elisa.models.add

Additive models.

class Band(alpha: 'Parameter' | float | None = None, beta: 'Parameter' | float | None = None, Ec: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntAdditive

Gamma-ray burst continuum developed by Band et al. (1993) [1].

\[\begin{split}N(E) = K \begin{cases} \bigl(\frac{E}{E_0}\bigr)^\alpha \exp\bigl(-\frac{E}{E_\mathrm{c}}\bigr), &\text{if } E < (\alpha-\beta) E_\mathrm{c}, \\\\ \bigl(\frac{E}{E_0}\bigr)^\beta \exp(\beta-\alpha) \left[ \frac{(\alpha-\beta)E_\mathrm{c}}{E_0} \right]^{\alpha-\beta}, &\text{otherwise,} \end{cases}\end{split}\]

where \(E_0\) is the reference energy fixed at 100 keV.

Parameters:

alphaParameter, optional

The low-energy power law index \(\alpha\), dimensionless.

betaParameter, optional

The high-energy power law index \(\beta\), dimensionless.

EcParameter, optional

The characteristic energy \(E_\mathrm{c}\), in units of keV.

KParameter, optional

The amplitude \(K\), in units of ph cm⁻² s⁻¹ keV⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

Ec	Band parameter \(E_\mathrm{c}\).
K	Band parameter \(K\).
alpha	Band parameter \(\alpha\).
beta	Band parameter \(\beta\).
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 additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

References

[1]

Band, D., et al. 1993, ApJ, 413, 281

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property Ec: Parameter

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

property K: Parameter

Band parameter \(K\).

property alpha: Parameter

Band parameter \(\alpha\).

property beta: Parameter

Band parameter \(\beta\).

class BandEp(alpha: 'Parameter' | float | None = None, beta: 'Parameter' | float | None = None, Ep: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntAdditive

Gamma-ray burst continuum developed by Band et al. (1993) [1], parametrized by the peak of \(\nu F_\nu\).

\[\begin{split}N(E) = K \begin{cases} \bigl(\frac{E}{E_0}\bigr)^\alpha \exp\left[-\frac{(2+\alpha)E}{E_\mathrm{p}}\right], &\text{if } E < \frac{(\alpha-\beta)E_\mathrm{p}}{2+\alpha}, \\\\ \bigl(\frac{E}{E_0}\bigr)^\beta \exp(\beta-\alpha) \left[ \frac{(\alpha-\beta)E_\mathrm{p}}{(2+\alpha)E_0} \right]^{\alpha-\beta}, &\text{otherwise}, \end{cases}\end{split}\]

where \(E_0\) is the reference energy fixed at 100 keV.

Warning

This model requires the low-energy power law index \(\alpha\) to be greater than -2.0.

Parameters:

alphaParameter, optional

The low-energy power law index \(\alpha\), dimensionless.

betaParameter, optional

The high-energy power law index \(\beta\), dimensionless.

EpParameter, optional

The peak energy \(E_\mathrm{p}\) of \(\nu F_\nu\), in units of keV.

KParameter, optional

The amplitude \(K\), in units of ph cm⁻² s⁻¹ keV⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

Ep	BandEp parameter \(E_\mathrm{p}\).
K	BandEp parameter \(K\).
alpha	BandEp parameter \(\alpha\).
beta	BandEp parameter \(\beta\).
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 additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

References

[1]

Band, D., et al. 1993, ApJ, 413, 281

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property Ep: Parameter

BandEp parameter \(E_\mathrm{p}\).

property K: Parameter

BandEp parameter \(K\).

property alpha: Parameter

BandEp parameter \(\alpha\).

property beta: Parameter

BandEp parameter \(\beta\).

class BentPL(alpha: 'Parameter' | float | None = None, Eb: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntAdditive

Bent power law.

\[N(E) = 2K \left[ 1 + \left(\frac{E}{E_\mathrm{b}}\right)^\alpha \right]^{-1}\]

Parameters:

alphaParameter, optional

The power law photon index \(\alpha\), dimensionless.

EbParameter, optional

The break energy \(E_\mathrm{b}\), in units of keV.

