Survey of Current Literature on Satellite Lifetimes

F the past three or four years numerous authors have obtained equations from which the lifetime of a satellite in orbit about the earth may be estimated. Because of the large amount of material available on this subject (e.g., see Kefs. 1-30), an interested person may find himself confronted with a maze of equations and the added confusion of choosing one author's favorite formula over another. It is in light of this fact that the present article has been written. The primary purpose here is to review the available literature and to present those equations which, in this reviewer's opinion, bear the most merit. When more than one equation falls into this category a comparison of the two will be made in order to determine the areas and degree of agreement. The mathematical manipulations employed by the various authors in deriving the formulas will be given in skeleton form only. In this way, attention can be centered on the end product rather than on the path taken in obtaining this end. The bulk of the material contained herein is concerned with the lifetime calculation. When, however, a subsidiary equation of considerable interest is uncovered, as, for example, the velocity in orbit, it will generally be noted. Only closed form solutions are to be considered. A satellite in orbit about the earth, in which at least part of the path is submerged within the atmosphere, will experience a force that tends to retard its motion. If the earth and its atmosphere are taken to be stationary and if the gravitational field is given by the simple inverse-square law, then the dynamics of the decay will be restricted to the planar or two-dimensional case. The atmospheric drag has a twofold effect on the orbit. First, the drag force causes a distortion in the shape of the orbit. The orbital parameters, i.e., eccentricity, angular momentum, semimajor axis, etc., which are constants on the nominal orbit, become continually varying functions of the time. Even more important, at least in the context of this article, is the fact that the continuous loss of energy from the satellite to the atmosphere, via the skin friction, causes the satellite orbit to drop in altitude. Thus, the orbit compensates for the change in kinetic energy by changing its potential energy. For initially circular orbits, the decay is approximately uniform over the entire orbit, and, therefore, the trajectory will remain very nearly circular. For elliptical orbits of high eccentricity, the drag in the vicinity of perigee is sufficiently higher than that experienced elsewhere on the orbit so that the effect is very nearly that of an impulse. This causes the apogee altitude to drop at a much greater rate than does perigee. In fact, it will be shown that the perigee altitude is constant for all practical purposes. As the eccentricity decreases to lower values, the orbit experiences a mean decay between the two just noted. Apogee still falls at a greater rate than perigee, but perigee is no longer constant. It is obvious, however, that a small change in perigee altitude will have a pronounced effect on the lifetime because of the large change in density for even small altitude variations. Once the orbit has decayed into the denser portions of the atmosphere, the satellite will be forced out of orbit and will plunge quickly thereafter into the ground. This latter phase takes place at a low velocity, and the trajectory is almost, vertical. It is at this point that the life of the satellite may be considered ended. The approximate range of eccentricities, where each of the conditions noted previously takes place, will be stipulated.