# A Simplified Pointing Model for Alt-Az Telescopes
## 1 Introduction  
Ground-based optical telescopes are key facilities in astronomy and planetary sciences. The telescopes are generally mounted on a stable concrete pier at a site. The electromechanical equipment (e.g., driven motors and rotation mechanisms) will drive the telescope to point toward a desired direction, such as a star, a planet, or a sky area. The basic requirement is that the target is in the field of view of the telescope. Usually, we need to stably trace a target for a long time (e.g., measuring the light curves of exoplanets, or long exposures for dark targets), during which time the position of the target in the image needs to be stationary. The telescope is required to overcome the rotation of the Earth and the movement of the target. As a result, the accuracy of the pointing, as achieved through the pointing model, becomes paramount for the success of these observations. The ability to point the telescope quickly and accurately towards the observational target is a crucial performance indicator in telescopes, both optical and radio. To realize this target, an accurate telescope pointing model is indispensable. It serves as a representation of the telescope's pointing accuracy, helping to determine the pointing deviations at different positions and orientations and providing means to calibrate these errors. The telescope pointing model is a mathematical representation used to describe the pointing errors of the telescope and the calibration methods. For an optical telescope, there are mainly two types of mounting, namely, equatorial mounting and alt-az mounting. For an equatorial mounting telescope, calibrating the pointing model is a relatively straightforward process. The primary advantage of an equatorial telescope is that it can easily compensate for the apparent motion of celestial objects by utilizing the uniform rotationof its right ascension axis. When observing celestial objects, it rotates at a constant speed around the polar axis in the direction and at the rate of the Earth's rotation, effectively canceling out the apparent motion of celestial objects caused by the Earth's rotation. This keeps the targeted celestial object within the field of view, and the positions of celestial bodies in the field of view do not undergo relative rotation. As a result, it allows for extended periods of observation and photography. The primary pointing errors of an equatorial mounting telescope usually result from one of the following issues: polar axis misalignment, zero offset on the declination circle, non-perpendicularity between the right ascension and the declination axes, bearing errors, or optical collimation errors. For an equatorial telescope, widely used pointing correction formulas are spherical trigonometry [1]. For an alt-az mounting telescope, the calibration of the pointing model is more complicated. The pointing of an alt-az telescope is realized through its azimuth axis and elevation axis. The azimuth motor drives the azimuth axis to point the telescope to different azimuth angles in the horizontal plane. The elevation motor drives the elevation axis to point the telescope to different elevation angles in a meridian plane determined by the azimuth angle. Usually, the elevation angle is defined with respect to the local horizontal plane, and the azimuth angle is defined with respect to a direction on the local horizontal plane, such as local north or local south. Therefore, the pointing errors of alt-az telescopes are mainly composed of two aspects, one is the inclination of the telescope's mounting plane with respect to the local horizontal plane, and the other is the rotation angle of the telescope in the mounting plane relative to local north. Other minor factors include the orthogonality between the azimuth and elevation axes, the mechanical accuracy of axes that are determined by the accuracy of the encoders, and atmospheric refraction [2]. The objective of calibrating the pointing model is to characterize these errors and correct them during telescope operation.