Alpha disk ($\alpha$-disk)

The alpha-disk prescription writes turbulent kinematic viscosity as $\nu =\alpha c_{s}H\simeq \alpha c_s^2/\Omega$, with dimensionless $\alpha \lesssim 1$. It reflects eddies no larger than the disk thickness and no faster than the sound speed.

Let $\mathcal{R}=k/({\mu }_m{m}_H)$, take constant opacity $\kappa$, and prescribe the dynamic viscosity $\mu =\alpha p_g/\Omega =\alpha \mathcal{R}\rho T/\Omega$ in a Keplerian disk. The pressure supporting the disk may contain both gas pressure and radiation pressure. Combining viscous dissipation and radiative diffusion gives ${\partial }_m^2{T}^4+A\Omega T=0$ in the vertical mass coordinate of a disk, where $A=27\alpha \kappa \mathcal{R}/(16\sigma )$. Set $\zeta =2m/\Sigma$, $T=T_{\ast }t(\zeta )$ and $T_{\ast }^3=A\Omega {\Sigma }^2/4$. With a radiative-zero disk surface, $t$ satisfies $(t^4{)}^{\prime \prime }+t=0$, $t^{\prime }(0)=t(1)=0$. Writing $I_t={\int }_0^{1}td\zeta$, the density-weighted mean kinematic viscosity is $\bar{\nu }=\alpha \mathcal{R}{T}_{\ast }{I}_t/\Omega$, giving the displayed formula. This calculation does not assume gas-pressure domination: the closure depends on gas pressure while hydrostatic support can be mixed.

Let $\mathcal{R}=k/({\mu }_m{m}_p)$ and choose $h^6=3\alpha \kappa {\Sigma }^2{\mathcal{R}}^4/(16\sigma {\Omega }^5)$. Set $z=h\zeta$, $\rho =(\Sigma /h)D$, $p=\Sigma {\Omega }^{2}hP$, $T=({\Omega }^2{h}^2/\mathcal{R})\Theta$ and $F=\alpha \Sigma {\Omega }^3{h}^{2}Q$. Vertical hydrostatic equilibrium, viscous dissipation, radiative diffusion and the ideal gas law reduce to $P^{\prime }=-D\zeta$, $Q^{\prime }=9P/4$, ${\Theta }^{\prime }=-DQ/{\Theta }^3$, $P=D\Theta$. Midplane symmetry gives $Q(0)=0$; a regular radiative-zero surface has $P=\Theta =0$ at a universal ${\zeta }_s$, with $2{\int }_0^{{\zeta }_s}Dd\zeta =1$. All physical parameters have disappeared. Uniqueness of this normalized boundary-value problem makes every physical profile a rescaling of one dimensionless solution.

// Target: astrophysics.bigb

For a constant-opacity, gas-pressure-supported radiative disk surface at $z=H$, imposing $p=T=0$ does not impose $F=0$. If the outgoing surface flux is finite and positive, dividing hydrostatic balance by the temperature gradient gives $dp/dT\sim 16\sigma {\Omega }^{2}H{T}^3/(3\kappa F_s)$. Consequently $p\propto T^4$, $\rho \propto T^3$, and the diffusion equation gives $T\propto H-z$. Thus the mass density also vanishes regularly while the surface radiates a finite flux. This ideal boundary replaces the finite-temperature photospheric transition; its accuracy is not controlled solely by the disk aspect ratio.

// Target: astrophysics.bigb

The density-weighted mean kinematic viscosity of the universal vertical profile is $\bar{\nu }=\alpha \Omega h^2{\int }_{-{\zeta }_s}^{{\zeta }_s}Pd\zeta$. Substitution of the height scale gives $\bar{\nu }=C{\alpha }^{4/3}(\kappa /\sigma {)}^{1/3}{\mathcal{R}}^{4/3}{\Sigma }^{2/3}{\Omega }^{-2/3}$. For Keplerian rotation, ${\Omega }^{-2/3}=(GM{)}^{-1/3}r$. The positive numerical constant depends only on the dimensionless vertical profile and the convention for $\alpha$.

// Target: astrophysics.bigb

For an optically thick alpha disk dominated by gas pressure, vertical hydrostatic equilibrium gives $H\propto T^{1/2}/\Omega$ and $\rho \propto \Sigma \Omega T^{-1/2}$. The specified opacity therefore scales as $\kappa \propto {\Sigma }^{-3/2}{\Omega }^{-3/2}{T}^2$. Viscous heating scales as $Q^{+}\propto \alpha \Sigma T\Omega$, while radiative diffusion gives $Q^{-}\propto T^4/(\kappa \Sigma )\propto {\Sigma }^{1/2}{\Omega }^{3/2}{T}^2$. Their balance gives $T\propto \alpha {\Sigma }^{1/2}{\Omega }^{-1/2}$, hence $\nu \propto {\alpha }^2{\Sigma }^{1/2}{\Omega }^{-3/2}$. Fixed $\alpha$ and Keplerian rotation yield the displayed law.

The local thermal stability of an accretion disk follows because, at fixed surface density and radius, cooling has logarithmic temperature slope two while heating has slope one. The viscous stability of an accretion disk follows from ${\partial }_{\Sigma }(\nu \Sigma )>0$ on the thermal-equilibrium branch. Both conclusions apply within this pressure, opacity and transport closure.

For the constant gas-radiation ratio vertical disc structure, the integrated alpha prescription $\bar{\nu }\Sigma =\int \mu dz=(\alpha /\Omega )\int pdz$ gives the displayed mean kinematic viscosity and $\lambda /(1+\lambda )=\alpha H\Omega \kappa \Sigma /(8c)$. Uniform dynamic viscosity and nonuniform pressure are compatible because this prescription is an integrated equality. Imposing the corresponding pointwise equality would not preserve the exact vertical solution.

For small positive radiation-to-gas ratio, constant opacity, fixed alpha and molecular constants, eliminate $\lambda$ between $\lambda \propto H\Omega \Sigma$ and $\lambda \Sigma \propto H^7{\Omega }^6$. This gives $H^6\propto {\Sigma }^2{\Omega }^{-5}$ and $\bar{\nu }\propto {\Sigma }^{2/3}{\Omega }^{-2/3}$. Keplerian rotation supplies the displayed radial power. Since $\bar{\nu }\Sigma \propto {\Sigma }^{5/3}$, its derivative with respect to surface density is positive and local viscous perturbations diffuse rather than amplify.

In the exact constant-opacity, pure-radiation pressure diffusion model, dynamic viscosity is height-independent. Imposing an additional exact pointwise relation $\mu \propto p(z)$ forces constant pressure, incompatible with nonzero vertical hydrostatic equilibrium away from the midplane. The characteristic-pressure alpha disk prescription can still be used for order-of-magnitude scalings; it is not an exact simultaneous vertical solution.

Accretion-disk self-gravity becomes important when the disk's own gravitational field is comparable with the central object's restoring field. The outer disk can then become gravitationally unstable, fragment, and form stars.

The Shakura--Sunyaev thin disk is a geometrically thin, optically thick, radiatively efficient accretion-disk model whose turbulent stress is parametrized by the alpha prescription.

A multitemperature blackbody disk is an optically thick disk approximated as concentric blackbody annuli of different effective temperatures. If $T\propto R^{-3/4}$, its intermediate-frequency spectrum satisfies $S_{\nu }\propto {\nu }^{1/3}$.

For a blackbody disk with $T\propto r^{-3/4}$, the substitution $x=h\nu /(k_{B}T)$ in $F_{\nu }\propto {\nu }^3\int r(e^x-1{)}^{-1}dr$ produces $F_{\nu }\propto {\nu }^{1/3}\int x^{5/3}(e^x-1{)}^{-1}dx$. When the inner and outer temperatures make the limits approximately zero and infinity, that convergent integral is independent of frequency. The law applies between the outer-edge turnover and the hot inner-disk cutoff.

For a finite multitemperature blackbody disk, the low-frequency Rayleigh-Jeans law gives $F_{\nu }\propto {\nu }^2\int rT(r)dr$. If $T\propto r^{-3/4}$ away from the inner edge, outer annuli dominate because the weight per logarithmic radial interval grows as $r^{5/4}$. A measure-zero zero-temperature inner edge does not invalidate the leading law when $rT(r)$ is integrable.

The standard thin-disk solution separates approximately into an inner radiation-pressure and electron-scattering zone, a middle gas-pressure and electron-scattering zone, and an outer gas-pressure and free-free-opacity zone.

The inner Shakura--Sunyaev thin disk is supported mainly by radiation pressure and has nearly constant electron-scattering opacity. Equating radiative diffusion with local dissipation and imposing vertical hydrostatic equilibrium gives the characteristic half-thicknessThe radiation-supported height approaches a plateau away from the inner boundary. The formal zero at that boundary is outside the reliable radiation-dominated approximation. At an Eddington accretion rate defined using efficiency $0.1$, this height can make the inner flow only marginally thin.

In a steady thin alpha disk, $|u_R|/c_s$ is of order $\alpha H/R\ll 1$, whereas $u_{\varphi }/c_s$ is of order $R/H\gg 1$. Matter therefore completes many nearly circular orbits during its slow viscous drift inward.

For a steady Keplerian disk with a zero-torque inner edge $R_{\mathrm{in}}$, the flux summed over both faces is

A Keplerian accretion disk with a zero-torque inner boundary condition and local blackbody cooling has $T_{\mathrm{eff}}^4=3GM\dot{M}(1-\sqrt{r_{\mathrm{in}}/r})/(8\pi \sigma r^3)$. The Stefan–Boltzmann law uses the flux through one face; the standard thin-disk dissipation flux includes both faces. The scale $T_{\mathrm{in}}=[3GM\dot{M}/(8\pi \sigma r_{\mathrm{in}}^3){]}^{1/4}$ is a normalization, not the actual temperature at the stress-free inner edge.

For the effective-temperature profile of a zero-torque disk, differentiation of $y^{-3}(1-y^{-1/2})$ with $y=r/r_{\mathrm{in}}$ gives $y_{\max }=49/36$. Thus $T_{\max }=(6/7{)}^{3/2}{7}^{-1/4}{T}_{\mathrm{in}}\simeq 0.488T_{\mathrm{in}}$. This distinguishes a genuine maximum effective temperature from the normalization used in a power-law approximation.

Integrating the standard thin-disk dissipation flux givesHalf of the Newtonian potential-energy decrease remains as orbital kinetic energy until the inner edge.

The magnetorotational instability destabilizes a magnetized differentially rotating flow when angular velocity decreases outward. Magnetic tension couples neighboring fluid elements, allowing the inner one to lose angular momentum and fall inward while the outer one gains angular momentum and moves outward.

For $0<A<2\Omega$ and equal kinematic viscosity and magnetic diffusivity, a vertical wavevector admits $\mathbf{v}=(V,V,0)$ and $\mathbf{b}=(i{\omega }_{A}V/A,-i{\omega }_{A}V/A,0)$ in Alfvén-speed units. Substitution into the magnetorotational instability equations gives the displayed frequency and growth rate. In a Keplerian disk, growth of this polarization requires $B_z^2/({\mu }_0\rho )>5\nu \Omega /4$. This is a bound for this selected channel and its wavelength, not a universal threshold for every MRI mode in an unbounded domain.

For complex velocity amplitudes $v_x=v_y=V$ and magnetic amplitudes $b_x=-b_y=B$ in Alfvén-speed units, phase averaging gives the displayed stress. For fixed horizontal velocity energy, equal radial and azimuthal components maximize their positive Reynolds stress. Opposite equal magnetic components maximize the outward tension contribution of the Maxwell stress tensor, since $-B_x{B}_y$ is then positive. These conclusions follow from $2|ab|\le |a{|}^2+|b{|}^2$. The vertical channel transports angular momentum outward and extracts energy from decreasing angular velocity.

For an incompressible vertical-field mode, frozen-in magnetic tension supplies a spring of frequency ${\omega }_A$. Its displacement satisfies ${\ddot{\xi }}_x-2\Omega {\dot{\xi }}_y-4\Omega A{\xi }_x=-{\omega }_A^2{\xi }_x$ and ${\ddot{\xi }}_y+2\Omega {\dot{\xi }}_x=-{\omega }_A^2{\xi }_y$. Their determinant gives the ideal magnetorotational dispersion relation. A weak spring transfers angular momentum from the faster inner element to the slower outer element, causing inward and outward migration that increase their separation when angular velocity decreases outward. A sufficiently strong spring instead supplies a restoring force.

// Target: physics.bigb

For ideal magnetohydrodynamics at constant mass density $\rho$, a shearing sheet with velocity $-q\Omega x{\mathbf{e}}_y+{\mathbf{v}}_h(z,t)$ and magnetic field ${\mathbf{B}}_h(z,t)+B_z{\mathbf{e}}_z$ obeysHorizontal invariance and vanishing vertical velocity eliminate nonlinear advection. These equations retain Coriolis acceleration, orbital shear and magnetic tension. The vertical pressure adjusts to balance gravity and magnetic pressure.

