Bayesian Inference of Gravity through Realistic 3D Modeling of Wide Binary Orbits: General Algorithm and a Pilot Study with HARPS Radial Velocities

When 3D relative displacement $\mathbf{r}$ and velocity $\mathbf{v}$ between the pair in a gravitationally-bound system are precisely measured, the six measured quantities at one phase can allow elliptical orbit solutions at a given gravitational parameter $G$. Due to degeneracies between orbital-geometric parameters and $G$, individual Bayesian inferences and their statistical consolidation are needed to infer $G$ as recently suggested by a Bayesian 3D modeling algorithm. Here I present a fully general Bayesian algorithm suitable for wide binaries with two (almost) exact sky-projected relative positions (as in the Gaia data release 3) and the other four sufficiently precise quantities. Wide binaries meeting the requirements of the general algorithm to allow for its full potential are rare at present, largely because the measurement uncertainty of the line-of-sight (radial) separation is usually larger than the true separation. As a pilot study, the algorithm is applied to 32 Gaia binaries for which precise HARPS radial velocities are available. The value of $Γ\equiv \log_{10}\sqrt{G/G_{\rm N}}$ (where $G_{\rm N}$ is Newton’s constant) is $-0.002_{-0.018}^{+0.012}$ supporting Newton for a combination of 24 binaries with Newtonian acceleration $g_{\rm N}>10^{-9}$m,s$^{-2}$, while it is $Γ=0.134_{-0.036}^{+0.056}$ ($0.143_{-0.041}^{+0.068}$) for 8 (6) binaries with $g_{\rm N}<10^{-9}$ ($<10^{-9.5}$) m,s$^{-2}$ representing $> 3.5σ$ discrepancy with Newton. However, one system (Stars HD189739 and HD189760) dominates the signal. Without it, the tension with Newton is significantly lessened with $Γ=0.063_{-0.041}^{+0.065}$. Thus, to verify the tentative signal, many such systems need to be discovered and their kinematic nature such as any possibility of hidden tertiary stars needs to be thoroughly addressed. The pilot study demonstrates the potential of the algorithm.

💡 Research Summary

The paper presents a fully general Bayesian framework for inferring the gravitational constant from the three‑dimensional (3‑D) relative displacement and velocity of wide binary (WB) stars. When the six components of the relative position vector r and velocity vector v are measured at a single epoch, the data are in principle sufficient to solve for an elliptical orbit given a value of the gravitational parameter G. In practice, the orbital geometry (semi‑major axis a, eccentricity e, argument of periastron φ₀, orbital phase φ, inclination i, and node angle θ) is strongly degenerate with G, so a Bayesian approach that treats all parameters probabilistically is required.

Algorithmic core
The method assumes that the sky‑projected separation (x′, y′) from Gaia DR3 is essentially error‑free, while the line‑of‑sight (LOS) separation (z′) and the three components of the relative velocity (vₓ′, v_y′, v_z′) are measured with finite uncertainties (HARPS radial velocities for v_z′, Gaia proper motions for vₓ′ and v_y′). The free parameters are therefore reduced to {e, φ₀, Δφ≡φ−φ₀, i, log₁₀f_M, Γ}, where f_M accounts for a possible 5 % mass‑scale error and Γ≡log₁₀√(G/G_N) quantifies any deviation from Newton’s constant. The semi‑major axis a and the node angle θ are analytically eliminated using the exact relations (2)–(7) that connect (x′, y′) to the orbital elements.

The posterior probability is
p(Θ|data) ∝ L(data|Θ) × ∏_l f_pr(Θ_l),
with a Gaussian likelihood based on the χ² of the four observables (z′, vₓ′, v_y′, v_z′). Priors encode physical expectations: inclination follows |sin i|, φ₀ is uniform, Δφ follows the Keplerian time‑uniform distribution (∝(1−e²)^{3/2}/(1+e cos Δφ)²), f_M is log‑normal (σ=5 %), and e is either a thermal distribution (2e) or flat. Γ is given a uniform prior in the interval