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 $\Gamma \equiv \log_{10}\sqrt{G/G_{\rm N}}$ (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 $\Gamma=0.134_{-0.036}^{+0.056}$ ($0.143_{-0.041}^{+0.068}$) for 8 (6) binaries with $g_{\rm N}<10^{-9}$ ($<10^{-9.5}$) m\,s$^{-2}$ representing $> 3.5\sigma$ discrepancy with Newton. However, one system (Stars HD189739 and HD189760) dominates the signal. Without it, the tension with Newton is significantly lessened with $\Gamma=0.063_{-0.041}^{+0.065}$. 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.