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 eccentricity e (< 0.2) was studied in part I on the assumption that atmospheric density $\rho$ varies exponentially with distance r from the earth's centre, so that the `density scale height' H, defined as -$\rho$/(d$\rho$/dr), is constant. In practice H varies with height in an approximately linear manner, and in the present paper the theory is developed for an atmosphere in which H varies linearly with r. 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-H equations of part I. The second (and usually better) method is to obtain the best constant value of H for use with the equations of part I. For example, it is found that the constant-H equations connecting perigee distance (or orbital period) and eccentricity can be used unchanged without loss in accuracy, if H is taken as the value of the variable H at a height $\gamma$ H above the mean perigee height during the time interval being considered, where $\gamma$ = $\frac{3}{2}$ for e > 0.02, and $\gamma$ decreases from $\frac{3}{2}$ towards zero as e decreases from 0.02 towards 0. Similarly the constant-H equations for air density at perigee can still be used if H is evaluated at a height $\xi H$ above perigee, where $\xi$ = $\frac{3}{4}$ for e > 0.01, and $\xi$ decreases towards zero as e decreases from 0.01 towards 0. For circular orbits the constant-H equations for radius in terms of time can still be used if H is evaluated at one scale height below the initial height. Variation of H with altitude has a small effect on the lifetime-about 3%-and on the curve of e against time.