The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Joel E. Tohline
tohline@rouge.phys.lsu.edu

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First Law of Thermodynamics

Following Chandrasekhar's (1967, Chapter I) detailed discussion of the laws of thermodynamics, we know1 that for an infinitesimal quasi-statical change of state, the change dQ in the total heat content Q of a fluid element is given by the following

Standard Form
of the First Law of Thermodynamics,

dQ = de + P dV,


[Equation I.C.1]
=

Ch67, Chapter II, Eq. (2)

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

Form A
of the First Law of Thermodynamics:

D e + P D(1/r) = DQ.


[Equation I.C.2]


From Form A of the ideal gas equation of state [II.A.3] we can write

(1/r)dP + P d(1/r) = (/m)dT,

[Equation I.C.3]

in other words,

VdP + PdV = (/m)dT.

[Equation I.C.4]

Combining this expression with the standard form of the First Law [I.C.1], we derive

Form B
of the First Law of Thermodynamics:

dQ = de + (/m)dT - VdP.


[Equation I.C.5]


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

Adiabatic Form
of the First Law of Thermodynamics:

De + P D(1/r) = 0.


[Equation I.C.6]

Throughout this H_Book, as we examine the stability and dynamical evolution of self-gravitating 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,


[Equation I.C.7]

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.

Multiply the adiabatic form of the First Law [I.C.6] through by r, then show that

Form C
of the First Law of Thermodynamics,

D(er) - (P + er) D(ln r) = 0,


[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

D(er) = g er D(ln r),

[Equation I.C.9]

or, dividing through by ger,

(1/g) Dln(er) = Dln r.

[Equation I.C.10]

This also may be written as,

Dln(er)1/g = Dln r.

[Equation I.C.11]

Now, from the standard Lagrangian representation of the continuity equation we know that,

Dln r = - v.

[Equation I.B.2]

Hence, from the previous expression [I.C.11] we derive,

Form D
of the First Law of Thermodynamics,

Dln(er)1/g = - v.

[Equation I.C.12]

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,

t (er)1/g

[Equation I.C.13]

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

Conservative Form
of the First Law of Thermodynamics,

tt + (tv) = 0,

[Equation I.C.14]

which may replace the adiabatic form of the First Law in our set of principal governing equations.

Using the standard form of the First Law [I.C.1], the relationship between ds and dQ [I.C.7], and Forms A & B of the ideal gas equation of state [II.A.3 & II.A.4], show that the specific entropy of an ideal gas, s, is related to the "entropy tracer per unit mass" (t/r) through the expression,

[(m/)(g - 1)/g] s = ln(t/r) + C0,

[Equation I.C.15]

where C0 is an arbitrary constant.


Footnotes

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


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