On Satellite Umbra/Penumbra Entry and Exit Positions
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On Satellite Umbra/Penumbra Entry and Exit Positions Deny Neta l and David Vallado2 Abstract The problem of computing Earth satellite entry and exit positions through the Earth's umbra and penumbra, for satellites in elliptical orbits, is solved without the use of a quartic equation. A condition for existence of a solution in the case of a cylindrical shadow is given. This problem is of interest in case one would like to include perturbation force resulting from solar radiation pressure. Most satellites (including geosynchronous) experience periodic eclipses behind the Earth. Of course when the satellite is eclipsed, it's not exposed to solar radiation pressure. When we need extreme accuracy, we must develop models that tum the solar radiation calculations "on" and "off," as appropriate, to account for these periods of inactivity.
Introduction The problem of computing Earth satellite (in elliptical orbits) entry and exit positions through the Earth's umbra and penumbra is a problem dating from the earliest days of the space age, but it is still of the utmost importance to many space projects for thermal and power considerations [1]. It's also important for optical tracking of a satellite. To a lesser extent, the satellite external torque history and the sensor systems are influenced by the time the satellite spends in the Earth's shadow. The umbra is the conical total shadow projected from the Earth on the side opposite the sun. In this region, the intensity of the solar radiation is zero. The penumbra is the partial shadow between the umbra and the full-light region (see Fig. 1). In the penumbra, the light of the sun is only partially cut off by the Earth, and the intensity is between 0 and 1. All textbooks discussing the problem (e.g. Geyling and Westerman [2] and Escobal [3]) and even the recent work by Mullins [1] suggest the use of a quartic equation analytic solution to determine umbra/penumbra boundaries. Because the quartic is a result of squaring the equation of interest, one must check all four solutions and discard the spurious ones. In this paper, we examine solving the original equation numerically. We will give a condition for the existence of a solution, discuss the initial guess for an 'Naval Postgraduate School, Department of Mathematics, Code MAlNd, Monterey, CA 93943. 2Lt. Col. and Orbital Analyst, Phillips Laboratory, Kirtland AFB, NM 87117. 91
Neta and Vallado
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FIG. 1. Earth Umbra and Penumbra.
iterative scheme, and compare the complexity of the two schemes (our scheme versus the analytic solution of the quartic [1]). The shadow problem has been solved in the past by assuming a cylindrical shadow behind the Earth [2], or a conical shadow which is more realistic [1,4]. The numerical solution will be discussed for each case.
Problem Formulation In this section, we formulate the problem using both cylindrical and conical shadow geometry. We'll see that the solution method is different in the two cases. Cylindrical Shadow
In this case the orbital geometry is given in Fig. 2 [3,5]. The ana
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