Windward Note

3.1 Speed and density

Equation (4) was evaluated from 8.0 to 15.0 km/s at intervals of 0.5 km/s, at the three densities of Table 2. Because β is constructed as a function of B and ρ only, speed cannot move a station out of the Hall regime. The cut retains a factor V/7600 through the interaction parameter.

Figure 1. Hall cut from equation (4) against freestream speed. The curves separate by density. Speed changes the slope and does not change the ordering.

A Hall parameter of 2, at 0.67 T, occurs only at 3.1×10⁻² kg/m³. That density is below the entry stations considered here. At orbital speed the steep density is the only case with a cut near 20%.

3.2 Distribution on the barrel

Figure 2 compares the selected station, LEO nominal, with the field removed and with a uniform 0.67 T face field. Both panels use the scale 0–7.2 MW/m². The model is invariant along the barrel, and the nose is outside a belly-flop cylinder, so the figure does not locate a nose peak. The loaded band is the keel. The field changes the magnitude and does not change the shape.

No field

Uniform 0.67 T

Figure 2. Convective flux on the addressed barrel for LEO nominal. Left of each panel is aft. The two panels share one scale.

Figure 3. The same comparison against clock angle. Ordinate in MW/m².

3.3 One coil in three

A uniform multiplier cannot represent a distributed tile. Here one loop occupies every third cell of a three-coloring, at 90,922 ampere-turns, so that an isolated loop would produce 0.67 T on axis. The face value is the summed field at 16.7 mm, and equation (4) is applied locally. Figure 4 shows the cut, on a scale of 0–20%. It is not on the flux scale of Figure 2.

Figure 4. Local Hall cut on a 12×8 patch at the keel, LEO nominal. Outlined cells contain the coil.

Return flux from neighboring windings reduces the coil-face field from 0.67 T to 0.63 T. The open neighbor, at 45 mT, lies under the 50 mT floor, so its cut is zero. Removing that floor leaves a cut of 2.7%. The hot cell is the unprotected cell. Two thirds of the area is unaltered, and the mean saving is one third of the uniform-field saving.

Figure 5 repeats the patch at the three stations on one flux scale. The coil positions do not move. Only the magnitude changes. The sequence is not a trajectory.

Figure 5. Keel patch at the three stations, common flux scale. Coil cells are outlined.

3.4 Angular distribution

Figure 6 places the bare wall, the uniform field, and the one-in-three mean on one abscissa, the clock angle from the keel. The cut fraction does not depend on that angle, so the curves share the geometric factor (cos φ)^1.5.

Figure 6. Flux against clock angle for LEO nominal. Ordinate in MW/m². The dashed curve removes the 50 mT floor.

At this station the bare keel is 2.40 MW/m², 1.59 MW/m² at 40°, and 0.76 MW/m² at 62°. Half of the keel value remains at 51° for every curve. The uniform field lies below the lattice mean at every angle. The difference is largest at the keel. The decline toward the chine is geometric.

3.5 Evaluation outside the anchor density

The closure was also evaluated at International Standard Atmosphere densities of 1.167 kg/m³ (500 m) and 1.112 kg/m³ (1 000 m), with speed held at 7.6 km/s and angle of attack at 70°. These are not flight states. They test whether the fit, taken far above its anchor density, produces a distinct thick-air regime.

Figure 7. Extrapolation to 500 m (solid) and 1 000 m (dashed). Ordinate in MW/m². Not a landing prediction.

The bare keel values are 112 and 109 MW/m². The uniform-field values are 22 and 22 MW/m². Both densities drive equation (4) to its cap of 0.80, so the field-on curves are one fifth of the field-off curves. The two altitudes differ by about 2%, consistent with the square-root density factor in equation (1). No additional regime appears. The cap, not a change in flow physics, sets the result.

3.6 Hall regime

The perpendicular conductivity retained in equation (4) is σ/(1+β²). Table 5 gives β, that factor, and the resulting cut at the stations of Table 2.

At every station β is between 10 and 16.5, so η is of order 10⁻² or smaller. The electrons are magnetized, and the Faraday current is the part the Hall term removes. A larger β produces a smaller cut. The density at which β = 2 is 3.1×10⁻² kg/m³. The cap f = 0.80 is already active near 1.5×10⁻² kg/m³ at 7.6 km/s, which is the region of Figure 7. At 0.67 T and 7.6 km/s the cut passes 5% near 5.8×10⁻⁴ kg/m³, 20% near 1.2×10⁻³ kg/m³, and 50% near 2.3×10⁻³ kg/m³.

