# LogParabolaSpectralModel¶

class gammapy.modeling.models.LogParabolaSpectralModel(**kwargs)[source]

Spectral log parabola model.

$\phi(E) = \phi_0 \left( \frac{E}{E_0} \right) ^ { - \alpha - \beta \log{ \left( \frac{E}{E_0} \right) } }$

Note that $$log$$ refers to the natural logarithm. This is consistent with the Fermi Science Tools and ctools. The Sherpa package, however, uses $$log_{10}$$. If you have parametrization based on $$log_{10}$$ you can use the from_log10() method.

Parameters
amplitudeQuantity

$$\phi_0$$

referenceQuantity

$$E_0$$

alphaQuantity

$$\alpha$$

betaQuantity

$$\beta$$

Attributes Summary

 alpha A model parameter. amplitude A model parameter. beta A model parameter. default_parameters e_peak Spectral energy distribution peak energy (Quantity). parameters Parameters (Parameters) reference A model parameter. tag

Methods Summary

 __call__(self, energy) Call self as a function. copy(self) A deep copy. create(tag, \*args, \*\*kwargs) Create a model instance. energy_flux(self, emin, emax, \*\*kwargs) Compute energy flux in given energy range. evaluate(energy, amplitude, reference, …) Evaluate the model (static function). evaluate_error(self, energy[, epsilon]) Evaluate spectral model with error propagation. from_dict(data) from_log10(amplitude, reference, alpha, beta) Construct from $$log_{10}$$ parametrization. integral(self, emin, emax, \*\*kwargs) Integrate spectral model numerically. inverse(self, value[, emin, emax]) Return energy for a given function value of the spectral model. plot(self, energy_range[, ax, energy_unit, …]) Plot spectral model curve. plot_error(self, energy_range[, ax, …]) Plot spectral model error band. spectral_index(self, energy[, epsilon]) Compute spectral index at given energy. to_dict(self) Create dict for YAML serialisation

Attributes Documentation

alpha

A model parameter.

Note that the parameter value has been split into a factor and scale like this:

value = factor x scale


Users should interact with the value, quantity or min and max properties and consider the fact that there is a factor and scale an implementation detail.

That was introduced for numerical stability in parameter and error estimation methods, only in the Gammapy optimiser interface do we interact with the factor, factor_min and factor_max properties, i.e. the optimiser “sees” the well-scaled problem.

Parameters
namestr

Name

factorfloat or Quantity

Factor

scalefloat, optional

Scale (sometimes used in fitting)

unitUnit or str, optional

Unit

minfloat, optional

Minimum (sometimes used in fitting)

maxfloat, optional

Maximum (sometimes used in fitting)

frozenbool, optional

Frozen? (used in fitting)

amplitude

A model parameter.

Note that the parameter value has been split into a factor and scale like this:

value = factor x scale


Users should interact with the value, quantity or min and max properties and consider the fact that there is a factor and scale an implementation detail.

That was introduced for numerical stability in parameter and error estimation methods, only in the Gammapy optimiser interface do we interact with the factor, factor_min and factor_max properties, i.e. the optimiser “sees” the well-scaled problem.

Parameters
namestr

Name

factorfloat or Quantity

Factor

scalefloat, optional

Scale (sometimes used in fitting)

unitUnit or str, optional

Unit

minfloat, optional

Minimum (sometimes used in fitting)

maxfloat, optional

Maximum (sometimes used in fitting)

frozenbool, optional

Frozen? (used in fitting)

beta

A model parameter.

Note that the parameter value has been split into a factor and scale like this:

value = factor x scale


Users should interact with the value, quantity or min and max properties and consider the fact that there is a factor and scale an implementation detail.

That was introduced for numerical stability in parameter and error estimation methods, only in the Gammapy optimiser interface do we interact with the factor, factor_min and factor_max properties, i.e. the optimiser “sees” the well-scaled problem.

Parameters
namestr

Name

factorfloat or Quantity

Factor

scalefloat, optional

Scale (sometimes used in fitting)

unitUnit or str, optional

Unit

minfloat, optional

Minimum (sometimes used in fitting)

maxfloat, optional

Maximum (sometimes used in fitting)

frozenbool, optional

Frozen? (used in fitting)

default_parameters = <gammapy.modeling.parameter.Parameters object>
e_peak

Spectral energy distribution peak energy (Quantity).

This is the peak in E^2 x dN/dE and is given by:

$E_{Peak} = E_{0} \exp{ (2 - \alpha) / (2 * \beta)}$
parameters

Parameters (Parameters)

reference

A model parameter.

Note that the parameter value has been split into a factor and scale like this:

value = factor x scale


Users should interact with the value, quantity or min and max properties and consider the fact that there is a factor and scale an implementation detail.

That was introduced for numerical stability in parameter and error estimation methods, only in the Gammapy optimiser interface do we interact with the factor, factor_min and factor_max properties, i.e. the optimiser “sees” the well-scaled problem.

Parameters
namestr

Name

factorfloat or Quantity

Factor

scalefloat, optional

Scale (sometimes used in fitting)

unitUnit or str, optional

Unit

minfloat, optional

Minimum (sometimes used in fitting)

maxfloat, optional

Maximum (sometimes used in fitting)

frozenbool, optional

Frozen? (used in fitting)

tag = 'LogParabolaSpectralModel'

Methods Documentation

__call__(self, energy)

Call self as a function.

copy(self)

A deep copy.

static create(tag, *args, **kwargs)

Create a model instance.

Examples

>>> from gammapy.modeling import Model
>>> spectral_model = Model.create("PowerLaw2SpectralModel", amplitude="1e-10 cm-2 s-1", index=3)
>>> type(spectral_model)
gammapy.modeling.models.spectral.PowerLaw2SpectralModel

energy_flux(self, emin, emax, **kwargs)

Compute energy flux in given energy range.

$G(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}} E \phi(E) dE$
Parameters
emin, emaxQuantity

Lower and upper bound of integration range.

**kwargsdict

Keyword arguments passed to func:integrate_spectrum

static evaluate(energy, amplitude, reference, alpha, beta)[source]

Evaluate the model (static function).

evaluate_error(self, energy, epsilon=0.0001)

Evaluate spectral model with error propagation.

Parameters
energyQuantity

Energy at which to evaluate

epsilonfloat

Step size of the gradient evaluation. Given as a fraction of the parameter error.

Returns
dnde, dnde_errortuple of Quantity

Tuple of flux and flux error.

classmethod from_dict(data)
classmethod from_log10(amplitude, reference, alpha, beta)[source]

Construct from $$log_{10}$$ parametrization.

integral(self, emin, emax, **kwargs)

Integrate spectral model numerically.

$F(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}} \phi(E) dE$

If array input for emin and emax is given you have to set intervals=True if you want the integral in each energy bin.

Parameters
emin, emaxQuantity

Lower and upper bound of integration range.

**kwargsdict

Keyword arguments passed to integrate_spectrum()

inverse(self, value, emin=<Quantity 0.1 TeV>, emax=<Quantity 100. TeV>)

Return energy for a given function value of the spectral model.

Calls the scipy.optimize.brentq numerical root finding method.

Parameters
valueQuantity

Function value of the spectral model.

eminQuantity

Lower bracket value in case solution is not unique.

emaxQuantity

Upper bracket value in case solution is not unique.

Returns
energyQuantity

Energies at which the model has the given value.

plot(self, energy_range, ax=None, energy_unit='TeV', flux_unit='cm-2 s-1 TeV-1', energy_power=0, n_points=100, **kwargs)

Plot spectral model curve.

kwargs are forwarded to matplotlib.pyplot.plot

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_range=(0.1, 100) * u.TeV)
ax.set_yscale('linear')

Parameters
axAxes, optional

Axis

energy_rangeQuantity

Plot range

energy_unitstr, Unit, optional

Unit of the energy axis

flux_unitstr, Unit, optional

Unit of the flux axis

energy_powerint, optional

Power of energy to multiply flux axis with

n_pointsint, optional

Number of evaluation nodes

Returns
axAxes, optional

Axis

plot_error(self, energy_range, ax=None, energy_unit='TeV', flux_unit='cm-2 s-1 TeV-1', energy_power=0, n_points=100, **kwargs)

Plot spectral model error band.

Note

This method calls ax.set_yscale("log", nonposy='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() explicitely 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', nonposy='clip') or plt.semilogy(nonposy='clip').

Parameters
axAxes, optional

Axis

energy_rangeQuantity

Plot range

energy_unitstr, Unit, optional

Unit of the energy axis

flux_unitstr, Unit, optional

Unit of the flux axis

energy_powerint, optional

Power of energy to multiply flux axis with

n_pointsint, optional

Number of evaluation nodes

**kwargsdict

Keyword arguments forwarded to matplotlib.pyplot.fill_between

Returns
axAxes, optional

Axis

spectral_index(self, energy, epsilon=1e-05)

Compute spectral index at given energy.

Parameters
energyQuantity

Energy at which to estimate the index

epsilonfloat

Fractional energy increment to use for determining the spectral index.

Returns
indexfloat

Estimated spectral index.

to_dict(self)

Create dict for YAML serialisation