Guide Scope & Off-Axis Guider Suitability Calculator

08How guiding suitability actually works

What the numbers mean, where they come from, and what to do about them.

Start with the question the usual advice gets wrong

Most guiding rules of thumb are about pixel scale. Keep the guide scale within three times the imaging scale; use a guide scope at least a third the imaging focal length. Those rules are not useless, but they rest on an assumption that turns out to be wrong: that an autoguider measures star position to a fixed fraction of a guide pixel, so that a coarser guide scale automatically means a proportionally worse measurement.

It does not. The astrometric literature is clear, and it has been for forty years: the precision with which you can locate a star is set by how wide the star is and how much signal it carries. In angular terms — arcseconds, the units that actually matter — it is very nearly independent of pixel scale, right up until the star gets so small on the sensor that it lands inside a single pixel. Then it falls off a cliff.

So this calculator does not assume a centroid precision. It computes one, from four things it works out first: how big the guide star really is, how much light it delivers, how that lands on the pixel grid, and how finely you need to measure it.

Step one: how wide is the guide star?

Three things set it, and they combine as a convolution rather than a simple sum:

star width ≈ ( seeing^1.5 + diffraction^1.5 + optics^1.5 )^1/1.5

Diffraction is the one people forget, and for a small guide scope it dominates. The Airy disc is 1.03 λ/D across, which at visual wavelengths works out to about 117 divided by the aperture in millimetres, in arcseconds. For a 30 mm mini guide scope that is 3.9″ — larger than the seeing disc. A 30 mm aperture is physically incapable of showing you a 2.5″ star, no matter how good the night is. At 60 mm it falls to 1.9″, and at 200 mm to 0.6″.

Optics. Fast short doublets have real chromatic and spherical residuals. Secondary spectrum in a common-glass achromat amounts to roughly a two-thousandth of the focal length of longitudinal colour error; the classical Sidgwick criterion asks for a focal ratio of at least three times the aperture in inches, which a 30 mm f/4 comfortably meets and a 60 mm f/3.8 does not. ED glass improves it by roughly a factor of three.

Seeing you supply. One tempting shortcut is worth warning against: it is true that an aperture smaller than the atmosphere's coherence length sees turbulence mostly as image motion rather than blur, and forms a sharp, dancing image. But the coherence length in 2.5″ seeing is only about 4.5 cm, and that dancing has a coherence time near a millisecond — so a two-second guide exposure averages a couple of thousand independent positions and every bit of that motion turns back into blur. Guide exposures are long exposures. There is no free lunch.

Step two: how much signal does it deliver?

This is the standard CCD equation, the same arithmetic professional photometry uses:

signal-to-noise = S ÷ √( S + n_pix × ( sky + dark + read² ) )

where S is the star's electrons, n_pix is how many pixels it covers, and the terms under the root are everything competing with it. Aperture, exposure, sky brightness, quantum efficiency and read noise all enter here rather than being asserted as fudge factors, which means the scaling comes out rather than going in.

Two results from that are worth knowing because they are commonly got wrong:

• Aperture is worth about five times the log of the diameter — the classical result. Doubling the guide scope's aperture buys about 1.5 magnitudes.
• Exposure time is worth much less than people assume. Because a star bright enough to guide on is already past the read-noise-limited regime, going from two seconds to eight buys about three quarters of a magnitude — roughly twice the available stars, not four times. Doubling the exposure is still the cheapest fix available; it is just not the miracle it is often described as.

Rather than ask you to guess a guide star's brightness, the calculator works out the magnitude at which your guide field would be expected to hold a couple of candidates, and uses that. A small field — an off-axis guider's, say — therefore lands on a fainter, noisier star automatically, which is exactly what happens in practice.

Step three: turning that into a pointing error

Two terms, added in quadrature.

random error ≈ 0.6 × star width ÷ signal-to-noise

This is the Cramér-Rao bound — the theoretical floor on how well any algorithm can locate a star — with a small allowance because a real centroider runs a little above the bound. Note what is absent: pixel scale. Over a fifteen-fold change in pixel scale the bound moves by under a quarter.

The second term is where sampling bites:

sampling error ≈ ( 0.14 e^−n² + 0.00014 n³ ) × guide scale,   n = pixels per star width

The first half is the pixel-phase or “S-curve” error. When a star is smaller than a pixel, a centre-of-mass measurement is systematically pulled toward the pixel centre, by an amount that depends on where in the pixel the star happens to sit. Because it follows a Gaussian in the sampling, it is a cliff, not a slope: at three pixels per star width it is 0.005 pixels and irrelevant; at one pixel it is 0.018; at 0.75 it is 0.065; below that it saturates around 0.15 pixels. And unlike random noise, it does not average away — when the mount is tracking well and the star sits still, it becomes a persistent offset.

The second half is the opposite problem. PHD2 measures the centroid inside a fixed seven-pixel radius, with its background annulus from seven to twelve pixels. A star much wider than about six pixels is being clipped by its own aperture and is contaminating its own background estimate. The fix is free: bin the guide camera.

Reading the verdict

The headline number is the guiding measurement error in arcseconds, and the verdict compares it to the width of the star in your finished image — not to your imaging pixel.

That distinction matters more than it sounds. A pointing error of a given angle trails the star by that same angle whether you are at 400 mm or 4000 mm. Whether anyone can see it depends on how it compares with the star's own width. Judging per pixel would punish you for sampling finely and reward you for binning, neither of which changes how round your stars are.

But measurement precision is only one of four scores. The verdict you see is whichever of these is worst, and the tool names it:

Guide scope, or off-axis guider?

