MCMC Analysis: S2 Orbital Models

WILL Relational Geometry vs General Relativity 1PN

24 walkers · 200 burn-in + 800 production · 174 observables · 9 parameters
willrg.com · Anton Rize · 2026-05-04

1. Summary

MCMC confirms the optimizer result: WILL RG and Hybrid achieve χ² ≈ 727, vs GR 1PN χ² ≈ 801. The Δχ² ≈ 74 is robust across the full posterior — not an optimizer artifact. The Hybrid match proves the advantage comes from ν-coupled precession phasing, not the WILL precession magnitude.

Metric	WILL RG	GR 1PN	Hybrid
MLE χ²	727.20	800.81	727.20
MAP χ²	727.92	801.40	727.78
Δχ² vs GR	-73.48	0.00	-73.63
Acceptance	0.440	0.438	0.436

Fig 1 — χ² from MLE and MCMC MAP estimates

2. Parameters (Median ± 1σ)

Parameter	WILL RG	GR 1PN	Hybrid
a_as [arcsec]	0.124138 +0.000157/−0.000178	0.124462 +0.000182/−0.000174	0.124122 +0.000177/−0.000187
e []	0.887184 +0.000257/−0.000285	0.887801 +0.000235/−0.000277	0.887182 +0.000272/−0.000265
i [rad]	2.376897 +0.001923/−0.001865	2.370964 +0.001895/−0.001930	2.377014 +0.002100/−0.002043
ω [rad]	1.162889 +0.001416/−0.001371	1.160573 +0.001555/−0.001545	1.162908 +0.001335/−0.001473
Ω [rad]	3.995273 +0.001685/−0.001857	3.987849 +0.001956/−0.001786	3.995300 +0.001624/−0.001847
P [days]	5862.502055 +0.534563/−0.540969	5862.663697 +0.582869/−0.517923	5862.468767 +0.535052/−0.473988
T₀ [MJD]	58257.797946 +0.105186/−0.102679	58257.648224 +0.101929/−0.099628	58257.784034 +0.096432/−0.106870
V₀ [km/s]	-15.659725 +2.896599/−2.750213	-15.168083 +2.895796/−2.660538	-15.744279 +2.772561/−2.672707
K [km/s]	2889.993264 +4.341304/−4.618635	2894.482989 +4.308747/−4.627340	2890.124355 +4.398206/−4.475936

Derived Quantities

Quantity	WILL RG	GR 1PN	Hybrid
β	0.00642460 +0.00002244/−0.00002049	0.00637886 +0.00002123/−0.00002050	0.00642585 +0.00002193/−0.00002097
R_s	12,817,420.73 +135,847.41/−121,665.65	12,546,111.72 +126,164.66/−120,689.89	12,824,976.40 +132,063.41/−125,545.04
M	4.3401 +0.0460/−0.0412	4.2482 +0.0427/−0.0409	4.3427 +0.0447/−0.0425
R₀	8,361.51 +40.04/−36.43	8,279.62 +38.63/−37.09	8,363.53 +40.06/−37.32

3. Posteriors

Fig 2 — Posterior overlap for key parameters

WILL RG

Fig 3a — WILL RG corner plot (68% & 95% contours)

GR 1PN

Fig 3b — GR 1PN corner plot

Hybrid

Fig 3c — Hybrid corner plot

4. Chain Convergence

Fig 4a — WILL RG traces

Fig 4b — GR 1PN traces

Fig 4c — Hybrid traces

5. Residuals

Normalized residuals at each model's MAP solution. Systematic phase-correlated structure in GR 1PN residuals — absent in WILL and Hybrid — confirms the ν-coupled parametrization reduces systematic error.

Fig 5 — Residuals (RA, Dec, RV)

6. Method

Data: 174 observables — S2 astrometry (GRAVITY/Keck) + radial velocities.

Sampler: emcee ensemble, 24 walkers, 200 burn-in + 800 production. Flat priors within physical bounds.

Key control: The Hybrid model uses GR 1PN precession magnitude but WILL's ν-coupled phasing. Its identical χ² to full WILL isolates the parametrization as the sole source of improvement.

WILL Relational Geometry · Open Research
Data: GRAVITY Collaboration (2018), Do et al. (2019)
willrg.com · github.com/AntonRize/WILL · DOI: 10.5281/zenodo.17115270