Thick disks - davidar/scholarpedia GitHub Wiki
- 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 <math>t_{dyn}</math> being the dynamical timescale in which pressure force adjusts to the balance of gravitational and centrifugal forces, <math>t_{the}</math> being the thermal timescale in which the entropy redistribution occurs due to dissipative heating and cooling processes, and <math>t_{vis}</math> 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, <math>T^i_{~k} = u^i\,u_k\,(P + \epsilon) - \delta^i_{~k}\,P\ .</math></div>
<div align="justify">From the equilibrium condition <math>\nabla^k\,T^i_{~k} = 0</math> one derives the von Zeipel condition that states that for barytropic fluids <math>\epsilon = \epsilon(P)\ ,</math> the surfaces of constant angular velocity and of constant angular momentum coincide, i.e. <math>\ell = \ell(\Omega)\ .</math> The functions <math>\epsilon = \epsilon(P)</math> and <math>\ell = \ell(\Omega)</math> are independent and may be separately assumed. When they are known, the analytic solution is given by,</div> <table><tr><td align="left" valign="middle">
- <math>
</td></tr></table></t_{vis}\>