A Simplified Geometrical Approach to Calculation of Solar Eclipses of a Planet’s Satellite in Solving Practical Problems of Venus Exploration

This study is focused on the problem of determining the position of a satellite at entry to and exit from the planet’s penumbra with the use of an analytical equation in the closed form. The approach taken is based on the geometric representation of the second-order curve that appeared when the plane of the satellite orbit cuts the conical surface formed by the intersection of sunlight rays and the boundaries of a central body. The time moments of the satellite’s entry to the penumbra and umbra of the planet are determined from the intersection of this curve with the satellite orbit. Based on these ideas, an analytical method for determining the duration of eclipses of a satellite by a planet has been developed. Its application to the analysis of the orbits of an artificial satellite of Venus has been demonstrated. It is ascertained that the method simplifies the search for orbits, the parameters of which satisfy the requirements for the duration of the shadow segment. It is shown that the method can be extended to the solution of the problem of determining the moments of time, at which the satellite passes the region obscured by the planetary atmosphere. It is illustrated on provided examples that the proposed approach can be applied to solving practically important problems in the study of Venus and its atmosphere.