# Handling the background in IXPE polarimetric data

Alessandro Di Marco INAF – IAPS, via Fosso del Cavaliere, 100, Rome, Italy
I-00133  Paolo Soffitta INAF – IAPS, via Fosso del Cavaliere, 100, Rome, Italy
I-00133  Enrico Costa INAF – IAPS, via Fosso del Cavaliere, 100, Rome, Italy
I-00133  Riccardo Ferrazzoli INAF – IAPS, via Fosso del Cavaliere, 100, Rome,
Italy I-00133  Fabio La Monaca INAF – IAPS, via Fosso del Cavaliere, 100,
Rome, Italy I-00133  John Rankin INAF – IAPS, via Fosso del Cavaliere, 100,
Rome, Italy I-00133  Ajay Ratheesh INAF – IAPS, via Fosso del Cavaliere, 100,
Rome, Italy I-00133  Fei Xie Guangxi Key Laboratory for Relativistic
Astrophysics, School of Physical Science and Technology,Guangxi University,
Nanning, China  INAF – IAPS, via Fosso del Cavaliere, 100, Rome, Italy I-00133
Luca Baldini Università di Pisa, Dipartimento di Fisica Enrico Fermi, Largo B.
Pontecorvo 3, Pisa, Italy, I-56127  INFN-Pisa, Largo B. Pontecorvo 3, Pisa,
Italy, I-56127  Ettore Del Monte INAF – IAPS, via Fosso del Cavaliere, 100,
Rome, Italy I-00133  Steven R. Ehlert NASA Marshall Space Flight Center,
Huntsville, AL 35812, USA  Sergio Fabiani INAF – IAPS, via Fosso del
Cavaliere, 100, Rome, Italy I-00133  Dawoon E. Kim INAF – IAPS, via Fosso del
Cavaliere, 100, Rome, Italy I-00133  Università di Roma “Tor Vergata”,
Dipartimento di Fisica, Via della Ricerca Scientifica, 1, Rome, Italy, I-00133
Università di Roma “La Sapienza”, Dipartimento di Fisica, Piazzale Aldo Moro
2, Rome, Italy I-00185  Fabio Muleri INAF – IAPS, via Fosso del Cavaliere,
100, Rome, Italy I-00133  Stephen L. O’Dell NASA Marshall Space Flight Center,
Huntsville, AL 35812, USA  Brian D. Ramsey NASA Marshall Space Flight Center,
Huntsville, AL 35812, USA  Alda Rubini INAF – IAPS, via Fosso del Cavaliere,
100, Rome, Italy I-00133  Carmelo Sgrò INFN-Pisa, Largo B. Pontecorvo 3, Pisa,
Italy, I-56127  Stefano Silvestri Università di Pisa, Dipartimento di Fisica
Enrico Fermi, Largo B. Pontecorvo 3, Pisa, Italy, I-56127  INFN-Pisa, Largo B.
Pontecorvo 3, Pisa, Italy, I-56127  Allyn F. Tennant NASA Marshall Space
Flight Center, Huntsville, AL 35812, USA  Martin C. Weisskopf NASA Marshall
Space Flight Center, Huntsville, AL 35812, USA

(Revised February 2 2022)

###### Abstract

IXPE (Imaging X-ray Polarimetry Explorer) is a Small Explorer mission by NASA
and ASI, launched on December 9th 2021, dedicated to investigating X-ray
polarimetry allowing angular-, time- and energy-resolved observations in the
2–8 keV energy band. IXPE is in Science Observation phase since January 2022;
it comprises of three identical telescopes with grazing-incidence mirrors,
each one having in the focal plane a Gas Pixel Detector (GPD). In this paper,
we present a possible guideline to obtain an optimal background selection in
polarimetric analysis, and a rejection strategy to remove instrumental
background. This work is based on the analysis of IXPE observations, aiming to
improve as much as possible the polarimetric sensitivity. In particular, the
developed strategies have been applied “as a case study” to the IXPE
observation of the 4U 0142+61 magnetar.

X-rays — polarimetry — gas detectors — background

††journal: AJ

##  1 Introduction

In X-ray astronomy, since the first observation of a source outside the Solar
system — Sco X-1 (Giacconi et al., 1962) — the background has had an important
role, in some instances limiting the exploitation of science data. During this
first detection, indeed, two other sources of counts were observed: a diffuse
emission coming from the whole sky, and an instrumental background. After
sounding rockets, X-ray experiments started to be flown on satellites, a few
examples were Uhuru (Giacconi et al., 1971), OSO-8 – which had a polarimeter
on board – (Kestenbaum et al., 1976), and HEAO1 (Bradt et al., 1992). These
had onboard X-rays detectors using collimators to achieve some degree of
angular resolution, but their sensitivity could be increased only by a larger
collecting area, which implies a higher instrumental background. Because of
this, the more advanced experiments of this kind were carefully designed to
minimize background intensity and/or maximize background reproducibility. A
good example of those advanced collimated detectors was the PDS experiment
(Frontera et al., 1997) on board BeppoSAX (Boella et al., 1997): source
observation was achieved by using two rocking collimators — alternatively
pointing the source and the background — so obtaining a continuous background
monitoring to control systematics associated to the single collimator and to
the background field. Despite half of the collecting area being used for
background monitoring, this strategy was quite successful because systematic
errors in the background were typically $<1$%. This allowed for an
unprecedented exploration of the spectral properties of several extra-galactic
sources.

Moreover, Einstein (Giacconi et al., 1979), adopting X-ray mirrors, changed
the picture considerably: telescopes, concentrating photons on a small spot,
significantly increased the ratio between source and background counting rates
for point-like sources. As long as the angular extent of the source is smaller
than the Point Spread Function (PSF), the better the angular resolution of the
telescope, the smaller the spot and the higher the ratio between source and
background counting rates. In the case of extended sources to be resolved,
this is obviously no longer true, but in light of the substantial advantage
offered by concentrators, in these instruments, significantly less attention
was given to background reduction and reproducibility. One exception to this
rule was the PSPC experiment (Pfeffermann et al., 1987) onboard ROSAT
(Aschenbach et al., 1981), an X-ray imaging detector featuring an anti-
coincidence system. Thanks to it, the achieved sensitivity to low surface
brightness emission has been unrivaled for over 25 years. Moreover, in the
last decades, Charge Coupled Device (CCD) detectors have substituted gas
detectors, because of the significant improvement in the spectral resolution
that could be achieved. This substantial improvement was obtained at a price:
the CCD detectors that have been flown are much slower than gas detectors,
implying that a reduction of instrumental background through active shielding
was no longer possible. This is one of the reasons why some X-ray experiments
show a high background and a limited sensitivity to low surface brightness
emission. Only in the last years, new technologies have led to CCD detectors
featuring anti-coincidence systems like NuSTAR (Harrison et al., 2013) and
Hitomi (Mitsuda et al., 2014).

Also, the orbit affects the background properties: High Earth Orbits allow
long uninterrupted observing windows, but suffer from a higher and far less
predictable background than Low Earth Orbits; these latter ones lead to a
lower and more stable background thanks to the shielding provided by Earth’s
magnetic field but suffer from source occultations.

