CALC-107

Pointing budget, and what retrodirection must deliver against mechanical pointing

REGISTERED 2026-09-14. Drafted by SBSP Lycean-EM, redrafted whole after CLAIM J was held on its arithmetic, and every figure below recomputed independently on registration from the stated relations rather than by re-running the author's working.

Purpose. Convert between residual phase tilt across the aperture, beam pointing angle, and ground miss at geostationary range; and establish the ratio between the pointing retrodirective control must achieve and the pointing a structure can be held to mechanically.

Relations. A linear phase gradient g (rad/m) tips the beam by delta_theta = g/k. Ground miss at range H is H . delta_theta. The rms of a linear ramp across an aperture depends on the aperture's SHAPE and its ILLUMINATION, and is A.F for edge value A and geometry factor F.

Inputs. H = 35,786 km; f = 2.45 GHz, k = 2.pi.f/c = 51.3482 rad/m. Aperture 1 km diameter, circular, with a 10 dB Gaussian edge taper - SRC-085's reference design, per CLM-0862. Target miss 250 m; mechanical tolerances 1 arcminute for the spacetenna and 3 arcminutes for individual subarrays, both from CLM-0160 via NASA RP-1076.

Step 1   delta_theta = 250 / 3.5786e7 = 6.9860e-6 rad = 1.4410 arcseconds
Step 2   g = k.delta_theta = 3.5872e-4 rad/m
         across 1 km: 0.35872 rad = 20.553 deg end to end, edge value A = 10.277 deg
         as differential range: delta_theta x 1000 m = 6.986 mm

Step 3, and the geometry factor is not optional.

THE APPLICABLE FIGURE IS 4.131 DEGREES. The superseded draft used A/sqrt(3), the one-dimensional uniform-line case, on an aperture that is neither one-dimensional nor uniformly illuminated, overstating the rms by 43.6 per cent. The taper weights down exactly the edges where a ramp is largest, so ignoring it ALWAYS overstates - the error has a direction and it is the flattering one. With amplitude taper exp(-alpha.u^2), u = r/R, exp(-alpha) = 10^(-10/20) for a 10 dB edge, the power weight is exp(-2.alpha.u^2) and F^2 = (1/2).integral_0^1 u^3.w du / integral_0^1 u.w du. Recomputed here by quadrature from that integral: 0.5774, 0.5000, 0.4020, the first two agreeing with their closed forms 1/sqrt(3) and 1/2.

Step 4, cross-check against a held record, independent of everything above. 1 arcminute = 2.9089e-4 rad gives 10.410 km; 3 arcminutes = 8.7266e-4 rad gives 31.229 km. CLM-0160 records 10 to 30 km for 1 to 3 arcminutes and both ends reproduce.

Step 5, the result. 1 arcmin / 1.4410 arcsec = 41.64 and 3 arcmin / 1.4410 arcsec = 124.92, confirmed by the independent route through ground distance, 10.410/0.250 and 31.229/0.250.

Retrodirective control must deliver pointing between 42 and 125 times better than the structure can be held to mechanically - between one and two orders of magnitude. THE RANGE IS NOT IMPRECISION. CLM-0160 gives two mechanical tolerances and does not say which the comparison is against, so quoting a single figure would assert a choice the source does not make.

WHAT THE SUPERSEDED DRAFT SAID, AND WHY BOTH ERRORS ARE RECORDED. It gave the ratio as 1,213 with the gloss "about three orders of magnitude", and the rms tilt as 5.93 deg. THE 1,213 HAS NO KNOWN MECHANISM AND NONE IS RECORDED. Three routes were tested - 180/0.1441 = 1249.1, 3600/1.441 = 2498.3, and a contrived 60/1.441 x 29.1 = 1211.7 - and none lands cleanly. An invented account of how an error happened is the same defect as an invented figure, and worse in one respect: it reads as established because it sits beside numbers that check out. THE 5.93 WAS FLAGGED BY NOBODY AND WAS FOUND ONLY BY REDRAFTING WHOLE, three lines above the result under investigation, in a step read past twice - once when OUT-003's ten figures were traced and once when CLAIM J was checked. It is a better argument for A68 than the case that produced A68. The gloss changed character rather than its number, because "three orders of magnitude" described the wrong figure rather than rounding the right one.

What this does not establish. That any system achieves the ratio - this states a requirement. Nor any pointing error from a source other than a linear phase tilt across the aperture. FURTHER STEP, ADDED 2026-09-14: WHERE THE CORRECTION STOPS HELPING, AND IT IS NOT AT ONE CORRELATION LENGTH. The same closed form (7) evaluated at larger spacings, not a second calculation. Solving g(t) = 1.5 - 2.exp(-(t/2)^2) + exp(-t^2)/2 = 1 by bisection gives the crossover at D/Lc = 2.3515, where the residual equals the uncorrected variance and the interpolation has stopped being worth anything.

Monte Carlo over 200,000 draws from the exact three-point covariance, seed 20260914: 0.9961 at the crossover against 1.0000, and 1.4553 at D/Lc = 4 against 1.4634. The crossover and the whole table were re-solved independently on registration and reproduce to four figures.

THE DRAFT OF CLM-0917 PUT THIS THRESHOLD AT ONE CORRELATION LENGTH. The asymptote is 1.5 and the parameter is D/Lc, so "beyond the correlation length" was written as the condition when the condition is "beyond 2.35 correlation lengths". The limit was computed and the threshold was assumed, and the error was PESSIMISTIC - it would have condemned a spacing at which interpolation removes seven eighths of the error. CLM-0912's own claim field carried the same loose phrasing and was corrected the same day.

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