KParameter, optional

The amplitude \(K\), in units of ph cm⁻² s⁻¹ keV⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

Eb	BentPL parameter \(E_\mathrm{b}\).
K	BentPL parameter \(K\).
alpha	BentPL parameter \(alpha\).
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 additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property Eb: Parameter

BentPL parameter \(E_\mathrm{b}\).

property K: Parameter

BentPL parameter \(K\).

property alpha: Parameter

BentPL parameter \(alpha\).

class Blackbody(kT: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntAdditive

Blackbody function.

\[N(E) = \frac{C K E^2}{(kT)^4 [\exp(E/kT)-1]},\]

where \(C=8.0525\) ph.

Parameters:

kTParameter, optional

The temperature \(kT\), in units of keV.

KParameter, optional

The amplitude \(K = L_{39}/D_{10}^2\), where \(L_{39}\) is the source luminosity in units of 10³⁹ erg s⁻¹ and \(D_{10}\) is the distance to the source in units of 10 kpc.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

K	Blackbody parameter \(K\).
eval	Get side-effect free component evaluation function.
kT	Blackbody parameter \(kT\).
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
name	Component name.
param_names	Component's parameter names.
type	Component type is additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property K: Parameter

Blackbody parameter \(K\).

property kT: Parameter

Blackbody parameter \(kT\).

class BlackbodyRad(kT: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntAdditive

Blackbody function with normalization proportional to the surface area.

\[N(E) = \frac{C K E^2}{\exp(E/kT)-1},\]

where \(C=1.0344 \times 10^{-3}\) ph cm⁻² s⁻¹ keV⁻³.

Parameters:

kTParameter, optional

The temperature \(kT\), in units of keV.

KParameter, optional

The amplitude \(K = R_\mathrm{km}^2/D_{10}^2\), where \(R_\mathrm{km}\) is the source radius in km and \(D_{10}\) is the distance to the source in units of 10 kpc.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

K	BlackbodyRad parameter \(K\).
eval	Get side-effect free component evaluation function.
kT	BlackbodyRad parameter \(kT\).
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
name	Component name.
param_names	Component's parameter names.
type	Component type is additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property K: Parameter

BlackbodyRad parameter \(K\).

property kT: Parameter

BlackbodyRad parameter \(kT\).

class BrokenPL(alpha1: 'Parameter' | float | None = None, Eb: 'Parameter' | float | None = None, alpha2: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None)

Bases: AnaIntAdditive

Broken power law.

\[\begin{split}N(E) = K \begin{cases} \left(\frac{E}{E_\mathrm{b}}\right)^{-\alpha_1}, &\text{if } E \le E_\mathrm{b}, \\\\ \left(\frac{E}{E_\mathrm{b}}\right)^{-\alpha_2}, &\text{otherwise}. \end{cases}\end{split}\]

Parameters:

alpha1Parameter, optional

The low-energy power law photon index \(\alpha_1\), dimensionless.

alpha2Parameter, optional

The high-energy power law photon index \(\alpha_2\), dimensionless.

EbParameter, optional

The break energy \(E_\mathrm{b}\), in units of keV.

KParameter, optional

The amplitude \(K\), in units of ph cm⁻² s⁻¹ keV⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

Eb	BrokenPL parameter \(E_\mathrm{b}\).
K	BrokenPL parameter \(K\).
alpha1	BrokenPL parameter \(alpha_1\).
alpha2	BrokenPL parameter \(alpha_2\).
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 additive.

Methods

integral(egrid, params)

Calculate the photon flux integrated over the energy grid.

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

Calculate the photon flux integrated over the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux integrated over egrid, in units of ph cm⁻² s⁻¹.

property Eb: Parameter

BrokenPL parameter \(E_\mathrm{b}\).

property K: Parameter

BrokenPL parameter \(K\).

property alpha1: Parameter

BrokenPL parameter \(alpha_1\).

property alpha2: Parameter

BrokenPL parameter \(alpha_2\).

class Compt(alpha: 'Parameter' | float | None = None, Ep: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntAdditive

Power law with high-energy exponential cutoff, parametrized by the peak of \(\nu F_\nu\).

\[N(E) = K \left(\frac{E}{E_0}\right)^{-\alpha} \exp \left[-\frac{(2-\alpha)E}{E_\mathrm{p}}\right],\]

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

Parameters:

alphaParameter, optional

The power law photon index \(\alpha\), dimensionless.

