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un:equations-of-state [2025/10/26 23:26] – created asadun:equations-of-state [2025/10/26 23:27] (current) – asad
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 ====== Equations of state ====== ====== Equations of state ======
  
-The **equation of state (EOS)** links the thermodynamic quantities that describe a gas — typically pressure (\(P\)), density (\(\rho\)), and temperature (\(T\)).   +The **equation of state (EOS)** defines the relationship between the pressure (\(P\)), density (\(\rho\)), and temperature (\(T\)) of matter.   
-It determines how matter responds to compression, heating, and cooling, and thus plays a central role in the structure and evolution of stars and planets.+It determines how gas responds to compression or heating and thus governs the **structure, stability, and evolution** of stars and planets.
  
-In stellar interiors, several distinct pressure regimes operate depending on temperature and density:   +Different physical processes dominate in different regimes of temperature and density.   
-ideal gas pressure, radiation pressure, and the quantum mechanical pressures of degenerate matter.+Four principal forms of pressure occur in astrophysical interiors:   
 +**ideal gas**, **radiation**, **nonrelativistic degeneracy**, and **relativistic degeneracy**.
  
 ===== Ideal gas and radiation pressure ===== ===== Ideal gas and radiation pressure =====
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 $$ $$
  
-where \(m_{av}\) is the average particle mass and \(k\) is the Boltzmann constant.   +where \(k\) is the Boltzmann constant and \(m_{av}\) is the average particle mass.   
-This equation of state applies to most main-sequence stars, where thermal motion dominates.+This law describes the interiors of most main-sequence stars, where thermal motion dominates the pressure.
  
 At even higher temperatures, photons contribute significantly to the total pressure.   At even higher temperatures, photons contribute significantly to the total pressure.  
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 where \(a\) is the radiation constant.   where \(a\) is the radiation constant.  
-In the hottest, most luminous stars, radiation pressure can rival or exceed gas pressure.+In massive, luminous stars, radiation pressure may equal or even exceed the gas pressure, driving **stellar winds** and influencing stability.
  
-===== Degenerate matter ======+===== Degenerate matter =====
  
-When matter becomes dense enough that quantum effects dominate, fermions such as electrons or neutrons fill nearly all low-energy quantum states.   +At very high densities and comparatively low temperatures, quantum effects dominate.   
-The resulting **degeneracy pressure** arises not from temperature but from the Pauli exclusion principle.   +Fermions such as electrons or neutrons fill nearly all available low-energy quantum states, creating a **degenerate gas**.   
-This pressure supports compact objects like **white dwarfs**, **neutron stars**, and the cores of giant planets.+Its pressure arises from the **Pauli exclusion principle**, not from thermal motion.   
 +This **degeneracy pressure** supports compact objects like **white dwarfs**, **neutron stars**, and the dense cores of giant planets.
  
 ===== Nonrelativistic degeneracy ===== ===== Nonrelativistic degeneracy =====
  
-For a nonrelativistic gas of electrons, the average energy per particle is+For a nonrelativistic electron gas, the average energy per particle is
  
 $$ $$
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 $$ $$
  
-where \(p_F\) is the **Fermi momentum** and \(m_e\) is the electron mass.   +where \(p_F\) is the **Fermi momentum** and \(m_e\) the electron mass.   
-The pressure is then+The pressure is obtained by integrating over all occupied momentum states:
  
 $$ $$
-P_e = \frac{2}{3} n_e E_{av}+P_e = \frac{2}{3}n_e E_{av}
 = \frac{1}{20}\left(\frac{3}{\pi}\right)^{2/3}\frac{h^2}{m_e} n_e^{5/3}, = \frac{1}{20}\left(\frac{3}{\pi}\right)^{2/3}\frac{h^2}{m_e} n_e^{5/3},
 $$ $$
  
-where \(n_e\) is the number density of electrons.   +where \(n_e\) is the electron number density.   
-Substituting \(n_e = \rho / (\mu_e m_p)\), with \(\mu_e\) the **electron molecular weight**, gives+Substituting \(n_e = \rho / (\mu_e m_p)\), where \(\mu_e\) is the **electron molecular weight**, gives
  
 $$ $$
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 $$ $$
  
-Thus, in a **nonrelativistic degenerate gas**, \(P_e \propto \rho^{5/3}\), and the pressure is **independent of temperature**.   +Thus, for **nonrelativistic degeneracy**, \(P_e \propto \rho^{5/3}\).   
-This law provides the main pressure support in white dwarfs of low to intermediate mass.+This pressure is **independent of temperature**, providing the main support for **white dwarfs** of low and intermediate mass.
  
