Gravitational Anomaly Measurement in Wide Binaries is Sensitive to Orbital Modeling

Abstract

Recent work by Chae et al. (2026) reported a gravitational anomaly in 36 wide-binary pairs, finding a gravity boost factor of $γ\equiv G_{\rm eff}/G_{\rm N} \approx 1.60_{-0.14}^{+0.17}$ at low accelerations, consistent with predictions from Modified Newtonian Dynamics (MOND). We reanalyze the same dataset using a hierarchical Bayesian model that infers a global $γ$ across all systems while fitting three-dimensional orbital elements. Our model yields $γ= 1.12^{+0.27}_{-0.22}$, consistent with Newtonian gravity ($γ= 1$) at the $\sim0.4σ$ level. To identify the source of the discrepancy, we perform a test using an approach similar to Chae et al. (2026), replacing the semi-major axis with a geometric de-projection of the observed projected separation. This test yields $γ= 1.56^{+0.21}_{-0.18}$, closely matching the result of Chae et al. (2026). This suggests that the inferred value of $γ$ is sensitive to how the three-dimensional orbital separation is modeled, and including an independent semi-major axis parameter can account for velocity excesses that would otherwise be attributed to non-Newtonian gravity.

Figures (4)

• Figure 1: Graphical representation of the hierarchical Bayesian model for inferring the gravity boost factor $\gamma$. The global parameter $\gamma$ (blue) is shared across all binary systems. The plate notation indicates that the enclosed structure is replicated for each of the $N = 36$ systems. For each system, we infer six orbital elements ($a$, $\alpha$, $\phi$, $i$, $\omega$, $\Omega$) and masses ($M_1$, $M_2$). These determine intermediate quantities (gray): true anomaly $\nu$, true separation $r_{\rm true}$, rotation matrix $R$, and component velocities $v_1$, $v_2$. The model predicts four observables (yellow): projected separation $r_\perp$, RV difference $\Delta v_r$, and differential proper motions $\Delta\mu_\alpha$, $\Delta\mu_\delta$. Red nodes denote probability distributions.
• Figure 2: Posterior distributions for the gravitational boost parameter. Left: posterior distributions of $\gamma$ for the two model variants. The blue solid curve shows the baseline model ($\gamma = 1.12$), which includes an independent semi-major axis parameter. The orange dashed curve shows the geometric de-projection model ($\gamma = 1.56$), in which $r_{\rm true}$ is derived directly from the observed projected separation. The vertical red dashed line marks the Newtonian prediction ($\gamma = 1$), while the vertical green dashed line indicates $\gamma \approx 1.6$ reported by Chae2026. Right: the same posteriors expressed in terms of $\Gamma \equiv \log_{10}\sqrt{\gamma}$. The horizontal dotted line shows the flat prior $\Gamma \sim \mathcal{U}(-1,1)$ used in the inference. The two models differ only in the treatment of the orbital separation, yet the inferred gravitational boost shifts substantially.
• Figure 3: Diagram showing how the two model variants diverge in their determination of the three-dimensional separation $r_{\rm true}$, while sharing all other components. At the branch point, baseline model treats the semi-major axis $a$ as a free parameter and computes $r_{\rm true}$ from Kepler's equation, yielding $\gamma = 1.12_{-0.22}^{+0.27}$, consistent with Newtonian gravity. Geometric model derives $r_{\rm true}$ directly from the observed projected separation $r_\perp$ through geometric de-projection, yielding $\gamma = 1.56_{-0.18}^{+0.21}$, consistent with the anomaly reported by Chae2026. All other steps, like velocity computation, coordinate rotation, and comparison to observables are identical between the two models.
• Figure 4: Posterior distributions comparing the effect of treating the projected separation as uncertain versus exact, both using the geometric de-projection for $r_{\rm true}$ (no independent semi-major axis). Left: posterior distributions of $\gamma$. The orange solid curve shows the geometric de-projection model with $r_\perp$ entering the likelihood with Gaussian uncertainty ($\gamma = 1.56$). The green solid curve shows the same model but with $r_\perp$ treated as exact ($\gamma = 1.59$). The vertical red dashed line marks the Newtonian prediction ($\gamma = 1$) and the vertical green line shows the result from Chae2026. Right: the same posteriors expressed in terms of $\Gamma \equiv \log_{10}\sqrt{\gamma}$. The horizontal dotted line shows the flat prior $\Gamma \sim \mathcal{U}(-1,1)$. The two curves are nearly identical, demonstrating that the $r_\perp$ treatment has a negligible effect ($\Delta\gamma \approx 0.03$) compared to the semi-major axis parameterization ($\Delta\gamma \approx 0.44$).