# A Simplified Pointing Model for Alt-Az Telescopes
## 4 Calibration Results for PAST  
By capturing stellar targets, we obtained the original absolute error map of the PAST telescope at different positions (Figure 3) when no pointing model was applied, as shown in Figure 5. For the PAST telescope, the azimuth angle is defined with respect to the local south, with \(\Phi_{\rm T}=0^{\circ}\) for the local south direction and \(\Phi_{\rm T}=180^{\circ}\) or \(-180^{\circ}\) for local north. The azimuth angle is negative when the telescope rotates eastward, and the azimuth angle is positive when the telescope rotates westward. The elevation angle is defined with respect to the mounting plane of the telescope, with \(\Theta_{\rm T}=0^{\circ}\) for horizon and \(\Theta_{\rm T}=90^{\circ}\) for zenith. From Figure 5, it can be observed that the overall absolute error in the azimuth axis of the PAST telescope is relatively large and irregular, while the absolute elevation axis error is smaller and more regular. The distributions of the errors at each elevation angle are similar to a sine function. Such distribution characteristics give us the inspiration to use trigonometric functions to fit the pointing model. Additionally, we have also generated a fitted three-dimensional error map of the PAST telescope, as depicted in Figure 6.  
Figure 7 displays the final residual error distributions of the azimuth and elevation axes of the PAST telescope after applying this pointing model. Comparing it to Figure 5, it is evident that the pointing accuracy of the PAST telescope has significantly improved. For both the azimuth and the elevation axes, the majority of the residual errors are within 10 arcs. After calibration, the root mean square errors (RMSE) of the azimuth axis and  
Figure 5: Original pointing error of the PAST in azimuth ((**left**), in unit of degree) and in elevation ((right), in unit of degree). The colors represent the value of the errors, with color bars shown at the top.  
elevation axis of the PAST telescope are 6.8 arcs and 3.8 arcs, respectively. Compared with previous models, our simple model achieved a high accuracy. The previous calibration methods and models have achieved a high pointing accuracy of less than 2 arcs but based initial pointing errors of less than 1 arcminute [4; 5]. The advantages of the novel method proposed in this paper are that there are no limitations on the initial pointing errors and that no prior knowledge of the properties of the initial pointing errors is needed. The field of view of the planet sensor is 2 arcminutes, and the pointing precision of the model is high enough to ensure that the target appears in the planet sensor, based on which the stabilizing mirror can stabilize the target to a fixed pixel, with an accuracy of 0.5 arcs.  
Figure 6: Original surface pointing error of the PAST in azimuth ((**left**), in unit of degree) and in elevation ((**right**), in unit of degree). The colors represent the value of the errors, with color bars shown at the top.  
Figure 7: The distribution of pointing calibration residuals. The azimuth axis and elevation axis errors (RMSE) of the PAST telescope are 6.8 arcs and 3.8 arcs, respectively.