CALC-007

Recovering the Duncan and Gordon relation, and evaluating its range

Feeds CLM-0068 and CLM-0069. This is the calculation authorised on 2026-09-02: evaluate the threshold across real conditions rather than at one nominal point.

Why the relation had to be recovered, not just copied

An extraction pass reported the relation as F_threshold = 2.8e-13 f^2 Tn^(1/2). Substituting f = 2.45 GHz and Tn = 200 K gives about 2.4e7 in the document's own units, against the document's stated 1.2e5. Wrong by a factor of about 198. So the relation as reported could not be right.

The document (SRC-108, section 3.2, equation 9) states two worked values:

> "Letting fo = 2.45 GHz and Tn = 200 K the thermal runaway threshold for the SPS microwave beam > can be estimated as F = 1.2 x 10^5 erg/cm2.s = 12 mW/cm2. The corresponding threshold for > fo = 5.8 GHz is approximately F = 67 mW/cm2."

Two worked values and one unknown exponent is a solvable problem.

Solving for the exponent

Trial 1, Tn^(+1/2) as reported:
  2.8e-13 x (2.45e9)^2 x 200^(+0.5)
  = 2.8e-13 x 6.0025e18 x 14.142
  = 1.6807e6 x 14.142
  = 2.376e7 erg/cm2.s          against the stated 1.2e5.  WRONG by ~198x

Trial 2, Tn^(-1/2):
  2.8e-13 x (2.45e9)^2 x 200^(-0.5)
  = 1.6807e6 x 0.070711
  = 1.1884e5 erg/cm2.s         against the stated 1.2e5.  MATCH

Cross-check at the second worked value, 5.8 GHz:
  2.8e-13 x (5.8e9)^2 x 200^(-0.5)
  = 2.8e-13 x 3.364e19 x 0.070711
  = 9.419e6 x 0.070711
  = 6.661e5 erg/cm2.s
  = 66.6 mW/cm2                against the stated "approximately 67 mW/cm2".  MATCH

The relation is therefore:

F_threshold [erg/cm2.s] = 2.8e-13 x f0[Hz]^2 x Tn[K]^(-1/2)

Both worked values reproduce. The exponent on neutral temperature is NEGATIVE one half. The extraction had it positive, which is a dropped minus sign in the text layer, and it inverts the physical behaviour: the threshold FALLS as the neutral atmosphere warms, it does not rise.

Unit conversion used throughout: 1 erg/cm2.s = 1e-7 J / 1e-4 m2 / s = 1e-3 W/m2. So 1.2e5 erg/cm2.s = 120 W/m2 = 12 mW/cm2, which is consistent with the document's own equivalence.

Evaluating the range

In W/m2, with f0 in Hz: F_th = 1.6807e3 x Tn^(-1/2) at 2.45 GHz, and 9.419e3 x Tn^(-1/2) at 5.8 GHz.

Neutral temperature at 100 km altitude spans roughly 170 to 210 K across season, solar activity and latitude. That range is provisional pending the profile data now being gathered, and the figures below will be re-run against it.

Result 1, and it is the opposite of what I expected. The runaway threshold barely varies with ionospheric conditions. Across a 40 K spread in neutral temperature it moves by about 11 per cent, because the dependence is only Tn^(-1/2). The threshold for this mechanism is almost entirely set by the design frequency, which is a choice, not by the state of the atmosphere.

So the premise of the calculation was wrong for this mechanism. The field's use of a single point value is defensible here. It is the SELF-FOCUSING threshold, with N_e cubed and T_e to the fourth (CLM-0057), where the spread will be enormous - and that is the calculation still to do.

Result 2, the factor-of-two band made explicit. The document states its uncertainty as a factor of two. Applying it to the Tn = 200 K case:

The 230 W/m2 design figure sits at 0.97 of the upper bound of the 2.45 GHz band. That is CLM-0066's finding, now derived rather than inherited: the adopted figure is at the extreme optimistic edge of its own source's stated uncertainty.

Result 3, the frequency case, from the document's own words. SRC-108 states that "the ohmic heating produced by an SPS microwave beam operating at 2.45 GHz with a power flux of 20 mW/cm2 is equivalent to the heating expected for a beam operating at 5.8 GHz and a power flux of approximately 112 mW/cm2". Check: 112/20 = 5.6, and (5.8/2.45)^2 = 5.60. Exact. So the f^2 inverse-square ohmic scaling is confirmed from the primary source in 1978, matching URSI in 2007 (CLM-0047), and CALC-005's ratio was right for ohmic heating even though CALC-006 corrected the exponent for the instability threshold.

What this does and does not establish

It establishes the relation, its units, and that the runaway threshold is insensitive to atmospheric state. It does NOT establish that the threshold is correct - the mechanism it describes was reportedly refuted in 1981 as a transport coefficient artefact (CLM-0056), which if true makes this whole relation moot except as history. That is the sharpest reason to verify CLM-0056 against Duncan 1981 directly.

Next

Evaluate the self-focusing relation from CLM-0057 once the electron density and temperature profiles arrive. That is where the range should be wide, and where a published range would be a genuine contribution.

POSTSCRIPT 2026-09-19. Equation (9)'s exponent is printed as -1/2 at the render of SRC-108 PDF p.19, printed 3-7 (read at the render): F_threshold = 2.8 x 10^-13 f_o^2 T_n^(-1/2), the minus on the page, where the text layer yields "2.8 x 10-13fo2Tn1/2" with the minus detached onto the next line. So the "dropped minus sign in the text layer" above is confirmed at the page, not inferred from the two worked values; the coefficient's reproduction from equations (6) and (8) is in CLM-1094 (CLM-1094, 2026-09-19).

Cite this