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3 Point spread functions and associates

For certain purposes of spatial analysis, the point spread function (PSF) of the instrument in operation during the observation is of paramount significance.

The PSF is defined to be the (normalized) photon distribution in the focal plane caused by a celestial X-ray point source at infinite distance. Parameters are thereby the direction of the X-ray point source relative to the instrument's optical axis and the energy of the incoming photons.

More technically, the photon distribution is conceived as (the density of) a probability distribution p in the detector plane. The implied random variable is the random event of the incidence of one photon in a small neighbourhood around a given point ${\bf x}$ in the detector plane. The variables and parameters of p are explained in detail in the sequel.

Analytical representations for the point spread functions of the ROSAT imagers XRT-PSPC, XRT-HRI, and XUV-WFC have been deduced by detailed estimations from in-flight data as well as from pre-launch calibration data.

For the PSPC, the PSF model adopted, its physical justification and the data sets used in the PSF estimation, ground calibration data and in-flight data are described in a series of three articles by Hasinger et al. (1992, 1993, 1994). The ROSAT mirror assembly is documented by Aschenbach (1988). See also the ROSAT spacecraft and instrumentation description in Trümper (1990) or Trümper (1991). The interested reader is referred to these original documents. In case of the HRI, the documentation is similar, see the report by David et al. (1999). The pertinent WFC documents are those by Barstow (1990), Brunner et al. (1993), Sansom (1990), Wells (1990), Sansom (1991), Willingale (1988) and again Trümper (1990) or Trümper (1991).

Imagine momentarily an ideal imaging system with complete focusing in a focal plane[*] and without stochastic influences. Then, in the geometrical ray approach, all photons having the same energy and coming from the same spatial direction strike the detector focal plane at a certain point, ${\bf s}$, called the source position. In a real imaging system, incomplete focusing as well as stochastic imaging processes caused by the micro-roughness of the mirror[*] and the detector physics are inevitable, and the assumed source point widens to an extended point spread function (PSF), viewed here as a two dimensional probability density over the detector plane ${\bf R}^2$ (in general) closely around the hypothetical source point ${\bf s}$. Suppose only photons of the same energy, E, and coming from the same spatial direction arrive at the detector plane. Then $p{\rm d}A$ is the expected photon count fraction falling into the area element ${\rm d}A$ around the point ${\bf x}$ in the detector plane. The quantity p has thus the dimension[*] (Photon Counts)/Area.

The general position, ${\bf x}$, in the detector plane will be described by means of two polar coordinate systems. The first, the optical axis system $\{{\cal O};\epsilon,\theta\}$, has its pole at the trace point, ${\cal O}$, of the optical axis of the mirror-detector system in the detector plane. The point ${\bf s}$ is referred to this system so that[*] $\epsilon:\,\,=\vert{\bf s}-{\cal O}\vert$, henceforth called off-axis angle, is the angular distance between the source position ${\bf s}$ and the optical axis' trace point ${\cal O}$. Further, the azimuthal angle $\theta$ is measured in positive, i.e. counter-clockwise, direction off the positive horizontal axis. The second system, the source system $\{{\bf s};r,\phi\}$, is a translate of the first one and attached to the source position ${\bf s}$, so that $r:\,\,=\vert{\bf x}-{\bf s}\vert$ is the source distance. The azimuthal angle $\phi$ of ${\bf x}-{\bf s}$ is measured in the same way off the related horizontal axis as in the optical axis system. Although known to exist at larger off-axis angles, no azimuthal dependence has been modelled so far. The models to follow (WFC excepted) represent the azimuthally averaged part of the observed PSF[*]. Consequently, the azimuthal angles $\theta,\phi$ do not occur in the parameterization of p. The remaining parameters of p are thus the photon energy E and the off-axis angle $\epsilon $. The source distance r is conceived as the variable.

Any ROSAT mirror-detector combination establishes a one-to-one correspondence, the so-called ray-trace relation, between the photon's arrival directions relative to the optical axis, forming the field of view, and the image of the field of view in the detector plane. So, the distances $\epsilon,r$ can be identified with angular distances from the related central positions ${\cal O},{\bf s}$ and are thus measured in angular units, namely the off-axis angle $\epsilon $ in arcmin and r in arcsec - the units of the arguments of the PSF p used together with E in keV.

The obtainment of an estimate, $\hat{\bf s}$, for the unknown source position ${\bf s}$ itself belongs to the tasks of the spatial analysis. Having found p, its mode (i.e. peak-) position serves for $\hat{\bf s}$. The subsequent notation stresses the dependency on the parameters.

