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.

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.

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 \(M(r)=\int_0^r 4\pi r'^2 \rho(r')\,dr'\) and \(M(R)\) 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 is 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, $$ dL = d\dot{E} \;\Rightarrow\; \frac{dL}{dr} = 4\pi r^2 \rho(r)\,\epsilon(\rho,T). $$

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 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 (composition dependent). Additional EOS terms (e.g., partial degeneracy or radiation pressure) can be included as needed.

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 sets the degree of ionization and hence \(\kappa(\rho,T)\) and \(\mu(\rho,T)\).

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 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.

1. Mass and luminosity equations follow directly from shell geometry: \(dV=4\pi r^2 dr\) and steady-state energy generation \(dL=\epsilon\,\rho\,dV\).
2. Hydrostatic equilibrium provides the mechanical backbone; EOS and Saha connect microphysics (composition, ionization) to macroscopic structure.
3. The steeper the required energy flux (large \(L\)) or the larger the opacity \(\kappa\), the steeper the radiative \(dT/dr\); convection sets in when this exceeds the adiabatic gradient.
4. Solving the seven equations yields internal profiles and the locations of radiative and convective zones for a given mass and composition.

1. Re-derive \(\frac{dM}{dr}=4\pi r^2\rho\) starting from a spherical shell of thickness \(dr\).
2. Show that steady energy generation in a shell implies \(\frac{dL}{dr}=4\pi r^2\rho\,\epsilon\).
3. Explain qualitatively why higher opacity \(\kappa\) steepens the radiative temperature gradient.
4. State the condition for onset of convection and interpret it physically.
5. How do the EOS and Saha equation together control \(\kappa(\rho,T)\) and the transition between radiative and convective zones?