Numerical Solutions of the External Field Effect on the Radial Acceleration in Disk Galaxies

Abstract

In MOND(modified Newtonian dynamics)-based theories the strong equivalence principle (SEP) is generically broken in an idiosyncratic manner, manifested in the action of an "external field effect (EFE)". The internal dynamics in a self-gravitating system is affected even by a constant external field in which the system is freely falling. In disk galaxies the EFE due to the cosmic large-scale structure can induce warps and modify the rotational speeds. Due to the non-linearity of MOND, it is difficult to derive analytic expressions of this important effect, especially since a disk in an inclined external field defines a three-dimensional geometry. Here we study the EFE numerically using a Miyamoto-Nagai model of disk galaxies, in two non-relativistic Lagrangian theories of MOND: the 'Aquadratic-Lagrangian' theory (AQUAL) and 'Quasilinear MOND' (QUMOND). We investigate particularly to what degree an external field modifies the quasi-flat part of rotation curves in disk systems. While our QUMOND results agree well with published numerical results in QUMOND, we find that AQUAL predicts weaker EFE than published AQUAL results. However, AQUAL still predicts stronger EFE than QUMOND, which demonstrates current theoretical uncertainties. We also illustrate how the MOND prediction on the rising part of the rotation curve, in the inner parts, depends largely on disk thickness but only weakly on a plausible external field for a fixed galaxy model. Finally, we summarize our results for the outer parts as an improved, approximate analytic expression. The results for the outer and inner parts will be useful for interpreting observed rotation curves.