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.

Of these seven, the first five are known as the primary equations, because they are differential relations that describe how the star’s physical variables vary with radius. The last two are called the secondary equations, because they provide the thermodynamic and compositional relations (the *equations of state*) needed to close the system.

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, \(g(r)=GM(r)/r^2\) the local gravitational acceleration, and \(G\) the gravitational constant. This equation ensures that the net force on a small mass element is zero in a stable star. It expresses mechanical equilibrium, the balance between gravity and pressure.

Statement.

$$ \frac{dM}{dr} = 4\pi r^2 \rho(r). $$

Derivation. Consider a thin spherical shell at radius \(r\) with thickness \(dr\). The shell’s volume is $$ dV = 4\pi r^2\,dr, $$ so the shell’s mass is $$ dM = \rho(r)\,dV = \rho(r)\,4\pi r^2\,dr. $$ Dividing by \(dr\) gives the radial mass accumulation rate $$ \frac{dM}{dr} = 4\pi r^2 \rho(r), $$ which is the local form of mass continuity in spherical symmetry. Integrating from the center yields \(M(r)=\int_0^r 4\pi r'^2 \rho(r')\,dr'\), giving the total mass \(M(R)\) at the surface.

Statement.

$$ \frac{dL}{dr} = 4\pi r^2 \rho(r)\,\epsilon(\rho,T), $$

where \(L(r)\) is the luminosity passing outward through radius \(r\), and \(\epsilon(\rho,T)\) is the energy generation rate per unit mass (W kg\(^{-1}\)) from nuclear reactions (composition dependence implied: \(\epsilon=\epsilon(\rho,T,\mathrm{C})\)).

Derivation. In steady state, the net increase of luminosity across a thin shell equals the power generated within that shell. For the same shell \(dV=4\pi r^2 dr\), the mass is \(dM=\rho(r)\,dV\). The energy produced per unit time in the shell is $$ d\dot{E} = \epsilon(\rho,T)\,dM = \epsilon(\rho,T)\,\rho(r)\,4\pi r^2 dr. $$ By energy conservation, this equals the outward increase in luminosity, $$ dL = d\dot{E} \;\Rightarrow\; \frac{dL}{dr} = 4\pi r^2 \rho(r)\,\epsilon(\rho,T). $$

This equation describes how luminosity builds up with radius as nuclear energy is released inside the star.

In regions where energy is transported primarily by radiation, the temperature decreases outward according to

$$ \frac{dT}{dr} = -\,\frac{3\,\kappa(\rho,T)\,\rho\, L}{16\pi a c\, T^3\, r^2}, $$

where \(\kappa\) is the opacity (m\(^2\) kg\(^{-1}\)), \(a\) 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. (We omit the detailed derivation here; see Heat transfer in stars for a full treatment.)

When the radiative gradient exceeds the adiabatic one, convection becomes efficient. The adiabatic temperature gradient can be written approximately as

$$ \left(\frac{dT}{dr}\right)_{\!ad} = -\left(1 - \frac{1}{\gamma}\right)\frac{\mu m_p g}{k}\,T, $$

where \(\gamma=C_P/C_V\) is the ratio of specific heats, \(\mu\) the mean molecular weight, \(m_p\) the proton mass, \(k\) Boltzmann’s constant, and \(g=GM(r)/r^2\). Convection dominates wherever \(\left|\frac{dT}{dr}\right|_{\!rad} > \left|\frac{dT}{dr}\right|_{\!ad}\). (We avoid the full derivation and mixing-length details here; see Heat transfer in stars.)

The equation of state (EOS) connects macroscopic variables — pressure, density, and temperature — describing the physical state of the stellar gas. For a fully ionized ideal gas,

$$ P = \frac{\rho kT}{m_{av}} = \frac{\rho kT}{\mu m_p}, $$

where \(m_{av}\) is the mean particle mass and \(\mu\) the mean molecular weight (composition dependent). In real stars, radiation pressure, degeneracy pressure, or Coulomb corrections can be added, but the above relation is the baseline form for most stellar interiors.

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 the electron density, \(E_i\) the ionization energy, and \(G_i\) the partition function of level \(i\). This equation determines the degree of ionization and thus sets \(\kappa(\rho,T)\) and \(\mu(\rho,T)\) for the opacity and equation of state.

The first five equations — for hydrostatic balance, mass conservation, luminosity, and the two temperature gradients — are primary because:

• They are differential equations that describe the structural variation of a star with radius \(r\).
• They express the fundamental conservation laws of physics:
  • (1) mechanical equilibrium,
  • (2) conservation of mass,
  • (3) conservation of energy, and
  • (4–5) energy transport by radiation or convection.
• Their solutions give the radial profiles of \(P(r)\), \(\rho(r)\), \(T(r)\), \(M(r)\), and \(L(r)\).

The last two equations — the equation of state and Saha ionization relation — are secondary, because:

• They are constitutive relations that specify how matter behaves under given physical conditions.
• They provide the closure needed to relate pressure, temperature, and density, allowing the primary equations to be solved.
• Without them, the system of five differential equations would contain more unknowns than equations and remain indeterminate.

Together, the primary and secondary equations form a closed system of seven equations with seven unknown functions.

These seven equations describe how \(P(r),\,\rho(r),\,T(r),\,L(r)\), and \(M(r)\) vary with radius in a spherically symmetric, non-rotating star. They can be summarized as:

A stellar model is obtained by solving these equations with suitable boundary conditions, e.g.,

$$ M(0)=0,\quad L(0)=0,\quad P(R)=0,\quad T(R)=T_{eff}, $$

given the total mass \(M(R)\) and composition of the star.

1. The primary equations describe the geometry and physical balances within a star; the secondary equations describe the state of the material composing it.
2. Mass and luminosity equations follow directly from shell geometry: \(dV=4\pi r^2 dr\) and energy generation \(dL=\epsilon\rho dV\).
3. Hydrostatic equilibrium provides the mechanical backbone; EOS and Saha supply the thermodynamic closure.
4. A steeper required flux (large \(L\)) or greater opacity \(\kappa\) makes the radiative \(dT/dr\) steeper; convection starts when this exceeds the adiabatic gradient.
5. Solving the seven coupled equations yields the full internal profiles of a star and defines its radiative and convective zones.

1. Which of the seven equations are differential and which are algebraic?
2. Why are the first five called primary and the last two secondary?
3. Derive \(\frac{dM}{dr}=4\pi r^2\rho\) and explain its physical meaning.
4. Explain how the equation of state and Saha relation “close” the system of stellar structure equations.
5. Discuss how different opacity or composition affects the balance between radiative and convective energy transport.