Accretion disk

An accretion disk is an orbiting flow in which stresses permit matter to move inward by transporting angular momentum to other matter or an outflow. Simply losing heat is insufficient for gas that retains too much angular momentum to reach the center. The stresses also dissipate orbital energy, potentially producing strong radiation. A spreading ring illustrates the exchange: some material can accrete while a smaller fraction moves outward carrying the required angular momentum.^[ref]

A hot, comparatively optically thin plasma region associated with an accretion flow. Thermal or nonthermal electrons can transfer energy to softer seed photons through inverse Compton scattering, producing a harder radiation component. Its geometry, heating and coupling to the cooler disc are model dependent; a hard radiation spectrum alone does not uniquely determine these properties.

Radiating material close to a compact object probes relativistic orbital motion, gravitational redshift and light propagation. A thin disk's inferred inner edge can be compared with the predicted innermost stable circular orbit, while iron-line transfer and thermal-continuum fitting probe complementary aspects of the same geometry. Agreement tests a combined spacetime-and-disk model. A black-hole spin measurement made after imposing a Kerr metric is not alone an independent verification of that metric tensor; geometry tests must vary gravitational predictions and astrophysical nuisance parameters together.

A geometrically thin, optically thick disk has a radially varying effective temperature. Relativistic transport and an atmosphere model convert its thermal spectrum into an inferred emitting area and inner radius. Black-hole mass, distance, inclination and spectral hardening are important external constraints. Selecting thermal-state observations with moderate luminosity limits contamination and the breakdown of the thin-disk approximation. Relating the fitted inner radius to the innermost stable circular orbit remains a physical modeling assumption.

An irradiated accretion disk can produce iron fluorescence with a known local energy scale. relativistic Doppler effect, gravitational redshift, beaming and light bending redistribute those photons into a broadened asymmetric line. The red wing is sensitive to the deepest emitting radii, while inclination and orbital motion influence the profile. Coronal illumination, ionization, absorption and the continuum model must be controlled before inferring an inner radius or testing a spacetime geometry.

For $p=K{\rho }^2$, a point-mass gravitational potential and a zero-density free surface at $z=\pm ϵr$, the barotropic cylindrical rotation theorem gives ${\Omega }^2=GM/(r^3\sqrt{1+{ϵ}^2})$. Vertical balance gives $\rho =GM[1/\sqrt{r^2+z^2}-1/(r\sqrt{1+{ϵ}^2})]/(2K)$. Integrating over the finite height produces the displayed constant surface density. This is a local radial equilibrium family; infinite radial extent would have infinite mass.

For smooth radial variation on scale $r$, $c_s\sim \Omega H$, $\nu \sim \alpha \Omega H^2$, $u_r\sim \nu /r$ and $u_z\sim H{u}_r/r$. The exact point-mass vertical gravity differs from ${\Omega }_K^{2}z$ by relative order $(H/r{)}^2$. Radial pressure support changes the rotation and dominant shear heating by the same order. Meridional viscous forces, radial radiative-flux divergence, thermal advection and compressional work are at most order $(H/r{)}^2$ relative to their retained vertical terms for $\alpha \lesssim 1$. These estimates require secular disk evolution and exclude sharp radial boundary layers or independently rapid vertical motion.

// Target: gravitational-instability-of-an-astrophysical-disk.bigb

For a Keplerian disk, let $S$ be the total loss through both faces per unit projected area and $\dot{M}>0$ the inward accretion rate. Annular mass conservation gives ${\dot{M}}^{\prime }=2\pi RS$. A circular orbit has specific orbital energy $-GM/(2R)$, so a marginally escaping wind requires additional energy $GM/(2R)$ per unit lost mass. Equating this sink to a fraction $f$ of the two-face viscous dissipation $(9/4)\nu \Sigma {\Omega }^2$ gives the displayed relation. If the wind carries the local specific angular momentum $h=\sqrt{GMR}$, conservation of angular momentum gives $(\dot{M}h-\mathcal{G}{)}^{\prime }=h{\dot{M}}^{\prime }$, with viscous torque in an accretion disk $\mathcal{G}=3\pi \nu \Sigma h$. Eliminating the torque yields $R^2{\dot{M}}^{\prime \prime }+\frac{3}{2}R{\dot{M}}^{\prime }=\frac{3}{2}f\dot{M}$.

