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Assessing the EPIC spectral calibration in the
hard band with a 3C 273 observation
Silvano Molendi
IASF/CNR, Via Bassini 15, I-20133 Milano, Italy
Steve Sembay
X-Ray Astronomy Group, Leicester University, LE1 7RH, UK
March 17, 2003
Abstract
We describe the analysis of a simultaneous XMM-Newton Bep-
poSAX observation of 3C273 carried out in June 2001. Our primary
aim is to asses spectral parameters for the high energy (above 3keV)
spectrum of EPIC MOS, EPIC PN and MECS and to compare them.
We nd that in the 3-10 keV band EPIC MOS and PN spectral slopes
are contained within a
= 0:03, with the exception of the MOS1
measurement, which is found to be signicantly atter. The EPIC
slopes are in good agreement with those obtained with the MECS
on-board BeppoSAX. These results represent a major improvement in
our understanding of the EPIC high energy response.
1 MOS Analysis
Event les were processed at LUX by S. Sembay using the publically available
SAS 5.4.1 S/W. Our rst concern is to determine if pileup is present and to
what extent. There are various ways to go about this. One possibility is
to use epatplot. We have run epatplot on various annuli centered on the
emission peak. In Fig. 1, 2 and 3 we present results for 3 annuli with
bounding radii 0 00 -6 00 , 6 00 -37.5 00 and 15.0 00 -37.5 00 . Clearly annuli including data
1

from the innermost regions are aected by pile-up to some extent. The
current version of epatplot is extremely useful to detect pile-up at a glance,
however it does not write to le the expected fractions and therefore does
not allow to perform a more quantitative analysis.
Moreover, as we shall verify further on, the large discrepancy observed
above 2 keV between predicted and measured pattern 0 fractions does not
imply that major spectral distortions of the pattern 0 spectrum are occurring
at energies greater than  2 keV. At moderate pile up rates such as those
occurring in this observation the main eect of pile-up is to take pattern 0
events and turn them into doubles. Piled up pattern 0 events are less likely
simply because the probability of producing a piled-up double is 4 times
larger than that of producing a piled up single. The major eect on pattern
0 events is a loss of events to higher order patterns (mainly doubles). Since
the probability of an event to be lost to pile-up is independent of its energy,
the loss of events will result in a lower normalization of the pattern 0 spectrum
and not in a distortion of its shape. In epatplot the predicted fractions are
computed under the assumption that none of the recorded events are due to
pile-up, if however, a certain number of double, triple and quadrupole events
are due to pile-up, then the assumed fraction of pattern 0 events will be
miscalculated simply because based on an incorrect number of total events.
A defect of pat. 0 events is observed above 2 keV because it is at these
energies that the eective area starts to drop rapidly and piled-up doubles
become a sizeable fraction of all doubles.
1.1 Pattern ratios
Another possibility to assess pile-up is to derive the ratio of doubles to sin-
gles spectra for dierent annuli, we shall indicate with R bivp0 (r min ; r max ), the
ratio of the spectra for vertical bipixels to pattern 0 from the annulus with
bounding radii r min ,r max and with R bihp0 (r min ; r max ) the ratio of the spectra
for horizontal bipixels to pattern 0 from the annulus with bounding radii
r min ,r max expressed in arcseconds. For regions unaected by pile-up the ra-
tio should be the same, while for regions where pile-up is important the ratio
should be larger, particularly for those energies where "true" bipixels are
expected to be few. To quantify somewhat the above dierences, we have
computed R bivp0 (0; 3)=R bivp0 (6; 37:5) and R bihp0 (0; 3)=R bihp0 (6; 37:5) , i.e. a
ratio of ratios, this is less complicated than it may appear to be.
2

Figure 1: epatplot for a circle with radius 6 00
3

Figure 2: epatplot for an annulus with bounding radii 6 00 and 37.5 00
4

Figure 3: epatplot for an annulus with bounding radii 15 00 and 37.5 00
5

Figure 4: the R bivp0 (0; 3)=R bivp0 (6; 37:5) and R bihp0 (0; 3)=R bihp0 (6; 37:5) ratio
of ratios, see text for details.
6