EpParameter, optional

The peak energy \(E_\mathrm{p}\) of \(\nu F_\nu\), in units of keV.

KParameter, optional

The amplitude \(K\), in units of ph cm⁻² s⁻¹ keV⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

Ep	Compt parameter \(E_\mathrm{p}\).
K	Compt parameter \(K\).
alpha	Compt parameter \(\alpha\).
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 additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property Ep: Parameter

Compt parameter \(E_\mathrm{p}\).

property K: Parameter

Compt parameter \(K\).

property alpha: Parameter

Compt parameter \(\alpha\).

class CutoffPL(alpha: 'Parameter' | float | None = None, Ec: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntAdditive

Power law with high-energy exponential cutoff.

\[N(E) = K \left(\frac{E}{E_0}\right)^{-\alpha} \exp \left(-\frac{E}{E_\mathrm{c}}\right),\]

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

Parameters:

alphaParameter, optional

The power law photon index \(\alpha\), dimensionless.

EcParameter, optional

The e-folding energy of exponential cutoff \(E_\mathrm{c}\), in units of keV.

KParameter, optional

The amplitude \(K\), in units of ph cm⁻² s⁻¹ keV⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

Ec	CutoffPL parameter \(E_\mathrm{c}\).
K	CutoffPL parameter \(K\).
alpha	CutoffPL parameter \(\alpha\).
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 additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property Ec: Parameter

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

property K: Parameter

CutoffPL parameter \(K\).

property alpha: Parameter

CutoffPL parameter \(\alpha\).

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

Bases: NumIntAdditive

Gaussian line profile.

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

Parameters:

ElParameter, optional

The line energy \(E_\mathrm{l}\), in units of keV.

sigmaParameter, optional

The line width \(\sigma\), in units of keV.

KParameter, optional

The total photon flux \(K\) of the line, in units of ph cm⁻² s⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

El	Gauss parameter \(E_\mathrm{l}\).
K	Gauss parameter \(K\).
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	Gauss parameter \(\sigma\).
type	Component type is additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property El: Parameter

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

property K: Parameter

Gauss parameter \(K\).

property sigma: Parameter

Gauss parameter \(\sigma\).

class Lorentz(El: 'Parameter' | float | None = None, sigma: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None)

Bases: AnaIntAdditive

Lorentzian line profile.

\[N(E) = \frac{K/\mathcal{F}}{(E - E_\mathrm{l})^2 + (\sigma/2)^2},\]

where

\[\mathcal{F}=\int_0^{+\infty} \frac{1}{(E - E_\mathrm{l})^2 + (\sigma/2)^2} \ \mathrm{d}E.\]

Parameters:

ElParameter, optional

The line energy \(E_\mathrm{l}\), in units of keV.

sigmaParameter, optional

The FWHM line width \(\sigma\), in units of keV.

KParameter, optional

The total photon flux \(K\) of the line, in units of ph cm⁻² s⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

Attributes

El	Lorentz parameter \(E_\mathrm{l}\).
K	Lorentz parameter \(K\).
eval	Get side-effect free component evaluation function.
latex	\(\LaTeX\) format of the component.
name	Component name.
param_names	Component's parameter names.
sigma	Lorentz parameter \(\sigma\).
type	Component type is additive.

Methods

integral(egrid, params)

Calculate the photon flux integrated over the energy grid.

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

Calculate the photon flux integrated over the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux integrated over egrid, in units of ph cm⁻² s⁻¹.

property El: Parameter

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

property K: Parameter

Lorentz parameter \(K\).

property sigma: Parameter

Lorentz parameter \(\sigma\).

class OTTB(kT: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntAdditive

Optically-thin thermal bremsstrahlung.

\[N(E) = K \left(\frac{E}{E_0}\right)^{-1} \exp\left(-\frac{E}{kT}\right) \exp\left(\frac{E_0}{kT}\right),\]

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

Parameters:

kTParameter, optional

The electron energy \(kT\), in units of keV.

