Numerical integration of Eq. (10.72) is needed to determine the equivalent free-space velo...

Numerical integration of Eq. (10.72) is needed to determine the equivalent free-space velocity increment required for constant-thrust interorbital transfers of long duration. Show that the equation may be written as three simultaneous first-order equations:

with initial conditions ρ =1, A = 0, andB = 1.

Beginning with low-altitude earth orbit (LEO) andv = 10−4, 10−3, 10−2 and 10−1, determine the time required and the equivalent free-space velocity increment ratio for escape. It may be helpful to use a standard Runge-Kutta routine, for example, Runge-Kutta-Merson, which integrates the simultaneous first-order differential equations with a step size varied according to an error estimate. Compare your results with the circumferential solution shown in Fig. 10.21.