# RMO95 — one fixed-upstream shock-polar sector

## Observer result

**A shock polar is not the shape of a front. Each point represents an allowed
downstream plasma velocity for a selected front normal, in one common frame.**
The curves below are in velocity space. Their resemblance to an arc, cone or
solar feature does not identify that feature or reconstruct its spatial shape.

R94 is reproduced as the central point of a newly calculated sector. With
the complete upstream state fixed, changing the field-to-normal angle from
89° to 81° changes the downstream flow deflection from about 2.56° to 1.03°.
Compression remains between 1.554878 and 1.560203 across the 321 computed
points. Every point receives a fast-shock classification from the unchanged
local diagnostic. Thus nearly equal compression can accompany different
flow directions within this particular model.

**Для наблюдателя:** почти одинаковое сжатие не означает одинакового поворота
потока. Здесь поле перед фронтом фиксировано. Это один модельный сектор
поляры, а не восстановленная форма EUV-фронта или полная карта решений.

| Field-normal angle | Normal rotation phi | Compression | u2x, km/s | u2y, km/s | Flow deflection |
|---:|---:|---:|---:|---:|---:|
| 89° | −4° | 1.554878 | −387.0931 | −17.3147 | 2.56114° |
| 85° | 0°; R94 anchor | 1.560000 | −384.6154 | −12.1849 | 1.81456° |
| 81° | +4° | 1.558214 | −384.5737 | −6.8990 | 1.02774° |

The largest sampled compression occurs inside the sector; the end points
alone do not give its complete sampled range. All quoted ranges here refer
to computed points, not a rigorous continuous-angle enclosure.

## Inputs, coordinates and the meaning of the axes

The exact adopted numerical inputs are the serialized R94 normalized upstream
state after one common frame change: rho1=1, p1=0.04056,
u1=(−1,0,0), B1=(0.05618818561896245,0.6422339004220036,0), mu0=1,
gamma=5/3. One velocity unit is 600 km/s. Density has an arbitrary fixed
normalization, so no new field in gauss is inferred. vA1≈386.812 km/s is fixed.
High-precision arithmetic does not add precision to these adopted inputs or
convert them into observations.

All candidate planes are stationary in this frame. The original x axis is
the outward normal of the R94 anchor; the original y axis is its chosen
tangent. Rotate n=(cos phi,sin phi,0), with t=(−sin phi,cos phi,0), for
−4°≤phi≤4°. The field direction stays fixed, so thetaBn≈85°−phi.
The figure always uses the original axes, not the rotating n,t components.

Adding +600 km/s in x returns to the anchor's solar frame: upstream plasma
then has zero velocity, while the candidate planes have normal speed
600 cos(phi) km/s. Holding their solar normal speed at 600 for every phi
would specify a different experiment. In neither representation is phi a
time coordinate or a reconstructed front surface.

Panel (a) plots (u2x,u2y), with equal Cartesian scales and a visible zoom.
Panel (b) plots |u2| against the signed angle from u1 to u2 in the same
frame. Positive delta is defined by atan2[(u1×u2)z,u1·u2]. The 81°, 85°
and 89° labels refer to the field-normal angle, not to delta.

## Calculation

Let w=u1·n in the stationary frame, b=p1/(rho1 w²),
h=Bn²/(rho1 w²), A=Bt1²/(rho1 w²). For gamma=5/3, the regular compression
equation used here is

    P(r) = 2(4−r−5br)(1−rh)² − Ar[r+5−2rh(4−r)] = 0.

At each sampled normal, bisection selects the root in the predeclared
bracket [1.3,1.9], surrounding the R94 anchor. The maximum of the quadratic
P' over this bracket is negative at each sampled normal, so this root is
unique inside this bracket at that normal. Other compression brackets and
other wave structures were not searched.

The linked state is recovered from Bt2=r(1−h)Bt1/(1−rh), w2=w/r,
ut2=ut1+Bn(Bt2−Bt1)/(rho1 w), and
p2=p1+rho1 w²(1−1/r)+(Bt1²−Bt2²)/2. Neither r=1.56 nor a family label
is imposed along the sector. Positive states and the regular denominator
are checked before applying entropy and characteristic tests.

The starting conservation laws are the ideal-MHD
[mass, stress, induction and total-energy jumps](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node102.html),
with the regular oblique relations discussed in
[Fitzpatrick's oblique-shock section](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node105.html).
Point-dependent de Hoffmann–Teller frames are not used for plotting this polar.

## What was checked

| Question | Check and outcome |
|---|---|
| Do computed states conserve the required quantities? | All 321 points pass mass, three momentum, two induction and total-energy jumps, plus normal-field continuity; maximum scaled Decimal residual 9.43×10⁻⁶⁷ |
| Are the jumps admissible ordinary fast shocks? | Positive states and entropy; all 321 diagnostic outputs are fast. Sampled upstream fast Mach 1.43508–1.43930; downstream 0.73691–0.73857; downstream flow remains above its normal Alfvén speed |
| Is the R94 point retained? | Central state agrees after the frame change to 9.51×10⁻¹⁷ in normalized scaled units |
| Does an independent formulation reproduce the state? | Three six-variable global-coordinate Newton solves of the original flux equations agree to 1.80×10⁻⁶⁶ or better; they do not use P(r) or its recovery formulas |
| Are the characteristic speeds consistent with the equations? | All seven eigenvalues of a conserved-variable flux Jacobian agree at both sides of the three controls; maximum binary64 difference 6.22×10⁻¹⁵ normalized |
| Does a coordinate or frame change alter the result? | Recomputed states after a rigid 37° rotation and a common velocity addition agree; moving-front F−SQ balances and classifications are preserved |
| Can a normal change be treated as a mere picture rotation? | No. Changing only the normal of the unchanged anchor pair violates the jumps and is rejected |
| Is the drawn curve resolved? | Midpoint discrepancy falls from 0.0001924 to 0.00004811 km/s when interpolation spacing decreases from 0.1° to 0.05°; improvement factor ≈4 |

These are separate formulations within this project, not an external solver
comparison or held-out solar benchmark. The 90-digit checks use dimensionless
scales; the existing binary64 classifier keeps its original tolerances.
The final grid spacing is 0.025°. The refinement test supports the drawn
interpolation, not a continuous-domain admissibility theorem.

## Scope for ApJ

The demonstrated result is a geometry-linked sector of allowable one-shock
transitions for a fixed upstream state. A full Riemann solution additionally
connects two prescribed far states through its complete wave pattern; a full
Riemann map varies a declared initial-data slice. This sector establishes
neither of those complete objects nor a unique solar-front identification.

The choice between ongoing driving, an impulsive excitation and later free
propagation requires a global time-dependent comparison. None is selected
by the word fast or by the shape of the velocity polar. A spatial front
reconstruction also needs geometric observations and global initial/boundary
conditions; local jump constraints can contribute to it but do not determine it.

Matched feature/time association, field/normal/plasma-flow constraints,
emission alternatives, uncertainty semantics and solver coverage remain as
recorded in the earlier claim ledger. No new raw data or global calculation
was used. No historical novelty claim is made for the classical polar concept.

## Reproduction and saved evidence

Run `python3 shock_polar/audit.py`, then `python3 shock_polar/plot.py`, from
the project root. Dependencies: Python standard library, NumPy and Matplotlib.
The protocol, numerical source hashes and runtime versions are saved. The
R94 JSON source hash is checked before calculation. Numerical outputs:
`RMO_shock_polar.json` and `verification.json`; vector figure:
`RMO_shock_polar.pdf` and `.svg`. No earlier audit is rerun by these commands.