An accretion-disk boundary layer connects nearly Keplerian rotation to a stellar surface rotating at a different angular speed. For a slowly rotating star, dissipation of the incoming orbital kinetic energy gives $L_{\mathrm{BL}}\simeq GM\dot{M}/(2r_{\ast })$, comparable to the standard thin-disk luminosity. If the emitting area scales as $r_{\ast }H$, its blackbody effective temperature scales as $T_{\mathrm{BL}}\sim (r_{\ast }/H{)}^{1/4}{T}_{\mathrm{in}}$. The coefficient depends on geometry, and stellar rotation reduces the available energy.

An accretion flow carries matter inward through a gravitationally bound system. In a disk, angular-momentum transport permits the inward drift while stresses carry angular momentum outward.

A spatially concentrated flow of transferred matter toward an accretor. In a binary it can be fed by overflow from a donor and can form an accretion-disk stream-impact hot spot when it encounters the disk.

In magnetically channelled accretion, a strong magnetic field guides an accretion flow along flux tubes. The component of the Lorentz force density parallel to the magnetic field vanishes, so the longitudinal flow can be treated hydrodynamically while the field controls the transverse geometry. A magnetic dipole field near a neutron star channels matter toward small polar caps.

An accretion column carries an inward accretion flow onto a localized region of a star. In the strong-field slender approximation, the magnetic field controls the flux-tube area while longitudinal pressure and gravity determine acceleration. The signed outward-coordinate velocity is negative; the inward speed and mass accretion rate are usually defined positive.

If both luminosity and spin-up torque are supplied by the same steady mass inflow, $L=GM\dot{M}/R$ and $N=\dot{M}\sqrt{GM{r}_m}$ imply $r_m=N^{2}GM/(L^2{R}^2)$. A model requiring $r_m<R$ cannot describe an exterior magnetosphere. Beaming, opposing magnetic torque, outflows and a different radiative efficiency change this inference and must be made explicit.

The spin at which the net material and magnetic torque on an accreting star vanishes. The corotation radius is typically comparable to the magnetospheric truncation radius, but the precise torque balance depends on the coupling and any outflow.

With magnetic spin equilibrium at the magnetospheric truncation radius, $T\propto R_M^{3/2}{M}^{-1/2}\propto B_{\ast }^{6/7}{R}_{\ast }^{18/7}{\dot{M}}^{-3/7}$ at fixed mass. Inserting the surface accretion Eddington rate changes the radius power to $15/7$. The relation assumes the boundary remains outside the star; very weak fields instead encounter surface-accretion and mass-shedding limits.

A rapidly spinning magnetosphere adds angular momentum to inflowing matter outside the corotation radius, inhibiting accretion and potentially ejecting gas. The centrifugal onset condition does not by itself ensure that every parcel escapes to infinity.

The inner radius at which a stellar magnetic dipole field disrupts a surrounding accretion disk. In a free fall pressure-balance estimate with fixed aspect ratio $ϵ=H/r$, $r_m\sim ({ϵ}^2{\mu }^4/(GM{\dot{M}}^2){)}^{1/7}$. Geometry and inflow structure change its numerical coefficient.

The accretion Alfvén radius estimates where the magnetic stress of an accreting star competes with inflow stress and disrupts an approximately Keplerian disk. It is distinct from the Alfvén critical radius of an outgoing stellar wind. A dipole and a free-fall pressure estimate give the characteristic $B_{\ast }^{4/7}{\dot{M}}^{-2/7}$ scaling; the numerical coefficient depends on flow geometry, and an inferred radius inside the star is outside the model's domain.

A dipole's magnetic pressure scales as $B_{\ast }^2{R}_{\ast }^6/({\mu }_0{r}^6)$. Spherical free fall with $\rho =\dot{M}/(4\pi r^{2}v)$ and $v\propto (GM/r{)}^{1/2}$ has ram pressure proportional to $\dot{M}(GM{)}^{1/2}{r}^{-5/2}$. Equating them gives the seventh-root scaling. Disc geometry changes the coefficient; the estimate ceases to describe an exterior magnetosphere if the resulting radius lies inside the star.

The dipolar flux-tube area has $A(r)=A_{\star }(r/R_{\star }{)}^3$. Applying transonic accretion in a power-law tube with $n=3$ and a polytropic equation of state givesHere $c_0$ and ${\rho }_0$ are the reservoir sound speed and mass density, and $c_s$ is the sound speed at the sonic point. The mass accretion rate isFor two equal polar caps of surface angular radius ${\theta }_{\star }\ll 1$, $A_{\star }\simeq 2\pi R_{\star }^2{\theta }_{\star }^2$. One loaded cap gives half the total rate. The finite-radius transonic branch exists for $1<\gamma <7/5$, must cross outside the star ($r_s>R_{\star }$), and requires ${\theta }_{\star }^2{r}_s/R_{\star }\ll 1$ for the sonic region to remain a narrow flux tube.

