# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Spectral models for Gammapy."""
import logging
import operator
import os
import warnings
from pathlib import Path
import numpy as np
import scipy.optimize
import scipy.special
import astropy.units as u
from astropy import constants as const
from astropy.table import Table
from astropy.utils.decorators import classproperty
from astropy.visualization import quantity_support
import matplotlib.pyplot as plt
from gammapy.maps import MapAxis, RegionNDMap
from gammapy.maps.axes import UNIT_STRING_FORMAT
from gammapy.modeling import Parameter, Parameters
from gammapy.utils.deprecation import GammapyDeprecationWarning
from gammapy.utils.integrate import trapz_loglog
from gammapy.utils.interpolation import (
ScaledRegularGridInterpolator,
interpolation_scale,
)
from gammapy.utils.roots import find_roots
from gammapy.utils.scripts import make_path
from .core import ModelBase
log = logging.getLogger(__name__)
__all__ = [
"BrokenPowerLawSpectralModel",
"CompoundSpectralModel",
"ConstantSpectralModel",
"EBLAbsorptionNormSpectralModel",
"ExpCutoffPowerLaw3FGLSpectralModel",
"ExpCutoffPowerLawNormSpectralModel",
"ExpCutoffPowerLawSpectralModel",
"GaussianSpectralModel",
"integrate_spectrum",
"LogParabolaNormSpectralModel",
"LogParabolaSpectralModel",
"NaimaSpectralModel",
"PiecewiseNormSpectralModel",
"PowerLaw2SpectralModel",
"PowerLawNormSpectralModel",
"PowerLawSpectralModel",
"scale_plot_flux",
"ScaleSpectralModel",
"SmoothBrokenPowerLawSpectralModel",
"SpectralModel",
"SuperExpCutoffPowerLaw3FGLSpectralModel",
"SuperExpCutoffPowerLaw4FGLDR3SpectralModel",
"SuperExpCutoffPowerLaw4FGLSpectralModel",
"TemplateSpectralModel",
"TemplateNDSpectralModel",
"EBL_DATA_BUILTIN",
]
EBL_DATA_BUILTIN = {
"franceschini": "$GAMMAPY_DATA/ebl/ebl_franceschini.fits.gz",
"dominguez": "$GAMMAPY_DATA/ebl/ebl_dominguez11.fits.gz",
"finke": "$GAMMAPY_DATA/ebl/frd_abs.fits.gz",
"franceschini17": "$GAMMAPY_DATA/ebl/ebl_franceschini_2017.fits.gz",
"saldana-lopez21": "$GAMMAPY_DATA/ebl/ebl_saldana-lopez_2021.fits.gz",
}
[docs]
def scale_plot_flux(flux, energy_power=0):
"""Scale flux to plot.
Parameters
----------
flux : `Map`
Flux map.
energy_power : int, optional
Power of energy to multiply flux axis with. Default is 0.
Returns
-------
flux : `Map`
Scaled flux map.
"""
energy = flux.geom.get_coord(sparse=True)["energy"]
try:
eunit = [_ for _ in flux.unit.bases if _.physical_type == "energy"][0]
except IndexError:
eunit = energy.unit
y = flux * np.power(energy, energy_power)
return y.to_unit(flux.unit * eunit**energy_power)
[docs]
def integrate_spectrum(func, energy_min, energy_max, ndecade=100):
"""Integrate one-dimensional function using the log-log trapezoidal rule.
Internally an oversampling of the energy bins to "ndecade" is used.
Parameters
----------
func : callable
Function to integrate.
energy_min : `~astropy.units.Quantity`
Integration range minimum.
energy_max : `~astropy.units.Quantity`
Integration range minimum.
ndecade : int, optional
Number of grid points per decade used for the integration.
Default is 100.
"""
# Here we impose to duplicate the number
num = np.maximum(np.max(ndecade * np.log10(energy_max / energy_min)), 2)
energy = np.geomspace(energy_min, energy_max, num=int(num), axis=-1)
integral = trapz_loglog(func(energy), energy, axis=-1)
return integral.sum(axis=0)
[docs]
class SpectralModel(ModelBase):
"""Spectral model base class."""
_type = "spectral"
[docs]
def __call__(self, energy):
kwargs = {par.name: par.quantity for par in self.parameters}
kwargs = self._convert_evaluate_unit(kwargs, energy)
return self.evaluate(energy, **kwargs)
@classproperty
def is_norm_spectral_model(cls):
"""Whether model is a norm spectral model."""
return "Norm" in cls.__name__
@staticmethod
def _convert_evaluate_unit(kwargs_ref, energy):
kwargs = {}
for name, quantity in kwargs_ref.items():
if quantity.unit.physical_type == "energy":
quantity = quantity.to(energy.unit)
kwargs[name] = quantity
return kwargs
def __add__(self, model):
if not isinstance(model, SpectralModel):
model = ConstantSpectralModel(const=model)
return CompoundSpectralModel(self, model, operator.add)
def __mul__(self, other):
if isinstance(other, SpectralModel):
return CompoundSpectralModel(self, other, operator.mul)
else:
raise TypeError(f"Multiplication invalid for type {other!r}")
def __radd__(self, model):
return self.__add__(model)
def __sub__(self, model):
if not isinstance(model, SpectralModel):
model = ConstantSpectralModel(const=model)
return CompoundSpectralModel(self, model, operator.sub)
def __rsub__(self, model):
return self.__sub__(model)
def _propagate_error(self, epsilon, fct, **kwargs):
"""Evaluate error for a given function with uncertainty propagation.
Parameters
----------
fct : `~astropy.units.Quantity`
Function to estimate the error.
epsilon : float
Step size of the gradient evaluation. Given as a
fraction of the parameter error.
**kwargs : dict
Keyword arguments.
Returns
-------
f_cov : `~astropy.units.Quantity`
Error of the given function.
"""
eps = np.sqrt(np.diag(self.covariance)) * epsilon
n, f_0 = len(self.parameters), fct(**kwargs)
shape = (n, len(np.atleast_1d(f_0)))
df_dp = np.zeros(shape)
for idx, parameter in enumerate(self.parameters):
if parameter.frozen or eps[idx] == 0:
continue
parameter.value += eps[idx]
df = fct(**kwargs) - f_0
df_dp[idx] = df.value / eps[idx]
parameter.value -= eps[idx]
f_cov = df_dp.T @ self.covariance @ df_dp
f_err = np.sqrt(np.diagonal(f_cov))
return u.Quantity([np.atleast_1d(f_0.value), f_err], unit=f_0.unit).squeeze()
[docs]
def evaluate_error(self, energy, epsilon=1e-4):
"""Evaluate spectral model with error propagation.
Parameters
----------
energy : `~astropy.units.Quantity`
Energy at which to evaluate.
epsilon : float, optional
Step size of the gradient evaluation. Given as a
fraction of the parameter error. Default is 1e-4.
Returns
-------
dnde, dnde_error : tuple of `~astropy.units.Quantity`
Tuple of flux and flux error.
"""
return self._propagate_error(epsilon=epsilon, fct=self, energy=energy)
@property
def pivot_energy(self):
"""Pivot or decorrelation energy, for a given spectral model calculated numerically.
It is defined as the energy at which the correlation between the spectral parameters is minimized.
Returns
-------
pivot energy : `~astropy.units.Quantity`
The energy at which the statistical error in the computed flux is smallest.
If no minimum is found, NaN will be returned.
"""
x_unit = self.reference.unit
def min_func(x):
"""Function to minimise."""
x = np.exp(x)
dnde, dnde_error = self.evaluate_error(x * x_unit)
return dnde_error / dnde
bounds = [np.log(self.reference.value) - 3, np.log(self.reference.value) + 3]
std = np.std(min_func(x=np.linspace(bounds[0], bounds[1], 100)))
if std < 1e-5:
log.warning(
"The relative error on the flux does not depend on energy. No pivot energy found."
)
return np.nan * x_unit
minimizer = scipy.optimize.minimize_scalar(min_func, bounds=bounds)
if not minimizer.success:
log.warning(
"No minima found in the relative error on the flux. Pivot energy computation failed."
)
return np.nan * x_unit
else:
return np.exp(minimizer.x) * x_unit
[docs]
def integral(self, energy_min, energy_max, **kwargs):
r"""Integrate spectral model numerically if no analytical solution defined.
.. math::
F(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}} \phi(E) dE
Parameters
----------
energy_min, energy_max : `~astropy.units.Quantity`
Lower and upper bound of integration range.
**kwargs : dict
Keyword arguments passed to :func:`~gammapy.modeling.models.spectral.integrate_spectrum`.
"""
if hasattr(self, "evaluate_integral"):
kwargs = {par.name: par.quantity for par in self.parameters}
kwargs = self._convert_evaluate_unit(kwargs, energy_min)
return self.evaluate_integral(energy_min, energy_max, **kwargs)
else:
return integrate_spectrum(self, energy_min, energy_max, **kwargs)
[docs]
def integral_error(self, energy_min, energy_max, epsilon=1e-4, **kwargs):
"""Evaluate the error of the integral flux of a given spectrum in a given energy range.
Parameters
----------
energy_min, energy_max : `~astropy.units.Quantity`
Lower and upper bound of integration range.
epsilon : float, optional
Step size of the gradient evaluation. Given as a
fraction of the parameter error. Default is 1e-4.
Returns
-------
flux, flux_err : tuple of `~astropy.units.Quantity`
Integral flux and flux error between energy_min and energy_max.
"""
return self._propagate_error(
epsilon=epsilon,
fct=self.integral,
energy_min=energy_min,
energy_max=energy_max,
**kwargs,
)
[docs]
def energy_flux(self, energy_min, energy_max, **kwargs):
r"""Compute energy flux in given energy range.