Two additional limits are imposed before the Hall expression. Below 2.9×10⁻⁶ kg/m³ the denominator of β is not allowed to decrease further. Below 50 mT the cut is zero. On the steep station the open neighbor is at 45 mT; without the floor its cut would be 6.8%. The coil axis is nearly wall-normal, the orientation associated with a Hall current rather than a streamwise Faraday current. Equation (4) does not distinguish the two.

3.7 Ion slip

The conductivity tensor used for MHD heat-shield calculations by Bisek, Boyd, and Poggie takes the mobility ratio μₑ/μᵢ = 400. For a weakly ionized gas the ion-slip coefficient is then s = βₑ²/400. The perpendicular factor becomes (1+s)/[(1+s)²+βₑ²], which reduces to equation (4) when s = 0. Substituting that factor into the Stuart number, without refitting the coefficient 12, gives Table 6. The Cowling limit 1/(1+s), appropriate to an insulating wall on which the Hall current has been cancelled, is a different boundary condition and is not applied to the figures.

The steep-station cut moves from 19.5% to 23.6%. The largest slip coefficient is at the lunar-class station, where β is largest, and that station remains the smaller cut. The ordering is unchanged, and the cap is not reached. Applying the Cowling factor in the same unfitted expression would saturate every station. That saturation is the boundary condition, not a measured ion-slip effect. The current closure around a buried coil is not resolved here. An independent criterion for a 0.5 T field on RAM C-II places the Hall effect above about 40 km and ion slip above about 70 km, which is the same direction: thinner air, larger β, larger s.

3.8 Integrated difference

Figure 8 is the bare flux minus the uniform-field flux. It is not the lattice of Figure 4. The difference occupies the keel band because the cut is a scalar at a given station.

Figure 8. Flux removed by a uniform 0.67 T field, LEO nominal. Same geometry as Figure 2.

For LEO nominal the barrel integral is 103.0 GJ with the field off and 97.4 GJ with the uniform field, a difference of 5.6 GJ. The one-in-three column is one third of the uniform column wherever the open tiles carry no cut. The sine column is not a trajectory. It bounds the sensitivity of equation (5) to the assumption that the peak flux persists for the whole dwell. Section 3.11 replaces the rectangles with one integrated glide. The steep station removes the most among the rectangles even with the shorter dwell, because it is the only station at which the cut is near 20%. Load scales as V³. Whether the field couples is controlled by density.

3.9 Sensitivity of equation (4)

Each coefficient in equation (4) was halved and doubled, one at a time, at the steep station (ρ = 1.2×10⁻³ kg/m³, 7.6 km/s, 0.67 T). The baseline cut is 19.5%. The same variations were repeated at the 500 m density of section 3.5, where the baseline sits on the cap.

At the steep station the cap and the floor do not move the cut. Both remain above or below the value that would change a 0.67 T result. The anchor Hall parameter moves the cut from 53.0% to 5.4%. The Stuart coefficient and the decay constant move it by the same amount, from 10.4% to 34.2%, because their product is still small and the exponential is nearly linear. The prefactor scales the cut from 9.7% to 39.0%. The baseline is not stable under these changes.

At 500 m the baseline is the cap, 80%. Halving the cap, or halving the prefactor, lowers that result to 40%. The other coefficients leave it on the cap. Section 3.5 is therefore a statement about the cap, not about entry.

3.10 Comparison with published ratios

Equation (4) was not fitted to either source below. The comparison asks whether the closure reproduces a published ratio. It does not.

The 0.67 T used throughout this note is the upper field of the tunnel magnet [11]. It is not a validated heat-flux ratio, and equation (4) remains an internal fit [10].

3.11 Flight 4, read off a published chart

The rectangles in section 3.8 hold one density and one speed for an assigned dwell. A public chart gives the alternative. Max Fagin plotted altitude and velocity against time for Starship Flight 4 from the webcast [14]. Wikipedia records atmospheric reentry at T+44:54 and transonic flight at T+01:03:17 [15], which is the same interval as that chart. This note reads the curves at one-minute marks, to about ±2 km and ±0.1 km/s. It does not have the samples, and it is not SpaceX telemetry. Angle of attack is held at 65° because the chart does not give attitude. Density at each altitude is the 1976 US Standard Atmosphere [13], which stops this integral near 85 km.

Figure 9. Keel flux along the Flight 4 reading, from about 85 km down. Ordinate in MW/m². The two curves nearly overlie because the cut is small where the flux is large.

