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- 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)?

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)?