# Open-field momentum integral at the banked base

### Isothermal terminal speed \(419\,\mathrm{km\,s^{-1}}\) · \(10\,\mathrm{G}\) does not enter · \(1\,\mathrm{km\,s^{-1}}\) stays the shake

**Date:** 2026-10-08
**Status:** **RECORD** · public page `/cosmology/coronal-hole` · no ledger row · no promote
**Parents:** measey.com `documents/a_geom_ft_stellar_condensation_cascade_chunk4.md` · `PASS_FT_CH_CURRENT_FIRST`

> **Along an open field the parallel Lorentz force is zero, so the \(10\,\mathrm{G}\) base field does not enter the speed. The isothermal integral at \(0.8\,\mathrm{MK}\) with area \(A\propto r^{2}\) reaches \(419\,\mathrm{km\,s^{-1}}\) at \(1\,\mathrm{AU}\). The same integral with \(A\propto r^{3}\) reaches \(526\,\mathrm{km\,s^{-1}}\). The smooth \(s=2\) solution has a base speed \(14.3\,\mathrm{m\,s^{-1}}\). A bulk speed of \(1\,\mathrm{km\,s^{-1}}\) at \(R_\odot\) turns sonic at \(1.55\,R_\odot\) and supplies no speed at \(1\,\mathrm{AU}\). The \(750\)–\(800\,\mathrm{km\,s^{-1}}\) class remains the \(2\,\mathrm{MK}\) row, \(752\,\mathrm{km\,s^{-1}}\), which is not the hole. The banked sentence stays the current-first channel.**

The steady momentum equation used here is the isotropic survivor

\[
\rho v\frac{dv}{ds}=-\frac{dP}{ds}-\rho\frac{GM_\odot}{r^{2}}\frac{dr}{ds},
\]

with \(\hat{\mathbf{b}}\cdot(\mathbf{j}\times\mathbf{B})=0\). The sound speed is \(c_s=\sqrt{2kT/m_p}\). The critical radius is \(r_c=GM_\odot/(s c_s^{2})\): \(7.22\,R_\odot\) at \(s=2\) and \(4.81\,R_\odot\) at \(s=3\). The speeds were read from that critical invariant and agree with a direct march of the differential equation. The constant-density reading \(v_\infty=v_A\) stays the illegal integral. A transonic nozzle is not opened. Gate 5 stays unpaid.

The \(1\,\mathrm{km\,s^{-1}}\) figure stays the footpoint shake. It is not retuned, and it is not the base speed of the smooth \(0.8\,\mathrm{MK}\) wind. Electrons stay near \(0.8\,\mathrm{MK}\). The public sentence remains footpoint shake, ion-circle handoff, and the outward mirror. The mirror is the banked gyrotropic term. This integral does not assign a new \(P_\perp-P_\parallel\). The live line stays §0bn1188. No chip. No promote. The wind bank stays `PASS_FT_CH_CURRENT_FIRST`.

Figure: `public/figures/coronal-hole/fig2_momentum.png`.