Telescope Focal Ratio Calculator

Background.

The focal ratio is one division — focal length divided by aperture — but almost everything an astrophotographer decides follows from it, and one of the most repeated claims about it is only half true. This calculator gives you the f-number in all three directions (solve for the ratio, for the focal length or for the aperture), then turns it into the numbers you actually plan a rig with: arcseconds per pixel, the size of the diffraction disc in micrometres, how many pixels span the star image, and exactly how many stops a focal reducer gained you.

Start with the caveat, because it belongs beside the number and not in a collapsed FAQ. A smaller focal ratio genuinely shortens the exposure for an extended object — a nebula, a galaxy, or the sky background — because the same collected light is concentrated onto fewer square millimetres of sensor. It does not make a star brighter. A star is a point source, and the total number of photons it delivers depends on the aperture and on nothing else. Bolt a 0.63× reducer onto a 200 mm telescope and the nebulosity speeds up by 1.33 stops while every star delivers exactly the same photons as before, merely repacked into fewer pixels. That is why this page prints the aperture inside the verdict every single time: 'f/4 is faster than f/8' is a true statement about nebulosity and a false one about point-source signal-to-noise at fixed aperture.

The plate scale is the bridge between optics and sensor. An object of angular size θ maps to a height f·θ in the focal plane, so a millimetre of focal plane covers 206264.806 divided by the focal length in millimetres, in arcseconds. Multiply by the pixel pitch and you have arcseconds per pixel, which is the number that determines whether your images will look sharp, blocky or merely bloated.

That is where the sampling verdict comes in, and where the reducer's bill arrives. Shannon's sampling theorem requires at least two samples across the narrowest feature you want to record; for imaging that means at least two pixels across the star image. From the ground the star image is set by the seeing, not by diffraction, so it is the seeing disc your pixels must sample. The default configuration on this page — a 1200 mm f/6 telescope with 3.76 micrometre pixels under two-arcsecond seeing — puts 3.09 pixels across the seeing disc, comfortably above the floor. Add the 0.63× reducer and the same rig drops to 1.95 pixels, below the Nyquist floor, and the detail lost there cannot be recovered by any amount of processing. The reducer bought 1.33 stops on the nebula and quietly sold the resolution to pay for them. Whether that is a good trade depends entirely on the target, which is a judgement the calculator leaves to you — but it will not let you make it unknowingly.

Units and conventions are declared throughout. Focal length and aperture are in millimetres, pixel pitch and the Airy disc in micrometres, wavelength in nanometres, seeing and both plate scales in arcseconds; focal ratios, factors and pixel counts are dimensionless; and the stops output is a base-2 logarithm of a flux ratio, signed so that positive means faster. One radian is 648000/π arcseconds, computed rather than typed. The plate scale uses the linear small-angle mapping that is standard for telescopes and is accurate to better than a tenth of a percent over any field a telescope images. The Airy disc assumes an unobstructed circular aperture and perfect optics; the sampling verdict assumes a seeing-limited star image, which is true for essentially every ground-based amateur system but not above the atmosphere or behind adaptive optics, where the Airy-disc pixel count the page also prints becomes the relevant one. Three significant figures is the honest precision: reducer factors are nominal and shift with back-focus spacing, and seeing changes minute to minute. Every input divides something somewhere, so any value of zero or less is rejected against the field that caused it.

What is telescope focal ratio calculator?

The focal ratio, written f/6 or f/6.0 and often called the f-number, is a telescope's focal length divided by its aperture. It is dimensionless: a 1200 mm focal length on a 200 mm aperture is f/6 whether you measure in millimetres or inches.

It is called the 'speed' of the system because, for an extended object, the surface brightness of the image falls as the square of the f-number. Going from f/8 to f/4 quarters the exposure needed for a nebula. It does nothing of the kind for a star, whose light is concentrated into a diffraction disc whose total flux is fixed by the aperture.

A focal reducer multiplies the effective focal length by a factor below one and so lowers the effective f-number; a Barlow or focal extender multiplies it above one and raises it. Both change the plate scale — arcseconds per millimetre in the focal plane — which is what converts an optical system into arcseconds per pixel for a given camera.

How to use this calculator.

1. 01Choose what you are solving for: the focal ratio itself, or the focal length or aperture that would reach a target ratio.
2. 02Enter the telescope's native focal length and aperture in millimetres.
3. 03Set the reducer or Barlow factor — 1 if you are using neither, 0.63 or 0.8 for a typical reducer, 2 for a Barlow.
4. 04Enter your camera's pixel pitch in micrometres; it is in the sensor specification.
5. 05Enter the seeing you typically get. Two arcseconds is an ordinary night; one arcsecond is excellent.
6. 06Read the image scale in arcseconds per pixel — this is the number to compare against other rigs.
7. 07Read the sampling verdict before buying a reducer. It will tell you if the speed gain costs you resolution you cannot get back.