KParameter, optional

The amplitude \(K\), in units of ph cm⁻² s⁻¹ keV⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

K	OTTB parameter \(K\).
eval	Get side-effect free component evaluation function.
kT	OTTB parameter \(kT\).
latex	\(\LaTeX\) format of the component.
method	Numerical integration method.
name	Component name.
param_names	Component's parameter names.
type	Component type is additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property K: Parameter

OTTB parameter \(K\).

property kT: Parameter

OTTB parameter \(kT\).

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

Bases: NumIntAdditive

Optically-thin thermal synchrotron [1].

\[N(E) = K \exp\left[-\left(\frac{E}{E_\mathrm{c}}\right)^{1/3}\right].\]

Parameters:

EcParameter, optional

The energy scale \(E_\mathrm{c}\), in units of keV.

KParameter, optional

The amplitude \(K\), in units of ph cm⁻² s⁻¹ keV⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

Ec	OTTS parameter \(E_\mathrm{c}\).
K	OTTS parameter \(K\).
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 additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

References

[1]

Liang, E. P., et al., 1983, ApJ, 271, 776

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property Ec: Parameter

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

property K: Parameter

OTTS parameter \(K\).

class PLEnFlux(emin, emax, alpha: 'Parameter' | float | None = None, F: 'Parameter' | float | None = None, latex: str | None = None)

Bases: PLFluxNorm

Power law function with energy flux used as normalization.

\[\begin{split}N(E) &= \mathcal{F}_\mathrm{en} \left[ \int_{E_\mathrm{min}}^{E_\mathrm{max}} \left(\frac{E}{E_0}\right)^{-\alpha} \, E \, \mathrm{d}E \right]^{-1} \left(\frac{E}{E_0}\right)^{-\alpha}\\ &= \mathcal{F}_\mathrm{en} (2 - \alpha) \left[ \left(\frac{E_\mathrm{max}}{E_0}\right)^{2 - \alpha} - \left(\frac{E_\mathrm{min}}{E_0}\right)^{2 - \alpha} \right]^{-1} \left(\frac{E}{E_0}\right)^{-\alpha},\end{split}\]

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

Parameters:

eminfloat or int

Minimum energy of the band to calculate the flux, in units of keV.

emaxfloat or int

Maximum energy of the band to calculate the flux, in units of keV.

alphaParameter, optional

Power law photon index \(\alpha\), dimensionless.

FParameter, optional

Energy flux \(\mathcal{F}_\mathrm{en}\), in units of erg cm⁻² s⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

Attributes

F	PLEnFlux parameter \(\mathcal{F}_\mathrm{en}\).
alpha	PLEnFlux 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 additive.

Methods

integral(egrid, params)

Calculate the photon flux integrated over the energy grid.

property F: Parameter

PLEnFlux parameter \(\mathcal{F}_\mathrm{en}\).

property alpha: Parameter

PLEnFlux parameter \(\alpha\).

class PLPhFlux(emin, emax, alpha: 'Parameter' | float | None = None, F: 'Parameter' | float | None = None, latex: str | None = None)

Bases: PLFluxNorm

Power law function with photon flux used as normalization.

\[\begin{split}N(E) &= \mathcal{F}_\mathrm{ph} \left[ \int_{E_\mathrm{min}}^{E_\mathrm{max}} \left(\frac{E}{E_0}\right)^{-\alpha} \, \mathrm{d}E \right]^{-1} \left(\frac{E}{E_0}\right)^{-\alpha}\\ &= \mathcal{F}_\mathrm{ph} (1 - \alpha) \left[ \left(\frac{E_\mathrm{max}}{E_0}\right)^{1 - \alpha} - \left(\frac{E_\mathrm{min}}{E_0}\right)^{1 - \alpha} \right]^{-1} \left(\frac{E}{E_0}\right)^{-\alpha},\end{split}\]

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

Parameters:

eminfloat or int

Minimum energy of the band to calculate the flux, in units of keV.

emaxfloat or int

Maximum energy of the band to calculate the flux, in units of keV.

alphaParameter, optional

Power law photon index \(\alpha\), dimensionless.

