Surface Density Rates of Star Formation in Disks

Class providing models of the radial profile of the star formation rate surface density \(\dot{\Sigma}_\star(r)\) (in \(\mathrm{M}_\odot \, \mathrm{Gyr}^{-1} \, \mathrm{Mpc}^{-2}\)) in disk components of galaxies. The surface density is a function of radius and depends on the local gas surface density, dynamical time, and molecular gas fraction according to the chosen star formation law. It is integrated radially over the disk to yield the total disk star formation rate, and is expected to vary strongly between implementations based on the assumed sub-grid physics.

Default implementation: starFormationRateSurfaceDensityDisksKrumholz2009

Methods

intervals → double precision, allocatable, dimension(:,:)

Return a set of integration intervals to use when integrating over the surface density of star formation rate.

• type (treeNode), intent(inout), target :: node

• double precision , intent(in ) :: radiusInner , radiusOuter

• logical , intent(inout), allocatable, dimension(:) :: intervalIsAnalytic

• double precision , intent(inout), allocatable, dimension(:) :: integralsAnalytic

unchanged → logical

Return true if the surface density rate of star formation is unchanged since the previous evaluation.

• type(treeNode), intent(inout) :: node

rate → double precision

Returns the star formation rate surface density (in \(\mathrm{M}_\odot\) Gyr\(^{-1}\) Mpc\(^{-2}\)) in the disk component of node at the given radius.

• type (treeNode), intent(inout) :: node

• double precision , intent(in ) :: radius

starFormationRateSurfaceDensityDisksBlitz2006

A star formation rate surface density class which assumes that the star formation rate is given by (Blitz and Rosolowsky, 2006):

\[\dot{\Sigma}_\star(R) = \nu_\mathrm{SF}(R) \Sigma_\mathrm{H_2, disk}(R),\]

where \(\nu_\mathrm{SF}\) is a frequency given by

\[\nu_\mathrm{SF}(R) = \nu_\mathrm{SF,0} \left[ 1 + \left({\Sigma_\mathrm{HI}\over \Sigma_0}\right)^q \right],\]

where \(q=\)[surfaceDensityExponent] and \(\Sigma_0=\)[surfaceDensityCritical] are parameters, the surface density of molecular gas \(\Sigma_\mathrm{H_2} = (P_\mathrm{ext}/P_0)^\alpha \Sigma_\mathrm{HI}\), where \(\alpha=\)[pressureExponent] and \(P_0=\)[pressureCharacteristic] are parameters, and the hydrostatic pressure in the disk plane assuming locally isothermal gas and stellar components is given by

\[P_\mathrm{ext} \approx {\pi\over 2} \G \Sigma_\mathrm{gas} \left[ \Sigma_\mathrm{gas} + \left({\sigma_\mathrm{gas}\over \sigma_\star}\right)\Sigma_\star\right]\]

where we assume that the velocity dispersion in the gas is fixed at \(\sigma_\mathrm{gas}=\)[velocityDispersionDiskGas] and, assuming \(\Sigma_\star \gg \Sigma_\mathrm{gas}\), we can write the stellar velocity dispersion in terms of the disk scale height, \(h_\star\), as

\[\sigma_\star = \sqrt{\pi \G h_\star \Sigma_\star}\]

where we assume \(h_\star/R_\mathrm{disk}=\)[heightToRadialScaleDiskBlitzRosolowsky].

Methods

• calculationReset — Reset memoized calculations.

• computeFactors — Compute various factors.

• pressureRatio — Compute the pressure ratio.

• integralFullyMolecular — Compute the integral of the star formation rate surface density in the fully-molecular regime.

• integralPartiallyMolecular — Compute the integral of the star formation rate surface density in the partially-molecular regime.

Parameters

• [velocityDispersionDiskGas] (default 10.0d0) — The velocity dispersion of gas in galactic disks (in km/s), used to compute the hydrostatic midplane pressure that determines the molecular-to-atomic gas ratio in the Blitz and Rosolowsky (2006) star formation model.

• [heightToRadialScaleDisk] (default 0.137d0) — The ratio of scale height to scale radius for disks in the Blitz and Rosolowsky (2006) star formation timescale calculation.

• [surfaceDensityCritical] (default 200.0d0) — The surface density (in units of \(\mathrm{M}_\odot\) pc\(^{-2}\)) in the Blitz and Rosolowsky (2006) star formation timescale calculation at which low-density truncation begins.

