Joel E. Tohline
tohline@rouge.phys.lsu.edu
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Apart from the independent variables t and x, the principal governing equations involve the vector velocity v, and the four scalar variables F, r, P, and e. Because the variables outnumber the equations by one, one (additional) supplemental relationship between the physical variables must be specified in order to close the set of equations.
Also, in order to complete the unique specification of a particular physical problem, either a steadystate flow field or initial conditions must be specified, depending on whether one is studying a timeindependent (structure) or timedependent (stability or dynamics) problem, respectively.
Throughout this H_Book, the following strategy will be adopted in order to complete the physical specification of each examined system:
where the ratio of specific heats g is assumed to be independent of both x and t. Simultaneously, Form A of the ideal gas equation of state [II.A.3] provides a relationship between the gas temperature T and the state variables P and r. 
See Tassoul (1978)  specifically the discussion associated with his Chapter 4, Eq. 13  for a more general statement related to a proper specification of the supplemental, equation of state relationship. 
Tassoul (1978) refers to it as a "geometrical" rather than a "structural" relationship; see the discussion associated with his Chapter 4, Eq. 14. 
Generally throughout this H_Book, we will assume that all timeindependent configurations can be described as barotropic structures. That is, we will assume that the gas pressure P is only a function of the gas density r throughout such structures. More specifically, we generally will adopt one of the following two analytically prescribable P(r) relationships.
In Polytropic Structures, 
where the polytropic index n and the coefficient K_{n} are assumed to be independent of both x and t .
where: c º ( r / b )^{1/3},
[Equation II.C.2]
and the coefficients a = ( 6.002 x 10^{22} dyne cm^{2} ) and b / m = ( 9.736 x 10^{22} g cm^{3} ) are assumed to be independent of both x and t .
What is the relationship, if any, between the polytropic index n and the ratio of specific heats g in an adiabatic system?

Because enthalpy has the same dimensional units as the gravitational potential F (i.e., energy per unit mass), it often proves to be a useful variable through which to describe the equilibrium structure of selfgravitating systems. It should be clear immediately, for example, that the above expression [III.F.16] can be used to simplify considerably the form of the righthandside of Euler's equation [I.A.1]. 
Assuming that H = 0 at r = 0, show that the analytical expression for the enthalpy H(r) of a polytropic gas is,
H = ( 1 + n ) K_{n} r^{(1/n)},
[Equation III.F.18] and that, in terms of the specific entropy of the gas s,
K_{n} = (1/n) exp[ s/c_{v} ].
[Equation III.F.19]


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