.. math::
G(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}} E \phi(E) dE
Parameters
----------
energy_min, energy_max : `~astropy.units.Quantity`
Lower and upper bound of integration range.
**kwargs : dict
Keyword arguments passed to :func:`~gammapy.modeling.models.spectral.integrate_spectrum`.
"""
def f(x):
return x * self(x)
if hasattr(self, "evaluate_energy_flux"):
kwargs = {par.name: par.quantity for par in self.parameters}
kwargs = self._convert_evaluate_unit(kwargs, energy_min)
return self.evaluate_energy_flux(energy_min, energy_max, **kwargs)
else:
return integrate_spectrum(f, energy_min, energy_max, **kwargs)
[docs]
def energy_flux_error(self, energy_min, energy_max, epsilon=1e-4, **kwargs):
"""Evaluate the error of the energy flux of a given spectrum in a given energy range.
Parameters
----------
energy_min, energy_max : `~astropy.units.Quantity`
Lower and upper bound of integration range.
epsilon : float, optional
Step size of the gradient evaluation. Given as a
fraction of the parameter error. Default is 1e-4.
Returns
-------
energy_flux, energy_flux_err : tuple of `~astropy.units.Quantity`
Energy flux and energy flux error between energy_min and energy_max.
"""
return self._propagate_error(
epsilon=epsilon,
fct=self.energy_flux,
energy_min=energy_min,
energy_max=energy_max,
**kwargs,
)
[docs]
def reference_fluxes(self, energy_axis):
"""Get reference fluxes for a given energy axis.
Parameters
----------
energy_axis : `MapAxis`
Energy axis.
Returns
-------
fluxes : dict of `~astropy.units.Quantity`
Reference fluxes.
"""
energy = energy_axis.center
energy_min, energy_max = energy_axis.edges_min, energy_axis.edges_max
return {
"e_ref": energy,
"e_min": energy_min,
"e_max": energy_max,
"ref_dnde": self(energy),
"ref_flux": self.integral(energy_min, energy_max),
"ref_eflux": self.energy_flux(energy_min, energy_max),
"ref_e2dnde": self(energy) * energy**2,
}
def _get_plot_flux(self, energy, sed_type):
flux = RegionNDMap.create(region=None, axes=[energy])
flux_err = RegionNDMap.create(region=None, axes=[energy])
if sed_type in ["dnde", "norm"]:
flux.quantity, flux_err.quantity = self.evaluate_error(energy.center)
elif sed_type == "e2dnde":
flux.quantity, flux_err.quantity = energy.center**2 * self.evaluate_error(
energy.center
)
elif sed_type == "flux":
flux.quantity, flux_err.quantity = self.integral_error(
energy.edges_min, energy.edges_max
)
elif sed_type == "eflux":
flux.quantity, flux_err.quantity = self.energy_flux_error(
energy.edges_min, energy.edges_max
)
else:
raise ValueError(f"Not a valid SED type: '{sed_type}'")
return flux, flux_err
[docs]
def plot(
self,
energy_bounds,
ax=None,
sed_type="dnde",
energy_power=0,
n_points=100,
**kwargs,
):
"""Plot spectral model curve.
By default a log-log scaling of the axes is used, if you want to change
the y-axis scaling to linear you can use::
>>> from gammapy.modeling.models import ExpCutoffPowerLawSpectralModel
>>> from astropy import units as u
>>> pwl = ExpCutoffPowerLawSpectralModel()
>>> ax = pwl.plot(energy_bounds=(0.1, 100) * u.TeV)
>>> ax.set_yscale('linear')
Parameters
----------
energy_bounds : `~astropy.units.Quantity`
Plot energy bounds passed to MapAxis.from_energy_bounds.
ax : `~matplotlib.axes.Axes`, optional
Matplotlib axes. Default is None.
sed_type : {"dnde", "flux", "eflux", "e2dnde"}
Evaluation methods of the model. Default is "dnde".
energy_power : int, optional
Power of energy to multiply flux axis with. Default is 0.
n_points : int, optional
Number of evaluation nodes. Default is 100.
**kwargs : dict
Keyword arguments forwarded to `~matplotlib.pyplot.plot`.
Returns
-------
ax : `~matplotlib.axes.Axes`, optional
Matplotlib axes.
"""
from gammapy.estimators.map.core import DEFAULT_UNIT
ax = plt.gca() if ax is None else ax
if self.is_norm_spectral_model:
sed_type = "norm"
energy_min, energy_max = energy_bounds
energy = MapAxis.from_energy_bounds(
energy_min,
energy_max,
n_points,
)
if ax.yaxis.units is None:
ax.yaxis.set_units(DEFAULT_UNIT[sed_type] * energy.unit**energy_power)
flux, _ = self._get_plot_flux(sed_type=sed_type, energy=energy)
flux = scale_plot_flux(flux, energy_power=energy_power)
with quantity_support():
ax.plot(energy.center, flux.quantity[:, 0, 0], **kwargs)
self._plot_format_ax(ax, energy_power, sed_type)
return ax
[docs]
def plot_error(
self,
energy_bounds,
ax=None,
sed_type="dnde",
energy_power=0,
n_points=100,
**kwargs,
):
"""Plot spectral model error band.
.. note::
This method calls ``ax.set_yscale("log", nonpositive='clip')`` and
``ax.set_xscale("log", nonposx='clip')`` to create a log-log representation.
The additional argument ``nonposx='clip'`` avoids artefacts in the plot,
when the error band extends to negative values (see also
https://github.com/matplotlib/matplotlib/issues/8623).
When you call ``plt.loglog()`` or ``plt.semilogy()`` explicitly in your
plotting code and the error band extends to negative values, it is not
shown correctly. To circumvent this issue also use
``plt.loglog(nonposx='clip', nonpositive='clip')``
or ``plt.semilogy(nonpositive='clip')``.
Parameters
----------
energy_bounds : `~astropy.units.Quantity`
Plot energy bounds passed to `~gammapy.maps.MapAxis.from_energy_bounds`.
ax : `~matplotlib.axes.Axes`, optional
Matplotlib axes. Default is None.
sed_type : {"dnde", "flux", "eflux", "e2dnde"}
Evaluation methods of the model. Default is "dnde".
energy_power : int, optional
Power of energy to multiply flux axis with. Default is 0.
n_points : int, optional
Number of evaluation nodes. Default is 100.
**kwargs : dict
Keyword arguments forwarded to `matplotlib.pyplot.fill_between`.
Returns
-------
ax : `~matplotlib.axes.Axes`, optional
Matplotlib axes.
"""
from gammapy.estimators.map.core import DEFAULT_UNIT
ax = plt.gca() if ax is None else ax
if self.is_norm_spectral_model:
sed_type = "norm"
energy_min, energy_max = energy_bounds
energy = MapAxis.from_energy_bounds(
energy_min,
energy_max,
n_points,
)
kwargs.setdefault("facecolor", "black")
kwargs.setdefault("alpha", 0.2)
kwargs.setdefault("linewidth", 0)
if ax.yaxis.units is None:
ax.yaxis.set_units(DEFAULT_UNIT[sed_type] * energy.unit**energy_power)
flux, flux_err = self._get_plot_flux(sed_type=sed_type, energy=energy)
y_lo = scale_plot_flux(flux - flux_err, energy_power).quantity[:, 0, 0]
y_hi = scale_plot_flux(flux + flux_err, energy_power).quantity[:, 0, 0]
with quantity_support():
ax.fill_between(energy.center, y_lo, y_hi, **kwargs)
self._plot_format_ax(ax, energy_power, sed_type)
return ax
@staticmethod
def _plot_format_ax(ax, energy_power, sed_type):
ax.set_xlabel(f"Energy [{ax.xaxis.units.to_string(UNIT_STRING_FORMAT)}]")
if energy_power > 0:
ax.set_ylabel(
f"e{energy_power} * {sed_type} [{ax.yaxis.units.to_string(UNIT_STRING_FORMAT)}]"
)
else:
ax.set_ylabel(
f"{sed_type} [{ax.yaxis.units.to_string(UNIT_STRING_FORMAT)}]"
)
ax.set_xscale("log", nonpositive="clip")
ax.set_yscale("log", nonpositive="clip")
[docs]
def spectral_index(self, energy, epsilon=1e-5):
"""Compute spectral index at given energy.
Parameters
----------
energy : `~astropy.units.Quantity`
Energy at which to estimate the index.
epsilon : float, optional
Fractional energy increment to use for determining the spectral index.
Default is 1e-5.
Returns
-------
index : float
Estimated spectral index.
"""
f1 = self(energy)
f2 = self(energy * (1 + epsilon))
return np.log(f1 / f2) / np.log(1 + epsilon)
[docs]
def spectral_index_error(self, energy, epsilon=1e-5):
"""Evaluate the error on spectral index at the given energy.
Parameters
----------
energy : `~astropy.units.Quantity`
Energy at which to estimate the index.
epsilon : float, optional
Fractional energy increment to use for determining the spectral index.
Default is 1e-5.
Returns
-------
index, index_error : tuple of float
Estimated spectral index and its error.
"""
return self._propagate_error(
epsilon=epsilon, fct=self.spectral_index, energy=energy
)
[docs]
def inverse(self, value, energy_min=0.1 * u.TeV, energy_max=100 * u.TeV):
"""Return energy for a given function value of the spectral model.
Calls the `scipy.optimize.brentq` numerical root finding method.
Parameters
----------
value : `~astropy.units.Quantity`
Function value of the spectral model.
energy_min : `~astropy.units.Quantity`, optional
Lower energy bound of the roots finding. Default is 0.1 TeV.
energy_max : `~astropy.units.Quantity`, optional
Upper energy bound of the roots finding. Default is 100 TeV.
Returns
-------
energy : `~astropy.units.Quantity`
Energies at which the model has the given ``value``.
