# PowerLaw¶

class gammapy.spectrum.models.PowerLaw(index=2.0, amplitude='1e-12 cm-2 s-1 TeV-1', reference='1 TeV')[source]

Spectral power-law model.

$\phi(E) = \phi_0 \cdot \left( \frac{E}{E_0} \right)^{-\Gamma}$
Parameters: index : Quantity $$\Gamma$$ amplitude : Quantity $$\phi_0$$ reference : Quantity $$E_0$$

Examples

This is how to plot the default PowerLaw model:

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

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


Attributes Summary

 amplitude index parameters Parameters (Parameters) reference

Methods Summary

 __call__(self, energy) Call evaluate method of derived classes copy(self) A deep copy. energy_flux(self, emin, emax) Compute energy flux in given energy range analytically. energy_flux_error(self, emin, emax, \*\*kwargs) Compute energy flux in given energy range analytically with error propagation. evaluate(energy, index, amplitude, reference) Evaluate the model (static function). evaluate_error(self, energy) Evaluate spectral model with error propagation. from_dict(val) Create from dict. integral(self, emin, emax, \*\*kwargs) Integrate power law analytically. integral_error(self, emin, emax, \*\*kwargs) Integrate power law analytically with error propagation. inverse(self, value) 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) Convert to dict.

Attributes Documentation

amplitude
index
parameters

Parameters (Parameters)

reference

Methods Documentation

__call__(self, energy)

Call evaluate method of derived classes

copy(self)

A deep copy.

energy_flux(self, emin, emax)[source]

Compute energy flux in given energy range analytically.

$G(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}}E \phi(E)dE = \left. \phi_0 \frac{E_0^2}{-\Gamma + 2} \left( \frac{E}{E_0} \right)^{-\Gamma + 2} \right \vert _{E_{min}}^{E_{max}}$
Parameters: emin, emax : Quantity Lower and upper bound of integration range.
energy_flux_error(self, emin, emax, **kwargs)[source]

Compute energy flux in given energy range analytically with error propagation.

Parameters: emin, emax : Quantity Lower and upper bound of integration range. energy_flux, energy_flux_error : tuple of Quantity Tuple of energy flux and energy flux error.
static evaluate(energy, index, amplitude, reference)[source]

Evaluate the model (static function).

evaluate_error(self, energy)

Evaluate spectral model with error propagation.

Parameters: energy : Quantity Energy at which to evaluate flux, flux_error : tuple of Quantity Tuple of flux and flux error.
classmethod from_dict(val)

Create from dict.

integral(self, emin, emax, **kwargs)[source]

Integrate power law analytically.

$F(E_{min}, E_{max}) = \int_{E_{min}}^{E_{max}}\phi(E)dE = \left. \phi_0 \frac{E_0}{-\Gamma + 1} \left( \frac{E}{E_0} \right)^{-\Gamma + 1} \right \vert _{E_{min}}^{E_{max}}$
Parameters: emin, emax : Quantity Lower and upper bound of integration range
integral_error(self, emin, emax, **kwargs)[source]

Integrate power law analytically with error propagation.

Parameters: emin, emax : Quantity Lower and upper bound of integration range. integral, integral_error : tuple of Quantity Tuple of integral flux and integral flux error.
inverse(self, value)[source]

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

Parameters: value : Quantity Function value of the spectral model.
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.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 ax : Axes, 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: 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 ax : Axes, optional Axis
spectral_index(self, 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. index : float Estimated spectral index.
to_dict(self)

Convert to dict.