In gure 4 we report the above ratio for MOS2, results for MOS1 are
very similar. In general the ratios lie above 1 indicating that some pileup is
present in the inner circle. Let us go in more detail and consider rst vertical
bipixels, below about 0.8 keV the ratio is close but slightly smaller than 1
indicating that pile-up is not very important and that the dominant eect
is that of a loss of events (most likely to higher order patterns). Around 2.2
keV we observe a jump, this is related to the gold edge. At energies larger
than 2.2 keV the eective area is reduced drastically and so is the probability
of producing a true bipixel, on the contrary the probability of producing a
bipixel through pile is unaected by the edge, consequently we observe a jump
in the ratio due to the increased fraction of piled up bipixels over true bipixels.
From about 2.2 keV to 9 keV the ratio remains essentially constant. Let us
now consider horizontal bipixels, as we know, the probability of producing
a true horizontal bipixel is smaller than that of producing a true vertical
bipixels. On the contrary the probability of producing a horizontal bipixel
from 2 piled up singles is identical to that of producing a vertical bipixel from
2 piled up singles, thus the fraction of pile up events we observe is larger for
horizontal bipixels than it is for vertical bipixels. Most interestingly the ratio
evidences a line-like structure emerging between 1.8 and 3 keV. This is due
to the combined eect of the Si edge at 1.8 keV and of the Au edge at 2.2
keV. Both these edges reduce drastically the amount of true bipixels while
leaving unaected the amount of piled-up bipixels, thus the ratio goes up.
Around 3 keV the ratio starts to drop because piled-up bipixels are now
being made in part from monopixels with energies larger than 1.8 keV. Up
to this point, the reader may have found the above results amusing but not
particularly important. The important part is that a mild pile up, such as
the one observed in the core of 3C 273 results in a substantial amount of
spurious bipixels in the energy range between 1.8-3 keV. Thus, when tting
pattern 0-12 spectra this will show up as an excess in the 1.8-3 keV energy
range. Since the energy range under exam is one where we observe the Si
and Au edges the naive calibrator may be lead to thinking that the problem
is related to a miss-calibration of the edge depth when it is infact a pile-up
problem.
7

1.2 Pattern 0 vs. patterns 0-12 spectral tting
Another way of assessing pile up is to t pattern 0 and pattern 0-12 spectra
from dierent annuli. For spectra including inner regions aected by pile-
up ts will yield dierent results for pattern 0 and pattern 0-12, for outer
regions where pile-up is absent they will yield consistent results. In Table
1 we report the result of spectral ts to MOS1 and MOS2 data in 2 annuli
with bounding radii 0 00 -6 00 , 6 00 -37.5 00 . Results are reported in the form of

where
 p0 12 p0
Table 1:
 p0 12 p0
Spectrum


3-10 keV 4-10 keV
MOS1 0 00 -6 00 0:08  0:04 0:17  0:06
MOS2 0 00 -6 00 0:06  0:04 0:10  0:06
MOS1 6 00 -37.5 00 0:02  0:03 0:00  0:05
MOS2 6 00 -37.5 00 0:01  0:03 0:02  0:04
In the inner ring the pattern 0-12 spectrum is atter than the p0 spec-
trum. This is due to piled up 0 patterns that are detected mostly as higher
order patterns with a spectrum that is harder than the real source spectrum
(the energy of the piled up event is the sum of the energies of the 2 or more
events that contribute). The pattern 0 spectrum is also aected by piled up
event which will harden the spectrum, however in this case the eect is sig-
nicantly smaller as the probability of producing a piled up valid double is 4
times larger than that of producing a piled up pattern 0. In the outer region
the eect of pile up is negligible and the pattern 0 and pattern 0-12 spectra
show consistent spectral slopes. It is interesting to note that the epatplot for
this outer region (see Fig. 2), show some evidence for pile up, however this
is insuфcient to introduce measurable spectral distortions in our ts.
Assuming that pattern 0 spectra are not substantially distorted even
within the core, we may compare the spectral ts for the two annuli. Since
each spectrum is analyzed using an eective area appropriate for the specic
choice of bounding radii, an agreement between spectral parameters would
indicate that the PSF correction is working properly. In Table 2 we report

