CALC-015

The coefficient confirmed against an independent source

This closes the last live dependency on the closed Perkins and Goldman paper.

The problem this solves

Every threshold figure in CALC-011 to CALC-014 uses Duncan's coefficient of 0.43. CLM-0093 established it by arithmetic agreement between two documents, but both were Duncan: his 1980 precursor and the 2004 review's quotation of his 1981 paper. If Duncan himself carried an error, both would carry it and the agreement would prove nothing. The original was closed access, so there was no way to check. register/layers.md recorded this as the live fragility.

The independent source

A forward-citation search on Perkins and Goldman 1981, never previously run, returns 72 to 74 citing papers. One of them applies the relation to the SPS case directly:

> SRC-124, Usui, "Interactions between Microwave Beam and Ionosphere", J. Vac. Soc. Jpn. 51(8) > 529-534, 2008, section 2.2.1, translated from the Japanese: the threshold of the microwave beam > power density at which beam thermal self-focusing occurs is expressed as a function of electron > density, electron temperature and microwave frequency. Calculating with an electron density of > 1e12 per cubic metre and an electron temperature of 1500 K, for the SPS model at 2.45 GHz the > threshold is 32 W/m2, and for the SPS model at 5.8 GHz the threshold is 425 W/m2.

It cites Perkins and Goldman directly, not Duncan. So it is independent of the chain under suspicion.

The check

Pth = 0.43 x (1e12/Ne)^3 x (Te/1000)^4 x (f_GHz)^3

Ne = 1e12 m^-3, Te = 1500 K:
  density term      (1e12/1e12)^3 = 1.0000     <- exactly the normalisation, so it drops out
  temperature term  (1500/1000)^4 = 5.0625
  frequency 2.45    2.45^3        = 14.706
  frequency 5.8     5.8^3         = 195.11

  2.45 GHz:  0.43 x 1 x 5.0625 x 14.706 = 32.01 W/m2     Usui states 32     diff +0.04 pct
  5.8  GHz:  0.43 x 1 x 5.0625 x 195.11 = 424.73 W/m2    Usui states 425    diff -0.06 pct

Both reproduce to better than 0.1 per cent.

Usui's chosen density of 1e12 m^-3 is exactly the relation's normalisation density, so the density term evaluates to one. That is fortunate: it means the test isolates the coefficient and the temperature and frequency exponents, with no contribution from the exponent most likely to hide an error. Two values at two frequencies, both matching, also test the f^3 exponent independently.

What this establishes, and what it does not

Establishes: - Duncan's 0.43 coefficient is correct, confirmed by a source independent of Duncan. - The correction factor C_F is unity. CALC-011 listed a non-unity C_F as one of three candidate explanations for the coefficient discrepancy and never resolved it. It is resolved. - The f^3 exponent is right, since one coefficient reproduces two frequencies. - The full text of Perkins and Goldman is no longer needed for any figure in the corpus.

Does not establish: the Ne^-3 exponent, because Usui evaluated at the normalisation density where that term is unity. That exponent still rests on the structural argument in CLM-0091 and on the abstract. It is the remaining untested part of the relation.

And an independent verdict on the design figure

SRC-124 states that at 2.45 GHz, with a design power density of 230 W/m2 against its 32 W/m2 threshold, thermal self-focusing may occur; and that at 5.8 GHz, against its 425 W/m2 threshold, it is considered not to occur.

This agrees in direction with CALC-014 at both frequencies, by a completely different route: a single-point evaluation at assumed conditions, against a distribution over 22,705 measured radar pairs. Note carefully what it means. A third party applying the original physics to the design figure finds the figure exceeds the threshold at 2.45 GHz. That is not support for 230 W/m2 as a limit. It is the opposite.

Caveat carried forward

The same source's section 2.1.4 states beam energy loss reaching "several per cent to about 10 per cent" under high solar activity, which is three to four orders of magnitude above CALC-013 and above SRC-105's independent full-path figure. Recorded as an unresolved contradiction in CLM-0107 and under investigation. It does not touch the threshold result above, which depends on a different relation, but it means SRC-124 is not to be treated as uniformly reliable.

Cite this