Accretion Disks

A class implementing circumnuclear accretion disks. Circumnuclear accretion disks surrounding supermassive black holes at the centers of galaxies influence the evolution of both the black hole (via accretion rates of mass and angular momentum and possibly by extracting rotational energy from the black hole) and the surrounding galaxy if they lead to energetic outflows (e.g. jets) from the nuclear region.

Default implementation: accretionDisksShakuraSunyaev

Methods

efficiencyRadiative → double precision

Returns the radiative efficiency of the accretion disk, defined as the fraction of accreted rest-mass energy \(\dot{M}\mathrm{c}^2\) that is emitted as radiation, given the black hole properties and accretion rate.

  • class (nodeComponentBlackHole ), intent(inout) :: blackHole

  • double precision , intent(in ) :: accretionRateMass

  • type (enumerationAccretionDiskTypeType), intent(in ), optional :: accretionDiskType

powerJet → double precision

Returns the power of the jet launched by the accretion disk in units of \(\mathrm{M}_\odot\) (km/s)\(^2\) Gyr\(^{-1}\).

  • class (nodeComponentBlackHole), intent(inout) :: blackHole

  • double precision , intent(in ) :: accretionRateMass

rateSpinUp → double precision

Returns the spin-up rate of the black hole due to accretion from the accretion disk.

  • class (nodeComponentBlackHole), intent(inout) :: blackHole

  • double precision , intent(in ) :: accretionRateMass

accretionDisksADAF

A circumnuclear accretion disk class, in which accretion is via an ADAF (Narayan and Yi, 1994) which is radiatively inefficient and geometrically thick. The radiative efficiency of the flow, which will be zero for a pure ADAF, is controlled by [efficiencyRadiationType]. If set to fixed, then the radiative efficiency is set to the value of the input parameter [efficiencyRadiation]. Alternatively, if set to thinDisk the radiative efficiency will be set to that of a Shakura-Sunyaev thin disk. The spin up rate of the black hole and the jet power produced as material accretes into the black hole are computed using the method of Benson and Babul (2009). The maximum efficiency of the jet (in units of the accretion power \(\dot{M} \mathrm{c}^2\)) is set by [efficiencyJetMaximum]—in the model of Benson and Babul (2009) the jet efficiency diverges as \(j\rightarrow 1\), setting a maximum is important to avoid numerical instabilities. The energy of the accreted material can be set equal to the energy at infinity (as expected for a pure ADAF) or the energy at the ISCO by use of the [energyOption] parameter (set to pureADAF or ISCO respectively). The ADAF structure is controlled by the adiabatic index, \(\gamma\), and viscosity parameter, \(\alpha\), which are specified via the [adiabaticIndex] and [viscosityOption] input parameters respectively. The field-enhancing shear, \(g\), is computed using \(g=\exp(\omega \tau)\) if [fieldEnhancementOption] is set to “exponential” where \(\omega\) is the frame-dragging frequency and \(\tau\) is the smaller of the radial inflow and azimuthal velocity timescales. If [fieldEnhancementOption] is set to “linear” then the alternative version, \(g=1+\omega \tau\) is used instead. [viscosityOption] may be set to “fit”, in which case the fitting function for \(\alpha\) as a function of black hole spin is used:

\[\begin{split}\alpha(j)=0.015+0.02 j^4 & \hbox{ if } g=\exp(\omega\tau) \hbox{ and } E=E_\mathrm{ISCO}, \\ \alpha(j)=0.025+0.08 j^4 & \hbox{ if } g=1+\omega\tau \hbox{ and } E=E_\mathrm{ISCO}, \\ \alpha(j)=0.010+0.00 j^4 & \hbox{ if } g=\exp(\omega\tau) \hbox{ and } E=1, \\ \alpha(j)=0.025+0.02 j^4 & \hbox{ if } g=1+\omega\tau \hbox{ and } E=1.\end{split}\]

Methods

  • height — Return the dimensionless height of the ADAF at a given radius.

  • velocity — Return the dimensionless velocity of the ADAF at a given radius.

