On the Proper 3D Modeling of Orbits in Wide Binary Gravity Tests

A gravitationally-bound system of wide binary stars with precise 3D relative displacement $\mathbf{r}$ and velocity $\mathbf{v}$ can allow a Keplerian orbit reconstruction for an assumed gravitational parameter $G$. A sufficiently large sample of such systems can also be used to directly infer $G$ specific to the sample. When the sky-plane angular coordinates are (essentially) exact and the distances to the binary stars are sufficiently precise, the orbit modeling can be readily done using two internal consistency relations provided by the ratio of angular separations in R.A.\ and decl.\ and the physical sky-plane separation $s$ that must be satisfied by any Keplerian orbits. A recent study argued that treating $a$ explicitly free with an empirical prior imposed on it can dramatically change the inference of $G$ in the Chae et al.\ sample of 36 wide binaries. Here I introduce and test a Bayesian MCMC algorithm that allows $a$ to be explicitly free, and find using the $α$-Cen AB - proxima system and the Chae et al.\ sample that the explicit-$a$ algorithm with an observational constraint is fully equivalent to the implicit-$a$ algorithm with an internal consistency relation involving $s$. For the $α$-Cen AB - proxima system, I present orbits in Milgromian (MOND) as well as Newtonian gravity models obtained with the verified algorithms. I show that a recent claim of Newtonian gravity from the Chae et al.\ sample may be a consequence of using arbitrarily enlarged errors for some of $\mathbf{r}$ and/or $\mathbf{v}$. I discuss how wide binaries can be properly analyzed and modeled to probe gravity at low acceleration and test gravity models, including the necessity of deriving individual-specific parameters, the importance of follow-up observations for anomalous systems, and the relevance of Bayesian MCMC approaches.