- 3.1 The point spread function for the ROSAT XRT-PSPC instrument
- 3.2 The point spread function for the ROSAT XRT-HRI
- 3.3 The point spread function for the ROSAT XUV-WFC
- 3.4 The ROSAT survey point spread function
- 3.5 Survey point spread function based on a Gauss approximation of the PSF

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
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, ,
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
(in general) closely around the hypothetical source point
.
Suppose only photons of the same energy, *E*, and coming from the same
spatial direction arrive at the detector plane. Then
is the expected photon
count fraction falling into the area element
around the point
in the
detector plane. The quantity *p* has thus the dimension^{}
(*Photon* *Counts*)/*Area*.

The general position, ,
in the detector plane will be described by means of
two polar coordinate systems. The first, the *optical axis system*
,
has its pole at the trace point, ,
of the optical axis of the mirror-detector system in the detector
plane.
The point
is referred to this system so that^{}
,
henceforth called
*off-axis angle*, is the angular distance between the source position
and the
optical axis' trace point .
Further, the azimuthal angle
is measured
in positive, i.e. counter-clockwise, direction off the positive horizontal axis. The second system, the
*source system*
,
is a translate of the first one and attached
to the source position ,
so that
is the *source distance*.
The azimuthal angle
of
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
do not occur in the parameterization of *p*. The remaining parameters
of *p* are thus the *photon energy* *E* and the *off-axis angle* .
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
can be identified with angular
distances from the related central positions
and are thus measured in
angular units, namely the off-axis angle
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,
,
for the unknown source position
itself
belongs to the tasks of the spatial analysis. Having found *p*, its mode (i.e. peak-) position
serves for
.
The subsequent notation stresses the dependency on the parameters.

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

This means that

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

is of relevance

The normalization (1) implies the limiting relation for .

We come to measures of spread for the PSF.
The *q*-quantile radius *r*_{q} is defined implicitly by
,
.
So, for *q*=1/2 the *median* *r*_{1/2} is obtained. The diameter 2*r*_{1/2} 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,
,
implicitly defined by

All analytical ROSAT point spread function models decrease with increasing

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 *r*_{1/2}, defined by
,
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^{}
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

in place of the time-independent source position . In (4), denotes the fractional part of the (dimensionless) real number

The correction vector is, finally, to be subtracted from the photons arriving at time

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 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*:
is parameterized by the photon energy, *E*,
and the off-axis angle, ,
as a three component additive mixture with energy and
off-axis angle dependent mixture proportions
,

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

The units used for are arcsec, keV, arcmin, respectively, and the numbers in (7) have the implied units, e.g. 39.95 arcseckeV in the case of

The PSPC field of view has a diameter
so that
arcsec. This, together with the energy range in which the energy dependencies of (10) hold,
gives for applications the domain of definition of
,

Figure 1 shows the radial dependence of the XRT-PSPC PSF density
in pointing
mode for a photon energy of 1 keV for six different off-axis angles
and 57^{} arcmin in the range
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
= 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.

Figure 4:
Cumulative PSPC Pointing PSF at energy E=1keV and off-axis angles
0, 12, 24, 36, 48 and 57arcmin |

Figure 6:
Cumulative Off-Axis PSPC Pointing PSF at
=30 arcmin for energies E= 0.1, 0.5, 0.9, 1.3,
1.7, 2keV |

The cumulative distribution
from (2) belonging to
from (6) with parameters from (7) is found to be

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 and 57 arcmin and

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
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, ,
and a detector component, ,
with energy dependent mixture
proportions *D*(*E*),*M*(*E*),

Since PSPC and HRI share the same telescope, the XRT, the functions

The units of

Notice that the quotient

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
as

Figure 7 shows the ROSAT XRT-HRI pointing PSF density for

Figure 7:
HRI Pointing PSF density for energy E=1keV and off-axis angles
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.

The curves for
*E*=0.1,0.5,0.9,1.3,1.7,2 keV are shown. The influence of the term containing
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
arcmin and photon energies
*E*=0.1,0.5,0.9,1.3,
1.7 and 2keV are shown.

