tt1-sat-01 · tt1-formation-01

Gravitational Waves

TT-1 is a three-spacecraft laser interferometer on heliocentric orbits, trailing Earth by 20° with 3-million-kilometre arms. It cannot be built on Mars — so the city exhibits it instead: a 1:1 spacecraft with its gold test mass laid open, and a formation display whose breathing motion is integrated orbital dynamics, not hand animation. Every number on this page passed a known-answer gate first.

Peak sensitivity

5.1e-21 /√Hz @ 7.55 mHz

Arm length

3.0 Gm

Test-mass noise

2.10 fm/s²/√Hz

Formation breathing

1.29 % pk-pk

Why gravitational waves, on Mars

A millihertz interferometer needs arms longer than ten thousand planets. The city can't host the instrument — it hosts the understanding of it.

Strain from a supermassive black-hole merger arrives at 10^-21: measuring it means resolving picometres across millions of kilometres, which is why TT-1 flies as three free-falling spacecraft in deep space. What stands in the science quarter is the teaching version: a full-scale engineering unit on a display cradle, its instrument bay cut open so the causal chain reads in one glance — telescope barrel → Zerodur optical bench → the 46 mm Au-Pt cube floating in its electrode cage. The cube is the whole story: a mass touching nothing, whose only remaining force is gravity.

Next to it, the formation exhibit compresses the constellation 2×10⁸:1 (1 Gm → 5 m) while keeping the spacecraft bodies 1:1. Six red beams close the triangle the way six heterodyne links close the real interferometer.

TT-1 mission poster: constellation geometry, spacecraft cutaway and sensitivity curve — TT-1 MISSION POSTER — 3.0 GM TRIANGLE, 20° EARTH-TRAILING, 60° PLANE TILT (EMERGENT, NOT CHOSEN)

1:1 TT-1 spacecraft exhibit on its display cradle in the city — TT1-SAT-01 AT (-404,-132): OSR RADIATORS FACING CAMERA, MGA GIMBAL — THE ONLY MOVING PART ON BOARD

Formation exhibit: three spacecraft on pylons joined by red laser beams above the Martian surface — TT1-FORMATION-01 AT (-430,-150): SIX BEAMS, APERTURE-TO-APERTURE, SCALE 1 GM = 5 M

How 5.1×10^-21 is made

One master equation, two budgets feeding it, and real solvers behind every allocation.

The whole design serves S_h(f) = (10/3L²) [P_OMS + 2(1+cos²(f/f∗)) P_acc/(2πf)⁴] [1+0.6(f/f∗)²]. Below a millihertz the (2πf)^-4 factor means only residual force on the test mass matters — a nine-term parasitic-force budget (gas damping, charge shot noise, patch potentials, magnetics, radiometer, radiation pressure, stiffness coupling, self-gravity) closes at 2.10 fm/s²/√Hz, gated against LISA Pathfinder flight data. At high frequency the optical chain takes over: an eight-term budget closes at 7.48 pm/√Hz per link.

The allocations are not typed in — they are solved for. Zemax ray tracing confirmed the confocal-paraboloid telescope is aberration-free on axis to 1.9×10^-10 waves; a COMSOL→Zemax thermo-elastic loop turned "3 pm of telescope pathlength" into material law (Zerodur survives at 220 µK/√Hz where aluminium needs an impossible 0.19 µK/√Hz); and a self-built two-beam overlap integral put the tilt-to-length coupling on a moment-integral footing, softening the alignment tolerance from 6.25 to 10.7 µm.

Final strain sensitivity curve with galactic foreground and source tracks — FINAL CURVE: 5.1E-21 @ 7.55 MHZ; SEED-BLACK-HOLE MERGER AT Z=10 GIVES SNR 103 (STANDARD CONVENTION)

Nine-term acceleration noise budget of the gravitational reference sensor — GRS BUDGET: STIFFNESS × PLATFORM JITTER OWNS 63% OF THE VARIANCE — LOW-F SENSITIVITY IS A PLATFORM PROPERTY

COMSOL bending mode of the telescope metering structure under a lateral thermal gradient — COMSOL 145K-DOF METERING TUBE, LATERAL-GRADIENT BENDING (2×10⁷× EXAGGERATED) — FEM HIT THE EXACT STRESS-FREE SOLUTION TO 0.04%

Five known-answer gates

No home-built model was allowed to speak about the design before reproducing something the world already knows.