A narrow dipolar flux-tube area $A(z)\propto z^3$ gives $A\rho v=\dot{M}$ for inward speed $v>0$. The isothermal magnetic Bernoulli integral in $\Phi =-GM/z$ then gives $(v-c_s^2/v)v\prime =3c_s^2/z-GM/z^2$. A smooth transonic branch crosses at $z_s=GM/(3c_s^2)$ with $v_s=c_s$, ${\rho }_s=e^{5/2}{\rho }_{\infty }$ and $\dot{M}={\rho }_s{c}_{s}A(z_s)$. The crossing must lie in the domain; subsonic branches need not attain it.

For transonic accretion in a dipolar flux tube, the Bernoulli equation at the sonic point contains $c_s^2[1/(\gamma -1)-5/2]$. A positive reservoir energy requires $\gamma <7/5$. Compression in the $r^3$ dipolar flux-tube area raises the specific enthalpy more rapidly than in the $r^2$ geometry of spherical Bondi accretion, whose corresponding upper index is $5/3$.

The accretion rate is the mass crossing an inward-oriented cylindrical surface per unit time. A common convention takes $\dot{M}>0$ for inward flow, so the outward radial mass flux is $-\dot{M}$.

A hyperaccreting black-hole disk has an accretion rate so large that photon cooling can be trapped and neutrino emission can dominate the cooling of its hot dense inner region. Such disks are models for the central engines of some gamma-ray bursts.

A neutrino-dominated accretion flow is a hot, dense accretion disk in which weak-interaction emission removes much of the dissipated energy. At extreme accretion rates, escaping photons can become inefficient at cooling the inner gas, while reactions involving nucleons and leptons emit neutrinos. Hyperaccreting black holes have therefore been investigated as engines for gamma-ray bursts. Whether cooling wins over inward energy transport depends on the flow parameters; the name describes a regime, not every rapidly accreting disk.^[ref]

Viscous evolution of an accretion disk follows the redistribution of surface density as internal stresses exchange angular momentum between rings. For an axisymmetric Keplerian disk without mass loss, combining mass and angular-momentum conservation gives ${\partial }_t\Sigma =(3/r){\partial }_r[r^{1/2}{\partial }_r(\nu \Sigma r^{1/2})]$. Here $\nu$ is an effective kinematic viscosity. The equation describes both inward accretion and outward spreading; the sign of local mass flow follows from the radial gradient, rather than viscosity alone.^[ref]

The Keplerian viscous diffusion equation with the displayed kinematic viscosity has two such nontrivial finite-support self-similar solutions: $a=1$, $R=C{t}^{2/5}$, and $a=5/4$, $R=C{t}^{1/2}$. Both have $\sigma =(15At{)}^{-1}$ for $A,C,t>0$. Matching powers of radius eliminates the third power only when $(5/2-2a)(2-2a)=0$. The first solution conserves disk angular momentum; the second conserves disk mass but requires an inner torque.

The $a=5/4$ compact self-similar disk with density-linear viscosity has outward drift $r/(2t)$ but zero limiting mass flux at the origin. Its viscous torque in an accretion disk has a positive finite inner limit for $t>0$, equal to $dJ/dt$. Thus disk mass conservation coexists with growth of angular momentum; an external central torque supplies that growth.

For the $a=1$ compact self-similar disk with density-linear viscosity, radial drift is $\dot{r}=r/(2t)-R/(10t)$. An integrating factor gives the displayed trajectories. From $0<r_i<R(t_i)$, the arrival time at the origin is $t_i[1-r_i/R(t_i){]}^{-10}$, finite for every interior point. The outer-edge trajectory has zero surface density of a disk and carries no mass.

Subtract local specific angular momentum times mass conservation from conservation of angular momentum including wind losses S,T. With outward radial mass flux $\mathcal{F}$ and outward viscous torque in an accretion disk $\mathcal{G}=-2\pi \bar{\nu }\Sigma r^3{\Omega }^{\prime }$, this gives $\mathcal{F}{h}^{\prime }+{\mathcal{G}}^{\prime }=2\pi r(Sh-T)$. For Keplerian rotation and $T=qSh$, substitution in mass conservation yields the displayed equation. A wind with q=1 removes local angular momentum but supplies no additional drift term; q>1 can drive accretion even with zero viscosity.

