surfbc.f

SURFBC provides the outer boundary condition of the interior solution, by the "envelope triangle." Envelopes are integrated at three points around the model's $(\log T_\mathrm{eff}, \log L)$; each gives $\log P$, $\log T$ and $\log r_\mathrm{f}$, the radius at the fitting point. A plane through the three points then gives $\log r_\mathrm{f}$, $\log L$ and $\log T_\mathrm{eff}$ as linear functions of $\log P$ and $\log T$ at the fitting point, which the Henyey solution uses as its outer boundary. It also switches between atmosphere tables and the gray atmosphere as $T_\mathrm{eff}$ changes.

The fitting point. It is the interior's outermost mass point, at $\log(m_\mathrm{f}/M)$ = SENV (SENV = HS(M) - HSTOT, set in CRRECT, HPOINT and STARIN; the documentation's standard is $-10^{-4}$, so about $2\times10^{-4}$ of the mass lies outside it). The layers above it are not on the Henyey grid but are integrated as a static envelope by ENVINT: the YREC paper (§2.2) explains that they hold little mass, typically one part in $10^4$ or less, but pressure and density fall steeply there, so leaving them out of the interior model is much faster, and the structure equations are more stable with a deeper fitting point. The outermost point has four unknowns ($\log P$, $\log T$, $\log r$, $L$); the plane fixes two relations among them, which are the two outer equations of the Henyey system in CRRECT (HSRF): $$\log r = c_1\log P + c_2\log T + c_3, \qquad \log L = c_4\log P + c_5\log T + c_6,$$ where $c_1, \dots, c_6$ are CFENV(1:6) and $P$, $T$, $r$, $L$ are the outermost point's; after the solution, CRRECT sets $\log T_\mathrm{eff}$ from the third plane, CFENV(7:9).

• Inputs: arguments from CRRECT: the current $\log T_\mathrm{eff}$ (TEFFL), $\log L/L_\odot$ (BL), the triangle (TRIT, TRIL, FTRI) and its envelopes' results (PS, TS, RS) from earlier calls, the surface $X$, $Z$, the rotation factors FPL, FTL, the pressure limit PLIM, and the mixing length CMIXL (passed on to ENVINT). The triangle's size TRIDT, TRIDL, the atmosphere choice (KTTAU, KTTAU0, LTTAU) and the spot parameters (SPOTF, SPOTX) come from NMLPAR_MOD; the Allard table's upper limit AL_TEFFLmax from ALATM05_MOD, set by ALTABINIT.
• Who calls it: only CRRECT.
• What it calls: ENVINT for each envelope to be recomputed.
• Outputs: CFENV(1:9), the plane's coefficients, which CRRECT uses for the boundary equations and to update $T_\mathrm{eff}$; the triangle and PS, TS, RS; LFLP (true if an envelope was recomputed); a table of the triangle on ISHORT.
• Modules: NMLPAR_MOD (the atmosphere flags, shared with ENVINT, PARMIN, SETUPS and others; this routine changes KTTAU and LTTAU), COMP_MOD (the total mass STOTAL), ALATM05_MOD, CONSTANTS_MOD, IOUNITS_MOD (ISHORT), JSON from DIMENSIONS_MOD.

1. Atmosphere switch (MHP 9/01, 06/13). With Kurucz tables (KTTAU = 3) above $\log T_\mathrm{eff} = 3.95$, or Allard tables (KTTAU = 4) above the table's maximum, the code switches to the gray atmosphere (KTTAU = 0), remembers it (LTTAU), and starts a new triangle; it switches back when $T_\mathrm{eff}$ drops below the limit again, and starts a new triangle then too, since the old vertices were integrated with the gray $T(\tau)$.

2. A new triangle (LNEW). With LRESET (from CRRECT) all three envelopes are recomputed without starting a new triangle; with LNEW the triangle is new too, and its vertices, with $T = \log T_\mathrm{eff}$, $L = \log L$, $$(T - \tfrac12\Delta T,\ L - \tfrac12\Delta L),\quad (T + \tfrac12\Delta T,\ L - \tfrac12\Delta L),\quad (T,\ L + \tfrac12\Delta L).$$ Otherwise the code checks that the point lies inside the triangle: for each vertex $i$, with the other two $j$, $k$, the sign of $$f\left[(L_j - L_k)(T - T_j) + (T_k - T_j)(L - L_j)\right]$$ ($f$ = FTRI, the orientation) must not be negative. If it is, vertex $i$ is replaced by $T_i \to T_j + T_k - T_i$, $L_i \to L_j + L_k - L_i$ (the triangle flips across that side, and its orientation changes), and the check repeats. After 1,000 reflections the run stops (STOP 1), with a message on the screen and ISHORT; the walk never ended for a degenerate triangle.

3. New envelopes. For each new vertex, ENVINT integrates at $L_i$ and $T_i$ (the vertex's $\log T_\mathrm{eff}$ is kept in the local VTEFFL), with $\log R = \tfrac12(\log L + \log L_\odot - 4\log T_\mathrm{eff} - \log 4\pi - \log\sigma)$ and $\log g = \log G + \log M - 2\log R$; for a spotted star whose outermost point is convective (LC(M)), at the ambient temperature $T_\mathrm{amb}$ of the unspotted surface, $$T_\mathrm{amb}^4 = \frac{T_\mathrm{eff}^4}{f x^4 + 1 - f},$$ where $f$ is the spot filling factor (SPOTF) and $x = T_\mathrm{spot}/T_\mathrm{amb}$ (SPOTX), so that spots and ambient surface together radiate $\sigma T_\mathrm{eff}^4$ (the same formula as in starin.f).

4. The plane. With $(P_i, T^\mathrm{f}_i)$ the fitting point's $\log P$ and $\log T$ of envelope $i$, each $Q \in \{\log r_\mathrm{f}, \log L, \log T_\mathrm{eff}\}$ is fitted exactly through the three envelopes: $$Q = c_1\,\log P + c_2\,\log T + c_3,$$ with CFENV(1:3) for $\log r_\mathrm{f}$, the radius at the fitting point, CFENV(4:6) for $\log L$, CFENV(7:9) for $\log T_\mathrm{eff}$. A degenerate fit stops the run (STOP 1), with the three fitting points on the screen and ISHORT: two points at the same $\log T$ ($|T^\mathrm{f}_2 - T^\mathrm{f}_1| \le 10^{-10}|T^\mathrm{f}_3 - T^\mathrm{f}_1|$), or a pivot DEN below $10^{-10}$ of the differences it is made of (three points collinear in $(\log P, \log T)$); either gave infinite or NaN coefficients before.

NUMENV here is a local counter (DATA NUMENV/0/), not the envelope's point count in ENVSTRUCT_MOD. The comment "IF LNEW0, REDO ALL 3 ENVELOPES EVERY MODEL" refers to the caller; LNEW0 is not used here.

Physics Options (Surface boundary conditions): the surface boundary condition is solved from three envelope models; TRIDT (default 0.01) is the spacing in $\log T_\mathrm{eff}$ between two of them, and TRIDL (0.08) the spacing in $\log L$ of the third, at their mean $T_\mathrm{eff}$. KTTAU selects the atmosphere (the documentation lists 0 Eddington, 1 Krishna Swamy, 3 Kurucz, 4 Allard).