According to the above definition, the point spread functions p are normalized so that

 \begin{displaymath}\int_{{\bf R}^2}p(r;E,\epsilon)r{\rm d}r{\rm d}\phi=1.
\end{displaymath} (1)

This means that p is the distribution density for one photon. As introduced above, r in arcsec is the angular distance of the area element ${\rm d}A:~=r{\rm d}r{\rm d}\phi$ in polar coordinates $r,\phi$ from the source position ${\bf s}$ in the detector plane ${\bf R}^2$. Moreover, E in keV is the energy of the photon registered and $\epsilon $ in arcmin the angular distance of ${\bf s}$from the trace point ${\cal O}$ of the optical axis in the detector plane.

Besides p, the cumulative point spread function, P, i.e. the radially and azimuthally integrated p,

 \begin{displaymath}P(r;E,\epsilon):\,\,=\int_0^{2\pi}{\rm d}\phi\int_0^r p(\rho;E,\epsilon)\rho{\rm d}\rho
\end{displaymath} (2)

is of relevance[*]. The quantity $P(r;E,\epsilon)$ is the fraction of photon counts which is expected within the circle $\vert{\bf x}-{\bf s}\vert=r$ of the detector plane when the photons with energy E hit the detector plane at a distance $\epsilon $ from the optical axis' trace point ${\cal O}$. In case of the WFC, the geometry of point spread function is more general; the level curves will be formed by ellipses with the shorter axis in radial direction.

The normalization (1) implies the limiting relation $P(r;E,\epsilon)\to1$ for $r\to\infty$.

We come to measures of spread for the PSF. The q-quantile radius rq is defined implicitly by $q=P(r_q;E,\epsilon)$, $0\le q<1$. So, for q=1/2 the median r1/2 is obtained. The diameter 2r1/2 is also called half-energy-width (HEW) in the context of monochromatic spectra. Associated with p is also the full-width-half-maximum (FWHM) function, $w(E,\epsilon)$, implicitly defined by

\end{displaymath} (3)

All analytical ROSAT point spread function models decrease with increasing r so that $w(E,\epsilon)$ is uniquely defined.

Several measures of spread of distributions are known and in use. The appropriate choice among them depends on the context. The FWHM characterizes the spread of a PSF density. The median radius r1/2, defined by $P(r_{1/2},E,\epsilon)=1/2$, is another measure of spread for the cumulative PSF[*].

Before describing the PSF in detail, recall again that two ROSAT observation modes were possible, pointed observation and all-sky survey observation. In contrast to survey observation, in most pointed observations a wobbling motion around the nominal pointing direction is carried out in order to lessen the detrimental shadow of the detector window support structure.

In this connection, some alerting words are in order. The attempt to verify the point spread functions given below from event files or images - as they are - may fail.

First, recall that the ROSAT attitude error[*] was specified to be up to 10 arcsec. Any attempt to stay below this specification requires an intimate instrument knowledge. Details of the achieved positional accuracy can be found in Voges et al. (1999).

Secondly, recall that uncorrected remainders of a wobbling motion in saw-tooth form S[u] and a possible systematic or stochastic attitude drift[*] ${\bf\delta}(t)$ may be superimposed to the nominal pointing direction of the telescope after attitude correction for wobbling remainders in the Standard Analysis Software System (SASS). This leads, after mapping onto the detector plane, to the adoption of the non-stationary, i.e. time-dependent, stochastic process model for the source position

{\bf s}(t):&={\bf s}_0+{\bf w}S\left[2\pi\nu(t...
4\{u\}-4, &3/4\le\{ u\}\le 1\end{array}\right.\\
\end{split}\end{displaymath} (4)

in place of the time-independent source position ${\bf s}$. In (4), $\{u\}$ denotes the fractional part of the (dimensionless) real number u. Moreover, ${\bf s}_0$ is the unknown true source position and ${\bf w}$ the constant wobbling vector with the nominal magnitude $\vert{\bf w}\vert=3$ arcmin. Deviations from the nominal value are known. After the ROSAT standard data processing, the remaining $\vert{\bf w}\vert$ is expected to be considerably smaller. At a time t=t0 in (or close to) the observation time interval [t1,t2], the wobbling motion vanishes. The wobbling frequency for ROSAT, $\nu$, has the nominal value $\nu=1/402$ Hz. Finally, $\delta(t)$ is at best a white noise process but is expected to have normally a systematic and/or stochastic additional drift component motion. The task is to find estimates $\hat{\bf w},\hat\nu,\hat t_0,\hat\delta(t)$ for the counterparts in (4) based on the observation at hand. Then (4) with these estimates is to be solved for

 \begin{displaymath}\hat{\bf s}_0:\,\,={\bf s}(t)-\hat{\bf w}S\left[2\pi\hat\nu(t-\hat t_0)\right]+\hat{\bf\delta}(t).
\end{displaymath} (5)

