Gaussian Fits

The LSF's are extracted from the simulations in the dispersion and cross-dispersion directions using the same 5.3 $^{\prime\prime}$ region used for the library files. These profiles are then fit with a Gaussian function with the center, width, and normalization free to vary. These fits are done using the IDL fitting routine curvefit. In general the Gaussian fits the central part of the line profile reasonably well but the outer parts of the profile, the wings, are systematically broader than a Gaussian (see figures 6,7). This can be seen in the fits and the $\chi ^2$ plots where very large values of $\chi ^2$ are seen in the outer regions. The absolute value of $\chi ^2$ is determined using only statistical errors of the simulation and ignore any calibration uncertainties for this calculation.

  

Figure: One dimensional on-axis dispersion and cross dispersion profiles of the HEG LSF order -1 at 1.49 keV. The left image is the dispersion profile and the right is the cross-dispersion profile. The lower plots show the $\chi ^2$ plot for the Gaussian fit to the data.

  

Figure: One dimensional on-axis dispersion and cross dispersion profiles of the HEG LSF order -1 at 8.05 keV. The left image is the dispersion profile and the right is the cross-dispersion profile. The lower plots show the $\chi ^2$ plot for the Gaussian fit to the data. The narrow, strong peak to the left of the fitted peak is from the offset mirror shell 6.

Plots of the Gaussian width vs Energy are given in Figure 8, 9, 10. The HEG and MEG data show clear trends of the Gaussian width with energy. The lowest energy (largest dispersion distance) shows the largest width which rapidly decreases to a fairly constant width for the MEG data. The HEG widths also decrease rapidly with increasing energy, show a minimum width at $\sim$ 6.4 keV, and then shows a mild increase up to $\sim$ 10 keV. The LETG widths (figure 10) show a steap decrease with increasing energy up to about 1 keV above which the widths are fairly constant.

Table 3.3.1 shows the resolution ( $\lambda/ \Delta\lambda$) (as defined in HETG Ground Calibration: version 2.0 where $\Delta\lambda = FWHM = 2\sqrt{2ln2}\sigma$) for some of the measured lines shown in figure 8. Not surprisingly, these values agree well with pre-flight design predictions and with earlier predictions (see http://space.mit.edu.ezproxyberklee.flo.org/HETG/res_power/res_power.html).

	MEG		HEG		LEG
Wavelength	$\sigma$	$\lambda/ \Delta\lambda$	$\sigma$	$\lambda/ \Delta\lambda$	$\sigma$	$\lambda/ \Delta\lambda$
Å	$\mu$m		$\mu$m		$\mu$m
67.71	$\cdots$	$\cdots$	$\cdots$	$\cdots$	11.70	1987
44.73	$\cdots$	$\cdots$	$\cdots$	$\cdots$	15.20	1010
23.42	28.8	1570	$\cdots$	$\cdots$	12.62	642
13.32	$\cdots$	$\cdots$	$\cdots$	$\cdots$	11.93	383
8.32	14.	767	20.5	991	11.74	242
3.54	10.9	352	14.8	443	$\cdots$	$\cdots$
2.75	10.5	266	14.4	340	$\cdots$	$\cdots$
1.94	9.65	196	14.1	234	11.74	122
1.54	10.6	146	14.9	177	$\cdots$	$\cdots$
1.24	$\cdots$	$\cdots$	15.5	136	$\cdots$	$\cdots$

 

  

Figure: The dispersion width ($\sigma$) of the Gaussian fits for the HEG and MEG are plotted vs energy and the symbols are the same as in figure 3.

  

Figure: The cross dispersion width ($\sigma$) of the Gaussian fits for the HEG and MEG are plotted vs energy

  

Figure: The width ($\sigma$) of the Gaussian fits for the LEG for the dispersion and cross dispersion are plotted vs energy

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