"""
eunit = "TeV"
energy_min = energy_min.to(eunit)
energy_max = energy_max.to(eunit)
def f(x):
# scale by 1e12 to achieve better precision
energy = u.Quantity(x, eunit, copy=False)
y = self(energy).to_value(value.unit)
return 1e12 * (y - value.value)
roots, res = find_roots(f, energy_min, energy_max, points_scale="log")
return roots
[docs]
def inverse_all(self, values, energy_min=0.1 * u.TeV, energy_max=100 * u.TeV):
"""Return energies for multiple function values of the spectral model.
Calls the `scipy.optimize.brentq` numerical root finding method.
Parameters
----------
values : `~astropy.units.Quantity`
Function values of the spectral model.
energy_min : `~astropy.units.Quantity`, optional
Lower energy bound of the roots finding. Default is 0.1 TeV.
energy_max : `~astropy.units.Quantity`, optional
Upper energy bound of the roots finding. Default is 100 TeV.
Returns
-------
energy : list of `~astropy.units.Quantity`
Each element contains the energies at which the model
has corresponding value of ``values``.
"""
energies = []
for val in np.atleast_1d(values):
res = self.inverse(val, energy_min, energy_max)
energies.append(res)
return energies
[docs]
class ConstantSpectralModel(SpectralModel):
r"""Constant model.
For more information see :ref:`constant-spectral-model`.
Parameters
----------
const : `~astropy.units.Quantity`
:math:`k`. Default is 1e-12 cm-2 s-1 TeV-1.
"""
tag = ["ConstantSpectralModel", "const"]
const = Parameter("const", "1e-12 cm-2 s-1 TeV-1")
const._is_norm = True
[docs]
@staticmethod
def evaluate(energy, const):
"""Evaluate the model (static function)."""
return np.ones(np.atleast_1d(energy).shape) * const
[docs]
class CompoundSpectralModel(SpectralModel):
"""Arithmetic combination of two spectral models.
For more information see :ref:`compound-spectral-model`.
"""
tag = ["CompoundSpectralModel", "compound"]
def __init__(self, model1, model2, operator):
self.model1 = model1
self.model2 = model2
self.operator = operator
super().__init__()
@property
def parameters(self):
return self.model1.parameters + self.model2.parameters
def __str__(self):
return (
f"{self.__class__.__name__}\n"
f" Component 1 : {self.model1}\n"
f" Component 2 : {self.model2}\n"
f" Operator : {self.operator.__name__}\n"
)
[docs]
def __call__(self, energy):
val1 = self.model1(energy)
val2 = self.model2(energy)
return self.operator(val1, val2)
[docs]
def to_dict(self, full_output=False):
dict1 = self.model1.to_dict(full_output)
dict2 = self.model2.to_dict(full_output)
return {
self._type: {
"type": self.tag[0],
"model1": dict1["spectral"], # for cleaner output
"model2": dict2["spectral"],
"operator": self.operator.__name__,
}
}
[docs]
def evaluate(self, energy, *args):
args1 = args[: len(self.model1.parameters)]
args2 = args[len(self.model1.parameters) :]
val1 = self.model1.evaluate(energy, *args1)
val2 = self.model2.evaluate(energy, *args2)
return self.operator(val1, val2)
[docs]
@classmethod
def from_dict(cls, data, **kwargs):
from gammapy.modeling.models import SPECTRAL_MODEL_REGISTRY
data = data["spectral"]
model1_cls = SPECTRAL_MODEL_REGISTRY.get_cls(data["model1"]["type"])
model1 = model1_cls.from_dict({"spectral": data["model1"]})
model2_cls = SPECTRAL_MODEL_REGISTRY.get_cls(data["model2"]["type"])
model2 = model2_cls.from_dict({"spectral": data["model2"]})
op = getattr(operator, data["operator"])
return cls(model1, model2, op)
[docs]
class PowerLawSpectralModel(SpectralModel):
r"""Spectral power-law model.
For more information see :ref:`powerlaw-spectral-model`.
Parameters
----------
index : `~astropy.units.Quantity`
:math:`\Gamma`.
Default is 2.0.
amplitude : `~astropy.units.Quantity`
:math:`\phi_0`.
Default is 1e-12 cm-2 s-1 TeV-1.
reference : `~astropy.units.Quantity`
:math:`E_0`.
Default is 1 TeV.
See Also
--------
PowerLaw2SpectralModel, PowerLawNormSpectralModel
"""
tag = ["PowerLawSpectralModel", "pl"]
index = Parameter("index", 2.0)
amplitude = Parameter(
"amplitude",
"1e-12 cm-2 s-1 TeV-1",
scale_method="scale10",
interp="log",
)
amplitude._is_norm = True
reference = Parameter("reference", "1 TeV", frozen=True)
[docs]
@staticmethod
def evaluate(energy, index, amplitude, reference):
"""Evaluate the model (static function)."""
return amplitude * np.power((energy / reference), -index)
[docs]
@staticmethod
def evaluate_integral(energy_min, energy_max, index, amplitude, reference):
r"""Integrate power law analytically (static function).
.. math::
F(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}}\phi(E)dE = \left.
\phi_0 \frac{E_0}{-\Gamma + 1} \left( \frac{E}{E_0} \right)^{-\Gamma + 1}
\right \vert _{E_{min}}^{E_{max}}
Parameters
----------
energy_min, energy_max : `~astropy.units.Quantity`
Lower and upper bound of integration range.
"""
val = -1 * index + 1
prefactor = amplitude * reference / val
upper = np.power((energy_max / reference), val)
lower = np.power((energy_min / reference), val)
integral = prefactor * (upper - lower)
mask = np.isclose(val, 0)
if mask.any():
integral[mask] = (amplitude * reference * np.log(energy_max / energy_min))[
mask
]
return integral
[docs]
@staticmethod
def evaluate_energy_flux(energy_min, energy_max, index, amplitude, reference):
r"""Compute energy flux in given energy range analytically (static function).
.. math::
G(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}}E \phi(E)dE = \left.
\phi_0 \frac{E_0^2}{-\Gamma + 2} \left( \frac{E}{E_0} \right)^{-\Gamma + 2}
\right \vert _{E_{min}}^{E_{max}}
Parameters
----------
energy_min, energy_max : `~astropy.units.Quantity`
Lower and upper bound of integration range.
"""
val = -1 * index + 2
prefactor = amplitude * reference**2 / val
upper = (energy_max / reference) ** val
lower = (energy_min / reference) ** val
energy_flux = prefactor * (upper - lower)
mask = np.isclose(val, 0)
if mask.any():
# see https://www.wolframalpha.com/input/?i=a+*+x+*+(x%2Fb)+%5E+(-2)
# for reference
energy_flux[mask] = (
amplitude * reference**2 * np.log(energy_max / energy_min)[mask]
)
return energy_flux
[docs]
def inverse(self, value, *args):
"""Return energy for a given function value of the spectral model.
Parameters
----------
value : `~astropy.units.Quantity`
Function value of the spectral model.
"""
base = value / self.amplitude.quantity
return self.reference.quantity * np.power(base, -1.0 / self.index.value)
@property
def pivot_energy(self):
r"""The pivot or decorrelation energy is defined as:
.. math::
E_D = E_0 * \exp{cov(\phi_0, \Gamma) / (\phi_0 \Delta \Gamma^2)}
Formula (1) in https://arxiv.org/pdf/0910.4881.pdf
Returns
-------
pivot energy : `~astropy.units.Quantity`
If no minimum is found, NaN will be returned.
"""
index_err = self.index.error
reference = self.reference.quantity
amplitude = self.amplitude.quantity
cov_index_ampl = self.covariance.data[0, 1] * amplitude.unit
return reference * np.exp(cov_index_ampl / (amplitude * index_err**2))
[docs]
class PowerLawNormSpectralModel(SpectralModel):
r"""Spectral power-law model with normalized amplitude parameter.
Parameters
----------
tilt : `~astropy.units.Quantity`
:math:`\Gamma`.
Default is 0.
norm : `~astropy.units.Quantity`
:math:`\phi_0`.
Default is 1.
reference : `~astropy.units.Quantity`
:math:`E_0`.
Default is 1 TeV.
See Also
--------
PowerLawSpectralModel, PowerLaw2SpectralModel
"""
tag = ["PowerLawNormSpectralModel", "pl-norm"]
norm = Parameter("norm", 1, unit="", interp="log")
norm._is_norm = True
tilt = Parameter("tilt", 0, frozen=True)
reference = Parameter("reference", "1 TeV", frozen=True)
[docs]
@staticmethod
def evaluate(energy, tilt, norm, reference):
"""Evaluate the model (static function)."""
return norm * np.power((energy / reference), -tilt)
[docs]
@staticmethod
def evaluate_integral(energy_min, energy_max, tilt, norm, reference):
"""Evaluate powerlaw integral."""
val = -1 * tilt + 1
prefactor = norm * reference / val
upper = np.power((energy_max / reference), val)
lower = np.power((energy_min / reference), val)
integral = prefactor * (upper - lower)
mask = np.isclose(val, 0)
if mask.any():
integral[mask] = (norm * reference * np.log(energy_max / energy_min))[mask]
return integral
[docs]
@staticmethod
def evaluate_energy_flux(energy_min, energy_max, tilt, norm, reference):
"""Evaluate the energy flux (static function)."""
val = -1 * tilt + 2
prefactor = norm * reference**2 / val
upper = (energy_max / reference) ** val
lower = (energy_min / reference) ** val
energy_flux = prefactor * (upper - lower)
mask = np.isclose(val, 0)
if mask.any():
# see https://www.wolframalpha.com/input/?i=a+*+x+*+(x%2Fb)+%5E+(-2)
# for reference
energy_flux[mask] = (
norm * reference**2 * np.log(energy_max / energy_min)[mask]
)
return energy_flux
[docs]
def inverse(self, value, *args):
"""Return energy for a given function value of the spectral model.