The correction vector ${\bf c}(t):\,\,=\hat{\bf w}S\left[2\pi\hat\nu(t-\hat t_0)\right]+\hat{\bf\delta}(t)$is, finally, to be subtracted from the photons arriving at time t in the observation interval $t_1\le t\le t_2$ to the effect that (a) systematic distortions due to wobbling are (greatly) removed and (b) the variances of photon clusters around source positions diminish. Use of attitude data may assist the described de-wobbling and "de-speckling'' procedure. The success of this effort may vary from observation to observation. Details, limitations and examples of the sketched method are beyond the scope of this paper.

Having done this, a good agreement between the point spread functions estimated that way and the ones below should be reached. The particular observations at hand may not allow a de-speckling. How to account for an attitude drift ${\bf\delta}(t)$ in such a case will be discussed in Sect. 4 in more detail.

3.1 The point spread function for the ROSAT XRT-PSPC instrument

The authors of the three papers by Hasinger et al. (1992, 1993, 1994) carried out the considerable amount of work related with the estimation of the PSF under consideration.

In the present case, p: $=p(r;E,\epsilon)$ is parameterized by the photon energy, E, and the off-axis angle, $\epsilon $, as a three component additive mixture with energy and off-axis angle dependent mixture proportions $p_1(E,\epsilon),p_2(E,\epsilon),p_3(E)$,

p(r;E,\epsilon):&=p_1(E,\epsilon)\cdot{{{\rm e...
...ght)^{\alpha(E)}, &r\ge r_2(E).
\end{split}\end{displaymath} (6)

The first addend in (6) stems from the random process taking place with the generation of primary electrons in the counter. The second term results from the finite penetration depth of the X-ray photons in the counter gas and the diffusion of the electron cloud. The last term is due to the mirror scattering. The estimates for the functions and parameters occuring in (6) are

...:&={861.9\over E},\\

\end{split}\end{displaymath} (7)

The units used for $r,E,\epsilon$ are arcsec, keV, arcmin, respectively, and the numbers in (7) have the implied units, e.g. 39.95 arcseckeV in the case of r1. Observe that the quotient r2(E)/r1(E) in (6) does not depend on E and is thus written without argument E. The above estimate of p3(E) was found by P. Predehl (private communication) and replaces since end 1995 the older estimate 0.075E1.43.

The PSPC field of view has a diameter $\approx 2^{\circ}$ so that $r\le7200$ arcsec. This, together with the energy range in which the energy dependencies of (10) hold, gives for applications the domain of definition of $p(r;E,\epsilon)$,

 \begin{displaymath}0\le r\le 7200,\quad0.07\le E\le3.0,\quad0\le\epsilon\le60.
\end{displaymath} (8)

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 1: PSPC Pointing PSF density for E=1keV,$\epsilon =$ 0, 12, 24, 36, 48 and 57arcmin

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 2: On-Axis PSPC Pointing PSF density forE= 0.1, 0.5, 0.9, 1.3, 1.7 and 2keV

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 3: PSPC Pointing PSF density for $\epsilon =30$arcmin, E= 0.1, 0.5, 0.9, 1.3, 1.7 and 2keV

$\!\!\!\! $Figure 1 shows the radial dependence of the XRT-PSPC PSF density $p(r;E,\epsilon)$ in pointing mode for a photon energy of 1 keV for six different off-axis angles $\epsilon=0(12)48$ and 57[*] arcmin in the range $0\le r\le3600$ arcsec in logarithmical scaling of the abscissa and ordinate.