IXPE (Imaging X-ray Polarimetry Explorer) is a new NASA Small Explorer (SMEX)
mission launched on 9th December 2021 by a Falcon 9; the mission operates at a
600 km altitude equatorial orbit and is fully dedicated to X-ray polarimetry
(Weisskopf et al., 2022). IXPE hosts three telescopes with a Gas Pixel
Detector (GPD) (Costa et al., 2001; Soffitta et al., 2021; Baldini et al.,
2021) in the focal plane. It started its observation plan in January 2022, and
during its first year, it is obtaining scientifically meaningful measurements
of the X-ray polarization in several sources, allowing us to improve our
knowledge of their geometry and emission mechanisms. Besides measuring X-ray
polarization, the telescopes provide moderate angular resolution over a useful
9’$\times$9’ field of view, moderate spectral resolution, and excellent
timing. Pointing-system capabilities were enhanced to support
“dithering”–controlled and small–amplitude oscillations of the pointing
direction to compensate for the low level of residual systematic.

In this paper, the IXPE detectors are briefly described to show the impact of
background on them. Moreover, background handling for polarimetric data
analysis is presented with particular care to its selection and rejection,
aiming to reach the best achievable sensitivity. Then the 4U 0142+61 magnetar
IXPE observation is analyzed to show the impact of this rejection+selection
strategy on the results.

##  2 The impact of background on IXPE focal plane detectors

Before IXPE, only one significant measurement of X-ray polarization had been
obtained, in 1975 by OSO-8. This mission had two graphite polarimeters
onboard, which allowed it to obtain a precise polarization measurement of the
X-ray flux from the Crab Nebula (Weisskopf et al., 1976, 1978). Only in 2018,
a new mission for X-ray polarimetry was launched: the CubeSat mission named
Polarimeter Light (PolarLight). Based on the GPD, PolarLight demonstrated the
potentiality of this detector, despite limitations due to the design typical
of CubeSat (small area, no optics), it obtained a new measurement of X-ray
polarization for the Crab Nebula (Feng et al., 2020) and Sco X-1 (Long et al.,
2022).

IXPE performances allow us to improve the polarimetric sensitivity, mainly
thanks to the imaging capabilities and the higher effective area obtained by
putting GPDs at the focus of X-ray mirrors. To estimate the sensitivity of a
polarimeter, the standard parameter to be used is the Minimum Detectable
Polarization (MDP) at a confidence level of 99%. It is defined as (Weisskopf
et al., 2010):

$MDP_{99}=\frac{4.29}{\mu S}\sqrt{\frac{S+B_{diff}+B_{res}}{T}},$ (1)

where:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int A(E)\epsilon(E)F(E)\,dE$
$\displaystyle B_{diff}$ $\displaystyle=$ $\displaystyle\int B_{diff}(E)\,dE$
$\displaystyle B_{res}$ $\displaystyle=$ $\displaystyle\int B_{res}(E)\,dE$
$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\int\mu(E)A(E)\epsilon(E)F(E)\,dE$

that are: $\mu$ the modulation factor, that is the response of the polarimeter
to a fully polarized source, and for photoelectric polarimeters it depends on
energy; $A$ the area of the source (for a focal plane instrument it is the
effective area of the X-ray telescope); $\epsilon$ is the quantum efficiency
of the detector; $F$ is the photon spectrum (photons$\cdot$s${}^{-1}\cdot$
cm${}^{-2}\cdot$ keV-1); $B_{diff}$ is the diffused X-ray background collected
by the instrument; $B_{res}$ is the residual background, that is the remaining
background after all the rejection techniques (described in the following)
have been applied; $T$ is the integration time (s). The smaller the $MDP_{99}$
is, the higher the reachable sensitivity is; reducing the background is
crucial to reach the highest possible sensitivity. Typically, for bright
point-like sources, for which it is possible to select the source region
spatially, the background can be assumed to be negligible:
($S+B_{diff}+B_{res}\simeq S$), but for faint or extended sources, even the
brightest ones, the background needs to be taken into account and properly
handled. Indeed, the background is not polarized, and it can dilute the
measured polarization degree ($P_{meas}$) of a source with respect to its true
polarization ($P_{src}$):

$P_{meas}=P_{src}\left(1+\frac{B_{res}+B_{diff}}{S}\right)^{-1}.$ (2)

From equations 1 and 2, the need to reduce the ratio between background and
source as much as possible with a proper background-rejection approach is
evident.

Typically, silicon detectors have a larger residual background than
proportional counters, mainly because a suitable background rejection cannot
be applied in silicon detectors. Proportional counters for background
rejection can make use, for example, of the pulse-shape discrimination (Ricker
& Gomes, 1969) or of anti-coincidence systems that allow reaching a background
rejection up to 99%. Pulse shape analysis is a synthetic parameter that
relates the extension of a track to the timing properties of the signal, and
it was successfully applied also in cylindrical proportional counters, where
it was used to detect the orientation of the track with respect to the anode
(Sanford et al., 1970) and therefore X-ray polarization, and in proportional
counters with a configuration with a constant drift field and a multi-wire
anode plane.

###  2.1 IXPE focal plane detectors: the GPDs

IXPE’s GPDs are able to obtain several information for each X-ray, such as
energy, polarization, and time of arrival, exploiting the photoelectric effect
in the gas. Summarizing, X-rays arrive through a beryllium window into the gas
cell where the photon, via the photoelectric effect, ionizes an atom of the
gas filling the GPD. The produced charge is driven by an electric field
parallel to the optical axis toward a GEM, where the ionization is amplified
before being read on the pixels of an Application Specific Integrated Circuit
(ASIC). The GPD final readout is an image of the ionization track (see the
example in Figure 1) containing the collected charge for each pixel.

Figure 1: An example of an ionization track resulting from absorption of a
5.89 keV X-ray, as amplified by the GEM and imaged onto the GPD’s pixelated
anode — a 300$\times$352 array of hexagonal pixels at 50 $\mu$m pitch (Sgrò,
2017).

The GPD gas cell is filled with Dimethyl Ether (DME) gas at 0.8 atm, which is
closed on the top by a Titanium frame and a Beryllium window, and on the
bottom by the ASIC. On the sides of the gas cell are Macor spacers. The DME
gas, used in the IXPE GPD, has a low diffusion coefficient and also selects
the energy range, of 2–8 keV. Photoelectron tracks collected by the GPD are
analyzed by a custom algorithm to extract relevant information. In this
algorithm, the photoelectron track direction is determined on the basis of a
two-step moment analysis that has been refined over the years (Bellazzini et
al., 2003a, b; Baldini et al., 2021; Di Marco et al., 2022a). In Di Marco et
al. (2022a), a summary of this algorithm is described.

In the GPD, the diffused X-ray background has a very low impact on the total
background due to the typical Half Energy Width (HEW) of the instrument; the
residual instrumental background is, therefore, much larger. The latter one
can be rejected only by using track parameters estimated in the track
reconstruction algorithm. An approach based on pulse shape discrimination
cannot be applied because of the lack of time distribution information for the
charge during the collection on the GPD ASIC; however, the time distribution
of the track is a proxy for its extension that is directly measured by the
GPD.

