Fitting a Model of the C-K edge

I extracted the observed count spectrum, n(E), for the PKS 2155-304 observation in June 2002 (obs ID 3669). Excluding the 0.28-1.0 keV region from the 2PL model fit gives N(E). The optical depth due to absorption is computed as -ln[n(E)/N(E)], plotted in Fig. 1.

Fig. 1. The optical depth of the contaminant as a function of energy, derived by adaptively binning the observed spectrum, n(E), fitting to a 2PL model, N(E), and computing tau(E) = -ln( n[E]/N[E] ). There are several features to notice. There is a detectable opacity beteween 0.285 keV and 0.287 keV that was not obvious previously. After the edge at 0.287 keV, there is a monotonic decline except for some possible structure near 0.5 keV (near the O-K edge) and 0.7 keV (where the F-K edge is).

The data from Fig. 1 were further rebinned in order to model the overall shape of the absorption (Fig. 2).

Fig. 2: Similar to the bottom half of Fig. 1 except the data have been rebinned to reduce statistical noise and only the region being fit for EXAFS is shown. The Henke model of the C-K edge (short dashed line) is a good match to the data in th 0.38 to 0.48 keV range, so the Henke optical constants are used above 0.48 keV. Below 0.38 and down to the edge, an damped oscillation adjustment is used.

Figure 2 shows the new model for the C-K edge that starts with the Henke constants from CXRO except that the optical depth was multiplied by a damped ripple adjustment factor:

1 + A exp( -x/dampscale ) * cos( freq * x )

where x = (lam_ck - lambda) / lam_ck. The constants of the adjustment were estimated (chi-by-eye): A = 0.4, dampscale = 4./lam_ck, and freq = 2 ¼ lam_ck/16. These provided a sharp increase in the opacity just shortward of the edge which damps out so that it is asympotically equivalent to the form from the CXRO web site. The oscillatory term accounts for the 'ripple' of the deviation.

Finally, the NEXAFS feature near 286 eV was added to the model as a new shallow edge. The new model of the C-K edge is folded with the instrument response and shown against the PKS 2155-304 data in Fig. 3.

Fig. 3: Comparison of the new C-K edge model with data. The top panel shows the ratio of the data to the model without including contamination so that the near edge structure is apparent and the new model is overlaid. The remaining panels show the model with the new absorption model folded with the effective area. The data fit the new model well with a slight systematic offset starting near 26 Å where there is a jump where one side goes from a BI chip to a FI chip, so the systematic deviation from 20 to 26 Å is primarily due to a slight error in the correction to bring the QEs of these chips into agreement. One can also begin to see the 18 Å edge due to F-K that is modeled using a Mk 421 observation and was ignored when fitting this C-K edge model.

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Herman Marshall
hermanm@space.mit.edu
Last updated July 6, 2003