Roche lobe shapes for testing MOND-like modified gravities

Dark Matter (DM) theories and mass-tracing-light theories like MOND are by construction nearly degenerate on galactic scales, but not when it comes to the predicted shapes of Roche Lobes of a two-body system (e.g., a globular cluster orbiting a host galaxy). We show that the flattening of the Roche lobe is sensitive to the function mu(g) in modification of the law of gravity. We generalise the analytical results obtained in the deep-MOND limit by Zhao (2005), and consider a binary in the framework of a MOND-like gravity modification function mu(g) or a general non-Keplerian gravity g proportional to R-zeta. We give analytical expressions for the inner Lagrange point and Robe lobe axis ratios. The Roche lobe volume is proven to scale linearly with the true mass ratio, which applies to any mu(g), hence mass-tracing light models would overpredict the Roche lobe of a DM-poor globular cluster in a DM-rich host galaxy, and underpredict the size of a DM-richer dwarf satellite. The lobes are squashed with the flattening similar to 0.4 in the strong gravity and similar to 0.6 in the weak gravity; a precise measurement of the flattening could be used to verify the anisotropic dilation effect which is generic to MOND-like gravity. We generalise these results for extended mass distribution, and compare predicted Roche radii with limiting radii of observed globular clusters and dwarf galaxy satellites.