CALC-011

The domain condition, and the threshold over the valid domain only

Supersedes CALC-008, CALC-009 and CALC-010's range figures. Those evaluated the threshold across all ionospheric conditions. This applies the domain condition first, which the earlier calculations did not have, and the range collapses.

The condition, from the original paper's abstract

Perkins and Goldman 1981, abstract recovered verbatim from OSTI record 6536431 after the full text proved closed access with no legitimate open copy anywhere. The threshold CONDITION is stated directly:

self-focusing will not occur unless   2cf / (pi * f_p^2 * l) < 1

  c   speed of light
  f   wave frequency
  f_p plasma frequency
  l   spatial growth length:  l ~ 25 km in the F region,  l ~ 1 km in the E region

Dimensionally checked: [m/s][1/s] / ([1/s^2][m]) is dimensionless, as a threshold condition must be.

This is the parameter the 2004 review says is unstated and that the 1981 restatement buries in its 0.43 coefficient. It is stated plainly in the original abstract: 25 km in the F region.

Solving for the critical density

Setting the condition to unity and solving, with f_p = 8.98 sqrt(Ne) Hz for Ne in m^-3:

Against the PyIRI modelled F-region range of 2.19e9 to 1.80e12 m^-3 with a median of 2.20e11:

- At 2.45 GHz the critical density is 2.32e11 m^-3, which is 1.05 times the modelled median. So self-focusing is possible in roughly the upper half of F-region conditions and cannot occur at all in the lower half, including at night. - At 5.8 GHz the critical density is 2.5 times the median, so the window is narrower still. - In the E region at 2.45 GHz the condition needs 5.80e12 m^-3, which exceeds real E-region densities by orders of magnitude. E-region self-focusing effectively cannot occur at this frequency.

The threshold over the valid domain

CALC-009 reported 10.6 orders at 2.45 GHz. Restricted to where the instability can physically occur, it is 4.8. Nearly six orders of the earlier spread came from densities where the condition is not met at all - at Ne of 2.19e9 m^-3 the condition evaluates to 105.9, far above unity, so there is no threshold there to compute. I was evaluating a threshold in a regime where the instability does not exist.

And the 230 W/m2 design figure sits INSIDE the valid range at both frequencies. It is not obviously wrong. It is within the physically possible span, towards the lower-threshold end.

An internal consistency check worth noting

At the critical density the threshold is the same at both frequencies, 506.8 W/m2 at Te = 1000 K. That is not a coincidence and it is a good check on the implementation: the critical density scales as f, so (1e12/Ne_crit)^3 scales as f^-3, which exactly cancels the relation's f^3 term. The threshold at the onset of validity is therefore frequency-independent. Neither source states this and it falls out of the algebra.

An unexplained discrepancy, flagged not resolved

The two relations are structurally identical for the F region: same 1e12 m^-3 density normalisation, same Te^4, same f^3, same Ne^-3. So the Duncan 1981 version quoted in the 2004 review IS the Perkins and Goldman F-region relation, renormalised from 15 MHz to 1 GHz.

But the coefficients disagree by a factor of about 1,034:

P&G:    1.5 mW/m2 at 15 MHz, renormalised by (1000/15)^3 = 296,296  ->  444.4 W/m2 at 1 GHz
Duncan: 0.43 W/m2 at 1 GHz
ratio:  1,034

Three candidate explanations, none established: the paper's C_F factor is not unity and absorbs it; Duncan used a different growth length than 25 km; or one restatement contains a milliwatt to watt slip. This must be resolved before any figure from either relation is published. All the threshold numbers above use Duncan's coefficient, so if his is the erroneous one, every value here moves by three orders of magnitude.

The E-region relation is a different relation

P_c,E ~ 1 mW/m2 x (f/15MHz)^3 x (1e5 cm^-3/n_e)^1 x C_E

Density to the power minus ONE, not minus three, and no electron temperature term at all. So the Duncan and 2004 version reproduces only the F-region form. Anyone applying it to E-region effects is using the wrong relation, and the 1978 report cited both regions.

Standing caveats

Electron temperature remains unsourced and enters to the fourth power. The correlation between density and temperature is still not accounted for, so the ranges above remain upper bounds on the spread. And the coefficient discrepancy above is unresolved. None of these touch the domain condition itself, which is the main result.

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