## P

From which the character of 1 p as a scale height is apparent. The coefficient p is evaluated at the initial, or reference, point indicated by subscript zero. b) The strictly exponential atmosphere. If p can be considered constant throughout the atmosphere, Eq. (1-7) holds for all r. In this case the reference level is commonly-taken to be the surface of the planet. If the temperature can be considered constant through an altitude interval of the atmosphere, dT dr 0, and 3 is given by Since for...

## Aerodynamic Heating

The various first-order solutions developed in Chapter 7 are now employed to analyze the thermal problems encountered in hypersonic flight. The study is of fundamental interest to scientists and engineers involved in design of space vehicles and in planning flight operations for a given mission. In space-flight, achieving a maximum payload is always a factor of prime consideration. A relatively important fraction of this payload is used in the protection and cooling process during entry if the...

## Dqo h

Cos I cos C3 cos log tan( + ) J + C where the C are constants of integration, n 14-3. 3. Asymptotic Matching and Composite Expansions The constants of integration C in the inner expansions will be determined by matching with the outer expansions. In this problem, matching is accomplished by expanding the inner solutions for large h , expressing the results in terms of the outer variables and matching with the outer solutions for small h . The outer solutions, Eqs. (14-21), become for small h On...

## Pr cos y Sln y

With the aid of Eq. (13-40), we consider the sum of the terms 2 , u cos y sin y 1 1 - X cos y tan y - --- -L The last step is obtained by applying Chapman's basic assumptions, Eqs. (13-33) and (13-34). We see that this is equivalent to neglecting the terms containing sin2 -y in Eq. (13-43). Thus, - d2z (dZ Z 1 - 2 4 I , 3 u _ - I - I __ cos y -y r * cos V Chapman derived this same equation for planar entry by repeatedly applying his two basic assumptions. Yaroshevskii's theory, (Ref. 5), is...

## Lift Modulation with Constraints on Speed and Flight Path Angle

STATE AND CONSTRAINT EQUATIONS The equations for flight with variable lift and bank control were derived in Chapter 16. For flight along a great circle, with the variables The rescaled lift coefficient X is used as a control and has been defined as where C * is the lift coefficient corresponding to maximum lift-to-drag ratio, E* (L D) . Hence, when X 1 the flight is at maximum lift-to-drag ratio. The lift coefficient is bounded by an upper limit and proportionally we have the constraint...

## I cos v

The integration is simple and we have the general solution where the c. are constants of integration. We see that s is equivalent to a and actually we only have 5 constants of integration. The last constant of integration is obtained by integrating the time equation, Eq. (15-16). In the first three equations (15-28), we evaluate the constants of integration by taking the origin of time at the time of passage through the periapsis. These three equations provide the link between the entry...

## M Y P

The integration of the equation requires two initial conditions on Z and Z dZ du at the initial time, u u. Z(u.) Z. , Z (u.) Z. (12-3) To evaluate Z , we use Eq. (11-12) reproduced here for convenience z(u.) Z. , z '(u.) + J& .r. sin y. (12-5) The kinematic elements at entry are r V and y_Hence, we can form the dimensionless entry speed V. V. Jg.r. _. Then u. V. cos y. , y. and the additional prescribed value Z. will provide sufficient initial conditions for the integration of the nonlinear...

## V2ciV

COMPARATIVE ANALYSIS OF THE PERFORMANCE OF HYPER VELOCITY VEHICLES Any detailed analysis of the performance would require variational theory and hence is not within the stated goal of this work. Nevertheless, since some of the performance criteria such as range, time of flight, speed and design parameters, such as convective heat and heat rate, are obtained in explicit form, it is possible to have some qualitative appraisal of the performance of a hypervelocity vehicle using different...

## Performance in Extra Atmospheric Flight

Beginning here, we shall analyze the performance of long-range hypervelocity vehicles. The flight is assumed to take place in the plane containing the great circle arc, between the take-off point and the landing point. The flight is thought of in two phases as illustrated in Fig. 3-1. a The powered phase, in which sufficient kinetic energy provided by the propulsion system is imparted to the vehicle to bring it, under a proper guidance, to a prescribed position and velocity in space. The...

