3.2 Thick discs - davidar/scholarpedia GitHub Wiki

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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 <math>\epsilon</math> or pressure <math>P</math> are independent on time <math>t</math> and the azimuthal angle <math>\phi\ .</math>
  • Matter moves on circular trajectories, i.e. the four velocity has the form <math>u^i = [u^t,]\ .</math> The angular velocity is defined as <math>\Omega = u^{\phi}/u^t\ ,</math> and the angular momentum as <math>\ell = - u_{\phi}/u_t\ ,</math>
  • <math></math>t_{dyn} \ll t_{the} <t_{vis}\> with &lt;math&gt;t_&#123;dyn&#125;&lt;/math&gt; being the dynamical timescale in which pressure force adjusts to the balance of gravitational and centrifugal forces, &lt;math&gt;t_&#123;the&#125;&lt;/math&gt; being the thermal timescale in which the entropy redistribution occurs due to dissipative heating and cooling processes, and &lt;math&gt;t_&#123;vis&#125;&lt;/math&gt; 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, &lt;math&gt;T^i_&#123;~\nu&#125; = u^&#123;\mu&#125;\,u_&#123;\nu&#125;\,(P + \epsilon) &#45; \delta^&#123;\mu&#125;_&#123;~\nu&#125;\,P\ .&lt;/math&gt;
Using this form of the stress-energy tensor, <a href="http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1978A%26A....63..221A" target="_blank">Abramowicz et al. 1978</a> have derived from the equilibrium condition &lt;math&gt;\nabla_&#123;\mu&#125;\,T^&#123;\mu&#125;_&#123;~\nu&#125; = 0&lt;/math&gt; the relativistic "Euler" equation,&lt;/div&gt;</t_{vis}\>
<math>
\frac{\nabla_{\mu} P}{\epsilon + P} = \frac{\nabla_{\mu}\,g_{tt} + 2\Omega\,\nabla_{\mu}\,g_{t\phi} + \Omega^2\,\nabla_{\mu}\,g_{\phi\phi}}{g_{tt} + 2\Omega\,g_{t\phi} + \Omega^2\,g_{\phi\phi}} = \nabla_{\mu}\ln A + \frac{\ell\,\nabla_{\mu}\Omega}{1 - \ell\,\Omega}, ~~~{\rm with}~~~A^2(r, \theta) = \frac{1}{g_{tt}(r,\theta) + 2\Omega\,g_{t\phi}(r,\theta) + \Omega^2\,g_{\phi\phi}(r,\theta)}. </math> 		<math>(3.1.1)</math>

Equipressure surfaces: analytic solution in a general case

The "equipressure" surfaces are defined by an implicit condition <math>P(r, \theta) = const\ ,</math> which may be solved to get the explicit form <math>\theta = \theta(r)\ .</math> Then, (3.2.1) implies that the function <math>\theta(r)</math> obeys,
<math>
\frac{d\theta}{dr} = - \frac{\partial_{r}\,g_{tt} + 2\Omega\,\partial_{r}\,g_{t\phi} + \Omega^2\,\partial_{r}\,g_{\phi\phi}} {\partial_{\theta}\,g_{tt} + 2\Omega\,\partial_{\theta}\,g_{t\phi} + \Omega^2\,\partial_{\theta}\,g_{\phi\phi}} = - \frac{\partial_{r}\,g^{tt} - 2\ell\,\partial_{r}\,g^{t\phi} + \ell^2\,\partial_{r}\,g^{\phi\phi}} {\partial_{\theta}\,g^{tt} - 2\ell\,\partial_{\theta}\,g^{t\phi} + \ell^2\,\partial_{\theta}\,g^{\phi\phi}}. </math> 		<math>(3.2.2)</math>
If <math>\ell = \ell(r, \theta)</math> or, which is equivalent, <math>\Omega = \Omega(r, \theta)</math> are known functions, the equation (3.2.2) for the equipressure surfaces takes the form of a standard ordinary differential equation <math>d\theta/dr = f(r, \theta)</math> with known rhs, and it may be directly integrated. This has been done first by Jaroszynski et al., 1980 and then by several other authors. In the Figure below, we show an example from a recent paper by Lei et al., 2008 who assumed the angular momentum distribution in the form that depends on three constant parameters (<math>\eta, \beta, \gamma</math>),
<math></math>
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