 ===== Relativistic degeneracy ===== ===== Relativistic degeneracy =====
  
-At very high densities, the electrons become relativistic (\(p_F \gtrsim m_e c\)), and their average energy per particle is approximately+At extremely high densities, the electrons’ momenta become relativistic (\(p_F \gtrsim m_e c\)).   
 +The average energy per particle approaches
  
 $$ $$
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 In this **relativistic regime**, \(P_e \propto \rho^{4/3}\).   In this **relativistic regime**, \(P_e \propto \rho^{4/3}\).  
-Because pressure now increases more slowly with density, the gas becomes **softer** — it cannot indefinitely resist gravitational compression.   +Because pressure rises more slowly with density, the gas becomes **softer** — it cannot indefinitely oppose gravity.   
-This softening leads directly to the **Chandrasekhar limit**, the maximum mass (\(\sim 1.4\,M_\odot\)) that can be supported by electron degeneracy pressure. +This softening leads to the **Chandrasekhar limit** (\(\approx 1.4\,M_\odot\)), above which electron degeneracy pressure fails and collapse ensues, forming a **neutron star** or, at still higher densities, a **black hole**.
- +
-In more massive remnants, electrons merge with protons to form neutrons, producing **neutron stars**, which are supported by neutron degeneracy and nuclear repulsion.   +
-At still higher densities, even these pressures fail, leading to the formation of **black holes**.+
  
 ===== Unified view of equations of state ===== ===== Unified view of equations of state =====
  
-{{:courses:ast301:eos.png?nolink&600|Different pressure regimes in temperature–density space.}}+{{:courses:ast301:eos.png?nolink&600|Temperature–density diagram showing dominant pressure regimes: radiation, ideal gas, and degeneracy. The solid lines mark approximate boundaries between these regimes.}}
  
-The figure above shows the dominant pressure regimes across temperature–density space:+The figure above shows the dominant **pressure regimes** across temperature–density space:
  
   - **Radiation pressure:** \(P = aT^4 / 3\)     - **Radiation pressure:** \(P = aT^4 / 3\)  
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   - **Relativistic degeneracy:** \(P \propto \rho^{4/3}\)   - **Relativistic degeneracy:** \(P \propto \rho^{4/3}\)
  
-At low density and high temperature, radiation and ideal gas laws dominate.   +At **low density and high temperature**, matter behaves as a **radiative or ideal gas**.   
-At high density and low temperature, degeneracy pressure takes over.   +At **high density and low temperature**, it becomes **degenerate**, with quantum mechanical pressure independent of temperature.   
-Between these extremes, both effects can coexist — for instance, in the cores of massive white dwarfs or brown dwarfs transitioning between ideal and degenerate conditions.+In **intermediate regions**, multiple contributions coexist — for example, in massive white dwarfs, both degeneracy and radiation pressures shape the stellar structure. 
 + 
 +As density increases along an isotherm, the effective equation of state transitions smoothly from \(P \propto \rho T\) to \(P \propto \rho^{5/3}\), and finally to \(P \propto \rho^{4/3}\).   
 +This sequence determines how stars evolve, collapse, and reach equilibrium at different stages of their life cycles.
  
 ===== Insights ===== ===== Insights =====
-  - The equation of state defines how pressure responds to changes in density and temperature.   +  - The equation of state defines how pressure responds to density and temperature, determining stellar structure.   
-  - Classical gases follow \(P \propto \rho T\), while degenerate matter obeys quantum power laws.   +  - Classical gases follow \(P \propto \rho T\); quantum-degenerate matter follows power laws with fixed exponents.   
-  - Nonrelativistic degeneracy gives \(P \propto \rho^{5/3}\); relativistic degeneracy gives \(P \propto \rho^{4/3}\).   +  - Nonrelativistic degeneracy yields \(P \propto \rho^{5/3}\); relativistic degeneracy yields \(P \propto \rho^{4/3}\).   
-  - Degeneracy pressure depends on density, not temperature, and stems from the Pauli exclusion principle.   +  - Degeneracy pressure arises from the exclusion principle and depends only on density, not temperature.   
-  - The Chandrasekhar limit arises from the softening of the relativistic EOS.   +  - The Chandrasekhar limit results from the softening of the relativistic EOS.   
-  - The transition between radiation, gas, and degeneracy pressures governs the life and death of stars.+  - Transitions between radiation, gas, and degeneracy regimes govern stellar formation, evolution, and endpoints.
  
 ===== Inquiries ===== ===== Inquiries =====
-  - Derive the relation \(P_e \propto \rho^{5/3}\) for a nonrelativistic degenerate gas.   +  - Derive \(P_e \propto \rho^{5/3}\) for a nonrelativistic degenerate gas.   
-  - Why does relativistic degeneracy yield \(P_e \propto \rho^{4/3}\)?   +  - Explain why relativistic degeneracy produces \(P_e \propto \rho^{4/3}\).   
-  - Explain how degeneracy pressure supports white dwarfs and defines the Chandrasekhar limit.   +  - How does the Chandrasekhar limit emerge from the relativistic EOS?   
-  - In what conditions does radiation pressure dominate over degeneracy pressure?   +  - Under what conditions does radiation pressure dominate over degeneracy pressure?   
-  - Describe how the equation of state determines the stability of stellar remnants.+  - Discuss how changes in the EOS determine the final states of stars.
  
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