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 \log F - \log E plane is a hyperbola.

The spectrum is given by:

F(E) = A E^{-\Gamma}e^{\eta(E)},

where E is the photon energy in keV, F the photon flux in units of 10^{44} ph \mathrm{s}^{-1}\mathrm{keV}^{-1}, and the function \eta(E) is given by

\eta(E) = \frac{r\xi + \sqrt{r^2\xi^2+b^2(1-r^2)} }{1-r^2},

with \xi \equiv \ln (E/E_0), and E_0, r and b adjustable parameters. For high energies, \xi becomes large and then \eta approaches 2r\xi/(1-r^2), while for low energies \xi approaches -\infty and as a consequence \eta goes to zero. Therefore the break \Delta\Gamma in the spectrum is \Delta\Gamma=2r\xi/(1-r^2). Inverting this we have

r = \frac{\sqrt{1+(\Delta\Gamma)^2} - 1 }{\vert \Delta\Gamma \vert}.

The parameter b gives the distance (in logarithmic units) from the interception point of the asymptotes of the hyperbola to the hyperbola. A value of b=0 therefore means a sharp break, while for larger values of b the break gets smoother.

The simple power law model is obtained by having \Delta\Gamma=0, or the break energy E_0 put to a very large value.

Warning

By default, the allowed range for the photon index \Gamma 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.

Warning

Note the sign of \Gamma: positive values correspond to spectra decreasing with energy. A spectrum with \Delta\Gamma>0 therefore steepens/softens at high energies, for \Delta\Gamma<0 it hardens.

As an extension, we allow for a different normalisation, namely the integrated luminosity L in a given energy band E_1E_2. 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 A is determined is outside your fit range, and the nominal error bars on A can be much larger than the actual flux uncertainty over the fitted range. Note that the parameters E_1 and E_2 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.

Warning

When you do spectral fitting, you must keep either A or L 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.

Warning

The conversion factor between L and A is calculated numerically and not analytically (because of the possible break). In the power law model, photon fluxes above the nominal limit (currently e^{34}=5.8\times 10^{14} 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 \Gamma=2 the results (including conversion factors) for E<10^{-7} keV become inaccurate.

Warning

Note that when you include a break, the value of \Gamma is the photon index at energies below the break. Also, the normalisation A 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 A of the power law, in units of 10^{44} ph \mathrm{s}^{-1} \mathrm{keV}^{-1} at 1 keV. Default value: 1. When \Delta\Gamma is not equal to 0, it is the asymptotic value at 1 keV of the low-energy branch.
gamm : The photon index \Gamma of the spectrum. Default value: 2. When \Delta\Gamma is not equal to 0, it is the slope of the low-energy branch.
dgam : The photon index break \Delta\Gamma of the spectrum. Default value: 0. and frozen. If no break is desired, keep this parameter 0 (and frozen!).
e0 : The break energy E_0 (keV) of the spectrum. Default value: 10^{10} and frozen.
b : Smoothness of the break b. Default: 0.
type : Type of normalisation. Type=0 (default): use A; type=1: use L.
elow : E_1 in keV, the lower limit for the luminosity calculation. Default value: 2 keV.
eupp : E_2 in keV, the upper limit for the luminosity calculation. Default value: 10 keV. Take care that E_2>E_1.
lum : Luminosity L between E_1 and E_2, in units of 10^{30} W.