Parameters
----------
value : `~astropy.units.Quantity`
Function value of the spectral model.
"""
base = value / self.norm.quantity
return self.reference.quantity * np.power(base, -1.0 / self.tilt.value)
@property
def pivot_energy(self):
r"""The pivot or decorrelation energy is defined as:
.. math::
E_D = E_0 * \exp{cov(\phi_0, \Gamma) / (\phi_0 \Delta \Gamma^2)}
Formula (1) in https://arxiv.org/pdf/0910.4881.pdf
Returns
-------
pivot energy : `~astropy.units.Quantity`
If no minimum is found, NaN will be returned.
"""
tilt_err = self.tilt.error
reference = self.reference.quantity
norm = self.norm.quantity
cov_tilt_norm = self.covariance.data[0, 1] * norm.unit
return reference * np.exp(cov_tilt_norm / (norm * tilt_err**2))
[docs]
class PowerLaw2SpectralModel(SpectralModel):
r"""Spectral power-law model with integral as amplitude parameter.
For more information see :ref:`powerlaw2-spectral-model`.
Parameters
----------
index : `~astropy.units.Quantity`
Spectral index :math:`\Gamma`.
Default is 2.
amplitude : `~astropy.units.Quantity`
Integral flux :math:`F_0`.
Default is 1e-12 cm-2 s-1.
emin : `~astropy.units.Quantity`
Lower energy limit :math:`E_{0, min}`.
Default is 0.1 TeV.
emax : `~astropy.units.Quantity`
Upper energy limit :math:`E_{0, max}`.
Default is 100 TeV.
See Also
--------
PowerLawSpectralModel, PowerLawNormSpectralModel
"""
tag = ["PowerLaw2SpectralModel", "pl-2"]
amplitude = Parameter(
name="amplitude",
value="1e-12 cm-2 s-1",
scale_method="scale10",
interp="log",
)
amplitude._is_norm = True
index = Parameter("index", 2)
emin = Parameter("emin", "0.1 TeV", frozen=True)
emax = Parameter("emax", "100 TeV", frozen=True)
[docs]
@staticmethod
def evaluate(energy, amplitude, index, emin, emax):
"""Evaluate the model (static function)."""
top = -index + 1
# to get the energies dimensionless we use a modified formula
bottom = emax - emin * (emin / emax) ** (-index)
return amplitude * (top / bottom) * np.power(energy / emax, -index)
[docs]
@staticmethod
def evaluate_integral(energy_min, energy_max, amplitude, index, emin, emax):
r"""Integrate power law analytically.
.. math::
F(E_{min}, E_{max}) = F_0 \cdot \frac{E_{max}^{\Gamma + 1} \
- E_{min}^{\Gamma + 1}}{E_{0, max}^{\Gamma + 1} \
- E_{0, min}^{\Gamma + 1}}
Parameters
----------
energy_min, energy_max : `~astropy.units.Quantity`
Lower and upper bound of integration range.
"""
temp1 = np.power(energy_max, -index.value + 1)
temp2 = np.power(energy_min, -index.value + 1)
top = temp1 - temp2
temp1 = np.power(emax, -index.value + 1)
temp2 = np.power(emin, -index.value + 1)
bottom = temp1 - temp2
return amplitude * top / bottom
[docs]
def inverse(self, value, *args):
"""Return energy for a given function value of the spectral model.
Parameters
----------
value : `~astropy.units.Quantity`
Function value of the spectral model.
"""
amplitude = self.amplitude.quantity
index = self.index.value
energy_min = self.emin.quantity
energy_max = self.emax.quantity
# to get the energies dimensionless we use a modified formula
top = -index + 1
bottom = energy_max - energy_min * (energy_min / energy_max) ** (-index)
term = (bottom / top) * (value / amplitude)
return np.power(term.to_value(""), -1.0 / index) * energy_max
[docs]
class BrokenPowerLawSpectralModel(SpectralModel):
r"""Spectral broken power-law model.
For more information see :ref:`broken-powerlaw-spectral-model`.
Parameters
----------
index1 : `~astropy.units.Quantity`
:math:`\Gamma1`.
Default is 2.
index2 : `~astropy.units.Quantity`
:math:`\Gamma2`.
Default is 2.
amplitude : `~astropy.units.Quantity`
:math:`\phi_0`.
Default is 1e-12 cm-2 s-1 TeV-1.
ebreak : `~astropy.units.Quantity`
:math:`E_{break}`.
Default is 1 TeV.
See Also
--------
SmoothBrokenPowerLawSpectralModel
"""
tag = ["BrokenPowerLawSpectralModel", "bpl"]
index1 = Parameter("index1", 2.0)
index2 = Parameter("index2", 2.0)
amplitude = Parameter(
name="amplitude",
value="1e-12 cm-2 s-1 TeV-1",
scale_method="scale10",
interp="log",
)
amplitude._is_norm = True
ebreak = Parameter("ebreak", "1 TeV")
[docs]
@staticmethod
def evaluate(energy, index1, index2, amplitude, ebreak):
"""Evaluate the model (static function)."""
energy = np.atleast_1d(energy)
cond = energy < ebreak
bpwl = amplitude * np.ones(energy.shape)
bpwl[cond] *= (energy[cond] / ebreak) ** (-index1)
bpwl[~cond] *= (energy[~cond] / ebreak) ** (-index2)
return bpwl
[docs]
class SmoothBrokenPowerLawSpectralModel(SpectralModel):
r"""Spectral smooth broken power-law model.
For more information see :ref:`smooth-broken-powerlaw-spectral-model`.
Parameters
----------
index1 : `~astropy.units.Quantity`
:math:`\Gamma_1`. Default is 2.
index2 : `~astropy.units.Quantity`
:math:`\Gamma_2`. Default is 2.
amplitude : `~astropy.units.Quantity`
:math:`\phi_0`. Default is 1e-12 cm-2 s-1 TeV-1.
reference : `~astropy.units.Quantity`
:math:`E_0`. Default is 1 TeV.
ebreak : `~astropy.units.Quantity`
:math:`E_{break}`. Default is 1 TeV.
beta : `~astropy.units.Quantity`
:math:`\beta`. Default is 1.
See Also
--------
BrokenPowerLawSpectralModel
"""
tag = ["SmoothBrokenPowerLawSpectralModel", "sbpl"]
index1 = Parameter("index1", 2.0)
index2 = Parameter("index2", 2.0)
amplitude = Parameter(
name="amplitude",
value="1e-12 cm-2 s-1 TeV-1",
scale_method="scale10",
interp="log",
)
amplitude._is_norm = True
ebreak = Parameter("ebreak", "1 TeV")
reference = Parameter("reference", "1 TeV", frozen=True)
beta = Parameter("beta", 1, frozen=True)
[docs]
@staticmethod
def evaluate(energy, index1, index2, amplitude, ebreak, reference, beta):
"""Evaluate the model (static function)."""
beta *= np.sign(index2 - index1)
pwl = amplitude * (energy / reference) ** (-index1)
brk = (1 + (energy / ebreak) ** ((index2 - index1) / beta)) ** (-beta)
return pwl * brk
[docs]
class PiecewiseNormSpectralModel(SpectralModel):
"""Piecewise spectral correction with a free normalization at each fixed energy nodes.
For more information see :ref:`piecewise-norm-spectral`.
Parameters
----------
energy : `~astropy.units.Quantity`
Array of energies at which the model values are given (nodes).
norms : `~numpy.ndarray` or list of `Parameter`
Array with the initial norms of the model at energies ``energy``.
Normalisation parameters are created for each value.
Default is one at each node.
interp : str
Interpolation scaling in {"log", "lin"}. Default is "log".
"""
tag = ["PiecewiseNormSpectralModel", "piecewise-norm"]
def __init__(self, energy, norms=None, interp="log"):
self._energy = energy
self._interp = interp
if norms is None:
norms = np.ones(len(energy))
if len(norms) != len(energy):
raise ValueError("dimension mismatch")
if len(norms) < 2:
raise ValueError("Input arrays must contain at least 2 elements")
parameters_list = []
if not isinstance(norms[0], Parameter):
parameters_list += [
Parameter(f"norm_{k}", norm) for k, norm in enumerate(norms)
]
else:
parameters_list += norms
self.default_parameters = Parameters(parameters_list)
super().__init__()
@property
def energy(self):
"""Energy nodes."""
return self._energy
@property
def norms(self):
"""Norm values"""
return u.Quantity([p.value for p in self.parameters])
[docs]
def evaluate(self, energy, **norms):
scale = interpolation_scale(scale=self._interp)
e_eval = scale(np.atleast_1d(energy.value))
e_nodes = scale(self.energy.to(energy.unit).value)
v_nodes = scale(self.norms)
log_interp = scale.inverse(np.interp(e_eval, e_nodes, v_nodes))
return log_interp
[docs]
def to_dict(self, full_output=False):
data = super().to_dict(full_output=full_output)
data["spectral"]["energy"] = {
"data": self.energy.data.tolist(),
"unit": str(self.energy.unit),
}
data["spectral"]["interp"] = self._interp
return data
[docs]
@classmethod
def from_dict(cls, data, **kwargs):
"""Create model from dictionary."""
data = data["spectral"]
energy = u.Quantity(data["energy"]["data"], data["energy"]["unit"])
parameters = Parameters.from_dict(data["parameters"])
if "interp" in data:
return cls.from_parameters(parameters, energy=energy, interp=data["interp"])
else:
return cls.from_parameters(parameters, energy=energy)
[docs]
@classmethod
def from_parameters(cls, parameters, **kwargs):
"""Create model from parameters."""
return cls(norms=parameters, **kwargs)
[docs]
class ExpCutoffPowerLawSpectralModel(SpectralModel):
r"""Spectral exponential cutoff power-law model.
