Multiplicative models

Expfac

Bases: MultiplicativeComponent

An exponential modification of a spectrum.

\[ \mathcal{M}(E) = \begin{cases}1 + A \exp \left(-fE\right) & \text{if $E>E_c$}\\1 & \text{if $E<E_c$}\end{cases} \]

Parameters

• \(A\) (A) \(\left[\text{dimensionless}\right]\) : Amplitude of the modification
• \(f\) (f) \(\left[\text{keV}^{-1}\right]\) : Exponential factor
• \(E_c\) (E_c) \(\left[\text{keV}\right]\): Start energy of modification

Source code in src/jaxspec/model/multiplicative.py

class Expfac(MultiplicativeComponent):
    r"""
    An exponential modification of a spectrum.

    $$
    \mathcal{M}(E) = \begin{cases}1 + A \exp \left(-fE\right) &
    \text{if $E>E_c$}\\1 & \text{if $E<E_c$}\end{cases}
    $$

    !!! abstract "Parameters"
        * $A$ (`A`) $\left[\text{dimensionless}\right]$ : Amplitude of the modification
        * $f$ (`f`) $\left[\text{keV}^{-1}\right]$ : Exponential factor
        * $E_c$ (`E_c`) $\left[\text{keV}\right]$: Start energy of modification

    """

    def __init__(self):
        self.A = nnx.Param(1.0)
        self.f = nnx.Param(1.0)
        self.E_c = nnx.Param(1.0)

    def factor(self, energy):
        return jnp.where(energy >= self.E_c, 1.0 + self.A * jnp.exp(-self.f * energy), 1.0)

FDcut

Bases: MultiplicativeComponent

A Fermi-Dirac cutoff model.

\[ \mathcal{M}(E) = \left( 1 + \exp \left( \frac{E - E_c}{E_f} \right) \right)^{-1} \]

Parameters

• \(E_c\) (Ec) \(\left[\text{keV}\right]\) : Cutoff energy
• \(E_f\) (Ef) \(\left[\text{keV}\right]\) : Folding energy

Source code in src/jaxspec/model/multiplicative.py

class FDcut(MultiplicativeComponent):
    r"""
    A Fermi-Dirac cutoff model.

    $$
        \mathcal{M}(E) = \left( 1 + \exp \left( \frac{E - E_c}{E_f} \right) \right)^{-1}
    $$

    ??? abstract "Parameters"
        * $E_c$ (`Ec`) $\left[\text{keV}\right]$ : Cutoff energy
        * $E_f$ (`Ef`) $\left[\text{keV}\right]$ : Folding energy
    """

    def __init__(self):
        self.Ec = nnx.Param(1.0)
        self.Ef = nnx.Param(3.0)

    def factor(self, energy):
        return (1 + jnp.exp((energy - self.Ec) / self.Ef)) ** -1

Gabs

Bases: MultiplicativeComponent

A Gaussian absorption model.

\[ \mathcal{M}(E) = \exp \left( - \frac{\tau}{\sqrt{2 \pi} \sigma} \exp \left( -\frac{\left(E-E_0\right)^2}{2 \sigma^2} \right) \right) \]

Parameters

• \(\tau\) (tau) \(\left[\text{dimensionless}\right]\) : Absorption strength
• \(\sigma\) (sigma) \(\left[\text{keV}\right]\) : Absorption width
• \(E_0\) (E0) \(\left[\text{keV}\right]\) : Absorption center

Note

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

Source code in src/jaxspec/model/multiplicative.py

class Gabs(MultiplicativeComponent):
    r"""
    A Gaussian absorption model.

    $$
        \mathcal{M}(E) = \exp \left( - \frac{\tau}{\sqrt{2 \pi} \sigma} \exp
        \left( -\frac{\left(E-E_0\right)^2}{2 \sigma^2} \right) \right)
    $$

    !!! abstract "Parameters"
        * $\tau$ (`tau`) $\left[\text{dimensionless}\right]$ : Absorption strength
        * $\sigma$ (`sigma`) $\left[\text{keV}\right]$ : Absorption width
        * $E_0$ (`E0`) $\left[\text{keV}\right]$ : Absorption center

    !!! note
        The optical depth at line center is $\tau/(\sqrt{2 \pi} \sigma)$.

