Year: 2026
Type: erratum
Abstract: This is an erratum to the paper ‘Strong constraints on the gravitational law from Gaia DR3 wide binaries’, published in MNRAS, 527 (3), 4573–4615 (2024), hereafter B24. Its fig. 15 presents an error in the |$\widetilde{\upsilon}$| calculations whereby the sky-projected separation |$r_{\rm sky}$| was assumed to equal the orbital semi-major axis a. The incorrect projection affects only the lines plotted on this figure, leaving its shaded grey bands and the rest of the original publication unaffected. However, part of the discussion involved comparing the results to the median |$\widetilde{\upsilon}$| expected in Keplerian dynamics at different |$\gamma$|, which parametrizes the orbital eccentricity distribution. We therefore briefly revisit how the corrected results in Fig. 1 impact our understanding of the rest of the results presented in B24. To clarify the discussion, we have plotted the Keplerian result alongside a version where results are scaled up by |$\sqrt{\eta }$|, where |$\eta$| is the angle-averaged boost to the inward radial gravity in Milgromian dynamics (MOND; M. Milgrom 1983) for wide binaries (WBs) in the Solar neighbourhood with asymptotically large separation. Following the original publication, we assume that |$\eta = 1.4228$| for the assumed Galactic external field. The solid red line shows the median |$\widetilde{\upsilon}$| for an ensemble of Keplerian orbits with a range of eccentricities viewed from a random viewing angle at a random orbital phase, without accounting for selection effects. This result is scaled up by |$\sqrt{\eta }$| to get the MOND prediction for local WBs with asymptotically large separation (dashed red line). The eccentricity distribution is governed by |$\gamma$| (equation 17 of B24). The vertical grey shaded band shows |$\gamma = 1.32 \pm 0.09$| based on observations of WBs with separations similar to those used for the WBT (H.-C. Hwang et al. 2022). The horizontal grey shaded band shows the median |$\widetilde{\upsilon}$| of the WB population, with the range giving the uncertainty arising from whether the |$\widetilde{\upsilon}$| distribution is truncated at 1.5, 2, or 2.5 when finding the median. The WBs are not further binned in |$r_{\rm {sky}}/r_{_M}$| as this has little impact on the median |$\widetilde{\upsilon}$| (fig. 11 of B24). The most immediate impact of fixing the projection issue is a much lower Keplerian result, as shown by the solid red line on Fig. 1. If we assume that |$\gamma \approx 2$| (as reported in table 3 of B24), the expected median |$\widetilde{\upsilon}$| now falls below the horizontal grey observed band. The deficit is presumably explained by undetected close binaries (CBs), which broaden the |$\widetilde{\upsilon}$| distribution (see their fig. 10). As argued repeatedly in B24, CBs were almost certainly present in a significant fraction of their WBs, so it is natural that CBs would somewhat increase the median |$\widetilde{\upsilon}$|. However, the higher CB-free prediction in MOND leaves almost no margin for inflation by CBs – even if we adopt a very high |$\gamma$|. In this respect, the original conclusions of B24 are not affected. The final analysis variant presented in table 3 of B24 assumed that CB contamination occurred at only half the level inferred by their nominal analysis, which is far lower than allowed by their inferred uncertainty and leads to a much poorer overall fit. With this reduced CB contamination, we might expect the B24 analysis to infer a lower |$\gamma$| from the same observed |$\widetilde{\upsilon}$| distribution as there is less scope for CBs to inflate the median |$\widetilde{\upsilon}$|. Indeed, this analysis variant does recover a much lower |$\gamma = 1.05^{+0.09}_{-0.14}$|. Such a low |$\gamma$| falls below the |$1.32 \pm 0.09$| reported by H.-C. Hwang, Y.-S. Ting & N. L. Zakamska (2022) based on the angle between the sky-projected separation and relative velocity of each WB. Since this estimate does not consider CBs, B24 argued that such an approach could underestimate |$\gamma$| because CBs randomise the angle by introducing hidden degrees of freedom, pushing the results closer to the isotropic case of |$\gamma = 1$| (the thermal eccentricity distribution). This suggests a higher |$\gamma$| and thus higher CB fraction than found with this unrealistic variant, as indeed inferred by the nominal B24 analysis. It is also helpful to compare our result in Fig. 1 to a subsequently published much cleaner sample of WBs, where CB contamination should be far less prevalent (S. A. Cookson et al. 2026, hereafter C26). Their fig. 5 shows that the median |$\widetilde{\upsilon} = 0.49$|. Although they do not state an uncertainty, it can be inferred from this figure that the uncertainty in the median |$\widetilde{\upsilon}$| is |$\lesssim 0.02$| because the highest of their 6 binned values is around 0.51 and all the rest are |$\lt 0.50$|. Assuming that the C26 sample is practically unaffected by CBs, this implies |$\gamma \gtrsim 2$| if gravity is Newtonian, as argued by those authors. This is very much in line with the nominal B24 result of |$\gamma = 2.04^{+0.27}_{-0.19}$| (see their table 3). If |$\gamma \approx 2$| as suggested by these studies, H.-C. Hwang et al. (2022) must have modestly underestimated |$\gamma$|, possibly due to the neglected impact of CBs on their WB sample.1 In the MOND case, we would need to assume very high values for |$\gamma$| in order to explain the low median |$\widetilde{\upsilon}$| reported by C26. This would require a much higher incidence of CBs to randomise the angle between WB separation and relative velocity, making it even more difficult to explain the low observed median |$\widetilde{\upsilon}$|. This complements the main argument of C26 against MOND based on the lack of any trend in the median |$\widetilde{\upsilon}$| with respect to how deep into the MOND regime each WB is, which they quantified using |$r_{\rm {sky}}/r_{_M}$|, where |$r_{_M}$| is the MOND radius of the WB. Those authors intentionally refrained from detailed modelling of the |$\widetilde{\upsilon}$| distribution, so they did not directly use its median value to draw conclusions regarding the gravity law. We stress that directly comparing the results in Fig. 1 to observed WB populations does not account for all the pertinent effects. Even if CBs do not play a role, we still need to account for selection effects. For instance, the B24 analysis was restricted to |$r_{\rm sky} \ge 2$| kAU. Since WBs with smaller a are much more common and systems spend more time near apocentre, the WBs at low |$r_{\rm sky}$| would typically have even smaller a and be observed near apocentre. Without any |$r_{\rm sky}$| floor, we would also observe these systems at pericentre, when they have much higher |$\widetilde{\upsilon}$|. However, the |$r_{\rm sky}$| floor means WBs are insufficiently separated at that phase to be included in the analysis. This causes a loss of systems with high |$\widetilde{\upsilon}$|, reducing its median value. The impact of a floor on |$r_{\rm sky}$| was carefully taken into account in the main analysis of B24, who explored a sufficiently wide range of a to capture all the WBs that might have |$r_{\rm sky} \ge 2$| kAU. Our results here suggest that although this effect should be included in a detailed analysis, it likely does not greatly impact the results given the small fraction of an eccentric orbit spent near pericentre. Indeed, Fig. 1 shows that the median |$\widetilde{\upsilon}$| declines as orbits become typically more eccentric – despite a greater eccentricity allowing a system to reach higher |$\widetilde{\upsilon}$|. The corrected results presented in this erratum thus refine our understanding of the studies mentioned above, strengthening the case that it is difficult to reconcile MOND with clean samples of observed WBs.
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,046
OpenAlex ID: W7224847440