Joel E. Tohline
tohline@rouge.phys.lsu.edu
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where e is the specific internal energy, P is the pressure, and V = 1/r is the specific volume of the fluid element. Now, if it is understood that the specified changes are occurring over a certain interval of time dt, then from this expression [I.C.1] we can directly derive what will henceforth be referred to as 
From Form A of the ideal gas equation of state [II.A.3] we can write 
Combining this expression with the standard form of the First Law [I.C.1], we derive 
If the state changes occur in such a way that no heat seeps into or leaks out of the fluid element, then DQ = 0 and the changes are said to have been made adiabatically. For an adiabatically evolving system, therefore, Form A of the First Law [I.C.2] assumes what henceforth will be referred to as the 
Throughout this H_Book, as we examine the stability and dynamical evolution of selfgravitating systems, we generally will confine our discussion to models that undergo strictly adiabatic changes. Hence, we have included this adiabatic form of the First Law in our set of principal governing equations.
It is important to note that, because the change in the specific entropy of a fluid element ds is related to dQ through the expression
ds = dQ/T,
then ds = 0 when dQ = 0. Hence, the adiabatic form of the First Law [I.C.6] also may be viewed as a statement of specific entropy conservation. 
[Equation I.C.8]
is an equally valid statement of the conservation of specific entropy in an adiabatic flow.

Combining Form B of the ideal gas equation of state [II.A.4] with Form C of the First Law [I.C.8], we find that 
Now, from the standard Lagrangian representation of the continuity equation we know that,
Hence, from the previous expression [I.C.11] we derive, 
Returning to the discussion associated with our derivation of the Lagrangian conservation expression [I.B.5] from the continuity equation, it is clear from Form D of the First Law that, 
must be the volume density of a conserved physical quantity. Henceforth, we will refer to t as the "entropy tracer" because, as the following homework problem demonstrates, there is a straightforward algebraic relationship between the "entropy tracer per unit mass" (t/r) and the specific entropy of the gas.
Finally, using the definition of t and Form D of the First Law [I.C.12], we can construct a 
which may replace the adiabatic form of the First Law in our set of principal governing equations.
[Equation I.C.15]
where C_{0} is an arbitrary constant.

^{1}Text in green is taken verbatim from Chapter II, § 1 of Chandrasekhar 1967.
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