# RMO92 — a bounded oblique extension of the central solar model

**Result in one sentence:** In this model, allowing a small normal magnetic field preserves the fast shock; the downstream flow also turns.

This extends the constructed central model associated with the 13 June 2010 event. It does not determine the magnetic angle or wave type of the observed solar front. The three numerical states below are model results, not three observations.

## Physical question and chosen inputs

Does the previously checked fast shock disappear immediately when the field is no longer exactly parallel to the front surface? Here the field-to-normal angle decreases from 90° (perpendicular propagation) to about 81.09° (oblique propagation). This is a deliberately chosen model interval, not an observational error bar.

We keep compression r = 1.56, upstream normal speed relative to the front U1 = 600 km/s, upstream sound speed c1 = 156 km/s, and gamma = 5/3 fixed. We use the prior composition and an initially stationary upstream plasma in the adopted solar frame. Field amplitude and all downstream quantities are reconstructed jointly. This is not a rotation at fixed magnetic strength and not the full scalar-error domain checked in RMO89.

In normalized units rho1 = U1 = mu0 = 1, p1 = b = 507/12500, and the front is at rest. The continuation parameter is h = Bn²/(mu0 rho1 U1²), with 0 <= h <= 1/100. The upstream tangential velocity is zero in this frame. At h = 0 the reconstruction returns the saved perpendicular central state exactly.

## What the picture means

The plasma enters faster than the upstream fast-mode speed and leaves slower than the downstream fast-mode speed, while remaining faster than the downstream normal Alfvén speed. Compression, magnetic amplification, the conservation relations and entropy increase also pass. These combined conditions support the fast-shock class for this constructed local family. A large image speed alone would not be sufficient.

On the right, the downstream plasma acquires a velocity along the front surface. The magnetic field changes the direction of the flow as well as its compression. This is a model prediction for the selected family.

| h | Field-to-normal angle (degrees) | Upstream fast Mach | Downstream fast Mach | Tangential flow after front (km/s) |
|---|---:|---:|---:|---:|
| 0 | 90.000000 | 1.439436 | 0.737023 | 0.000 |
| 1/400 | 85.551185 | 1.439261 | 0.736994 | 10.839 |
| 1/100 | 81.090455 | 1.438728 | 0.736906 | 21.773 |

## Conservation-linked reconstruction

Let K = gamma/(gamma-1), q = Bt2/Bt1 and A = Bt1² in normalized units. The coplanar construction is

```
q = r(1-h)/(1-rh)
C = (h+K/r)(1-q²)/2 + q²/r
A = [1/(2r²) + (K/r)(b+1-1/r) - (1/2+Kb)] / (1-C)
rho2 = r; un2 = 1/r; Bn1 = Bn2 = sqrt(h)
Bt1 = sqrt(A); Bt2 = q sqrt(A)
ut2 = sqrt(h A)(q-1)
p2 = b+1-1/r + A(1-q²)/2
```

The tangential momentum equation gives ut2, tangential induction gives q, normal momentum gives p2, and total energy gives A. Mass and normal-field continuity are built into the parameterization. The code verifies the remaining rational conservation identities exactly, after clearing denominators whose positivity is checked on the chosen domain. Negative Bn with the corresponding reversed tangential flow is also checked at the three control states.

## Continuous-domain check and independent controls

All 32 adjacent intervals [j/3200,(j+1)/3200], j=0,...,31, are checked with outward-rounded 50-digit decimal interval arithmetic. They cover the entire chosen h domain. Reconstruction denominators, pressure, field strength and the fast-shock inequalities retain the required signs. The entropy increase divided by cv has a lower enclosure above 0.16277. Conservative rounded enclosures for the fast Mach numbers are 1.435–1.443 upstream and 0.733–0.741 downstream; the unrounded interval records are in the JSON.

The sign of C'(h) is checked throughout. With a positive constant numerator in A and 1-C positive, this implies A decreases; consequently h/A increases and the field-to-normal angle decreases continuously. The displayed endpoint angles are numerical evaluations. At exactly h=0 the separate perpendicular diagnostic applies; the nonzero-h states use the existing oblique diagnostic.

At h = 0, 1/400 and 1/100, an independent four-variable Newton reconstruction solves the full flux equations without using the continuation formulas. It agrees with the constructed states to better than 1e-60 in normalized units. Direct 80-digit flux differences are below 1e-65. A separate magnetosonic matrix calculation agrees with the diagnostic's characteristic speeds. Common Galilean boosts preserve classification, and a 1% inconsistent change of downstream pressure is rejected.

The diagnostic receives states and front speed without a family label. The construction intentionally follows the saved branch, so this is a local continuation check, not a blind search over all solution families. The 201 plotted samples visualize the curve; they do not replace the continuous-domain check. 97 recorded checks pass. Their count includes the covering intervals and controls, not independent solar events.

## Why energy cannot be replaced by constant entropy

RMO checks conservation of mass, momentum and **total energy**, together with induction and normal-field continuity. Total energy contains thermal, kinetic and magnetic terms. Entropy provides a separate admissibility condition: it need not remain constant through a shock and increases in this model.

As a deliberate negative control we impose p2 = p1 r^gamma, then reconstruct the field to satisfy normal momentum at h=0. Mass, momentum and induction still agree, and the entropy change is essentially zero. However the energy-flux difference is about -0.01390614 in the chosen normalized units. The unchanged diagnostic rejects these states and assigns no wave family. This is an illustration of an incorrect closure, not an admissible shock or a reconstruction of Riemann's historical gas calculation.

The historical distinction concerns an early shock treatment; the modern MHD Riemann problem uses total-energy conservation. Riemann's wave construction and the later correct shock relations are complementary parts of the modern framework.

## Limits and next dependency

The result does not establish global uniqueness, enumerate a full Riemann fan, determine the driver (blast/piston), fit EUV emission, or exclude non-wave image interpretations. It does not combine h uncertainty with all earlier scalar input ranges. A matched local field-to-normal angle, upstream plasma motion, radio/EUV/thermal association and joint observational uncertainty remain needed before treating this as a diagnosis of the selected solar front. No new raw data or new observed angle was used.

## Sources

- [Takahashi & Yamada, MHD jump relations and admissibility, Section 2.2](https://arxiv.org/html/1310.2330v1#S2.SS2).
- [Fitzpatrick, Oblique MHD Shocks](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node105.html).
- [Ma et al. (2011), event and perpendicular model context](https://arxiv.org/html/1106.6056v1).
- [Kozarev et al. (2011), event and magnetic-model context](https://arxiv.org/html/1406.2372v1).
- [Salas (2007), history of shock-wave theory](https://doi.org/10.1007/s00193-007-0084-z).

The current source audit and prior numerical records are preserved separately. RMO92 code and outputs are stored in `oblique_continuation/` and `results/oblique_continuation/`.