## Flight with Lift Modulation

Until now we have considered entry trajectories with constant angle of attack and constant bank angle. A space vehicle entering a planetary atmosphere and having control devices allowing modulation of the lift coefficient and the bank angle wrill have more flexibility in the selection of the appropriate trajectory. For a highly maneuverable vehicle, lift and bank controls can guide the vehicle along a prescribed trajectory to a correct presentation for making the final approach and landing....

## Yaroshevskiis Theory for Entry into Planetary Atmospheres

The analytical development in Chapters 7, 8 and 9 presents the classical theories for entry into a planetary atmosphere. The simple results given are adequate for the purpose of a preliminary estimate of the variations of the trajectory variables along an entry flight path and the different physical characteristics expressed as functions of these entry variables. Beginning in this chapter, we shall present various modern theories for planetary entry. First, with the physical understanding of...

## Powered Phase

In this chapter, we shall analyze the trajectory of the vehicle from the launching pad, point A, to the burnout position, point B . During the thrusting phase, the energy provided by the propellant is transformed into potential energy through the increase in the altitude of the vehicle, and kinetic energy through its increase in speed. Also, a part of the energy provided by the propulsion system is dissipated in the form of heat by action of the aerodynamic drag. The powered phase is the phase...

## Info

And the square root can be approximated by In this case, the critical flight path angle, as given by Eq. (7-32), becomes If the flight path angle is not too large, we have the approximate relation In this case, we obtain from Eq. (7-28) the critical speed where maximum deceleration occurs The solution obtained by Lees applies to circular speed entry. For supercircular speed entry, it has been generalized by Ting (Ref. 4) In this case, the second term on the right-hand side of Eq. (7-5) is not...

## Chapmans Theory for Entry into Planetary Atmospheres

Compared with Yaroshevskii's theory for the entry trajectory, Chapman's theory offers a higher degree of sophistication. Using some simplifying assumptions, Chapman derived a relatively simple nonlinear differential equation of the second-order, free of the characteristics of the vehicle (Ref. 1). This is made possible by introducing a set of completely nondimensionalized variables. Chapman's reduced equation includes various terms, certain of which represent the gravity force, the centrifugal...

## Return to the Atmosphere

In Chapter 3, the trajectory for flight outside the atmosphere, assumed to be spherical with a finite radius R , was considered. We have seen that, if the burnout position B , (Fig. 3-1), is outside the atmosphere, the resulting trajectory is a Keplerian conic until the atmosphere is reencountered. In the case of an elliptic trajectory, if the periapsis distance r is less than the radius R of the atmosphere, a condition expressed by the inequality (3-71), the trajectory will intersect the...

## Analysis of First Order Planetary Entry Solutions

To help in the understanding of the basic physical phenomena encountered by a vehicle during its descent through a planetary atmosphere, in this chapter we shall derive several first-order solutions for planetary entry by making, separately for each case, the necessary physical assumptions. Each type of trajectory will be analyzed in detail. In particular, we shall be concerned with the variations during the entry of the altitude, the speed, and the acceleration of the vehicle. Other physical...

## Equations for Flight Over a Spherical Planet

In this chapter we shall derive the equations of motion of a vehicle considered as a point mass of mass m flying inside a planetary atmosphere. The motion of the vehicle is defined by At each instant, it is subject to a total force F composed of the gravitational force mg , the aerodynamic force A and a thrusting force T provided by the propulsion system. With respect to an inertial system, we have the vector equation Consider a fixed system 0 X Y Z , and another system Oxyz which is rotating...

## 1

Hence, it will require that V. is not near the circular speed. The assumption is good if Since -y. is the order of 0. 1 , the use of F as a similarity parameter for gntry into different planetary atmoipheres is restricted to about V. > 1. 1 , or v. > 1.05. Next although theoretically a skip trajectory is a trajectory leading to Z Z. 0 , in constructing the diagrams Chapman qualified an overshoot trajectory such that the exit speed exceeds the circular speed, that is, a trajectory such that...