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Equations of stellar structure
A star is a self-gravitating sphere of plasma in hydrostatic and thermal equilibrium. Its internal structure is governed by a set of seven fundamental equations that link the distributions of pressure, density, temperature, luminosity, and mass with radius. These equations express conservation of mass, momentum, and energy, together with the laws of heat transfer and the thermodynamic state of stellar matter.
1. Hydrostatic equilibrium
The inward pull of gravity is balanced by the outward pressure gradient at every layer inside the star:
$$ \frac{dP}{dr} = -\rho(r)\, g(r) = -\rho(r)\, \frac{G M(r)}{r^2}. $$
Here \(P(r)\) is the local pressure, \(\rho(r)\) the density, \(M(r)\) the enclosed mass, and \(G\) the gravitational constant. This equation ensures that the net force on a small mass element is zero in a stable star.
2. Conservation of mass
The rate at which mass increases with radius equals the density times the area of the spherical shell:
$$ \frac{dM}{dr} = 4\pi r^2 \rho(r). $$
Integrating from the center to the surface gives the total stellar mass \(M(R)\).
3. Conservation of energy (luminosity distribution)
The energy generated inside a shell of thickness \(dr\) adds to the outward luminosity \(L(r)\):
$$ \frac{dL}{dr} = 4\pi r^2 \rho(r)\, \epsilon(\rho,T), $$
where \(\epsilon(\rho,T)\) is the energy generation rate (in W kg\(^{-1}\)), mainly from nuclear reactions. At the stellar surface \(L(R)\) equals the total luminosity of the star.
4. Temperature gradient: radiative transfer
In regions where energy is transported primarily by radiation, the temperature decreases outward according to
$$ \frac{dT}{dr} = -\,\frac{3\kappa\rho L}{16\pi a c T^3 r^2}, $$
where \(\kappa\) is the opacity (m\(^2\) kg\(^{-1}\)), \(a\) is the radiation constant, and \(c\) the speed of light. A higher luminosity or greater opacity requires a steeper temperature gradient to carry the same energy flux. (This is the full radiative transfer equation—its derivation and detailed discussion are given separately in Heat transfer in stars.)
5. Temperature gradient: convective transfer
When the radiative gradient exceeds the adiabatic one, convection becomes efficient. The adiabatic temperature gradient is approximately
$$ \left(\frac{dT}{dr}\right)_{\!ad} = -\left(1 - \frac{1}{\gamma}\right)\frac{\mu m_p g}{k}\,T, $$
where \(\gamma = C_P/C_V\) is the ratio of specific heats, \(\mu\) the mean molecular weight, and \(m_p\) the proton mass. Convection dominates wherever the surrounding radiative gradient is steeper than this adiabatic value. (Full derivations and mixing-length formulations are covered in the article Heat transfer in stars.)
6. Equation of state
The gas pressure relates density and temperature through the ideal-gas law:
$$ P = \frac{\rho kT}{m_{av}} = \frac{\rho kT}{\mu m_p}, $$
where \(m_{av}\) is the mean particle mass and \(\mu\) the mean molecular weight. This equation links the thermodynamic state of the plasma with its composition.
7. Ionization balance (Saha equation)
The ionization state of the stellar gas determines both opacity and mean molecular weight. For an element in thermal equilibrium, the Saha equation gives the ratio of consecutive ionization stages:
$$ \frac{n_{i+1}}{n_i} = \frac{2}{n_e} \left(\frac{2\pi m_e kT}{h^2}\right)^{3/2} \frac{G_{i+1}}{G_i} e^{-E_i/(kT)}, $$
where \(n_i\) and \(n_{i+1}\) are number densities of successive ions, \(n_e\) is electron density, \(E_i\) the ionization energy, and \(G_i\) the partition function of level \(i\). This determines the degree of ionization and hence the opacity \(\kappa(\rho,T)\).
Interdependence of the seven equations
These seven equations describe how pressure, mass, luminosity, and temperature vary with radius in a spherically symmetric, non-rotating star. They can be summarized as:
| Equation | Physical principle | Variable(s) linked |
| ———– | ——————– | ——————- |
| (1) \(dP/dr\) | Hydrostatic balance | \(P, \rho, M\) |
| (2) \(dM/dr\) | Mass continuity | \(M, \rho\) |
| (3) \(dL/dr\) | Energy generation | \(L, \rho, T\) |
| (4) \(dT/dr\) | Radiative transport | \(T, L, \rho, \kappa\) |
| (5) \(dT/dr\) | Convective transport | \(T, g, \gamma, \mu\) |
| (6) \(P=\rho kT/\mu m_p\) | Equation of state | \(P, \rho, T\) |
| (7) Saha equation | Ionization equilibrium | \(\kappa, \mu, T, \rho\) |
Together, they form a closed system that defines a stellar model once the boundary conditions are specified:
$$ M(0) = 0, \quad L(0) = 0, \quad P(R) = 0, \quad T(R) = T_{eff}. $$
By solving them numerically—using observed mass \(M\) and composition \(C\)—one can determine a star’s radius \(R\), luminosity \(L\), internal profiles of \(P(r)\), \(T(r)\), and \(\rho(r)\), and the boundaries between its radiative and convective zones.
Insights
- The seven equations together express mass, momentum, and energy conservation within a self-gravitating plasma.
- Hydrostatic equilibrium ensures mechanical stability; the two energy-transport equations govern thermal equilibrium.
- The EOS and ionization balance connect microscopic particle physics to macroscopic structure.
- Radiative and convective transport determine where a star has radiative cores or convective envelopes.
- Solving these equations numerically yields the radius, luminosity, and internal structure for any given stellar mass and composition.
Inquiries
- Write down the physical meaning of each of the seven stellar structure equations.
- What boundary conditions are required to solve them numerically?
- Why is the radiative temperature gradient steeper in high-luminosity stars?
- Under what condition does convection replace radiation as the dominant heat-transfer mechanism?
- Explain how the EOS and Saha equation together determine the opacity profile of a star.