###  2.2 Extraction of polarimetric information from IXPE data

IXPE is providing us information on imaging, energy, time, and polarization
from several categories of X-ray astronomical sources. It is sensitive to
linear X-ray polarization, and it is optimized to operate in the nominal 2–8
keV energy band.

IXPE data are distributed on HEASARC in two levels/formats: event_l1 and
event_l2. The event_l1 files consist of raw data containing unfiltered events,
including data from in-flight calibrations, and they provide all the
information about the collected tracks. The event_l2 files are produced from
event_l1 after calibration/correction procedures. In particular, they are
cleaned of in-flight calibration data and occultation/SAA passages. After
this, energy calibration/equalization is applied, and the Stokes parameters
are estimated from the photoelectron emission angle, $\phi_{k}$, for every
single k-th event, as explained in Kislat et al. (2015)

$\displaystyle i_{k}$ $\displaystyle=$ $\displaystyle 1$ $\displaystyle q_{k}$
$\displaystyle=$ $\displaystyle 2\cos 2\phi_{k}$ (3) $\displaystyle u_{k}$
$\displaystyle=$ $\displaystyle 2\sin 2\phi_{k},$

and a weight to optimize the polarimetric response is calculated (Di Marco et
al., 2022a). To take into account systematic effects due to spurious
modulation, this latter one is removed from the Stokes parameters on an event-
by-event approach, as explained in (Rankin et al., 2022). After this latter
correction, photons and their Stokes parameters are referenced to sky
coordinates, removing dithering pattern and boom motion effects.

The IXPE nominal 2–8 keV energy band is constrained by the effective area of
the Mirror Module Assemblies (MMA), and the efficiency of the Detector Unit
(DU) placed in its focal plane (Soffitta et al., 2021; La Monaca et al.,
2022). The effective area of the MMAs has slight variations in the 2–7 keV
energy band ($\sim 168$ cm2 at 2.3 keV and $\sim 195$ cm2 at 4.5 keV), while
at 1 keV and at 8 keV it drops to $\sim 50$ cm2. Also, the DU efficiency, that
is the convolution of the GPD quantum efficiency and the X-ray transmission of
an ion-UV filter (La Monaca et al., 2021) mounted on top of the DU, has a
maximum ($\sim 20$%) at 2 keV with a quick drop at lower energies ($\sim 0.1$%
at 1 keV), and a slower decrease up to 8 keV where it reaches a value of $\sim
1$%. IXPE telescopes are numbered from 1 to 3 to be identified, the same
nomenclature is applied to MMAs, and DUs included in them.

In the following sections, a background optimal selection approach is shown on
the basis of studies of the background induced polarimetric signal as
estimated from event_l2 files. To identify a possible background rejection
approach, we started from the event_l1 files containing several parameters for
each track, then the best set of them is identified to obtain a set of
constraints to tag background events. Each photon in the event_l2 data is also
present in the event_l1, and they can be identified by using the TIME
information. Thus, we can select the background events in event_l1 files, then
they can be removed from the TIME identifier from the event_l2 photon list to
obtain a background rejected data set to be characterized, as shown in the
following. In the end, to show the effects on one of the IXPE’s observations,
this approach is applied to data analysis of 4U 014+61 magnetar.

##  3 Background selection

In IXPE, the observing strategy is ”point-and-stare“, which means that one
source at a time is observed, best-centered with respect to the sensitive area
of the three detectors. In the case of point-like sources, the source can be
spatially selected in a circular region with a radius that is at least equal
to the IXPE PSF (HEW$\simeq$ 30”). Similarly, the background can be selected
in a circle or an annular region outside the source region. In this section,
the IXPE observation of the 4U 0142+61 magnetar, because of its high exposure
($\sim 2$ weeks) and its low flux (unabsorbed flux $\sim 7\times 10^{-11}$
erg$\cdot$s${}^{-1}\cdot$cm-2 in the 2–10 keV, Taverna et al. (2022)), is used
as a typical case study to characterize the background behavior when these
selections are applied. The 4U 0142+61 observation was performed from January
31 at 07:23 UTC until February 14 at 23:44 UTC and from February 25 at 04:38
UTC until February 27 at 18:46 UTC.

Particles, inducing instrumental background, don’t have polarization, but they
can induce a modulation mimicking the X-ray polarization signal. Being not
possible in IXPE to obtain particle identification, we are not able to
distinguish between the X-ray polarimetric signal and the fake modulation
induced by the particle background; thus, we will treat this fake signal as a
background induced polarization in the following. To estimate this effect
along the radial distance from the point-like source, we spatially selected
the source first in a circular region with a 40” radius, then in an annular
region with a 40” of difference between the inner and the outer radius. We
increased the external radius (and consequently the internal one) up to 440”.
The polarization as a function of the mean radius is shown in Figure 2 for
events in the IXPE nominal energy band of 2–8 keV.

Figure 2: Polarization degree (left) and angle (right) in the IXPE nominal
energy band 2–8 keV as a function of the radius for the IXPE observation of 4U
0142+61, in orange, and the corresponding MDP99, in dotted blue line: the
source is significantly polarized (first point within a circular region with
radius 40”), and the background induced polarization signal (see text) is
below the MDP99 up to $\sim 350$ arcsec, where it starts to be significant
because of geometrical effects due to the shape of the IXPE detectors.

The first data point is, by the fact, the source polarization in a small
circular region with a 40” radius; the source shows a significant polarization
as described in the IXPE discovery paper (Taverna et al., 2022). The
polarization degree reaches values compatible with zero starting from
$\sim$100”, up to 300” from the center. The polarization angle shows values
compatible with the source up to $\sim$100”, from which it starts to be
compatible with zero up to $\sim$350”. The regions more distant than
$\sim$350” from the center show polarization above the MDP, as shown in Figure
2. The MDP value increases because of the lower counting rate in the annular
regions up to 100”, then starts to decrease due to larger regions starting to
include more events up to $\sim$350”, where edge effects begin. To have a safe
annular selection of the background, it should have an inner radius of 150”,
(i.e. 2.5 arcmin, which is about 5 times the IXPE HEW diameter, reducing the
probability to have source events) and an outer radius of 300” (5 arcmin) to
avoid geometrical edge effects due to the detectors’ shape. For the source
selection, circular regions up to $\sim$100” should include more events from
the source than the background, allowing to select a region as large as
possible to decrease the MDP99 as much as possible, avoiding dilution effects
due to background.

A second possible choice in the background selection is the use of circular
regions, but from the previous considerations, to avoid contribution from the
source and/or from detector edge effects, such circular regions can have a
maximum radius of $\sim$100”. In Figure 3, the annular region obtained with
the previous prescriptions is compared with 4 possible circular selections for
which the background induced polarization is estimated, and in Figure 4, the
background induced polarization degree arising from these regions is compared:
a circular selection has a larger uncertainty on the background estimation
with respect to the annular region; moreover the mean values, despite being
below the MDP99, are typically higher than the one obtained from the annular
selection, and this means that in background subtraction, we are associating
to the background a higher induced polarization with higher uncertainties.

