Modified Newtonian Dynamics - The Mathematics of MOND

The Mathematics of MOND

In non-relativistic Modified Newtonian Dynamics, Poisson's equation,

(where is the gravitational potential and is the density distribution) is modified as

where is the MOND potential. The equation is to be solved with boundary condition for . The exact form of is not constrained by observations, but must have the behaviour for (Newtonian regime), for (Deep-MOND regime). In the deep-MOND regime, the modified Poisson equation may be rewritten as

\nabla \cdot \left = 0

and that simplifies to

\frac{\left\| \nabla\Phi \right\|}{a_0} \nabla\Phi - \nabla\Phi_N = \nabla \times \mathbf{h}.

The vector field is unknown, but is null whenever the density distribution is spherical, cylindrical or planar. In that case, MOND acceleration field is given by the simple formula

\mathbf{g}_M = \mathbf{g}_N \sqrt{\frac{a_0}{\left\| \mathbf{g}_N \right \|}}

where is the normal Newtonian field.

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