Alt-Azimuth Mount Tracking

What is modelled

The sky turns once per sidereal day (86164.09 s). A star at declination dec and hour angle H appears at an altitude and azimuth set by the observer latitude. An alt-azimuth mount has to turn both axes at changing speeds to follow it, and the camera field also rotates about the line of sight. Near the zenith the azimuth rate grows without limit, so a stepper-driven worm wheel runs out of step rate and a blind spot opens. The 3D page shows the worm drives and bearings of the mount; their rates and the stepper numbers are computed here.

Equations

• sin h = sinφ sinδ + cosφ cosδ cos H; azimuth from north through east A = atan2(-cosδ sin H, sinδ cosφ - cosδ sinφ cos H).
• Altitude rate dh/dt = Ω cosφ sin A.
• Azimuth rate dA/dt = Ω (sinφ - cosφ cos A tan h).
• Field rotation rate -Ω cosφ cos A / cos h, with Ω = 2 pi / 86164.09 s = 15.041 arcsec/s.
• Microsteps per axis turn N x 200 x m; step rate = azimuth rate divided by the angle of one microstep.

Checks run in the page

• Spherical formulas against a rotation of the star unit vector into the horizon frame.
• Closed-form rates against central differences of the computed positions.
• Integral of the altitude rate against the altitude difference.
• Worm-drive step count over one sidereal day against N x 200 x m.

Honest simplifications

Perfect polar alignment of the sky model, no refraction, no mount flexure, no periodic worm error, a 200 step per turn motor, no extra gear stage after the worm wheel, and a simple constant step-rate limit instead of a torque-speed curve.

• Press Near the zenith and watch the azimuth rate and the step-rate graph climb toward the motor limit.
• Press Polaris-like circumpolar: the star barely moves in altitude but the field still rotates.
• Press Fast wheel, low limit to see the blind spot grow when the step-rate limit is low.