FParameter, optional

Photon flux \(\mathcal{F}_\mathrm{ph}\), in units of ph cm⁻² s⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

Attributes

F	PLPhFlux parameter \(\mathcal{F}_\mathrm{ph}\).
alpha	PLPhFlux 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 additive.

Methods

integral(egrid, params)

Calculate the photon flux integrated over the energy grid.

property F: Parameter

PLPhFlux parameter \(\mathcal{F}_\mathrm{ph}\).

property alpha: Parameter

PLPhFlux parameter \(\alpha\).

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

Bases: AnaIntAdditive

Power law function.

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

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

Parameters:

alphaParameter, optional

Power law photon index \(\alpha\), dimensionless.

KParameter, optional

Amplitude \(K\), in units of ph cm⁻² s⁻¹ keV⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

Attributes

K	PowerLaw parameter \(K\).
alpha	PowerLaw 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 additive.

Methods

integral(egrid, params)

Calculate the photon flux integrated over the energy grid.

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

Calculate the photon flux integrated over the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux integrated over egrid, in units of ph cm⁻² s⁻¹.

property K: Parameter

PowerLaw parameter \(K\).

property alpha: Parameter

PowerLaw parameter \(\alpha\).

class SmoothlyBrokenPL(alpha1: 'Parameter' | float | None = None, Eb: 'Parameter' | float | None = None, alpha2: 'Parameter' | float | None = None, rho: 'Parameter' | float | None = None, K: 'Parameter' | float | None = None, latex: str | None = None, method=None)

Bases: NumIntAdditive

Smoothly broken power law.

\[N(E) = K \left( \frac{E}{E_0} \right) ^ {-\alpha_1} \left\{ 2\left[ 1 + \left(\frac{E}{E_\mathrm{b}} \right)^{\left( \alpha_2 - \alpha_1 \right) / \rho} \right] \right\}^{-\rho},\]

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

Parameters:

alpha1Parameter, optional

The low-energy power law index \(\alpha_1\), dimensionless.

EbParameter, optional

The break energy \(E_\mathrm{b}\), in units of keV.

alpha2Parameter, optional

The high-energy power law index \(\alpha_2\), dimensionless.

rhoParameter, optional

The smoothness parameter \(\rho\), dimensionless.

KParameter, optional

The amplitude \(K\), in units of ph cm⁻² s⁻¹ keV⁻¹.

latexstr, optional

\(\LaTeX\) format of the component. Defaults to class name.

method{‘trapz’, ‘simpson’}, optional

Numerical integration method. Defaults to ‘trapz’.

Attributes

Eb	SmoothlyBrokenPL parameter \(E_\mathrm{b}\).
K	SmoothlyBrokenPL parameter \(K\).
alpha1	SmoothlyBrokenPL parameter \(alpha_1\).
alpha2	SmoothlyBrokenPL parameter \(alpha_2\).
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.
rho	SmoothlyBrokenPL parameter \(\rho\).
type	Component type is additive.

Methods

continuum(egrid, params)

Calculate the photon flux at the energy grid.

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

Calculate the photon flux at the energy grid.

Parameters:

egridndarray

Photon energy grid in units of keV.

paramsdict

Parameter dict for the model.

Returns:

jax.Array

The photon flux at egrid, in units of ph cm⁻² s⁻¹ keV⁻¹.

property Eb: Parameter

SmoothlyBrokenPL parameter \(E_\mathrm{b}\).

property K: Parameter

SmoothlyBrokenPL parameter \(K\).

property alpha1: Parameter

SmoothlyBrokenPL parameter \(alpha_1\).

property alpha2: Parameter

SmoothlyBrokenPL parameter \(alpha_2\).

property rho: Parameter

SmoothlyBrokenPL parameter \(\rho\).