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--- un:degenerate-matter [2025/10/26 13:00] – created asad
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-====== Degenerate matter ======
-When a gas of fermions becomes dense enough that nearly all low-energy quantum states are filled, it is said to be **degenerate**.
-The resulting pressure — the **degeneracy pressure** — arises not from thermal motion but from quantum mechanical exclusion.
-Such matter occurs naturally in **white dwarfs**, **neutron stars**, and the cores of massive planets.
-{{:courses:ast301:fds.png?nolink&700|Fermi sphere in momentum space for degenerate electrons.}}
-===== Fermi energy and degeneracy pressure =====
-At zero temperature, electrons occupy all momentum states up to the **Fermi momentum** \(p_F\):
-$$
-n_e = \frac{8\pi p_F^3}{3h^3} \Rightarrow
-p_F = h\left(\frac{3n_e}{8\pi}\right)^{1/3}.
-$$
-The corresponding Fermi energy is
-$$
-E_F = \frac{p_F^2}{2m_e} \quad \text{(nonrelativistic)}, \qquad
-E_F = \sqrt{p_F^2 c^2 + m_e^2 c^4} - m_e c^2 \quad \text{(relativistic)}.
-$$
-===== Nonrelativistic equation of state =====
-The pressure of a degenerate gas can be found by integrating over all filled states:
-$$
-E_{av} = \frac{3}{5}\frac{p_F^2}{2m_e}, \quad
-P_e = \frac{2}{3} n_e E_{av}.
-$$
-Hence,
-$$
-P_e = \frac{1}{20}\left(\frac{3}{\pi}\right)^{2/3}
-\frac{h^2}{m_e} n_e^{5/3}.
-$$
-Replacing \(n_e = \rho / (\mu_e m_p)\), where \(\mu_e\) is the **electron molecular weight**, gives
-$$
-P_e = \frac{1}{20}\left(\frac{3}{\pi}\right)^{2/3}
-\frac{h^2}{m_e}\left(\frac{\rho}{\mu_e m_p}\right)^{5/3}.
-$$
-Thus, in a **nonrelativistic degenerate gas**, \(P_e \propto \rho^{5/3}\) and is independent of temperature.
-===== Relativistic equation of state =====
-When the electrons’ momentum becomes relativistic (\(p_F \gtrsim m_e c\)), the average energy per particle is \(E_{av} = \tfrac{3}{4}cp_F\).
-Substituting into the pressure equation:
-$$
-P_e = \frac{1}{8}\left(\frac{3}{\pi}\right)^{1/3}
-ch\left(\frac{\rho}{\mu_e m_p}\right)^{4/3}.
-$$
-In this **relativistic degeneracy** regime, \(P_e \propto \rho^{4/3}\).
-Because pressure grows more slowly with density, the gas becomes **softer** — eventually unable to resist gravity at high mass.
-===== Astrophysical significance =====
-  - In **white dwarfs**, nonrelativistic electron degeneracy pressure supports the star.
-  - As mass increases, the central density rises and electrons become relativistic.
-  - The limiting case is the **Chandrasekhar limit** (\(\sim 1.4\,M_\odot\)), above which degeneracy pressure can no longer balance gravity, leading to collapse into a neutron star.
-  - In **neutron stars**, neutron degeneracy and nuclear repulsion provide the main support.
-  - At even higher densities, all forms of pressure fail, resulting in a **black hole**.
-===== Unified view of equations of state =====
-{{:courses:ast301:eos.png?nolink&600|Different pressure regimes in temperature–density space.}}
-Four major pressure regimes govern stellar interiors:
-  - **Radiation pressure:** \(P = aT^4 / 3\)
-  - **Ideal gas pressure:** \(P = \rho kT / m_{av}\)
-  - **Nonrelativistic degeneracy:** \(P \propto \rho^{5/3}\)
-  - **Relativistic degeneracy:** \(P \propto \rho^{4/3}\)
-At low density and high temperature, radiation dominates;
-at high density and low temperature, degeneracy dominates.
-The ideal gas law applies between these extremes.
-===== Insights =====
-  - Degenerate matter obeys quantum, not thermal, pressure laws.
-  - Nonrelativistic degeneracy yields \(P \propto \rho^{5/3}\); relativistic degeneracy yields \(P \propto \rho^{4/3}\).
-  - The Chandrasekhar limit arises from the softening of relativistic degeneracy.
-  - Degeneracy pressure depends on particle density, not temperature.
-  - The transition between pressure regimes defines the life and death of stars.
-===== Inquiries =====
-  - Derive the expression \(P_e \propto \rho^{5/3}\) for nonrelativistic degeneracy.
-  - Why does relativistic degeneracy lead to a softer EOS?
-  - Explain how degeneracy pressure supports white dwarfs.
-  - How does the Chandrasekhar limit emerge from the balance between gravity and degeneracy pressure?
-  - In what conditions does radiation pressure dominate over degeneracy pressure?