The cumulative HRI point spread function is

Figure 10 shows the cumulative ROSAT XRT-HRI PSF for the same parameters as in Fig. 7, namely, for

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 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 . 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
has to be introduced. The *x*-axis points in the direction of the
radius vector from
to .
Rotating the *x*-axis by
about the
source position
yields the orientation of the *y*-axis. In this coordinate
system, the said ellipses have the representation

where

is
a measure of eccentricity^{}. Thus, the *x*-axis is aligned with the minor axis
of the ellipse. The relation
means a radial squeezing of the PSF distribution.

The *ad hoc* model of the WFC PSF is an additive mixture of two components,
and ,
with energy *E* and off-axis angle
dependent mixture proportions
and with
,

Again, the units of

The upper estimate matrix in (17) belongs to the lower energy

Denote by
,
the five off-axis angles in the first row of
the matrices in (17) in increasing order. Let *q*_{k} be the parameter of
the vectors in (17) from the (*k*+1)th row, *k*=1(1)6, and
the matrix entry at energy *E*_{l} and off-axis
angle
.
Then interpolation with respect to
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

The six functions to are thus formally defined for all but the inequalities

must be satisfied.

The diameter of the field of view of the WFC is with
in zoom
mode and
without zoom. This, together with the energy
range gives for applications the domain of definition of
as

The energy interval from (20) corresponds to the 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 defines for in -plane the hyperbolic region

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 -plane is fully contained in (21).

Figure 13 shows the ROSAT XUV-WFC pointing PSF density
for *E*=150eV at the off-axis angles
arcmin,
*k*=0,28,56,84,112 and 133, in the range
arcsec in the radial cross-section along the major axis, i.e. for *y*=0 with the UV filter.

Figure 13:
WFC Pointing PSF density for E=0.15keV,
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.

Figure 14:
WFC Pointing PSF density for
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

A good approximation and upper bound for

The component in (23) is exact, and is a sufficiently precise approximation for small off-axis angles arcmin and again exact for the remaining off-axis angles .

The exact expression for
allows the representation

The stable numerical evaluation of , as used for the EXSAS command (37), poses no problem. For the estimates from (17), it follows the enclosure

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

Figure 15:
Cumulative WFC Pointing PSF for E=0.15keV,
0, 28, 56, 84, 112, 133arcmin, radial cross-section,
UV Filter |

The differences to the approximation to from (23) are so small that no deviations are visible at the scale of Fig. 15.

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
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 of the corresponding PSF in effect in pointed observation mode.

We will make the following modelling assumptions:

- The detector's field of view is a circular disk of radius arcmin in the detector plane with the trace point of the optical axis as centre;
- The scanning angular velocity in survey motion is constant in the field of view;
- The distance, arcmin, between consecutive scanning tracks is small ( arcmin), and we assume the limiting case .

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 .
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
be the effective area function of the mirror-detector system under consideration. Then

is the energy and off-axis angle dependent vignetting function for the mirror-detector unit in operation. In EXSAS, 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 and 60. Then the vignetting corrected PSF density, , in pointing mode observation is

Taking instead of

Figure 17 shows the PSPC vignetting function for fourteen off-axis angles arcmin in the energy range [0,3]keV. The fact that decreases in allows the identification of the off-axis angle in Fig. 17.

Figure 17:
PSPC vignetting function, Vig, for 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
and 60.
Since it is based on amplitudes rather than energies, it is not to be used for
.
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.

The cumulative survey PSF

is shown in Fig. 19 for the energies

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,

Figure 20 shows the PSPC Survey PSF density, , based on the Gauss approximation as defined in (30) in the interval [1,3600] arcsec for the energies

Figure 20:
PSPC survey PSF
based on the Gauss approximation for
E=0.1,0.5,0.9,1.3,1.7,2keV for
[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.

Figure 21:
Cumulative PSPC survey PSF
based on Gauss approximation
for E= 0.1, 0.5, 0.9, 1.3, 1.7, 2keV for
arcsec |

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