A new line on the wide binary test of gravity

The relative velocity distribution of wide binary (WB) stars is sensitive to the law of gravity at the low accelerations typical of galactic outskirts. I consider the feasibility of this wide binary test using the `line velocity' method. This involves considering only the velocity components along the direction within the sky plane orthogonal to the systemic proper motion of each WB. I apply this technique to the WB sample of Hernandez et. al. (2019), carefully accounting for large-angle effects at one order beyond leading. Based on Monte Carlo trials, the uncertainty in the one-dimensional velocity dispersion is $\approx 100$ m/s when using sky-projected relative velocities. Using line velocities reduces this to $\approx 30$ m/s because these are much less affected by distance uncertainties. My analysis does not support the Hernandez et. al. (2019) claim of a clear departure from Newtonian dynamics beyond a radius of $\approx 10$ kAU, partly because I use $2\sigma$ outlier rejection to clean their sample first. Nonetheless, the uncertainties are small enough that existing WB data are nearly sufficient to distinguish Newtonian dynamics from Modified Newtonian Dynamics. I estimate that $\approx 1000$ WB systems will be required for this purpose if using only line velocities. In addition to a larger sample, it will also be important to control for systematics like undetected companions and moving groups. This could be done statistically. The contamination can be minimized by considering a narrow theoretically motivated range of parameters and focusing on how different theories predict different proportions of WBs in this region.