The formula.

An object of angular size θ radians forms an image of height f·θ in the focal plane, so one millimetre of focal plane spans 1/f radians, which is 206264.806247/f arcseconds with f in millimetres. Multiply by the pixel pitch expressed in millimetres and you have arcseconds per pixel. The reducer or Barlow enters by multiplying f, which is why it changes the plate scale and the effective focal ratio together.

Work through the default configuration. A 1200 mm focal length on a 200 mm aperture is f/6. With no reducer the effective focal length stays 1200 mm, so the plate scale is 206264.806247 ⁄ 1200 = 171.8873385392 arcseconds per millimetre, and 3.76 micrometre pixels give 0.6462963929 arcseconds per pixel. The Airy disc is 2.44 × 0.55 µm × 6 = 8.052 micrometres, spanning 2.1414893617 pixels. The two-arcsecond seeing disc spans 2 ⁄ 0.6462963929 = 3.09 pixels, so the rig is slightly oversampled — nothing is lost, but the same photons are divided among more pixels. With no reducer the stops output is exactly zero.

Now fit the 0.63× reducer. The effective focal length becomes 756 mm and the effective ratio f/3.78. The plate scale widens to 272.8370453004 arcseconds per millimetre and 1.0258672903 arcseconds per pixel. The Airy disc shrinks to 5.07276 micrometres, spanning only 1.3491382979 pixels. And the seeing disc now spans 2 ⁄ 1.0258672903 = 1.95 pixels — below the Nyquist floor of two. The exposure gain is 2 log₂(6 ⁄ 3.78) = 1.3331525325 stops on extended objects. So the reducer bought 1.33 stops of nebulosity and sold enough sampling to drop below the theoretical floor. A 2× Barlow does the mirror image: the ratio doubles to f/12 and the output reads exactly −2 stops, because doubling the f-number quarters the surface brightness.

ROUNDING STAGE. Nothing is rounded part-way through; all arithmetic runs at forty significant digits and the single rounding is at the return boundary, to ten decimal places. Figures inside the verdict sentence are display roundings of those same values.

SIGNIFICANT FIGURES. Three. Reducer factors are nominal and shift with back-focus spacing, seeing is a fluctuating statistic, and manufacturers' focal lengths are round numbers.

APPROXIMATION REGIME. The plate scale uses the linear mapping y = f·θ, which is the convention for telescope plate scales and is accurate to better than 0.1 % over any field a telescope images. The Airy disc assumes an unobstructed circular aperture and perfect optics; a central obstruction moves light into the rings. The sampling verdict assumes the star image is seeing-limited, which fails in space or behind adaptive optics — in those cases read the Airy-disc pixel count instead.

SAMPLING THRESHOLDS, AND WHICH ONE IS A THEOREM. The floor of two pixels per resolution element is Shannon's sampling theorem. The upper bound of three is a practical convention that leaves margin for a non-Gaussian star profile and for dithering; it is not a theorem, and this page labels it as a convention rather than dressing it up as one.

INVALID DOMAIN. Focal length, aperture, target ratio, reducer factor, pixel size, wavelength and seeing must all be strictly positive; every one of them divides something. Each zero or negative value raises a labelled error against its own field rather than returning NaN or Infinity.

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: Focal Ratio (f/#) = 'Objective focal length / objective diameter'. Open access. Retrieved 2026-07-29.
2. [2]Shannon, C.E. (1949). 'Communication in the presence of noise.' Proceedings of the IRE 37(1):10–21 (reprinted in Proceedings of the IEEE 86(2):447–457, 1998). Source of the two-samples-per-resolution-element floor used as the undersampling threshold. The three-sample upper bound used on this page is a practical convention and is labelled as such, not attributed to Shannon. Peer-reviewed; the IEEE reprint is paywalled and its landing page returns no content to automated retrieval, so this is shipped as a bibliographic reference rather than a link that reads as dead.
3. [3]Born, M. & Wolf, E. (1999). Principles of Optics, 7th (expanded) edition, §8.5.2, Cambridge University Press. The Airy pattern and the 1.22 coefficient that doubles to the 2.44 used for the disc diameter. Print / bibliographic reference; the publisher's page refuses automated retrieval, so no URL is shipped.
4. [4]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 1.22 coefficient. Print / bibliographic reference.