CALC-083

timings, node count, and the Rayleigh-distance caveat

Calculation · Cited by CLM-0864

What this is

This is the project's own arithmetic, published in full so it can be checked rather than taken on trust. It is not a source. Every input it uses is cited to a record, and the conclusion it reaches only carries as far as those inputs do.

Supports CLM-0864. A calculation is required because that claim's stated value is arithmetic performed here. Independently confirmed by PyPO 1.2.1 (SRC-358). Constants re-derived by the registrar before landing: lambda = 0.122364269 m, alpha = 1.151292546, half-wavelength element count 2.098e8, Rayleigh distance 16,345 km at D = 1 km and 65,379 km at D = 2 km. All reproduce exactly as stated.

wall clock, single CPU core, numpy + scipy only
  4001-point radial profile, 0 to 20 km    3.610 s
  encircled-power integral to 5 km        11.534 s
  TOTAL                                   15.144 s
  result: peak 248.20 W/m2; encircled inside 5 km 94.76 pct

node count against element count, 2.45 GHz, 1 km aperture
  half-wavelength elements  pi * 500^2 / (lambda/2)^2      = 2.098e8
  quadrature nodes used                                    = 4.0e3
  ratio                                                    = 5.2e4

Rayleigh distance 2D^2/lambda at 2.45 GHz
  D = 1 km   16,300 km    GEO slant range 3.58e7 m  -> just OUTSIDE
  D = 2 km   65,000 km    GEO slant range 3.58e7 m  -> still INSIDE

Scripts: emsim/sbsp-beam/po_beam.py, emsim/sbsp-beam/focused_results.py.

CALC-083 reproducible specification

An input nobody else can rebuild is grounds to doubt the result, not merely a documentation gap. My own CLM-0860 lattice is the argument for this section existing.

Physical case


Aperture	circular, radius 500.0 m (1 km diameter), planar, normal along the beam axis
Illumination	10 dB Gaussian amplitude taper, `E(r) = exp(-alpha * (r/a)^2)` with `alpha = -ln(10^(-10/20)) = 1.151292546`
Taper convention	10 dB is edge POWER below centre power, so edge AMPLITUDE is `10^(-10/20) = 0.316228`. The alternative reading, edge amplitude `10^(-10/10)`, gives a different answer and is NOT used
Radiated power	6.72e9 W, total from the aperture
Frequency	2.45e9 Hz; `lambda = c/f` with `c = 2.99792458e8`, giving 0.122364269 m. NOT 0.1224 rounded
Range	35,800,000 m, sub-satellite altitude, NOT a slant range
Phase condition	FOCUSED (retrodirective), i.e. the quadratic Fresnel phase is cancelled. In the script this is `fresnel=False`

Numerical scheme


Formulation	scalar Rayleigh-Sommerfeld far-field aperture integral reduced to a zero-order Hankel transform by circular symmetry: `F(rho) = integral_0^a E(r) J0(k r rho / R) r dr`
Quadrature	Gauss-Legendre, n = 4000 nodes, mapped to `r` in `[0, a]` via `numpy.polynomial.legendre.leggauss`
Normalisation	`S(0) = P * A_eff / (lambda^2 R^2)` with `A_eff = \	integral E dA\	^2 / integral \	E\	^2 dA`; pattern normalised to the IDEAL on-axis value so any Fresnel on-axis loss appears in the result
Bessel function	`scipy.special.j0`
Profile grid	4001 points, `numpy.linspace(0.0, 20000.0, 4001)` metres
Encircled integral	Gauss-Legendre, m = 6000 nodes over `rho` in `[0, 5000]` m, integrand `S(rho) * rho`, times `2*pi`, divided by 6.72e9

Machine and software


Machine	Windows 11, single CPU core - the code is not parallelised and uses no BLAS-threaded call in the timed path
Python	3.14
Libraries	numpy 2.3.5, scipy 1.16.3, pinned in `emsim/.venv-po`. No other dependency.
Script	`emsim/sbsp-beam/po_beam.py`, `profile()` and `encircled_fraction()`
Independent confirmation	PyPO 1.2.1, SRC-358, commit `e75c79a809db6e21a6a4a1f19a90761f18fdb938`, CPU double-precision path (`libpypocpu.dll`), run by a separate party

Results to be reproduced

4001-point radial profile, 0 to 20 km     3.610 s
encircled-power integral to 5 km         11.534 s
TOTAL                                    15.144 s

peak on axis                            248.20 W/m2
encircled inside 5 km radius             94.76 pct

PyPO (SRC-358), same case                247.60 W/m2 collimated / 248.20 focused, 94.56 pct

Timings are indicative and machine-dependent; the PHYSICAL results are not. A verifier who reproduces 248.20 W/m2 and 94.76 per cent on any hardware has reproduced the claim. The 15.144 s is offered as evidence of tractability, not as a benchmark, and a reproduction within an order of magnitude on a single core supports the claim equally well.

What a verifier should attack

1. The focused-versus-collimated branch. These differ by 0.25 per cent at 2.45 GHz and by 1.15 per cent at 5.8 GHz, and an earlier comparison mistook one for the other and called a mismatch agreement. Confirm which branch is being run before comparing anything. 2. The taper convention. The alternative reading of "10 dB" gives 181.4 W/m2 rather than 248.20. 3. Whether n = 4000 is converged. Convergence was checked from 500 to 8000 nodes and the peak was identical to six decimal places, but that check was mine.

CALC-083: the reproduction criterion, stated explicitly

THE TIMING IS EVIDENCE OF TRACTABILITY, NOT A BENCHMARK. THE REPRODUCTION CRITERION IS THE PHYSICAL RESULT.

Quantity	Is it the assertion?	Reproduction test
Peak 248.20 W/m2	YES	Must reproduce, on any hardware
Encircled 94.76 pct inside 5 km	YES	Must reproduce, on any hardware
15.144 s wall clock	NO	Indicative only; within an order of magnitude on a single core supports the tractability point equally well

A timing is a property of the machine. The peak and the encircled fraction are properties of the physics. Holding a verifier on faster or slower hardware to a wall-clock number that was never the assertion would record a failure against the wrong quantity - a label defect dressed as rigour, and precisely what VF1 says to keep apart.

Cite this

energyaperture.com/calc/calc-083

CALC-083. The working above, not a source document.