For zero viscosity, $S=f(r)\Sigma$ and constant $q>1$, set $\beta =2(q-1)$. On intervals where f>0, the wind-loaded Keplerian disk evolution equation becomes ${\Sigma }_t=\beta rf{\Sigma }_r+[\beta (2f+r{f}^{\prime })-f]\Sigma$. Multiplication by $f{r}^{2-1/\beta }$ and the displayed travel-time coordinate reduce it to constant-speed advection. Its profile is $g^{\prime }=F(x^{\prime }+t)$ and its characteristics move inward with $dr/dt=-\beta rf$. The weight accounts for mass loss as well as radial compression.

In a steady Keplerian accretion disk with $\Sigma ={\Sigma }_0{R}^{-2}$, $\nu ={\nu }_0{R}^2$, and tracer diffusivity $k=\zeta \nu$, the trace passive scalar concentration obeys the displayed constant-coefficient advection-diffusion equation in logarithmic radius. Contaminant mass is proportional to $CdY$, because $2\pi R\sigma dR=2\pi {\Sigma }_{0}CdY$. A radial impulse $C_0\delta (R-R_0)$ therefore has logarithmic amplitude $C_0/R_0$ and evolves by the Gaussian heat kernel centered at $\ln R_0-vt$. The full-line solution idealizes radii far from the disk boundaries.

For $\nu ={\nu }_0(\Sigma /{\Sigma }_0{)}^{1/2}(R/R_0{)}^{9/4}$, the variables $x=\sqrt{R/R_0}$, $S=(\Sigma /{\Sigma }_0)x^3$ and $\tau =3{\nu }_{0}t/(4R_0^2)$ reduce the source-free Keplerian viscous diffusion equation to $S_{\tau }=(x{S}^{3/2}{)}_{xx}$. The displayed self-similar solution has an outer edge $R_e=36R_0\tau$ and a surface-density cusp proportional to $R^{-3/2}$ at the origin. The positive-part cutoff is essential: beyond the edge, $S^{3/2}$ is the cube of the absolute value, so continuing the squared polynomial without truncation does not solve the same equation.

The disk mass is $8\pi {\Sigma }_0{R}_0^2{\tau }^{-1/2}$ and its angular momentum is $12\pi {\Sigma }_0{R}_0^2\sqrt{GM{R}_0}$, independent of time. It therefore describes accretion with outward spreading and conserved disk angular momentum, under the formal zero-torque condition at the origin. The inward mass flux changes sign at $R=9R_0\tau /4$, so the outer portion moves outward while the inner portion accretes.

Within an instantaneous vertically averaged transport closure, the local Keplerian viscous diffusion equation has effective diffusion coefficient $3{\partial }_{\Sigma }(\bar{\nu }\Sigma )$. A positive coefficient smooths short radial Fourier modes in the allowed thin-disc wavelength range; a negative coefficient gives the negative-diffusion criterion for viscous disk instability. A zero coefficient is linearly degenerate and does not establish strict decay. This local response criterion alone does not settle thermal, self-gravitating or boundary-driven stability.

In an axisymmetric thin disc with no vertical mass or torque flux, integrate mass and conservation of angular momentum over height. With fixed height-independent $l=r^2\Omega$, the equations are ${\partial }_t\Sigma +r^{-1}{\partial }_r(r\Sigma v_r)=0$ and ${\partial }_t(\Sigma l)+r^{-1}{\partial }_r(r\Sigma v_{r}l-r^3\bar{\nu }\Sigma {\Omega }^{\prime })=0$. Their difference gives $r\Sigma v_r{l}^{\prime }=(r^3\bar{\nu }\Sigma {\Omega }^{\prime }{)}^{\prime }$. The density-weighted viscosity of a disk supplies the integrated stress.

For cylindrical rotation independent of height, write $W_{r\varphi }=\int T_{r\varphi }dz$. Define the effective kinematic viscosity by $W_{r\varphi }=\bar{\nu }\Sigma r{\Omega }^{\prime }$, where $\Sigma$ is the surface density of a disk. For Keplerian rotation, $\bar{\nu }=-2W_{r\varphi }/(3\Sigma \Omega )$, and the outward viscous torque in an accretion disk is $-2\pi r^2{W}_{r\varphi }=3\pi \bar{\nu }\Sigma \sqrt{GMr}$. A Newtonian fluid stress tensor gives the density-weighted viscosity of a disk; the stress definition can also parameterize an effective turbulent or magnetic transport stress.

