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

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

uestion:

What is the relationship, if any, between the polytropic index n and the ratio of specific heats g in an adiabatic system?

nswer:

From Form C of the First Law of Thermodynamics [I.C.8] we know that, for an adiabatic evolution,

and from

Form B of the Ideal Gas Equation of State [II.A.4] we know that

Hence,

or,

Therefore, in general for adiabatic evolutions, the gas pressure P must be related to the gas density r through the relation,

where K is independent of time. Now, for a polytropic gas of index n, the following relationship between P and r holds:

From these last two expressions, we conclude that

uestion:

If the mean molecular weight m of the gas is defined such that 1/m gives the number of free particles per proton mass mp, what is the algebraic expression relating r and N? Utilizing this relationship, show that the above two forms of the ideal gas equation of state [II.A.1] and [II.A.3] provide equivalent expressions for the gas pressure.

nswer: