LogParabola

class gammapy.spectrum.models.LogParabola(amplitude='1e-12 cm-2 s-1 TeV-1', reference='10 TeV', alpha=2, beta=1)

Bases: gammapy.spectrum.models.SpectralModel

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:

amplitude : Quantity

\(\phi_0\)

reference : Quantity

\(E_0\)

alpha : Quantity

\(\alpha\)

beta : Quantity

\(\beta\)

Examples

This is how to plot the default LogParabola model:

from astropy import units as u
from gammapy.spectrum.models import LogParabola

log_parabola = LogParabola()
log_parabola.plot(energy_range=[0.1, 100] * u.TeV)
plt.show()

Attributes Summary

e_peak

Spectral energy distribution peak energy (Quantity).

Methods Summary

__call__(energy)	Call evaluate method of derived classes
copy()	A deep copy.
energy_flux(emin, emax, **kwargs)	Compute energy flux in given energy range.
energy_flux_error(emin, emax, **kwargs)	Compute energy flux in given energy range with error propagation.
evaluate(energy, amplitude, reference, …)	Evaluate the model (static function).
evaluate_error(energy)	Evaluate spectral model with error propagation.
from_dict(val)	Create from dict.
from_log10(amplitude, reference, alpha, beta)	Construct LogParabola from \(log_{10}\) parametrization
integral(emin, emax, **kwargs)	Integrate spectral model numerically.
integral_error(emin, emax, **kwargs)	Integrate spectral model numerically with error propagation.
inverse(value[, emin, emax])	Return energy for a given function value of the spectral model.
plot(energy_range[, ax, energy_unit, …])	Plot spectral model curve.
plot_error(energy_range[, ax, energy_unit, …])	Plot spectral model error band.
spectral_index(energy[, epsilon])	Compute spectral index at given energy.
to_dict()	Convert to dict.

Attributes Documentation

e_peak

Spectral energy distribution peak energy (Quantity).

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

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

Methods Documentation

__call__(energy)

Call evaluate method of derived classes

copy()

A deep copy.

energy_flux(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, emax : Quantity

Lower and upper bound of integration range.

**kwargs : dict

Keyword arguments passed to func:integrate_spectrum

energy_flux_error(emin, emax, **kwargs)

Compute energy flux in given energy range with error propagation.

\[G(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}} E \phi(E) dE\]

Parameters:

emin, emax : Quantity

Lower bound of integration range.

**kwargs : dict

Keyword arguments passed to integrate_spectrum()

Returns:

energy_flux, energy_flux_error : tuple of Quantity

Tuple of energy flux and energy flux error.

static evaluate(energy, amplitude, reference, alpha, beta)

Evaluate the model (static function).

evaluate_error(energy)

Evaluate spectral model with error propagation.

Parameters:

energy : Quantity

Energy at which to evaluate

Returns:

flux, flux_error : tuple of Quantity

Tuple of flux and flux error.

classmethod from_dict(val)

Create from dict.

classmethod from_log10(amplitude, reference, alpha, beta)

Construct LogParabola from \(log_{10}\) parametrization

integral(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, emax : Quantity

Lower and upper bound of integration range.

**kwargs : dict

Keyword arguments passed to integrate_spectrum()

integral_error(emin, emax, **kwargs)

Integrate spectral model numerically with error propagation.

Parameters:

emin, emax : Quantity

Lower adn upper bound of integration range.

**kwargs : dict

Keyword arguments passed to func:integrate_spectrum

Returns:

integral, integral_error : tuple of Quantity

Tuple of integral flux and integral flux error.

inverse(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:

value : Quantity

Function value of the spectral model.

emin : Quantity

Lower bracket value in case solution is not unique.

emax : Quantity

Upper bracket value in case solution is not unique.

Returns:

energy : Quantity

Energies at which the model has the given value.

plot(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.spectrum.models import ExponentialCutoffPowerLaw
from astropy import units as u

pwl = ExponentialCutoffPowerLaw()
ax = pwl.plot(energy_range=(0.1, 100) * u.TeV)
ax.set_yscale('linear')

Parameters:

ax : Axes, optional

Axis

energy_range : Quantity

Plot range

energy_unit : str, Unit, optional

Unit of the energy axis

flux_unit : str, Unit, optional

Unit of the flux axis

energy_power : int, optional

Power of energy to multiply flux axis with

n_points : int, optional

Number of evaluation nodes

Returns:

ax : Axes, optional

Axis

plot_error(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:

ax : Axes, optional

Axis

energy_range : Quantity

Plot range

energy_unit : str, Unit, optional

Unit of the energy axis

flux_unit : str, Unit, optional

Unit of the flux axis

energy_power : int, optional

Power of energy to multiply flux axis with

n_points : int, optional

Number of evaluation nodes

**kwargs : dict

Keyword arguments forwarded to matplotlib.pyplot.fill_between

Returns:

ax : Axes, optional

Axis

spectral_index(energy, epsilon=1e-05)

Compute spectral index at given energy.

Parameters:

energy : Quantity

Energy at which to estimate the index

epsilon : float

Fractional energy increment to use for determining the spectral index.

Returns:

index : float

Estimated spectral index.

to_dict()

Convert to dict.