2.1 Equations of Stellar Structure

Stellar structure equations form the foundation of understanding how stars function. These mathematical relationships describe the delicate balance of forces and energy within a star, from its dense core to its outer layers.

Hydrostatic equilibrium, mass conservation, energy conservation, and the equation of state work together to paint a complete picture of a star's interior. These equations help us unravel the mysteries of stellar evolution and behavior across the cosmos.

Fundamental Equations

Hydrostatic Equilibrium and Mass Conservation

Top images from around the web for Hydrostatic Equilibrium and Mass Conservation

• 

  Evolution from the Main Sequence to Red Giants | Astronomy View original

  Is this image relevant?

• 

  The Solar Interior: Theory | Astronomy View original

  Is this image relevant?

• 

  Evolution from the Main Sequence to Red Giants | Astronomy View original

  Is this image relevant?

• 

  The Solar Interior: Theory | Astronomy View original

  Is this image relevant?

1 of 2

Top images from around the web for Hydrostatic Equilibrium and Mass Conservation

• 

  Evolution from the Main Sequence to Red Giants | Astronomy View original

  Is this image relevant?

• 

  The Solar Interior: Theory | Astronomy View original

  Is this image relevant?

• 

  Evolution from the Main Sequence to Red Giants | Astronomy View original

  Is this image relevant?

• 

  The Solar Interior: Theory | Astronomy View original

  Is this image relevant?

1 of 2

• Hydrostatic equilibrium describes the balance between gravitational force and pressure gradient in a star
• Expressed mathematically as $\frac{dP}{dr} = -\frac{GM_r\rho}{r^2}$
• $P$ represents pressure, $r$ denotes radius, $G$ stands for gravitational constant, $M_r$ signifies mass within radius $r$, and $\rho$ indicates density
• Crucial for maintaining stellar stability prevents collapse or expansion
• Mass conservation equation ensures mass remains constant within spherical shells
• Represented by $\frac{dM_r}{dr} = 4\pi r^2 \rho$
• Allows calculation of mass distribution throughout the star's interior

Energy Conservation and Equation of State

• Energy conservation governs energy flow through stellar layers
• Expressed as $\frac{dL_r}{dr} = 4\pi r^2 \rho \epsilon$
• $L_r$ denotes luminosity at radius $r$, $\epsilon$ represents energy generation rate per unit mass
• Accounts for energy produced by nuclear reactions and gravitational contraction
• Equation of state relates pressure, density, and temperature in stellar material
• For ideal gas, expressed as $P = \frac{\rho k_B T}{\mu m_H}$
• $k_B$ represents Boltzmann constant, $T$ indicates temperature, $\mu$ denotes mean molecular weight, $m_H$ stands for mass of hydrogen atom
• Varies depending on stellar composition and physical conditions (degenerate matter, radiation pressure)

Stellar Structure

Pressure and Temperature Gradients

• Pressure gradient describes how pressure changes with radius inside a star
• Determined by hydrostatic equilibrium equation
• Steeper gradients indicate stronger gravitational compression
• Temperature gradient represents change in temperature with radius
• Influenced by energy transport mechanisms (radiation, convection, conduction)
• In radiative zones, described by $\frac{dT}{dr} = -\frac{3}{4ac} \frac{\kappa \rho L_r}{4\pi r^2 T^3}$
• $\kappa$ denotes opacity, $a$ represents radiation constant, $c$ stands for speed of light
• Adiabatic temperature gradient occurs in convective zones

Density Profile and Stellar Layers

• Density profile shows how density varies from core to surface
• Typically decreases outward due to gravitational stratification
• Core density can reach $10^5$ kg/m³ in main sequence stars
• Envelope density may be as low as $10^{-6}$ kg/m³
• Stellar structure divided into distinct layers (core, radiative zone, convective zone, photosphere)
• Each layer characterized by different physical processes and energy transport mechanisms
• Layer boundaries determined by changes in opacity, temperature gradient, or chemical composition

Energy Transport and Opacity

Radiative and Convective Energy Transport

• Radiative transport involves energy transfer through photon diffusion
• Dominates in stellar interiors where material highly ionized
• Efficiency depends on temperature gradient and opacity
• Convective transport occurs through bulk motion of stellar material
• Becomes important when temperature gradient exceeds adiabatic gradient (Schwarzschild criterion)
• Convection more efficient in cooler, outer layers of low-mass stars
• Energy transport mechanism affects stellar structure and evolution

Opacity Sources and Rosseland Mean Opacity

• Opacity measures resistance of stellar material to radiation passage
• Major opacity sources include bound-free absorption (photoionization), free-free absorption (bremsstrahlung), electron scattering
• Opacity varies with temperature, density, and chemical composition
• Rosseland mean opacity provides frequency-averaged measure of opacity
• Expressed as $\frac{1}{\kappa_R} = \frac{\int_0^\infty \frac{1}{\kappa_\nu} \frac{\partial B_\nu}{\partial T} d\nu}{\int_0^\infty \frac{\partial B_\nu}{\partial T} d\nu}$
• $\kappa_\nu$ represents frequency-dependent opacity, $B_\nu$ denotes Planck function
• Crucial for determining radiative energy transport and stellar structure calculations