⚛️ general relativity

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

This paper presents a general Bayesian algorithm for inferring the gravitational constant from wide binary orbits using 3D modeling and applies it in a pilot study of 32 Gaia systems with HARPS radial velocities, finding tentative evidence for a deviation from Newtonian gravity that is heavily influenced by a single outlier system and requires further verification.

Original authors: Kyu-Hyun Chae

Published 2026-02-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kyu-Hyun Chae

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Testing Gravity in the "Slow Lane"

Imagine gravity as a rulebook for how objects move. For over 300 years, we've trusted Isaac Newton's rulebook. It says that if you know where two stars are and how fast they are moving, you can predict exactly how they will dance around each other.

But there's a mystery in the universe. In the slow, quiet outskirts of galaxies, stars seem to move faster than Newton's rulebook predicts. Some scientists think this means Newton's rules break down when things move very slowly (low acceleration). They propose a new rulebook called MOND (Modified Newtonian Dynamics), which suggests gravity gets a little "boost" in these slow zones.

This paper is a pilot study to see if we can finally catch Newton and MOND in a fight using Wide Binary Stars.

The Analogy: The Slow-Motion Dance

Think of a wide binary star system as two dancers holding hands, spinning very slowly in a vast, empty ballroom.

Newton's Prediction: If you know their positions and speeds, Newton says, "They should be dancing in a perfect, predictable ellipse."
The Problem: We usually only see them from the side (like watching a movie on a flat screen). We can see how far apart they are left-to-right, but we can't see how far apart they are towards or away from us (depth). It's like trying to figure out a 3D dance routine from a 2D shadow.

Because of this "flat screen" problem, we can't be sure if the dancers are following Newton's rules or a different set of rules.

The New Tool: The "3D Glasses" Algorithm

The author, Kyu-Hyun Chae, has built a new Bayesian algorithm. Think of this as a pair of high-tech 3D glasses combined with a super-smart detective.

The Data: The study uses data from the Gaia satellite (which gives us the 2D "shadow" positions) and HARPS (a super-precise telescope that measures how fast the stars are moving toward or away from us).
The Method: Instead of guessing one single answer, the algorithm runs millions of simulations. It asks: "If gravity works like Newton says, what does the dance look like? If gravity works like MOND says, what does it look like?"
The Goal: It tries to find the "best fit" for the gravitational constant ( $G$ ). If the data fits Newton perfectly, the answer is zero. If gravity is boosted (like MOND suggests), the answer will be positive.

The Pilot Study: A Small Sample of 32 Couples

The author tested this new "3D glasses" on 32 pairs of stars that have very precise speed measurements.

The Results:

The Fast Dancers (High Acceleration):
For the 24 pairs of stars that are moving relatively "fast" (or are closer together), the results were perfect. They followed Newton's rules exactly. No surprises here. Newton wins in the fast lane.
The Slow Dancers (Low Acceleration):
For the 8 pairs of stars that are very far apart and moving very slowly, things got interesting.
- The Signal: The data suggested that gravity was slightly stronger than Newton predicted. It looked like a "boost," which supports the MOND idea.
- The "Star" of the Show: However, the study found that one single pair of stars (HD189739 and HD189760) was doing all the heavy lifting. This specific pair was moving so fast that, under Newton's rules, they should have flown apart and never been a pair at all. But under the "boosted gravity" rules, they fit perfectly.

The Twist: Is the Star a Fluke?

When the author removed that one "rebellious" star pair from the calculation, the evidence for the "gravity boost" dropped significantly. The remaining stars were still a little weird, but not enough to definitively say Newton is wrong.

So, what's the verdict?

Scenario A: That one rebellious star pair is real, and it proves that Newton's rules break down in the slow lane. This would be a massive discovery for physics.
Scenario B: That one rebellious star pair is actually a "fake couple." Maybe they aren't dancing together at all; maybe they just happened to pass each other by by pure chance (a "chance association"), or maybe one of them has a hidden third partner (a secret third star) messing up the speed measurements.

