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 the phenomena associated with an entry trajectory as presented in the previous chapters, scientists and engineers are led to formulating very general assumptions which are valid for nearly all types of entry trajectories. This results in a set of equation of motion valid for all types of entry trajectories of practical interests. Furthermore, if these equations can be presented in dimensionless form, in the same way as has been done in the previous chapters for first-order analysis, the results obtained can be applied to any entry vehicle regardless of its physical characteristics. This type of approach is illustrated by Yaroshevskii's theory presented in this chapter, and Chapman's theory in Chapters 11 and 12.
Next, if the restrictive assumptions introduced are removed while the universal character of the entry equations is still preserved, the exact dimensionless equations of motion for planetary entry are obtained and they are valid for all types of entry trajectories of any entry vehicle regardless of its mass, size and aerodynamic characteristics. Furthermore, the trajectory considered can be completely immersed inside the atmosphere or can be partly outside of it in the form of Keplerian arcs. This also will include orbital motion of satellites at very high altitude subject to Newtonian gravitational attraction and infinitesimally small atmospheric drag. This analysis will be considered in the following chapters.
Finally, one may consider the case where the aerodynamic controls in the form of the lift-to-drag ratio and the bank angle are not constant but can be modulated according to a certain law in order to achieve a specific purpose. This type of trajectories will be analyzed in the last chapters.
Yaroshevskii's theory for entry trajectory is a semi-analytical theory. Using some simplifying assumptions, he derived a nonlinear, second-order differential equation which can be integrated analytically by using series expansions. To some extent, Yaroshevskii's theory is a special case (Refs. 1,2) of a more sophisticated theory developed by Chapman (Ref. 3). Because his theory has some features of merit, we shall present it in this chapter. Chapman's theory will be developed in the next chapter, and the connection between the two theories will be examined.
10-2. SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATION FOR ENTRY TRAJECTORY
Consider the basic equations for planar entry derived in Chapter
dt 2m pSC V2 2
sin y vÈL L i v i v —= —-- - g - - cos y dt 2m r
Strictly speaking, as was mentioned in Section 6-2, the lift and the drag coefficients are functions of the angle of attack a , of the Mach number M , and the Reynolds number R . For constant angle of attack, Yaroshevskii assumed that the lilt c oefficient CT , and the drag coefficient C are functions of the Mach number. For an isothermal atmosphere, this is just a function of the speed, V
If, in the equation for V , we neglect the tangential component of the gravity force, and in the equation for y and r we use the approximation of small flight path angle, we can write the Eqs. (10-1)
The first of these equations can be used to change the independent variable to V :
To derive his second-order nonlinear differential equation for entry into a planetary atmosphere, Yaroshevskii used an independent variable x , and a dependent variable y defined as x
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