Engineering
Telescope Rayleigh Diffraction Limit, Dawes' Limit & Focal Ratio formulas
Calculate Airy-disk angular resolution in arcseconds theta = DEGREES(1.22 * lambda / D) * 3600, Dawes' empirical double-star limit 116 / D_mm, f-number, and magnification.
Ideal aperture resolution and visual magnification
For an ideal unobstructed circular aperture, the small-angle Rayleigh criterion is approximately 1.22λ/D radians. The rounded coefficient and small-angle approximation do not give exact real-world resolution. Here wavelength is entered in nanometres and diameter in millimetres, so multiply wavelength by 10^-6 before dividing.
Dawes’ 116/Dmm estimate is an empirical visual double-star criterion. It does not incorporate wavelength as an input and is not interchangeable with a general imaging resolution guarantee. Atmospheric seeing, aberrations, obstruction, alignment, sampling and target contrast affect practical results.
Focal ratio is objective focal length divided by aperture. Visual magnification is objective focal length divided by eyepiece focal length, assuming no extra optics. Higher magnification does not itself improve the diffraction limit. The 2,400-mm column is an ideal aperture calculation, not a prediction of Hubble instrument performance.
| Optical Instrument Metric | Formula | 200 mm f/10 Scope (550 nm, 25 mm Eyepiece) | Ideal 2,400-mm aperture at 550nm |
|---|---|---|---|
| Rayleigh Limit (arcsec) | DEGREES(1.22 * (B2*1e-6 / B3)) * 3600 | 0.692 arcseconds | 0.058 arcsec (ideal model) |
| Dawes' Binary Limit (arcsec) | 116 / B3 | 0.580 arcseconds | 0.048 arcseconds |
| Focal Ratio & Magnification | B4 / B3 & B4 / B5 | f/10 | 80x magnification | Requires focal lengths |
Rayleigh Diffraction-Limited Angular Resolution (Arcseconds)
Multiplies wavelength B2 (nm) by 1e-6 to convert to mm, divides by aperture B3 (mm), and converts radians to arcseconds via DEGREES(...) * 3600.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(MIN(B2:B5)<=0,"Positive dimensions required",ROUND(DEGREES(1.22 * ((B2 * 0.000001) / B3)) * 3600, 3)))Dawes’ empirical double-star limit
Estimate angular separation in arcseconds as 116 divided by aperture in millimetres. This formula does not calculate magnification.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(MIN(B2:B5)<=0,"Positive dimensions required",ROUND(116 / B3, 3)))Focal ratio
Objective focal length and aperture must both be in millimetres.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(MIN(B2:B5)<=0,"Positive dimensions required",B4/B3))Visual magnification
Objective focal length divided by eyepiece focal length, both in millimetres.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(MIN(B2:B5)<=0,"Positive dimensions required",B4 / B5))Try the example
Use the original values or change them to test your understanding.