elisa.models.mul

Multiplicative models.

class Constant(f: 'Parameter' | float | None = None, latex: str | None = None)

Bases: AnaIntMultiplicative

Energy-independent multiplicative factor.

\[M(E) = f.\]

Parameters:

• f (Parameter, optional) – The multiplicative factor \(f\), dimensionless.

• latex (str, optional) – \(\LaTeX\) format of the component. Defaults to class name.

Attributes

eval	Get side-effect free component evaluation function.
f	Constant parameter \(f\).
latex	\(\LaTeX\) format of the component.
name	Component name.
param_names	Component's parameter names.
type	Component type is multiplicative.

Methods

integral(egrid, params)

Calculate the average value of the model between the grid.

static integral(egrid: JAXArray, params: NameValMapping) → JAXArray

Calculate the average value of the model between the grid.

Parameters:

• egrid (Array) – Photon energy grid in units of keV.

• params (dict[str, Array]) – Parameter dict for the model.

Returns:

The model value, dimensionless.

property f: Parameter

Constant parameter \(f\).

class Edge(Ec: 'Parameter' | float | None = None, D: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntMultiplicative

Absorption edge.

\[\begin{split}M(E) = \begin{cases} \exp\left[-D \bigl(\frac{E}{E_\mathrm{c}}\bigr)^3\right], &\text{if } E \ge E_\mathrm{c}, \\\\ 1, &\text{otherwise.} \end{cases}\end{split}\]

Parameters:

• Ec (Parameter, optional) – The threshold energy \(E_\mathrm{c}\), in units of keV.

• D (Parameter, optional) – The absorption depth \(D\) at the threshold energy, dimensionless.

• latex (str, optional) – \(\LaTeX\) format of the component. Defaults to class name.

• method ({'trapz', 'simpson'}, optional) – Numerical integration method. Defaults to ‘trapz’.

Attributes

D	Edge parameter \(D\).
Ec	Edge parameter \(E_\mathrm{c}\).
eval	Get side-effect free component evaluation function.
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
name	Component name.
param_names	Component's parameter names.
type	Component type is multiplicative.

Methods

continuum(egrid, params)

Calculate the model value at the energy grid.

static continuum(egrid: Array, params: dict[str, Array]) → Array

Calculate the model value at the energy grid.

Parameters:

• egrid (Array) – Photon energy grid in units of keV.

• params (dict[str, Array]) – Parameter dict for the model.

Returns:

The model value at egrid, dimensionless.

property D: Parameter

Edge parameter \(D\).

property Ec: Parameter

Edge parameter \(E_\mathrm{c}\).

class ExpAbs(Ec: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntMultiplicative

Low-energy exponential rolloff.

\[M(E) = \exp\left(-\frac{E_\mathrm{c}}{E}\right).\]

Parameters:

• Ec (Parameter, optional) – The e-folding energy \(E_\mathrm{c}\) for the absorption, in units of keV.

• latex (str, optional) – \(\LaTeX\) format of the component. Defaults to class name.

• method ({'trapz', 'simpson'}, optional) – Numerical integration method. Defaults to ‘trapz’.

Attributes

Ec	ExpAbs parameter \(E_\mathrm{c}\).
eval	Get side-effect free component evaluation function.
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
name	Component name.
param_names	Component's parameter names.
type	Component type is multiplicative.

Methods

continuum(egrid, params)

Calculate the model value at the energy grid.

static continuum(egrid: Array, params: dict[str, Array]) → Array

Calculate the model value at the energy grid.

Parameters:

• egrid (Array) – Photon energy grid in units of keV.

• params (dict[str, Array]) – Parameter dict for the model.

Returns:

The model value at egrid, dimensionless.

property Ec: Parameter

ExpAbs parameter \(E_\mathrm{c}\).

class ExpFac(A: 'Parameter' | float | None = None, f: 'Parameter' | float | None = None, Ec: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntMultiplicative

Exponential modification.

\[\begin{split}M(E) = \begin{cases} 1 + A \exp\bigl(-\frac{f E}{E_0}\bigr), &\text{if } E \ge E_\mathrm{c}, \\\\ 1, &\text{otherwise,} \end{cases}\end{split}\]

where \(E_0\) is the pivot energy fixed at 1 keV.