The point spread function becomes wider as the off-axis angle increases. The energy dependence of the on-axis, PSPC pointing PSF is exhibited in Fig. 2 for the energies E=0.1,0.5,0.9,1.3,1.7 and 2 keV. Figure 2 is supplemented by Fig. 3 showing the PSPC PSF density at a large off-axis angle $\epsilon $ = 30 arcmin and the same photon energies as in Fig. 2, E=0.1,0.5,0.9,1.3,1.7 and 2 keV.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 4: Cumulative PSPC Pointing PSF at energy E=1keV and off-axis angles $\epsilon =$ 0, 12, 24, 36, 48 and 57arcmin

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 5: Cumulative On-Axis PSPC Pointing PSF for energies E= 0.1, 0.5, 0.9, 1.3, 1.7 and 2keV

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 6: Cumulative Off-Axis PSPC Pointing PSF at $\epsilon $ =30 arcmin for energies E= 0.1, 0.5, 0.9, 1.3, 1.7, 2keV

The cumulative distribution $P(r;E,\epsilon)$ from (2) belonging to $p(r;E,\epsilon)$ from (6) with parameters from (7) is found to be

...ight)^2\right]}}, &r\ge r_2(E).
\end{split}\end{displaymath} (9)

Figures 4 to 6 display the corresponding cumulative counterparts to Figs. 1 to 3. Thus Fig. 4 shows the ROSAT XRT-PSPC cumulative pointing PSF for the same parameters as in Fig. 1, namely, for $\epsilon=0,12,24,36,48$ and 57 arcmin and E=1 keV in the same range $0\le r$ $\le$ 3600 arcsec in double logarithmical representation.

The larger the off-axis angle, the lower the initial slope at r=0. Figure 5 shows the cumulative counterpart to Fig. 2.

The cumulative counterpart of Fig. 3 is Fig. 6 with a large off-axis angle $\epsilon =30$arcmin and photon energies E=0.1,0.5,0.9,1.3,1.7,2keV. For further details and reference, consult the articles and documents compiled below.

3.2 The point spread function for the ROSAT XRT-HRI

The presently used PSF without the mirror term was determined by David et al. (1999)[*]. The mirror contribution was added by P. Predehl (private communication).

This point spread function is modelled as an additive two-component mixture of a mirror contributions, $p_{\rm M}$, and a detector component, $p_{\rm D}$, with energy dependent mixture proportions D(E),M(E),

p(r;E,\epsilon):&=\hskip-2pt D(E)\cdot p_{\rm ...
...right)^2}+A_3{\rm e}^{-{r\over\sigma_3}}\right),\\

p_{\rm M}(r;E):&=\hskip-2pt {1\over{2\pi\left[...
...ght)^{\alpha(E)}, &r\ge r_2(E).
\end{split}\end{displaymath} (10)

Since PSPC and HRI share the same telescope, the XRT, the functions m(r,E) in (6) and $p_{\rm M}(r;E)$ in (10) are identical. The estimates for the parameters in (10) and the function $\sigma_2(\epsilon)$ of the $p_{\rm D}$ component are

A_1:&=0.9638, \hskip-2mm \sigma_1:=2.1858, \\ ...
...n^3, \\
A_3:&=0.0009, \hskip-2mm \sigma_3:=31.69.
\end{split}\end{displaymath} (11)

The units of r,E and $\epsilon $ in (11) are again arcsec, keV and arcmin, respectively. Since PSPC and HRI share the same telescope mirrors, also the estimates for the parameters r1(E), r2(E), $\alpha(E)$ of the mirror component $p_{\rm M}$ in (10) are the same as in (7),

r_1(E):&={39.95\over E},\\
\end{split}\end{displaymath} (12)

Notice that the quotient r2(E)/r1(E) does not depend on E. Thus the argument E is dropped in (10).

The diameter of the field of view for the HRI is 38 arcmin. This combined with the permissible energy range gives the domain of definition of $p(r;E,\epsilon)$ as

 \begin{displaymath}0\le r\le2400,\quad0.07\le E\le3.0,\quad0\le\epsilon\le20.
\end{displaymath} (13)

Figure 7 shows the ROSAT XRT-HRI pointing PSF density $p(r;E,\epsilon)$ for E=1 keV and off-axis angles $\epsilon=0(4)16,19$ arcmin in the range $0\le r\le3600$ arcsec.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 7: HRI Pointing PSF density for energy E=1keV and off-axis angles $\epsilon =$ 0, 4, 8, 12, 16 and 19arcmin

The point spread function becomes wider as the off-axis angle increases. The energy dependence of the PSF is due to the mirror component and is moderate, as Fig. 8 exhibits.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36,clip]{}\end{figure} Figure 8: On-Axis HRI Pointing PSF density for E= 0.1, 0.5, 0.9, 1.3, 1.7 and 2keV

