# A Simplified Pointing Model for Alt-Az Telescopes
## 1 Introduction
The orthogonality and the mechanical accuracy can be well measured during laboratory tests, and atmosphere refraction can be well corrected, given the geographical location of the telescope and a reference atmosphere model. Therefore, the main task of pointing model calibration at a telescope mounting site is to determine the mounting errors. By establishing a precise pointing model, the telescope's observational results can be enhanced in terms of accuracy and reliability, ensuring the effective implementation of astronomical observations and research. This is of significant importance in fields such as astronomy, aerospace, and planetary sciences. Many articles have introduced the principles and methods of the telescope pointing model [2; 3; 4; 5; 6]. [PERSON] et al. [3] achieved high-precision pointing calibration for the Haystack antenna by measuring the radiation from cosmic radio sources. The pointing errors were fitted by the method of least squares to determine seven instrumental parameters that take into account the axis-alignment errors and the gravitational effects on the antenna structure. In the end, they achieved highly accurate pointing with an elevation axis error of 12.42 arcs and an azimuth axis error of 10.44 arcs. [PERSON] et al. [2] expanded the linear calibration model with Fourier expansion in order to establish a nonlinear calibration model for a 40-m radio telescope of alt-az mounting and achieved an overall pointing accuracy of 18.9 arcs. [PERSON] et al. [4] introduced a novel telescope pointing calibration model that utilized a semi-parametric regression model to compensate for the nonlinear errors in telescope pointing. Ultimately, they achieved pointing errors of 0.99 arcs in the elevation axis and 1.40 arcs in the azimuth axis. However, the original errors in the elevation axis and azimuth axis of the telescope (without any pointing model) were 39.57 arcs and 5.81 arcs, respectively. [PERSON] et al. [5] employed an Allan-variance-based semi-parametric model to perform pointing corrections on a 1-m diameter alt-az telescope. They achieved a pointing accuracy of 1.31 arcs in the elevation axis and 1.13 arcs in the azimuth axis, starting from the initial errors of 18.15 arcs in the elevation axis and 26.35 arcs in the azimuth axis. [PERSON] et al. [6] employed an optimized parameter model for the Moving-Platform Electro-Optical Telescope. A complex transformation matrix, with 15 dimensions, was adopted. Ultimately, they achieved a pointing accuracy with mean errors of 17.55 arcs in the elevation axis and 17.99 arcs in the azimuth axis. The above-mentioned telescope pointing calibration methods and models all adopted complex functions with multiple parameters (e.g., 14 parameters in [PERSON] et al. [2] and 15 parameters in [PERSON] et al. [5]). A simplified and accurate calibrating model that can be quickly applied to calibrate the telescope's pointing is necessary. The Planetary Atmospheric Spectroscopic Telescope (PAST) was an ultraviolet (UV) -visible telescope with a diameter of 0.8 m. It was constructed through the strategic priority research program of the Chinese Academy of Sciences (CAS), in collaboration with the near-space science experiment system [7; 8; 9]. The primary scientific objectives of the telescope were to investigate the orbital motion characteristics, atmospheric and plasma distribution patterns, and spectral radiation attributes of the celestial bodies situated within the orbital path of Jupiter in our solar system. The telescope was designed for dual operation--one on a balloon-based platform and the other at a ground station. The PAST telescope was successfully installed at Saishiteng Mountain, Lenghu Town, Qinghai Province in the year 2022. The accurate pointing and stabilization of the PAST is realized in two steps. The first is the rough pointing realized by the azimuth axis and elevation axis. The second is accurate pointing and image stabilization realized by a focusing and stabilizing mirror [9].