Parameters:

• A (Parameter, optional) – The amplitude of effect \(A\), dimensionless.

• f (Parameter, optional) – The exponential factor \(f\), dimensionless.

• Ec (Parameter, optional) – The start energy of modification \(E_\mathrm{c}\), in units of keV.

• latex (str, optional) – \(\LaTeX\) format of the component. Defaults to class name.

• method ({'trapz', 'simpson'}, optional) – Numerical integration method. Defaults to ‘trapz’.

Attributes

A	ExpFac parameter \(A\).
Ec	ExpFac parameter \(E_\mathrm{c}\).
eval	Get side-effect free component evaluation function.
f	ExpFac parameter \(f\).
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
name	Component name.
param_names	Component's parameter names.
type	Component type is multiplicative.

Methods

continuum(egrid, params)

Calculate the model value at the energy grid.

static continuum(egrid: Array, params: dict[str, Array]) → Array

Calculate the model value at the energy grid.

Parameters:

• egrid (Array) – Photon energy grid in units of keV.

• params (dict[str, Array]) – Parameter dict for the model.

Returns:

The model value at egrid, dimensionless.

property A: Parameter

ExpFac parameter \(A\).

property Ec: Parameter

ExpFac parameter \(E_\mathrm{c}\).

property f: Parameter

ExpFac parameter \(f\).

class GAbs(El: 'Parameter' | float | None = None, sigma: 'Parameter' | float | None = None, tau: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntMultiplicative

Gaussian absorption line.

\[M(E) = \exp\left[ -\frac{\tau}{\sqrt{2\pi} \sigma} \exp\left[ -\frac{\left(E - E_\mathrm{l}\right)^2}{2 \sigma^2} \right] \right].\]

The optical depth at line center is \(\frac{\tau}{\sqrt{2\pi}\sigma}\).

Parameters:

• El (Parameter, optional) – The line energy \(E_\mathrm{l}\), in units of keV.

• sigma (Parameter, optional) – The line width \(\sigma\), in units of keV.

• tau (Parameter, optional) – The line depth \(\tau\), in units of keV.

• latex (str, optional) – \(\LaTeX\) format of the component. Defaults to class name.

• method ({'trapz', 'simpson'}, optional) – Numerical integration method. Defaults to ‘trapz’.

Attributes

El	GAbs parameter \(E_\mathrm{l}\).
eval	Get side-effect free component evaluation function.
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
name	Component name.
param_names	Component's parameter names.
sigma	GAbs parameter \(\sigma\).
tau	GAbs parameter \(\tau\).
type	Component type is multiplicative.

Methods

continuum(egrid, params)

Calculate the model value at the energy grid.

static continuum(egrid: Array, params: dict[str, Array]) → Array

Calculate the model value at the energy grid.

Parameters:

• egrid (Array) – Photon energy grid in units of keV.

• params (dict[str, Array]) – Parameter dict for the model.

Returns:

The model value at egrid, dimensionless.

property El: Parameter

GAbs parameter \(E_\mathrm{l}\).

property sigma: Parameter

GAbs parameter \(\sigma\).

property tau: Parameter

GAbs parameter \(\tau\).

class HighECut(Ec: 'Parameter' | float | None = None, Ef: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntMultiplicative

High-energy cutoff.

\[\begin{split}M(E) = \begin{cases} \exp\bigl(\frac{E_\mathrm{c}-E}{E_\mathrm{f}}\bigr), &\text{if } E \ge E_\mathrm{c}, \\\\ 1, &\text{otherwise.} \end{cases}\end{split}\]

Parameters:

• Ec (Parameter, optional) – The cutoff energy \(E_\mathrm{c}\), in units of keV.

• Ef (Parameter, optional) – The e-folding energy \(E_\mathrm{f}\), in units of keV.

• latex (str, optional) – \(\LaTeX\) format of the component. Defaults to class name.

• method ({'trapz', 'simpson'}, optional) – Numerical integration method. Defaults to ‘trapz’.

Attributes

Ec	HighECut parameter \(E_\mathrm{c}\).
Ef	HighECut parameter \(E_\mathrm{f}\).
eval	Get side-effect free component evaluation function.
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
name	Component name.
param_names	Component's parameter names.
type	Component type is multiplicative.

Methods

continuum(egrid, params)

Calculate the model value at the energy grid.

static continuum(egrid: Array, params: dict[str, Array]) → Array

Calculate the model value at the energy grid.

Parameters:

• egrid (Array) – Photon energy grid in units of keV.

• params (dict[str, Array]) – Parameter dict for the model.

Returns:

The model value at egrid, dimensionless.

property Ec: Parameter

HighECut parameter \(E_\mathrm{c}\).

property Ef: Parameter

HighECut parameter \(E_\mathrm{f}\).

class PLAbs(alpha: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None)

Bases: AnaIntMultiplicative

Absorption as a power-law in energy. Useful for things like dust.

\[M(E) = K \left(\frac{E}{E_0}\right)^{-\alpha},\]

where \(E_0\) is the pivot energy fixed at 1 keV.

Parameters:

• alpha (Parameter, optional) – The power law index \(\alpha\), dimensionless.

• K (Parameter, optional) – The coefficient \(K\), dimensionless.

• latex (str, optional) – \(\LaTeX\) format of the component. Defaults to class name.

Attributes

K	PLAbs parameter \(K\).
alpha	PLAbs parameter \(\alpha\).
eval	Get side-effect free component evaluation function.
latex	\(\LaTeX\) format of the component.
name	Component name.
param_names	Component's parameter names.
type	Component type is multiplicative.

Methods

integral(egrid, params)

Calculate the average value of the model between the grid.

static integral(egrid: Array, params: dict[str, Array]) → Array

Calculate the average value of the model between the grid.

Parameters:

• egrid (Array) – Photon energy grid in units of keV.

• params (dict[str, Array]) – Parameter dict for the model.

Returns:

The model value, dimensionless.

property K: Parameter

PLAbs parameter \(K\).

property alpha: Parameter

PLAbs parameter \(\alpha\).

class PhAbs(nH: 'Parameter' | float | None = None, latex: str | None = None, abund=None, xsect=None, method=None)

Bases: PhotonAbsorption

Photoelectric absorption.

\[M(E) = \exp \left[ -\eta_\mathrm{H}\ \sigma(E) \right],\]

where \(\sigma(E)\) is the photo-electric cross-section, NOT including Thomson scattering.

Parameters:

• nH (Parameter, optional) – The equivalent hydrogen column density \(\eta_\mathrm{H}\), in units of 10²² cm⁻².

• latex (str, optional) – \(\LaTeX\) format of the component. Defaults to class name.

• abund (str, optional) – Abundance table to use. Available options are:

Attributes

abund	Current abundance table.
eval	Get photon absorption model function.
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
nH	PhAbs parameter \(\eta_\mathrm{H}\).
name	Component name.
param_names	Component's parameter names.
type	Component type is multiplicative.
xsect	Current photon cross-section.

Methods

abund_list()	Get available abundance list.
continuum(egrid, params, abs_model, abund, xsect)	Photon absorption model.
xsect_list()	Get available photon cross-section list.

References

[1]

Anders & Grevesse 1989, Geochimica et Cosmochimica Acta, 53, 1, 197-214

[2]

Asplund et al. 2009 ARAA, 47, 481

[3]

Feldman 1992, Phys. Scr. 46, 202

[4]

Anders & Ebihara 1982, Geochimica et Cosmochimica Acta, 46, 11, 2363-2380

[5]

Grevesse & Sauval 1998, Space Science Reviews, 85, 161–174

[6]

Wilms et al 2000, ApJ, 542, 914

[7]

Lodders 2003, ApJ, 591, 1220

[8]

Lodders et al. 2009

[9]

Balucinska-Church & McCammon 1992, ApJ, 400, 699

[10]

Yan et al. 1998, ApJ, 496, 1044

[11]

Verner et al. 1996, ApJ, 465, 487

static abund_list() → list[str]

Get available abundance list.

static xsect_list() → list[str]

Get available photon cross-section list.

property nH: Parameter

PhAbs parameter \(\eta_\mathrm{H}\).

class TBAbs(nH: 'Parameter' | float | None = None, latex: str | None = None, abund=None, xsect=None, method=None)

Bases: PhotonAbsorption

The Tuebingen-Boulder ISM absorption model.

