β = P_{g} / P

, with

P = P_{g} + P_{rad}

u = \frac{3}{2} R T + \frac{a T ^{4}}{ρ}, h = u + \frac{P}{ρ} = \frac{5}{2} R T + \frac{4 a T ^{4}}{3 ρ} .

At fixed total pressure, differentiating the equation of state gives

0 = β d lo g ρ + (4 - 3 β) d lo g T, (\frac{\partial l o g ρ}{\partial l o g T})_{P} = - \frac{4 - 3 β}{β} .

In particular,

β

c_{P} = (\partial h / \partial T)_{P}

, so

c_{P} = \frac{5}{2} R + \frac{4 a T ^{3}}{3 ρ} (4 + \frac{4 - 3 β}{β}) .

Since

a T^{3} / (3 ρ) = R (1 - β) / β

, this simplifies to

c_{P} = \frac{R}{2 β ^{2}} (32 - 24 β - 3 β^{2}) = \frac{k}{2 μ H β ^{2}} (32 - 24 β - 3 β^{2}) .

The pure-gas limit is

5 R /2

. For a gas with unspecified molecular degrees of freedom, replace

3 R /2

u

c_{V, g}

: then

c_{P} = c_{V, g} + R + 4 R (1 - β) (4 + β) / β^{2}

. The boxed formula uses the conventional monatomic stellar-gas interpretation.

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.