For more information see :ref:`exp-cutoff-powerlaw-spectral-model`.
Parameters
----------
index : `~astropy.units.Quantity`
:math:`\Gamma`.
Default is 1.5.
amplitude : `~astropy.units.Quantity`
:math:`\phi_0`.
Default is 1e-12 cm-2 s-1 TeV-1.
reference : `~astropy.units.Quantity`
:math:`E_0`.
Default is 1 TeV.
lambda_ : `~astropy.units.Quantity`
:math:`\lambda`.
Default is 0.1 TeV-1.
alpha : `~astropy.units.Quantity`
:math:`\alpha`.
Default is 1.
See Also
--------
ExpCutoffPowerLawNormSpectralModel
"""
tag = ["ExpCutoffPowerLawSpectralModel", "ecpl"]
index = Parameter("index", 1.5)
amplitude = Parameter(
name="amplitude",
value="1e-12 cm-2 s-1 TeV-1",
scale_method="scale10",
interp="log",
)
amplitude._is_norm = True
reference = Parameter("reference", "1 TeV", frozen=True)
lambda_ = Parameter("lambda_", "0.1 TeV-1")
alpha = Parameter("alpha", "1.0", frozen=True)
[docs]
@staticmethod
def evaluate(energy, index, amplitude, reference, lambda_, alpha):
"""Evaluate the model (static function)."""
pwl = amplitude * (energy / reference) ** (-index)
cutoff = np.exp(-np.power(energy * lambda_, alpha))
return pwl * cutoff
@property
def e_peak(self):
r"""Spectral energy distribution peak energy (`~astropy.units.Quantity`).
This is the peak in E^2 x dN/dE and is given by:
.. math::
E_{Peak} = \left(\frac{2 - \Gamma}{\alpha}\right)^{1/\alpha} / \lambda
"""
reference = self.reference.quantity
index = self.index.quantity
lambda_ = self.lambda_.quantity
alpha = self.alpha.quantity
if index >= 2 or lambda_ == 0.0 or alpha == 0.0:
return np.nan * reference.unit
else:
return np.power((2 - index) / alpha, 1 / alpha) / lambda_
[docs]
class ExpCutoffPowerLawNormSpectralModel(SpectralModel):
r"""Norm spectral exponential cutoff power-law model.
Parameters
----------
index : `~astropy.units.Quantity`
:math:`\Gamma`.
Default is 1.5.
norm : `~astropy.units.Quantity`
:math:`\phi_0`.
Default is 1.
reference : `~astropy.units.Quantity`
:math:`E_0`.
Default is 1 TeV.
lambda_ : `~astropy.units.Quantity`
:math:`\lambda`.
Default is 0.1 TeV-1.
alpha : `~astropy.units.Quantity`
:math:`\alpha`.
Default is 1.
See Also
--------
ExpCutoffPowerLawSpectralModel
"""
tag = ["ExpCutoffPowerLawNormSpectralModel", "ecpl-norm"]
index = Parameter("index", 1.5)
norm = Parameter("norm", 1, unit="", interp="log")
norm._is_norm = True
reference = Parameter("reference", "1 TeV", frozen=True)
lambda_ = Parameter("lambda_", "0.1 TeV-1")
alpha = Parameter("alpha", "1.0", frozen=True)
def __init__(
self, index=None, norm=None, reference=None, lambda_=None, alpha=None, **kwargs
):
if index is None:
warnings.warn(
"The default index value changed from 1.5 to 0 since v1.2",
GammapyDeprecationWarning,
)
if norm is not None:
kwargs.update({"norm": norm})
if index is not None:
kwargs.update({"index": index})
if reference is not None:
kwargs.update({"reference": reference})
if lambda_ is not None:
kwargs.update({"lambda_": lambda_})
if alpha is not None:
kwargs.update({"alpha": alpha})
super().__init__(**kwargs)
[docs]
@staticmethod
def evaluate(energy, index, norm, reference, lambda_, alpha):
"""Evaluate the model (static function)."""
pwl = norm * (energy / reference) ** (-index)
cutoff = np.exp(-np.power(energy * lambda_, alpha))
return pwl * cutoff
[docs]
class ExpCutoffPowerLaw3FGLSpectralModel(SpectralModel):
r"""Spectral exponential cutoff power-law model used for 3FGL.
For more information see :ref:`exp-cutoff-powerlaw-3fgl-spectral-model`.
Parameters
----------
index : `~astropy.units.Quantity`
:math:`\Gamma`.
Default is 1.5.
amplitude : `~astropy.units.Quantity`
:math:`\phi_0`.
Default is 1e-12 cm-2 s-1 TeV-1.
reference : `~astropy.units.Quantity`
:math:`E_0`.
Default is 1 TeV.
ecut : `~astropy.units.Quantity`
:math:`E_{C}`.
Default is 10 TeV.
"""
tag = ["ExpCutoffPowerLaw3FGLSpectralModel", "ecpl-3fgl"]
index = Parameter("index", 1.5)
amplitude = Parameter(
"amplitude",
"1e-12 cm-2 s-1 TeV-1",
scale_method="scale10",
interp="log",
)
amplitude._is_norm = True
reference = Parameter("reference", "1 TeV", frozen=True)
ecut = Parameter("ecut", "10 TeV")
[docs]
@staticmethod
def evaluate(energy, index, amplitude, reference, ecut):
"""Evaluate the model (static function)."""
pwl = amplitude * (energy / reference) ** (-index)
cutoff = np.exp((reference - energy) / ecut)
return pwl * cutoff
[docs]
class SuperExpCutoffPowerLaw3FGLSpectralModel(SpectralModel):
r"""Spectral super exponential cutoff power-law model used for 3FGL.
For more information see :ref:`super-exp-cutoff-powerlaw-3fgl-spectral-model`.
.. math::
\phi(E) = \phi_0 \cdot \left(\frac{E}{E_0}\right)^{-\Gamma_1}
\exp \left( \left(\frac{E_0}{E_{C}} \right)^{\Gamma_2} -
\left(\frac{E}{E_{C}} \right)^{\Gamma_2}
\right)
Parameters
----------
index_1 : `~astropy.units.Quantity`
:math:`\Gamma_1`.
Default is 1.5.
index_2 : `~astropy.units.Quantity`
:math:`\Gamma_2`.
Default is 2.
amplitude : `~astropy.units.Quantity`
:math:`\phi_0`.
Default is 1e-12 cm-2 s-1 TeV-1.
reference : `~astropy.units.Quantity`
:math:`E_0`.
Default is 1 TeV.
ecut : `~astropy.units.Quantity`
:math:`E_{C}`.
Default is 10 TeV.
"""
tag = ["SuperExpCutoffPowerLaw3FGLSpectralModel", "secpl-3fgl"]
amplitude = Parameter(
"amplitude",
"1e-12 cm-2 s-1 TeV-1",
scale_method="scale10",
interp="log",
)
amplitude._is_norm = True
reference = Parameter("reference", "1 TeV", frozen=True)
ecut = Parameter("ecut", "10 TeV")
index_1 = Parameter("index_1", 1.5)
index_2 = Parameter("index_2", 2)
[docs]
@staticmethod
def evaluate(energy, amplitude, reference, ecut, index_1, index_2):
"""Evaluate the model (static function)."""
pwl = amplitude * (energy / reference) ** (-index_1)
cutoff = np.exp((reference / ecut) ** index_2 - (energy / ecut) ** index_2)
return pwl * cutoff
[docs]
class SuperExpCutoffPowerLaw4FGLSpectralModel(SpectralModel):
r"""Spectral super exponential cutoff power-law model used for 4FGL-DR1 (and DR2).
See equation (4) of https://arxiv.org/pdf/1902.10045.pdf
For more information see :ref:`super-exp-cutoff-powerlaw-4fgl-spectral-model`.
Parameters
----------
index_1 : `~astropy.units.Quantity`
:math:`\Gamma_1`. Default is 1.5.
index_2 : `~astropy.units.Quantity`
:math:`\Gamma_2`. Default is 2.
amplitude : `~astropy.units.Quantity`
:math:`\phi_0`. Default is 1e-12 cm-2 s-1 TeV-1.
reference : `~astropy.units.Quantity`
:math:`E_0`. Default is 1 TeV.
expfactor : `~astropy.units.Quantity`
:math:`a`, given as dimensionless value but
internally assumes unit of :math:`E_0^{-\Gamma_2}`.
Default is 1e-2.
"""
tag = ["SuperExpCutoffPowerLaw4FGLSpectralModel", "secpl-4fgl"]
amplitude = Parameter(
"amplitude",
"1e-12 cm-2 s-1 TeV-1",
scale_method="scale10",
interp="log",
)
amplitude._is_norm = True
reference = Parameter("reference", "1 TeV", frozen=True)
expfactor = Parameter("expfactor", "1e-2")
index_1 = Parameter("index_1", 1.5)
index_2 = Parameter("index_2", 2)
[docs]
@staticmethod
def evaluate(energy, amplitude, reference, expfactor, index_1, index_2):
"""Evaluate the model (static function)."""
if isinstance(index_1, u.Quantity):
index_1 = index_1.to_value(u.one)
if isinstance(index_2, u.Quantity):
index_2 = index_2.to_value(u.one)
pwl = amplitude * (energy / reference) ** (-index_1)
cutoff = np.exp(
expfactor / reference.unit**index_2 * (reference**index_2 - energy**index_2)
)
return pwl * cutoff
[docs]
class SuperExpCutoffPowerLaw4FGLDR3SpectralModel(SpectralModel):
r"""Spectral super exponential cutoff power-law model used for 4FGL-DR3.
