Deep-MOND polytropes

Abstract

Working within the very-low-acceleration limit of modified Newtonian dynamics (MOND)---the deep-MOND limit (DML)---I describe spherical, self-gravitating systems governed by a polytropic equation of state, $P=\\mathcal{K}{\\ensuremath{\\rho}}^{\\ensuremath{\\gamma}}$ ($P$ is the pressure, and $\\ensuremath{\\rho}$ the density). As self-consistent structures, such idealized systems can serve as heuristic models for DML, astronomical systems, such as dwarf spheroidal galaxies, low-surface-density elliptical galaxies and star clusters, and diffuse galaxy groups. They can also serve as testing ground for various theoretical MOND inferences. In dimensionless form, the equation satisfied by the radial density profile $\\ensuremath{\\zeta}(y)$ is (for $\\ensuremath{\\gamma}\\ensuremath{\\ne}1$) $[{\\ensuremath{\\int}}_{0}^{y}\\ensuremath{\\zeta}{\\overline{y}}^{2}d\\overline{y}{]}^{1/2}=\\ensuremath{-}yd({\\ensuremath{\\zeta}}^{\\ensuremath{\\gamma}\\ensuremath{-}1})/dy$. Or, ${\\ensuremath{\\theta}}^{n}(y)={y}^{\\ensuremath{-}2}[(y{\\ensuremath{\\theta}}^{\\ensuremath{ }}{)}^{2}{]}^{\\ensuremath{ }}$, where $\\ensuremath{\\theta}={\\ensuremath{\\zeta}}^{\\ensuremath{\\gamma}\\ensuremath{-}1}$, and $n\\ensuremath{\\equiv}(\\ensuremath{\\gamma}\\ensuremath{-}1{)}^{\\ensuremath{-}1}$. I discuss general properties of the solutions, contrasting them with those of their Newtonian analogues---the Lane--Emden polytropes. Due to the stronger MOND gravity, all DML polytropes have a finite mass, and for $nl\\ensuremath{\\infty}$ ($\\ensuremath{\\gamma}g1$) all have a finite radius (Lane--Emden spheres have a finite mass only for $n\\ensuremath{\\le}5$). I use the DML polytropes to study DML scaling relations. For example, they satisfy a universal relation (for all $\\mathcal{K}$ and $\\ensuremath{\\gamma}$) between the total mass, $M$, and the mass-average velocity dispersion $\\ensuremath{\\sigma}$: $MG{a}_{0}=(9/4){\\ensuremath{\\sigma}}^{4}$. However, the relation between $M$ and other measures of the velocity dispersion, such as the central, projected one, $\\overline{\\ensuremath{\\sigma}}$, does depend on $n$ (but not $\\mathcal{K}$), defining a ``fundamental surface in the $[M,\\overline{\\ensuremath{\\sigma}},n]$ space. I also describe the generalization to anisotropic polytropes, which also all have a finite radius (for $\\ensuremath{\\gamma}g1$) and which all satisfy the above universal $M\\ensuremath{-}\\ensuremath{\\sigma}$ relation. This more extended class of models exhibits the central-surface-densities relation: a tight relation between the baryonic and the dynamical central-surface-densities predicted by MOND.