Astronomy tools · Tracked imaging
Polar alignment drift and sub length.
Give your polar alignment error, pixel pitch and focal length. Get the worst-case declination drift, in arcseconds and in pixels, and the longest sub-exposure before it reaches the drift you accept.
A polar scope or plate-solve routine ends with a number such as “3.2 arcminutes of error,” and the next question is what that does to a 5-minute sub on an unguided mount. The answer is a short piece of geometry: a misaligned axis makes the target drift in declination, at no more than the error times the Earth’s rotation rate. This page does that sum for your pixel size and focal length and states the limit as a sub length.
Short answer: A 5 arcminute polar error drifts a star at most 1.31 arcseconds per minute. With 3.76 µm pixels on a 400 mm scope, one pixel is 1.94 arcseconds, so a 1 pixel drift limit is reached after 89 seconds.
Why a polar error makes stars drift
An equatorial mount turns about an axis meant to point at the celestial pole. If the axis is off by an angle, tracking at the sidereal rate still moves the telescope around the wrong axis, and the target slides slowly north or south in the frame. Wikipedia’s polar alignment article notes that any error in the polar alignment will show up as drift of the stars, and gives the two-star relation Δδ = Δe·cos h + Δa·cos Φ·sin h, where Δe is the altitude error, Δa the azimuth error, h the hour angle, Φ the site latitude and δ the declination. The rate of change of that is at most the Earth’s angular rate times the total error, which is the bound this page uses.
The arithmetic
Wikipedia gives the sidereal day as about 86,164.09 seconds, so the Earth turns 2π ÷ 86,164.09 = 7.292 × 10-5 radians per second. Multiply by the error in radians and you have the worst-case drift in radians per second; convert to arcseconds. For 1 arcminute of error that is 60 × 7.292 × 10-5 = 0.00438 arcseconds per second, or 0.26 arcseconds per minute. Wikipedia’s plate scale page gives the image scale as 206,265 × pixel size ÷ focal length in arcseconds per pixel, which converts arcseconds into pixels. The longest sub is the drift you accept divided by the drift rate.
What it ignores, and why that is fine
- It is a worst case: the real drift depends on where in the sky you point and how the error splits between altitude and azimuth, and can be much smaller.
- It counts declination drift only. Field rotation, which grows away from the pole, and the right-ascension component that Wikipedia’s formula gives (it grows with the tangent of declination) are not included.
- Periodic error in the gears, flexure and wind add their own movement. If you guide in declination, this limit does not apply to you.
How good an alignment is realistic
Wikipedia says a polar scope can reach within an arcminute or two, and plate solving within 0.1 arcminutes. Plug 1 arcminute and 0.1 into the tool to see how much longer each lets a sub run.
Worked example
These are illustrative numbers, not a specific rig. Error 5 arcminutes = 300 arcseconds; drift rate 300 × 7.292 × 10-5 = 0.02188 arcseconds per second = 1.31 arcseconds per minute. Pixel pitch 3.76 µm on a 400 mm scope gives 206,265 × 0.00376 ÷ 400 = 1.94 arcseconds per pixel, so the drift is 0.68 pixels per minute. A limit of one pixel is therefore reached after 1.94 ÷ 0.02188 = 89 seconds. At 1 arcminute of error the same limit is 443 seconds, and at 0.1 arcminute it is 4,430 seconds, at which point other errors will dominate.
Frequently asked questions
How much does polar alignment error affect exposure time?
Worst-case declination drift is the error times the Earth’s rotation rate, about 0.26 arcseconds per minute for each arcminute of error. Divide the drift you accept by that to get the longest sub.
How accurate does polar alignment need to be?
It depends on your image scale and sub length. A coarse scale of 3 or 4 arcseconds per pixel tolerates minutes of error, a fine scale needs far less. The tool shows the sub length for the error you enter.
Does this apply if I use an autoguider?
Not for declination drift. A guider corrects it, so polar error then matters mainly through field rotation, which this tool does not calculate.
Why is the result a worst case?
Drift depends on where you point and how the error splits between altitude and azimuth. The tool assumes the least favourable combination, so your real drift is usually smaller.
What should I enter for allowed drift?
One pixel is a common, strict choice. A looser limit of two or three pixels gives longer subs but visible elongation on a close look; choose by how much you will crop or print.
How is this different from the NPF rule?
NPF is for a fixed camera with no tracking, where the Earth’s whole rotation trails the stars. This tool is for a tracking mount, where only the drift from polar misalignment is left.
Sources and how it is calculated
Retrieved
The drift bound (error × the Earth’s angular rate) and the conversion to a sub length are this page’s own arithmetic on the cited sidereal day and plate scale. It is a worst case and covers declination drift only. Pages were read on 26 October 2026.
- Wikipedia, Polar alignment: error shows as star drift; the two-star declination formula; polar scope and plate-solve accuracy
- Wikipedia, Sidereal time: sidereal day of about 86164.0905 s
- Wikipedia, Plate scale: arcseconds per pixel = 206265 × pixel size ÷ focal length
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A ceiling, not a promise
Real tracking has periodic error, flexure and wind on top of polar drift. Take a test sub at the length you choose and inspect the stars at 100%.