  • temperature — Return the dimensionless temperature of the ADAF at a given radius.

  • enthalpy — Return the dimensionless enthalpy of the ADAF at a given radius.

  • enthalpyAngularMomentumProduct — Return the product of dimensionless enthalpy and angular momentum of the ADAF at a given radius.

  • angularMomentum — Return the dimensionless angular momentum of the ADAF at a given radius.

  • gammaRadial — Return the radial part of the relativistic boost factor in the ADAF at a given radius.

  • gammaAzimuthal — Return the azimuthal part of the relativistic boost factor in the ADAF at a given radius.

  • gamma — Return the relativistic boost factor in the ADAF at a given radius.

  • viscosityParameter — Return the viscosity parameter, \(\alpha\), in the ADAF.

  • fluidAngularVelocity — Return the dimensionless angular velocity of the ADAF fluid at the given radius.

  • fieldEnhancement — Return the magnetic field enhancement factor in the ADAF at the given radius.

  • jetPowerBlackHole — Return the power of the jet launched by the black hole for the ADAF.

  • jetPowerDisk — Return the power of the jet launched from the disk for the ADAF.

  • jetPowerDiskFromBlackHole — Return the power of the jet launched from the disk which is derived from the black hole.

Parameters

  • [efficiencyRadiationType] (default var_str('thinDisk')) — Specifies the specific energy of material at the inner edge of an ADAF. pureADAF makes the specific energy equal to 1 (i.e. all energy is advected with the flow); ISCO makes the specific energy equal to that for the innermost stable circular orbit.

  • [efficiencyRadiation] (default 0.01d0) — Specifies the radiative efficiency of an ADAF (i.e. the fraction of \(\dot{M}\clight^2\) that is emitted in radiation).

  • [energyOption] (default var_str('pureADAF')) — Specifies the specific energy of material at the inner edge of an ADAF. pureADAF makes the specific energy equal to 1 (i.e. all energy is advected with the flow); ISCO makes the specific energy equal to that for the innermost stable circular orbit.

  • [fieldEnhancementOption] (default var_str('exponential')) — Controls how the field enhancing shear is determined. exponential will cause the form \(g=\exp(\omega t)\) (Benson and Babul, 2009) to be used, while linear will cause \(g=1+\omega t\) to be used instead. The functional form of \(\alpha(j)\) (if used) will be adjusted to achieve a sensible spin-up function in each case.

  • [adiabaticIndex] (default adafAdiabaticIndexDefault(enumerationAdafFieldEnhancementEncode(char(fieldEnhancementOption),includesPrefix=.false.))) — Specifies the effective adiabatic index \(\gamma\) of gas in an ADAF, which controls the thermodynamic response of the accreting gas and influences the structure and radiative efficiency of the flow.

  • [viscosityOption] (default var_str('fit')) — Controls how the viscosity parameter \(\alpha\) in an ADAF is determined. fit will cause \(\alpha\) to be computed using the fitting function of Benson and Babul (2009); fixed will cause \(\alpha=\)[adafViscosityFixedAlpha] to be used.

  • [viscosityAlpha] (default 0.1d0) — The value for the viscosity parameter \(\alpha\) in an ADAF to be used if [adafViscosityOption]\(=\)fixed.

  • [efficiencyJetMaximum] (default 2.0d0) — The maximum efficiency allowed for ADAF-driven jets (in units of the accretion power).

accretionDisksEddingtonLimited

A circumnuclear accretion disk class in which accretion is always Eddington-limited. This class does not assume any physical model for the accretion disk, but merely assumes that jets are powered at a fixed fraction [efficiencyJet] of the Eddington luminosity. The radiative efficiency is similarly set at a fixed value of [efficiencyRadiation]. Since no physical model for the disk is assumed, the black hole spin up rate is always set to zero.

Parameters

  • [efficiencyRadiation] (default 0.1d0) — The radiative efficiency of the Eddington-limited accretion disk.

