CALC-010

The finished calculation: what actually sets this threshold

Supersedes CALC-008 and CALC-009 as the authoritative treatment. Those used the 1981 relation, which buries its criterion in a numerical constant. This uses the 1978 relation, which exposes it, and the result is that the criterion is the dominant term.

The relation, units recovered

From SRC-108 page 3-10, captioned as the threshold power density for underdense heating, transcribed from a rendered image:

Pth [W/m2] = 1.2e23 x Te^5 x fo^2 x fe^-6 x lambda_perp^-4 x sin^2(phi)

  Te          electron temperature, K
  fo          wave frequency, Hz
  fe          plasma frequency, Hz
  lambda_perp perpendicular striation scale, m
  phi         angle, degrees

Units recovered by back-solving against the document's own worked case and cross-checked against a companion overdense relation on page 3-15 that states W/m2 outright. Failing conventions miss by 8 to 30 orders of magnitude, so this is not a coincidental fit.

Implementation verified against the document's own figure

The document states Figure 3.2 was generated from this equation with Pth = 20 mW/cm2, which is 200 W/m2, Te = 1200 K and fe = 8 MHz. Solving for lambda_perp:

Figure 3.2 is described as two panels: lambda_perp against phi in KILOMETRES over 0 to 90 degrees, and a zoomed inset in METRES over 0 to 10 degrees. The computed values land in kilometres at large angles and hundreds of metres at small ones. The implementation matches the figure's own structure.

Sanity check on fe. fe = 8.98 x sqrt(Ne) Hz with Ne in m^-3. For Ne = 1e12 that gives 8.98 MHz, close to the document's 8 MHz, corresponding to Ne of about 7.9e11 m^-3. Consistent with the F-region. So fe and Ne are interchangeable inputs and Madrigal density data feeds straight in.

The finding: lambda_perp is an OUTPUT, so the threshold is a CHOICE

The document states lambda_perp "is determined by the ionospheric conditions and the power density of the heating radiation". It never uses the words growth length or scale length, only "striation width". So this relation does not yield a threshold. It is a locus relating threshold power to striation width, and to extract a single threshold you must decide which striation width matters.

The document's only concrete values for that are 5 km and then 1 km, and both are engineering cutoffs tied to the SPS beam footprint, not physically derived.

Because Pth goes as lambda_perp^-4, that choice dominates everything:

Moving the criterion from 5 km to 1 km changes the threshold by a factor of 625, at either frequency, because the lambda_perp term carries no frequency dependence.

And the 1978 report's own 20 mW/cm2 self-focusing figure implies a striation tolerance of about 2.4 km at 90 degrees. Nobody states that. It is the hidden assumption inside the number, made explicit here for the first time, so far as the registered sources show.

The sensitivity budget

Each input varied alone about the document's case, baseline Te 1200 K, fe 8 MHz, lambda_perp 1 km, phi 90 degrees, fo 2.45 GHz, giving 6,837 W/m2.

What this establishes

The largest single source of variation in this threshold is a human choice about which striation width matters, not any physical quantity. It exceeds the electron density term by about 2.4 times, the temperature term by 40 times, and the frequency term by nearly 1,800 times.

And the frequency term is the SMALLEST of the five, at 5.6. The entire space solar power literature debates 2.45 against 5.8 GHz, a factor of 5.6 on this threshold, while the criterion choice alone spans four orders of magnitude and is never stated.

This is why the 1981 version of the relation gives a clean-looking single number: it fixed the criterion inside its 0.43 coefficient and stopped mentioning it. And it is why the 2004 review's only caveat on that version concerns "the arbitrary growth-rate criterion used to define threshold". That caveat is the whole story, and it is one line in a 28-page review.

What this does NOT establish

It does not give a threshold. It shows that a single threshold cannot be given without stating a striation-width criterion, and that no source among the 115 registered sources states one. It does not establish that any design exceeds a real physical limit, and it does not establish that self-focusing at 2.45 GHz matters at all - CLM-0083 records that nobody has bounded the relation's validity at fo/fpe of about 200, and no experiment has ever verified underdense self-focusing.

Consequence for the Madrigal work

The measured density and temperature pairs still matter and still improve the calculation, but they are now the second and third terms rather than the first. Dropping the independence assumption (CLM-0081) will narrow the combined Ne and Te contribution from its current 4,096 and 243 towards something smaller. It will not touch the 1.0e4 from the criterion, because that is not a measurable quantity. So the honest conclusion holds regardless of what the radar data says.

CALC-010 ADDENDUM, 2026-09-02, after independent verification

Verification confirmed all six claims. The transcription was right this time, checked against a 300 dpi render. The units were independently re-derived, with MHz, GHz and unconverted mW/cm2 all failing by factors of 1e6, 1e9 and 1.78, and cross-checked against the page 3-15 companion relation which also only reproduces its own stated result with frequencies in Hz and lengths in metres. The solved lambda_perp values were reproduced exactly and checked against a read of Figure 3.2 itself rather than its description. And the central claim was strengthened: Table 3.1 on page 3-11 shows the document treats lambda_perp as an engineering cutoff, and page 3-10 states the locus reading almost verbatim.

Two defects were found, and the first is not a new error but a repeat of one I had already identified.

Defect 1: the sensitivity budget varies correlated inputs independently.

The budget sweeps plasma frequency while holding electron temperature fixed, and vice versa. Both are ionospheric state variables and are physically correlated. This is exactly the error I identified in CALC-009, wrote a dated caveat about, and then committed again one calculation later. So the caveat was recorded and not carried forward, which is the same propagation failure that let CLM-0014 hold CLM-0012's defect for a day.

The consequence is bounded and knowable. The factors of 4,096 for plasma frequency and 243 for temperature must not be read as clean independent contributions. Correcting for correlation will narrow their combined contribution, probably substantially. It cannot touch the 1.0e4 from the criterion choice, because a criterion is not an ionospheric state variable and does not correlate with either. So the conclusion that the criterion dominates survives, and in fact strengthens: if the physical terms narrow and the criterion term does not, the gap widens.

The corrected budget therefore reads: criterion 1.0e4, physical inputs combined less than the product of their independent sweeps, and wave frequency 5.6. The ordering is unchanged.

Defect 2: claim 6 was presented as a fresh derivation and is not.

The statement that the 1978 report's 20 mW/cm2 figure implies a striation tolerance of about 2.4 km is arithmetically correct, but it is literally the phi = 90 degree row already computed in the verification of Figure 3.2. Presenting it as a separate finding inflates one result into two. It stands as a true observation and it is not independent evidence.

Recorded rather than corrected silently. The numbers above stand as computed with these two caveats attached. What changes is how they may be stated: the physical-input factors are upper bounds, not estimates, and claim 6 is a restatement rather than a second line of evidence.

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