Publication date: Nov 2016
Abstract:
Spherically symmetric equilibrium configurations of perfect fluid
obeying a polytropic equation of state are studied in spacetimes with a
repulsive cosmological constant. The configurations are specified in
terms of three parameters—the polytropic index $n$, the ratio of
central pressure and central energy density of matter $sigma$, and the
ratio of energy density of vacuum and central density of matter
$lambda$. The static equilibrium configurations are determined by two
coupled first-order nonlinear differential equations that are solved by
numerical methods with the exception of polytropes with $n=0$
corresponding to the configurations with a uniform distribution of
energy density, when the solution is given in terms of elementary
functions. The geometry of the polytropes is conveniently represented by
embedding diagrams of both the ordinary space geometry and the optical
reference geometry reflecting some dynamical properties of the geodesic
motion. The polytropes are represented by radial profiles of energy
density, pressure, mass, and metric coefficients. For all tested values
of $n>0$, the static equilibrium configurations with fixed parameters
$n$, $sigma$, are allowed only up to a critical value of the
cosmological parameter
$lambda_{mathrm{c}}=lambda_{mathrm{c}}(n,sigma)$. In the case of
$n>3$, the critical value $lambda_{mathrm{c}}$ tends to zero for
special values of $sigma$. The gravitational potential energy and the
binding energy of the polytropes are determined and studied by numerical
methods. We discuss in detail the polytropes with an extension
comparable to those of the dark matter halos related to galaxies, i.e.,
with extension $ell > 100,mathrm{kpc}$ and mass $M >
10^{12},mathrm{M}_{odot}$. …
Authors:
Stuchlík, Zdeněk; Hledík, Stanislav; Novotný, Jan;