In the steady state of the horizontally invariant magnetized shearing-sheet equations, let $B_x(\pm H)=\pm B_{+}$ and $B_y(\pm H)=0$. For $q>0$ and $B_z\ne 0$, the symmetric equilibrium hasThis follows by combining azimuthal induction with radial magnetic tension balance to obtain $B_x^{\prime \prime }+K^2{B}_x=0$. Nonzero $B_{+}$ requires $\sin (KH)\ne 0$; even homogeneous additions at $\cos (KH)=0$ are excluded by imposing midplane symmetry.

For nonzero imposed surface inclination, magnetic bending in an incompressible disk is nonmonotonic when $KH>\pi /2$: the first maximum of $|B_x|$ occurs at $z=\pi /(2K)<H$, followed by a decrease. This is exactly the finite-thickness magnetorotational instability criterion for the same slab. The zero-inclination equilibrium can still be unstable, so nonmonotonic bending is a property of the forced profile, rather than a condition on every possible unstable equilibrium.

For a vertical magnetic field in a homogeneous shearing sheet, a horizontal normal mode with growth rate $s$ and vertical Alfvén frequency ${\omega }_a=k{v}_{Az}$ satisfiesThe two roots for $s^2$ are real. For $q>0$, a growing magnetic mode exists precisely when $0<{\omega }_a^2<2q{\Omega }^2$. Magnetic tension is strong enough to couple radially displaced elements, yet weak enough to allow their separation to increase. Ogilvie's lecture on magnetorotational instability derives this relation and its finite-thickness boundary conditions.

For local cylindrical shear $q=-d\ln \Omega /d\ln r$ with $0<q<2$, maximizing the positive squared-growth root of the ideal magnetorotational dispersion relation gives the displayed formulas. For a Keplerian disk these are $s_{\max }=3\Omega /4$ and ${\omega }_A^2=15{\Omega }^2/16$. Field strength selects the fastest wavelength, while the maximum ideal rate is set by shear. Finite domains and magnetic diffusion can exclude that wavelength.

In the local incompressible weak-field limit, take an axisymmetric wavevector $(k_r,0,k_z)$ and ${\omega }_A=\mathbf{k}\cdot \mathbf{B}/\sqrt{{\mu }_0\rho }$. Eliminating the pressure projects the radial force by $k_z^2/k^2$. This yields the displayed dispersion relation, where $\kappa$ is the radial epicyclic frequency. For a hydrodynamically stable rotation profile with outward-decreasing angular velocity, instability requires $0<{\omega }_A^2<-(k_z^2/k^2)d{\Omega }^2/d\ln r$. A purely toroidal field has no axisymmetric local field-line-bending frequency and therefore no ordinary weak-field MRI in this approximation.

For an axisymmetric vertical mode of a shearing sheet with a vertical magnetic field, radial buoyancy frequency $N_r$, and $A=k^2{v}_A^2>0$, the nonstationary amplitudes obey ${\sigma }^4+(2A+N_r^2+{\Omega }^2){\sigma }^2+A(A+N_r^2-3{\Omega }^2)=0$. The two roots for ${\sigma }^2$ are real. A growing mode exists exactly when $A+N_r^2<3{\Omega }^2$. If the advected buoyancy displacement variable is retained as an independent amplitude, the full characteristic polynomial also has a stationary factor $\sigma$.

For $-5{\Omega }^2<N_r^2<3{\Omega }^2$ and a continuum of vertical wavenumbers, the growing branch of the radially stratified magnetorotational dispersion relation is maximized at $k^2{v}_A^2={\Omega }^2-(N_r^2+{\Omega }^2{)}^2/(16{\Omega }^2)$, with ${\sigma }_{\max }=(3{\Omega }^2-N_r^2)/(4\Omega )$. For $N_r^2\le -5{\Omega }^2$ the optimum moves to the $k\to 0$ boundary and the limiting rate is $\sqrt{-N_r^2-{\Omega }^2}$. For $N_r^2\ge 3{\Omega }^2$ there is no exponential instability. Boundaries that discretize the allowed wavenumbers can alter the actual maximum.

