## 1 Introduction  
External payload platforms outside the space station support multidisciplinary space science research, encompassing Earth observation, space astronomy, radiation biology, material exposure, and new technology tests [1]. Earth observation from the space station, characterized by low orbital altitude, multiple time windows, high spatial resolution, and varying lighting conditions, serves various purposes such as remote sensing, resource surveying, disaster assessment, weather forecasting, and climate change monitoring [2]. Constrained by the orbital altitude, payload field of view, and target imaging conditions, the continuous observation of targets is limited, achievable only when the field of view intersects with targets, known as the visible time window (VTW) [3]. For large-scale space science payloads, it is imperative to calculate their VTW to determine operational modes. Therefore, VTW calculation is crucial for payload mission planning, setting the foundation for mission activities [4; 5]. Research on VTW calculation is significant for rapid determination, particularly for emergency and real-time planning, and accurate VTWs enhance planning efficiency by reducing planning margins [6; 7]. The crux of the VTW calculation problem is determining the relationships among spatial points, lines, and areas. Research has progressed from the tracking and propagation (TP) method to more rapid algorithms. TP, a classical approach, involves continuously tracking a trajectory at fixed time intervals [8]. This involves computing the orbit positions and determination of the payload's visibility to the target at each sampled moment, yielding the VTW. While TP provides accurate results, its time-consumingnature poses challenges, especially for time-constrained planning tasks. Nonetheless, it remains a benchmark for its accuracy. To expedite computations and address various targets, rapid algorithms are proposed, mainly including numerical and geometric methods. The former focuses on solving the intersection information as a function of time, while the latter calculates the geometric relationship between the projection and the targets. Numerical methods derive analytical formulas for VTWs from spacecraft-target relationships, solving transcendental equations based on [PERSON]'s theory. These methods employ approximation techniques like [PERSON]'s method to reduce computation time and avoid multiple solutions to [PERSON]'s equation. Key methods include the eccentricity function method, visibility function method, sub-point trajectory method, and Poincare mapping method. [PERSON] [9] introduced a transcendental equation using the eccentricity function, obtaining a closed solution for two-body motion through interpolation. [PERSON] [8] proposed a visibility function method based on the Fourier series, further improved by [PERSON] et al. [10] using parabolic hybrid technology. [PERSON] [11] proposed a visibility function transcendental equation for satellite coverage, and [PERSON] [12] defined a general expression of the function for a single satellite concerning ground position. [PERSON] et al. [13] proposed the sub-point trajectory method, approximating it within the orbit period using the same orbit great circle. [PERSON] et al. [14] proposed an analytical form based on the Poincare mapping method, while [PERSON] et al. [15] enhanced it with adaptive interpolation technology. [PERSON] et al. [16] proposed a rigorously adaptive Hermite interpolation method for function curve fitting, and [PERSON] et al. [17] transformed the problem into a multi-peak function root-finding problem, approximating the visibility function based on various modeling techniques. Geometric methods employ orbit ergodic search to determine the geometric relationship between the field of view and targets to obtain the VTW, with a simple model and low complexity. [PERSON] et al. [18; 19] proposed a coarse and precise search method, using a two-body and complex models to obtain an approximate and precise time, respectively.