Equatorial Mount

What is modelled

The sky turns once in a sidereal day about the celestial pole, which stands above the horizon at the observer latitude. An equatorial mount has one axis parallel to the pole, so a single steady rotation at the sidereal rate cancels the motion of any star. If the axis is a little off the pole, the telescope and the star rotate about different axes and slowly part. The page rotates the pointing vector about the true and the actual axis with Rodrigues rotations and measures the angle between them.

Equations

• Horizon coordinates sin h = sin δ sin φ + cos δ cos φ cos H, hour angle H = H0 + Ω t, Ω = 2π / 86164.09 s = 15.041 arcsec/s.
• Rotation v' = v cos θ + (k × v) sin θ + k (k · v)(1 − cos θ).
• First-order drift ε |m × s(t) − R m × s(0)| for an axis error ε about the direction m.
• Focus x = F2 / (s − F), plate scale 206.265 p / F arcsec per pixel, Rayleigh limit 1.22 λ / D.

Checks run in the page

• Hour angle and declination to altitude and azimuth and back for 400 random positions, and the altitude swing of Polaris.
• The sidereal period and rate and one full turn of the sky.
• The exact drift against the first-order formula, and zero drift for a perfect alignment.
• The thin-lens drawtube extension and the optical resolution limit.

Honest simplifications

Atmospheric refraction, flexure, periodic gear error, precession and the exact refractor geometry are left out, the sidereal day is treated as constant and the mount axis error is a fixed small angle. It illustrates the geometry, it is not a pointing model.

• Press Well aligned, tracking and then Typical polar error: watch the pointing error of the second grow minute by minute.
• Press Drive off and see the star leave the field at 15 arcsec per second.
• Press Terrestrial focus, 1 km and read how far the drawtube must travel.