The Stellar Blueprint

Part 3: The Four Equations — The Stellar Blueprint

We now write the four stellar structure equations in their standard spherical form.

Equation 1: Mass conservation

Here $M(r)$ is the mass enclosed inside radius $r$ and $\rho(r)$ is the local density. This equation is geometry. A thin shell of radius $r$ and thickness $dr$ has volume $dV = 4\pi r^2\,dr$, so its mass is $dM = \rho\,dV = 4\pi r^2 \rho\,dr$. Divide by $dr$ and you get the differential form above.

Equation 2: Hydrostatic equilibrium

Here $P(r)$ is the pressure and $G$ is Newton’s gravitational constant. This is force balance: the pressure must decrease outward so that the pressure gradient can support the weight of the overlying gas against gravity.

Equation 3: Energy generation

Here $L(r)$ is the luminosity passing through radius $r$, and $\epsilon(r)$ is the nuclear energy generation rate per unit mass, with units $\mathrm{erg\ g^{-1}\ s^{-1}}$. This is energy conservation inside the star: as you move outward through a shell, the luminosity increases by the energy produced in that shell.

Equation 4: Radiative transport

From Reading 4, the radiative-diffusion luminosity comes from the flux

where $a$ is the radiation constant, $c$ is the speed of light, and $\kappa$ is the opacity. Since luminosity is flux times area, $L(r) = 4\pi r^2 F_{\rm rad}$, the radiative temperature gradient can be written as

Using $a = 4\sigma/c$, this is equivalently

This equation tells you how steep the temperature gradient must be if radiation alone carries the luminosity.

At this point, you might think: “I have four equations — I can just use them.” But each equation contains quantities defined by the others. That means none of them can be solved independently. This is not four problems. This is one coupled problem. Do not try to memorize these equations; focus on what each one controls in the star.