The contraction of satellite orbits under the influence of air drag IV. With scale height dependent on altitude

Abstract
The effect of air drag on satellite orbits of small eccentricitye(< 0.2) was studied in part I on the assumption that atmospheric densitypvaries exponentially with distancerfrom the earth’s centre, so that the ‘density scale height’H, defined as —p/(dp/dr), is constant. In practiceHvaries with height in an approximately linear manner, and in the present paper the theory is developed for an atmosphere in whichHvaries linearly withr. Equations are derived which show how perigee distance and orbital period vary with eccentricity, and how eccentricity varies with time. Expressions are also obtained for the lifetime and air density at perigee in terms of the rate of change of orbital period. The main results are presented graphically. The results are formulated in two ways. The first is to specify the extra terms to be added to the constant-Hequations of part I. The second (and usually better) method is to obtain the best constant value ofHfor use with the equations of part I. For example, it is found that the constant-Hequations connecting perigee distance (or orbital period) and eccentricity can be used unchanged without loss in accuracy, ifHis taken as the value of the variableHat a height yHabove the mean perigee height during the time interval being considered, where y — 3/2 fore> 0.02, and y decreases from § towards zero as e decreases from 0.02 towards 0. Similarly the constant-Hequations for air density at perigee can still be used ifHis evaluated at a height above perigee, where £ = 3/4 fore>0.01, and £ decreases towards zero as e decreases from 0.01 towards 0. For circular orbits the constant-Hequations for radius in terms of time can still be used ifHis evaluated at one scale height below the initial height. Variation ofHwith altitude has a small effect on the lifetime—about 3 %— and on the curve of e against time. 1 *

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