Below 85 km the reading lasts 937 s. The keel flux peaks at 0.71 MW/m² near 70 km and 7.05 km/s, on the high-speed shelf. The addressed barrel carries 57.0 GJ with the field off and 55.8 GJ with the uniform field, a difference of 1.17 GJ. The keel accumulates 399 MJ/m² without the field and 391 MJ/m² with it. Moving every altitude up or down by 2 km changes the saving from 0.67 GJ to 2.03 GJ.

The nominal rectangle removes 5.6 GJ and the steep rectangle removes 18 GJ. The Flight 4 reading removes 1.17 GJ. On the chart the vehicle is still near 7 km/s at about 70 km, where the Hall parameter is large and the cut is small, and it has slowed to about 2 km/s by about 42 km, where equation (1) has already collapsed with V³. The steep station is not a stretch of this entry. It is a state the vehicle does not hold.

3.12 Chemistry and the sheath

Table 10 evaluates equations (7)–(12) at 0.67 T. The enthalpy matches V²/2. The Hall parameter in the parcel is equation (11). The Hall parameter in the last column is equation (4).

At these states the parcel temperature is 5919–7097 K and the electron density is 10¹³–10¹⁴ cm⁻³. The parcel Hall parameter is 8.2, 1.0, 0.5. Equation (4) gives 38, 14, 10. The fit is larger because it scales with the freestream density. The parcel uses the hot, compressed gas, where the collision frequency is higher and β is of order one. A Hall parameter of order one does not suppress the perpendicular conductivity to one percent. That does not authorize a new heat-flux map. The Stuart number inside equation (4) is not σB²L/(ρV), and this note does not pass the parcel conductivity through that formula.

The floating potential is 2.3–2.8 V. The electron energy flux at the sheath is 0.9%, 4.3%, 6.9% of equation (1). It is not the convective load. The electrical boundary condition for an insulating wall remains the one already used in section 3.7: the normal current is zero. The sheath voltage does not replace that condition and does not change Figures 2–8.

The time to build the equilibrium NO⁺ density at the Park rate, once the atoms are present, is 1.39, 0.15, 0.07 μs in Table 10. A layer a few centimetres thick, crossed at a post-shock speed near 1 km/s, lasts tens of microseconds. On that comparison the ionization step is not frozen. The time to produce the atoms is not computed. The Saha ratio n(N⁺)/n(NO⁺) is 0.16, 0.49, 0.51. At the nominal and steep stations the omitted atomic ion is the same order as NO⁺, so the electron density there is low. The lunar-class station is not in the table. Its enthalpy is 61 MJ/kg, and the same parcel only closes if N⁺ is ignored. The Saha check then exceeds NO⁺ by orders of magnitude, so a seven-species electron density would be the wrong one.

3.13 Mechanism of the sheath

Equation (12) is an electrostatic floating sheath on a dielectric. The electrons are Maxwellian at the parcel temperature, with Tₑ set equal to the heavy-particle temperature. NO⁺ is the only ion. There is no secondary emission, no electron reflection, and no magnetic field inside the sheath itself. Under those restrictions the random electron flux at the sheath edge exceeds the cold-ion Bohm flux by √(m_NO⁺ / (2π mₑ)), a factor of about 93. A wall held at the edge potential would draw a net electron current and charge negative.

The wall potential relative to the edge falls until the Boltzmann factor cuts the electron flux down to the ion flux. That potential is φ_f in equation (12). It is not the potential relative to the undisturbed parcel. Reaching the Bohm speed takes a presheath drop of order kTₑ/(2e), and the density at the edge is then nₛ = nₑ exp(−1/2). The wall-to-parcel drop is φ_f plus that presheath drop, about 5.0 kTₑ/e. Only φ_f enters the energy flux. The ions are given the cold-ion speed cₛ = √(kTₑ / m_NO⁺) even though the parcel has Tᵢ = Tₑ. Replacing cₛ by √(k(Tₑ + Tᵢ)/mᵢ) would multiply the flux by √2. Adding the ion thermal energy 2 kTᵢ, which equation (12) leaves out, would raise the energy per ion by 2 kTₑ against the 2 kTₑ + eφ_f already counted, a factor of about 1.3. Neither correction is applied.

