###### Abstract  
Fast determination of satellite visibility with respect to a target area is important for satellite navigation and positioning. In this paper, we propose an adaptive interpolation algorithm based on vertex protection to solve the satellite visibility period problem more accurately and quickly, where \"vertex\" refers to the local extremum point. The algorithm can avoid the error in the visibility period calculation caused by skimming the vertices when fitting the multi-hump visibility function under certain fitting accuracy requirements with the traditional adaptive interpolation method. The algorithm does not need to construct a cubic polynomial in each subinterval to determine whether the satellite is visible or not; it only constructs a cubic polynomial to solve the problem if the visibility function of that subinterval is judged to have a solution from the existence theorem of zero points, which can improve the computational efficiency. For the lunar navigation problem, a solution to satellite-Moon visibility calculations based on a vertex-protected adaptive interpolation is given, and the experimental results show that the computation time of the algorithm can be reduced by approximately 98% compared with the brute force method and by approximately 30% compared with the traditional adaptive interpolation algorithm. satellite-to-Moon visibility; rapid determination; adaptive interpolation; vertex protection; lunar navigation 202  
used the parabolic blending technique to construct the waveform of the visibility function at equal intervals and used the root of the local cubic polynomial to represent the rise and set times of the satellite. The algorithm is applicable to all satellite orbit types. [PERSON] et al. [9] improved [PERSON]'s method and proposed an adaptive Hermite interpolation method in order to solve the satellite visibility problem. This method determines the interpolation step by checking the consistency of the second derivative and comparing the extreme value variance. Based on this work, [PERSON] et al. [5] derived an adaptive Hermite interpolation technology in a strict sense, which uses the fourth derivative to control the approximation error and achieve a better balance between the accuracy and efficiency. [PERSON] et al. [10] used a radial basis function to fit the satellite visibility function and accelerated the calculation of satellite visibility according to the interval contraction strategy. [PERSON] et al. [11] designed a metamodeling framework based on adaptive interpolation and used different metamodel technologies, such as the radial basis function, Kriging, and support vector regression, to solve the satellite visibility problem. The return to the Moon program has been proposed [12], and it has been realized that precise Moon position information is of great importance for an in-depth exploration of the Moon. In addition, the Magnetospheric Multiscale (MMS) mission proposed by NASA has also confirmed that satellite signals can acquire the position information of spacecraft located beyond the Earth's orbit [13]. In recent years, the application of Global Navigation Satellite System (GNSS) technology to lunar navigation has attracted the attention of scientists. [PERSON] et al. [14] simulated lunar navigation with GPS, GLONASS, Galileo, etc., using weak satellite signals for positioning with an accuracy of 200-300 m. The European Space Agency's proposed Moonlight initiative will carry an advanced satellite signal receiver and perform the first satellite navigation and positioning mission in lunar orbit [15]. In order to carry out navigation and positioning on the Moon, the visibility of the Moon from satellites is an issue that must be analyzed and studied. Both the ESA and NASA have performed detailed analyses of the expected visibility of GNSS signals at Moon altitude [16]. A high-precision and efficient method for determining satellite-to-Moon visibility can provide a theoretical basis for achieving navigation and positioning on the Moon.