Understand the relationship
The reasoning behind the result
A circular unobstructed aperture diffracts light
θR≈1.22λ/D
For an ideal circular clear aperture of diameter D, the central diffraction maximum extends to its first minimum at approximately 1.22 times wavelength λ divided by diameter, in radians. The Rayleigh criterion places the centre of one ideal point-source pattern at the first minimum of the other. The familiar 1.22 coefficient is an approximation used consistently here.
Wavelength and aperture must share a length unit. Nanometres are converted to millimetres before division; radians are converted to arcseconds using 180×3,600/π. The supported paraxial domain requires aperture to be at least 100 wavelengths across; this is a modeling boundary, not an instrument recommendation. Longer wavelength gives a larger diffraction angle for a fixed aperture, while increasing aperture reduces it.
Dawes is an empirical comparison, not the same criterion
θD≈116/Dmm arcseconds
The conventional Dawes value uses aperture in millimetres and returns arcseconds. It comes from empirical close double-star resolution under favorable visual conditions, rather than specifying the first diffraction minimum at an arbitrary wavelength.
It is therefore shown as a separate wavelength-independent convention and never substituted into the wavelength-specific Rayleigh inverse. Neither value predicts whether a particular unequal, low-contrast or extended target will be resolved. Magnification changes apparent size, not the aperture's underlying diffraction scale.
Angular radius and focal-plane diameter have different units
rAiry≈FθR; dAiry≈2FθR=2.44λ(F/D)
Multiplying the small diffraction angle by effective focal length F gives the central Airy-pattern radius at the focal plane. Doubling gives the diameter between the first minima. The output labels this as a diameter, and converts the focal-plane millimetres to micrometres.
At fixed wavelength and f-number F/D, this ideal linear diameter stays the same even if aperture and focal length both change. Angular resolution does change because the larger focal length maps the same physical spot to a smaller sky angle. The result is not a seeing FWHM, pixel pitch or complete point-spread-function prediction.
An inverse target describes a model requirement
Drequired≈1.22λ/θtarget
A target angle and wavelength determine the clear-aperture diameter for that Rayleigh criterion. The chosen effective focal length then determines the f-number and physical Airy size for the resulting ideal geometry.
The required diameter is not a recommendation or a guarantee of observed performance. Aberrations, central obstruction, atmosphere, alignment, stability, focus, detector response and target structure require additional evidence. No actual observing-resolution limit is inferred by taking a maximum or minimum of unrelated estimates.
Follow the numbers
Aperture and wavelength in one consistent unit system
- At 550 nm, wavelength is 0.00055 mm. For a 200 mm clear aperture, θR≈1.22×0.00055/200=0.000003355 radians, about 0.692018425 arcseconds.
- The separate Dawes value is 116/200=0.58 arcseconds. It is not a second estimate of the same wavelength-specific first-minimum angle.
- At 1,000 mm effective focal length the Airy radius is 1,000×0.000003355=0.003355 mm. The diameter is 0.00671 mm=6.71 µm. Doubling wavelength doubles the Rayleigh angle and Airy diameter in this model.
The angular and focal-plane results reconcile without being interchangeable measures of real image quality.
Quick guide
How to use this calculator
- Choose known aperture with focal length or f-number, or an inverse Rayleigh-angle target.
- Enter the observing wavelength explicitly in nanometres. Apertures and focal lengths use millimetres.
- Compare angular quantities in arcseconds separately from focal-plane spot diameter in micrometres.
- Read the ideal-model boundary. These estimates do not include atmospheric seeing, focus, obstruction, optical quality, target contrast or a detector's sampling.
Calculation method
Calculation and interpretation
Keep angular diffraction, empirical double-star separation and focal-plane spot diameter distinct.
θRayleigh≈1.22λ/D radians; θDawes≈116/Dmm arcseconds; Airy diameter≈2.44λ·(F/D); Dtarget≈1.22λ/θtarget.
Worked example
Aperture and wavelength in one consistent unit system
The angular and focal-plane results reconcile without being interchangeable measures of real image quality.
θRayleigh≈1.22λ/D radians; θDawes≈116/Dmm arcseconds; Airy diameter≈2.44λ·(F/D); Dtarget≈1.22λ/θtarget.
Supported inputs
Precision and limits
Ideal, paraxial, unobstructed optics
The Rayleigh and Airy formulas assume a clear circular aperture with diameter much larger than wavelength. They omit obstruction, aberrations, atmospheric seeing, focus error, motion and the detailed point-spread function.
No actual resolving-performance guarantee
Dawes and Rayleigh are different criteria. The outputs do not establish visibility of a target, detector sampling adequacy, useful magnification or a certified optical design.
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