Image scale calculator: arcseconds per pixel

The formula

A pixel of size p at the focal plane of a telescope with focal length f subtends an angle of p ÷ f radians, the plate-scale relation given in standard telescope optics course notes. One radian is 206,264.8 arcseconds (180 ÷ π × 3600). With the pixel size in micrometres and the focal length in millimetres, the unit conversion moves the decimal point three places:

image scale (″/px) = 206.265 × pixel size (µm) ÷ focal length (mm)

The approximation p ÷ f is excellent at this scale: across a single pixel, the angle is millions of times smaller than a radian, so the difference from the exact arctangent is far below anything a camera can measure.

Examples with real cameras and telescopes

Camera pixel	Telescope	Focal length	Image scale
3.76 µm (IMX571)	William Optics Cat 51	250 mm	3.10″
3.76 µm (IMX571)	Askar FRA400	400 mm	1.94″
3.76 µm (IMX571)	Sky-Watcher Esprit 100EDX	550 mm	1.41″
3.76 µm (IMX533)	Celestron C8 with f/6.3 reducer	1280 mm	0.61″
3.76 µm (IMX533)	Celestron EdgeHD 8	2032 mm	0.38″
2.9 µm (IMX585)	Celestron EdgeHD 8	2032 mm	0.29″
3.75 µm (ASI120MM Mini)	120 mm guide scope	120 mm	6.45″

The same 3.76 µm pixel spans about eight times the angle through a 250 mm Petzval as through a 2032 mm SCT, which is about 66 times the area of sky. Whether either is a good match depends on the sky, not on the camera alone; the sampling calculator makes that comparison with your seeing.

Binning

Binning combines neighbouring pixels into one: 2×2 binning turns four 3.76 µm pixels into one 7.52 µm pixel, doubling the image scale. The calculator lists 1×1 to 4×4. On many CMOS cameras binning is done in software after readout, which keeps the image scale arithmetic the same while the noise behaviour differs from the hardware binning of CCDs; the gain and read noise guide covers that difference.

Reducers, Barlows and colour cameras

A reducer or Barlow changes the effective focal length, so the image scale changes by the inverse of the factor: a 0.8× reducer makes the arcseconds per pixel 25 % larger, so each pixel covers about 56 % more area of sky. Use the factor box, or pick an official reducer from the list when the telescope has one in our data.

For one-shot colour cameras the calculator reports the scale of the individual photosites. Each colour channel is sampled by only some of them (a Bayer filter places red, green and blue over different pixels), so the effective resolution per colour is coarser than the number suggests; the mono vs colour guide explains how much.

The aperture’s own limit

The calculator also shows the telescope’s resolution limit for its aperture: Dawes’ empirical limit, which Dawes gave in 1867 as 4.56″ divided by the aperture in inches (116 ÷ D in mm), and the Rayleigh criterion, 1.22 λ ÷ D, about 138 ÷ D(mm) at 550 nm. For an 80 mm refractor these are 1.45″ and 1.73″; for a 203 mm SCT, 0.57″ and 0.68″. On long-exposure deep-sky images the atmosphere usually blurs stars more than this, which is why seeing rather than aperture sets the useful image scale for most imagers. The resolution limits guide explains both criteria and where they come from.