See equations (2) and (3) of https://arxiv.org/pdf/2201.11184.pdf
For more information see :ref:`super-exp-cutoff-powerlaw-4fgl-dr3-spectral-model`.
Parameters
----------
index_1 : `~astropy.units.Quantity`
:math:`\Gamma_1`. Default is 1.5.
index_2 : `~astropy.units.Quantity`
:math:`\Gamma_2`. Default is 2.
amplitude : `~astropy.units.Quantity`
:math:`\phi_0`. Default is 1e-12 cm-2 s-1 TeV-1.
reference : `~astropy.units.Quantity`
:math:`E_0`. Default is 1 TeV.
expfactor : `~astropy.units.Quantity`
:math:`a`, given as dimensionless value. Default is 1e-2.
"""
tag = ["SuperExpCutoffPowerLaw4FGLDR3SpectralModel", "secpl-4fgl-dr3"]
amplitude = Parameter(
name="amplitude",
value="1e-12 cm-2 s-1 TeV-1",
scale_method="scale10",
interp="log",
)
amplitude._is_norm = True
reference = Parameter("reference", "1 TeV", frozen=True)
expfactor = Parameter("expfactor", "1e-2")
index_1 = Parameter("index_1", 1.5)
index_2 = Parameter("index_2", 2)
[docs]
@staticmethod
def evaluate(energy, amplitude, reference, expfactor, index_1, index_2):
"""Evaluate the model (static function)."""
# https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/source_models.html#PLSuperExpCutoff4
pwl = amplitude * (energy / reference) ** (-index_1)
cutoff = (energy / reference) ** (expfactor / index_2) * np.exp(
expfactor / index_2**2 * (1 - (energy / reference) ** index_2)
)
mask = np.abs(index_2 * np.log(energy / reference)) < 1e-2
ln_ = np.log(energy[mask] / reference)
power = expfactor * (
ln_ / 2.0 + index_2 / 6.0 * ln_**2.0 + index_2**2.0 / 24.0 * ln_**3
)
cutoff[mask] = (energy[mask] / reference) ** power
return pwl * cutoff
[docs]
class LogParabolaSpectralModel(SpectralModel):
r"""Spectral log parabola model.
For more information see :ref:`logparabola-spectral-model`.
Parameters
----------
amplitude : `~astropy.units.Quantity`
:math:`\phi_0`. Default is 1e-12 cm-2 s-1 TeV-1.
reference : `~astropy.units.Quantity`
:math:`E_0`. Default is 10 TeV.
alpha : `~astropy.units.Quantity`
:math:`\alpha`. Default is 2.
beta : `~astropy.units.Quantity`
:math:`\beta`. Default is 1.
See Also
--------
LogParabolaNormSpectralModel
"""
tag = ["LogParabolaSpectralModel", "lp"]
amplitude = Parameter(
"amplitude",
"1e-12 cm-2 s-1 TeV-1",
scale_method="scale10",
interp="log",
)
amplitude._is_norm = True
reference = Parameter("reference", "10 TeV", frozen=True)
alpha = Parameter("alpha", 2)
beta = Parameter("beta", 1)
[docs]
@classmethod
def from_log10(cls, amplitude, reference, alpha, beta):
"""Construct from :math:`log_{10}` parametrization."""
beta_ = beta / np.log(10)
return cls(amplitude=amplitude, reference=reference, alpha=alpha, beta=beta_)
[docs]
@staticmethod
def evaluate(energy, amplitude, reference, alpha, beta):
"""Evaluate the model (static function)."""
xx = energy / reference
exponent = -alpha - beta * np.log(xx)
return amplitude * np.power(xx, exponent)
@property
def e_peak(self):
r"""Spectral energy distribution peak energy (`~astropy.units.Quantity`).
This is the peak in E^2 x dN/dE and is given by:
.. math::
E_{Peak} = E_{0} \exp{ (2 - \alpha) / (2 * \beta)}
"""
reference = self.reference.quantity
alpha = self.alpha.quantity
beta = self.beta.quantity
return reference * np.exp((2 - alpha) / (2 * beta))
[docs]
class LogParabolaNormSpectralModel(SpectralModel):
r"""Norm spectral log parabola model.
Parameters
----------
norm : `~astropy.units.Quantity`
:math:`\phi_0`. Default is 1.
reference : `~astropy.units.Quantity`
:math:`E_0`. Default is 10 TeV.
alpha : `~astropy.units.Quantity`
:math:`\alpha`. Default is 0.
beta : `~astropy.units.Quantity`
:math:`\beta`. Default is 0.
See Also
--------
LogParabolaSpectralModel
"""
tag = ["LogParabolaNormSpectralModel", "lp-norm"]
norm = Parameter("norm", 1, unit="", interp="log")
norm._is_norm = True
reference = Parameter("reference", "10 TeV", frozen=True)
alpha = Parameter("alpha", 0)
beta = Parameter("beta", 0)
def __init__(self, norm=None, reference=None, alpha=None, beta=None, **kwargs):
if alpha is None:
warnings.warn(
"The default alpha value changed from 2 to 0 since v1.2",
GammapyDeprecationWarning,
)
if beta is None:
warnings.warn(
"The default beta value changed from 1 to 0 since v1.2",
GammapyDeprecationWarning,
)
if norm is not None:
kwargs.update({"norm": norm})
if beta is not None:
kwargs.update({"beta": beta})
if reference is not None:
kwargs.update({"reference": reference})
if alpha is not None:
kwargs.update({"alpha": alpha})
super().__init__(**kwargs)
[docs]
@classmethod
def from_log10(cls, norm, reference, alpha, beta):
"""Construct from :math:`log_{10}` parametrization."""
beta_ = beta / np.log(10)
return cls(norm=norm, reference=reference, alpha=alpha, beta=beta_)
[docs]
@staticmethod
def evaluate(energy, norm, reference, alpha, beta):
"""Evaluate the model (static function)."""
xx = energy / reference
exponent = -alpha - beta * np.log(xx)
return norm * np.power(xx, exponent)
[docs]
class TemplateSpectralModel(SpectralModel):
"""A model generated from a table of energy and value arrays.
For more information see :ref:`template-spectral-model`.
Parameters
----------
energy : `~astropy.units.Quantity`
Array of energies at which the model values are given
values : `~numpy.ndarray`
Array with the values of the model at energies ``energy``.
interp_kwargs : dict
Interpolation keyword arguments passed to `scipy.interpolate.RegularGridInterpolator`.
By default all values outside the interpolation range are set to zero.
If you want to apply linear extrapolation you can pass `interp_kwargs={'fill_value':
'extrapolate', 'kind': 'linear'}`. If you want to choose the interpolation
scaling applied to values, you can use `interp_kwargs={"values_scale": "log"}`.
meta : dict, optional
Meta information, meta['filename'] will be used for serialisation.
"""
tag = ["TemplateSpectralModel", "template"]
def __init__(
self,
energy,
values,
interp_kwargs=None,
meta=None,
):
self.energy = energy
self.values = u.Quantity(values, copy=False)
self.meta = {} if meta is None else meta
interp_kwargs = interp_kwargs or {}
interp_kwargs.setdefault("values_scale", "log")
interp_kwargs.setdefault("points_scale", ("log",))
if len(energy) == 1:
interp_kwargs["method"] = "nearest"
self._evaluate = ScaledRegularGridInterpolator(
points=(energy,), values=values, **interp_kwargs
)
super().__init__()
[docs]
@classmethod
def read_xspec_model(cls, filename, param, **kwargs):
"""Read XSPEC table model.
The input is a table containing absorbed values from a XSPEC model as a
function of energy.
TODO: Format of the file should be described and discussed in
https://gamma-astro-data-formats.readthedocs.io/en/latest/index.html
Parameters
----------
filename : str
File containing the XSPEC model.
param : float
Model parameter value.
Examples
--------
Fill table from an EBL model (Franceschini, 2008)
>>> from gammapy.modeling.models import TemplateSpectralModel
>>> filename = '$GAMMAPY_DATA/ebl/ebl_franceschini.fits.gz'
>>> table_model = TemplateSpectralModel.read_xspec_model(filename=filename, param=0.3)
"""
filename = make_path(filename)
# Check if parameter value is in range
table_param = Table.read(filename, hdu="PARAMETERS")
pmin = table_param["MINIMUM"]
pmax = table_param["MAXIMUM"]
if param < pmin or param > pmax:
raise ValueError(f"Out of range: param={param}, min={pmin}, max={pmax}")
# Get energy values
table_energy = Table.read(filename, hdu="ENERGIES")
energy_lo = table_energy["ENERG_LO"]
energy_hi = table_energy["ENERG_HI"]
# set energy to log-centers
energy = np.sqrt(energy_lo * energy_hi)
# Get spectrum values (no interpolation, take closest value for param)
table_spectra = Table.read(filename, hdu="SPECTRA")
idx = np.abs(table_spectra["PARAMVAL"] - param).argmin()
values = u.Quantity(table_spectra[idx][1], "", copy=False) # no dimension
kwargs.setdefault("interp_kwargs", {"values_scale": "lin"})
return cls(energy=energy, values=values, **kwargs)
[docs]
def evaluate(self, energy):
"""Evaluate the model (static function)."""
return self._evaluate((energy,), clip=True)
[docs]
def to_dict(self, full_output=False):
data = super().to_dict(full_output)
data["spectral"]["energy"] = {
"data": self.energy.data.tolist(),
"unit": str(self.energy.unit),
}
data["spectral"]["values"] = {
"data": self.values.data.tolist(),
"unit": str(self.values.unit),
}
return data
[docs]
@classmethod
def from_dict(cls, data, **kwargs):
data = data["spectral"]
energy = u.Quantity(data["energy"]["data"], data["energy"]["unit"])
values = u.Quantity(data["values"]["data"], data["values"]["unit"])
return cls(energy=energy, values=values)
[docs]
@classmethod
def from_region_map(cls, map, **kwargs):
"""Create model from region map."""
energy = map.geom.axes["energy_true"].center
values = map.quantity[:, 0, 0]
return cls(energy=energy, values=values, **kwargs)
[docs]
class TemplateNDSpectralModel(SpectralModel):
"""A model generated from a ND map where extra dimensions define the parameter space.
