High-Precision Differential Radial Velocities of C3PO Wide Binaries: A Test of Modified Newtonian Dynamics (MOND)

Wide-binary stars, separated by thousands of AU, reside in low-acceleration regimes where Modified Newtonian Dynamics (MOND) predicts deviation from Newtonian gravity. This study introduces a technique to measure differential radial velocities of wide binary stars using high resolution, high signal-to-noise spectra and applies it to 100 wide-binaries from the C3PO survey, achieving precisions of ∼8–15 m s−1 per binary pair—a ∼24× improvement over Gaia DR3—to test MOND against the canonical acceleration scale.

Measurement Technique and Precision Improvement

The paper introduces a technique to measure differential RVs between wide-binary components by employing a forward-modeling approach that accounts for the pixel-integrated nature of echelle spectroscopy. This method involves designating one star as a template and fitting a cubic spline to compute a cumulative flux function, which allows for the evaluation of model fluxes at trial velocity shifts. The resulting chi-squared statistic is calculated over a velocity grid, and the best-fit RV shift minimizes this statistic. This procedure yielded one differential RV measurement per echelle order. By combining across ∼30—50 valid echelle orders, the final differential RV precisions reached ∼8—15 m s−1 per binary pair, representing a median ∼ 24× improvement (median ∼ 24×) over Gaia DR3.

MOND Formulation and External Field Effects

The MOND formulation is described using a generalized interpolating function of the form, µ(x) = x / (1 + x b)(1/b), where x = a/a0 is the acceleration in units of a0, and b is the sharpness parameter. The study tests two commonly used forms:

1. The simple interpolating function: b = 1, µ(x) = x / (1 + x).

2. The standard form: b = 2, µ(x) = x / √ (1 + x 2).

The analysis also incorporates the external gravitational field from the galaxy. Because MOND acceleration depends on the total acceleration, a modified effective acceleration equation is used: aeff = µ squared tot a squared int + (µ tot − µ ext) squared a squared ext + 2 µ tot(µ tot − µ ext)a int aext cos θ. The external field is set to aext = 2.1× 10−10 m s−2 for the solar neighborhood, and the angle θ between internal and external acceleration vectors is computed based on the relative position of the stars.

Hierarchical Bayesian Modeling

A hierarchical Bayesian model was constructed to infer both the MOND parameter a0 and all orbital elements for 100 binary systems simultaneously. The prior distribution for log10 a0 was chosen as a uniform prior spanning from −12 to −8, specifically chosen so that the canonical MOND value of log10 a0 ≈ −9.92 sits in the mid-range of the uniform prior. The model parameters include six Keplerian orbital elements (semi-major axis 'a', eccentricity 'e', inclination 'i', longitude of ascending node 'omega', argument of periastron 'ω', and mean anomaly 'M') for each pair, as well as stellar masses (M1 and M2). The likelihood functions combine constraints from observed projected separations, differential radial velocity, and differential proper motion.

Inference Results and Tension with MOND

The inference was performed using Hamiltonian Monte Carlo (HMC) with the No-U-Turn Sampler (NUTS). The results for the two interpolating functions were compared:

1. For b = 1, the posterior on log10 a0 has a median of −11.32, and the canonical MOND value log10 a0 = −9.92 lies at the 99.8th percentile, excluded at 3.1σ.

2. For b = 2, the posterior has a median of −10.97 and the canonical value lies at the 94.5th percentile, excluded at 1.9σ.

These results indicate tension with MOND at the presently accepted a0 value, with the degree of tension depending on the assumed interpolating function choice, suggesting that constraints on a0 from different astrophysical systems may not be directly comparable without accounting for this degeneracy. The analysis also verified robustness to prior choices, showing consistent exclusion levels across different prior ranges.

Based on a rigorous analysis of this scientific paper, here are the specific improvements that can be implemented in AI systems and what those improved systems will be capable of:

) 1. Precision Kinematic Modeling for Low-Acceleration Regimes (MOND Testing)

The core technical improvement lies in moving beyond 2D kinematic constraints to incorporate high-precision, differential radial velocities (RVs). The AI system would be upgraded from a simple data fitter to a full Hierarchical Bayesian Inference engine capable of handling complex orbital dynamics under non-Newtonian gravity.

Specific improvements:

1. Implement the novel pixel-integrated spectral fitting technique for echelle spectroscopy, which accounts for instrumental line-spread functions and order-level jitter terms, to achieve differential RV precisions of 8–15 m s−1.

2. Integrate a full three-dimensional orbital model that explicitly incorporates the external gravitational field effect (Equation 5), where the total acceleration depends on both internal and external components, accounting for the varying angle between these vectors.

3. Develop a robust Hamiltonian Monte Carlo (HMC) framework using NUTS sampling to simultaneously infer:

4. The global MOND acceleration scale parameter, a0 (as a free parameter rather than assuming its canonical value), across all systems.