Figure 3: IXPE observation of 4U 0142+61 overimposed with four circular
regions, in black, with radius 100” and an annular region, dotted line in red,
with inner radius 2.5’ and outer radius 5’. These regions are used to test and
compare different background selections in this analysis.

Figure 4: Induced polarization degree (left) and angle (right) from the
background as measured in the different selection regions shown in Figure 3,
compared with the source polarization: the polarization degree induced by the
background is better estimated in the annular region.

In conclusion, the best selection for the background from a polarimetric point
of view is obtained by using an annular region with an inner radius of 2.5’
and an outer radius of 5’. This is somewhat expected because annular regions
exploit the azimuthal averaging of possible modulated components of the
background.

##  4 Background rejection

IXPE is designed in order to avoid that X-ray photons in the nominal range of
energy can arrive at the sensitive parts of the detectors only through the
optics. Photons from the diffuse cosmic ray background not reflected by the
optics are stopped by the opaque structures: the mirror block, the long
collimators and, aiming to complete the coverage, dedicated skirts around the
mirrors. Therefore we can assume that the photons detected in the IXPE nominal
energy range are from the field of view and belong either to the source of
interest or to the surrounding X-Ray background.

Background rejection methods are based on differences between the signal of
interest, in our case X-rays, and other detected signals produced from other
particles. In the case of IXPE GPDs, the possibility to disentangle the events
due to “good” X-ray photons, from those constituting the background, is based
on the different morphology of the recorded tracks derived from processes
following the different interactions, i.e. “good” photons are detected by the
photoelectrons. These interactions are always detected in the GPDs in the form
of ionization tracks. In most cases, such as electrons, positrons, muons, or
protons, the sources of background arrive from outside or are produced in the
satellite’s passive structures. Their energy is high, and consequently, their
linear ionization loss is lower than that of photoelectrons. They release
energy in the IXPE range because they lose part of their energy outside the
active volume, where, instead, they induce tracks typically more extended,
straight, and with a lower charge density than photoelectrons. However, the
detectors are not imaging the ionization tracks in 3 dimensions, but only
their projection on the sensing plane. When a particle crosses the detector at
a small angle from the perpendicular, the detected track can be as long as
that of a photoelectron but more straight and with constant ionization. A last
complication is due to the triggering logic. In order to contain the dead
time, the GPD ASIC only routes to the output the content of the Region of
Interest that, because of pixels clustering and of its fixed size, can leave
out the content of some pixels.

Following these considerations, we started a systematic study of possible
selection criteria to separate good tracks from background ones. We aim, in
this first stage, to a simple and robust method. In this spirit, given the
extreme variability of the shape of good tracks, we haven’t yet applied any
analysis based on the track shape.

###  4.1 IXPE pre-launch background study

A preliminary study of IXPE background was performed using a Geant4 simulator
in Xie et al. (2021): in this paper, the origin and the phenomenology of the
background were briefly discussed, and the relative fluxes have been estimated
(see Table 1 for a summary). The main background contribution in the IXPE
nominal energy band arises from the primary protons, primary $\alpha$s, and
secondary cosmic rays, while $\gamma$-origin background (including cosmic
X-ray and albedo $\gamma$) is not significant. The total counting rate for the
instrumental background results to be 6.31$\times$10-4 counts/sec/arcmin2.

Component | Expected rate in 2–8 keV (counts/s)
---|---
Cosmic X-ray | 1.73$\times 10^{-3}$
Albedo Gamma | 1.24$\times 10^{-3}$
Albedo Neutron | 2.97$\times 10^{-4}$
Primary Proton | 3.16$\times 10^{-2}$
Primary Electron | 2.39$\times 10^{-4}$
Primary Positron | 1.91$\times 10^{-5}$
Primary Alpha | 1.09$\times 10^{-2}$
Secondary Proton | 1.41$\times 10^{-2}$
Secondary Electron | 1.11$\times 10^{-2}$
Secondary Positron | 3.36$\times 10^{-2}$
Total | 1.05$\times 10^{-1}$
Table 1: Count rates from all the background components estimated for IXPE in
Xie et al. (2021).

In Xie et al. (2021), a parametrization of background tracks to identify and
reject them with respect to X-rays tracks induced by a source has been
proposed at the cost of also removing a relatively high fraction of good
events ($\simeq 20$%). The main track properties estimated in the IXPE track
reconstruction algorithm, see also Di Marco et al. (2022a), that were
considered in Xie et al. (2021) are:

  1. 1.

Pulse Invariant (PI): gain equalized sum of the charge collected in the track,
which is proportional to the detected energy.

  2. 2.

Track size: number of ASIC pixels above the threshold in the main cluster,
that is the largest group of contiguous pixels of the event; typically, the
background events produce a larger track size than photoelectrons at the same
energy.

  3. 3.

Skewness: the third standardized moment, refers to the asymmetry of the energy
distribution in the track along the major axis; typically, the background
tracks are less skewed (straighter) than the photoelectron tracks.

  4. 4.

Elongation: defined as the ratio of the longitudinal over transverse
components of the second moment of the track; typically the background tracks,
being straighter, have a larger ratio.

  5. 5.

Charge density: defined as the energy (PI) divided by track size; typically,
the background induced by (minimum) ionizing tracks has a smaller charge
density.

  6. 6.

Cluster number: number of clusters after applying clustering algorithm in the
ROI; typically X-rays have only one cluster while minimum ionizing particles
(MIPs) produce multiple clusters.

  7. 7.

Border pixels: number of pixels in the track which are at the edge of the
track ROI; typically background tracks intercept the border pixel.

The resulting best-rejection approach of Xie et al. (2021) is based on
constraining a few parameters as a function of energy to select only source
events, the constraints to be applied are summarized in Table 2.

Energy bin | Track size | Skewness | Elongation | Charge density
---|---|---|---|---
2.0 – 3.5 keV | 26–67 | -0.383–0.383 | 1.023–1.506 | 116.702–315.914
3.5 – 5.0 keV | 34–92 | -0.620–0.620 | 1.044–2.414 | 140.186–315.930
5.0 – 8.0 keV | 48–156 | -1.013–1.010 | 1.112–4.386 | 134.936–386.334
Table 2: Constraints on tracks parameters to be applied for background
rejection as estimated in Xie et al. (2021).

This approach rejects about 75% of background events at the cost of removing
$\simeq 20$% of the source events. This approach, based on MonteCarlo
simulations, when applied to flight data produced discouraging results because
of the large fraction of source events removed ($\simeq 50$%) despite
$\simeq$80% of rejected background events. Such a different performance
between MonteCarlo simulations pre-launch and flight data can be ascribed to
different reasons: a not optimal MonteCarlo physics-list, different gain
during flight operations with respect to the simulated ones, etc…

###  4.2 A new rejection approach based on flight data

From Xie et al. (2021), background events are expected to have longer tracks
in IXPE GPDs, with a smaller charge collection per pixel because of a nearly
constant (and minimum) energy loss and a larger probability of having active
pixels in the track border. In this study, all the available IXPE track
parameters were considered in order to identify the best ones to decouple
background and source contributions. These parameters are not available in the
IXPE final photon list, distributed in event_l2 files, but they can be
retrieved from raw data included in the event_l1 files, both files are
distributed on HEASARC. To identify a photon in the event_l2 files inside the
event_l1 files, it is possible to use the keyword ’TIME’.

