The fundamental plane of elliptical galaxies with modified Newtonian dynamics

The modified Newtonian dynamics (MOND), suggested by Milgrom as an alternative to dark matter, implies that isothermal spheres with a fixed anisotropy parameter should exhibit a near-perfect relation between the mass and velocity dispersion of the form M proportional to sigma(4). This is consistent with the observed Faber-Jackson relation for elliptical galaxies: a luminosity-velocity dispersion relation with large scatter. However, the observable global properties of elliptical galaxies comprise a three-parameter family; they lie on a 'fundamental plane' in a logarithmic space consisting of central velocity dispersion, effective radius (r(e)) and luminosity. The scatter perpendicular to this plane is significantly less than that about the Faber-Jackson relation. I show here that, in order to match the observed properties of elliptical galaxies with MOND, models must deviate from being strictly isothermal and isotropic; such objects can be approximated by high-order polytropic spheres with a radial orbit anisotropy in the outer regions. MOND imposes boundary conditions on the inner Newtonian regions which restrict these models to a dynamical fundamental plane of the form M proportional to sigma (alpha)r(e)(gamma) where the exponents may differ from the Newtonian expectations (alpha = 2, gamma = 1). Scatter about this plane is relatively insensitive to the necessary deviations from homology.