### Question --- What is the Shakura-Sunyaev (alpha-disk) model for accretion disks? What are the assumptions that go into its derivation? ### Answer --- ##### What is the Shakura-Sunyaev (alpha-disk) model for accretion disks? In accreting sources, the conservation of angular momentum causes the accreting matter to form a disk-like structure. In order accrete onto the central object, the matter in the disk must lose angular momentum. This loss in energy causes the mass to be transferred inwards and a commensurate increase in angular momentum to be transferred outwards. *That is, for accretion to occur, angular momentum must be transferred from inner parts of the disk to the outer parts of the disk.* $\ell = v r = r^{2} \Omega = \sqrt{G M r} \hspace{1cm} \Rightarrow \hspace{1cm} \substack{\text{leads to spreading} \\ \text{the accretion material into a ring}}$ The mechanism for this transport is the viscosity, but phenomenologically, we need turbulence-enhanced viscosity to explain what we see. However, the microphysics is not understood and is an active area of research. The **$\mathbf{\alpha}$-disk model** is a steady-state model for a geometrically thin accretion disk with constant mass transfer ($\dot{M}$) that is parameterized by a single parameter ($\alpha$) that encodes our ignorance of the viscosity microphysics. $ \nu = \alpha c_{\rm s} H \hspace{1cm} \text{where} \hspace{1cm} \begin{aligned} \nu &\equiv \text{disk visocity} \\ \alpha &\equiv \text{parameter} \in \left[ 0, \, 1 \right]\\ c_{\rm s} &\equiv \text{speed of sound} \\ H &\equiv \text{scale height} \end{aligned} $ > [!note]- Viscosity is Scaled with Pressure > > The viscous shear stress (force in $\varphi$ direction, per unit area in $r$ direction) is directly proportional to pressure. > $ > \begin{align} > \left| T_{r\varphi} \right| &= \rho \nu r \left| \frac{\mathrm{d} \Omega}{\mathrm{d} r} \right| \\ > &= \rho \nu r \left| \frac{\mathrm{d} }{\mathrm{d} r} \left( \sqrt{ \frac{G M}{r^{3/2}}} \right) \right| \\ > &= \frac{3 \rho \nu r \sqrt{G M}}{2 r^{5/2}} \\ > &= \frac{3}{2} \rho \nu \Omega > \end{align} > $ > Combining with the vertical pressure balance... > $\frac{\mathrm{d} P}{\mathrm{d} z} = - \Omega^{2} z \rho$ > $\int_{P}^{0} \mathrm{d} P = \int_{0}^{H} - \Omega^{2} z \rho \; \mathrm{d} z \hspace{1cm} \Rightarrow \hspace{1cm} H = \sqrt{\frac{2 P}{\rho \Omega^{2}}} \sim \frac{c_{\rm s}}{\Omega}$ > ...and the pressure of an ideal gas... > $P = c_{\rm s}^{2} \rho$ > ...we can express the shear stress in terms of pressure. > $\left| T_{r\varphi} \right| \sim \frac{3}{2} \left(\frac{P}{c_{\rm s}^{2}}\right) \left( \alpha c_{\rm s} H \right) \left(\frac{c_{\rm s}}{H}\right) \sim \frac{3}{2} \alpha P \sim \alpha' P$ For this model, there are 6 governing equations: 1) *Conservation of Mass* $\dot{M} = - 2 \pi r \Sigma v_{\rm r} \hspace{1cm} \text{where} \hspace{1cm} \Sigma = \int_{-\infty}^{+\infty} \rho \; \mathrm{d} z = 2 H \bar{\rho} \equiv \text{surface density}$ 2) *Conservation of Angular Momentum* ($\times 3$) $\widetilde{\tau}_{\rm net} = \underbrace{\left(\frac{1}{(2 \pi r) (2 H)}\right)}_{\text{surface area of disk}} \frac{\mathrm{d} }{\mathrm{d} r} \Big[ 2 \pi r (\Sigma \Omega r^{2}) v_{\rm r} \Big] \hspace{1cm} \text{where} \hspace{1cm} \begin{aligned} \widetilde{\tau}_{\rm net} &\equiv \text{net torque} \\ \Sigma \Omega r^{2} &\equiv \text{ang. mom. density} \end{aligned}$ 3) *Conservation of Energy* $\underbrace{\dot{Q} = \sigma \, T_{\rm eff}^{4}}_{\rm blackbody\ radiation} \hspace{1cm} \text{where} \hspace{1cm} \dot{Q} \equiv \text{power per unit surface area}$ 4) *Radiation Transport* $\frac{\mathrm{d} T}{\mathrm{d} r} = -\frac{3 \kappa_{\rm R} \rho L}{16 \pi a c T^{3} r^{2}}$ Solving these allow us to model the accretion disk's **temperature, pressure and density profiles.** To do this, we use the zero torque boundary conditions. - [[Black Hole#Innermost Stable Circular Orbit (ISCO)|ISCO]] for [[Black Hole|BH]] - [[Question 64|Magnetospheric Radius]] for [[Neutron Star|NS]] - Surface of star To go further and infer more about the disk, we need to consider the... 5. *Equation of State* (pressure or radiation dominated) 6. *[[Optical Depth#Kramer's Opacity Law]]* > [!space] Accretion Luminosity > Accretion assumes in-falling objects start from rest and convert gravitational potential energy into luminosity. > > $L_{\rm acc} \propto \underbrace{\frac{\mathrm{d} T}{\mathrm{d} t} = - \frac{1}{2} \frac{\mathrm{d} U}{\mathrm{d} t}}_{\rm Virial\ Theorem} = - \frac{1}{2} \frac{\mathrm{d} }{\mathrm{d} t} \left( - \frac{G M \; \mathrm{d} M}{R} \right) = \frac{G M \dot{M}}{2 R}$ > > *(See [[Question 54]])* ##### What are the assumptions that go into its derivation? 1. Subsonic turbulence 2. Geometrically thin 3. Optically thick 4. Disk acts as a [[Blackbody Radiation|blackbody]] - Local Thermodynamic Equilibrium (LTE) - Efficiently radiates heat 5. Steady state (constant $\dot{M}$) 6. Self-gravity of disk neglected