    """

    def __init__(self):
        self.tau = nnx.Param(1.0)
        self.sigma = nnx.Param(1.0)
        self.E0 = nnx.Param(1.0)

    def g(self, E):
        return (
            2
            / (
                self.sigma
                * jnp.sqrt(2 * jnp.pi)
                * (1 - jsp.erf(-self.E0 / (self.sigma * jnp.sqrt(2))))
            )
            * jnp.exp(-((E - self.E0) ** 2) / (2 * self.sigma**2))
        )

    def factor(self, energy):
        return jnp.exp(-self.tau * self.g(energy))

Highecut

Bases: MultiplicativeComponent

A high-energy cutoff model.

\[ \mathcal{M}(E) = \begin{cases} \exp \left( \frac{E_c - E}{E_f} \right)& \text{if $E > E_c$}\\ 1 & \text{if $E < E_c$}\end{cases} \]

Parameters

• \(E_c\) (Ec) \(\left[\text{keV}\right]\) : Cutoff energy
• \(E_f\) (Ef) \(\left[\text{keV}\right]\) : Folding energy

Source code in src/jaxspec/model/multiplicative.py

class Highecut(MultiplicativeComponent):
    r"""
    A high-energy cutoff model.

    $$
        \mathcal{M}(E) = \begin{cases} \exp
        \left( \frac{E_c - E}{E_f} \right)& \text{if $E > E_c$}\\ 1 & \text{if $E < E_c$}\end{cases}
    $$

    !!! abstract "Parameters"
        * $E_c$ (`Ec`) $\left[\text{keV}\right]$ : Cutoff energy
        * $E_f$ (`Ef`) $\left[\text{keV}\right]$ : Folding energy
    """

    def __init__(self):
        self.Ec = nnx.Param(1.0)
        self.Ef = nnx.Param(1.0)

    def factor(self, energy):
        return jnp.where(energy <= self.Ec, 1.0, jnp.exp((self.Ec - energy) / self.Ef))

MultiplicativeConstant

Bases: MultiplicativeComponent

A multiplicative constant

Parameters

• \(K\) (norm) \(\left[\text{dimensionless}\right]\): The multiplicative constant.

Source code in src/jaxspec/model/multiplicative.py

class MultiplicativeConstant(MultiplicativeComponent):
    r"""
    A multiplicative constant

    !!! abstract "Parameters"
        * $K$ (`norm`) $\left[\text{dimensionless}\right]$: The multiplicative constant.

    """

    def __init__(self):
        self.norm = nnx.Param(1.0)

    def factor(self, energy):
        return self.norm

Phabs

Bases: MultiplicativeComponent

A photoelectric absorption model.

Parameters

• \(N_{\text{H}}\) (nh) \(\left[\frac{\text{atoms}~10^{22}}{\text{cm}^2}\right]\) : Equivalent hydrogen column density

Source code in src/jaxspec/model/multiplicative.py

class Phabs(MultiplicativeComponent):
    r"""
    A photoelectric absorption model.

    !!! abstract "Parameters"
        * $N_{\text{H}}$ (`nh`) $\left[\frac{\text{atoms}~10^{22}}{\text{cm}^2}\right]$ : Equivalent hydrogen column density

    """

    def __init__(self):
        self._energy, self._sigma = _load_xsect("xsect_phabs_aspl.fits")
        self.nh = nnx.Param(1.0)

    def factor(self, energy):
        sigma = jnp.interp(energy, self._energy, self._sigma, left=1e9, right=0.0)

        return jnp.exp(-self.nh * sigma)

Tbabs

Bases: MultiplicativeComponent

The Tuebingen-Boulder ISM absorption model. This model calculates the cross section 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.

\[ \mathcal{M}(E) = \exp^{-N_{\text{H}}\sigma(E)} \]

Parameters

• \(N_{\text{H}}\) (nh) \(\left[\frac{\text{atoms}~10^{22}}{\text{cm}^2}\right]\) : Equivalent hydrogen column density

Note

Abundances and cross-sections \(\sigma\) can be found in Wilms et al. (2000).