• [surfaceDensityExponent] (default 0.4d0) — The exponent for surface density in the Blitz and Rosolowsky (2006) star formation timescale calculation at in the high density regime.

• [starFormationFrequencyNormalization] (default 5.25d-10) — The star formation frequency (in the low-density limit and in units of yr\(^{-1}\)) in the Blitz and Rosolowsky (2006) star formation timescale calculation.

• [pressureCharacteristic] (default 4.54d0) — The characteristic pressure (given as \(P_0/k_\mathrm{B}\) in units of K cm\(^{-3}\)) in the scaling relation of molecular hydrogen fraction with disk pressure in the Blitz and Rosolowsky (2006) star formation timescale calculation.

• [pressureExponent] (default 0.92d0) — The exponent in the scaling relation of molecular hydrogen fraction with disk pressure in the Blitz and Rosolowsky (2006) star formation timescale calculation.

• [assumeMonotonicSurfaceDensity] (default .false.) — If true, assume that the surface density in disks is always monotonically decreasing.

• [useTabulation] (default .false.) — If true, then use tabulated solutions to the integrated star formation rate.

starFormationRateSurfaceDensityDisksExtendedSchmidt

A star formation rate surface density class implementing the extended Schmidt law (Shi et al., 2011):

\[\dot{\Sigma}_\star = A \left(x_\mathrm{H} {\Sigma_\mathrm{gas}\over \mathrm{M}_\odot \hbox{pc}^{-2}}\right)^{N_1} \left({\Sigma_{\star}\over \mathrm{M}_\odot \hbox{pc}^{-2}}\right)^{N_2}\]

where \(A=\)[normalization], \(N_1=\)[exponentGas] and \(N_2=\)[exponentStars] are parameters.

Methods

• calculationReset — Reset memoized calculations.

Parameters

• [normalization] (default 0.5248d-10) — The normalization of the extended Schmidt star formation law [\(\mathrm{M}_\odot\) yr\(^{-1}\)pc\(^{-2}\)].

• [exponentGas] (default 1.0000d+0) — The exponent of gas surface density in the extended Schmidt star formation law.

• [exponentStars] (default 0.4800d+0) — The exponent of stellar surface density in the extended Schmidt star formation law.

starFormationRateSurfaceDensityDisksKennicuttSchmidt

A star formation rate surface density class which assumes that the Kennicutt-Schmidt law holds (Kennicutt, 1998, Schmidt, 1959):

\[\dot{\Sigma}_\star = A \left({\Sigma_\mathrm{H} \over \mathrm{M}_\odot \hbox{pc}^{-2}} \right)^N,\]

where \(A=\)[normalization] and \(N=\)[exponent] are parameters. Optionally, if the [truncate] parameter is set to true, then the star formation rate is truncated below a critical surface density such that

\[\begin{split}\dot{\Sigma}_\star = \left\{ \begin{array}{ll} A \left({\Sigma_\mathrm{H} \over \mathrm{M}_\odot \hbox{pc}^{-2}} \right)^N & \hbox{ if } \Sigma_\mathrm{gas,disk} > \Sigma_\mathrm{crit} \\ A \left({\Sigma_\mathrm{H} \over \mathrm{M}_\odot \hbox{pc}^{-2}} \right)^N \left(\Sigma_\mathrm{gas,disk}/\Sigma_\mathrm{crit}\right)^\alpha & \hbox{ otherwise.} \end{array} \right.\end{split}\]

Here, \(\alpha=\)[exponentTruncated] and \(\Sigma_\mathrm{crit}\) is a critical surface density for star formation which we specify as

\[\Sigma_\mathrm{crit} = {q_\mathrm{crit} \kappa \sigma_\mathrm{gas} \over \pi \G},\]

where \(\kappa\) is the epicyclic frequency in the disk, \(\sigma_\mathrm{gas}\) is the velocity dispersion of gas in the disk and \(q_\mathrm{crit}=\)[toomreParameterCritical] is a dimensionless constant of order unity which controls where the critical density occurs. We assume that \(\sigma_\mathrm{gas}\) is a constant equal to [velocityDispersionDiskGas] and that the disk has a flat rotation curve such that \(\kappa = \sqrt{2} V/R\).

Methods

• calculationReset — Reset memoized calculations.

Parameters

• [normalization] (default 0.147d0) — The normalization of the Kennicutt-Schmidt star formation law [\(\mathrm{M}_\odot\) Gyr\(^{-1}\)pc\(^{-2}\)].