The curves for E=0.1,0.5,0.9,1.3,1.7,2 keV are shown. The influence of the term containing $\sigma_2(\epsilon)$ is pronounced when the source distance r ranges in [5,10] arcsec, say. Figure 9 extends Fig. 8 in that the HRI PSF density for a large off-axis angle $\epsilon =15$arcmin and photon energies E=0.1,0.5,0.9,1.3, 1.7 and 2keV are shown.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 9: HRI Pointing PSF density for $\epsilon =15$arcmin, E= 0.1, 0.5, 0.9, 1.3, 1.7 and 2keV

The cumulative HRI point spread function is

P(r;E,\epsilon):&=D(E)\cdot P_{\rm D}(r;\epsil...
...[1+\left({r_1\over r_2}\right)^2\right]}}}}\cdot\\

 \begin{displaymath}\left\{\begin{array}{*2{l}}{1\over2}\ln\left[1+\left(r\over r...
...\over r_2\right)^2\right]}}, &r\ge r_2(E).
\end{displaymath} (14)

Figure 10 shows the cumulative ROSAT XRT-HRI PSF for the same parameters as in Fig. 7, namely, for E=1 keV and $\epsilon=0(2)10$ arcmin, in the same range $0\le r\le3600$arcsec. The rule that the initial slope becomes smaller with increasing off-axis angle $\epsilon $ is confirmed in a qualitative way. Figure 11 shows the cumulative counterpart of Fig. 8. Finally, Fig. 12 is the cumulative counterpart of the large off-axis HRI PSF density of Fig. 9 with the same off-axis-angle $\epsilon =15$arcmin and photon energies E=0.1,0.5,0.9,1.3,1.7,2 keV. For further information and reference see the documents listed below.
  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 10: Cumulative HRI Pointing PSF for E=1keV, $\epsilon =$ 0, 4, 8, 12, 16 and 19arcmin

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 11: Cumulative On-Axis HRI Pointing PSF for E= 0.1, 0.5, 0.9, 1.3, 1.7 and 2keV

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 12: Cumulative HRI Pointing PSF for $\epsilon =15$arcmin, E= 0.1, 0.5, 0.9, 1.3, 1.7 and 2keV

3.3 The point spread function for the ROSAT XUV-WFC

The original work was done by Sansom (1990) and Wells (1990).

As already mentioned, the Wide Field Camera has a separate telescope mirror, and the detector is a microchannel plate detector. The relative large field of view of $5^{\circ}$ diameter results in larger distortions for sources near the border of the field of view. It was found that the geometry of the level curves of the observed PSF changes remarkably with the off-axis angle $\epsilon $. In contrast to the PSPC and HRI detectors, not only the azimuthally averaged part of the observed PSF is modelled. The level curves of the PSPC and HRI PSF models were circles for all off-axis angles. In the WFC case, ellipses replace the circles.

In order to introduce them, a Cartesian source coordinate system $\{{\bf s},x,y\}$ has to be introduced. The x-axis points in the direction of the radius vector from ${\cal O}$ to ${\bf s}$. Rotating the x-axis by $+\pi/2$ about the source position ${\bf s}$ yields the orientation of the y-axis. In this coordinate system, the said ellipses have the representation

 \begin{displaymath}\rho(x,y):=\sqrt{\left({x\over e}\right)^2+y^2}=r,\qquad r\ge0,
\end{displaymath} (15)


$e:\,\,=\{\hbox{length of minor axis}\}/\{\hbox{length of major axis}\}\le1$ is a measure of eccentricity[*]. Thus, the x-axis is aligned with the minor axis of the ellipse. The relation $e\le1$ means a radial squeezing of the PSF distribution.

The ad hoc model of the WFC PSF is an additive mixture of two components, $p_{\rm A}$ and $p_{\rm B}$, with energy E and off-axis angle $\epsilon $ dependent mixture proportions $A(E,\epsilon),B(E,\epsilon)$ and with $\alpha(E,\epsilon)>1$,

p(x,y;E,\epsilon):&=A(E,\epsilon)\cdot p_{\rm ...
\end{split}\end{displaymath} (16)

