Stellar structure and hydrostatic equilibrium

A star as a self-gravitating gas

A star is a massive sphere of hot plasma held together by gravity. Gravity pulls every layer inward, while pressure from hot gas and radiation pushes outward. For most of a star's life, these effects nearly balance. This balance is called hydrostatic equilibrium.

Without pressure, a star would collapse under its own gravity. Without gravity, it would expand into space.

Hydrostatic equilibrium

Consider a thin spherical shell inside a star at radius $r$. The pressure below the shell must be slightly higher than the pressure above it to support the weight of overlying material. The equation of hydrostatic equilibrium is

ho(r)}{r^2}$$ Here $P$ is pressure, $ ho$ is density, and $m(r)$ is the mass enclosed within radius $r$. The negative sign means pressure decreases outward. ## Mass conservation The enclosed mass increases with radius according to $$rac{dm}{dr}=4pi r^2 ho(r)$$ Together, hydrostatic equilibrium and mass conservation describe how gravity and pressure are distributed inside the star. ## Equation of state To close the system, we need an equation of state relating pressure, density, and temperature. For an ideal gas, $$P=rac{ ho k_BT}{mu m_H}$$ where $mu$ is the mean molecular weight and $m_H$ is the mass of a hydrogen atom. In very hot stars, radiation pressure can also be important: $$P_{rad}=rac{1}{3}aT^4$$ where $a$ is the radiation density constant. ## Energy generation Stars shine because they generate energy in their cores, mainly through nuclear fusion. The luminosity passing through radius $r$ changes according to $$rac{dL}{dr}=4pi r^2 hoepsilon$$ where $epsilon$ is the energy generation rate per unit mass. In main-sequence stars, fusion is concentrated near the core because temperature and density are highest there. ## Energy transport Energy moves outward by radiation, convection, or conduction. In radiative transport, photons diffuse outward through repeated absorption and reemission. In convection, hot material rises and cooler material sinks. The temperature gradient determines which mechanism dominates. If the temperature gradient is steep enough, convection begins because rising fluid elements remain warmer and less dense than their surroundings. ## Stellar timescales The dynamical timescale estimates how quickly a star would collapse if pressure support vanished: $$t_{dyn}simsqrt{rac{R^3}{GM}}$$ For the Sun, this is about an hour. The Sun survives for billions of years because pressure and gravity are balanced and nuclear fusion supplies energy. ## The big idea A star is a self-gravitating plasma in hydrostatic equilibrium. Its structure is determined by gravity, pressure, mass conservation, energy generation, and energy transport. These principles explain why stars are stable, why their cores are hot, and why their observable properties depend strongly on mass.