Image scale explained: from s = Fθ to arcseconds per pixel

A pixel is an angle

A camera behind a telescope records directions, not distances. Each point in the focal plane corresponds to one direction on the sky, so each pixel collects light from a small patch of sky with a definite angular width. Image scale is that width, usually quoted in arcseconds per pixel (″/px). One arcsecond is 1/3600 of a degree. For a sense of size, the Moon’s mean apparent diameter of 1896″, from NASA’s Moon Fact Sheet, would span about 978 pixels at 1.94″ per pixel (computed).

From s = Fθ to 206.265

Starlight arrives as parallel rays. The objective brings rays that arrive at an angle θ from the optical axis to a point a distance s from the axis in the focal plane. For small angles, the University of Sheffield’s optics notes write this as:

s = F × θ   (θ in radians, F = focal length)

Rearranged, θ ÷ s = 1 ÷ F: each millimeter of focal plane holds 1/F radians of sky when F is in millimeters. A radian is 180 × 3600 ÷ π = 206,264.8 arcseconds, and the same course’s worked problem turns that into the plate scale:

plate scale (″/mm) = 206,265 ÷ F (mm)

A pixel p micrometers wide is p/1000 mm, which gives the per-pixel form printed in the AAVSO photometry guide and used by the image scale calculator:

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

The small-angle step costs nothing at this size. A pixel subtends a few millionths of a radian, where an angle and its tangent agree far beyond anything a camera can measure. The approximation only starts to matter for whole-field angles with short camera lenses, which the field of view guide covers.

How big an arcsecond is at your focus

Plate scale tells you the physical size of an arcsecond in the focal plane before any camera is involved (all values computed):

Focal length	Plate scale	1″ in the focal plane
250 mm	825″ per mm	1.21 µm
400 mm	516″ per mm	1.94 µm
2032 mm	102″ per mm	9.85 µm

At 400 mm one arcsecond of sky is about half the width of a 3.76 µm pixel. At 2032 mm the same arcsecond is wider than two such pixels. Sheffield’s second problem works the same arithmetic backwards: a 20 cm f/8 telescope (1600 mm, 129″ per mm) has a diffraction limit of 0.69″ at 550 nm, which is 0.005 mm, or about 5 µm, in its focal plane.

Pixel size and focal length trade one for one

The two inputs enter as a ratio. Halving the pixel size or doubling the focal length gives the same image scale, but the pictures differ. A longer focal length shrinks the field of view on the same sensor; smaller pixels keep the field and put more pixels across it. Camera and telescope choice therefore form one decision, and the sampling and seeing guide compares the result with the blur your sky adds.

Focal ratio does not appear in the formula. Two 600 mm telescopes give identical arcseconds per pixel at f/4 and at f/8. The Sheffield notes link focal ratio to how much light lands on a given area of the focal plane, a separate question from how much sky each pixel covers.

Where real setups land

Our camera data covers 42 cooled deep-sky cameras from ZWO, QHYCCD, Player One and ToupTek. Their pixels run from 2.4 µm to 5.94 µm, and 27 of the 42 use 3.76 µm pixels (IMX571, IMX533, IMX455 and IMX461 sensors; the ZWO ASI2600 page is one example). Leaving out visual Dobsonians and Maksutovs, our telescope data has 111 tubes with native focal lengths from 178 mm to 3910 mm and a median of 600 mm.

With a 3.76 µm pixel (all computed, native focal lengths, no reducers):

• the lower quartile, median and upper quartile focal lengths (408, 600 and 983 mm) give 1.90″, 1.29″ and 0.79″ per pixel;
• 50 of the 111 telescopes land between 1″ and 2″ per pixel, 21 above 2″ and 40 below 1″;
• the extremes in the data are 6.88″ (5.94 µm at 178 mm) and 0.13″ (2.4 µm at 3910 mm).

The cluster around 1–2″ per pixel matches the rule of thumb that Atik describes. Your seeing decides whether that range suits your sky.

Binning

Binning adds a block of pixels into one value. With 2×2 binning, a sensor with 3.76 µm pixels behaves like one with 7.52 µm pixels, so the image scale doubles: 0.38″ becomes 0.76″ at 2032 mm (computed). The AAVSO guide separates two ways of doing the sum. CCDs can combine charge before the analog-to-digital converter, which lowers read noise per binned pixel. Binning after the converter, as CMOS cameras do it, “offers no noise advantage over unbinned images” (AAVSO, section 3.3.7) but shrinks the files. The geometry is the same either way.

Color sensors

A one-shot color camera has a filter over every photosite. The preferred layout in Bayer’s 1976 patent puts green filters on every other position and red and blue on the remaining positions in alternate rows. Green therefore takes half the photosites, and red and blue a quarter each. In the red or blue channel, neighboring samples sit two pixels apart in each direction, so their effective scale is twice the calculator’s figure (derived from the layout). As Frédéric Michaud’s notes on star images put it, the camera rebuilds the full-color image in software from neighboring photosites, which is why the mono vs color guide treats color resolution separately. Atik notes a visible side effect at coarse scales: when most of a star’s light falls on one photosite, the star takes that filter’s color.

The calculator reports the scale of the photosites themselves.

What image scale leaves out

Image scale measures how finely the camera samples the image. The detail available to sample comes from the aperture and the atmosphere. Sheffield’s visual-use notes give the Rayleigh limit as about 140″ ÷ D(mm) at 550 nm; for a 72 mm refractor the exact form, 1.22 λ ÷ D, gives 1.92″ (computed). Once the smallest blur in the image is well sampled, finer pixels record that blur in more pixels without revealing more of the object; the sampling guide puts numbers on “well sampled”, and the resolution limits guide covers the aperture side.