CALC-009

Both thresholds re-run against modelled profile data

Supersedes the provisional figures in CALC-007 and CALC-008. Those calculations stated their input ranges were provisional pending profile data. The data arrived, from models actually run rather than tabulated, so both are re-run rather than adjusted. CALC-007 and CALC-008 remain in place unedited; their numbers are superseded by those below.

The input data, and how it was obtained

Raw data saved as iono-msis2-tn-90-100-110km.csv and iono-pyiri-ne-200-250-300km.csv.

Two acquisition caveats recorded because they bear on confidence. The official IRI Fortran binary could not be executed in the agent's sandbox, so PyIRI was substituted; it uses the same coefficient maps but is a different implementation. And no working route to IRI's electron temperature sub-model was found, with the relevant paper behind a publisher wall - so T_e remains unsourced.

Runaway threshold re-run

F_th(W/m2) = 1.6807e3 x Tn^(-1/2)   at 2.45 GHz
             9.419e3 x Tn^(-1/2)    at 5.8 GHz

Spread at 2.45 GHz: 122.0 to 125.4 W/m2, which is 2.8 per cent. CALC-007 estimated 11 per cent from a wider assumed range. The real spread is four times narrower.

So the conclusion strengthens. The runaway threshold is effectively a constant of about 124 W/m2 at 2.45 GHz, set by the design frequency and almost independent of the atmosphere. Using a single point value for this mechanism is not merely defensible, it is correct.

Self-focusing threshold re-run

Spread at 2.45 GHz: 2.25 W/m2 to 9.04e10 W/m2. A factor of 4.0e10, or 10.6 orders of magnitude.

CALC-008 estimated 7.2 orders from assumed ranges. The real spread is over three orders wider, because the modelled electron density minimum of 2.19e9 m^-3 is 23 times lower than the 5e10 I assumed, and density enters cubed.

Against the 230 W/m2 design figure:

Note the low-threshold extreme is less severe than CALC-008 suggested, 102 times rather than 980, because the real modelled density maximum of 1.80e12 m^-3 is lower than the 3e12 I assumed. The two errors ran in opposite directions.

What the re-run changes, and what it does not

It changes the numbers and it sharpens both conclusions. It does not touch the four caveats in CALC-008, and one of them is now more important, not less: an independent verification pass established that Duncan and Gordon explicitly state no fixed self-focusing threshold exists in the usual sense, and that the 20 mW/cm2 figure often associated with it was an illustrative graphing input rather than a derived threshold (CLM-0072). So the originators of this mechanism declined to state a fixed number for it, and this calculation shows why: across real conditions there is no single number to state.

The honest summary. Of the two mechanisms behind the inherited 23 mW/cm2 figure, one gives a threshold that is effectively constant and about 124 W/m2 at 2.45 GHz, and was reportedly refuted in 1981 as a transport coefficient artefact (CLM-0056). The other varies by ten and a half orders of magnitude with time of day and solar cycle, and its own originators declined to state a fixed threshold for it. Neither supports a single design constant, and NASA's own 1981 report says the figure "is not valid" (CLM-0073).

Do not publish these numbers yet. T_e is unsourced and enters to the fourth power, which is the dominant remaining uncertainty after N_e. Re-run once a real electron temperature model is available, ideally IRI's own sub-model outside a sandbox.

CALC-009 CAVEAT ADDED 2026-09-02, before any re-run

An assumption was made and not flagged: that N_e and T_e vary independently.

CALC-008 and CALC-009 both take the extremes of each input separately and pair them - highest electron density with lowest electron temperature, and lowest density with highest temperature - to bound the range. That is only valid if the two vary independently.

They probably do not. A hot daytime ionosphere is generally also a dense one, so N_e and T_e are likely positively correlated. If they are, then the pairs producing both extremes of my range describe conditions that do not physically occur, and the stated 10.6 orders of magnitude is an overestimate.

The direction of the error is knowable in advance. The relation goes as N_e^-3 and T_e^+4. A positive correlation means high density comes with high temperature, so the two exponents partly cancel, which compresses the range. So the true spread is narrower than 10.6 orders, possibly much narrower.

This is now being fixed properly rather than estimated: measured, time-matched N_e and T_e pairs from incoherent scatter radar are being obtained from the Madrigal database, so that the range can be computed from the real joint distribution instead of from independently assumed bounds. Arecibo is the preferred instrument, because the 1977-81 self-focusing observations this whole question derives from were made there.

The quantity that actually matters is not either variable but the combination N_e^3 / T_e^4, and its measured range is what the next calculation will report.

Recorded here rather than corrected silently. The numbers in CALC-009 stand as computed, with this caveat attached, and CALC-010 will supersede them once measured pairs are in hand. The methodological point is worth keeping on the record: an unflagged independence assumption between two correlated inputs, in a relation with exponents of opposite sign, inflates an uncertainty range. It is a quieter error than a wrong exponent and it survived two of my own calculations.

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