Stellar Interior Structure: How Stars Support Their Own Weight

Hydrostatic equilibrium: pressure versus weight

Take a thin spherical shell of stellar material at radius r. Its weight pulls inward; the difference in pressure across it pushes outward. Setting the two forces equal gives the single equation that underlies all of stellar structure theory:

dP/dr = −G·M(r)·ρ(r) / r²        (hydrostatic equilibrium)

Sun's centre:  P ≈ 2.5×10¹⁶ Pa  (≈250 billion atmospheres)
               T ≈ 1.5×10⁷ K

Pressure must decrease outward — the interior of a star is crushed to that pressure precisely because it has to support everything sitting above it. A consequence of this balance, the virial theorem, states 2E_th + E_grav = 0: when a proto-star contracts and radiates energy away, half the released gravitational energy heats the interior and the other half escapes as light. Stars get hotter as they lose energy — they have negative heat capacity.

Four equations, one star

A complete spherically-symmetric stellar model needs four coupled first-order equations: mass continuity (dM/dr = 4πr²ρ), hydrostatic balance, energy generation (dL/dr = 4πr²ρε, from nuclear burning), and energy transport, either radiative or convective depending on local conditions. Closing the system also requires an equation of state P(ρ,T) and prescriptions for opacity κ(ρ,T) and the nuclear energy generation rate ε(ρ,T) — with boundary conditions P = L = ρ = 0 at the surface and M = L = 0 at the centre.

The Lane-Emden equation and polytropes

Before computers, astronomers needed analytic shortcuts. Assuming a polytropic equation of state P = Kρ^(1+1/n) collapses the four structure equations into one dimensionless second-order ODE, the Lane-Emden equation, solved outward from θ(0)=1 until the dimensionless density θ first hits zero at the stellar surface.

θ″ + (2/ξ)·θ′ + θⁿ = 0        θ(0)=1,  θ′(0)=0

n=0 (incompressible):  θ = 1 − ξ²/6,        ξ₁ = √6 ≈ 2.449
n=1:                    θ = sin(ξ)/ξ,        ξ₁ = π
n=5:                    θ = (1+ξ²/3)^(−1/2), ξ₁ → ∞ (infinite radius, finite mass)

n = 3/2 (γ = 5/3) describes fully convective stars and non-relativistic white dwarfs; n = 3 (γ = 4/3) is Eddington's standard model for radiation-pressure-dominated massive stars, and the same index marks the Chandrasekhar mass limit M_Ch = 5.83 M☉/μₑ² for relativistic white dwarfs. Eddington showed in 1926 that the Sun itself is well approximated by n = 3, predicting a central temperature near 15 million K and central density near 150 g/cm³ — figures helioseismology later confirmed to high precision.

Opacity, convection, and the Schwarzschild criterion

Radiation carries energy outward as photons random-walk through the plasma, scattering off electrons and being absorbed and re-emitted by ions — a journey from the Sun's core to the base of its convection zone that takes roughly 170,000 years. When opacity κ gets too high, the radiative temperature gradient needed to push the required luminosity outward becomes steeper than a rising gas parcel's own adiabatic cooling rate — the Schwarzschild criterion, ∇_rad > ∇_ad — and convection takes over instead. In the Sun that happens above about 0.72 R☉, where partial ionisation of helium spikes the opacity; the resulting convective envelope moves energy the rest of the way to the surface in about a month, not 170,000 years, and produces the granulation visible on the solar surface.

Helioseismology: proof from inside the Sun

The Sun oscillates in millions of acoustic modes whose frequencies, measured as Doppler shifts at the photosphere, encode the sound-speed profile at every depth. Inverting those frequencies reconstructs the interior to better than 0.1% accuracy, confirming the radiative-convective boundary at exactly 0.713 R☉. The same technique — asteroseismology — now runs on thousands of other stars via Kepler and TESS. It even helped resolve the decades-old solar neutrino problem: measurements found only a third of the predicted neutrino flux until 2001, when the Sudbury Neutrino Observatory proved electron neutrinos were oscillating into other flavours en route to Earth — work that won Art McDonald the 2015 Nobel Prize.

Frequently asked questions

Try it live

Everything above runs in your browser — open Stellar Interior Structure, pick a polytropic index n, and drag the central density and mass sliders to watch the density, pressure, temperature and luminosity profiles rebuild in real time. Nothing is installed, nothing is uploaded.