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evigweight (evigweight-1.8) [xmmsas_20211130_0941-20.0.0]

Spectral analysis

One can then define (via evselect called with withzcolumn=Y withzerrorcolumn=N) a `corrected' spectrum $O'(I)$ within region $Reg$ (usually in sky coordinates) and its associated error $\sigma(O'(I))$:

$\displaystyle O'(I)$ $\textstyle =$ $\displaystyle \sum_j \frac{w_j}{T_{exp}(CCD_j)}
~~~~~~~~(\alpha_j,\delta_j) \in Reg; \;
E_I-\Delta E_I/2 \: < \: E_j \: < \: E_I+\Delta E_I/2$ (3)
$\displaystyle \sigma^2(O'(I))$ $\textstyle =$ $\displaystyle \sum_j \frac{w_j^2}{T^2_{exp}(CCD_j)}$ (4)
where $T_{exp}(CCD_j)$ is the exposure time for the CCD/node where the event was detected and $\Delta E_I$ the width of bin $I$. If the region $Reg$ extends over a single CCD, the exposure time may be taken out of the sums. $O'(I)$ is an estimate of the spectrum one would get if the detector was flat.

In terms of the usual `uncorrected' spectrum:

$\displaystyle O'(I)$ $\textstyle =$ $\displaystyle \langle{w_j}\rangle(I) \: O(I)$ (5)
$\displaystyle \sigma^2(O'(I))$ $\textstyle =$ $\displaystyle \langle{w_j^2}\rangle(I) \: O(I)$ (6)
where the $\langle\rangle$ denote the average over the $O(I)$ photons in bin $I$.

The corresponding source spectrum $S(E)$ is also obtained by summing over the region, and the model spectrum $M'(I)$ (to be compared to the data) by multiplying with the central effective area and applying the response matrix.

$\displaystyle S(E)$ $\textstyle =$ $\displaystyle \sum_{\alpha,\delta \in Reg} s_{\alpha,\delta}(E)$ (7)
$\displaystyle M'(I)$ $\textstyle =$ $\displaystyle \int_e \: R_{Reg}(e,I) \: A_{0,0}(e) \: S(e) \: \mathrm d e$ (8)
Of course the response matrix should be taken (via rmfgen) in the true detector region (not at the center) associated with the sky region $Reg$. The central effective area may be obtained by calling arfgen with special settings:
arfgen arfset=your_arf spectrumset=your_spectrum withbadpixcorr=N modelee=N  \
       withdetbounds=Y filterdss=N detmaptype=flat detxbins=1 detybins=1  \
       withsourcepos=Y sourcecoords=tel sourcex=0 sourcey=0

Model fitting may be performed via XSPEC, entering $O'(I)$ as RATE, and $\sigma(O'(I))$ as STAT_ERR, with $A_{0,0}(e)$ and $R_{Reg}(e,I)$ in the Ancillary Response File and Redistribution Matrix File, respectively.

Note that the weighting procedure is incompatible with using the Poisson model (C-statistic) in XSPEC (the $\chi^2$ formula must be used). This means that care must be taken to have enough counts per spectral bin.

XMM-Newton SOC -- 2021-11-30