CALC-109

Scaling-law bands for transmit-array deformation and correlation length

Approach 1 of synthesis/structural-phase-modelling.md: is the phase-error problem centimetres or metres? Bands, not numbers, from two closed forms with every input cited to a held source or marked ASSUMED. Producer-computed by the frequency stream 2026-09-17 and barred from verifying its own work; the landing coordinator reproduced every headline figure independently before this landed (recorded at the end).

Provenance. The project's working file struct_scaling_bands.py, 9,004 bytes, sha256 D52A2B2B00476E20, 146 lines. Runs with no arguments and prints every table below.

The two forms, each cited for its form. First bending frequency of a beam, f_i = (beta_i^2 / 2 pi) sqrt(EI / (rho L^4)) - stated in that notation on SRC-449 PDF p.200 (printed 178), Fig. 4 of a 1980 SPS large-space-system dynamics paper; beta_1 = 4.730 free-free. Thermal bow of a member with a through-depth gradient: curvature kappa = alpha dT / h, cantilever tip deflection kappa L_half^2 / 2, tip slope kappa L_half.

Inputs, with sources.

EI and rho are stated nowhere for any candidate geometry (CLM-0907). They are BUILT here from held depth, length and material class through the truss relation r = h/2, sqrt(EI/rho) = (h/2) sqrt(E/rho_mat) sqrt(f_cap). The two ASSUMED rows are where most of the band width comes from.

Check against the held document before anything else. Setting the tip slope to SRC-451's 1 arc-min at its own 4 K, h = 40 m and L_half = 500 m gives alpha = 5.8e-6/K - between the graphite target and aluminium, in the range of a 1975 quasi-isotropic graphite/epoxy layup. The form reproduces the document's own figure with a physically sensible alpha. Tip deflection at that slope: 7.3 cm. The document's assembly-tolerance figure, 0.17 degree on a 500 m half-span: 1.48 m, static, if the angle is read as a straight line from the centre; read as the tip slope of a uniform bow, the reading used above for the 1 arc-min, it is 0.74 m. The page does not say which. (Corrected 2026-10-22: this line gave 1.48 m only.)

Band 1 - thermal bow at the tip of a 500 or 750 m half-span.

Band 2 - first bending frequency and correlation length. Primary structure, h = 40 m: f_1 = 0.088 to 0.69 Hz (periods 1.5 to 11 s). Whole band including the 3 m secondary: 0.007 to 0.69 Hz. That is 4,400 to 34,000 times SRC-085 p.A-44's gravity-gradient disturbance of about 2e-5 Hz, and it sits inside CLM-0907's uncited working guess of 0.01 to 1 Hz - from held geometry rather than a guess. Correlation length by mode, Lc_n = sqrt(2) L / beta_n from matching a Gaussian autocorrelation to a sinusoid at small lag: for L = 1 km, 299 m (mode 1), 82 m (mode 5), 43 m (mode 10); for 1.5 km, 448 / 123 / 64 m. Mode frequencies scale as (beta_n/beta_1)^2: mode 5 at 13.3 f_1, mode 10 at 48.6 f_1.

Band 3 - what CLM-0912 then permits for detector spacing, L = 1 km.

Reference subarray 10 m (SRC-085 p.A-28); bays 18 m (SRC-451 Vol III p.39). Only the metre case on modes above about 5 drives detector spacing below the subarray size.

What this establishes. The 1975 design put itself in the centimetre regime by choosing 40 m depth and a 4 K cap gradient, and left a 0.7 to 1.5 m static build error to be absorbed by phase control - Vol III p.25 says the electronic control system is "shown to desensitize the tolerance on structural assembly accuracy". A static error is calibratable and does not move; the CLM-0912 relation is for a field that moves.

What it does not establish, and the limits every downstream record must carry. (1) Dynamic amplitude is not determined by scaling laws. These give frequency and shape; amplitude needs a forcing model - gravity gradient, control forces, thermal snap at the 87 eclipses a year SRC-449 p.300 records - and none is held. (2) sqrt(E/rho) and f_cap are ASSUMED; halving f_cap moves f_1 by 30 per cent. (3) The cantilever form is a bound; a free-free plate with two-layer grid structure will bow less at the edge and more in the interior. (4) Lc_n from a sinusoid-to-Gaussian match at small lag is a convention, not a derivation; a different match moves Lc by a factor of order 1.5. (5) No stiffness data for the CASSIOPeiA geometry exists in the register - a patent and a cost study give geometry and mass, not stiffness - so the 1.5 km rows inherit the 1975 design's depth as an assumption.

INDEPENDENTLY REPRODUCED ON LANDING, 2026-09-17, without the script: graphite 2e-7 at 4 K on a 500 m half-span, 2.50 mm; aluminium 2.3e-5, 0.287 m; the alpha that gives SRC-451's 1 arc-min at 4 K, 5.82e-6/K, with a 7.3 cm tip; 0.17 degree on 500 m, 1.48 m; f_1 at graphite, f_cap 1, 1 km, 0.69 Hz and at aluminium, f_cap 0.3, 1.5 km, 0.088 Hz; Lc for modes 1 and 10 at 1 km, 299 m and 43 m. All match the tables to the stated precision. The script runs clean and its digest matches the one declared. The three SRC-451 Vol III quotations were tested against the text layer: p.34 matches at the normalised level, and both p.28 sentences match only after letterspacing is squashed - so they are transcribed from the render, as the stream said, and the squashed match confirms they are there.

POSTSCRIPT 2026-09-18. The CDEP corpus supplies what this calc assumed. Stated beam sections and moduli for two contractors' designs (CLM-0996: Boeing cap EI 3.39e8 and 1.80e8 N cm2 at 7.48 and 4.11 kg/m; Rockwell girder EI 1.22e10 N m2, alpha 0.18e-6 per degC) replace the assumed specific stiffness and cap mass fraction, and the graphite CTE is confirmed at 0.18e-6. Stated first modes exist and are for other structures: 0.0012 Hz for the whole 10 km satellite and a 0.0016 Hz minimum required of a 1 km tension-web antenna (CLM-0995); this calc's 0.088 to 0.69 Hz band is for a 1 km, 40 m deep antenna and is not in conflict, but any reader comparing them must carry the length. A third stated cap gradient, 24 degC (CLM-0997), sits between the 4 K and 50 to 100 K rows.

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