Solid or Gas?

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Solid or Gas?#

Let’s compare the Coulomb energy to \(kT\) to determine if a star is a solid or a gas.

We’ll defined the quantity:

\[\Gamma = \frac{E_\mathrm{Coul}}{kT}\]

as a measure of the importance of the Coulomb energy to the thermal energy. If \(\Gamma \gg 1\), then we are a solid.

The Coulomb energy (in CGS) is:

\[E_\mathrm{Coul} = \frac{(Ze)^2}{r}\]

where \(Z\) is the proton number of the nucleus we are dealing with. We can find the separation, \(r\), from the average density, \(d \sim n^{-1/3}\) with

\[n = \frac{\rho}{A m_u}\]

so

\[d \sim \left ( \frac{A m_u}{\rho} \right )^{1/3} \sim \left ( \frac{4\pi A m_m}{3 M} \right )^{1/3} R\]

where we assumed that \(\rho = \bar{\rho}\), the average density of the star.

From the Virial theorem, we found that

\[k T \sim \frac{1}{3} \frac{GM A m_u}{R}\]

so together, we have:

\[\Gamma \sim \frac{Z^2 e^2}{A^{4/3} m_u^{4/3} G M^{2/3}}\]

Sun#

We’ll keep track of units using the unyt package.

import unyt

e = unyt.qe.in_cgs()
m_u = (1.0 * unyt.amu).in_cgs()
G = unyt.G.in_cgs()
M = unyt.Msun.in_cgs()

Notice that these have units:

m_u

unyt_quantity(1.66053892e-24, 'g')

Let’s just take \(A = 1\) and \(Z = 1\) (hydrogen)

Z = 1
A = 1

Now we can compute \(\Gamma\)

gamma = (Z * e)**2 / ((A * m_u)**(4./3.) * G * M**(2./3.))
gamma

unyt_quantity(0.01111733, '(dimensionless)')

This gives \(\Gamma \sim 0.01 \ll 1\), so the Sun is not a solid.

Earth#

For Earth, let’s use iron.

A = 56
Z = 26
M = unyt.Mearth.in_cgs()

gamma = (Z * e)**2 / ((A * m_u)**(4./3.) * G * M**(2./3.))
gamma

unyt_quantity(167.13713567, '(dimensionless)')

Now we see that \(\Gamma \gg 1\), so Earth is a solid.

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