How good does your polar alignment need to be?

A polar-aligned mount turns your camera about one axis to cancel the Earth's rotation. If that axis is off by a small angle, two separate things go wrong, and which one hurts you depends entirely on whether you are guiding.

Unguided, the whole picture slides. Nothing is watching, so the error accumulates for the whole frame. This is brutal: it scales directly with focal length and with exposure time, and no amount of sensor cleverness helps.

Guided, the slide is corrected but a slow twist is not. Guiding pins one star. The rest of the field turns very slowly around that star, so the middle stays sharp and the corners smear. How much smear you get depends on how far the corner is from the guide star — which is why an on-axis guider is always the best case. A guide scope is usually the worst, though on a short focal length a nearby guide star can beat an off-axis prism.

The practical upshot. For most guided setups away from the pole, a few arcminutes of polar error is genuinely fine, and chasing arcseconds is wasted time. Unguided is the opposite — it is unforgiving, and long focal lengths make it worse in direct proportion.

With the polar axis off by an angle θ, the mount drives every star along a slightly wrong circle. Over an exposure of t seconds the whole field walks:

That reproduces Hook's figure of 0.262 arcseconds per minute for every arcminute of misalignment. It is the worst case: the real figure depends on where you happen to be pointing relative to the direction of your error, and is often smaller. Sizing to the worst case means the answer holds wherever you point.

When you guide, that drift is corrected at the guide star, and what is left is a slow rotation of the field about it:

In other words: the corner smear is the unguided drift, shrunk by the ratio d/F and amplified as you approach the pole. On a 530 mm scope with an APS-C sensor and an off-axis guider, d/F is about 1/15, which is where the factor-of-a-hundred difference between guided and unguided comes from.

The distance d is measured from the guide star, not from the centre of your picture. That is the one thing people most often get wrong, and it is worth about three times the answer.

Declination is the sky's version of latitude. The celestial equator is 0°, the north celestial pole is +90°, the south celestial pole is −90°. Every catalogued object has one fixed declination that never changes with the time or with where you are standing. This is the target's declination, not yours.

Sign does not matter here — the maths uses the cosine, so −60° and +60° give the same answer.

Why it changes the result. Field rotation carries a 1/cos(δ) term, so it grows steeply near either pole. At the celestial equator there is no penalty at all. At 60° the effect doubles. At 80° it is about six times worse. Close to the pole it runs away entirely. This is why imaging near Polaris demands alignment discipline that the same rig would never need on Orion.

Unguided drift, by contrast, does not depend on declination in the worst-case model used here — which is why the tool does not ask for it unless you are guiding.

Guiding holds one star perfectly still. Everything else turns slowly around it, and the further from it you are, the more you smear. So the only distance that matters is from the guide star to the furthest corner of your picture.

• On-axis guider. The guide star is in the middle of the frame, so the far corner is half a sensor diagonal away. Best case by a wide margin.
• Off-axis guider. The prism sits just outside the illuminated circle, so the far corner is roughly a full diagonal away — about two and a half to three times worse than on-axis. The tool works this out from your sensor size, so you don't have to measure anything.
• Separate guide scope. The star is in a different telescope entirely and is not on your sensor at all, so there is no distance to measure — only an angle. The tool projects that angle through your imaging focal length to find where the star would land. The same guide star 1° away sits 8.7 mm from centre on a 500 mm scope but 34.9 mm on a 2000 mm scope. Four times further out, four times the smear.

For scale: the full Moon is about half a degree wide, and your closed fist at arm's length covers about ten degrees. One degree is a small hop.

• Periodic error, seeing, flexure and differential flexure — any of which can dwarf polar misalignment
• Atmospheric refraction, which causes its own drift near the horizon no matter how well you are aligned
• Guiding performance itself — a mount guiding at 1.5″ RMS will not give you half-pixel stars however well it is polar aligned

Treat the answer as your alignment budget, not a guarantee of round stars.

The drift figure comes from R. N. Hook, “Polar axis alignment requirements for astronomical photography”, Journal of the British Astronomical Association, vol. 99 no. 1, February 1989. Ours reproduces his result of 0.262 arcseconds per minute for every arcminute of misalignment.

The field rotation figure is the standard small-angle relation for a misaligned polar axis. It falls straight out of the geometry: the residual twist about the guide star is the drift rate scaled by d/F, with a 1/cos(δ) term that grows as you approach the pole. It is long-established, uncontroversial and reproduced in the literature in several forms.

We sanity-check ours against the standard worked case: a 1′ polar error over a one-hour exposure at declination 50° should smear a point 38 mm from the guide star by about 4.5 µm at the focal plane, and a point 12 mm out by about 1.4 µm. That is exactly what this calculator produces.