Source code in src/jaxspec/model/multiplicative.py

class Tbabs(MultiplicativeComponent):
    r"""
    The Tuebingen-Boulder ISM absorption model. This model calculates the cross section 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.

    $$
        \mathcal{M}(E) = \exp^{-N_{\text{H}}\sigma(E)}
    $$

    !!! abstract "Parameters"
        * $N_{\text{H}}$ (`nh`) $\left[\frac{\text{atoms}~10^{22}}{\text{cm}^2}\right]$ : Equivalent hydrogen column density


    !!! note
        Abundances and cross-sections $\sigma$ can be found in Wilms et al. (2000).

    """

    def __init__(self):
        self._energy, self._sigma = _load_xsect("xsect_tbabs_wilm.fits")
        self.nh = nnx.Param(1.0)

    def factor(self, energy):
        sigma = jnp.exp(
            jnp.interp(
                jnp.log(energy),
                jnp.log(self._energy),
                jnp.log(self._sigma),
                left="extrapolate",
                right="extrapolate",
            )
        )

        return jnp.exp(-self.nh * sigma)

Tbpcf

Bases: MultiplicativeComponent

Partial covering model using the Tuebingen-Boulder ISM absorption model (for more details, see Tbabs).

\[ \mathcal{M}(E) = f \exp^{-N_{\text{H}}\sigma(E)} + (1-f) \]

Parameters

• \(N_{\text{H}}\) (nh) \(\left[\frac{\text{atoms}~10^{22}}{\text{cm}^2}\right]\) : Equivalent hydrogen column density
• \(f\) (f) \(\left[\text{dimensionless}\right]\) : Partial covering fraction, ranges between 0 and 1

Note

Abundances and cross-sections \(\sigma\) can be found in Wilms et al. (2000).

Source code in src/jaxspec/model/multiplicative.py

class Tbpcf(MultiplicativeComponent):
    r"""
    Partial covering model using the Tuebingen-Boulder ISM absorption model (for more details, see `Tbabs`).

    $$
        \mathcal{M}(E) = f \exp^{-N_{\text{H}}\sigma(E)} + (1-f)
    $$

    !!! abstract "Parameters"
        * $N_{\text{H}}$ (`nh`) $\left[\frac{\text{atoms}~10^{22}}{\text{cm}^2}\right]$ : Equivalent hydrogen column density
        * $f$ (`f`) $\left[\text{dimensionless}\right]$ : Partial covering fraction, ranges between 0 and 1

    !!! note
        Abundances and cross-sections $\sigma$ can be found in Wilms et al. (2000).

    """

    def __init__(self):
        self._energy, self._sigma = _load_xsect("xsect_tbabs_wilm.fits")
        self.nh = nnx.Param(1.0)
        self.f = nnx.Param(0.2)

    def factor(self, energy):
        # Interpolate the shared Wilms table in log-log space.
        sigma = jnp.exp(
            jnp.interp(
                jnp.log(energy),
                jnp.log(self._energy),
                jnp.log(self._sigma),
                left="extrapolate",
                right="extrapolate",
            )
        )

        return self.f * jnp.exp(-self.nh * sigma) + (1.0 - self.f)

Wabs

Bases: MultiplicativeComponent

A photo-electric absorption using Wisconsin (Morrison & McCammon 1983) cross-sections.

Parameters

• \(N_{\text{H}}\) (nh) \(\left[\frac{\text{atoms}~10^{22}}{\text{cm}^2}\right]\) : Equivalent hydrogen column density

Source code in src/jaxspec/model/multiplicative.py

class Wabs(MultiplicativeComponent):
    r"""
    A photo-electric absorption using Wisconsin (Morrison & McCammon 1983) cross-sections.