In order to obtain a new rejection strategy, we adopted the safer approach to
maximize the fraction of X-ray source acceptance avoiding a background
rejection that removes a large fraction of source events. At this aim, we used
Cyg X-2 data — the brightest point-like source observed by IXPE at the time of
this work with flux $\sim 10^{-8}$ erg$\cdot$cm${}^{-2}\cdot$s-1 — to chose
the parameters to use and for their parametrization. This source has been
observed by IXPE from 2022 May 2nd up to 2022 May 3rd. Spatial selections, as
described in Section 3, allowing to identify the source (central circular
region with radius 60”) and the background (annular region with inner radius
2.5’ and outer radius 5’) were applied.

For each track property, the background and source populations were compared,
aiming to identify parameters having strongly different behaviors. We
identified three promising track parameters that are presented in the
following: (i) the number of pixels in the track; (ii) the energy fraction in
the main cluster over the total charge collected for the single event; (iii)
the number of border pixels. We discuss each one of them in the following.

The various methods of background rejection that we discuss in the next
sections are suited to be applied all together or separately, or not applied
depending on the source brightness, angular structure, or spectrum. Most
rejection methods, finely trimmed to maximize the increase of sensitivity,
would imply a change of the convolution matrix, to be recomputed every time.
To allow for flexible use of the rejection methods, while avoiding this
cumbersome task, we try to identify rejection filters that have a negligible
impact on the efficiency and do not depend on the spectrum. In other terms,
filters totally compatible with the standard software of polarimetric analysis
and the existing response functions.

####  4.2.1 Number of pixels

The number of pixels is a parameter growing with energy for X-rays, this is
due to the fact that typically tracks at higher energy are bigger than the
ones at lower energy. Background events produce bigger tracks than X-rays when
the same PI is measured. In Figure 5–left, it is possible to observe that for
events in the IXPE nominal 2–8 keV energy band, source, and background events
show different behaviors: in particular for a number of pixels higher than 250
the background starts to be dominating.

Figure 5: Number of pixels distributions for the source (orange) and the
background (light blue) events from Cyg X-2 in IXPE DU1, as a representative
case. Left: comparison between the two distributions in the IXPE nominal
energy band, the black dotted line shows the threshold (250 pixels) above
which background is dominating on source events in both plots. Right: scatter
plot showing the number of pixels as a function of the energy for the source
(orange) and the background (light blue) populations for $2\times 10^{5}$
events from Cyg X-2. Vertical black dashed lines show the IXPE nominal energy
band. The threshold values on the number of pixels as a function of energy
following function 4 with best-fit parameters from Figure 6 (purple dashed
line) and the approximated linear behavior (red line) are very similar in the
2–8 keV energy band.

In Figure 5–right, it is possible to see that the constant threshold of 250
for the number of pixels can be improved with an energy dependent threshold.
Repeating the comparison of Figure 5–left at different energies within the
0–10 keV energy interval with 1 keV steps, it is possible to estimate such an
energy dependent threshold value.

Figure 6 shows the threshold values estimated in different energy intervals
from distributions similar to the one of Figure 5–left. The threshold as a
function of the energy, $E$, is well described by the function reported in
Muleri et al. (2010):

$\textrm{Number of pixels}=k_{a}+k_{b}\times\left(\frac{\textrm{E}}{\textrm{1
keV}}\right)^{k_{c}},$ (4)

with best-fit ($\chi^{2}/dof=0.82$) values:

  * •

$k_{a}=67.1\pm 5.6$

  * •

$k_{b}=25.3\pm 4.6$

  * •

$k_{c}=1.108\pm 0.084$

Figure 6: Threshold on the number of pixels — for which the source events
above it are negligible — as a function of energy (black points). The purple
dashed line shows a best-fit in the 0–10 keV energy band for the function 4,
and the red line is a simplified linear function valid in the 2–8 keV IXPE
nominal energy band.

In the IXPE 2–8 keV nominal energy band, this equation can be simplified with
a linear function ($\chi^{2}/dof=0.37$):

$\textrm{Number of pixels}=70+30\times E.$ (5)

The scatter plot of Figure 5–right shows a comparison between the different
threshold functions, in the IXPE nominal 2–8 keV energy band, the two are
consistent, while the fixed threshold at 250 number of pixels is not effective
like these energy dependent cuts.

####  4.2.2 Energy fraction

A readout track from the GPD can have more clusters of pixels, depending on
the energy density and the track threshold. Only the main cluster, the larger
one, is considered for the event reconstruction Baldini et al. (2021), as
shown in Figure 7, where the raw track is represented on the left panel and
the main cluster, considered for the track reconstruction, on the right one.

Figure 7: Example of a track with more clusters (left), where only the one
reported in the right panel is considered for the track reconstruction by the
IXPE algorithm.

The parameter here considered, the energy fraction, takes into account the
ratio between the energy/charge collected in the main cluster and the one
collected in all the detected clusters.

Figure 8: Energy fraction in the main cluster for source (orange) and
background (light blue) events in IXPE DU1 as a representative case. Left:
comparison between the two distributions in the IXPE nominal energy band, the
black line shows the threshold above which background is dominating on source
events in both plots. Right: scatter plots showing the energy fraction in the
main cluster as a function of energy for $2\times 10^{5}$ events from Cyg X-2.
Vertical black dashed lines show the IXPE nominal energy band 2–8 keV. The
threshold values on the energy fraction in the main cluster from the physical
condition $\leq 1$ and from the energy dependent function estimated from fit
(red lines) are reported. For this bright source, it is possible to identify
also a population of events with a small energy fraction $\leq 0.25$ probably
due to events produced by simultaneous tracks (pile-up).

The energy fraction definition shows that X-rays sources can have an energy
fraction $\leq 1$, but from Figure 8, it is possible to observe a small tail
of events with a higher energy fraction, probably due to noisy pixels. These
events having energy fraction $>1$ must be removed from the analysis. It is
possible to observe, for this bright source, the presence of a population of
events with a small energy fraction, $<0.25$. These are probably due to the
detection of simultaneous tracks (pile-up). Cyg X-2 is a very bright point-
like source, which means that photons are focused in a small region on the GPD
sensitive area. In this particular case, it is possible to observe
simultaneous detection of 2 X-rays in the track ROI, but only one will be
included in the main cluster producing events having energy fraction $<0.5$
also in case of source events.

