LogParabola¶

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

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

`alpha`
`amplitude`
`beta`
`e_peak`	Spectral energy distribution peak energy (`Quantity`).
`parameters`	Parameters (`Parameters`)
`reference`

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

alpha¶

amplitude¶

beta¶

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¶

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)[source]¶: 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)[source]¶: 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.