On satellite umbra/penumbra entry and exit positions

The problem of computing Earth satellite entry and exit positions through t he Earth's umbra and penumbra, for satellites in elliptical orbits, is solv ed without the use of a quartic equation. A condition for existence of a so lution in the case of a cylindrical shadow is given. This problem is of int erest in case one would like to include perturbation force resulting from s olar radiation pressure. Most satellites (including geosynchronous) experie nce periodic eclipses behind the Earth. Of course when the satellite is ecl ipsed, it's not exposed to solar radiation pressure. When we need extreme a ccuracy, we must develop models that turn the solar radiation calculations "on" and "off," as appropriate, to account for these periods of inactivity.