Gate	Reproduced	Verdict
Sensitivity master equation	LISA public curve: f∗ = 19.09 mHz; asymptotic slopes −3 / −2 / +1	pass
Waveform & SNR machinery	aLIGO BNS range 220 vs ~190 Mpc; f_ISCO(2.8 M_☉) 1570 vs 1571 Hz; D_L(z=1) to 0.03%	pass
Optical link	LISA received power: 297 pW vs ~300 pW public	pass
Constellation dynamics	LISA orbital elements e=0.0048, i=0.48°; 60.00° plane tilt emerges from i=√3e	pass
Acceleration-noise model	LISA Pathfinder in-flight band 1.2–2.5 fm/s²/√Hz	pass, flagged weak — spectral shape not reproduced below 1 mHz, so a 2.0 fm/s² unmodelled reserve is carried

The same discipline runs deeper than these five: the FEM was gated on the exact stress-free solution of a linear thermal field (hit to 0.00–0.47%), the wave-optics overlap on a closed-form complex-Gaussian integral (0.00%), and the transmit-aperture efficiency on the literature optimum 0.81 at w=0.89R — reproduced as 0.8145 at 0.890R. A gate that passes too easily is itself a finding: one gate window had to be widened after it was caught hiding a real bug (see 06).

Breathing from real orbits

The exhibit's motion loop is a baked year of Sun + Earth-perturbation integration — 1.29% arm breathing, 0.21° yaw wobble, zero keyframes.

Three spacecraft on one shared semi-major axis, eccentricity and inclination vectors rotated 120° apart with i = √3 e: the first-order relative orbit is a circle, so the triangle stays equilateral — and the 60° tilt of its plane is algebra, not a choice. Earth then spoils the ideal: a DOP853 propagation with Earth's point mass shows the formation destroyed at 0° trailing (274% arm flexing) and settling to 2.13% at 20° — the trailing angle is a stability-versus-comm-distance compromise you can read straight off the scan.

The display asset samples year two of that propagation into 120 frames, blends the tail into the head for a closed 36 s loop, and drives each craft's yaw along the baked arm-bisector. Any time offset is a valid entry point; three instances could run phase-shifted. What you see wobbling above the regolith is the actual celestial mechanics that caps the arm length of every space interferometer.

Arm breathing, Doppler beat requirement and formation cost scan versus arm length — TWO-BODY IDEAL 0.48% → 2.13% WITH EARTH; 84% OF THE DOPPLER COMES FROM THE TRAILING ANGLE, NOT THE ARM

The ledger

Every number above, and which run produced it.

