CALC-100
Phase tolerance at 2.45 GHz, and the RP-1076 cross-check
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.
Method. Ruze tolerance relation for random phase errors across an aperture,
G/G0 = exp(-sigma^2), with sigma the rms phase error in radians. Scattered power fraction is
1 - exp(-sigma^2). Path error converts as delta = sigma * lambda / 2pi.
Inputs. c = 299,792,458 m/s exactly. f = 2.45 GHz. Hence
lambda = c/f = 0.1223642 m = 122.3642 mm.
Step 1, invert for a given loss. sigma = sqrt(-ln(10^(-L/10))) for loss L in dB.
| L (dB) | sigma (rad) | sigma (deg) | delta (mm) | lambda/n | scattered |
|---|---|---|---|---|---|
| 0.10 | 0.15175 | 8.69 | 2.955 | lambda/41.4 | 2.3 pct |
| 0.25 | 0.24004 | 13.75 | 4.673 | lambda/26.2 | 5.6 pct |
| 0.50 | 0.33931 | 19.44 | 6.608 | lambda/18.5 | 10.9 pct |
| 1.00 | 0.47985 | 27.49 | 9.345 | lambda/13.1 | 20.6 pct |
| 3.00 | 0.83110 | 47.62 | 16.186 | lambda/7.6 | 49.9 pct |
Step 2, the cross-check. Forward, not inverted. sigma = 10 degrees = 0.1745329 rad.
sigma^2 = 0.0304617. 1 - exp(-0.0304617) = 0.0300025 = 3.0002 per cent.
CLM-0160 records RP-1076 stating main-beam power loss "under 3%" at a 10 degree RMS budget.
Agreement to four significant figures, nothing fitted.
Peak gain cost at the same point: -10*log10(exp(-0.0304617)) = 0.1323 dB.
As path error: 0.1745329 * 122.3642 / (2*pi) = 3.399 mm.
Step 3, the open-loop scale. 1 m / 0.1223642 m = 8.172 wavelengths = 2,942.0 degrees.
Coherent-to-incoherent loss 10*log10(N): 23.9 dB at N = 243, 60.0 dB at N = 1e6, 90.0 dB at
N = 1e9.
Step 4, global versus local. 2,000 m / 2.955 mm = 676,780. Holding the 0.1 dB tolerance
across the whole structure would be 1 part in 676,780.
Provenance. cassiopeia-phase-tolerance.py, sha256
3b96b1485af2def6, 5,335 bytes. No dependencies. Invocation: python
cassiopeia-phase-tolerance.py from any directory. Run 2026-09-13 under Python 3.14.0.
What this CALC does not establish. That any real structure meets or misses these figures. There is no CASSIOPeiA structural model here, no modal analysis, and no detector spacing - the one parameter that would turn the tolerance into a pass or fail is the parameter SRC-088 does not state.
Cite this
energyaperture.com/calc/calc-100
CALC-100. The working above, not a source document.