Statistical physics, using the ideas of Fermi Energies, etc. for a star

[Benlaww]

Homework Statement

a) Look up typical values for the mass and radius of a neutron star and find the corresponding Fermi energy and the Fermi temperature using E_f = (hbar^2 / 2m) (3(pi^2)N / V)^(2/3)
Typical core temperatures are 10^8 − 10^9 K for a mature neutron star. Does this agree
with your value? If not, how do you explain the discrepancy? What does this tell you
about the occupation of energy levels below the Fermi energy?

b) The gravitational self-energy of a spherical mass of radius R and total mass M is given
by: U_g = -(3/5)(G M^2 / R).
Derive an expression for the gravitational pressure assuming that entropy remains constant during a gravitational collaps.
Hint: Start from dU = T dS − pdV to find the pressure.

c) Use the expression for the internal energy of the neutron gas to derive an expression for
the pressure of this neutron gas. You again may assume that entropy remains constant.
This pressure is also referred to as the Pauli pressure. U_F = (3/5) N E_F [1 + 5(pi^2)/12 (T/T_F)^2]
You may use your earlier findings to justify working in the limit T → 0. This will
simplify the expression for U_F by removing the temperature dependence.

d) Use your results from parts b) and c) to derive an expression for the radius of a neutron
star as a function of its mass.

Relevant Equations

E_f = (hbar^2 / 2m) (3(pi^2)N / V)^(2/3)
U_g = -(3/5)(G M^2 / R).
dU = T dS − pdV
U_F = (3/5) N E_F [1 + 5(pi^2)/12 (T/T_F)^2]

a) V=(4/3)pi(r^3)
N=M/m_n (M=mass of neutron star, m_n=mass of neutron)
Subbed into E_f = (hbar^2 / 2m) (3(pi^2)N / V)^(2/3).
T_F = E_F / k_B

b) dU = (dU/dS)_s dS + (dU/dV)_s dV
p = -(dU/dV)_s dV
V=(4/3)pi(r^3) -> r = cubedroot(3V/4pi)
subbed into U_g = -(3/5)(G M^2 / r)
take (dU/dV)
plug into p = -(dU/dV)_s dV

c)?

d)?

 

[ergospherical]

As you wrote, the gravitational energy is $U_{\mathrm{G}} = - 3 GM^2/5R$ and the gravitational pressure is given by\begin{align*}
p_{\mathrm{G}} = - \dfrac{\partial U_{\mathrm{G}}}{\partial V} \bigg{|}_{S} = - \dfrac{\partial U_{\mathrm{G}}}{\partial R} \bigg{|}_S \dfrac{\partial R}{\partial V} = \dfrac{3GM^2}{20 \pi R^4}
\end{align*}It is given that we are in low temperature regime $T \sim 0$. The energy of the neutron state $n$ is
\begin{align*}
\mathcal{E}(n) = \dfrac{\hbar^2}{2m V^{2/3}} n^2
\end{align*}and the Fermi energy is $\mathcal{E}_{\mathrm{F}} \equiv \mathcal{E}(n_{\mathrm{F}})$ where $n_{\mathrm{F}} = (3N/\pi)^{1/3}$ is the highest filled neutron state. The internal energy of the Fermi gas is an integral over the state space domain $\mathcal{D} = \{ \mathbf{n} \in \mathbf{R}^3 : |\mathbf{n}| \leq n_{\mathrm{F}} : n_x, n_y, n_x \geq 0 \}$, which is a quadrant of a ball.
\begin{align*}
U_{\mathrm{F}} = 2\int_{\mathcal{D}} \mathcal{E}(n) d^3 n = \dfrac{\hbar^2}{mV^{2/3}} \int_{\mathcal{D}} n^4 \ dn \ d\Omega &= \dfrac{\pi \hbar^2}{10mV^{2/3}} n_{\mathrm{F}}^5 \\
&= \dfrac{\pi \hbar^2}{10mV^{2/3}} (3N/\pi)^{5/3} \\
&= \dfrac{3}{5} N\mathcal{E}_{\mathrm{F}}
\end{align*}This is what the expression for $U_{\mathrm{F}}$ given in the problem goes over to in the regime $T \sim 0$. The degeneracy pressure is\begin{align*}
p_{\mathrm{F}} =-\dfrac{\partial U_{\mathrm{F}}}{\partial V} \bigg{|}_S = \dfrac{N\hbar^2}{5m V^{5/3}} n_{\mathrm{F}}^2 = \dfrac{2N}{5V} \mathcal{E}_{\mathrm{F}}
\end{align*}What is the condition relating the gravitational pressure $p_{\mathrm{G}}$ and the degeneracy pressure $p_{\mathrm{F}}$ in an equilibrium state?