Analytic solution for vertical launch into orbit

An approximate analytic solution is presented for the motion and heating of nonlifting spacecraft launched vertically for ascending flight through the atmosphere into earth orbit. The solution is based on Chapman's Z-function transformation of the equations of motion with the assumption of an exponential atmosphere. The transformed equations have been modified to include the effects of thrusting with mass loss. It is assumed that the vehicle negotiates a gravity turn during ascent so that the thrust vector is aligned in the flight-path direction. Numerical results are presented to document the analytic solution and provide evidence of the useful information generated by it.