// Target: astrophysics.bigb

Use $l=\sqrt{G{M}_{\ast }r}$ and ${\Omega }^{\prime }=-(3/2)\Omega /r$ in the vertically averaged viscous disk equations. The radial drift is $v_r=-3(\Sigma \sqrt{r}{)}^{-1}{\partial }_r(\sqrt{r}\bar{\nu }\Sigma )$. Conservation of mass gives this nonlinear diffusion equation. It assumes the central rotation law remains fixed and excludes wind/surface torque terms.

For an axisymmetric thin disc supplied at $R_0$ with the local Keplerian rotation specific angular momentum, the displayed source augments the Keplerian viscous diffusion equation. The radial velocity remains $v_R=-3(\Sigma R^{1/2}{)}^{-1}{\partial }_R(\nu \Sigma R^{1/2})$: subtracting local specific angular momentum times conservation of mass from conservation of angular momentum cancels the source term. The normalization of the Dirac delta function makes the integrated supplied mass per unit time equal to $\dot{M}(t)$.

For a steady zero-mass-flux background and constant viscous transport response exponent, this change removes the first derivative in the radial diffusion operator. If ${\Sigma }_0\propto r^{-p}$, the kinematic viscosity scales as $r^{p-1/2}$, so the transformed diffusivity scales as $x^{2p-3}$. It is constant at $p=3/2$.

For source-free Keplerian viscous diffusion equation, the variables $y=\sqrt{r}$ and $g=r^{3/2}\Sigma$ give $g_t=(3{\nu }_0/4)g_{yy}$ when $\bar{\nu }={\nu }_{0}r$. Rescaling y gives the displayed heat equation. Initial and radial boundary conditions must also be transformed; the reduction does not specify them.

With $x=2\sqrt{r/(3A)}$ and $g=A{r}^{3/2}\Sigma$, the half-line heat equation solution $g=C{t}^{-1/2}{e}^{-x^2/(4t)}$ has $g_x(0)=0$ but $g(0)>0$. It gives $\Sigma \propto r^{-3/2}{t}^{-1/2}{e}^{-r/(3At)}$, constant disk mass and angular momentum proportional to $t^{1/2}$. Its radial velocity is $r/t$, and a central torque proportional to $t^{-1/2}$ supplies the increasing angular momentum. It is an outward-spreading, torque-driven disk, not a zero-torque accreting solution.

// Target: astrophysical-fluid-dynamics.bigb

For $\bar{\nu }=Ar$ with $A>0$, use $x=2\sqrt{r/(3A)}$ and $g=A{r}^{3/2}\Sigma$. The Keplerian viscous diffusion equation becomes $g_t=g_{xx}$. The zero-torque inner boundary condition is $g(0,t)=0$, and the positive solution $g=Cx{t}^{-3/2}{e}^{-x^2/(4t)}$ gives $\Sigma \propto r^{-1}{t}^{-3/2}{e}^{-r/(3At)}$. Its mass is proportional to $t^{-1/2}$, angular momentum is constant and inward mass flow at the origin is proportional to $t^{-3/2}$. Its radial velocity is $r/t-3A/2$, exhibiting inner accretion and outer spreading.

// Target: astrophysics.bigb

Two integrations of the steady Keplerian viscous diffusion equation give this general transported quantity. $A$ fixes the mass accretion rate, while $B$ fixes the additive angular-momentum flux or inner torque. A zero-torque inner boundary condition gives $B=-A\sqrt{r_{\mathrm{in}}}$; a nonaccreting constant-torque disk has $A=0$.

For steady injection into a Keplerian disk between two zero-torque boundaries, conservation of mass gives $\dot{M}={\dot{M}}_{\mathrm{in}}+{\dot{M}}_{\mathrm{out}}$. Conservation of angular momentum gives $\dot{M}{h}_0={\dot{M}}_{\mathrm{in}}{h}_{\mathrm{in}}+{\dot{M}}_{\mathrm{out}}{h}_{\mathrm{out}}$, where $h(R)=\sqrt{GMR}$. Eliminating the outward mass flux gives the displayed inward fraction. This fraction does not depend on the viscosity law: the two boundary fluxes must carry away the injected mass and angular momentum. Equivalently, continuity of the viscous torque in an accretion disk joins $\nu \Sigma \sqrt{R}={\dot{M}}_{\mathrm{in}}(\sqrt{R}-\sqrt{R_{\mathrm{in}}})/(3\pi )$ inside the source to ${\dot{M}}_{\mathrm{out}}(\sqrt{R_{\mathrm{out}}}-\sqrt{R})/(3\pi )$ outside.

