This article shows how to build a static stellar model by numerically integrating the seven equations of stellar structure with the program STATSTAR (Python-3 version). You will (i) supply basic global inputs, (ii) let the code step inward through thin spherical shells using Runge–Kutta, and (iii) inspect the shell-by-shell output to judge whether the model is physically consistent. :contentReference[oaicite:0]{index=0}

STATSTAR integrates the primary structure ODEs with radius \(r\):

1. Hydrostatic equilibrium: \(\displaystyle \frac{dP}{dr}=-\rho\,\frac{G M}{r^2}\)
2. Mass conservation: \(\displaystyle \frac{dM}{dr}=4\pi r^2\rho\)
3. Energy generation (luminosity): \(\displaystyle \frac{dL}{dr}=4\pi r^2\rho\,\epsilon\)
4. Temperature gradient by radiation *or* by convection (picked locally)

and closes them with the secondary relations:

1. Equation of state (ideal-gas baseline, extendable with radiation/degeneracy)
2. Ionization balance (Saha; affects \(\mu\) and \(\kappa\))

The code walks inward shell-by-shell:

1) compute auxiliary quantities in the current shell (\(\rho,\ \kappa,\ \epsilon\)),  
2) evaluate derivatives \(dM/dr,\ dL/dr,\ dT/dr,\ dP/dr\),  
3) advance to the next shell (4th-order Runge–Kutta),  
4) repeat until a termination criterion is reached. :contentReference[oaicite:1]{index=1}

When run, STATSTAR prompts for five global inputs:

1. total mass \(M\) (in \(M_\odot\))
2. total luminosity \(L\) (in \(L_\odot\))
3. effective temperature \(T_{\!eff}\) (K)
4. hydrogen mass fraction \(X\)
5. metal mass fraction \(Z\) (with helium \(Y=1-X-Z\))

These define the composition and the approximate outer boundary conditions for the integration that proceeds inward. :contentReference[oaicite:2]{index=2}

The course distribution includes FORTRAN and Python translations; use statstar_python3.py for modern environments (Python 3). Directory listings and the Python-3 script are hosted with the teaching materials; the code header states it “calculates a static stellar model using the equations developed in the text.” Output is written to starmodl.dat (shell-by-shell structure) and a brief statstar.out summary. :contentReference[oaicite:3]{index=3}

In a typical outer shell, the state vector is \((M_r,L_r,T,P)\). From these, STATSTAR obtains \(\rho\) via the EOS, \(\kappa\) from opacity formulae/tables, and \(\epsilon\) from nuclear-energy fits. Those “auxiliary sources” provide the microphysics that the differential system itself does not specify. Then the program evaluates the radial derivatives and steps inward to the next shell. :contentReference[oaicite:4]{index=4}

Primary equations used each step (no detailed derivations here):

1. \(dP/dr=-\rho GM/r^2\) (mechanical balance)
2. \(dM/dr=4\pi r^2\rho\) (geometry + continuity)
3. \(dL/dr=4\pi r^2\rho\,\epsilon\) (energy conservation)
4. \(dT/dr\) chosen as

\[ \left(\frac{dT}{dr}\right)_{\!\rm rad}=-\frac{3\kappa\rho L}{16\pi a c\,T^3 r^2} \qquad\text{or}\qquad \left(\frac{dT}{dr}\right)_{\!\rm ad}\;(\text{if convective}), \]

  with the switch ruled by the convective instability condition. (Full derivations are deferred to [[un:heat-transfer-in-stars|Heat transfer in stars]].)

Secondary relations used inside each step:

1. EOS (ideal-gas baseline) \(\Rightarrow\) \(\rho(P,T,\mu)\)
2. Saha ionization \(\Rightarrow\) \(\mu(\rho,T)\), \(\kappa(\rho,T)\)

*Initialization.* STATSTAR constructs an outermost layer from your \(L,\ T_{\!eff}\) and composition, then begins stepping inward (\(dr<0\)). The integrator is 4th-order Runge–Kutta for accuracy and stability. :contentReference[oaicite:5]{index=5}

*Termination.* A run ends either in success (“CONGRATULATIONS, I THINK YOU FOUND IT!” printed) when a sensible center is reached, or in failure if a sanity check is violated (e.g., nonphysical central values, premature termination). Either way, inspect starmodl.dat. :contentReference[oaicite:6]{index=6}

