Measurements of the Low-Acceleration Gravitational Anomaly from the Normalized Velocity Profile of Gaia Wide Binary Stars and Statistical Testing of Newtonian and Milgromian Theories

Low-acceleration gravitational anomaly is investigated with a new method of exploiting the normalized velocity profile $\tilde{v}\equiv v_p/v_c$ of wide binary stars as a function of the normalized sky-projected radius $s/r_{\rm{M}}$ where $v_p$ is the sky-projected relative velocity between the pair, $v_c$ is the Newtonian circular velocity at the sky-projected separation $s$, and $r_{\rm{M}}$ is the MOND radius. With a Monte Carlo method Gaia observed binaries and their virtual Newtonian counterparts are probabilistically distributed on the $s/r_{\rm{M}}$ versus $\tilde{v}$ plane and a logarithmic velocity ratio parameter $\Gamma$ is measured in the bins of $s/r_{\rm{M}}$. With three samples of binaries covering a broad range in size, data quality, and implied fraction of hierarchical systems including a new sample of 6389 binaries selected with accurate distances and radial velocities, I find a unanimous systematic variation from the Newtonian flat line. With $\Gamma=0$ at $s/r_{\rm{M}}\lesssim 0.15$ or $s\lesssim 1$~kilo astronomical units (kau), I get $\Gamma=0.068\pm 0.015$ (stat) $_{-0.015}^{+0.024}$ (syst) for $s/r_{\rm{M}} \gtrsim 0.7$ or $s\gtrsim 5$~kau. The gravitational anomaly (i.e.\ acceleration boost) factor given by $\gamma_g = 10^{2\Gamma}$ is measured to be $\gamma_g = 1.37_{-0.09}^{+0.10}$ (stat) $_{-0.09}^{+0.16}$ (syst). With a reduced $\chi^2$ test of Newtonian and Milgromian nonrelativistic theories, I find that Newtonian gravity is ruled out at $5.8\sigma$ ($\chi^2_\nu=9.4$) by the new sample (and $9.2\sigma$ by the largest sample used). The Milgromian AQUAL theory is acceptable with $0.5\lesssim \chi^2_\nu\lesssim 3.1$. These results agree well with earlier results with the "acceleration-plane analysis" for a variety of samples and the "stacked velocity profile analysis" for a pure binary sample.