A Keplerian disk with surface density $\Sigma \sim r^{-p}$ at infinity has finite outer mass for $p>2$ and finite outer angular momentum for $p>5/2$, since the relevant integrals are $\int \Sigma rdr$ and $\int \Sigma r^{3/2}dr$. The borderline $p=5/2$ has logarithmically divergent angular momentum even though its mass converges.

For a Keplerian accretion disk with $\bar{\nu }\propto r^{5/2}$, dimensionless $g=\bar{\nu }\Sigma /({\bar{\nu }}_0{\Sigma }_0)$ satisfies $g_{\tau }=4x^{5/2}{g}_{xx}+6x^{3/2}{g}_x$. In $y=x^{-1/2}$ this is $g_{\tau }=y{g}_{yy}$. The ansatz $g=G(y/\tau )$ gives $G=A+B{e}^{-y/\tau }$; finite disk mass selects $A=0$. Its nonzero profile has logarithmically infinite disk angular momentum on an infinite radial domain.

For a Keplerian disk with evolution ${\partial }_t\Sigma =(2\pi r{)}^{-1}{\partial }_r\mathcal{F}+S$ and $\mathcal{F}=6\pi r^{1/2}{\partial }_r(r^{1/2}\bar{\nu }\Sigma )$, take zero viscous torque at $r_1>0$ and zero flux outside the source region at $r_2>r_1$. The steady state gives ${\mathcal{F}}^{\prime }=-2\pi rS$, and its outer condition gives the displayed flux. Integrating once more from the zero-torque inner boundary condition yieldsThe equation determines the viscosity-density product, so a thermodynamic viscosity closure can be imposed afterward. For nonnegative integrable $S$, it is nonnegative and the inner accretion rate is exactly the total supplied mass per unit time.

A narrow surface density of a disk source approaching the outer edge from inside, with $S=S_0\delta (r-r_2)$ normalized to have full weight, gives $\dot{M}=2\pi r_2{S}_0$ and the displayed steady interior profile. This follows by replacing the inner source integral by $r_2{S}_0$ and integrating $u^{-1/2}$. The flux jumps from $\dot{M}$ just inside to zero outside the injecting edge. Thus a zero exterior flux is consistent, whereas a continuous zero interior boundary trace is not. A half-weight convention for a symmetric endpoint Dirac delta function represents a different total injected mass; it must not be silently mixed with the full-weight boundary source.

For inward accretion rate $\dot{M}>0$, the steady conservation of angular momentum equation is $\mathcal{G}=\dot{M}(h-h_{\mathrm{in}})$ when the inner torque vanishes. The viscous torque in an accretion disk is $\mathcal{G}=2\pi qh\bar{\nu }\Sigma$, where $h=r^2\Omega$ and $q$ is the orbital shear parameter. ThusThe familiar Keplerian accretion disk follows when $q=3/2$ and $h\propto r^{1/2}$. The general form also applies to the non-Keplerian Paczyński-Wiita circular orbit rotation law.

The viscous timescale is the radial diffusion time $t_{\nu }\sim R^2/\nu$ associated with a kinematic viscosity $\nu$ on length scale $R$.

The viscous torque in an accretion disk is the rate of outward angular momentum transport by an internal shear stress. For an axisymmetric thin disk rotating with angular velocity $\Omega (r)$, a local Newtonian viscous stress has $T_{r\varphi }=\rho \nu rd\Omega /dr$. Integrating that stress over a cylinder and multiplying by the lever arm givesHere $\bar{\nu }$ is the density-weighted mean kinematic viscosity, with units of length squared per time. The vertically integrated quantity is $\bar{\nu }\Sigma$. For decreasing $\Omega$, the sign convention makes $\mathcal{G}>0$, describing outward angular momentum transport. More general internal stresses can be represented by an effective $\bar{\nu }$ through the same integrated torque relation.^[ref]

