2.2.3 In-flight Point Spread Functions (PSF)

The PSF describes the surface brightness distribution of imaged point sources. There are three significant components: the XRT PSF, the HRI PSF, and a residual blurring attributable to the aspect correction (often discernible as a small ellipsoidal component).

Although the XRT PSF has a central core with a FWHM of 3''(as determined by ground test), microroughness in the mirror surfaces leads to a slight but measurable scattering of X rays and produces a small energy-dependent tail or wing in the PSF. As determined by ground test measurements, the fraction of imaged photons in the tail (beyond 10'') of the image of a point source is given in Table 1.

**Table 1:** Energy dependence of the mirror-scattering
E	% photons outside
(keV)	a 10'' radius
0.18	0.7
0.28	1.3
0.93	6.0
1.49	11.0
1.70	13.3

The HRI PSF has a central core with a FWHM of 1.7'' (as determined in the laboratory). In the absence of the electrostatic shield, the PSF has a near-Gaussian shape, but the electrostatic shield (introduced to reduce the background count rate) has broadened the PSF, creating a so-called ``halo''. X-rays interacting in the interchannel web in front of the MCPs produce electrons that are ``captured'' in microchannel pores a considerable distance from the initial event site, initiating an electron avalanche leading to a false position. Although the region between the shield and the MCP is, to first order, field free, fringe electrostatic fields deflect the web electrons back into the MCP. The magnitude of the effect is predicted to decrease with increasing energy, whereas mirror scattering increases with energy.

The in-flight, on-axis PSF, which includes aspect solution effects, has been determined from a 9.1 ks exposure of HZ43 which yielded 109,000 net counts. Figure 1 shows the azimuthally averaged surface brightness profile of this observation. The HRI halo is obvious in Figure 1 and dominates the PSF beyond approximately 15''. The solid curve is a parametric representation of the in-flight, on-axis, PSF and consists of the sum of two Gaussians and an exponential. This functional form for the azimuthally averaged PSF, provides a good description of the PSF out to a radius of 2', and is given by:

Note the arbitrary normalization of the above expression for the azimuthally-averaged surface brightness. Figure 1 shows that there is an additional component in the PSF beyond 2', but this component will be well below the background in most observations. We have not fit this additional component, since we do not know at present how strongly it depends on energy.

The azimuthally averaged surface brightness for on-axis observations of HZ43, AR Lac, and LMC X-1 is shown in Figure 2. Since these sources represent a wide range of mean X-ray energies, this figure demonstrates that the combined PSF (HRI/XRT) is largely energy independent within 2'.

The integrated surface brightness (encircled energy function) for the long HZ43 observation is shown in Figure 3. The extended HRI halo is evident in this figure and shows that approximately 15% of the total power is scattered beyond 10'', 7% beyond 1', and less than 1% beyond 5'. Figure 4 shows the encircled energy function for HZ43, Ar Lac, and LMC X-1. This figure shows that the encircled energy function is consistent within the statistical errors for these three sources, and is not strongly dependent on the photon energy.

**Figure 1:** The point spread function from a long exposure of HZ43. The surface brightness is in units of counts per pixel. To improve the statistics, the data were binned in increasingly larger annuli (0.5, 1, 2, 4, 10, and 60 arcseconds). The dashed line is the background rate in this observation determined between 8' and 12' off-axis. Also shown is the parametric form for the PSF.
$\includegraphics[width=.8\textwidth]{hz43_psf.ps}$

**Figure 2:** A comparison of the point spread functions of HZ43, LMC X-1, and AR Lac with arbitrary normalization. To improve the statistics, the data were binned in increasingly larger annuli (0.5, 1, 2, 5, 10, and 30 arcseconds). Also shown is the parametric form for the PSF normalized to the AR Lac observation.
$\includegraphics[width=0.9\textwidth]{surf_brite.ps}$

**Figure 3:** The on-axis encircled energy function from a long exposure of HZ43. The profile has been normalized to unity at 8'. This figure shows that approximately 15% of the power is scattered beyond 10'', 7% is scattered beyond 1', and less than 1% is scattered beyond 5'.
$\includegraphics[totalheight=.85\textheight]{hz43_encirc.ps}$

**Figure 4:** A comparison of the on-axis encircled energy functions for observations of HZ43, LMC X-1, and AR Lac. Some extended emission is evident in the profile of LMC X-1. The error bars shown in the figure were all evaluated at 1', but were offset slightly on the graph for clarity.
$\includegraphics[width=\textwidth]{encirc.ps}$

The on-axis PSF given above is based on a single long observation of HZ43 with excellent photon statistics. Such an observation is required in order to accurately parameterize the HRI halo. We have also looked at a number of shorter (nearly on-axis) observations of HZ43, AR Lac, LMC X-1, and the Meaty Source taken throughout the mission in order to determine the reproducibility of the on-axis PSF. In general, there is no difference (within the statistics) beyond 10''. This indicates that the HRI halo is stable and approximately energy independent. However, due to random errors in the aspect solution, the width of the core of the HRI images can vary. Fitting the above functional form of the PSF to this sample (treating the width and normalizations of the two gaussians as free parameters) shows that the best fit value of S1 varies from 1.9'' to 2.5'', and the best fit value of S2 varies from 3.5'' to 4.1'' (see Fig. 5). Variations in S1 and S2 between observations of the same source (e.g., the Meaty Source) span almost the entire range of values shown in Figure 5.