Again, the units of x,y,E and $\epsilon $ are arcsec, arcsec, keV and arcmin, respectively. The six functions $\sigma_{\rm A}(E,\epsilon)$, $e_{\rm A}(E,\epsilon)$, $\sigma_{\rm B}(E,\epsilon)$, $e_{\rm B}(E,\epsilon)$, $\alpha(E,\epsilon)$ and $b(E,\epsilon)$ in (16) are obtained by iterated one-dimensional linear interpolation (or extrapolation) with respect to off-axis angle $\epsilon $ and energy E from the corresponding estimates from (17) estimated at five discrete off-axis angles $\epsilon_j,\,j=1(1)5$, arcmin and at two energies, E1=0.0454 keV and E2=0.1834 keV,

\begin{pmatrix}\epsilon\\ \sigma_{\rm A}\\ e_{\...
...4 \\
0.265 & 0.0 & 0.0 & 0.0 & 0.0 \end{pmatrix}.
\end{split}\end{displaymath} (17)

The upper estimate matrix in (17) belongs to the lower energy E1, the lower one to the higher E2. Notice that $b(E,\epsilon)>0$ only for the on-axis case $\epsilon =0$ in (17). This setting was caused by the insufficient amount of data available at the time of PSF analysis (Sansom 1990). This applies also to $e_{\rm B}$.

Denote by $\epsilon_j,\,j=1(1)5$, the five off-axis angles in the first row of the matrices in (17) in increasing order. Let qk be the parameter of the vectors in (17) from the (k+1)th row, k=1(1)6, and $q_k\left(E_l,\epsilon_{j(\epsilon)}\right)$ the matrix entry at energy El and off-axis angle $\epsilon_j$. Then interpolation with respect to $\epsilon $ followed by that with respect to E based on the estimates of (17) yields the continuous parameter functions for k=1(1)6 as well as l=1,2

4, &j(\epsilon)=5.
\end{split}\end{displaymath} (18)

The six functions $\sigma_{\rm A}(E,\epsilon)$ to $b(E,\epsilon)$ are thus formally defined for all $\epsilon\ge0,\,E\ge0$ but the inequalities

\sigma_{\rm A}(E,\epsilon)&>0,\quad\sigma_{\rm...
\end{split}\end{displaymath} (19)

must be satisfied.

The diameter of the field of view of the WFC is with $\approx2.5^{\circ}$ in zoom mode and $\approx5^{\circ}$ without zoom. This, together with the energy range gives for applications the domain of definition of $p(r;E,\epsilon)$ as

 \begin{displaymath}0\le r\le18000,\quad0.017\le E\le 0.210,\quad0\le\epsilon\le150.
\end{displaymath} (20)

The energy interval from (20) corresponds to the $10\%$ fall-off value of the appropriate WFC filter, i.e. the one for the highest energy. The analysis of the system of inequalities (19) shows that all inequalities from (19) are everywhere satisfied in the domain (20). The region $\alpha(E,\epsilon)>1$ defines for $\epsilon\ge98.4$ in $(E,\epsilon)$-plane the hyperbolic region

\end{displaymath} (21)

The set (21) contains the domain of definition (20). The arc of the boundary of (21) connecting the point (0.017,249.14) with (0.0416,300) forms the curvilinear boundary arcs of the domain (21). So, the domain (20) in the $(E,\epsilon)$-plane is fully contained in (21).

Figure 13 shows the ROSAT XUV-WFC pointing PSF density $p(r;E,\epsilon)$ for E=150eV at the off-axis angles $\epsilon:\,\,=k$ arcmin, k=0,28,56,84,112 and 133, in the range $0\le r\le7200$ arcsec in the radial cross-section along the major axis, i.e. for y=0 with the UV filter.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 13: WFC Pointing PSF density for E=0.15keV, $\epsilon =$ 0, 28, 56, 84, 112 and 133arcmin, radial cross-section, UV filter

The difference to the cross-section along the major axis, i.e. for x=0, is so small that no changes are visible at the scale of Fig. 13. Therefore, no plot of the transversal profile is shown. Figure 14 shows the energy dependence of the on-axis WFC density in radial cross-section with the UV filter, as in Fig. 13.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 14: WFC Pointing PSF density for $\epsilon =0$arcmin, E= 0.02, 0.06, 0.1, 0.14, 0.18 and 0.21keV, radial cross-section, UV filter

The cumulative pointing point spread function is conveniently defined in the present case by

 \begin{displaymath}P(r;E,\epsilon):\,\,=\int_{\rho_{\rm A}(x,y)\le r}p(x,y;E,\epsilon){\rm d}x{\rm d}y.
\end{displaymath} (22)

A good approximation and upper bound for P under the data from (17) is the fully explicit expression with $\alpha(E,\epsilon)>1$

P(r;E,\epsilon):&=A(E,\epsilon)\cdot P_{\rm A}...
...ight){\rm e}^{-{r\over\sigma_{\rm B}(E,\epsilon)}}.
\end{split}\end{displaymath} (23)

The $P_{\rm A}$ component in (23) is exact, and $P_{\rm B}$ is a sufficiently precise approximation for small off-axis angles $0<\epsilon<37.8$ arcmin and again exact for the remaining off-axis angles $\epsilon\ge37.8$.