where this time
 p0 (0 00 6 00 ) p0 (6 00 37:5 00 )
8

Table 2:
 p0 (0 00 6 00 ) p0 (6 00 37:5 00 )
Detector


3-10 keV 4-10 keV
MOS1 0:21  0:04 0:10  0:06
MOS2 0:21  0:04 0:10  0:05
Results for MOS1 and MOS2 are similar, for both detectors we nd that
the spectrum extracted from the core is signicantly steeper than the one
extracted from the wings. The obvious conclusion is that the PSF correction
does not work properly. The reader is cautioned that this does not imply that
the PSF correction does not work in general. The current PSF calibration is
the result of a detailed analysis of the radial proles of many dierent sources
observed with EPIC (Ghizzardi 2002) and for the typical case of a non-piled
up source with an extraction radius of 20 00 40 00 and no hole in the center it
does work adequately.
1.3 Diagonal Patterns
Yet another way of assessing the role of pile up is to consider diagonal bipixels
(patterns 26-29). The main advantage of diagonal bipixels is that they are
produced virtually only from the pile up of 2 single events. Consequently
diagonals can be used to estimate the importance of pile-up. In Fig. 5
we compare the image of 3C 273 in diagonal patterns with the one derived
for 0-12 patterns. The source is clearly visible in diagonals and the surface
brightness more peaked than in the standard patterns indicating, once again,
that some pile-up is present.
The eect of pile up can be split into two dierent parts. There is a gain
of events, and a loss of events, generally speaking the events which are gained
have energies larger than those that are lost as the former are made from the
sum of the latter.
Let us now for simplicity's sake concentrate on pattern 0 events and as-
sume that the only form of pile-ups are two photon pile-ups (i.e. we neglect
pile-ups due to 3 or more events). Following the above arguments the spec-
trum of observed singles S obs (E) per frame per pixel may be written as:
S obs (E) = S(E) S lost (E) + S gain (E); (1)
9

Figure 5: image of 3C 273. Left panel image in diagonal patterns; right
panel image in 0-12 patterns
where S(E) is the pattern 0 spectrum in the no pile-up limit, S lost (E) is
the spectrum \lost" to singles and higher order pattern pile-ups and S gain (E)
is the spectrum gained from single-single pile-ups. Both the spectrum of
events gained S gain (E) and lost S lost (E) from pile-up can be estimated from
the spectrum of diagonal events (patterns 26-29). To avoid confusion we
shall call S dia (E) the spectrum observed in diagonal patterns and S dia p0 (E)
the spectrum of pattern 0 reconstructed by splitting each diagonal event into
two pattern 0 events. It can be shown that
S gain (E) = 1
4 S dia (E); (2)
similarly S lost (E) may be related to S dia p0 (E) through the equation:
S lost (E) = 1
4 1
S dia p0 (E); (3)
where 1 = 9 + 3 2 + 6 3 + 7 4 and  1 ,  2 ,  3 and  4 are the fraction of
events giving rise to monopixels, bipixels tripixels and quadripixels in the no
10

pile-up limit. A derivation of Eq. (2) and (3) will be given in Appendix A.
As shown in Appendix B Eq. (2) and (3), which are derived in the case of
a source illuminating the MOS CCDs homogeneously, can be extended to a
generic point source with a surface brightness described by the MOS PSF.
Thus we may write:

