An Analytic Solution for Entry into Planetary Atmospheres

Approximate closed form solutions for inclination angle and density or altitude as a func- tion of velocity are obtained for a body entering a planetary atmosphere. The method of solu- tion used assumes that the atmospheric density may be expanded in a power series in terms of the logarithm of the velocity. Termination of the series after the first three terms is shown to be equivalent to a result obtained by Loh on the basis of observations that a certain combination of the entry variables may be considered constant. The approximate solution obtained here is compared with the exact solution, and other approximate solutions obtained by Loh and Alien and Eggers, over wide regions of lift to drag ratio, entry velocity, and entry inclination angle. Good agreement is found over all regions except those corresponding to certain skipping-type trajectories. For this critical case, the solution of Loh is also inac- curate. In general, the solution obtained yields the same accuracy as the "second order theory" of Loh, although it is somewhat simpler to apply. Alien and Eggers1 obtained an approximate solution to these equations for a ballistic vehicle by assuming that: 1) the gravitational component of the force tangent to the entry trajectory is negligible when compared to the drag force along the path; 2) the inverse scale height ft is a constant, which implies an isothermal atmosphere in which the atmospheric density is an exponential function of altitude; and 3) the path inclination angle remains a constant. The solution, although quite useful, is limited to large entry angles. Eggers, Alien, and Neice2 extended these re- sults to a lifting vehicle; however, their analysis was mostly concerned with the range of the vehicle. Norman3 and others have shown that the effect of the first assumption is small. This fact is especially true at such points of interest as that of maximum deceleration. Work3' 4 has also been done which shows the effect of the second as- sumption. The forementioned improvements increase the accuracy of the Alien and Eggers solution, but do not extend its limited range of application. Chapman5 studied the problem of entry by numerically solving the equations of motion. The technique used was that of combining the equations into a single equation govern- ing the entry variables, with small quantities being neg- lected. Boltz6 extended the work of Chapman to include the effects of the neglected quantities. Eggers7 also used a single equation describing the relationship of the entry variables to obtain a solution that is useful for analyzing supercircular entry of a skipping vehicle. Loh8"12 has developed an ap- proximate analytical solution and has shown its agreement over wide regions of lift conditions, entry angles, and entry ve- locities. However, comparison of Lon's solution with an exact numerical solution shows it to be inaccurate for certain skipping-type trajectories. Many results obtained using Loh's method along with other methods and exact results are given in the book by Loh.13 The present work attempts to clarify some of the assump- tions required for the derivation of Loh's results. The method of solution developed provides accurate analytical results that are valid over many entry conditions. The solu- tion form is such that the atmospheric density, or altitude, and flight path angle are found as explicit functions of the vehicle velocity.