The exact expression for $P_{\rm B}$ allows the representation

P_{\rm B}(r;E,\epsilon):&=1-{1\over2\pi}\int_0...
..._B^2(E,\epsilon)\over e_A^2(E,\epsilon)}-1\right)}.
\end{split}\end{displaymath} (24)

The stable numerical evaluation of $P_{\rm B}$, as used for the EXSAS command (37), poses no problem. For the estimates from (17), it follows the enclosure

&\hskip-5pt 1-\left(1+{re_{\rm A}(E,\epsilon)\...
...ight){\rm e}^{-{r\over\sigma_{\rm B}(E,\epsilon)}}.
\end{split}\end{displaymath} (25)

The rightmost expression in (25) is that of (23). Figure 15 shows the cumulative ROSAT XUV-WFC pointing PSF for the same parameters as in Fig. 13, namely for E = 150 eV and $\epsilon=28,56,84,112$ and 133 arcmin in the same range $0\le r\le600$ arcsec with the exact $P_{\rm B}$ from (24).

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 15: Cumulative WFC Pointing PSF for E=0.15keV, $\epsilon =$ 0, 28, 56, 84, 112, 133arcmin, radial cross-section, UV Filter

The differences to the approximation to $P_{\rm B}$ from (23) are so small that no deviations are visible at the scale of Fig. 15.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 16: Cumulative on-axis WFC Pointing PSF for E= 0.02, 0.06, 0.1, 0.14, 0.18 and 0.21keV, radial cross-section, UV filter

The component $p_{\rm A}$ in (16) is sometimes called a Moffat distribution, (see Moffat 1969, Eq. (7)). It is a generalization of the so called King distribution (King 1983, Eq. (10)).

For further details and reference, consult the related documents below.

3.4 The ROSAT survey point spread function

Following the trace curve of a celestial point source in the detector's field of view while in survey mode explains that the survey point spread function is a weighted mean with respect to the off-axis angle $\epsilon $of the corresponding PSF in effect in pointed observation mode.

We will make the following modelling assumptions:

  1. The detector's field of view is a circular disk of radius $\epsilon_0$ arcmin in the detector plane with the trace point of the optical axis as centre;
  2. The scanning angular velocity in survey motion is constant in the field of view;
  3. The distance, $\epsilon_{\rm s}$ arcmin, between consecutive scanning tracks is small ( $0\le\epsilon_{\rm s}\le4$ arcmin), and we assume the limiting case $\epsilon_{\rm s}/\epsilon_0=0$.
In an observation under analysis, not all scanning tracks may be present. In critical cases, the use of the attitude file frees from accepting assumption 2. Similarly, one can use the more correct weighting distribution corresponding to the distance $\epsilon_{\rm s}$ actually used in survey motion.

Critical cases in the above sense exist. In a neighbourhood of 1 degree latitude of the ecliptic poles, assumption 3 is violated. Assumption 2 is not obeyed in fields in which the survey observation was interrupted due to the earth's radiation belts or due to the South Atlantic Anomaly.

Based on the above assumptions, the EXSAS implementation of the survey PSF density and the cumulative survey PSF with vignetting correction for the PSPC detector will be described and represented in this section.

3.4.1 The vignetting corrected ROSAT survey point spread function

The reflectivity of the gold surface decreases with increasing incidence angle on the reflective surface. The projection of the gold-coated reflective viewable area in direction perpendicular to the infalling bundle of X-rays multiplied by the reflectivity is called effective area. It decreases with increasing off-axis angle $\epsilon $. This energy dependent degradation of the mirror assembly is termed vignetting degradation, or in short, vignetting. All previous EXSAS PSFs were not vignetting corrected.

