When the characteristic time of variability of the mass flux along the accretion disk is longer than the relaxation time of the local disk equilibrium, it is possible to use the approximation of local equilibrium Shakura (1972), see also Bisnovatyi-Kogan (2011), to calculate the transient disk structure. The equilibrium along a radius of the accretion disk around a star with a mass M is determined by the Keplerian rotational velocity ΩK(1)\begin{equation} \Omega=\Omega_{\rm K}=\left(\frac{GM}{r^3}\right)^{1/2}\cdot \label{omega} \end{equation}Ω=ΩK=GMr31/2·Writing the equation of the vertical equilibrium in approximate algebraic form, we obtain (2)\begin{equation} h=\sqrt 2 \frac{v_{\rm s}}{\Omega}, \label{h} \end{equation}h=2vsΩ,where \hbox{$v_{\rm s}=\sqrt{P/\rho}$}vs=P/ρ is the speed proportional to the sound velocity, P and ρ are the (gas + radiation) pressure and density at the symmetry plane of the accretion disk, and h is the semi-thickness of the accretion disk. The specific angular momentum l of the matter in the accretion disk is connected to the rotation velocity as(3)\begin{equation} l=r\,v_\phi=r^2\Omega. \label{l} \end{equation}l=r vφ=r2Ω.The mass flux through the disk at radius r is connected to the radial velocity vr as (4)\begin{equation} \dot M=-4\pi h\rho r v_r, \quad \dot M>0,\quad v_r<0. \label{mflux} \end{equation}Ṁ=−4πhρrvr, Ṁ>0, vr0.We use an α approximation for the turbulent viscosity (Shakura 1972) when the (rφ) component of the stress tensor trφ is written as (5)\begin{equation} t_{r\phi}=\alpha\, P, \label{trphi} \end{equation}trφ=α P,where the phenomenological non-dimensional parameter α ≤ 1. The condition of stationarity of the angular momentum, in which the outward viscous radial flux of the angular momentum is balanced by the angular momentum of the inward flux of the mass, is written as (see, e.g., Bisnovatyi-Kogan 2011) (6)\begin{equation} r^2 h \alpha P=\frac{\dot M}{4\pi}(l-l_{\rm in}). \label{angmom} \end{equation}r2hαP=Ṁ4π(l−lin).The main input into the time lag comes from the outer regions of the disk with l ≫ lin. Then we have from Eqs. (4) and (6) the expression for the radial velocity in the form (7)\begin{equation} v_r=-\alpha\frac{v_{\rm s}^2}{v_\phi}\cdot \label{vr} \end{equation}vr=−αvs2vφ·We also define the surface density Σ, and write Eq. (6) in light of Eq. (3), using condition l ≫ lin, in the form (8)\begin{equation} \Sigma=2\rho h,\quad \dot M \Omega=4\pi \alpha P h. \label{sigma} \end{equation}Σ=2ρh, ṀΩ=4παPh.The equation of the local thermal balance in the accretion disk, when the heat produced by viscosity Q+ is entirely emitted through the sites of the optically thick accretion disk with a total flux Q−, at l ≫ lin is written as (see, e.g., Bisnovatyi-Kogan 2011)(9)\begin{equation} \frac{3}{2}\dot M \Omega^2=\frac{16\pi ac T^4}{3\varkappa \Sigma}\cdot \label{theq} \end{equation}32ṀΩ2=16πacT43ϰΣ·Here T is the temperature in the symmetry plane of the accretion disk, a is the constant of the radiation energy density, c is the speed of light, and ϰ is the Thompson (scattering) opacity of the matter.
Citation: Shakura (1972)
Citation function: Uses; Uses
Citation: Shakura 1972
Citation function: Uses; Uses