Telescope Mount: Periodic Error and Guiding

What is modelled

An equatorial mount follows the stars by turning one axis at the sidereal rate, about 15 arcseconds per second. The drive is a worm screw meshing with a large worm wheel, and the wheel turns once in a sidereal day. No gear is perfect, so the telescope speeds up and slows down a little in each turn of the worm, which gives a periodic error that repeats every sidereal day divided by the number of teeth. During a long exposure the star wanders by the change of this error, and the image is a short trail instead of a point. An autoguider measures the position of a star on a second small camera every few seconds and tells the mount to correct part of the measured error; this is a feedback loop that removes slow errors well but can make fast errors worse if it is too strong or too slow.

Equations

• Worm period T = 86164 s / N; sidereal rate 360 × 3600 / 86164 = 15.04 arcsec/s.
• Error δ(t) = A [sin(2 π t / T) + h sin(4 π t / T + φ)] + r t; trail = largest change of δ during the exposure.
• Image scale 206.265 p / f arcsec per pixel.
• Guiding loop ck = ck−1 + g (δk − ck−1) with one sample of delay; the error left at frequency ω is |1 − z−1 g / (1 − (1 − g) z−1)|, z = ei ω τ.

Checks run in the page

• The worm period and the sidereal rate.
• The longest trail from a numerical scan against the closed form for a pure sine.
• The guiding loop simulated against its theoretical error reduction.
• The image scale from a tangent against the small-angle formula.

Honest simplifications

A single axis, a smooth two-harmonic error, a constant extra rate error, a guiding loop with a fixed delay of one sample and no measurement noise, flexure or backlash. A real mount also has random gear errors, and a real star centroid has noise that limits the gain of the loop, so these numbers show the principle and are not an imaging plan.

• Press Unguided, 2 minute exposure and see the stars stretch; then press Short exposure, no guiding.
• Press Autoguided and compare the trail with the unguided case.
• Press Guiding too slow and too strong and look at the last graph: a loop can amplify an error it cannot follow.