setscv.f

Builds the equation-of-state table for the envelope's mixture from the SCV (Saumon–Chabrier–Van Horn, by the module's name SCVEOS_MOD, "MHP 5/97") tables of pure hydrogen, pure helium and metals, which SETUPS has read. It adds the entropy of mixing, and computes by finite differences the derivatives the structure equations need. (The header comment, "evaluate the EOS with the new tables and call eqstat … and compare," describes an earlier purpose; the routine calls neither.)

• Who calls it: STARIN, once the starting model's envelope composition is known.
• Inputs, through modules: SCVEOS_MOD (the tables TABLEX (hydrogen), TABLEY (helium), TABLEZ (metals) on a grid of temperatures TLOGX(63) and up to 76 pressures NPTSX(I)), and COMP_MOD (XENV, ZENV).
• What it calls: INTER3, three-point Lagrange interpolation weights (FT) and their derivatives (FTD).
• Outputs, through SCVEOS_MOD: column 12 of TABLEX and TABLEY, SMIX (the entropy of mixing), and TABLENV(63, 76, 12), the mixture's table; also the interpolation indices IDTSCV (named IDT at rp_1) and IDP left from the last point. It is read by EQSCVE and EQSCVG, through EQSTAT, when LSCV is true.

The table columns are used, in the code's notation, as $\log P$ (1), the ionization fractions (2, 3), $\log\rho$ (4), $\log S$ (5), $\partial\ln\rho/\partial\ln T$ (7), $\partial\ln\rho/\partial\ln P$ (8) and $\partial\ln S/\partial\ln T$ (9); we read this from the formulas.

1. $dU/dT$ of each pure table (column 12; the loop's comment, "READ IN EQUATION OF STATE TABLES FOR HYDROGEN AND HELIUM" (line 37↗), is left from the reading, now done in SETUPS, setups.f, lines 368–393↗), from the two expressions for $c_P$ in the comments, $$c_P = S\left(\frac{\partial\ln S}{\partial\ln T}\right)_P = \frac{dU}{dT} + \frac{P}{\rho T}\,\frac{(\partial\ln\rho/\partial\ln T)^2}{\partial\ln\rho/\partial\ln P}.$$

2. Entropy of mixing. With $Y = 1 - X$ (XENV), the mixture's density $\rho_\mathrm{mix} = 1/((1-Y)/\rho_X + Y/\rho_Y)$, the number densities of hydrogen and helium particles and of their electrons from the ionization fractions, and $\beta = (m_H/m_{He})\,Y/(1-Y)$, $\gamma$ the ratio of particles per nucleus, the code computes $$S_\mathrm{mix} = \frac{k}{m_H}\,\frac{2(1-Y)}{1+x_2+3x_1}\left[\ln(1+\beta\gamma) - \frac{n_{e,H}}{n}\ln(1+\delta) + \beta\gamma\ln\Bigl(1+\frac{1}{\beta\gamma}\Bigr) - \frac{n_{e,He}}{n}\ln\Bigl(1+\frac1\delta\Bigr)\right],$$ with $n$ = TNUM the sum of the hydrogen and helium number densities (HNUM + HENUM), $n_{e,H}$, $n_{e,He}$ their electron number densities (HNUME, HENUME, each floored at 1), $\delta$ (DE) the ratio of electrons from helium to those from hydrogen, scaled by $\beta\gamma$ (the last term uses DEI, which is $1/\delta$ when both are positive); $x_1$, $x_2$ are TABLEX(I,J,2) and (I,J,3). Constants: $m_H$ = CMH, $m_{He}$ = CMHE (g), $k$ = CBOLTZ (erg K$^{-1}$). This is the code's formula; its source is not cited.

3. The mixture at each $(T, P)$, with $X$, $Y = 1 - X - Z$, $Z$ from COMP_MOD: densities add as volumes, $$\frac1\rho = \frac{X}{\rho_X} + \frac{Y}{\rho_Y} + \frac{Z}{\rho_Z},\qquad \frac{\partial\ln\rho}{\partial\ln P} = \sum_k X_k\frac{\rho}{\rho_k}\left(\frac{\partial\ln\rho}{\partial\ln P}\right)_k,$$ and the same for $\partial\ln\rho/\partial\ln T$; the entropy is $S = X S_X + Y S_Y + S_\mathrm{mix}$, and $c_P$ follows, with the metals' term added. The radiation pressure PRAD $= aT^4/3$ is added to P and PL (lines 122–126↗), but neither is used afterwards (unused?), nor is QPT0 (line 144↗, unused?): The table stays on the gas-pressure grid (TABLENV(I,J,1) is TABLEX(I,J,1)), and EQSCVE adds radiation itself, looking the table up at the gas pressure $\beta P$ (settled from the code, Nov 9, 2026: eqscve.f, lines 52–53↗, eqscve.f, lines 159–160↗). Stored: TABLENV(·,·,1..6) = $\log P$, $\log\rho$, $\partial\ln\rho/\partial\ln T$, $\partial\ln\rho/\partial\ln P$, $c_P$, $dU/dT$.

4. Derivatives, by three-point Lagrange differentiation in $\ln T$ and $\ln P$: of $\partial\ln\rho/\partial\ln T$ in $\ln P$ and $\ln T$, of $\partial\ln P/\partial\ln\rho$ in $\ln\rho$, of $\ln c_P$ in $\ln T$, and of $dU/dT$ in $\ln P$ and $\ln T$; stored in columns 7–12. The derivative of $\ln c_P$ in $\ln P$, QCPP, is computed but not stored (line 218↗, unused?). Of the table, EQSCVE reads columns 1–6 and EQSCVG column 1; columns 7–12 are read nowhere at 7be4133.

   The comment's list of derivatives is stale. The list before the loop (lines 173–179↗) names D LN CP / D LN P, "D LN DEL(AD)/D LN P," and "D LN DEL(AD)/D LN T," but columns 7–12 hold QDTP, QDTT, QPDD, QCPT, QAP, and QAT (lines 245–250↗): The derivatives of $dU/dT$ (QAP, QAT: lines 225–236↗) take the place of those of $\nabla_\mathrm{ad}$, and QCPP, $d\ln c_P/d\ln P$, is computed (line 218↗) but not stored.