5. The six Keplerian orbital elements for every wide binary pair, including the mass ratio and semi-major axis.

Improved AI System Capability:

This system will be capable of performing gravity testing on astrophysical systems with unprecedented precision. It can rigorously test Modified Newtonian Dynamics (MOND) against Newtonian gravity by determining if a single universal acceleration scale (a0) can consistently explain the observed kinematics of wide binaries, accounting for the complex interplay between internal orbital motion and the external galactic field. It will be able to identify tension not just in aggregate statistics, but at the level of individual systems or specific interpolating functions (b=1 vs b=2).

) 2. Uncertainty Propagation and Degeneracy Resolution

The paper emphasizes constructing a hierarchical Bayesian model specifically to propagate uncertainties from orbital degeneracies into the final constraints on a0.

Specific improvements:

1. Develop automated uncertainty propagation routines that quantify how uncertainties in orbital elements (like semi-major axis 'a' or eccentricity 'e') translate into uncertainty in the inferred global parameter 'a0'.

2. Implement rigorous prior sensitivity analysis, testing the robustness of the results against alternative prior choices for log10 a0 (e.g., U(-13, -7) vs U(-12, -8)) and interpolating functions (b=1 vs b=2).

3. Integrate outlier rejection mechanisms based on the ratio of measurement uncertainties (σGaia/σmeas), allowing the system to selectively weight high-precision data while maintaining statistical validity.

Improved AI System Capability:

This system will move beyond simply reporting a single value for 'a0'. It will provide a full probabilistic map of 'a0', quantifying the cost (in terms of tension) associated with different assumptions about the underlying MOND interpolation function or prior distributions. It can definitively state whether the observed discrepancy is due to systematic uncertainty, sample limitations, or an actual breakdown of MOND at the canonical scale.

) 3. Multi-Parameter Joint Constraint Optimization

The AI system must manage a high-dimensional parameter space efficiently (orbital elements for 100 systems + global parameters).

Specific improvements:

1. Leverage advanced sampling techniques like HMC/NUTS to effectively navigate the high-dimensional posterior landscape, ensuring thorough exploration of complex degeneracies between orbital parameters (e.g., 'a' and 'e') and the external field angle 'θ'.

2. Develop a streamlined data pipeline capable of efficiently integrating disparate datasets: high-resolution differential RVs, Gaia astrometry (proper motions), and projected separations into a single likelihood function.

Improved AI System Capability:

The system will act as a sophisticated astrophysical constraint optimizer. It can simultaneously solve for the entire kinematic state of 100 wide binaries while determining the single global parameter 'a0'. This capability allows for the identification of outlier systems—those whose parameters are highly improbable under a given gravity model—which is crucial for refining physical theories.

) 4. Automated Model Comparison and Interpretation Module

The system needs to interpret why one interpolating function (b=1 vs b=2) yields different constraints on 'a0'.

Specific improvements:

1. Implement a module that automatically compares the posterior results for different MOND formulations, identifying which form provides tighter constraints and quantifying the resulting tension difference (e.g., 3.1σ vs 1.9σ exclusion).

2. Develop a qualitative reasoning engine to suggest physical interpretations of the tension: whether it points toward an environmental dependency of a0, a need for more complex MOND formulations in external fields, or potential unresolved tertiary companions (as discussed in Section 5.2).

Improved AI System Capability:

This system will serve as an expert consultant. It won't just give numbers; it will explain the physics behind the tension. For example, if the results favor b=2 but still show tension, the system can suggest investigating relativistic extensions like TeVeS or testing for spatial variations in a0 across different regions of the solar neighborhood.

Wide-binary stars, separated by thousands of AU, reside in low-acceleration regimes where Modified Newtonian Dynamics (MOND) predicts deviation from Newtonian gravity. However, Gaia radial velocities (RVs) lack the precision to resolve the small velocity differences expected in these systems, limiting previous MOND analyses to two-dimensional kinematics. In this paper, we introduce a technique to measure differential RVs of wide binary stars using high resolution, high signal-to-noise spectra. We apply this method to measure differential RVs of 85 wide-binaries from the C3PO survey and achieved precisions of about 8-15 m/s per binary pair, a about 10 - 100 times improvement (median about 24 times) over Gaia DR3. After applying a selection criterion based on the scaled velocity, we retain 57 gravitationally bound systems for further analysis. Combining these measurements with Gaia astrometry, we construct a hierarchical Bayesian model to infer the orbital elements of all wide-binary pairs and the global MOND acceleration scale (a 0). We test two commonly used interpolating functions in MOND formulation: the simple form (b=1, μ= x/(1+x)) and the standard form (b=2, μ= x/sqrt 1+x squared). Our results indicate tension with MOND at the presently accepted a 0 value: for b=1, the canonical value is excluded at 2.5σ, while for b=2, the exclusion is at 1.5σ. We discuss how systematic uncertainties in the external field effect treatment, particularly in the transition regime, may affect these conclusions.