The Conclusion: Keep Dancing, Keep Watching

The paper concludes that while the new algorithm is a powerful tool, we need more data. We need to find more of these "slow dancers" to see if the rebellious behavior is a common trend or just a fluke.

In simple terms:
The author built a better calculator to test gravity. When they ran it on a small group of stars, it hinted that Newton might be wrong in the slow zones. But the hint was mostly driven by one very strange star pair. Until we find more stars like that, we can't be 100% sure if the universe is rewriting its rulebook or if we just found one weird outlier.

The Takeaway: The tool works, the method is sound, but we need a bigger sample size to settle the debate between Newton and the new theories.

1. Problem Statement

The paper addresses the challenge of testing gravitational theories in the low-acceleration regime ( $\lesssim 10^{-9} \, \text{m s}^{-2}$ ), where Modified Newtonian Dynamics (MOND) predicts a deviation from standard Newtonian gravity. While wide binary stars are ideal laboratories for this test because they are isolated from galactic dark matter effects, previous studies have been limited by:

Data Limitations: Precise 3D relative positions ( $\mathbf{r}$ ) and velocities ( $\mathbf{v}$ ) are difficult to obtain. Specifically, the line-of-sight (radial) separation is often measured with uncertainties larger than the separation itself, and radial velocities (RVs) have historically been imprecise.
Degeneracies: There is a fundamental degeneracy between orbital geometric parameters (eccentricity, inclination, phase) and the gravitational parameter ( $G$ ). Different gravity theories can fit the same instantaneous displacement and velocity data with different orbital solutions.
Methodological Gaps: Previous Bayesian approaches (e.g., Chae 2025a) used simplified models that suppressed certain orientation parameters due to insufficient data constraints, potentially introducing bias.

2. Methodology

The author presents a fully general Bayesian algorithm designed to model wide binary orbits using six measured quantities (3D position and 3D velocity) to infer the gravitational parameter.

General 3D Model: Unlike previous "simple" models, this algorithm treats all six components of $\mathbf{r}$ and $\mathbf{v}$ as observables. It utilizes four free parameters: eccentricity ( $e$ ), argument of periastron ( $\phi_0$ ), orbital phase ( $\phi$ ), and inclination ( $i$ ). The semi-major axis ( $a$ ) and rotation angle ( $\theta$ ) are derived mathematically from these and the observed sky-projected separation.
Bayesian Framework:
- Likelihood: Defined by the $\chi^2$ difference between observed quantities (sky separation, radial separation, proper motions, and RVs) and model predictions.
- Priors: Logical priors are imposed on inclination ( $i$ ), periastron ( $\phi_0$ ), and true anomaly ( $\Delta\phi$ ) based on the randomness of orbital orientations and Keplerian time-averaging. A log-normal prior is applied to the mass factor ( $f_M$ ).
- Gravitational Parameter: The core parameter of interest is $\Gamma \equiv \log_{10}\sqrt{G/G_N}$ . The algorithm allows $\Gamma$ to vary freely for each binary to test for deviations from Newton's constant ( $G_N$ ).
Implementation: The posterior distributions are derived using Markov Chain Monte Carlo (MCMC) via the emcee Python package. The study utilizes a sample of 32 wide binaries from the Saglia et al. (2025) dataset, which features extremely precise HARPS radial velocities.

3. Key Contributions

Algorithm Development: Introduction of a rigorous, general Bayesian algorithm capable of handling full 3D kinematic data without suppressing orientation parameters, providing a more physically complete framework than previous iterations.
Pilot Study Application: The first application of this general algorithm to a high-precision sample (HARPS RVs), demonstrating its capability to constrain orbital parameters and test gravity even when radial separation uncertainties are high.
Statistical Rigor: Implementation of robust statistical tests, including Bayesian Information Criterion (BIC) comparisons and Anderson-Darling (A-D) tests, to evaluate the consistency of orbital phase distributions with theoretical expectations under different gravity models.