Create an input file (or type values interactively). Example inputs:

• Mass (M/M⊙)*: 1.0
• Luminosity (L/L⊙)*: 0.86
• Effective temperature (K)*: 5500
• X*: 0.70
• Z*: 0.008

These values reproduce the lecture’s “near-solar” demonstration. The model proceeds inward through several hundred shells, writing a column table per shell (radius, \(M_r, L_r, T, P, \rho, \kappa, \epsilon,\dots\)). :contentReference[oaicite:7]{index=7}

A viable model has:

1. \(r\) ranging smoothly to (near) zero,
2. \(T\) rising to \(\sim10^7\)–\(10^7.5\) K in the center for a solar-type star,
3. \(P,\ \rho\) monotonic inward increases,
4. no unphysical spikes at the innermost gridpoints.

Runs with slightly different \(T_{\!eff}\) can “miss” the center (e.g., wrong mass at \(r\to0\)) or produce implausible central temperatures; these are cues to adjust outer inputs and re-run. :contentReference[oaicite:8]{index=8}

STATSTAR solves static (time-independent) stellar structure for homogeneous main-sequence-like models: composition is fixed through the star, rotation and magnetic fields are neglected, and the program chooses radiative or convective transport locally. Only certain \((L,\,T_{\!eff})\) pairs are consistent for a given \(M\) and composition, reflecting the classic constraints on ZAMS models. :contentReference[oaicite:9]{index=9}

1. Inputs: \(M,\ L,\ T_{\!eff}, X, Z\). (Derive \(Y=1-X-Z\).) :contentReference[oaicite:10]{index=10}
2. Primary ODEs stepped with RK4: \(dP/dr,\ dM/dr,\ dL/dr,\ dT/dr\) (rad/conv). :contentReference[oaicite:11]{index=11}
3. Secondary closures each shell: EOS \(\Rightarrow \rho\); Saha \(\Rightarrow \mu,\ \kappa\). :contentReference[oaicite:12]{index=12}
4. Sanity checks: central \(T\) high enough for the adopted \(L\); monotonic inward trends; \(M(R)=M_{\rm input}\), \(L(R)=L_{\rm input}\).
5. Outputs: starmodl.dat (full structure), statstar.out (summary). :contentReference[oaicite:13]{index=13}
6. Success message: “CONGRATULATIONS, I THINK YOU FOUND IT!” (still verify physically). :contentReference[oaicite:14]{index=14}

Families of STATSTAR-type models reproduce the qualitative behavior of real stars on the H–R diagram: luminosity rises steeply with mass on the main sequence, central temperature correlates with mass, and radii grow mildly with mass (since \(L\) increases a bit faster than \(T^4\)). Such trends echo classic evolutionary sequences (e.g., Iben 1967) and Gaia observations. :contentReference[oaicite:15]{index=15}

1. Primary vs. secondary: the four ODEs (+ one transport choice) control the *geometry and balances*; EOS+Saha close the *microphysics* each step. :contentReference[oaicite:16]{index=16}
2. Numerics matter: RK4 stepping and robust start/stop criteria are essential; small changes to \(T_{\!eff}\) can make or break a run. :contentReference[oaicite:17]{index=17}
3. Outputs need physics checks: center too cool for the specified \(L\) ⇒ inconsistent model; revisit inputs or microphysics. :contentReference[oaicite:18]{index=18}

1. Starting from your \(M, L, T_{\!eff}, X, Z\), run STATSTAR and plot \(T(r), P(r), \rho(r)\). Is the model sensible at \(r\to0\)? Why? :contentReference[oaicite:19]{index=19}
2. For fixed \(M\), vary \(T_{\!eff}\) slightly. How do the central values and the success of the run change? Explain using the transport equations. :contentReference[oaicite:20]{index=20}
3. Using starmodl.dat, estimate \(R\) and compare \(L\) to \(4\pi R^2\sigma T_{\!eff}^4\). Discuss consistency.
4. Replace the EOS with a partially degenerate term in the deep interior. Predict qualitative changes to \(P(r)\) and \(T(r)\).