On Table 10, φ_f is 2.31–2.77 V. With nₑ taken from NO⁺ alone, λ_D = √(ε₀ kTₑ / (nₑ e²)) is 1.24, 0.38, 0.26 μm. The ratio of the NO⁺ thermal gyroradius at 0.67 T to that Debye length is 481, 1,696, 2,491. The electrostatic sheath is thinner than the ion gyromotion by those factors. It is not a shock standoff. The energy flux is 0.9, 4.3, 6.9% of equation (1), and it is not added to the tiles.

The result used later is the boundary condition, not the voltage. Zero net current means the Hall current does not close through the ceramic. A scalar model of that wall therefore uses σ/(1 + β²) rather than σ. Equation (12) does not evaluate σ, and it does not replace section 3.7.

The same boundary condition is incomplete once the field is oblique. Ions then cross a magnetic presheath of thickness comparable to the ion sound gyroradius before they enter the Debye sheath. Equation (12) has no angle between B and the wall normal, and no such layer. No term from it is added to equation (1).

3.14 What the published ratios require

Equation (4) multiplies the flux on every heated tile by the same factor. The two rows of Table 9 do not have that shape. Fujino’s calculation lowered the stagnation point less than it lowered the surface integral. Oswald, Lani, and Herdrich lowered the total heat flux and raised the stagnation point. A single factor cannot do either.

The smallest statement that can is a two-zone split of the bare barrel. Tiles with clock angle inside ±15° are called the keel band. They are assigned a ratio r_k. The remaining tiles, the flank, receive the single ratio r_f that makes the load-weighted mean equal a published area ratio r_A:

f_k r_k + (1 − f_k) r_f = r_A

f_k is the bare-load fraction of the keel band under equation (2), not a solved shock layer. The ±15° edge is a definition. r_f is then determined. It is not a prediction. If r_f is negative, the published pair removes more heat than this split can move onto the keel.

On the nominal barrel the ±15° band carries 28% of the bare load. Imposing Fujino’s pair leaves the flank at 60% of its bare flux: both zones fall, and the flank falls farther. Imposing the Oswald 181 kW pair raises the keel band to 124% and leaves the flank at 37%. Heat has to leave the flank and arrive on the keel. Equation (4) has no term that can do that. The 227 kW setting, an 83% drop in total heat flux, did not come with a reported stagnation ratio, so no flank ratio is inferred for it. A 24% rise on a band that held more than 14% of the load would make r_f negative at an area ratio of 0.17. That would no longer be a rearrangement of the same heat.

A stagnation solution in the class of Bush’s increases the shock standoff and lowers the stagnation flux as the interaction parameter rises. That sign agrees with Fujino and disagrees with Oswald. The parcel of section 3.12 supplies the interaction parameter σB²R/(ρV), with R = 0.5 m: 13.8, 2.6, 1.5 on the three rows of Table 10. Those numbers are not inserted into a standoff solution here. Doing so would still be a stagnation correction, not a map of where the displaced heat goes.

3.15 Pattern from the shock-layer crossflow

Equation (13) is integrated from the keel to the chine at 0.67 T. The perpendicular conductivity is 5.33e-1, 2.74e+1, 4.73e+1 S/m on the rows of Table 10. Figure 10 is the absolute heating ratio. Figure 11 is the share. Figure 12 puts that ratio on the nominal convective flux from equation (2). Table 12 reports the bands.

Figure 10. Absolute heating ratio from equation (13). Values below 1 are a reduction. The ratio falls from the keel toward the chine.

Figure 11. Share of the remaining wall load. The bare-weighted mean is 1. Above 1, that clock angle takes a larger fraction of the heat that still arrives.

Figure 12. Nominal arc. Equation (2) is the bare convective flux. The second curve multiplies it by the absolute ratio in Figure 10. The heat that remains is shifted toward the keel. The keel flux itself is lower than the bare curve.

The flow has two results, and they answer different questions. The absolute ratios in Figure 10 are below 1 everywhere: on the nominal arc the keel is 0.49 and the flank is 0.39, against an arc mean of 0.41. Both zones lose heat, and the flank loses more. That ordering matches Fujino’s imposed pair in Table 11 and not its magnitude. The share in Figure 11 removes that mean. On the nominal arc the keel’s share of the heat that still reaches the wall is 1.18, and the flank’s share is 0.95. Relative to the surviving load, heat has moved toward the keel. The absolute keel flux has not risen, so this is not the Oswald pattern. The Flight 4 shelf, where σ_⊥ is small, stays near an absolute ratio of 0.83 and a keel share of 1.04. The Lees values are not substituted for equation (1), and Figures 2–8 are unchanged.