For a steady Keplerian accretion disk with inward accretion rate $\dot{M}$ and vanishing inner torque, the outward angular-momentum torque is $\mathcal{G}=\dot{M}(\sqrt{GMr}-\sqrt{GM{r}_{\mathrm{in}}})$. It transports energy outward at rate $\Omega \mathcal{G}$. The binding-energy acquisition rate between radii $b$ and $a$ is $B_{a,b}=GM\dot{M}(1/a-1/b)/2$. Integrating the vertical viscous dissipation and then the annulus givesSince $\Omega \mathcal{G}=GM\dot{M}[r^{-1}-\sqrt{r_{\mathrm{in}}}{r}^{-3/2}]$, subtraction proves the displayed energy balance. Far from the inner edge $D_{a,b}\simeq 3B_{a,b}$, whereas the boundary torque powers vanish for the complete disk and $D=B=GM\dot{M}/(2r_{\mathrm{in}})$. Local dissipation is therefore not supplied solely by local changes in orbital binding energy.

A steady Keplerian disk can transport angular momentum by viscous stress with zero mass flux. The radial drift vanishes, but the torque is constant and nonzero. An inner source and outer sink must supply its boundary angular-momentum exchange. This is distinct from a decretion flow, which transports mass outward.

Integrating the Keplerian accretion disk diffusion equation gives $dJ/dt=6\pi \sqrt{G{M}_{\star }}[r^{3/2}{\partial }_r(\bar{\nu }\Sigma ){]}_{\mathrm{inner}}^{\mathrm{outer}}$ when the angular-momentum integral and boundary limits are legitimate. An inner zero-torque inner boundary condition alone does not eliminate the outer angular-momentum flux. For divergent total $J$, use finite-radius cutoffs before making a conservation claim.

The zero-torque inner boundary condition sets the internal disk torque to zero at a chosen inner radius. It still permits a nonzero accretion rate: the gas carries its own specific angular momentum across the boundary. For a steady Keplerian disk, it gives $\nu \Sigma =\dot{M}[1-\sqrt{r_{\mathrm{in}}/r}]/(3\pi )$. A rapid plunge near the innermost stable circular orbit motivates this approximation, but magnetic coupling or a stellar boundary layer requires examination of the actual stresses.^[ref]

For constant kinematic viscosity $\nu$, use the phenomenological viscous torque in an accretion disk, specific angular momentum $h=rc\sqrt{m/(r-3m)}$, and $h_0=\sqrt{12}mc$ at the innermost stable circular orbit. Neglect radial kinetic motion in the clock relation and neglect differential light travel time. The Schwarzschild time to drift formally from $r_1\ge 6m$ to $6m$ is the displayed expression, withIndeed $\mathcal{T}(2)=0$ and ${\mathcal{T}}^{\prime }(X)=6X[X-2\sqrt{X-1}]/(X-1)\ge 0$ for $X\ge 2$. The zero-torque inner boundary condition makes the formal drift speed diverge as $r\downarrow 6m$, so the nearly circular approximation must give way to the plunging region of a black-hole accretion disk sufficiently close to that orbit. This formula describes the stipulated approximation, not an exact relativistic flow solution.

The plunging region begins near the innermost stable circular orbit, where nearly circular motion becomes unstable and radial inflow becomes rapid. If the inflow time is shorter than the stress-transport time, the gas approximately conserves its specific angular momentum there.

A thin disk with alpha disk viscosity $\nu \sim \alpha H^2\Omega$ has a viscous timescale much longer than its dynamical infall time in the plunging region of a black-hole accretion disk. At fixed nonzero accretion rate, rapid inflow reduces its surface density of a disk. The resulting viscous torque in an accretion disk is smaller by the displayed factor than in the slowly drifting circular disk. This estimate concerns radii outside the narrow matching layers.

A magnetic field crossing the two faces of an accretion disk exerts a surface torque through its azimuthal and vertical Maxwell stress. This torque can remove angular momentum without relying on radial internal stress.

