Astrophysics: Stellar Structure & the HR Diagram
Hydrostatic equilibrium, energy transport, and where stars live on the HR diagram.
A star is a self-gravitating ball of plasma in long-term equilibrium. Four governing equations:
$$\frac{dP}{dr} = -\frac{G m(r) \rho}{r^2}, \qquad \frac{dm}{dr} = 4\pi r^2 \rho,$$ $$\frac{dL}{dr} = 4\pi r^2 \rho \epsilon, \qquad \frac{dT}{dr} = -\frac{3\kappa \rho L}{16\pi a c\, r^2 T^3}.$$(The last line assumes radiative diffusion; convection has its own equation.) Energy generation $\epsilon$ comes from nuclear burning, primarily the pp-chain and CNO cycle. Together with an equation of state and opacity table, these specify $\rho(r), T(r), L(r), m(r)$.
The Hertzsprung–Russell diagram plots luminosity $L$ vs surface temperature $T_{\rm eff}$. Main-sequence stars (burning hydrogen) form a diagonal band; red giants populate the upper right, white dwarfs the lower left. Stellar lifetimes scale as $\tau \sim M/L \propto M^{-2.5}$ — high-mass stars are short-lived.