Exact solutions and approximations of MOND fields of disc galaxies

Abstract

We consider models of thin discs (with and without bulges) in the Bekenstein-Milgrom formulation of MOND as a modification of Newtonian gravity. Analytic solutions are found for the full gravitational fields of Kuzmin discs, and of disc-plus-bulge generalizations of them. For all these models a simple relation between the MOND potential field,ψ, and the Newtonian potential, ψN, holds everywhere outside the disc:|$\mu(|\nabla\psi|/a_\text o)\nabla\psi=\nabla \varphi_\text N$|⁠. We give exact expressions for the rotation curves for these models. We also find that this algebraic relation is a very good approximation for exponential discs. The algebraic relation outside the disc is then extended into the disc to derive an improved approximation for the MOND rotation curve of disc galaxies, requiring only knowledge of the Newtonian curve and the surface density.