From the point of view of the background, MIPs, characteristic of cosmic rays
impinging on the detector, are expected to have longer tracks in the IXPE
detectors. This means that they are able to produce more clusters along their
passage through the detector, giving rise to a smaller energy fraction in the
main track with respect to X-rays, like in the example of Figure 7. From
Figure 8–left, it is possible to observe that a mean energy fraction in the
main cluster below 0.65 guarantees that background is dominant with respect to
source events. In Figure 8–right, similarly to the case of the number of
pixels, the energy fraction in the main cluster threshold can be optimized as
a function of energy. In Figure 9, the threshold has been estimated at
different energies within the 0–10 keV energy interval with 1 keV steps.

Figure 9: Threshold on the energy fraction as a function of energy for which
source events below it are negligible (light blue points). The red line shows
a best-fit in the 0–10 keV energy band.

The energy dependent threshold on the energy fraction in the main cluster,
such that background events are dominating below it, is well described by the
function:

$\textrm{Fraction of
energy}=k_{a}\times[1+e^{-\frac{E+k_{b}}{k_{c}}}]+k_{d}\times E$ (6)

with best-fit ($\chi^{2}/dof=0.01$) parameters:

  * •

$k_{a}=(0.814\pm 0.004)\sim 0.8$

  * •

$k_{b}=(0.254\pm 0.010)\textrm{ keV}\sim 0.25$ keV

  * •

$k_{c}=1.12\pm 0.02\textrm{ keV}\sim 1.1$ keV

  * •

$k_{d}=(0.0037\pm 0.0005)\textrm{ keV}^{-1}\sim 0.004$ keV-1.

The scatter plot for the populations of source and background events of Figure
8–right shows the threshold lines obtained from the previous fit and from the
physical condition that this quantity must be $\leq 1$ compared with the
fixed, energy independent, threshold from Figure 8–left, showing that energy
dependent threshold is more effective.

####  4.2.3 Border pixels

Background events induced by particles, as said in the previous sections, are
capable of producing longer tracks and more clusters, meaning that they are
also characterized by tracks with a higher probability of having pixels on the
ROI edges. Moreover, border tracks are induced by events having part of the
energy released outside the ROI.

Figure 10: Comparison of source (orange) and background (light blue)
distributions spatially selected by using Cyg X-2 IXPE data for the number of
pixels in the border. The black dotted line shows the threshold value above
which the background is dominating with respect to the source. See text.

In Figure 10, the number of border pixels for events in the IXPE energy band
is shown. An energy dependent behavior for this parameter has not been
observed: at every energy, the distributions are similar; thus, the condition
for background rejection is given by a cut for border pixels $<2$.

####  4.2.4 Rejection strategy

In conclusion, this analysis repeated for all the three IXPE detectors and for
more observed sources, shows that source events must simultaneously satisfy
the following conditions in the IXPE nominal energy band 2–8 keV:

  * •

Number of pixels $<70+30\times E$

  * •

$0.8\times[1-e^{-\frac{E+0.25}{1.1}}]+0.004\times E<$ Energy fraction $<1$

  * •

Border pixels $<2$

In case of future IXPE energy band extensions, see e.g. (La Monaca et al.,
2022), the condition for the number of pixels should be changed with the one
arising from the fit of function 4.

##  5 Background rejection application to flight data

In this section, we apply our background-rejection strategy to IXPE flight
data in order to validate its effectiveness and its negligible impact on the
response matrices. The fraction of events removed in the background region
will be estimated, for point-like sources observed by IXPE, together with the
counting rate in the background annular region. With the aim to estimate the
effect on the response matrices from this rejection approach, we applied it to
in-flight calibration data and to images of extended sources.

###  5.1 Background rejected fraction

The rejection efficiency has been estimated for all the point-like sources
observed by IXPE up to August 2022. In particular, the observed sources are
reported in Table 3. Some of them have been observed in different times and
states — as Cen X-3, Mrk 421, … — in these cases in Table 3 the different
observations are analyzed separately, and they correspond to different raws of
counting rates.

Source name | Rate (cps) |  | Source name | Rate (cps)
---|---|---|---|---
Cen X-3 | 0.14 |  | Mrk 421 | 0.32
| 4.3 |  |  | 0.59
4U 0142+61 | 0.26 |  |  | 1.0
Cen A | 0.092 |  | BL Lac | 0.024
Her X-1 | 1.20 |  |  | 0.043
Mrk 501 | 0.27 |  | Cyg X-1 | 12
| 0.49 |  |  | 14
| 0.26 |  | 3C 454.3 | 0.013
4U 1626-67 | 0.34 |  | 3C 273 | 0.089
GS 1826-238 | 9.2 |  | 3C 279 | 0.020
S5 0716+714 | 0.0062 |  | GX 301-2 | 0.42
Vela X-1 | 0.41 |  | Background |  $\sim$0.003
Cyg X-2 | 15 |  |  |
Table 3: List of point-like sources observed by IXPE that have been considered
in this analysis with their counting rate compared to the mean estimated
background counting rate. Source counting rates are estimated per DU and in a
central circular region with a radius 1’; the background value has been
normalized for the same area.

The rejection approach has been applied to the background region of all these
sources. The background is selected as in Section 3. In Figure 11, the
rejected fraction of events as a function of the source rate, extracted in a
circular region with radius 1’, is shown.

Figure 11: Fraction of background events rejected as a function of the source
counting rate.

Bright sources (inducing more than 2 cps/arcmin2) show a slightly different
behavior with respect to fainter ones. This is due to the fact that these
bright sources, because of the IXPE PSF, contaminate the background region:
this is visible from Figure 12, where the rate in the background region is
shown as a function of the rate in the source region. These plots show that
background events are overwhelmed by source events also in the outer region,
making rejection/subtraction of events not effective. Therefore, the
background is well negligible for bright sources in IXPE.

Figure 12: Rate in the annular region where the background is estimated as a
function of the source counting rate, the right panel shows a zoom for sources
with lower counting rates. Both counting rates are normalized in a $\pi$
arcmin2 spatial region.

The IXPE background is estimated, considering the faintest sources, to range
from $(8.74\pm 0.18)\times 10^{-4}$ counts per second per arcmin2 for S5
0716+714 (RA 07h 21m 53.45s Dec. 71 20’ 36”) up to $(12.10\pm 0.37)\times
10^{-4}$ cps/arcmin2 for 3C 279 (RA 12h 56m 11.1s Dec. -05 47’ 22”). These
values include both particle and X-ray background, and they are 1.3-1.9 times
higher than the one estimated from simulations in Xie et al. (2021), where the
Cosmic X-Ray Background arriving through the optics — negligible with respect
to particle background — was not included in the simulator. When higher values
of the counting rate in the background region are observed, these events are
mainly due to the observed source. As an example, if one estimates for Cyg
X-1, the polarization in the central circular region obtain $(3.81\pm 0.31)$%
with polarization angle $(46.6\pm 2.0)^{\circ}$, while the polarization in the
background outer region is estimated to be $(4.0\pm 1.7)$% with angle $(30\pm
14)^{\circ}$. The difference between the source and the background region is
$\Delta$PD$=-(2.0\pm 1.7)\%$ for polarization degree and $\Delta$PA$=(17\pm
14)^{\circ}$ for the polarization angle, both compatible with a null
difference, confirming that source events are also dominating in the annular
region where the background is selected.

