Eclipse analysis for small-eccentricity orbits using analytical model

Abstract

The orbital eclipse is an old but fundamental problem in astrodynamics, and it has been fully studied in many papers and books. However, the properties of the orbital eclipse under the critical situation and the effects of the eccentricity on the eclipse have not been clearly revealed. Applying the first-order approximation to the eccentricity, the simple analytical solution to the conical eclipse of the elliptical orbit with a small eccentricity is developed, which is similar to the circular-orbit solution in form. The proposed model is validated by numerical simulation, and the properties of the critical situation between the cases with and without orbital eclipses and the effects of the eccentricity on the eclipse are both analyzed. The results show that when the eccentricity is less than 0.01, the proposed analytical model agrees well with the numerical method in common situations. When the orbital sun angle is close to the critical value defined by the ratio of the Earth radius to the semimajor axis, the precision of the proposed analytical model would be degraded. In that situation, the penumbra arc length would be no longer a small value, and the statement in previous studies that the penumbra arc was short and neglectable would not always be right. The difference between the eclipse of the small-eccentricity orbit and that of the circular orbit increases as the eccentricity increases. Moreover, when the argument of perigee is in the same or the opposite direction with the projection direction of the Sun in the orbital plane, the variation in the eclipse length caused by the eccentricity reaches the maximum value. When the above two directions are perpendicular to each other, the variation in the eclipse arc center caused by the eccentricity reaches the maximum value.