StochasticDustEmissionCalculator Class Reference

#include <StochasticDustEmissionCalculator.hpp>

Public Member Functions
	~StochasticDustEmissionCalculator ()
size_t	allocatedBytes () const
Array	emissivity (const Array &Jv) const
void	precalculate (SimulationItem *item, const Array &lambdav, const Array &sigmaabsv, string grainType, double bulkDensity, double meanMass, const StoredTable< 1 > &enthalpy)

Private Attributes
Array	_Bcmbv
vector< const SDE_Calculator * >	_calculatorsA
vector< const SDE_Calculator * >	_calculatorsB
vector< const SDE_Calculator * >	_calculatorsC
Array	_emlambdav
vector< string >	_grainTypes
const SDE_TemperatureGrid *	_gridA
const SDE_TemperatureGrid *	_gridB
const SDE_TemperatureGrid *	_gridC
vector< double >	_maxEnthalpyTemps
vector< double >	_meanMasses
Array	_rfdlambdav
Array	_rflambdav

Detailed Description

StochasticDustEmissionCalculator is a helper class to calculate the combined emissivity spectrum ${\epsilon }_{\lambda }$ of a set of representative dust grain populations embedded in a given radiation field $J_{\lambda }$, taking into account the temperature distribution of stochastically heated dust grains (i.e. without assuming that the dust grains are in local thermal equilibrium) where appropriate.

The class handles multiple representative grains that together describe a dust mixture. These are called bins because the calculator is typically used to handle the various grain size bins in a dust mix. The calculator assumes that, for each material type, the size distribution is discretized into a sufficient number of size bins. The calculator also requires access to enthalpy data for the various dust grain materials in the dust mix.

A client of the class must first call the precalculate() function for each bin to supply the absorption cross sections and enthalpy data for the representative grain population corresponding to that bin. The emissivity() function can then be used to obtain the combined emissivity spectrum for the representative grains in all bins. The embedding radiation field is specified by the mean intensities $(J_{\lambda }{)}_k$, which must be discretized on the simulation's radiation field wavelength grid as returned by the Configuration::radiationFieldWLG() function. The emissivity spectrum $({\epsilon }_{\lambda }{)}_{\ell }$ produced by the second function is discretized on the wavelength grid returned by the Configuration::dustEmissionWLG() function.

Using the discretization of the dust composition and size distribution into a range of representative grains (size bins) and the output wavelength grid described above, in addition to a specialized temperature grid constructed by this class, the emissivity in an interstellar radiation field $J_{\lambda }$ can be written as

$${\epsilon }_{\lambda }=\sum _{b=0}^{N_{\text{size}}-1}{\varsigma }_{\lambda ,b}^{\text{abs}}\sum _{i=0}^{N_{\text{temp}}-1}{P}_{b,i}{B}_{\lambda }(T_i)$$

with ${\varsigma }_{\lambda ,b}^{\text{abs}}$ the absorption cross section of the $b$'th size bin, $T_i$ the $i$'th temperature grid point, and $P_{b,i}$ the probability of finding a grain of the $b$'th size bin in the $i$'th temperature bin.

The probabilities $P_{b,i}$ are calculated following a scheme based on Guhathakurta & Draine (ApJ 1989), Draine & Li (ApJ 2001), Kruegel (book, 2003), and Misselt et al. (arXiv, 2008). The scheme is also nicely described by Verstappen (PhD thesis, 2013, section 2.5) and is summarized by Camps et al. (A&A 2015). An overview is presented below. To simplify the notation we focus on a single representative dust grain, dropping the index $b$. Also note that the value of the radiation field is now obtained from the input wavelength grid.

We define a transition matrix $A_{f,i}$ describing the probability per unit time for a grain to transfer from initial temperature bin $i$ to final temperature bin $f$. The transition matrix elements in the case of heating $(f>i)$ are given by

$$A_{f,i}=4\pi {\varsigma }_{{\lambda }_{fi}}^{\text{abs}}{J}_{{\lambda }_{fi}}\frac{hc\mathrm{\Delta }{H}_f}{(H_f-H_i{)}^3}$$

where $H_f$ and $H_i$ are the enthalpies of the final and initial temperature bins, $\mathrm{\Delta }{H}_f$ is the width of the final temperature bin, and ${\lambda }_{fi}$ is the transition wavelength which can be obtained from

$${\lambda }_{fi}=\frac{hc}{H_f-H_i}.$$

If the simulation's configuration requests the inclusion of the cosmic microwave background (CMB) as an additional source for dust heating, the corresponding term is added to the heating matrix element, similar to the treatment of the energy balance equation in the EquilibriumDustEmissionCalculator class. We assume that cooling transitions occur only to the next lower level, so that $A_{f,i}=0$ for $f<i-1$ and

$$A_{i-1,i}=\frac{4\pi }{H_i-H_{i-1}}{\int }_0^{\mathrm{\infty }}{\varsigma }_{\lambda }^{\text{abs}}{B}_{\lambda }(T_i)\text{d}\lambda .$$

The diagonal matrix elements are defined as

$$A_{i,i}=-\sum _{f\ne i}{A}_{f,i}$$

however as we will see below there is no need to explicitly calculate these values.

