Inter-satellite laser communication demands a precise pointing, acquisition, and tracking (PAT) system. Establishing a thorough and efficient digital model for the acquisition and tracking system is crucial. It provides a robust groundwork for designing and developing high-precision coarse-fine compound-axis tracking systems. However, previous digital models for compound-axis systems have been overly generalized and lacking in accuracy, manifesting four key shortcomings. Firstly, the coarse and fine loops have been oversimplified, with fewer specific mechanism models and unclear relative motion relationships between components. It results in a significant disconnect from real-world physical contexts. Secondly, devices such as detectors and feedback systems exhibit varying sampling rates, delays, and errors, yet quantitative analyses of these aspects are sparse. Thirdly, error sources are singularly considered, with incomplete identification of their application points. Fourthly, models for friction torque and inertial torque are distorted. To aid in the design of high-precision PAT systems, this paper has tackled these issues by establishing a comprehensive digital model for the compound-axis tracking system. Using this model, simulations have been conducted to elucidate the characteristics and primary-secondary relationships of various error sources. Furthermore, focusing on specific in-orbit laser link scenarios like stationary tracking, satellite attitude adjustments and satellite maneuvers, the causes of tracking accuracy decline and link interruptions have been scrutinized. Additionally, recommendations for optimizing the design of PAT systems and selecting parameters for various devices and controllers are provided in this paper.Commencing with the foundational theories of the coarse and fine subsystems, and integrating the physical characteristics of actual in-orbit products, a comprehensive digital model for compound-axis tracking has been established (Fig.1). Through an examination of the mechanics behind each error source and merging it with existing data, the mechanisms of satellite-body micro-vibrations, detector noise, static and dynamic friction torques, inertial torques, feedback errors, and gear transmission noise have been quantified (Fig.2). These error sources are all applied at corresponding positions in the digital model. Utilizing this model, a Simulink simulation system has been constructed to delineate tracking errors, servo bandwidth, and other dynamic characteristics. Subsequently, by conducting simulations, specific excitations and boundary conditions have been introduced for in-orbit laser link scenarios encompassing stationary tracking, satellite attitude adjustments, and satellite maneuvers. The time and frequency characteristics of tracking errors have been thoroughly analyzed.Building upon the proposed digital model, a sophisticated simulation system has been implemented. This system characterizes the open-loop and closed-loop response curves of the coarse tracking subsystem, the fine tracking subsystem, and the compound-axis system (Fig.3-Fig.4). The simulation yielded a servo bandwidth of 201 Hz for the compound-axis system, which closely aligns with actual in-orbit operational conditions. Tracking errors under different satellite-body micro-vibration models were simulated (Fig.5). Contributions of each error source were calculated (Tab.1), with satellite-body micro-vibrations identified as the primary cause of performance degradation. For the stationary tracking scenario, an analysis unveiled that the "spike"-type decline in tracking accuracy stemmed from the influence of friction torque on the turntable. The rapid oscillation of the friction torque between positive and negative extremes left the turntable in a slow crawl state, leading to abrupt spikes in tracking errors (Fig.7). Regarding satellite attitude adjustments, it was discerned that high-frequency harmonic vibrations of the satellite body primarily undermined tracking performance (Fig.8). In satellite maneuvers, diverse constraints on mechanisms were found to be the cause of laser communication link interruptions (Fig.9). Specific strategies were proposed for each of the aforementioned scenarios.Drawing on a comprehensive assessment of disturbances impacting the onboard laser communication terminal, this paper undertakes a thorough and accurate digital modeling of the compound-axis PAT system. Using the digital model, all key performance has been characterized. Moreover, through simulations and analyses of stationary tracking, satellite attitude adjustments and satellite maneuvers, this paper has pinpointed the factors contributing to the decline in inter-satellite laser communication tracking precision and link disruptions. Tailored enhancement strategies have been devised for each scenario. This study holds significant implications for the design, development, and test of compound-axis control systems for onboard laser communication payloads.