Deep-MOND polytropes. (arXiv:2012.11412v3 [astro-ph.GA] UPDATED)
<a href="http://arxiv.org/find/astro-ph/1/au:+Milgrom_M/0/1/0/all/0/1">Mordehai Milgrom</a>
Working within the deep-MOND limit (DML), I describe spherical,
self-gravitating systems governed by a polytropic equation of state,
$P=mathcal{K}rho^gamma$. As self-consistent structures, such 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 $zeta(y)$ is (for $gammanot=1$) $[int_0^y zeta
bar y^2 dbar y]^{1/2}=-yd(zeta^{gamma-1})/dy$. Or,
$theta^n(y)=y^{-2}[(ytheta’)^2]’$, where $theta=zeta^{gamma-1}$, and
$nequiv (gamma-1)^{-1}$. I discuss 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
$n<infty$ ($gamma>1$) all have a finite radius. (Lane-Emden spheres have a
finite mass only for $nle 5$.) I use the DML polytropes to study DML scaling
relations. For example, they satisfy a universal relation (for all
$mathcal{K}$ and $gamma$) between the total mass, $M$, and the mass-average
velocity dispersion $sigma$: $MGa_0=(9/4)sigma^4$. However, the relation
between $M$ and other measures of the velocity dispersion, such as the central,
projected one, $barsigma$, does depend on $n$ (but not $mathcal{K}$),
defining a `fundamental surface’ in the $[M,~barsigma,~n]$ space. I also
describe the generalization to anisotropic polytropes, which also all have a
finite radius (for $gamma>1$), and all satisfy the above universal $M-sigma$
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.
Working within the deep-MOND limit (DML), I describe spherical,
self-gravitating systems governed by a polytropic equation of state,
$P=mathcal{K}rho^gamma$. As self-consistent structures, such 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 $zeta(y)$ is (for $gammanot=1$) $[int_0^y zeta
bar y^2 dbar y]^{1/2}=-yd(zeta^{gamma-1})/dy$. Or,
$theta^n(y)=y^{-2}[(ytheta’)^2]’$, where $theta=zeta^{gamma-1}$, and
$nequiv (gamma-1)^{-1}$. I discuss 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
$n<infty$ ($gamma>1$) all have a finite radius. (Lane-Emden spheres have a
finite mass only for $nle 5$.) I use the DML polytropes to study DML scaling
relations. For example, they satisfy a universal relation (for all
$mathcal{K}$ and $gamma$) between the total mass, $M$, and the mass-average
velocity dispersion $sigma$: $MGa_0=(9/4)sigma^4$. However, the relation
between $M$ and other measures of the velocity dispersion, such as the central,
projected one, $barsigma$, does depend on $n$ (but not $mathcal{K}$),
defining a `fundamental surface’ in the $[M,~barsigma,~n]$ space. I also
describe the generalization to anisotropic polytropes, which also all have a
finite radius (for $gamma>1$), and all satisfy the above universal $M-sigma$
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.
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