The Structure, Stability, and Dynamics
of Self-Gravitating Systems
Joel E. Tohline
tohline@rouge.phys.lsu.edu
[ Download PDF file dated
]
Joel E. Tohline
tohline@rouge.phys.lsu.edu
[ Download PDF file dated
]
An ideal gas containing N free particles per unit volume will exert on its surroundings an isotropic pressure (i.e., a force per unit area) P given by the followingProperty #1
if the gas is in thermal equilibrium at a temperature T .
The internal energy per unit mass e of an ideal gas is a function only of
the gas temperature T , that is,
Property #2:
| Throughout most of this H_Book, we will define the relative degree of compression of a gas in terms of its mass density r rather than in terms of its number density N. Hence, in place of the standard form of the ideal gas equation of state [II.A.1], we more commonly will adopt the following expression which will be referred to as |
where  º k/mp is the "gas constant," and m is the mean molecular weight of the gas. See § 3 of Chapter VII (p. 254) in Chandrasekhar (1967) for a clear discussion of how to calculate the mean molecular weight of a gas.
| NOTE: When Chandrasekhar (1967) first introduces the gas constant in his Chapter II (p. 38), he assumes that m = 1. In order to generalize his presentation so that it matches ours, his "Â" must everywhere be replaced by the expression "Â/m." Note also that in his presentation, as well as in the presentation of Landau & Lifshitz (1975), "V" does not identify the volume of the gas but rather the volume per unit mass; i.e. , V º specific volume = 1/r. |
|
From the accompanying discussion of specific heats, we know that, for any ideal gas, the universal gas constant  is related to the specific heat at constant pressure of the gas cP and to the specific heat at constant volume of the gas cV through the simple expression,
and the specific internal energy of the gas is related to the gas temperature through the expression,
e = cVT.
[Equation II.B.4]
=
Landau & Lifshitz '75, Chapter IX, Eq. (80.10)
| Combining these two relationships with Form A of the ideal gas equation of state [II.A.3], we derive what will be referred to as |
where the ratio of specific heats,
| Throughout this H_Book we will most frequently use Form B of the ideal gas equation of state [II.A.4] to supplement the principal governing equations. |
| Chapter II of Chandrasekhar (1967) contains a very thorough discussion along these lines. |
1Text in green is taken verbatim from Chapter II, § 1 of Chandrasekhar 1967.
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