output as a physics result rather than an instrument specification.
