4.1.29. Pow: power law model¶
The power law spectrum as given here is a generalization of a simple power law with the possibility of a break, such that the resultant spectrum in the plane is a hyperbola.
The spectrum is given by:
where is the photon energy in keV, the photon flux in units of ph , and the function is given by
with , and , and adjustable parameters. For high energies, becomes large and then approaches , while for low energies approaches and as a consequence goes to zero. Therefore the break in the spectrum is . Inverting this we have
The parameter gives the distance (in logarithmic units) from the interception point of the asymptotes of the hyperbola to the hyperbola. A value of therefore means a sharp break, while for larger values of the break gets smoother.
The simple power law model is obtained by having , or the break energy put to a very large value.
By default, the allowed range for the photon index is (-10,10). If you manually increase the limits, you may run the risk that SPEX crashes due to overflow for very large photon indices.
Note the sign of : positive values correspond to spectra decreasing with energy. A spectrum with therefore steepens/softens at high energies, for it hardens.
As an extension, we allow for a different normalisation, namely the integrated luminosity in a given energy band –. If you choose this option, the parameter “type” should be set to 1. The reason for introducing this option is that in several cases you may have a spectrum that does not include energies around 1 keV. In that case the energy at which the normalisation is determined is outside your fit range, and the nominal error bars on can be much larger than the actual flux uncertainty over the fitted range. Note that the parameters and act independently from whatever range you specify using the “elim” command. Also, the luminosity is purely the luminosity of the power law, not corrected for any transmission effects that you may have specified in other spectral components.
When you do spectral fitting, you must keep either or a fixed parameter! The other parameter will then be calculated automatically whenever you give the calculate or fit command. SPEX does not check this for you! If you do not do this, you may get unexpected results / crashes.
The conversion factor between and is calculated numerically and not analytically (because of the possible break). In the power law model, photon fluxes above the nominal limit (currently in unscaled units) are put to the maximum value in order to prevent numerical overflow. This implies that you get inaccurate results for low energies, for example for a simple power law with the results (including conversion factors) for keV become inaccurate.
Note that when you include a break, the value of is the photon index at energies below the break. Also, the normalisation is the nominal normalisation of this low-energy part. In such a case of a break, the true flux at 1 keV may be different from the value of A! Of course, you can always calculate the flux in a given band separately.
The parameters of the model are:
norm: Normalisation of the power law, in units of ph at 1 keV. Default value: 1. When is not equal to 0, it is the asymptotic value at 1 keV of the low-energy branch.
gamm: The photon index of the spectrum. Default value: 2. When is not equal to 0, it is the slope of the low-energy branch.
dgam: The photon index break of the spectrum. Default value: 0. and frozen. If no break is desired, keep this parameter 0 (and frozen!).
e0: The break energy (keV) of the spectrum. Default value: and frozen.
b: Smoothness of the break . Default: 0.
type: Type of normalisation. Type (default): use ; type: use .
elow: in keV, the lower limit for the luminosity calculation. Default value: 2 keV.
eupp: in keV, the upper limit for the luminosity calculation. Default value: 10 keV. Take care that .
lum: Luminosity between and , in units of W.