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courses:ast402:stellar-structure [2026/06/07 22:04] – shuvocourses:ast402:stellar-structure [2026/06/07 22:08] (current) – [Stellar Models and Simulation] shuvo
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 **Timescales:** **Timescales:**
  
-Three primary timescales characterize stellar life:+Three primary timescales characterize stellar life:\\
 1.  **Dynamic (Free-fall) Timescale:** The time for a cloud to collapse under gravity if pressure is removed: $t_{ff} = \sqrt{\frac{3\pi}{32} \frac{1}{G \rho_0}}$.\\ 1.  **Dynamic (Free-fall) Timescale:** The time for a cloud to collapse under gravity if pressure is removed: $t_{ff} = \sqrt{\frac{3\pi}{32} \frac{1}{G \rho_0}}$.\\
 2.  **Thermal (Kelvin-Helmholtz) Timescale:** The time to radiate away a star's total gravitational energy: $t_{KH} \approx \frac{3}{10} \frac{GM^2}{RL}$.\\ 2.  **Thermal (Kelvin-Helmholtz) Timescale:** The time to radiate away a star's total gravitational energy: $t_{KH} \approx \frac{3}{10} \frac{GM^2}{RL}$.\\
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 Nucleosynthesis is the sequence of nuclear reactions that transform elements: Nucleosynthesis is the sequence of nuclear reactions that transform elements:
 Hydrogen Burning: Occurs via the **proton-proton (pp) chains** (dominant in low-mass stars) or the **CNO cycle** (dominant in massive stars).\\ Hydrogen Burning: Occurs via the **proton-proton (pp) chains** (dominant in low-mass stars) or the **CNO cycle** (dominant in massive stars).\\
-Helium Burning: Occurs via the **triple-alpha process** ($3\alpha \to ^{12}C$) at temperatures $\sim 10^8$ K. +Helium Burning: Occurs via the **triple-alpha process** ($3\alpha \to ^{12}C$) at temperatures $\sim 10^8$ K.\\ 
-Advanced Burning:** Successive stages (carbon, oxygen, neon, and silicon burning) produce elements up to the **iron peak**. Elements heavier than iron are produced via the **s-process** (slow neutron capture) or **r-process** (rapid neutron capture).+Advanced Burning: Successive stages (carbon, oxygen, neon, and silicon burning) produce elements up to the **iron peak**. Elements heavier than iron are produced via the **s-process** (slow neutron capture) or **r-process** (rapid neutron capture).
  
 ===== Energy Transport and Thermodynamics ===== ===== Energy Transport and Thermodynamics =====
  
 Energy is transported from the core to the surface via **radiation, convection, or conduction**. Energy is transported from the core to the surface via **radiation, convection, or conduction**.
-*   **Radiative Transport:** Driven by the radiation pressure gradient:+**Radiative Transport:** Driven by the radiation pressure gradient:
 $$\frac{dT}{dr} = -\frac{3}{4ac} \frac{\kappa \rho}{T^3} \frac{L_r}{4\pi r^2}$$ $$\frac{dT}{dr} = -\frac{3}{4ac} \frac{\kappa \rho}{T^3} \frac{L_r}{4\pi r^2}$$
 where $\kappa$ is the opacity. where $\kappa$ is the opacity.
-*   **Convection:** Occurs when the temperature gradient is **superadiabatic**. The **Schwarzschild criterion** for convection (neglecting composition changes) is:+ 
 +**Convection:** Occurs when the temperature gradient is **superadiabatic**. The **Schwarzschild criterion** for convection (neglecting composition changes) is:
 $$\left| \frac{dT}{dr} \right|_{act} > \left| \frac{dT}{dr} \right|_{ad}$$ $$\left| \frac{dT}{dr} \right|_{act} > \left| \frac{dT}{dr} \right|_{ad}$$
 where the adiabatic gradient for an ideal gas is $\frac{dT}{dr}|_{ad} = -\left(1 - \frac{1}{\gamma}\right) \frac{\mu m_H}{k} \frac{GM_r}{r^2}$. where the adiabatic gradient for an ideal gas is $\frac{dT}{dr}|_{ad} = -\left(1 - \frac{1}{\gamma}\right) \frac{\mu m_H}{k} \frac{GM_r}{r^2}$.
-*   **Thermodynamics:** The **first law** ($dU = dQ - dW$) relates internal energy changes to heat and work. For an adiabatic process, $P V^\gamma = \text{constant}$.+ 
 +**Thermodynamics:** The **first law** ($dU = dQ - dW$) relates internal energy changes to heat and work. For an adiabatic process, $P V^\gamma = \text{constant}$.
  
 ===== Stellar Models and Simulation ===== ===== Stellar Models and Simulation =====
  
-**Stellar models** are constructed by numerically solving the four fundamental differential equations (hydrostatic equilibrium, mass conservation, energy generation, and energy transport).  +Stellar models are constructed by numerically solving the four fundamental differential equations (hydrostatic equilibrium, mass conservation, energy generation, and energy transport). \\ 
-*   **Numerical Modeling:** The star is divided into **concentric shells (zones)**, and differential equations are replaced by **difference equations**. Solutions require matching **boundary conditions** at the center ($M_r \to 0, L_r \to 0$) and surface ($P \to 0, T \to 0$). + 
-*   **Vogt-Russell Theorem:** States that a star's mass and composition uniquely determine its structure and evolution. +**Numerical Modeling:** The star is divided into **concentric shells (zones)**, and differential equations are replaced by **difference equations**. Solutions require matching **boundary conditions** at the center ($M_r \to 0, L_r \to 0$) and surface ($P \to 0, T \to 0$).\\ 
-*   **Polytropes:** Simplified analytical models where pressure is a power of density ($P = K\rho^{(n+1)/n}$); they are solved using the **Lane-Emden equation**:+ 
 +**Vogt-Russell Theorem:** States that a star's mass and composition uniquely determine its structure and evolution.\\ 
 +**Polytropes:** Simplified analytical models where pressure is a power of density ($P = K\rho^{(n+1)/n}$); they are solved using the **Lane-Emden equation**:
 $$\frac{1}{\xi^2} \frac{d}{d\xi} \left( \xi^2 \frac{dD_n}{d\xi} \right) = -D_n^n$$ $$\frac{1}{\xi^2} \frac{d}{d\xi} \left( \xi^2 \frac{dD_n}{d\xi} \right) = -D_n^n$$
 where $\xi$ is a dimensionless radius and $D_n$ is a dimensionless density function. where $\xi$ is a dimensionless radius and $D_n$ is a dimensionless density function.
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