If the horizontal magnetic field perturbations vanish at $z=\pm H$, their allowed vertical wavenumbers are $k_m=m\pi /(2H)$, $m=1,2,\dots$. The lowest wavenumber is unstable exactly whenIf that magnetic mode is stable, all higher magnetic modes are stable as well. A weak field admits longer unstable wavelengths that fit inside the slab; sufficiently strong magnetic tension stabilizes every allowed wavelength.

For density proportional to ${\mathrm{sech}}^2(z/H)$, the magnetic vertical equation $F^{\prime \prime }+k^2{\mathrm{sech}}^2(z/H)F=0$ becomes the Legendre differential equation under $\xi =\tanh (z/H)$. Bounded solutions at both surfaces are Legendre polynomials, with $k_n=\sqrt{n(n+1)}/H$. The $n=0$ constant function has no magnetically coupled finite wavenumber. The smallest MRI-relevant eigenvalue is $k_1^2=2/H^2$, which makes sufficiently strong magnetic tension stabilize all allowed modes.

The growing root of the ideal magnetorotational dispersion relation is maximized continuously at $a=v_A^2{k}^2/{\Omega }^2=15/16$, with growth $3\Omega /4$. Discrete vertical modes instead have $a_n=2n(n+1)/\beta$. The fastest one is found by comparing admissible integers on either side of the continuous maximum, rather than rounding a vertical wavenumber blindly. At $\beta =24$, $n=3$ gives $s=\Omega \sqrt{(\sqrt{17}-3)/2}$.

For a vertical magnetic field in a Keplerian disk, axisymmetric vertical MRI modes are unstable when $k^2{v}_A^2<3{\Omega }^2$. The shortest unstable wavelength is ${\lambda }_{\mathrm{crit}}=2\pi v_A/(\sqrt{3}\Omega )$.

A local thermal equilibrium is stable when a small temperature increase raises cooling faster than heating at fixed surface density. The standard total-pressure alpha prescription makes a radiation-pressure-dominated, Thomson-opacity disk thermally unstable.

Partial hydrogen ionization changes disk cooling and permits cold and hot stable thermal branches separated by unstable equilibria. A dwarf nova can cycle between a cold matter-storing disk and a hot, rapidly accreting disk. The cycle is described by the evolution of local surface density and temperature, coupled to viscous evolution of an accretion disk.

At fixed disk radius, balancing local heating and cooling gives thermal equilibria parameterized by surface density and temperature. In the hydrogen-ionization disk instability, this locus has cold and hot stable branches joined by unstable equilibria. Crossing the turning points drives the state changes of a dwarf nova. Thermal stability of an accretion disk tests the local heating and cooling response, while viscous evolution of an accretion disk changes the surface density.

For a gas-supported alpha disk with $\kappa \propto {\rho }^{1/3}{T}^{10}$, compare rates at fixed surface density of a disk, radius, $\alpha$ and composition on the thermal timescale of an accretion disk. Assume that the vertical dynamical timescale of a disk is shorter than that thermal time, and that the viscous timescale is longer; the separation is controlled for $\alpha \ll 1$ and small disk aspect ratio. Vertical hydrostatic equilibrium gives $H\propto T^{1/2}$, $\rho \propto T^{-1/2}$, so $\kappa \propto T^{59/6}$. Viscous heating satisfies $Q^{+}\propto T$ and radiative cooling satisfies $Q^{-}\propto T^4/(\kappa \Sigma )\propto T^{-35/6}$. At equilibrium a positive temperature perturbation changes the net heating by $(41/6)(Q_0/T)\delta T>0$. With positive effective heat capacity, it grows rather than relaxes. This is a local thermal statement and uses the fixed-column, rather than fixed-volume-density, comparison.

The thermal timescale compares the internal energy stored in a disc annulus with its local heating or cooling. In a gas-pressure-supported alpha disk, $t_{\mathrm{th}}\sim (\alpha {\Omega }_K{)}^{-1}$, whereas the viscous timescale is longer by approximately $(R/H{)}^2$. A local thermal perturbation is therefore studied at fixed surface density of a disk, rather than fixed steady-state mass accretion rate.

At fixed surface density, a total-pressure alpha disk supported by radiation pressure has viscous heating $Q^{+}\propto T_c^8$ but radiative cooling $Q^{-}\propto T_c^4$ when electron scattering supplies a nearly constant opacity. Heating therefore wins after a positive temperature perturbation.