S(E) = 
S obs (E) + 1
4 1

S dia p0 (E) 1
4

S dia (E); (4)
where 
S indicates the spectrum per frame per pixel averaged over the
source surface brightness distribution of the source. Since  1 ,  2 ,  3 and  4
depend upon energy, mean values where derived by averaging over the whole
MOS spectral range using a 3C 273 spectrum accumulated in the wings
of the PSF where no pile-up is present. Derived values are  1 = 0:8208,
 2 = 0:1623,  3 = 0:0069 and  4 = 0:0100.
Before applying Eq. (4) to our data we need to consider one last technical
issue, when running emchain with the default parameters diagonal patterns
are recognized and split into pattern 0 events, they are however agged and
consequently 
S dia p0 (E) can be easily extracted from the MOS event les.
On the contrary to derive 
S dia (E) diagonal patterns must not be split. An
easy way to do this is to run the emchain in the following fashion: emchain
emevents:splitdiagonals=N.
In Table 3 we report the results of spectral ts to MOS1 and MOS2
spectra extracted from a circle with a radius of 37.5 00 . For each detector
we report results for 3 spectra, the rst is accumulated from patterns 0-12,
the second and third are accumulated from pattern 0 only. Of the pattern
0 spectra the rst is not corrected for pile-up eects, while the second is
corrected using the pattern 26-29 spectrum as indicated in Eq. (4).
Clearly there is a dierence between pattern 0-12 spectra and pattern
0 spectra, this is due to piled-up events present in the patterns 1-12 which
harden the pattern 0-12 spectrum. The dierence between the uncorrected
and the corrected pattern 0 spectra is small indicating that at the moderate
counting rate of 3C 273 pattern 0 spectra are only mildly distorted from
pile-up.
11

Table 3: comparison of spectral ts to MOS1 and MOS2 data
Detector
3-10 keV 4-10 keV
MOS1 pat0-12 1:50  0:02 1:50  0:03
MOS1 pat0 1:54  0:02 1:62  0:03
MOS1 pat0 diacor 1:57  0:02 1:67  0:04
MOS2 pat0-12 1:57  0:02 1:52  0:03
MOS2 pat0 1:62  0:02 1:63  0:03
MOS2 pat0 diacor 1:65  0:03 1:70  0:04
2 PN
PN event les were generated by M. Freyberg using the version of epchain
available at MPE in November 2002. Spectra and response matrices were
produced by F. Haberl, the response matrices are coded as PN6.3 dec02.
we have veried a posteriori that running SAS 5.4.1 on the ODF les we
derive consistent results.
The nominal frame-time for the PN small window is 5.67 ms. Taking into
account that the PN pixels are about 16 times larger than the MOS pixels,
that the MOS frame-time in SW is 0.3 s and that the MOS2 and PN count
rates are respectively 17.5 and 70 cts/s, we nd that the PN counts per frame
per pixel are about the same of those of either MOS1 or MOS2. Since the
MOS pile-up on singles is small we expect that this will also be the case for
PN singles; a modest pile-up may aect PN doubles which are of course more
sensitive than singles. This conclusion is conrmed by the epatplot analysis.
In Fig. 6 we report the epatplot for a circle with a 40 00 radius centered
on the emission peak. The small discrepancy seen below 600eV is related
to a PN calibration issue currently under investigation and not related to
pileup. A real pile up problem would show up as a double excess (and single
defect) above 1-2 keV, no such eect is seen. Some discrepancy is observed
above 8 keV, however similar discrepancies have been observed before and
are probably not related to pile up.
We have performed power-law ts in the 3-10 keV and 4-10 keV band
for singles and doubles for 3 dierent extraction regions: a circle with a 40 00
radius and two annuli with bounding radii 5 00 -40 00 and 10 00 -40 00 . Results are
12

Figure 6: epatplot for a circle with radius 40 00
13

reported in Table 4.
Table 4: Fits to PN spectra.
Spectrum
3-10 keV 4-10 keV
PN S 0 00 -40 00 1:65  0:01 1:67  0:01
PN D 0 00 -40 00 1:62  0:01 1:62  0:01
PN S 5 00 -40 00 1:67  0:01 1:72  0:02
PN D 5 00 -40 00 1:63  0:01 1:66  0:02
PN S 10 00 -40 00 1:67  0:01 1:72  0:02
PN D 10 00 -40 00 1:63  0:01 1:66  0:02
As can be seen there is a dierence in the spectral slope recovered from
singles and doubles. The dierence is larger in the 4-10 keV ts than in the
3-10 keV ts. Inspection of the residuals shows that it is related to an excess
in the doubles spectrum in the 8-10 keV band. Pile-up is unlikely to be a
cause for this as the excess is observed in all the doubles spectra reported in
Table 5. Interestingly, F. Haberl has recently reported that a similar eect
has been observed in other sources, a likely origin could be a small error in
the doubles to singles relative quantum eфcency in the 8-10 keV range.
3 MECS
A BeppoSAX observation was carried out simultaneously with the XMM-
Newton observation. The MECS instrument on-board BeppoSAX operating
in the 2-10 keV band is well suited for a comparison with the EPIC MOS and
PN hard bands. The MECS is a GSPC which has been extensively calibrated
both on ground and in- ight.
We have performed spectral ts to the combined MECS2 and MECS3
spectrum produced by the standard pipeline at the BeppoSAX SDC in Rome.
The extraction radius is 4 0 . The energy band used are as for EPIC 3-10 keV
and 4-10 keV.
14

