Thick discs: assumptions
For "thick discs" models of accretion discs one assumes that:
- Matter distribution is stationary and axially symmetric, i.e. matter quantities such as density \(\epsilon\) or pressure \(P\) are independent on time \(t\) and the azimuthal angle \(\phi\ .\)
- Matter moves on circular trajectories, i.e. the four velocity has the form \(u^i = [u^t, u^{\phi}, 0, 0]\ .\) The angular velocity is defined as \(\Omega = u^{\phi}/u^t\ ,\) and the angular momentum as \(\ell = - u_{\phi}/u_t\ ,\)
- \(t_{dyn} \ll t_{the} <t_{vis}\ ,\) with \(t_{dyn}\) being the dynamical timescale in which pressure force adjusts to the balance of gravitational and centrifugal forces, \(t_{the}\) being the thermal timescale in which the entropy redistribution occurs due to dissipative heating and cooling processes, and \(t_{vis}\) being the viscous timescale in which angular momentum distribution changes due to torque caused by dissipative stresses. Mathematically, this is equivalent to assume the stress energy tensor in the form, \(T^i_{~k} = u^i\,u_k\,(P + \epsilon) - \delta^i_{~k}\,P\ .\)
Thick discs: analytic solution in the barytropic case (Polish doughnuts)
From the equilibrium condition \(\nabla^k\,T^i_{~k} = 0\) one derives the von Zeipel condition that states that for barytropic fluids \(\epsilon = \epsilon(P)\ ,\) the surfaces of constant angular velocity and of constant angular momentum coincide, i.e. \(\ell = \ell(\Omega)\ .\) The functions \(\epsilon = \epsilon(P)\) and \(\ell = \ell(\Omega)\) are independent and may be separately assumed. When they are known, the analytic solution is given by,
|
\[
W(P) \equiv \int \frac{dP}{\epsilon(P) + P} = \ln A + \int \frac{d\Omega}{1 - \Omega\,\ell(\Omega)} \equiv F(r, \theta),
~~~{\rm with}~~~A = \left[ g_{tt}(r,\theta) + 2\Omega\,g_{t\phi}(r,\theta) + \Omega^2\,g_{\phi\phi}(r,\theta)\right]^{-1/2}.
\]
| \((3.1.1)\) |
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