Assuming a steady state situation, the probabilities $P_i$ can be obtained from the transition matrix by solving the set of $N$ linear equations

$$\sum _{i=0}^{N-1}{A}_{f,i}{P}_i=0f=0,...,N-1$$

along with the normalization condition

$$\sum _{i=0}^{N-1}{P}_i=1,$$

where $N$ is the number of temperature bins. Because the matrix values for $f<i-1$ are zero these equations can be solved by a recursive procedure of computational order $\mathcal{O}(N^2)$. To avoid numerical instabilities caused by the negative diagonal elements, the procedure employs a well-chosen linear combination of the original equations. This leads to the following recursion relations for the adjusted matrix elements $B_{f,i}$, the unnormalized probabilities $X_i$, and finally the normalized probabilities $P_i$:

$$\begin{array}{rlr}{B}_{N-1,i}& =A_{N-1,i}& i=0,\dots ,N-2\\ B_{f,i}& =B_{f+1,i}+A_{f,i}& f=N-2,\dots ,1;i=0,\dots ,f-1\\ X_0& =1\\ X_i& =\frac{\sum _{j=0}^{i-1}{B}_{i,j}{X}_j}{A_{i-1,i}}& i=1,\dots ,N-1\\ P_i& =\frac{X_i}{\sum _{j=0}^{N-1}{X}_j}& i=0,\dots ,N-1\end{array}$$

Constructor & Destructor Documentation

~StochasticDustEmissionCalculator()

StochasticDustEmissionCalculator::~StochasticDustEmissionCalculator

(

)

The destructor destructs the data structures allocated by the precalculate() function.

Member Function Documentation

allocatedBytes()

size_t StochasticDustEmissionCalculator::allocatedBytes

(

)

const

This function returns the size of the memory, in bytes, allocated by the precalculate() function so far. This information can be used for logging purposes.

emissivity()

Array StochasticDustEmissionCalculator::emissivity

(

const Array &

Jv

)

const

This function returns the emissivity spectrum per hydrogen atom $({\epsilon }_{\lambda }{)}_{\ell }$ of the dust mix (or rather of the corresponding mixture of representative grain populations) when embedded in the radiation field specified by the mean intensities $(J_{\lambda }{)}_k$, taking into account the temperature distribution of stochastically heated dust grains. The input and output arrays are discretized on the wavelength grids returned by the Configuration::radiationFieldWLG() and Configuration::dustEmissionWLG() functions, repectively. If the precalculate() function has not been called for at least one bin, the behavior of this function is undefined.

precalculate()

void StochasticDustEmissionCalculator::precalculate	(	SimulationItem *	item,
		const Array &	lambdav,
		const Array &	sigmaabsv,
		string	grainType,
		double	bulkDensity,
		double	meanMass,
		const StoredTable< 1 > &	enthalpy
	)

This function precalculates and stores information used to calculate the emissivity spectrum for a particular bin (i.e. representative grain) to be handled by the calculator. It must be called once for each bin.

When it is first called, the function uses its first argument to obtain the simulation's radiation field and dust emission wavelength grids, and it builds a set of temperature grids for use in the calculator: a coarse grid for quickly determining the appropriate temperature range, and medium and fine grids for performing the actual probability calculations.

The second and third function arguments specify the absorption cross sections ${\varsigma }_i^{\text{abs}}$ for the representative grain corresponding to the current bin on some fine wavelength grid ${\lambda }_i$. The remaining arguments specify an identifier for the type of grain material in the bin, the bulk density of the grain material, the average mass of a dust grain in the current bin, and a stored table containing the specific enthalpy (per unit volume) for the grain material as a function of temperature.

The function stores the absorption cross sections interpolated on the radiation field and dust emission wavelength grids and it precalculates Planck-integrated absorption cross sections on each of the constructed temperature grids through integration over the fine wavelength grid specified as an argument. Furthermore, the function precalculates the heating and cooling rates used for the stochastic probability calculations, barring the input radiation field dependency, again on each of the constructed temperature grids.

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The documentation for this class was generated from the following file:

• StochasticDustEmissionCalculator.hpp