Air drag effect on a satellite orbit described by difference equations in the revolution number
For a satellite on an orbit which is affected by air drag, difference equations are derived whose solutions express changes in orbital size and shape and give the satellite's time behavior as functions of revolution number. The results are obtained from a first order perturbation theory using a small air drag parameter. Introduction. The behavior of a satellite orbit under the effects of air drag have been discussed by Petersen [l]1, the present author [2], and others. Using the method of variation of parameters, I derived some approximate expressions for the decay of eccentricity with radius, the decay of radius with revolution number, and the growth of time with revolution number. Those results appear to be especially useful for eccentricities which are quite small, say e < 0.01, although a complete investigation of their range of validity has not been investigated. An alternative approach is available which may be preferable for larger eccentricities. It consists of a perturbation method in a small parameter, carried only to the first approximation, followed by the derivation of an exact relationship between values at the beginning and the end of any orbital revolution. Thus one obtains a set of difference equations in the values of the problem's dependent variables which occur at each full orbital revolution. Because the approach leads to difference equations, it appears well suited to numerical treatment. It offers a significant advantage even when a high speed digital computer is available, for it obviates the numerical integration around each orbital revolution which can impose a significant burden on machine storage capacity and result in a prohibitively long solution time. It is clear that in a conventional numerical solution in which the satellite is followed step-wise around each revolution, even the finest practicable integration interval and the greatest care in handling errors may not avoid a significant accumulation of error over tens of thousands of revolutions. On the other hand, the difference equation method developed here allows the errors for any complete revolution to be made as small as desired without excessive difficulty. Outline of the method. Like [2], this treatment begins with the drag law and equations of motion discussed by Petersen [l]2. The following notation is used: r radial distance from center of earth to satellite 13 angular advance of satellite from arbitrary interially-fixed reference line in plane of orbit K product of earth mass and constant of gravitation •Received March 28, 1957; revised manuscript received August 27, 1957. 'Numbers in square brackets refer to the bibliography at the end of the paper. These apply only to the case where the aerodynamic force acts only as a drag along the line of the velocity vector. Lift forces are assumed absent. Other assumptions are given with the notation below. 132 ROBERT E. ROBERSON [Vol. XVI, No. 2 CD drag coefficient, assumed constant A satellite projected area on plane normal to velocity vector, assumed constant3 m satellite mass a equatorial radius of earth h altitude, r — a (neglecting oblateness of earth) p(h) standard stable atmospheric density function In these terms, the equations of motion are P" _ ^ = -£ ~ P(h)rY2 + r^T', (1) r/3" + 2r'f}' = P{hW(r2 + r°/3"2),/2. (2) It is convenient to introduce auxiliary dimensionless variables £ = a/r, (3) V = Ka/m\ (4) and to regard them as functions of 0. Primes henceforth denote differentiation with respect to 0 as the new independent variable. Also define the parameter.