To further confirm this effect, during the IXPE ground calibration, one
telescope was calibrated, and the PSF was estimated to be composed by a
gaussian plus a King function plus a power-law term (see Figure 13).

Figure 13: Example of PSF function as estimated during IXPE telescope ground
calibrations (left). And the three different regions of selection.

From this mean estimate obtained for the spare telescope, having the best PSF,
similar to IXPE telescope 1, $\sim$97% of events is included in a circle
having radius 1.5’ and $\sim$1.1% of events is included in the background
annular region. This means that for Cyg X-2, having a rate in DU1 of 15.7 cps
in the source region, we expect in the background annular region $\sim 0.2$
cps.

###  5.2 Effects of rejection on calibration data

IXPE adopts a point and stare observing strategy, so that when the source
cannot be observed because of Earth occultation, one detector at a time is
calibrated by using onboard calibration sources (Ferrazzoli et al., 2020; Di
Marco et al., 2022b). The rejection approach proposed here has been applied to
data from in-flight calibration with the unpolarized sources called Cal C, a
55Fe source having a peak at 5.89 keV, and Cal D, a 55Fe source illuminating a
Silicon target producing a peak at 1.7 keV.

Figure 14: Energy spectrum from Cal C in the three detectors of IXPE, from
left to right: DU1, DU2, DU3. In each plot, the spectrum without rejection
(blue line) with label all is compared with the one of events tagged as source
(src) after rejection (orange line) and the rejected events tagged background
(bkg, green line).

In Figure 14, the energy spectra observed by the three IXPE detectors when
exposed to Cal C source are shown and compared with events identified by this
rejection approach to be source or background. It is possible to estimate that
in these data sets, the rejection approach removes only 1–2% of events, and it
is able to remove the high energy tail, not due to the calibration source
itself.

In the case of Cal D, whose energy peak is below the IXPE nominal energy band,
the fraction of events rejected is 1–2% in the three detectors, as shown in
Figure 15.

Figure 15: Energy spectrum from Cal D in the three detectors of IXPE, from
left to right: DU1, DU2 and DU3. In each plot, the spectrum without rejection
(blue line) with label all is compared with the one of events tagged as source
(src) after rejection (orange line) and the rejected events tagged background
(bkg, green line).

A small spill-out of source events identified to be background is present but
very small. To quantify this effect, the fraction of residual events between
the energy spectra of calibration sources before and after the rejection are
shown in Figure 16: the effect is that the energy spectrum at maximum differs
of 0.05%, that is well below the uncertainty on the IXPE response functions.
As a result, the effects of this rejection approach on the IXPE response
matrices are well negligible. As an example, in the Cal D spectra of Figure
15, the 55Fe line is not visible in the background spectra because its source
counting rate ($\sim 1000$ counts/0.04 keV) give rise to a spill-out $\leq$0.5
events/0.04 keV.

Figure 16: Percentage residuals of the energy spectrum from Cal C (left) and
Cal D (right) in one of the three detectors (DU1), as a case study, but
results in the other DUs are comparable. It is possible to observe that the
residuals deviate at maximum at 0.05% from zero.

###  5.3 Effects of rejection on extended sources

The IXPE imaging capabilities allow studying spatially resolved polarization
in extended sources like Supernova remnants, Pulsar Wind Nebula, jet
structures, and Galaxy clusters. The first extended source observed by IXPE is
Cas A, observed from January 11 to January 29, 2022, for a total exposure time
of $\sim$900 ks (Vink et al., 2022). In the analysis of this source, reported
in Vink et al. (2022), a preliminary version of this rejection approach was
applied to allow to improve the significance of the polarimetric result. Such
a preliminary approach used a fixed cut on the number of pixels, the energy
fraction in the main cluster, and the border pixels. Figure 17 shows the IXPE
image of Cas A before (left) and after (center) the application of this new
rejection approach using the selection of Section 4.2.4.

Figure 17: Cas A data images from the original data (left), after the
background rejection (center), and for the rejected events (right). The same
color scale was applied.

The rejection is not affecting imaging capabilities. Moreover, in Figure
17–right, the image of rejected events shows that they are mainly localized on
the edge of the field of view of the three IXPE telescopes.

##  6 4U 0142+61, a case study

The IXPE observation of 4U 0142+61 was obtained from two segments: the first
started on 2022-01-31 at 07:37:07 UTC and ended on 2022-02-14 at 23:44:12 UTC,
the second started on 2022-02-25 at 04:38:09 UTC and ended on 2022-02-27 at
18:46:09 UTC, for 835719 seconds of total live-time in the three IXPE
telescopes Taverna et al. (2022). Here, the original data – available on the
IXPE archive – have been filtered as indicated in Section 4.2.4, and results
from the data analysis before and after the background rejection are compared.

In this analysis, after background rejection, the residual background has also
been subtracted because the source is faint. To make this, the source and
background selection have been obtained following the selection prescriptions
of Section 3, by using the SAOIMAGEDS9 software (Joye & Mandel, 2003) to
produce the regions and FTOOLs included in HEASoft 6.30.1 to extract selected
data.

A preliminary overall analysis is here performed with the ixpeobssim analysis
tool (Baldini et al., 2022), developed for IXPE, considering events in the
whole IXPE 2–8 keV nominal energy band. The polarization before and after the
background rejection is reported in Figure 18 and compared with the
polarization estimated for the events removed from the rejection because
tagged as background, which is well compatible with zero; polarization in the
original data-set results to be PD = $(12.2\pm 0.8)$% with PA = $(48\pm
2)^{\circ}$, well compatible with the value obtained after background
rejection PD = $(12.1\pm 0.8)$% with PA = $(48\pm 2)^{\circ}$.

Figure 18: Contours enclosing regions at the confidence level of 68.3% (solid
lines) and 99.7% (dotted lines) in case of original unrejected data (light
blue), after the rejection of Section 4.2.4 (orange) and for the events
rejected because identified as background (green).

A spectral analysis was also carried out by using xspec (Arnaud, 1996) version
12.12.1 released with HEASoft 6.30.1, which includes polarimetric models.
Following the analysis reported in Taverna et al. (2022); Rea et al. (2007),
the standard 4U 0142+61 spectral model: TBabs*(BBody+PowerLaw) has been used.
The interstellar absorption was taken into account by using the TBabs with
abundances from (Wilms et al., 2000) fixing the column density value
$n_{H}=0.57\times 10^{22}$ cm-2 (den Hartog et al., 2008). The IXPE spectra
from the three telescopes have been simultaneously fitted, including a cross-
normalization factor for the second and the third telescope.

Figure 19 shows the energy distributions of source events (colored crosses)
and of the mean background in the three detectors black (solid histogram)
before (left) and after (right) the background rejection.

