Fermion Gas Problems: Avg Speed & Density Comparison

[ben133]

Homework Statement

Consider the collapse of the Sun into a white dwarf. For the Sun, $$M = 2 \times 10^{30} kg$$,
$$R = 7 \times 10^{8}m$$, $$V = 1.4 \times 10^{27}m^{3}$$ .

(c) What is the average speed of the electrons in the fermion gas?

(d) What is the density of the electron gas? Compare it with the density of water?

Homework Equations

(Note: For a
star with mass greater than a critical value of 1.44 times the mass of the Sun, the collapse
continues beyond the white dwarf stage as gravitational attraction overcomes the electron
pressure. The result is a very dense neutron star that eventually ends in a supernova
explosion.) ($$0.39 kg m^{3}$$)

The Attempt at a Solution

I must admit, I haven't a clue. I've read through my textbook (Huang) and can't find any hints on were to start off. Any help with a starting point folks?

 

[mark726]

I would first like to clarify the question and make sure I understand the context correctly. The forum post is discussing the collapse of the Sun into a white dwarf, and the question is asking for the average speed of electrons in the fermion gas and the density of the electron gas. The given values for mass, radius, and volume are for the Sun, and the note mentions a critical value for the mass of a star that determines whether it will collapse into a white dwarf or continue collapsing into a neutron star. Is this correct?

Assuming my understanding is correct, we can start by looking at the properties of a white dwarf. A white dwarf is a highly dense star made up of degenerate fermions, mainly electrons. This means that the electrons are packed so closely together that they are no longer able to occupy the same energy states due to the Pauli exclusion principle.

To find the average speed of the electrons in the fermion gas, we can use the formula for Fermi velocity: v = (hc/4π)(3n/8π)^1/3, where h is Planck's constant, c is the speed of light, and n is the number density of electrons. To find n, we can use the formula for number density: n = N/V, where N is the total number of electrons and V is the volume of the gas.

Plugging in the given values for the Sun, we get:
n = (M/m_e)(1/V) = (2 x 10^30 kg)/(9.11 x 10^-31 kg)(1.4 x 10^27 m^3) = 1.74 x 10^33 m^-3

Using this value for n in the formula for Fermi velocity, we get:
v = (hc/4π)(3(1.74 x 10^33)/8π)^1/3 = 2.53 x 10^6 m/s

This is the average speed of the electrons in the fermion gas in a white dwarf.

To find the density of the electron gas, we can use the formula for density: ρ = m/V, where m is the mass of the gas and V is the volume. In this case, the mass of the gas is the same as the mass of the Sun. So, we get:
ρ = (2 x 10^30 kg)/(1.4 x