The External Field Dominated Solution In QUMOND & AQUAL: Application To Tidal Streams

The standard $\Lambda$CDM paradigm seems to describe cosmology and large scale structure formation very well. However, a number of puzzling observations remain on galactic scales. An example is the anisotropic distribution of satellite galaxies in the Local Group. This has led to suggestions that a modified gravity theory might provide a better explanation than Newtonian gravity supplemented by dark matter. One of the leading modified gravity theories is Modified Newtonian Dynamics (MOND). For an isolated point mass, it boosts gravity by an acceleration-dependent factor of $\nu$. Recently, a much more computer-friendly quasi-linear formulation of MOND (QUMOND) has become available. We investigate analytically the solution for a point mass embedded in a constant external field of $\mathbf{g}_{ext}$. We find that the potential is $\Phi = - ~ \frac{GM \nu_{ext}}{r}\left(1 + \frac{K_0}{2} \sin^2 \theta \right)$, where $r$ is distance from the mass $M$ which is in an external field that `saturates' the $\nu$ function at the value $\nu_{ext}$, leading to a fixed value of $K_0 \equiv \frac{\partial Ln ~ \nu}{\partial Ln ~ g_{ext}}$. In a very weak gravitational field $\left(\left| \mathbf{g}_{ext} \right| \ll a_0 \right)$, $K_0 = -\frac{1}{2}$. The angle $\theta$ is that between the external field direction and the direction towards the mass. Our results are quite close to the more traditional aquadratic Lagrangian (AQUAL) formulation of MOND. We apply both theories to a simple model of the Sagittarius tidal stream. We find that they give very similar results, with the tidal stream seeming to spread slightly further in AQUAL.