F-X3 · The pressure inside a star

Inside a Star in Hydrostatic Balance

What holds a star up against its own gravity? Pressure rises toward the center, so each layer feels a stronger push from below than from above. That pressure difference balances the inward pull of the mass enclosed by the layer.

Open a stellar cross-section and drag a shell from the core to the surface. Compare the inward pull of gravity with the outward push from the pressure gradient.

Central / mean density $1$

Enclosed mass $1$

Shell pressure $1$

Shell gravity $1$

The model solves the spherical Newtonian Lane–Emden equation for polytropic indices from zero to four. Mass and radius set the density and pressure scales; neither is inferred from an observation. Density and pressure colors use separate normalized scales. Equal arrows show the two opposing accelerations at the selected radius. Their common length is normalized to the strongest interior gravity. This static model contains no nuclear burning, opacity, radiation transport, composition changes, rotation, magnetic fields or relativistic gravity. A polytropic index describes a pressure–density relation; it does not give a stellar age. The zero-index case is an incompressible mathematical reference. The cross-section depicts density, not a photograph or a temperature map.

Numerical hydrostatic solution

How the star changes with depth

Density and pressure

r/R,\quad \rho/\rho_c,\quad P/P_c

Gold: density. Pink: pressure. Each is divided by its own central value. The bright marker follows the selected shell.

How much mass lies inside a shell

r/R,\quad m(r)/M

Teal: the mass enclosed by radius. The surface contains the whole stellar mass.

Read the discoveries behind these models →

Start with the essentials

Focus question: What holds a star up against its own gravity?
One-sentence intuition: A pressure difference between the two faces of each layer balances its weight.

Core mathematical model

Balance at each radius

\frac{dP}{dr}=-\rho(r)\frac{Gm(r)}{r^2}

Pressure P decreases outward. Density is rho, the mass inside radius r is m, and G is the gravitational constant. The pressure gradient supplies an outward force per volume equal to the inward gravitational force.

A polytropic pressure law

P=K\rho^{1+1/n}

The index n sets the shape of the pressure–density relation. The constant K follows from the selected mass and radius. The uniform reference is the incompressible limit at index zero.

Common difficulties

Pressure and force

Typical misconceptionHigh pressure pushes the gas outward by itself.

Better mental modelPressure acts from every direction. The outward force comes from the difference between the stronger pressure below a layer and the weaker pressure above it.

Run the experiment

01
Look below the surface
Choose Broad interior and drag the bright shell toward the center.
What to observe: The pressure and enclosed mass change smoothly. The two arrows stay equal because every layer is in equilibrium.
02
Concentrate the mass
Choose Dense core. Compare its density profile with Broad interior at the same mass and radius.
What to observe: Much more mass sits near the center. The central density rises relative to the mean density.
03
Compress the star
Reduce Radius in Suns while leaving the mass fixed.
What to observe: The central pressure rises steeply. A smaller sphere needs stronger internal pressure to support the same mass.