    ??? abstract "Parameters"
        * $N_{\text{H}}$ (`nh`) $\left[\frac{\text{atoms}~10^{22}}{\text{cm}^2}\right]$ : Equivalent hydrogen column density
    """

    def __init__(self):
        self._energy, self._sigma = _load_xsect("xsect_wabs_angr.fits")
        self.nh = nnx.Param(1.0)

    def factor(self, energy):
        sigma = jnp.interp(energy, self._energy, self._sigma, left=1e9, right=0.0)

        return jnp.exp(-self.nh * sigma)

Zedge

Bases: MultiplicativeComponent

A redshifted absorption edge model.

\[ \mathcal{M}(E) = \begin{cases} \exp \left( -D \left(\frac{E(1+z)}{E_c}\right)^3 \right) & \text{if $E > E_c$}\\ 1 & \text{if $E < E_c$}\end{cases} \]

Parameters

• \(E_c\) (Ec) \(\left[\text{keV}\right]\) : Threshold energy
• \(D\) (D) \(\left[\text{dimensionless}\right]\) : Absorption depth at the threshold
• \(z\) (z) \(\left[\text{dimensionless}\right]\) : Redshift

Source code in src/jaxspec/model/multiplicative.py

class Zedge(MultiplicativeComponent):
    r"""
    A redshifted absorption edge model.

    $$
        \mathcal{M}(E) = \begin{cases} \exp \left( -D \left(\frac{E(1+z)}{E_c}\right)^3 \right)
        & \text{if $E > E_c$}\\ 1 & \text{if $E < E_c$}\end{cases}
    $$

    !!! abstract "Parameters"
        * $E_c$ (`Ec`) $\left[\text{keV}\right]$ : Threshold energy
        * $D$ (`D`) $\left[\text{dimensionless}\right]$ : Absorption depth at the threshold
        * $z$ (`z`) $\left[\text{dimensionless}\right]$ : Redshift
    """

    def __init__(self):
        self.Ec = nnx.Param(1.0)
        self.D = nnx.Param(1.0)
        self.z = nnx.Param(0.0)

    def factor(self, energy):
        return jnp.where(
            energy <= self.Ec,
            1.0,
            jnp.exp(-self.D * (energy * (1 + self.z) / self.Ec) ** 3),
        )

zTbabs

Bases: MultiplicativeComponent

The redshifted Tuebingen-Boulder ISM absorption model. See Tbabs for more details. From Xspec manual: This model assumes that 20% of the hydrogen is molecular and that there is NO MATERIAL IN GRAINS.

\[ \mathcal{M}(E) = \exp^{-N_{\text{H}}\sigma(E)} \]

Parameters

• \(N_{\text{H}}\) (nh) \(\left[\frac{\text{atoms}~10^{22}}{\text{cm}^2}\right]\) : Equivalent hydrogen column density
• \(z\) (z) \(\left[\text{dimensionless}\right]\) : Redshift

Note

Abundances and cross-sections \(\sigma\) can be found in Wilms et al. (2000).

Source code in src/jaxspec/model/multiplicative.py

class zTbabs(MultiplicativeComponent):
    r"""
    The redshifted Tuebingen-Boulder ISM absorption model. See `Tbabs` for more details.
    From Xspec manual:
    This model assumes that 20% of the hydrogen is molecular
    and that there is NO MATERIAL IN GRAINS.

    $$
        \mathcal{M}(E) = \exp^{-N_{\text{H}}\sigma(E)}
    $$

    !!! abstract "Parameters"
        * $N_{\text{H}}$ (`nh`) $\left[\frac{\text{atoms}~10^{22}}{\text{cm}^2}\right]$ : Equivalent hydrogen column density
        * $z$ (`z`) $\left[\text{dimensionless}\right]$ : Redshift


    !!! note
        Abundances and cross-sections $\sigma$ can be found in Wilms et al. (2000).

    """

    def __init__(self):
        self._energy, self._sigma = _load_xsect("xsect_tbabs_wilm.fits")
        self.nh = nnx.Param(1.0)
        self.z = nnx.Param(0.0)

    def factor(self, energy):
        z = jnp.asarray(self.z)
        sigma = jnp.exp(
            jnp.interp(
                jnp.log(energy) + jnp.log1p(z),
                jnp.log(self._energy),
                jnp.log(self._sigma),
                left="extrapolate",
                right="extrapolate",
            )
        )

        return jnp.exp(-self.nh * sigma)