A Keplerian accretion disk orbits a dominant central mass $M_{\ast }$ with angular velocity $\Omega =(G{M}_{\ast }/r^3{)}^{1/2}$ and specific angular momentum $l=r^2\Omega$. In an axisymmetric steady thin disk without wind losses, take the inward mass accretion rate $\dot{M}>0$ to be constant. Conservation of angular momentum givesThe zero-torque inner boundary condition at $r_{\mathrm{in}}$ fixes this constant to $-\dot{M}{l}_{\mathrm{in}}$. Since Keplerian rotation gives $d\Omega /dr=-3\Omega /(2r)$, the viscous torque in an accretion disk is $\mathcal{G}=3\pi \bar{\nu }\Sigma l$. Equating the two expressions yieldsFar outside the inner edge this approaches $\dot{M}/(3\pi )$. Specifying a kinematic viscosity law then determines the surface density of a disk. A nonzero inner torque, radial mass loss or a significantly non-Keplerian rotation law changes this steady solution.^[ref]

For a positive inner magnetic torque $T_m$, define $\lambda =T_m/(\dot{M}{h}_{\mathrm{in}})$. Integrating steady conservation of angular momentum gives $\mathcal{G}=T_m+\dot{M}(h-h_{\mathrm{in}})$. With Keplerian rotation, this determines the surface density of a disk in the displayed formula.

When the applied inner torque dominates the advective angular-momentum flux, the constant-kinematic viscosity inner profile resembles a decretion disk. This resemblance does not alone imply outward mass transport.

The negative-diffusion criterion for viscous disk instability is ${\partial }_{\Sigma }(\bar{\nu }\Sigma )<0$. A surface density enhancement then transports angular momentum less effectively and grows. Freezing the Keplerian viscous diffusion equation locally gives growth proportional to the square of radial wavenumber, within the thin-disc transport range; wavelengths must still be sufficiently long compared with the disk scale height for that closure. The formal short-wave limit is a backward heat equation.

Freezing coefficients locally in the Keplerian viscous diffusion equation gives ${\Sigma }_{1,t}\simeq 3{\partial }_{\Sigma }(\bar{\nu }\Sigma ){\Sigma }_{1,rr}$. A radial Fourier mode grows at rate $-3{\partial }_{\Sigma }(\bar{\nu }\Sigma )k_r^2$. Negative response reverses ordinary smoothing and gives a backward heat equation, within the wavelength range where the thin disc transport closure applies.

The response to a small surface density perturbation is $\delta (\bar{\nu }\Sigma )=q\bar{\nu }\delta \Sigma$. For an instantaneous law $\bar{\nu }\propto r^a{\Sigma }^b$, $q=b+1$. This exponent is not the orbital shear exponent or a satellite-to-star mass ratio, which also often use the letter $q$.

In a characteristic vertical model of a radiation-pressure-dominated alpha disk with constant opacity, $\mu \sim \alpha p/\Omega \sim c/\kappa$ and $p\sim \Sigma {\Omega }^{2}H$ imply $H\propto {\Sigma }^{-1}$ and $\bar{\nu }\Sigma \propto {\Sigma }^{-1}$. The resulting $d(\bar{\nu }\Sigma )/d\Sigma <0$ gives a viscous instability of an accretion disk, distinct from a thermal instability of a radiation-pressure-dominated alpha disk.

In a local Keplerian sheet, slow viscous drift gives ${\partial }_t\Sigma =3{\partial }_x^2(\nu \Sigma )$. For $\nu =A{\Sigma }^2$, the compact profile $\Sigma =\sigma \sqrt{1-x^2/w^2}$ preserves its shape, conserves $\sigma w$, and spreads with $w\propto t^{1/4}$ at late times.

A decretion disk is supplied near its inner edge and transports matter outward. Internal stresses redistribute the supplied angular momentum through a nearly orbiting flow. Classical Be stars provide an example: their gaseous circumstellar disks are modeled as viscously spreading, approximately Keplerian disks. A disk carrying an outward torque is not automatically a decretion disk; it must actually have outward mass transport.^[ref]

A Be star, in the classical sense used here, is a rapidly rotating B-type star that shows or has shown emission in Balmer series lines from circumstellar gas. Its decretion disk is approximately Keplerian rotation and differs from a young star's dust-bearing protoplanetary disk. Disk growth and loss can change the emission spectrum. The viscous disk model describes the gas after injection; it does not by itself explain how material is lifted from the stellar surface.^[ref]

An alpha disk uses a dimensionless parameter $\alpha$ to represent unresolved transport by an effective kinematic viscosity, commonly $\nu =\alpha c_{s}H$. With gas-supported hydrostatic equilibrium, this gives $\nu \sim \alpha c_s^2/\Omega$ and a viscous timescale much longer than an orbital time when $\alpha (H/r{)}^2\ll 1$. The prescription makes disk evolution calculable without resolving every turbulent motion. A value of $\alpha$ is a closure assumption, rather than an identification of the physical instability producing the stress.^[ref]