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 \propto \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 \propto { \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 .