Why are telescopes in domes

Rotation of the dome with the telescope

Mathematics helps in solving the problem. Basically it is just a transformation of different coordinate systems: one has the celestial pole as its center and shows us the celestial sphere with the stars above us - the other has the mounting axis in the observatory as its center and rotates around a vertical axis that is in the Zenit shows: it has the horizon as a reference point.

One coordinate system is called "Equatorial", the other "Azimuthal". The systems are named after the orientation of their 0 ° orientation: with the first mentioned it is the celestial equator (hence the name equatorial) and with the second it is the horizon. To add to the confusion, the equatorial system usually specifies the angle in hours and the horizontal system in degrees of the azimuth (an Arabic term for horizon angle).

The transformation from one coordinate system to the other with a little trigonometry helps:
Phi denotes the geographical latitude of the observer
DE is the declination of a star or object
SW is the hour angle (24 h corresponds to 360 °)
H is the height of an object
A is the azimuth angle of an object
H (height) = arc sin (sin Phi * sin DE + cos Phi * cos DE * cos SW)
A (zimut) = 2 * arc tan ((cos H + cos Phi * sin DE - sin Phi * cos DE * cos SW) / (cos DE * sin SW))

We are looking for the temporal change in the azimuth angle with the movement of the object in the starry sky, which is determined by the declination and hour angle. From this result, the motor is controlled, which always moves the narrow dome gap of 60 cm wide in front of the telescope so that the telescope cannot see the dome, but through the dome gap at the sky!