4. Key Results

A. Statistical Testing of Gravity Models

Orbital Phase ( $\Delta\phi$ ) Distribution:
- Under Newtonian gravity ( $\Gamma=0$ ), the distribution of the true anomaly ( $\Delta\phi$ ) shows a significant discrepancy with the expected distribution derived from Keplerian dynamics (BIC difference $\Delta\text{BIC} \approx 65.5$ ; A-D test $p \approx 1.9 \times 10^{-3}$ ).
- Under Generalized Gravity (where $\Gamma$ is free) or a MOND-type toy model (boosted gravity for low acceleration), the $\Delta\phi$ distribution aligns much better with expectations.
- This tension is most pronounced in the low-acceleration subsample ( $g_N < 10^{-9} \, \text{m s}^{-2}$ ).

B. Inference of the Gravitational Anomaly ( $\Gamma$ )

The study consolidates individual posterior PDFs of $\Gamma$ for two subsamples:

High Acceleration ( $g_N > 10^{-9} \, \text{m s}^{-2}$ ):
- Result: $\Gamma = -0.002^{+0.012}_{-0.018}$ .
- Conclusion: Consistent with Newtonian gravity ( $\Gamma=0$ ), validating standard dynamics down to $\sim 10^{-9} \, \text{m s}^{-2}$ .
Low Acceleration ( $g_N < 10^{-9} \, \text{m s}^{-2}$ ):
- Result (8 binaries): $\Gamma = 0.134^{+0.056}_{-0.036}$ .
- Significance: Represents a $\gtrsim 3.5\sigma$ discrepancy with Newton, suggesting a boost in gravity consistent with MOND predictions.
- Result (6 binaries with $g_N < 10^{-9.5}$ ): $\Gamma = 0.143^{+0.068}_{-0.041}$ .

C. The Dominance of Binary #24 (HD189739/HD189760)

A critical finding is that the statistical significance of the gravity boost in the low-acceleration regime is driven almost entirely by a single system, Binary #24.

Newtonian Tension: In Newtonian gravity, this system appears unbound ( $v_{obs} > v_{esc}$ ) and requires an unphysically large semi-major axis ( $a \approx 51$ kau vs. projected separation $s \approx 5.8$ kau) and a mass increase of $\sim 8\%$ .
MOND Consistency: Under MOND-type gravity, the system is bound ( $v_{obs} < v_{esc}$ ) with a natural orbital configuration.
Exclusion Test: When Binary #24 is excluded, the significance of the anomaly drops drastically ( $\Gamma = 0.063^{+0.065}_{-0.041}$ ), becoming consistent with Newton within $1.5\sigma$ and only weakly favoring MOND.

5. Significance and Outlook

Validation of Method: The pilot study proves that the general Bayesian 3D modeling algorithm is a powerful tool for testing gravity, capable of extracting meaningful constraints even from data with imperfect radial separations.
The "Binary #24" Anomaly: The results highlight the critical importance of individual systems in small samples. While Binary #24 strongly supports modified gravity, its nature must be verified. The author rules out chance association (probability $\lesssim 10^{-4}$ ) but notes the possibility of kinematic contamination by a hidden tertiary companion, which would require further high-resolution imaging or long-baseline RV monitoring to rule out.
Future Prospects: The paper argues that while current samples are small, the accumulation of high-precision 3D velocity data (from HARPS, ESPRESSO, etc.) will allow for decisive tests of gravity. Future large samples will reduce the statistical weight of individual outliers and provide a definitive distinction between Newtonian and non-standard gravity theories in the low-acceleration regime.

In summary, this work provides a robust mathematical framework for 3D wide binary analysis and presents tentative evidence for a deviation from Newtonian gravity at low accelerations, heavily influenced by a single, anomalous system that requires further observational scrutiny.