Using a recent homogeneous sample of 40 high quality velocity dispersion profiles for Galactic globular clusters, we study the low gravitational acceleration regime relevant to the outskirts of these systems. We find that a simple empirical profile having a central Gaussian component and a constant large radius asymptote, $sigma_{infty}$, accurately describes the variety of observed velocity dispersion profiles. We use published population synthesis models, carefully tailored to each individual cluster, to estimate mass to light ratios from which total stellar masses, $M$, are inferred. We obtain a clear scaling, reminiscent of the galactic Tully-Fisher relation of $sigma_{infty}( km s^{-1})= 0.084^{+0.075}_{-0.040} (M/M_{odot})^{0.3 pm 0.051} $, which is interesting to compare to the deep MOND limit of $sigma_{infty} (km s^{-1})=0.2(M/M_{odot})^{0.25}$. Under a Newtonian interpretation, our results constitute a further restriction on models where initial conditions are crafted to yield the outer flattening observed today. Within a modified gravity scheme, as the globular clusters studied are not isolated objects in the deep MOND regime, the results obtained point towards a modified gravity where the external field effect of MOND does not appear, or is much suppressed.
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