4 Comparing EPIC MOS, EPIC PN and MECS
In Table 5 we compare the results of EPIC PN, EPIC MOS and MECS
spectral ts in the 3-10 keV and 4-10 keV. For MOS we report the pattern 0
spectra corrected for pile-up according to the procedure reported in Section
1; for PN we report PN singles and doubles spectra for the circle with a
40 00 extraction radius and for MECS the ts to the combined MECS2 and
MECS3 spectrum produced by the standard pipeline at the BeppoSAX SDC
in Rome.
Table 5: comparison between ts obtained with dierent instruments
Spectrum
3-10 keV 4-10 keV
MOS1 pat0 diacor 1:57  0:02 1:67  0:04
MOS2 pat0 diacor 1:65  0:03 1:70  0:04
PN S 0 00 -40 00 1:65  0:01 1:67  0:01
PN D 0 00 -40 00 1:62  0:01 1:62  0:01
MECS23 1:63  0:02 1:64  0:03
Let us rst address the issue of the MOS PN cross-calibration. In the
3-10 keV band all measurements, except the MOS1, cluster within a
of
0.03 and are consistent with one another at the 2.2 level. The MOS1 slope
is signicantly harder than all others,
 0:07. Interestingly when we go
to the 4-10 keV band all measurements, including the MOS1, are contained
within a
of 0.08. Each measurement is consistent with any other at the
1 level, with the exception of the PN singles and PN doubles measurements
which dier at the 3.5  level (for more details on this see the PN section).
As far as the comparison with the BeppoSAX MECS is concerned, we nd
that in the 3-10 keV band all EPIC measurements, except for the MOS1, are
in good agreement with the MECS measurement. For the 4-10 keV the MECS
slope is consistent with all EPIC slopes, indeed the MECS 1 condence re-
gion encompasses all the EPIC measurements. Considering that BeppoSAX
and ASCA are no longer operational and that Chandra's eective area drops
sharply above 6 keV, the observation at hand is likely one of the best we have
to perform a hard band cross-calibration between XMM-Newton EPIC and
another X-ray experiment on a dierent observatory. Thus the fact that the
15

MECS measurements in the 3-10 keV and 4-10 keV band are consistent with
all EPIC measurements, except of course the MOS1 measurement in the 3-10
keV band, implies that cross-calibration has been achieved to the available
statistical level and that it is unlikely that BeppoSAX, or any other satellite
for that matter, will be able to place more stringent constraints on spectral
parameters derived with EPIC in the hard band. From now on eorts will
have to be concentrated on improving the MOS / PN cross calibration. Here
we are left with two problems: the MOS1 response in the 3-10 keV band and
the PN singles and doubles inconsistency in the 4-10 keV band. Given these
problems we may quantify the residual systematic indetermination on the
spectral slope  to be 0.02 in the 3-10 keV band (if we exclude the MOS1
result) and 0.04 in the 4-10 keV.
5 Summary
The main results of this Report may be summarized in the following bullets.
 MOS Spectra
{ At the observed rate of  17 cts/s MOS1 and MOS2 spectra show
signicant pile-up in the PSF core. This is borne out by the
standard epatplot analysis, the pattern ratio analysis described in
section 1.1, the pattern 0 vs. pattern 0-12 spectral tting (section
1.2) and the diagonal pattern analysis (section 1.3).
{ In section 1.3 (see Appendices A and B for details ) we provide a
formula to correct pattern 0 spectra from pile-up. The formula can
be used for any source provided that pile-up is relatively modest,
(i.e. pile-ups involving 3 or more photons are neglected). Appli-
cation of this formula to the MOS data shows that in the case at
hand pattern 0 spectra are only mildly distorted by pile-up.
{ Pattern 0-12 spectra are substantially distorted by pile-up, mea-
sured spectral indices are atter than they would be in the absence
of pile-up and features appear at the position of the Si and Au
and edges, these are due to piled up events which ll-up the sharp
drops expected at these energies.
 PN Spectra
16

