Guiding math: guide image scale, RMS in arcseconds and the 1:4 rule

Two image scales

A guiding setup has two cameras and two focal lengths, so it has two image scales. Both follow the formula of the image scale calculator:

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

PHD2 needs exactly these two inputs. Its manual describes the guide focal length and pixel size as the parameters it uses to compute the image scale and report guiding in arcseconds, and warns that a wrong focal length also leaves some default guiding parameters set incorrectly. The log-analysis tutorial shows why the unit matters. A 300 mm guide scope with 5 µm pixels samples 3.4″ per pixel, so 0.3 px of star motion is about 1.0″. Switch to an off-axis guider at 2000 mm and the scale becomes 0.52″ per pixel: 2 px of motion is again about 1.0″. The mount did not change; the units did. The tutorial calls pixel-based comparisons of guiding “completely meaningless”.

What PHD2 asks for

The PHD2 Best Practices slides say to judge performance in arcseconds, not pixels, and to try initially for a guiding RMS of around 1″. The log tutorial calls an RMS of 1″ or lower “a rough starting point”, with the achievable figure set mainly by seeing and the mount, and warns that RA and Dec RMS values differing by 2–3× tend to produce elongated stars. Guiding corrects slow errors such as drift, periodic error and refraction. It cannot correct most seeing effects, which is why the slides suggest 2–4 s guide exposures for mounts with good RA tracking. The slides end on a useful test: judge the guiding by what your main camera records.

Sub-pixel centroids

A guider measures a star’s position far more finely than one pixel. The tutorial’s own example reads motions of 0.3 px, and PHD2’s multi-star mode averages the centroids of several stars with weights. How finely depends on how the star is sampled. In a 1989 U.S. Naval Observatory study, Ronald Stone simulated Gaussian star images and found that, for most of the methods tested, position errors caused by coarse pixels stayed negligible with at least about four pixels across the FWHM; a Gaussian fit held up to about two, and errors grew below that. The study also confirms that poor seeing and a bright background degrade centering. Its context was a transit circle rather than an autoguider, but the sampling geometry is the same.

So the star’s width in guide pixels matters, not only the arcseconds per pixel. On a 120 mm guide scope with 3.75 µm pixels, a star blurred by 3″ of seeing alone would span 0.47 px (computed). Real guide stars are wider, because the guide optics and focus add their own blur in quadrature, as Stan Moore describes for any image.

The 1:4 rule

Forum advice often says the guide scale should be no more than about four times the imaging scale. PHD2’s documentation does not say this. The Best Practices, the log tutorial and the manual give no ratio. The closest statement is in the manual’s section on binning: binning reduces the guider’s image scale, which is “not likely to be a problem” when the unbinned scale is below 1″ per pixel, while guiding “may suffer” when the unbinned scale is above 2.5″ per pixel, and the outcome also depends on the main camera’s image scale. The basic-use page adds that PHD2 shows a warning when the image scale is very small and suggests binning. Treat 1:4 as lore and the ratios below as information, not pass or fail.

Worked example: guide scopes from 120 to 240 mm

ZWO’s guide camera manual lists 3.75 µm pixels for the ASI120MM Mini and 4 µm for the ASI220MM Mini. The imaging side uses 3.76 µm pixels on the Askar FRA400 (400 mm, 1.94″), Esprit 100ED (550 mm, 1.41″), and the C8 with its 0.63× reducer and the EdgeHD 8 from Celestron’s reducer table (1280 mm, 0.61″; 2032 mm, 0.38″). Guide scale divided by imaging scale (all computed):

Guide setup	Guide scale	FRA400	Esprit 100ED	C8 + 0.63×	EdgeHD 8
120 mm, ASI120MM Mini	6.45″	3.3	4.6	10.6	16.9
120 mm, ASI220MM Mini	6.88″	3.5	4.9	11.3	18.0
180 mm, ASI120MM Mini	4.30″	2.2	3.0	7.1	11.3
240 mm, ASI120MM Mini	3.22″	1.7	2.3	5.3	8.4
240 mm, ASI220MM Mini	3.44″	1.8	2.4	5.7	9.0

Every guide scope in the table falls within 1:4 for the FRA400. None does for the long SCTs. All the guide scales are above PHD2’s 2.5″ figure, which in the manual’s wording argues against binning these guiders further.

Off-axis guiding at the main focal length

An off-axis guider shares the main telescope’s focal length. On the EdgeHD 8 at 2032 mm, an ASI220MM Mini samples 0.41″ per pixel and an ASI174MM Mini (5.86 µm) 0.59″ (computed). Both are below 1″, where PHD2’s manual expects binning to do no harm; 2×2 binning puts the ASI220MM Mini at 0.81″ and raises the guide star’s signal-to-noise ratio, which the manual describes as especially helpful when you are limited to faint stars. The full ASI220MM Mini sensor (7.68 × 4.32 mm) covers only 13.0′ × 7.3′ of sky there (computed), which is why faint stars are the usual case. The EdgeHD and SCT page lists the focal lengths and reducers for each tube.

What 1″ RMS means for your images

The same RMS lands differently on different imaging scales: 1″ is 0.52 px on the FRA400 and 2.62 px on the EdgeHD 8 (computed). For a rough sense of the blur, assume the errors are Gaussian and equal on both axes. A total RMS of 1″ is then 0.71″ per axis, a tracking FWHM of about 1.67″ (2.355 × σ), and combined in quadrature with 3″ of seeing the stars grow to about 3.43″. At 0.5″ RMS they come out near 3.11″ (all derived estimates). Feed your measured FWHM into the sampling guide’s comparison, and see the guide camera table for pixel sizes.