Let $A(E,\epsilon)$ be the effective area function of the mirror-detector system under consideration. Then

 \begin{displaymath}V(E,\epsilon):\,\,={A(E,\epsilon)\over A(E,0)}
\end{displaymath} (26)

is the energy and off-axis angle dependent vignetting function for the mirror-detector unit in operation. In EXSAS, $A(E,\epsilon)$ for the detector PSPC_C with which the survey was performed is represented, by the calibration table EXSAS_CAL:effarea_pspcc.tbl with entries for 729 non-equidistant energy values in the interval [0.0713,3.005] keV and $\epsilon=0(5)55,57.5$ and 60. Then the vignetting corrected PSF density, $p^{\rm V}$, in pointing mode observation is

 \begin{displaymath}p^{\rm V}(r;E,\epsilon):\,\,=V(E,\epsilon)\cdot p(r;E,\epsilon).
\end{displaymath} (27)

Taking $p^{\rm V}$ instead of p in (28) leads to the definition for the vignetting corrected survey PSF density, $p_{\rm S}^{\rm V}$

 \begin{displaymath}p_{\rm S}^{\rm V}(r;E):\,\,=\displaystyle{{\int_0^{\epsilon_0...{\int_0^{\epsilon_0}V(E,\epsilon)2\epsilon{\rm d}\epsilon}}.
\end{displaymath} (28)

Figure 17 shows the PSPC vignetting function $V(E,\epsilon)$ for fourteen off-axis angles $\epsilon=0(5)55,57.5,60$ arcmin in the energy range [0,3]keV. The fact that $V(E,\epsilon)$ decreases in $\epsilon $ allows the identification of the off-axis angle in Fig. 17.

  \begin{figure}\includegraphics[bb=0 60 290 232,angle=0,scale=0.84]{}\end{figure} Figure 17: PSPC vignetting function, Vig, for $\epsilon =$ 0(5)55, 57.5 and 60 arcmin (top to bottom) in the energy range [0,3] keV

In EXSAS exists yet another table, EXSAS_CAL:vignet_pspc.tbl, with vignetting values. It has entries for the pulse-height values Amplitude=1(1)300 and the off-axis angles $\epsilon=0(5)55,57.5$ and 60. Since it is based on amplitudes rather than energies, it is not to be used for $V(E,\epsilon)$. The survey PSF density is shown in Fig. 18 for the energies E=0.1,0.5,0.9,1.3,1.7 and 2.0keV.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 18: PSPC Survey PSF density for E= 0.1, 0.5, 0.9, 1.3, 1.7, 2.0keV

The cumulative survey PSF

 \begin{displaymath}P_{\rm S}^{\rm V}(r;E):\,\,=2\pi\int_0^r p_{\rm S}^{\rm V}(\rho;E)\rho{\rm d}\rho.
\end{displaymath} (29)

is shown in Fig. 19 for the energies E = 0.1, 0.5, 0.9, 1.3, 1.7 and 2.0keV.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 19: Cumulative PSPC Survey PSF for E= 0.1, 0.5, 0.9, 1.3, 1.7, 2.0keV

3.5 Survey point spread function based on a Gauss approximation of the PSF

For the sake of a fast function evaluation, a Gauss approximation of the pointed mode PSPC PSF is used in the EXSAS Maximum Likelihood Source Estimation,

p_{\rm G}(r;E,\epsilon)&:\,\,={1\over{2\pi\sig...
\end{split}\end{displaymath} (30)

Figure 20 shows the PSPC Survey PSF density, $p_{\rm S}^{\rm G}(r;E)$, based on the Gauss approximation $p_{\rm G}(r;E)$ as defined in (30) in the interval [1,3600] arcsec for the energies E=0.1,0.5,0.9,1.3,1.7 and 2.0 keV. Within the scaling of Fig. 20 no larger energy dependence is seen. The agreement with the full PSF model from (28)in the interval [0,500] arcsec is acceptable but the tails of $p_{\rm G}^{\rm V}$ fall off too steeply in comparison with Fig. 18. The energy dependence is weak enough that, at the scale of Fig. 20, the curves $p_{\rm S}^{\rm G}$ fall close together.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 20: PSPC survey PSF  $p_{\rm S}^{\rm G}(r;E)$ based on the Gauss approximation for  E=0.1,0.5,0.9,1.3,1.7,2keV for $r\in $ [1,3600] arcsec

The cumulative counter-part of Fig. 20 is shown in Fig. 21. We see that the finer model underlying Fig. 19 leads to about the same weak energy dependence.

  \begin{figure}\includegraphics[bb=40 70 530 745,angle=270,scale=0.36]{}\end{figure} Figure 21: Cumulative PSPC survey PSF $P_{\rm S}^{\rm G}(r;E)$ based on Gauss approximation for E= 0.1, 0.5, 0.9, 1.3, 1.7, 2keV for $r\in [1,3600]$arcsec

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