Parameters
----------
map : `~gammapy.maps.RegionNDMap`
Map template.
meta : dict, optional
Meta information, meta['filename'] will be used for serialisation.
interp_kwargs : dict
Interpolation keyword arguments passed to `gammapy.maps.Map.interp_by_pix`.
Default arguments are {'method': 'linear', 'fill_value': 0}.
"""
tag = ["TemplateNDSpectralModel", "templateND"]
def __init__(self, map, interp_kwargs=None, meta=None, filename=None):
self._map = map.copy()
self.meta = dict() if meta is None else meta
if filename is not None:
filename = str(make_path(filename))
self.filename = filename
parameters = []
has_energy = False
for axis in map.geom.axes:
if axis.name not in ["energy_true", "energy"]:
unit = axis.unit
center = (axis.bounds[1] + axis.bounds[0]) / 2
parameter = Parameter(
name=axis.name,
value=center.to_value(unit),
unit=unit,
scale_method="scale10",
min=axis.bounds[0].to_value(unit),
max=axis.bounds[-1].to_value(unit),
interp=axis.interp,
)
parameters.append(parameter)
else:
has_energy |= True
if not has_energy:
raise ValueError("Invalid map, no energy axis found")
self.default_parameters = Parameters(parameters)
interp_kwargs = interp_kwargs or {}
interp_kwargs.setdefault("values_scale", "log")
self._interp_kwargs = interp_kwargs
super().__init__()
@property
def map(self):
"""Template map as a `~gammapy.maps.RegionNDMap`."""
return self._map
[docs]
def evaluate(self, energy, **kwargs):
coord = {"energy_true": energy}
coord.update(kwargs)
pixels = [0, 0] + [
self.map.geom.axes[key].coord_to_pix(value) for key, value in coord.items()
]
val = self.map.interp_by_pix(pixels, **self._interp_kwargs)
return u.Quantity(val, self.map.unit, copy=False)
[docs]
def write(self, overwrite=False):
"""
Write the map.
Parameters
----------
overwrite: bool, optional
Overwrite existing file.
Default is False, which will raise a warning if the template file exists already.
"""
if self.filename is None:
raise IOError("Missing filename")
elif os.path.isfile(self.filename) and not overwrite:
log.warning("Template file already exits, and overwrite is False")
else:
self.map.write(self.filename)
[docs]
@classmethod
def from_dict(cls, data, **kwargs):
data = data["spectral"]
filename = data["filename"]
m = RegionNDMap.read(filename)
model = cls(m, filename=filename)
for idx, p in enumerate(model.parameters):
par = p.to_dict()
par.update(data["parameters"][idx])
setattr(model, p.name, Parameter(**par))
return model
[docs]
def to_dict(self, full_output=False):
"""Create dictionary for YAML serilisation."""
data = super().to_dict(full_output)
data["spectral"]["filename"] = self.filename
data["spectral"]["unit"] = str(self.map.unit)
return data
[docs]
class ScaleSpectralModel(SpectralModel):
"""Wrapper to scale another spectral model by a norm factor.
Parameters
----------
model : `SpectralModel`
Spectral model to wrap.
norm : float
Multiplicative norm factor for the model value.
Default is 1.
"""
tag = ["ScaleSpectralModel", "scale"]
norm = Parameter("norm", 1, unit="", interp="log")
def __init__(self, model, norm=norm.quantity):
self.model = model
self._covariance = None
super().__init__(norm=norm)
[docs]
def evaluate(self, energy, norm):
return norm * self.model(energy)
[docs]
def integral(self, energy_min, energy_max, **kwargs):
return self.norm.value * self.model.integral(energy_min, energy_max, **kwargs)
[docs]
class EBLAbsorptionNormSpectralModel(SpectralModel):
r"""Gamma-ray absorption models.
For more information see :ref:`absorption-spectral-model`.
Parameters
----------
energy : `~astropy.units.Quantity`
Energy node values.
param : `~astropy.units.Quantity`
Parameter node values.
data : `~astropy.units.Quantity`
Model value.
redshift : float
Redshift of the absorption model.
Default is 0.1.
alpha_norm: float
Norm of the EBL model.
Default is 1.
interp_kwargs : dict
Interpolation option passed to `~gammapy.utils.interpolation.ScaledRegularGridInterpolator`.
By default the models are extrapolated outside the range. To prevent
this and raise an error instead use interp_kwargs = {"extrapolate": False}.
"""
tag = ["EBLAbsorptionNormSpectralModel", "ebl-norm"]
alpha_norm = Parameter("alpha_norm", 1.0, frozen=True)
redshift = Parameter("redshift", 0.1, frozen=True)
def __init__(self, energy, param, data, redshift, alpha_norm, interp_kwargs=None):
self.filename = None
# set values log centers
self.param = param
self.energy = energy
self.data = u.Quantity(data, copy=False)
interp_kwargs = interp_kwargs or {}
interp_kwargs.setdefault("points_scale", ("lin", "log"))
interp_kwargs.setdefault("values_scale", "log")
interp_kwargs.setdefault("extrapolate", True)
self._evaluate_table_model = ScaledRegularGridInterpolator(
points=(self.param, self.energy), values=self.data, **interp_kwargs
)
super().__init__(redshift=redshift, alpha_norm=alpha_norm)
[docs]
def to_dict(self, full_output=False):
data = super().to_dict(full_output=full_output)
param = u.Quantity(self.param)
if self.filename is None:
data["spectral"]["energy"] = {
"data": self.energy.data.tolist(),
"unit": str(self.energy.unit),
}
data["spectral"]["param"] = {
"data": param.data.tolist(),
"unit": str(param.unit),
}
data["spectral"]["values"] = {
"data": self.data.value.tolist(),
"unit": str(self.data.unit),
}
else:
data["spectral"]["filename"] = str(self.filename)
return data
[docs]
@classmethod
def from_dict(cls, data, **kwargs):
data = data["spectral"]
redshift = [p["value"] for p in data["parameters"] if p["name"] == "redshift"][
0
]
alpha_norm = [
p["value"] for p in data["parameters"] if p["name"] == "alpha_norm"
][0]
if "filename" in data:
if os.path.exists(data["filename"]):
return cls.read(
data["filename"], redshift=redshift, alpha_norm=alpha_norm
)
else:
for reference, filename in EBL_DATA_BUILTIN.items():
if Path(filename).stem in data["filename"]:
return cls.read_builtin(
reference, redshift=redshift, alpha_norm=alpha_norm
)
raise IOError(f'File {data["filename"]} not found')
else:
energy = u.Quantity(data["energy"]["data"], data["energy"]["unit"])
param = u.Quantity(data["param"]["data"], data["param"]["unit"])
values = u.Quantity(data["values"]["data"], data["values"]["unit"])
return cls(
energy=energy,
param=param,
data=values,
redshift=redshift,
alpha_norm=alpha_norm,
)
[docs]
@classmethod
def read(cls, filename, redshift=0.1, alpha_norm=1, interp_kwargs=None):
"""Build object from an XSPEC model.
Parameters
----------
filename : str
File containing the model.
redshift : float, optional
Redshift of the absorption model.
Default is 0.1.
alpha_norm: float, optional
Norm of the EBL model.
Default is 1.
interp_kwargs : dict, optional
Interpolation option passed to `~gammapy.utils.interpolation.ScaledRegularGridInterpolator`.
Default is None.
"""
# Create EBL data array
filename = make_path(filename)
table_param = Table.read(filename, hdu="PARAMETERS")
# TODO: for some reason the table contain duplicated values
param, idx = np.unique(table_param[0]["VALUE"], return_index=True)
# Get energy values
table_energy = Table.read(filename, hdu="ENERGIES")
energy_lo = u.Quantity(
table_energy["ENERG_LO"], "keV", copy=False
) # unit not stored in file
energy_hi = u.Quantity(
table_energy["ENERG_HI"], "keV", copy=False
) # unit not stored in file
energy = np.sqrt(energy_lo * energy_hi)
# Get spectrum values
table_spectra = Table.read(filename, hdu="SPECTRA")
data = table_spectra["INTPSPEC"].data[idx, :]
model = cls(
energy=energy,
param=param,
data=data,
redshift=redshift,
alpha_norm=alpha_norm,
interp_kwargs=interp_kwargs,
)
model.filename = filename
return model
[docs]
@classmethod
def read_builtin(
cls, reference="dominguez", redshift=0.1, alpha_norm=1, interp_kwargs=None
):
"""Read from one of the built-in absorption models.
Parameters
----------
reference : {'franceschini', 'dominguez', 'finke'}, optional
Name of one of the available model in gammapy-data. Default is 'dominquez'.
redshift : float, optional
Redshift of the absorption model. Default is 0.1.
alpha_norm : float, optional
Norm of the EBL model. Default is 1.
interp_kwargs : dict, optional
Interpolation keyword arguments. Default is None.