Figure 19: 4U 0142+61 data fitted with xspec before (left) and after (right) the background rejection; the considered spectral model is TBabs*(BBody+PowerLaw). At the bottom, residuals with respect to the best-fit model are shown. On the right panel, the background before and after rejection are compared, and it is possible to appreciate its reduction after the rejection procedure. Parameter | Before rejection | After rejection
---|---|---
$n_{\rm H}$ | $0.57^{\dagger}$
$kT_{\rm BB}$ | 0.471$\pm 0.004$ | 0.472$\pm 0.004$
norm | 0.00111$\pm 0.00004$ | 0.00109$\pm 0.00004$
$\Gamma$ | 3.68$\pm 0.05$ | 3.70$\pm 0.05$
norm | 0.121$\pm 0.008$ | 0.123$\pm 0.008$
f | 1.0†
f2 | 0.965$\pm 0.003$ | 0.965$\pm 0.003$
f3 | 0.858$\pm 0.003$ | 0.858$\pm 0.003$
$\chi^{2}$ | 520.0 with 441 d.o.f. | 529.5 with 441 d.o.f.
Table 4: Results of Xspec fit of 4U 0142+61 data before and after the
rejection, by using the spectral model: TBabs*(BBody+PowerLaw), with
normalization factors for the IXPE telescopes two and three. †Parameters value
fixed in the spectral fit.

The distributions are superimposed with the best-fit results obtained taking
into account a background-subtraction in the IXPE nominal energy interval 2–8
keV: at higher energy, the background is dominant with respect to the source,
but after rejection, this effect is strongly reduced. A comparison of the
background level before and after the rejection procedure is shown in Figure
19–right. The fit results are reported in Table 4 for both original and
background-rejected data sets. The fit before and after background rejection
show very well compatible results, moreover the reduced chi-square is also
very similar $\chi^{2}/dof=1.18$ before the rejection and 1.20 after it.

However, since the main scientific results for IXPE arise from the measurement
of polarization, a joint spectro-polarimetric analysis of the observed data is
performed by building a binned (40 eV) spectra for the Stokes parameters I, Q
and U, which are simultaneously fitted with the usual xspec procedure of
forward folding and applying a background subtraction. Polarization in a given
energy interval was calculated by assuming the spectral model for I equal to
TBABS*(BBODY+POWERLAW) with parameters fixed at the ones of Table 4 and
convolving it with the constant polarization model POLCONST provided by XSPEC.
Results from this procedure show a polarization PD = $(13.7\pm 1.1)\%$ with PA
= $(48\pm 2)^{\circ}$ before the background rejection and PD = $(13.6\pm
1.1)\%$ with PA = $(48\pm 2)^{\circ}$ after it.

Figure 20: Polarization degree (top) and angle (bottom) obtained for 4U
0142+61 data fitted with xspec in different energy bins before (light blue)
and after (orange) the background rejection. It is possible to observe that in
the last energy bin, where the background dominates, the rejection strategy
produces a more significant polarization degree because the dilution effect is
partially washed out, while the polarization angle is well compatible in the
two cases.

Repeating this analysis for different energy bins produces the result of
Figure 20, where it is possible to confirm that results are well compatible. A
remarkable result arising from the background rejection is the improvement of
significance for the polarization degree in the last energy bin (6–8 keV),
where the background dominates. After the rejection, the background dilution
effect is partially washed out, and the polarization degree increases from
$\sim 36$% up to $\sim 43$%. Polarization degrees and angles are summarized in
Table 5.

| 2–3 keV | 3–4 keV | 4–5 keV | 5–6 keV | 6–8 keV | 2–8 keV
---|---|---|---|---|---|---
Unrejected | PD (%) | $16.1\pm 1.0$ | $14.2\pm 1.4$ | $5\pm 3$ | $22\pm 6$ | $36\pm 11$ | $13.7\pm 1.1$
| PA (∘) | $46.9\pm 1.8$ | $52\pm 3$ | $36\pm 17$ | $-59\pm 8$ | $-32\pm 8$ | $48\pm 2$
| $\sigma$ | 16 | 10 | 0.54 | 3.6 | 1.5 | 12
Rejected | PD (%) | $16.1\pm 1.0$ | $14.1\pm 1.4$ | $6\pm 3$ | $21\pm 6$ | $43\pm 10$ | $13.6\pm 1.1$
| PA (∘) | $47.0\pm 1.8$ | $52\pm 3$ | $35\pm 15$ | $-56\pm 8$ | $-30\pm 6$ | $48\pm 2$
| $\sigma$ | 16 | 10 | 0.78 | 3.6 | 2.8 | 12
Table 5: Polarization degree and angle obtained fitting by using xspec 4U
0142+61 data in different energy intervals within the IXPE nominal energy
band. The significance ($\sigma$) is obtained in a non-polarized test
hypothesis, and its estimate is based on the standard normal distribution;
hence the corresponding P value means the probability that such a polarized
signal is generated by a non-polarized source.

##  7 Conclusion

The IXPE detectors allow to select background events thanks to imaging
capabilities, with the best selection being an annular region with an inner
radius of 2.5’ and an outer radius of 5’. Larger selections or circular ones
can introduce systematics in the polarization because of the detector’s
geometrical effects.

The IXPE background is at a level of 0.003 cps/arcmin2. In this paper, a
possible rejection approach is presented, allowing to remove up to 40% of
background events. Such a rejection approach is more effective for faint
sources because, for the brighter ones, the background region is dominated by
source events scattered by mirrors. Because of this, three possible background
treatments can be performed:

  * •

Bright sources (rate $>2$ cps/armin2): background is negligible, rejection can
be applied, but it is neither effective nor useful, and subtraction from the
same field should not be performed because the background region is dominated
by the source itself;

  * •

Faint sources (rate $<1$ cps/armin2): background rejection is recommended and
effective: after rejection, the residual background contribution should be
subtracted in the analysis;

  * •

Intermediate sources: background rejection is recommended and effective, after
the rejection, the background region is still dominated by source events,
which means that subtraction in the analysis should be avoided unless using a
template background extracted from different sources.

As an example, the 4U 0142+61 data have been analyzed using prescriptions on
the background obtained in this study. The result shows that, when the
background impact is negligible, the rejection results are well in agreement
with the ones arising from unrejected data, but when the background is
comparable or higher than the source counting rate, the background rejection
becomes crucial to improve the significance of the result.

##  8 Acknowledgements

The Imaging X-ray Polarimetry Explorer (IXPE) is a joint US and Italian
mission. The US contribution is supported by the National Aeronautics and
Space Administration (NASA) and led and managed by its Marshall Space Flight
Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C). The
Italian contribution is supported by the Italian Space Agency (Agenzia
Spaziale Italiana, ASI) through contract ASI-OHBI-2017-12-I.0, agreements ASI-
INAF-2017-12-H0 and ASI-INFN-2017.13-H0, and its Space Science Data Center
(SSDC) with agreements ASI-INAF-2022-14-HH.0 and ASI-INFN 2021-43-HH.0, and by
the Istituto Nazionale di Astrofisica (INAF) and the Istituto Nazionale di
Fisica Nucleare (INFN) in Italy. This research used data products provided by
the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional
software tools by the High-Energy Astrophysics Science Archive Research Center
(HEASARC), at NASA Goddard Space Flight Center (GSFC).

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