{ PN spectra are not signicantly piled-up, this is borne out by the
comparison of singles and doubles spectral ts and by the epatplot
analysis.
{ PN doubles spectra are atter (
 0:04 ) than PN singles
spectra, the most likely cause for this dierence is a an error in
the doubles to singles relative quantum eфcency in the 8-10 keV
range.
 MOS / PN cross calibration: in the 3-10 keV band MOS2, PN singles
and PN doubles measurements are all found within a
 0:03, only
MOS1 returns a signicantly atter spectral slope.
 EPIC / MECS cross calibration: EPIC measurements, with the ex-
ception of the MOS1 measurement in the 3-10 keV band, are in good
agreement with the MECS measurements.
Acknowledgments
It is a pleasure to acknowledge the many colleagues that have helped
us. Thanks are due to to F. Haberl and M. Freyberg for the production of
datasets and response matrices. M. Freyberg and B. Altrieri are thanked for
their comments to a preliminary version of this report, and M. Turner for
stimulating discussions on pile-up. Finally we are in debt with J.Ballet for a
scrupulous reading of the manuscript, appendices included, which uncovered
errors in some of the formulae.
A Appendix A
Let us consider the case of a source illuminating the MOS CCDs homoge-
neously. Let us also assume that the only form of pile-ups are two photon
pile-ups (i.e. we neglect pile-ups due to 3 or more events). The spectrum of
singles in the no pile-up limit S(E) per frame per pixel may be written as:
S(E) = S obs (E) + S lost (E) S gain (E); (A1)
where S obs (E) is the observed pattern 0 spectrum, S lost (E) is the spec-
trum \lost" to singles and higher order pattern pile-ups and S gain (E) is the
spectrum gained from single-single pile-ups. The spectrum of events gained
17

from pile-up S gain (E) can be expressed as a convolution of the non-piled up
spectrum with itself.
S gain (E) = 1
2
Z
E1
S(E 1 )
S(E E 1 ) dE 1 : (A2)
To derive the lost spectrum we must rst recall that the loss of events
does not result in a distortion of the spectra shape. Indeed the probability
of an event to be piled-up is independent of its energy. Thus the loss will
simply result in a reduction of the normalization of the spectrum which can
be readily estimated from the formulae reported in Ballet (1999). The rate
of recorded monopixels events per frame per pixel can be expressed as:
 1 = (e 1 1)e 1  (A3);
where:  is the in-coming X-ray ux/pixel/frame, 1 = 9+3 2 +6 3 +7 4 and
 1 ,  2 ,  3 and  4 are the fraction of events giving rise to monopixels, bipixels
tripixels and quadripixels in the no pile-up limit, Since the above parameters
vary with energy, their actual values are computed by averaging over the
whole spectral range. Equation (A3) is easily understood, if we expand the
rst term in a Taylor series we nd  1  + ( 1 ) 2 =2+ ( 1 ) 3 =3+ ::: the rst
term in the expansion is the expected rate in the no pile-up limit, the second
term accounts for the gain from 2 photon pile-up, the third term for the gain
from 3 photon pile-up and so on. The second term in Eq. (A3), e 1  , is a
loss term, in the no pile-up limit it reduces to 1. For modest pile-up it may
be expressed as (1 1 ), where 1  accounts for the events which have been
lost to form piled up singles doubles tripixels and quadripixels. Under this
condition the rate of lost events may be written as  1 lost = 1  1  2 . Similarly
it can be shown that the spectrum of lost events may be expressed as:
S lost (E) = 1
 1
( 1 ) S(E); (A4)
where  1  = R
E S(E)dE and R
E S lost (E)dE =  1 lost .
Both the spectrum of gained events, S gain (E), and lost events, S lost (E),
can be estimated from the spectrum of diagonal events (patterns 26-29).
To avoid confusion we shall call S dia (E) the spectrum observed in diagonal
patterns and S dia p0 (E) the spectrum of pattern 0 reconstructed by splitting
each diagonal event into two pattern 0 events. As far as the S gain (E) term
18

