# RMO89: does the perpendicular solar model survive the input bounds?

**Result in one sentence:** the perpendicular magnetic model remains a fast
shock throughout the tested speed, sound-speed and radio bounds, if the upstream
plasma is at rest and the stated measurements describe the same front.

This is a conditional model calculation for the **13 June 2010 EUV event**.
Ma et al. (2011) already interpreted this event as a shock and used a perpendicular
magnetic model. RMO extends the model check across an explicit, linked input
domain; it does not claim a new solar discovery or an independent measurement of
the front's MHD family.

## Read the picture

The figure compares plasma flow relative to the front with the local fast-wave
speed. Each plotted number is a ratio, so 1 is the dividing line.

- **Before the front: 1.148–1.863.** The incoming plasma is faster than the local fast
  wave throughout the tested domain.
- **After the front: 0.630–0.879.** The outgoing plasma is slower than the local fast
  wave throughout the tested domain.
- The reconstructed density and tangential field both increase by the same
  factor, 1.199–2.009; entropy increases and conservation laws hold.

This super-fast to sub-fast transition supports a **fast shock in this model**.
It describes flow through one front, not how the front accelerates over time.
The ratio bars enclose all model outputs in the domain. They are not measured
Mach-number confidence intervals; upstream and downstream values remain linked.

## Published inputs and provenance

| Input | Value used | Source / condition |
| --- | --- | --- |
| Event and time | 13 June 2010, 05:40 UT | Ma et al. (2011), Section III.2 |
| Front speed | 495.77–667.03 km/s | RMO88 evaluation of every Kozarev Table 1 fit at 180 s from 05:37 UT |
| Upstream sound speed | 126–186 km/s | Ma: 156 ± 30 km/s; a supplied thermal-model constraint |
| Lower radio frequency | 127–137 MHz | Ma: 132 ± 5 MHz |
| Upper radio frequency | 150–180 MHz | Ma: 165 ± 15 MHz |
| Density ratio r | 22500/18769 to 32400/16129 | Derived from (upper/lower radio frequency)^2 |
| Ratio of specific heats | 5/3 | Fixed model choice |
| Magnetic direction | Bn = 0 exactly | Assumed perpendicular geometry, not observed here |
| Upstream plasma velocity | 0 in the frame of the adopted front speed | Assumed, not measured |

The plus/minus quantities are treated as simultaneous bounds for this calculation.
Their statistical definitions and joint covariances are not supplied by the
extracted records, so no confidence level is assigned. The outer set is deliberately
an enclosure of the scalar combinations; it need not describe the original fit's
attainable corners. Published profiles and their averages are not independent
pieces of evidence.

The sound speed is the supplied thermal input. A separate temperature value is
not varied independently alongside it. Derived density, pressure, field, flow
speed, Mach numbers and downstream temperature are not independent measurements.
The radio interpretation assumes paired upstream/downstream emission from the
same discontinuity and fixed composition. EUV is not radio emission.

## Linked construction, not independent noise on downstream states

Use one front-rest frame with positive flow from state 1 to state 2. Normalize
rho1 = U1 = mu0 = 1 separately for each model; the dimensional velocity scale is
the adopted U in km/s. Bn and tangential velocities vanish. Define

\[
b = p_1 = \frac{3}{5}\left(\frac{c_1}{U}\right)^2,\qquad
A = B_{t1}^2 = \frac{8-2r-10br}{r(r+5)}.
\]

Conservation fixes the other state:

\[
\rho_2=r,\quad U_2=1/r,\quad B_{t2}=r\sqrt A,\quad
p_2=b+1-1/r+\frac A2(1-r^2).
\]

The same b, r and A are used in every expression. None of the derived inputs is
perturbed independently. Magnetic-field strength is reconstructed under the shock
model, not independently measured. This is a restricted perpendicular family of
single discontinuities, not a search over full Riemann fans, oblique branches or
non-wave explanations.

## What was proved over the entire domain?

The following are algebraic identities or outward-rounded interval bounds over
the full scalar enclosure. The finite control states below are additional checks,
not a substitute for this whole-domain argument.

