Analytical determination of eclipse entry and exit points considering a conical shadow and oblate Earth

This paper presents a new analytical procedure to model the umbra generated during an eclipse considering an oblate ellipsoid of rotation as occulting body and a conical shadow. The method is based on purely geometrical considerations and results in the analytical definition of the entry and exit points from the conical shadow starting from the knowledge of the Sun position vector, the occulting body position vector and the orbital elements of the spacecraft orbiting the occulting body. The conical shadow also permits analytical definition of the entry and exit points of the penumbra region, which cannot be defined by using the classic cylindrical approach. Some numerical applications are proposed to test the effectiveness of the analytical formulations and to check the error in the prediction of the time spent in the shadow by the satellite. Finally, a discussion between the new conical shadow model and the classic cylindrical eclipse is carried out to see the improvements introduced by the refined geometry and the effects on space missions focusing on the cumulative error when multiple revolutions are considered.