**Figure 5:** The best fit values of S1 and S2 to a sample of point sources observed nearly on-axis (within 3' ). The pair of values in the on-axis PSF (eq. 1) is shown as an open square. Observations of the Meaty Source are indicated by open circles.
$\includegraphics[width=0.8\textwidth]{psf_s1s2.ps}$

Residual errors in the aspect solution also occasionally produce asymmetrical features in the PSF. Figure 6 displays images of the so-called ``Meaty Source'' (a white dwarf star discovered with the WFC) illustrating the inner core's appearance when the aspect solution is correct and when there are errors leading to an ellipsoidal image. The major axis of the ellipsoidal images are not aligned with the wobble direction. The asymmetry is the strongest between 5'' and 10'' from the centroid of the image and can have amplitudes up to 30%. Even more extensive and bizarre distributions have been produced by aspect errors for sources known to be unresolved because they are highly variable. When possible, the user should use the image of a known point source within 5' of their target as a template for the HRI PSF.

WARNING: In addition to aspect errors which alter the effective PSF for each observation, the user is reminded that eq. 1 is an analytical fit to an actual observation. Although the fit is quite good, it is not precise and if used unwisely, can lead to erroneous conclusions. Note particularly the difference between the fit and the data between 10'' and 20'' in figure 1.

The actual two-dimensional PSF has a 16-fold-symmetrical pattern (``radial spokes'') caused by the shadowing of mirror-scattered X rays by the mirror-support structure, as well as an ellipsoidal component due to an imperfect aspect solution. The radial spokes should only appear in images with large numbers of detected photons in observations which are not ``wobbled''.

**Figure 6:** Comparison of several nearly on-axis (within a few arcminutes) HRI images of point sources. While the core of the images are symmetrical, the surface brightness between 5'' - 8'' can exhibit some asymmetry. This feature is randomly oriented (i.e., it is not in the direction of the wobble) and depends on the date of the observation. The three images at the bottom of the figure show the WFC ``Meaty Source'' observed at three different times. Notice that the asymmetry is present in the first and last observation, but not in the intermediate observation.
$\includegraphics[width=\textwidth]{core.ps}$

**Figure 7:** HRI images of HZ43 at nine different angles off-axis.
$\includegraphics[width=\textwidth]{hz43x9.ps}$

**Figure 8:** HRI images of LMC X-1 at nine different angles off-axis.
$\includegraphics[width=\textwidth]{lmcx1x9.ps}$

The spatial resolution of the HRI off-axis is dominated by the XRT PSF which degrades rapidly with increasing off-axis angle. Figures 7 and 8 display several off-axis pointings of HZ43 and LMC X-1, and illustrate how the images change with off-axis angle. Figure 9 shows contours of the HZ43 images. The images were first smoothed with a gaussian filter with $\sigma$ = 1''( $\theta$ $\leq$ 6.0'), $\sigma$ = 2'' ( 8.6' $\leq$ $\theta$ $\leq$ 12.0'), and $\sigma$ = 4'' ( 13.5' $\leq$ $\theta$ ). These images show that the PSF starts to degrade beyond 5' off-axis. Beyond 12', the images become very asymmetrical. A central ``bar'' in the images well off-axis is perpendicular to a line connecting the optical axis and the image centroid.

**Figure 9:** Contour plots of the HZ43 images. The contours are at 1, 5, 10, 30, 50, and 90% of the peak value, with the 50% contour level shown in bold.
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The 50% power radius was determined for a series of off-axis pointings of HZ43, AR Lac, and LMC X-1. The asymmetry at large angles has been ignored and the 50% power radius has been computed from radial binning about the image centroids. Figure 10 is a plot of the 50% power radius as a function of off-axis angle for these sources. The solid curve is the best-fit to the following parametric representation of the 50% power radius.

r₅₀ = $\displaystyle {\frac{2.35}{2}}$ [ $\displaystyle \sigma_{HRI}^{2}$ + $\displaystyle \sigma_{aspect}^{2}$ + ( $\displaystyle \sigma_{mirror}^{}$ + a $\displaystyle \theta^{b}_{}$ )²]^0.5 arcsec

(2)

It is best to use a circle of radius r₅₀ when extracting counts of a source detected off-axis. The total net count rate (corrected for scattering) is then simply twice the net count rate within r₅₀.

**Figure 10:** The 50% power radius in images of HZ43, LMC X-1, and AR Lac as a function of angle off-axis. The parametric expression given in the text is also shown for comparison.
$\includegraphics[width=\textwidth]{r50.ps}$

A₁ = 0.9638	S₁ = 2.1858 arcsec
A₂ = 0.1798	S₂ = 4.0419 arcsec
A₃ = 0.00090	S₃ = 31.69 arcsec

$\sigma_{HRI}^{}$	=	0.74''
$\sigma_{aspect}^{}$	=	1.0''
$\sigma_{mirror}^{}$	=	1.3''
a	=	0.0205
b	=	2.349