1. **Conservation:** mass and tangential induction follow from U2 = 1/r and
   Bt2 = r Bt1. With the prescribed zero components, the other tangential fluxes
   vanish. Exact rational polynomial cancellation proves normal momentum and
   energy conservation for all r, b with nonzero denominators.
2. **Physical states:** A > 0 everywhere. A positive expression for downstream
   pressure avoids a cancellation-sensitive sign test:

\[
p_2=\frac{br(5r+1)}{r+5}+\frac{(r-1)^3}{r(r+5)}>0.
\]

3. **Fast crossing:** cf1^2 = 5b/3 + A and
   cf2^2 = (5p2/3 + r^2 A)/r. The exact factorizations are

\[
1-c_{f1}^2=\frac{(r-1)[3(r+8)-5br]}{3r(r+5)}>0,
\]

\[
c_{f2}^2-1/r^2=
\frac{(r-1)[(10-r)(r+2)-5br^2]}{3r^2(r+5)}>0.
\]

Outward interval evaluation gives strictly positive lower bounds for both
expressions on the complete domain. No point is certified using only a central
state or a Monte Carlo fraction.

4. **Entropy:** put k = A/(2b) > 0. Conservation also gives

\[
\frac{p_2}{b}=\frac{4r-1+k(r-1)^3}{4-r}.
\]

Thus entropy/cv is bounded below by
ln[(4r−1)/(4−r)] − (5/3) ln r. Its derivative is
20(r−1)^2/[3r(4r−1)(4−r)] > 0 for 1 < r < 4.
At the lower compression bound its outward-rounded lower value is
**0.0014871**, strictly positive.

5. **Mach envelopes:** for cf1^2 the r derivative has numerator
(2+10b)r^2−16r−40 < 0; the b derivative is 5(r−1)/[3(r+5)] > 0.
For r^2 cf2^2 the r derivative has numerator
−2(1+5b)r^3−(6+70b)r^2+(90+50b)r+80 > 0,
and its b derivative is −5r^2(r−1)/[3(r+5)] < 0.
Every sign is interval-checked over the domain. These monotonicities give the
Mach endpoints in the figure, with outward rounding rather than an unproven
corner-extremum assumption.

## Independent controls and practical limits

**56 focused checks pass.** Eight scalar corners and the previously used
central model (r = 1.56, U = 600 km/s, c1 = 156 km/s) are separately evaluated
with 75-digit Decimal arithmetic. Direct mass, induction, momentum and energy
residuals are below 1e−65 in the normalized calculation. The separately evaluated
published compression relation, Ma Eq. 5, recovers r to the same tolerance.
These small residuals describe arithmetic, not observational precision.

The unchanged RMO77 perpendicular diagnostic receives the nine complete model
state pairs and front speed **without a family label**. Every pair passes its
existing conservation, entropy and characteristic-speed criteria and is classified
as a fast shock. Choosing the perpendicular construction itself already restricts
the candidate family; this is not a blind competition between solar fast and slow
interpretations. Controls at and beyond the gas boundary confirm that A becomes
zero or negative, rather than assigning a magnetic shock to arbitrary inputs.

The existing solver, uncertainty assessors, source audit and earlier results are
unchanged. This calculation runs separately and its checked result is embedded in
QuickLook. Loading the figure does not start a fresh browser-to-Python calculation.
Native browser acceptance remains open; PDF rendering and content preservation
are checked separately.

## What is still needed for a solar diagnosis?

The observed local front normal and field direction, upstream plasma motion, and
association of EUV, radio and thermal information remain unresolved dependencies.
The perpendicular geometry and zero flow are held fixed here. The RMO88 result
already shows why allowing a sufficient upstream flow can change the gas/magnetic
comparison. A positive result in the present slice does not certify those extra
dimensions of uncertainty.

The separate RMO84/85 EUV-delay comparison also remains open. This result does not
repair that comparison, fit an emission history, independently measure the field,
exclude oblique/slow/non-wave alternatives for the actual event, establish
stability or enumerate all Riemann solutions.

**Next useful step:** constrain the same-front geometry and plasma-flow assumption,
then test compatible observational constraints and competing interpretations. The
published shock interpretation is credited; RMO has verified conditional physical
consistency and robustness, not independently confirmed the solar type.

## Sources and reproducibility

- [Ma et al. (2011), Sections III.2–III.3, Eqs. 3 and 5](https://arxiv.org/html/1106.6056v1),
  [published article](https://doi.org/10.1088/0004-637X/738/2/160).
- [Kozarev et al. (2011), Table 1 and its note](https://arxiv.org/html/1406.2372v1),
  [published article](https://doi.org/10.1088/2041-8205/733/2/L25).
- [Fitzpatrick, Perpendicular MHD Shocks, Eqs. 7.269–7.277](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node104.html).

Source numbers are read from the unchanged saved RMO88 calculation and its source
record; no new observational data or atomic-rate files were acquired.
Run `python solar_magnetic_bounds/audit.py`, then `report.py` and `plot.py` from
that directory path in the project root. The JSON preserves exact domain fractions,
outward decimal bounds, source hash and the nine input/output control records.
The prior PDF/report and all previous QuickLook sections remain available.