solid.f

SOLID makes a range of shells, JSTART to JEND, rotate as a solid body while conserving their total angular momentum. Given each shell's angular momentum per unit mass, it returns the common rotation rate $\Omega$ and the shells' moments of inertia. Its comment adds that it also gives $r_0$ and $\eta_2$ (R0, ETA2), which FPFT needs.

• Inputs: arguments only: the shells' masses (HS2, probably in grams, since the angular momentum is the sum of HS2*HJM), specific angular momenta HJM, the previous rates OMEGA, and the structure (HD, HR, HS), passed on to the shape routines.
• Who calls it: the rotation routines that impose solid-body rotation on a zone: CHECKJ, GETROT, MIDMOD, SECULR and WCZIMP (for convection zones, by its name; WCZ, removed since rp_1, was another).
• What it calls: SHAPE (the distortion of the shells, ETA2 and R0) and MOMI (each shell's moment of inertia HI and QIW, which the comments call $dI/d\Omega$ but which is $\tfrac12\,\Omega\,dI/d\Omega$: suspected bug 2).
• Outputs: OMEGA, HI, QIW, ETA2, R0 and an updated HJM, all through the arguments.
• Modules: ERRMOM_MOD (the tolerance TOLERI, $10^{-12}$, used only here and in CHECKJ: CHECKJ saves it (TOLER0 = TOLERI), relaxes it to MAX(DJOK*1.0D-2,TOLER0) around its call to GETROT, and restores it); DISK_MOD (the age SAGE, shared with GETW, SECULR, MDOT and others); NMLPAR_MOD (the disk-locking parameters: the locking time TDISK, the locked rate PDISK and the switch LDISK, namelist parameters read by PARMIN); DIMENSIONS_MOD (JSON).

Let $m_j$ be HS2(J) and $j_j$ be HJM(J). The target is the zone's total angular momentum,

$$J = \sum_{j=J_\mathrm{start}}^{J_\mathrm{end}} m_j\, j_j .$$

The first guess for $\Omega$ is the mean of the previous rates. With every shell at $\Omega$, SHAPE and MOMI give the moments of inertia $I_j(\Omega)$ and $I'_j$, taken as $dI_j/d\Omega$ (QIW, which is in fact $\tfrac12\,\Omega\,dI_j/d\Omega$; suspected bug 2). With $I = \sum I_j$ (CZI) and $I' = \sum I'_j$ (CZQIW), a Newton step solves $\Omega\, I(\Omega) = J$:

$$\Delta\Omega = \frac{J - \Omega I}{I + \Omega I'} ,$$

repeated until $|J - \Omega I|/J \le$ TOLERI, or 20 times (lines 68–95↗). Finally each shell's specific angular momentum is reset to $j_j = I_j\,\Omega / m_j$ (lines 97–99↗).

Disk locking. For the surface convection zone (JEND = M, more than one shell), when LDISK is set and the age SAGE is at most TDISK, $\Omega$ is fixed to PDISK with no iteration, and the angular momentum follows from it ("MHP 9/94," lines 58–67↗).

PDISK is an angular rate. Under disk locking, WGUESS = PDISK (line 66↗), which is used as $\Omega$, and NMLPAR_MOD's initial value 7.2722D-6 (nmlpar_mod.f, line 210↗) is $2\pi/(10\ \mathrm{d})$ in rad/s, a rate for a period of 10 days. The comment calls it the "PERIOD AT WHICH LOCKING IS SET" (lines 29–30↗), and the Physics namelist page gives the unit rad/s but calls it a "Disk rotation period," so both could say "rate."

Physics Options (Initializing rotation): LDISK enforces disk locking until the age TDISK (Gyr; $10^{-4}$, i.e. $10^5$ yr, is the deuterium birthline, and $10^{-4}$–$10^{-2}$ the recommended range), at the disk's rate PDISK. The documentation calls PDISK the "disk rotation period (rad/s)": it is an angular velocity, used as such here; the default 7.2722D-6 is a 10-day period, while the documentation suggests 8 days (9.0902D-6), the median for disked solar-mass stars in star-forming regions.