Claim	Value	Produced by
Peak strain sensitivity	5.1e-21 /√Hz @ 7.55 mHz	gravity-wave/sim/07_performance_science.py
Seed-BH merger SNR (z=10, std convention)	103	gravity-wave/sim/07 + sim/gwlib.py
LISA gate: f∗ and slopes	19.09 mHz; −3/−2/+1	gravity-wave/sim/01_sensitivity_core.py
aLIGO gate: BNS range	220 Mpc (ref ~190)	gravity-wave/sim/02_gate_snr.py
Constellation elements; plane tilt	e=0.00580, i=0.5735°; 60.00°	gravity-wave/sim/04_constellation_orbit.py
Test-mass acceleration budget	2.10 fm/s²/√Hz	gravity-wave/sim/05_acceleration_budget.py
Per-link displacement budget	7.48 pm/√Hz	gravity-wave/sim/06_optical_budget.py
Telescope on-axis WFE; exit pupil	1.9e-10 waves; 5.000000 mm	gravity-wave/opt/01_telescope.py (Zemax ZOS-API)
Thermal soak allowance: Zerodur vs Al	220 vs 0.19 µK/√Hz	gravity-wave/opt/03_thermal_opl.py
Gradient / absorbed-power allowance	214 µK/√Hz; 9.9 µW/√Hz	gravity-wave/fem/tel_grad.java + fem/analyze_fem.py (COMSOL 145k DOF)
TTL lever correction; alignment tolerance	η=0.586; 6.25→10.7 µm	gravity-wave/opt/06_ttl_wave.py
TX aperture efficiency (lit. gate 0.81@0.89R)	0.8145 @ 0.890R	gravity-wave/opt/06_ttl_wave.py
Coating phase drift (closed with evidence)	0.0036 nm/K	gravity-wave/opt/04_coating_lumerical.py (stackrt)
Thruster-noise allowance (binding point)	0.201 µN/√Hz @ 24 mHz	gravity-wave/sim/09_dfacs_loop.py
Formation breathing; yaw wobble	1.29% pk-pk; 0.21°	gravity-wave/model/bake_formation.py (DOP853, year 2)
Received optical power at 3 Gm	651 pW	gravity-wave/sim/03_arm_trade.py (LISA-calibrated link)

What broke

A two-agent independent review corrected fourteen defects. Four of them are worth retelling.

A fit parameter copied 10⁵× too small — and the gate that covered for it

The galactic-foreground fit's κ and γ went in as 0.00521 and 0.00713 instead of 521 and 1680. Both carry units of Hz^-1, so κ·f must be O(1) — a dimensional sniff test that was skipped. The high-frequency cutoff vanished and the foreground stayed 7.9×10⁵ too high at 10 mHz. Worse: the validation gate's window ended at 5 mHz, right before the bug became visible. Both were fixed; the gate now checks that the foreground is invisible at 10 mHz, which any public LISA plot shows.

The arm-length argument, stated backwards

The design doc originally argued "shot-noise-limited strain is independent of arm length, so longer arms buy nothing at high frequency." The algebra is right — but this design is not shot-noise limited: the length-independent noise floor (5.69 pm) exceeds shot noise (4.85 pm), so h ∝ 1/L holds and longer arms pay everywhere. The best argument for the chosen 3 Gm had been written as an argument against it. The plot's own caption contradicted its own curves, 12× apart, on the same page.

102% error with a perfect twin next to it

The FEM's relative mirror decenter missed the exact analytic value by 102% while the relative tilt matched to 0.03%. Fingerprint of a frame-invariance bug: u(M2)−u(M1) is not invariant under the rigid-body rotation that the solver's motion suppression injects — the tilt difference is. Correcting to the M1-frame quantity, dDec = (u₂−u₁) + tilt₁·z₂, landed on the analytic answer to 0.04%. One quantity right and its sibling wrong is diagnostic gold.

Six lasers all pointing at the centroid

First render of the formation exhibit: every beam converged on the triangle's centre — too tidy to be random. The aperture-lookup function used two scratch vectors internally while its callers used the same two vectors to hold both endpoints; the second call overwrote the first result with a unit vector near the origin. Shared mutable scratch versus output containers. Suspiciously regular wrongness is a shared-state signature.

Try it

Both exhibits are live city assets with bilingual knowledge cards.

Open the viewer with viewer/index.html?colony=1&inspect=tt1-formation-01 or walk to the science quarter around (-430,-150). Eleven POI cards (EN/中文 toggle in the viewer) carry the same ledger numbers as this page — 651 pW, 2.10 fm/s²/√Hz, 220 µK/√Hz all resolve to the runs listed above.

Things worth catching: the gold cube behind the cutaway wall of the 1:1 spacecraft; the beams landing exactly on the telescope apertures while the triangle breathes (watch ~36 s for a full baked year); the MGA gimbal — deliberately the only moving part, because a drag-free platform tolerates no reaction wheels; and the spec plaque's claim that the whole thermal design collapses to one line: 2×10^-4 K/√Hz along the metering path.