Here is where the honest answer diverges from the usual one. It is commonly said that long focal lengths need an off-axis guider because the pixel scale arithmetic demands it. That is not really why.

Differential flexure — and a correction to the usual story

If you guide with a separate scope, the guider assumes the two tubes point in exactly the same direction for the whole exposure. Any relative movement is invisible to it and lands straight in the image.

The calculator converts your tolerance into the physical movement you are asking the hardware not to make, and the number is sobering: half an imaging pixel at 1000 mm with 3.76 µm pixels is 0.39″, which across a 200 mm mounting baseline is a relative shift of 0.38 µm — about a hundredth of a human hair, held through every temperature change and every meridian side.

But note something important: flexure produces an angular error, and an angular error trails the star by the same amount at any imaging focal length. Flexure does not inherently get worse as you go longer. What changes is that you notice.

Mirror flop — the actual reason SCTs use off-axis guiders

On a Schmidt-Cassegrain or Maksutov that focuses by moving its primary mirror, the mirror can shift slightly as the telescope tracks across the sky. A guide scope cannot see this, because it happens downstream of where the guide scope is looking: the guide star sits perfectly still while your image walks away. The guider never sees the error, so it never corrects it.

An off-axis guider picks its star off after the mirror, so any flop moves the guide star too and gets corrected like anything else. This, not pixel scale, is the categorical argument for off-axis guiding on these telescopes — and it is why the calculator raises it as a flat warning rather than folding it into a score.

Off-axis guiding: what actually goes wrong

1. The guide star is aberrated

The prism picks off a star 10–20 mm off the optical axis, where the telescope's off-axis aberrations apply. How bad that is depends entirely on the design, and the spread is enormous. Working from third-order aberration theory, at 13 mm off axis:

A standard Schmidt-Cassegrain is about 4.6 times worse in coma than a classical Cassegrain of the same focal ratio — the corrector plate sits well in front of the mirrors, and the resulting system behaves like a much faster paraboloid off-axis. So a single flat penalty for “using an off-axis guider” is meaningless: it is a tenth of a magnitude on an RC and over two magnitudes on a bare fast Newtonian.

2. The prism under-fills at fast focal ratios

The prism sits in a converging cone some distance in front of focus. How wide that cone is where it crosses the prism is simply:

beam diameter at the prism (mm) = distance from prism to focal plane (mm) ÷ focal ratio

Counter-intuitively, fast telescopes are harder for off-axis guiders, not easier. An 8×8 mm prism with a filter wheel and a cooled camera sits about 54 mm from focus. At f/7 the cone there is 7.7 mm across and the prism catches essentially all of it. At f/4 the same cone is 13.5 mm across and the prism catches under half. On an obstructed telescope it is worse still, because the cone is an annulus whose dark core sits neatly inside the prism — the prism loses the wide outer ring and keeps nothing extra for it.

There is a second, less-known limit: an uncoated right-angle prism relies on total internal reflection at 45°, and a converging cone spreads the angle of incidence. Below about f/5 part of the beam falls under the critical angle and is simply lost — around 4% at f/5 and 11% at f/4. Prisms with an aluminised hypotenuse are unaffected.

3. The telescope has already vignetted before the prism sees anything

This one is usually left out entirely, and it is often the largest term. Cassegrain baffles are sized for the on-axis cone, so these telescopes vignette from the centre of the field outward. Celestron's own figures put an EdgeHD 8 at 84% relative illumination at the corner of an APS-C sensor — 13.3 mm out, which is exactly where an off-axis guider's prism lives. A classic Schmidt-Cassegrain, with smaller baffles, is worse.

The calculator estimates this, but treat it as an estimate. If you have a flat frame, read the illumination off it at the guide star's radius — that is the exact answer and it costs nothing.

4. The prism has to clear the imaging sensor

The distance it has to clear is the sensor's half-dimension on the axis the prism advances along — not the half-diagonal. A prism entering over one edge occupies a band across the sensor; the corners sit further out, but in a different direction, and are irrelevant. For APS-C along the short axis that puts the prism centre about 13 mm off axis, needing a 27 mm circle for the guide star — comfortably inside the 44 mm most flatteners deliver.

One subtlety the calculator accounts for: the star the prism collects is not at the prism's own radius. Because the bundle for an off-axis point is centred on its chief ray rather than the optical axis, a prism at 13 mm on a Schmidt-Cassegrain actually collects a star at about 14.2 mm — 9% further out, into slightly worse aberration and vignetting. Behind a reducer the effect reverses and grows.

5. Back focus

Reducers and flatteners need a specific distance to the sensor — 55 mm is close to universal — and the guider body, filter wheel, adapters and camera all have to add up to it within about half a millimetre. The ledger in the results does the arithmetic.

If the verdict is poor, here is what actually helps

• Look at which score is worst — the four are fixed by different things and the remedies do not overlap.
• Sampling too coarse? A longer guide scope, or a smaller guide pixel. The results panel gives the shortest focal length that works and the one we would call comfortable, both scored against all four axes rather than just one.
• Sampling too fine? Bin the guide camera. It is a software setting and it costs nothing.
• Signal too low? A longer exposure first, then a more sensitive camera. Mono is worth about 0.6 magnitudes over colour, because a Bayer matrix throws away most of the light and you gain nothing from colour you do not use.
• Not enough stars? Longer exposure, larger guide aperture, or a bigger pick-off. The Milky Way has roughly ten times the star density of the Virgo–Coma region, so a setup that never misses in Cygnus can fail repeatedly in the spring galaxies.
• Guiding fine but stars still trail? Flexure or mirror flop. Neither is visible to the guide graph, and neither is fixed by buying a longer guide scope.