Finally, the differential equations for y and ip > written in terms of the dimensionless variables, are dy _ 2
ds cos y
These dimensionless equations are exact. In particular, they reduce to the equations for Keplerian motion when Z -» 0 .
In Eq. (13-10) for Z , if a strictly exponential atmosphere is used p = constant, and dp /dr = 0 . On the other hand, if an isothermal atmosphere is considered, (3 /g = constant, and (l/2p2)(dp /dr) = - 1/pr . In both cases, in the equations of motion, the variables p and r enter as the product pr . For the Earth, for altitudes below 120 kilometers, the mean value is large. In this same region pr varies from a low of about 7 50 to a high of about 1300 . It is, however, a better assumption to use a mean value for pr than simply to put p constant and use the simple exponential atmosphere in the computation. This development will follow Chapman's lead and put pr constant. Also, because of the large value for pr , the quantity inside the brackets in Eq. (13-10) is practically unity. This minor assumption concerning the product pr does not alter the asymptotic behavior of the trajectory at very high altitudes where the equation in Z becomes inoperative.
In summary, the equations of motion for three-dimensional entry trajectories are dZ ds du ds
The equations (13-13) were first derived by Vinh and Brace (Ref. 3). In view of the definition (13-5) of Z , they are restricted to flight at constant lift-to-drag ratio, C^J C^ = constant , and for flight with a completely free modulation in the bank angle. Extension of these equations to the case of free modulation in the coefficients C^ and C^ and in the bank angle tr for the study of three-dimensional optimal trajectories in atmospheric, hypervelocity flight has been obtained by Vinh, Busemann and Culp (Ref. 4).
The equations derived can be considered as the exact equations for entry into a planetary atmosphere. Just as Chapman's simplified equations, they are completely free of the characteristics of the vehicle. Hence, they can be used to analyze the motion of an arbitrary vehicle regardless of its weight, size and shape. The characteristics of the atmosphere enter the equations in the form of the parameter (3r .
Once the atmosphere has been specified through pr , for any prescribed lift-to-drag ratio, Q^J C^ , and bank angle, <r , and with a prescribed set of initial conditions, the universal function Z can be generated, and different physical quantities during entry can be evaluated and analyzed exactly as in Chapman's theory, described in Chapters 11 and 12. It may be thought at first glance that, to integrate Chapman's simplified equation, Eq. (11-16) of Chapter 11, only the product ^/pr (C. / C^) need be prescribed and not pr and (C^J Cj^) separately. That is, Chapman's analysis appears to apply to any arbitrary atmosphere. But this is not rigorously true since in evaluating the flight path angle y , using Chapman's first equation,
Eq. (11-12) of Chapter 11, the parameter VPr needs to be prescribed. A normalizing technique to obtain a similarity solution for an arbitrary atmosphere requires sacrificing the accuracy in evaluating the universal Z function and the flight path angle y , and restricting the analysis to a small class of entry trajectories.
The equations derived are the universal equations in the sense of Chapman since they produce the universal Z functions for analyzing the motion, deceleration and heating of an arbitrary vehicle. Furthermore, they are the exact equations for flight of a vehicle in a Newtonian gravitational field subject to aerodynamic force. In particular, they provide the Keplerian solution for flight in a vacuum and all other classical solutions when appropriate assumptions are introduced. These particular solutions can be obtained as follows.
For flight in the vacuum, let Z — 0 . The first of Eqs. (13-13) is inoperative. It is replaced by Eq. (13-8). Using this equation to change the independent variable from s to r , we rewrite the other Eqs. (13-13)
dcj> _ sin dr r tan y di = _ cos 4, tan 4, dr r tan y
Integrating the first of these equations yields u = - (13-16)
r where p is a constant of integration. Next, combining the first two equations to eliminate r gives
By the change of variable
cos y this becomes the linear equation
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