References
----------
.. [1] Franceschini et al. (2008), "Extragalactic optical-infrared background radiation, its time evolution and the cosmic photon-photon opacity", # noqa: E501
`Link <https://ui.adsabs.harvard.edu/abs/2008A%26A...487..837F>`__
.. [2] Dominguez et al. (2011), " Extragalactic background light inferred from AEGIS galaxy-SED-type fractions" # noqa: E501
`Link <https://ui.adsabs.harvard.edu/abs/2011MNRAS.410.2556D>`__
.. [3] Finke et al. (2010), "Modeling the Extragalactic Background Light from Stars and Dust"
`Link <https://ui.adsabs.harvard.edu/abs/2010ApJ...712..238F>`__
.. [4] Franceschini et al. (2017), "The extragalactic background light revisited and the cosmic photon-photon opacity"
`Link <https://ui.adsabs.harvard.edu/abs/2017A%26A...603A..34F/abstract>`__
.. [5] Saldana-Lopez et al. (2021) "An observational determination of the evolving extragalactic background light from the multiwavelength HST/CANDELS survey in the Fermi and CTA era"
`Link <https://ui.adsabs.harvard.edu/abs/2021MNRAS.507.5144S/abstract>`__
"""
return cls.read(
EBL_DATA_BUILTIN[reference],
redshift,
alpha_norm,
interp_kwargs=interp_kwargs,
)
[docs]
def evaluate(self, energy, redshift, alpha_norm):
"""Evaluate model for energy and parameter value."""
absorption = np.clip(self._evaluate_table_model((redshift, energy)), 0, 1)
return np.power(absorption, alpha_norm)
[docs]
class NaimaSpectralModel(SpectralModel):
r"""A wrapper for Naima models.
For more information see :ref:`naima-spectral-model`.
Parameters
----------
radiative_model : `~naima.models.BaseRadiative`
An instance of a radiative model defined in `~naima.models`.
distance : `~astropy.units.Quantity`, optional
Distance to the source. If set to 0, the intrinsic differential
luminosity will be returned. Default is 1 kpc.
seed : str or list of str, optional
Seed photon field(s) to be considered for the `radiative_model` flux computation,
in case of a `~naima.models.InverseCompton` model. It can be a subset of the
`seed_photon_fields` list defining the `radiative_model`. Default is the whole list
of photon fields.
nested_models : dict
Additional parameters for nested models not supplied by the radiative model,
for now this is used only for synchrotron self-compton model.
"""
tag = ["NaimaSpectralModel", "naima"]
def __init__(
self,
radiative_model,
distance=1.0 * u.kpc,
seed=None,
nested_models=None,
use_cache=False,
):
import naima
self.radiative_model = radiative_model
self.radiative_model._memoize = use_cache
self.distance = u.Quantity(distance)
self.seed = seed
if nested_models is None:
nested_models = {}
self.nested_models = nested_models
if isinstance(self.particle_distribution, naima.models.TableModel):
param_names = ["amplitude"]
else:
param_names = self.particle_distribution.param_names
parameters = []
for name in param_names:
value = getattr(self.particle_distribution, name)
parameter = Parameter(name, value)
parameters.append(parameter)
# In case of a synchrotron radiative model, append B to the fittable parameters
if "B" in self.radiative_model.param_names:
value = getattr(self.radiative_model, "B")
parameter = Parameter("B", value)
parameters.append(parameter)
# In case of a synchrotron self compton model, append B and Rpwn to the fittable parameters
if self.include_ssc:
B = self.nested_models["SSC"]["B"]
radius = self.nested_models["SSC"]["radius"]
parameters.append(Parameter("B", B))
parameters.append(Parameter("radius", radius, frozen=True))
self.default_parameters = Parameters(parameters)
self.ssc_energy = np.logspace(-7, 9, 100) * u.eV
super().__init__()
@property
def include_ssc(self):
"""Whether the model includes an SSC component."""
import naima
is_ic_model = isinstance(self.radiative_model, naima.models.InverseCompton)
return is_ic_model and "SSC" in self.nested_models
@property
def ssc_model(self):
"""Synchrotron model."""
import naima
if self.include_ssc:
return naima.models.Synchrotron(
self.particle_distribution,
B=self.B.quantity,
Eemax=self.radiative_model.Eemax,
Eemin=self.radiative_model.Eemin,
)
@property
def particle_distribution(self):
"""Particle distribution."""
return self.radiative_model.particle_distribution
def _evaluate_ssc(
self,
energy,
):
"""
Compute photon density spectrum from synchrotron emission for synchrotron self-compton
model, assuming uniform synchrotron emissivity inside a sphere of radius R (see Section
4.1 of Atoyan & Aharonian 1996).
Based on :
https://naima.readthedocs.io/en/latest/examples.html#crab-nebula-ssc-model
"""
Lsy = self.ssc_model.flux(
self.ssc_energy, distance=0 * u.cm
) # use distance 0 to get luminosity
phn_sy = Lsy / (4 * np.pi * self.radius.quantity**2 * const.c) * 2.24
# The factor 2.24 comes from the assumption on uniform synchrotron
# emissivity inside a sphere
if "SSC" not in self.radiative_model.seed_photon_fields:
self.radiative_model.seed_photon_fields["SSC"] = {
"isotropic": True,
"type": "array",
"energy": self.ssc_energy,
"photon_density": phn_sy,
}
else:
self.radiative_model.seed_photon_fields["SSC"]["photon_density"] = phn_sy
dnde = self.radiative_model.flux(
energy, seed=self.seed, distance=self.distance
) + self.ssc_model.flux(energy, distance=self.distance)
return dnde
def _update_naima_parameters(self, **kwargs):
"""Update Naima model parameters."""
for name, value in kwargs.items():
setattr(self.particle_distribution, name, value)
if "B" in self.radiative_model.param_names:
self.radiative_model.B = self.B.quantity
[docs]
def evaluate(self, energy, **kwargs):
"""Evaluate the model.
Parameters
----------
energy : `~astropy.units.Quantity`
Energy to evaluate the model at.
Returns
-------
dnde : `~astropy.units.Quantity`
Differential flux at given energy.
"""
self._update_naima_parameters(**kwargs)
if self.include_ssc:
dnde = self._evaluate_ssc(energy.flatten())
elif self.seed is not None:
dnde = self.radiative_model.flux(
energy.flatten(), seed=self.seed, distance=self.distance
)
else:
dnde = self.radiative_model.flux(energy.flatten(), distance=self.distance)
dnde = dnde.reshape(energy.shape)
unit = 1 / (energy.unit * u.cm**2 * u.s)
return dnde.to(unit)
[docs]
def to_dict(self, full_output=True):
# for full_output to True otherwise broken
return super().to_dict(full_output=True)
[docs]
@classmethod
def from_dict(cls, data, **kwargs):
raise NotImplementedError(
"Currently the NaimaSpectralModel cannot be read from YAML"
)
[docs]
@classmethod
def from_parameters(cls, parameters, **kwargs):
raise NotImplementedError(
"Currently the NaimaSpectralModel cannot be built from a list of parameters."
)
[docs]
class GaussianSpectralModel(SpectralModel):
r"""Gaussian spectral model.
For more information see :ref:`gaussian-spectral-model`.
Parameters
----------
amplitude : `~astropy.units.Quantity`
:math:`N_0`. Default is 1e-12 cm-2 s-1.
mean : `~astropy.units.Quantity`
:math:`\bar{E}`. Default is 1 TeV.
sigma : `~astropy.units.Quantity`
:math:`\sigma`. Default is 2 TeV.
"""
tag = ["GaussianSpectralModel", "gauss"]
amplitude = Parameter("amplitude", 1e-12 * u.Unit("cm-2 s-1"), interp="log")
amplitude._is_norm = True
mean = Parameter("mean", 1 * u.TeV)
sigma = Parameter("sigma", 2 * u.TeV)
[docs]
@staticmethod
def evaluate(energy, amplitude, mean, sigma):
return (
amplitude
/ (sigma * np.sqrt(2 * np.pi))
* np.exp(-((energy - mean) ** 2) / (2 * sigma**2))
)
[docs]
def integral(self, energy_min, energy_max, **kwargs):
r"""Integrate Gaussian analytically.
.. math::
F(E_{min}, E_{max}) = \frac{N_0}{2} \left[ erf(\frac{E - \bar{E}}{\sqrt{2} \sigma})\right]_{E_{min}}^{E_{max}}
Parameters
----------
energy_min, energy_max : `~astropy.units.Quantity`
Lower and upper bound of integration range
""" # noqa: E501
# kwargs are passed to this function but not used
# this is to get a consistent API with SpectralModel.integral()
u_min = (
(energy_min - self.mean.quantity) / (np.sqrt(2) * self.sigma.quantity)
).to_value("")
u_max = (
(energy_max - self.mean.quantity) / (np.sqrt(2) * self.sigma.quantity)
).to_value("")
return (
self.amplitude.quantity
/ 2
* (scipy.special.erf(u_max) - scipy.special.erf(u_min))
)
[docs]
def energy_flux(self, energy_min, energy_max):
r"""Compute energy flux in given energy range analytically.
.. math::
G(E_{min}, E_{max}) = \frac{N_0 \sigma}{\sqrt{2*\pi}}* \left[ - \exp(\frac{E_{min}-\bar{E}}{\sqrt{2} \sigma})
\right]_{E_{min}}^{E_{max}} + \frac{N_0 * \bar{E}}{2} \left[ erf(\frac{E - \bar{E}}{\sqrt{2} \sigma})
\right]_{E_{min}}^{E_{max}}
Parameters
----------
energy_min, energy_max : `~astropy.units.Quantity`
Lower and upper bound of integration range.
""" # noqa: E501
u_min = (
(energy_min - self.mean.quantity) / (np.sqrt(2) * self.sigma.quantity)
).to_value("")
u_max = (
(energy_max - self.mean.quantity) / (np.sqrt(2) * self.sigma.quantity)
).to_value("")
a = self.amplitude.quantity * self.sigma.quantity / np.sqrt(2 * np.pi)
b = self.amplitude.quantity * self.mean.quantity / 2
return a * (np.exp(-(u_min**2)) - np.exp(-(u_max**2))) + b * (
scipy.special.erf(u_max) - scipy.special.erf(u_min)
)
