VbvAstE-001

averaged temperature
and averaged particle density
∫
V ̂T =
T dV
V
∼ T 0
(
R0
R ⋆
)
(6.2)
̂n ≈ NA
(6.3)
R 3 0
After it by means of thermodynamical methods, one can find relation between parameters
of a star.
6.1.1 The c P /c V ratio
At a movement of particles according to the theorem of the equidistribution, the energy
kT/2 falls at each degree of freedom. It gives the heat capacity c v = 3/2k.
According to the virial theorem [12, 22], the full energy of a star should be equal
to its kinetic energy (with opposite sign)(Eq.(5.3)), so as full energy related to one
particle
E = − 3 kT (6.4)
2
In this case the heat capacity at constant volume (per particle over Boltzman’s constant
k) by definition is
( ) dE
c V = = − 3 (6.5)
dT
V
2
The negative heat capacity of stellar substance is not surprising. It is a known fact and
it is discussed in Landau-Lifshitz course [12]. The own heat capacity of each particle
of star substance is positive. One obtains the negative heat capacity at taking into
account the gravitational interaction between particles.
By definition the heat capacity of an ideal gas particle at permanent pressure [12]
is
where W is enthalpy of a gas.
As for the ideal gas [12]
c P =
( dW
dT
)
, (6.6)
P
W − E = NkT, (6.7)
and the difference between c P and c V
c P − c V = 1. (6.8)
Thus in the case considered, we have
c P = − 1 2 . (6.9)
Supposing that conditions are close to adiabatic ones, we can use the equation of the
Poisson’s adiabat.
40