is concerned we recall that diagonals are produced from the pile-up of two
pattern 0 events just as piled-up singles. Thus the dierence is only in the
normalization, which for diagonals is 4 times larger to account for the 4
possible positions of the second event giving rise to the diagonal. We may
then write:
S gain (E) = 1
4 S dia (E): (A5)
Similarly S lost (E) may be related to S dia p0 (E), if we recall that lost single
events are either lost to permitted singles, doubles, tripixels and quadripixels
or to diagonals, i.e.: S lost (E) = S per p0 (E) + S dia p0 (E), where S per p0 (E) is
the spectrum lost to permitted patterns and S dia p0 (E) is the spectrum lost
to diagonals. S per p0 (E) and S dia p0 (E) are related to S(E) as follows:
S per p0 (E) = 1 p S(E); S dia p0 (E) = 1 d S(E);
where the loss coeфcient 1 d , which may be derived from Eq. (A3) or
(A4) of Ballet (1999) is 1 d = 4 1 and 1 p is obtained from the relation
1 p = 1 1 d . From the above it follows that S lost (E) may be easily
re-written in terms of S dia p0 (E):
S lost (E) = 1
4 1
S dia p0 (E): (A6)
B Appendix B
The equations derived in Appendix A refer to the case of a source illuminating
the MOS CCDs homogeneously. In this Appendix we shall see how they may
be extended to a generic point source with a surface brightness described by
the MOS PSF. Let us start by dening S(E; r) as the spectrum per pixel
per frame at the radial distance r from the center of the surface brightness
distribution in the no pile-up limit.
Since the size of the MOS pixels is smaller than the PSF of the XMM
telescope the equations derived in Appendix A are all applicable locally. For
instance we may write:
S(E; r) = S obs (E; r) + S lost (E; r) S gain (E; r): (B1)
19

In the case of S gain (E; r) we may write:
S gain (E; r) = 1
2
Z
E 1
S(E 1 ; r)
S(E E 1 ; r) dE 1 : (B2)
We may dene a mean spectrum as follows:

S gain (E) 
R rmax
r min
2rdrS gain (E; r)
R rmax
r min
2rdr : (B3)
As for other equations derived in Appendix A Eq. (A5) applies locally.
i.e.:
S gain (E; r) = 1
4 S dia (E; r): (B4)
Substituting Eq. (B4) in (B3) and dening

S dia (E) 
R rmax
r min
2rdrS dia (E; r)
R rmax
r min
2rdr ; (B5)
we obtain the generalization of Eq. (A5):

S gain (E) = 1
4

S dia (E): (B6)
Similarly if we dene

S lost (E) 
R rmax
r min
2rdrS lost (E; r)
R rmax
r min
2rdr ; (B7)
assume Eq. (A6) to be valid locally, i.e.
S lost (E; r) = 1
4 1
S dia p0 (E; r); (B8)
and dene

S dia p0 (E) 
R rmax
r min
2rdrS dia p0 (E; r)
R rmax
r min
2rdr ; (B9)
we derive the generalization of Eq. (A6):

S lost (E) = 1
4 1

S dia p0 (E): (B10)
20

Integrating Eq. (B1) over the source surface brightness we derive

S(E) = 
S obs (E) + 
S lost (E) 
S gain (E); (B11)
nally substituting Eq. (B6) and (B8) in (B11) we obtain

S(E) = 
S obs (E) + 1
4 1

S dia p0 (E) 1
4

S dia (E): (B12)
21