• [exponent] (default 1.400d0) — The power-law exponent \(N\) in the Kennicutt-Schmidt star formation law \(\dot{\Sigma}_\star \propto \Sigma_\mathrm{H}^N\), with a default value of 1.4 from Kennicutt (1998).

• [truncate] (default .true.) — Specifies whether or not to truncate star formation below a critical surface density in disks.

• [exponentTruncated] (default 6.0d0) — The exponent of the \(\Sigma_\mathrm{gas}/\Sigma_\mathrm{crit}\) term used in truncating the Kennicutt-Schmidt star formation law.

• [velocityDispersionDiskGas] (default 10.0d0) — The velocity dispersion of gas in galactic disks (in km/s), used to compute the critical gas surface density \(\Sigma_\mathrm{crit}\) for Toomre stability and truncation of the Kennicutt-Schmidt star formation law at low surface densities.

• [toomreParameterCritical] (default 0.4d0) — The dimensionless critical Toomre stability parameter \(q_\mathrm{crit}\) that sets the threshold gas surface density \(\Sigma_\mathrm{crit} = q_\mathrm{crit}\kappa\sigma_\mathrm{gas}/(\pi G)\) below which star formation is suppressed in disks.

starFormationRateSurfaceDensityDisksKrumholz2009

A star formation rate surface density class implementing the model of (Krumholz et al., 2009):

\[\begin{split}\dot{\Sigma}_\star(R) = \nu_\mathrm{SF} f_\mathrm{H_2}(R)\Sigma_\mathrm{HI, disk}(R) \left\{ \begin{array}{ll} (\Sigma_\mathrm{HI}/\Sigma_0)^{-1/3}, & \hbox{ if } \Sigma_\mathrm{HI}/\Sigma_0 \le 1 \\ (\Sigma_\mathrm{HI}/\Sigma_0)^{1/3}, & \hbox{ if } \Sigma_\mathrm{HI}/\Sigma_0 > 1 \end{array} \right. ,\end{split}\]

where \(\nu_\mathrm{SF}=\)[frequencyStarFormation] is a frequency and \(\Sigma_0=85 \mathrm{M}_\odot \hbox{pc}^{-2}\). The molecular fraction is given by

\[f_\mathrm{H_2} = 1 - \left( 1 + \left[ { 3 s \over 4 (1+\delta)} \right]^{-5} \right)^{-1/5},\]

where

\[\delta = 0.0712 \left[ 0.1 s^{-1} + 0.675 \right]^{-2.8},\]

and

\[s = {\ln(1+0.6\chi+0.01\chi^2) \over 0.04 \Sigma_\mathrm{comp,0} Z^\prime},\]

with

\[\chi = 0.77 \left[ 1 + 3.1 Z^{\prime 0.365} \right],\]

and \(\Sigma_\mathrm{comp,0}=c \Sigma_\mathrm{HI}/\mathrm{M}_\odot \hbox{pc}^{-2}\) where \(c=\)[clumpingFactorMolecularComplex] is a density enhancement factor relating the surface density of molecular complexes to the gas density on larger scales. Alternatively, if [molecularFractionFast] is set to true, the molecular fraction will be computed using the faster (but less accurate at low molecular fraction) formula

\[f_\mathrm{H_2} = 1 - { 3s/4 \over (1 + s/4)}.\]

(Default implementation)

Methods

• calculationReset — Reset memoized calculations.

• computeFactors — Compute constant factors required.

• surfaceDensityFactors — Compute surface density factors required.

• molecularFraction — Compute the molecular fraction.

Parameters

• [frequencyStarFormation] (default 2.36d0) — The characteristic timescale for star formation in nuclear star clusters (in units of Gyr), corresponding to the normalization factor \((2.36~\mathrm{Gyr})^{-1}\) in the Krumholz et al. (2009) star formation rate law.

• [frequencyStarFormation] (default 0.385d0) — The star formation frequency (in units of Gyr\(^{-1}\)) in the “Krumholz-McKee-Tumlinson” star formation timescale calculation.

• [clumpingFactorMolecularComplex] (default 5.0d0) — The density enhancement (relative to mean disk density) for molecular complexes in the “Krumholz-McKee-Tumlinson” star formation timescale calculation.

• [molecularFractionFast] (default .false.) — Selects whether the fast (but less accurate) fitting formula for molecular hydrogen should be used in the “Krumholz-McKee-Tumlinson” star formation timescale calculation.

• [assumeMonotonicSurfaceDensity] (default .false.) — If true, assume that the surface density in disks is always monotonically decreasing.