  • [efficiencyJet] (default 0.1d0) — The jet efficiency of the Eddington-limited accretion disk, defined as the fraction of the accreted rest-mass energy \(\dot{M}\mathrm{c}^2\) that is channeled into a relativistic jet outflow.

accretionDisksShakuraSunyaev

A circumnuclear accretion disk class, in which the accretion disks are always described by a radiatively efficient, geometrically thin accretion disk as described by Shakura and Sunyaev (1973). The radiative efficiency of the flow is computed assuming that material falls into the black hole without further energy loss from the ISCO, while the spin-up rate of the black hole is computed assuming that the material enters the black hole with the specific angular momentum of the ISCO (i.e. there are no torques on the material once it begins to fall in from the ISCO; Bardeen 1970). For these thin disks, jet power is computed, using the expressions from Meier (2001; his equations 4 and 5).

(Default implementation)

accretionDisksSwitched

A circumnuclear accretion disk class in which accretion switches between radiatively efficient (Shakura-Sunyaev) and inefficient (ADAF) modes. This is intended to crudely model the fact that accretion disks are expected to be radiatively inefficient at both high accretion rates (i.e. as they approach the Eddington luminosity the radiation pressure from a radiatively efficient flow would begin to disrupt the flow itself by definition), while accretion flows at low accretion rates (\(\dot{M}_\mathrm{acc} < \alpha^2 \dot{M}_\mathrm{Edd}\), where \(\alpha\sim 0.1\) is the usual parameter controlling the rate of angular momentum transport in a Shakura and Sunyaev (1973) accretion disk) are also radiatively inefficient as radiative processes are too inefficient at the associated low densities to radiate energy at the rate it is being liberated. A more thorough discussion is given by Begelman (2014).

The properties of the switched disk (e.g. radiative efficiency, jet power), are a linear combination of those of the Shakura-Sunyaev and ADAF modes, with the ADAF fraction being given by:

(2)\[f_\mathrm{ADAF} = [1+\exp(y_\mathrm{min})]^{-1} + [1+\exp(y_\mathrm{max})]^{-1},\]

where

\[\begin{split}y_\mathrm{min} & = +\log(x/x_\mathrm{min})/\Delta_{x}, \\ y_\mathrm{max} & = -\log(x/x_\mathrm{max})/\Delta_{x},\end{split}\]

and,

\[x = \dot{M} / \dot{M}_\mathrm{Eddington}.\]

Here, \(x_\mathrm{min}=\)[accretionRateThinDiskMinimum], \(x_\mathrm{max}=\)[accretionRateThinDiskMaximum], and \(\Delta_{x}\)=[accretionRateTransitionWidth]. If either [accretionRateThinDiskMinimum] or [accretionRateThinDiskMaximum] is set to “none” then the corresponding term in eqn. ((2)) is excluded.

Additionally, if [scaleADAFRadiativeEfficiency] is set to true then the radiative efficiency of the ADAF component is reduced by a factor \(x/x_\mathrm{min}\) when \(x<x_\mathrm{min}\).

Methods

  • fractionADAF — Return the fraction of the accretion flow to be represented as an ADAF.

  • efficiencyRadiativeScalingADAF — Return the scaling of radiative efficiency of the ADAF component in a switched accretion disk.

Parameters

  • [modelParameterName1] — Names of parameters to be initialized by initializer number 1.

  • [modelParameterName2] — Names of parameters to be initialized by initializer number 2.

  • [accretionRateThinDiskMinimum] (default '0.01d0') — The accretion rate (in Eddington units) below which a switched accretion disk becomes an ADAF.

  • [accretionRateThinDiskMaximum] (default '0.3d0') — The accretion rate (in Eddington units) above which a switched accretion disk becomes an ADAF.

  • [accretionRateTransitionWidth] (default 0.1d0) — The width (in \(\ln[\dot{M}/\dot{M}_\mathrm{Eddington}]\)) over which transitions between accretion disk states occur.

  • [scaleADAFRadiativeEfficiency] (default .true.) — Specifies whether the radiative efficiency of the ADAF component in a switched accretion disk scales with accretion rate.