Paczyński and his collaborators developed, in the late 1970s and early 1980s, a very general method of constructing perfect fluid equilibria of matter orbiting around a Kerr black hole [139, 236, 235, 234]. They assumed for the stress energy tensor and four velocity,
and derived from In real flows, the function is determined by dissipative processes that have timescales much
longer than the dynamical timescale, and are not yet fully understood. Paczyński realized that instead of
deriving
from unsure assumptions about viscosity that involve a free function fixed
ad hoc (e.g., by assuming
const), one may instead assume the result, i.e., assume
. Assuming
is not self-consistent, but neither is assuming
const.
In Boyer–Lindquist coordinates, the equation for the equipressure surfaces, const, may
be written as
, with the function
given by
Another useful way to think about thick disks is from the relativistic analog of the Newtonian effective
potential ,
Before leaving the topic of Polish doughnuts, we should point out that, starting with the work of
Hawley, Smarr, and Wilson [125], this simple, analytic solution has been the most commonly used starting
condition for numerical studies of black hole accretion.
Komissarov [156] was able to extend the Polish doughnut solution by adding a purely azimuthal magnetic
field to create a magnetized torus. This is possible because a magnetic field of this form only enters the
equilibrium solution as an additional pressure-like term. For example, the extended form of Eq. (81
) is
http://www.livingreviews.org/lrr-2013-1 |
Living Rev. Relativity 16, (2013), 1
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