5.1-2

Question 1

Degeneracy Pressure in Dense Matter (5)

maybe a better title: degenerate matter

Answer 1

• In white dwarf stars, matter is degenerate
• Electrons are not bound to individual nuclei
• Average electron separation is smaller than in heavy atoms
• Density ρ » 10 g/cm³
• Laws for an ideal electron gas do not apply

Question 2

Heisenberg Uncertainty Principle

Answer 2

For a 6-dimensional phase space, the size of a quantum state is defined by ΔV Δ³p ≥ h³, where h is Planck’s constant.

Question 3

Pauli Exclusion Principle

Answer 3

• Electrons (fermions) cannot occupy the same quantum states
• Each element of phase space has at most 2 fermions (spin up and down)

Question 4

Degeneracy Pressure in High Density

Answer 4

• At high densities, fermions are forced to higher momentum (p) states
• Degeneracy pressure arises when all p-states are occupied up to p_max: ΔV Δ³p = h³

Question 5

Equipartition of Kinetic Energy (2)

what does it tell us?

Answer 5

• p_i² / 2m_i
• Lighter particles have lower p for a given temperature T, less p-states occupied

-> Electrons degenerate first

i guess the first bullet point is just the kinetic energy, not really relevant for the conclusion i think

Question 6

Pressure in a Gas

Answer 6

• Pressure (P) is momentum transfer on the walls of a container
• For an ideal gas: P = n k_B T
• P -> 0 as T -> 0

(where n is particle density, k_B is Boltzmann’s constant, and T is temperature)

Question 7

Number of Electron States in Phase Space

Answer 7

• For a volume V, the number of states is (V/h³) d³p · 2 spin states
• Occupancies are determined by Fermi-Dirac statistics

Question 8

Fermi Momentum

Answer 8

p_F = (3h³/8π)^1/3 n_e^1/3 ∝ n_e^1/3
where n_e is electron density.

good to know that it goes as the thrid root of electron density, whether he cares about the proportionality constant is debatable in my opinon

Question 9

Pressure of Degenerate Electrons (Non-Relativistic)

Answer 9

P = C₁ n_e^5/3

where C₁ = (3^3/2 h²) / (20 π^2/3 m_e).

I dont think he would be intersted in more than that it goes as n_e^5/3

Question 10

Pressure of Degenerate Electrons (Relativistic)

Answer 10

P ∝ n_e^4/3

here i just changed it from (reference to another footnote): **P = C₂ n_e^(4/3)**, where C₂ = (2²/₃ hc) / (8 π¹/₃).

Question 11

Mass-Radius Relation for Degenerate Stars (3)

two types of degenerate asked

Answer 11

• Equilibrium between degeneracy pressure (P_e) and gravitational pressure (P_G)
• Non-relativistic case: R is proportional to 1/M^1/3
• Relativistic electron gas: no stable solution - leads to a maximum mass limit for stable white dwarfs.
• occurs if M ≈ 1.4 M_O

Question 12

Chandrasekhar Mass Limit

Answer 12

The maximum mass for a stable white dwarf star is ≈ 1.4 M_☉.

Question 13

What relations are there (roughly) for initial masses of stars to their end state configuration?

Answer 13

• stars less massive than about 4M_⊙ eventually become white dwarfs
• stars with initial masses in range 4M_⊙ to 10M_⊙ are believed to end up as neutron stars (typically after undergoing supernova explosion)
• stars with initial masses more than 10M_⊙ probabaly cannot shed enough mass to become white dwarfs or neutron stars.
• go on contracting until gravitational attraction so stron not even light can escape = black hole configuration