This model calculates the cross-sections for X-ray absorption by the ISM as the sum of the cross-sections for X-ray absorption due to the gas-phase ISM, the grain-phase ISM, and the molecules in the ISM.

Parameters:

• nH (Parameter, optional) – The equivalent hydrogen column density \(\eta_\mathrm{H}\), in units of 10²² cm⁻².

• latex (str, optional) – \(\LaTeX\) format of the component. Defaults to class name.

• abund (str, optional) – Abundance table to use. Available options are:

Attributes

abund	Current abundance table.
eval	Get photon absorption model function.
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
nH	TBAbs parameter \(\eta_\mathrm{H}\).
name	Component name.
param_names	Component's parameter names.
type	Component type is multiplicative.
xsect	Current photon cross-section.

Methods

abund_list()	Get available abundance list.
continuum(egrid, params, abs_model, abund, xsect)	Photon absorption model.
xsect_list()	Get available photon cross-section list.

References

[1]

Anders & Grevesse 1989, Geochimica et Cosmochimica Acta, 53, 1, 197-214

[2]

Asplund et al. 2009 ARAA, 47, 481

[3]

Feldman 1992, Phys. Scr. 46, 202

[4]

Anders & Ebihara 1982, Geochimica et Cosmochimica Acta, 46, 11, 2363-2380

[5]

Grevesse & Sauval 1998, Space Science Reviews, 85, 161–174

[6]

Wilms et al 2000, ApJ, 542, 914

[7]

Lodders 2003, ApJ, 591, 1220

[8]

Lodders et al. 2009

[9]

Verner et al. 1996, ApJ, 465, 487

static abund_list() → list[str]

Get available abundance list.

static xsect_list() → list[str]

Get available photon cross-section list.

property nH: Parameter

TBAbs parameter \(\eta_\mathrm{H}\).

class WAbs(nH: 'Parameter' | float | None = None, latex: str | None = None, abund=None, xsect=None, method=None)

Bases: PhotonAbsorption

A photo-electric absorption using Wisconsin cross-sections [1].

\[M(E) = \exp \left[ -\eta_\mathrm{H}\ \sigma(E) \right],\]

where \(\sigma(E)\) is the photo-electric cross-section, NOT including Thomson scattering.

Warning

The WAbs model is obsolete and is only included for comparison with historical results. The TBAbs model should be used for the ISM or PhAbs for general photoelectric absorption.

Parameters:

• nH (Parameter, optional) – The equivalent hydrogen column density \(\eta_\mathrm{H}\), in units of 10²² cm⁻².

• latex (str, optional) – \(\LaTeX\) format of the component. Defaults to class name.

• abund (str, optional) – Always use abundance table 'aneb' [2].

• xsect (str, optional) – Always use Wisconsin cross-sections [1].

• method ({'trapz', 'simpson'}, optional) – Numerical integration method. Defaults to ‘trapz’.

Attributes

abund	Current abundance table.
eval	Get photon absorption model function.
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
nH	WAbs parameter \(\eta_\mathrm{H}\).
name	Component name.
param_names	Component's parameter names.
type	Component type is multiplicative.
xsect	Current photon cross-section.

Methods

abund_list()	Get available abundance list.
continuum(egrid, params, abs_model, abund, xsect)	Photon absorption model.
xsect_list()	Get available photon cross-section list.

References

[1] (1,2)

Morrison & McCammon 1983, ApJ, 270, 119

[2]

Anders & Ebihara 1982, Geochimica et Cosmochimica Acta, 46, 11, 2363-2380

property nH: Parameter

WAbs parameter \(\eta_\mathrm{H}\).

static abund_list() → list[str]

Get available abundance list.

static xsect_list() → list[str]

Get available photon cross-section list.