Measuring energy dependent polarization in soft gamma-rays using Compton
  scattering in PoGOLite
Measuring energy dependent polarization in soft gamma-rays using Compton
scattering in PoGOLite
M. Axelsson a,∗, O. Engdeg̊ard b,a, F. Ryde b,a, S. Larsson a, M. Pearce b, L. Hjalmarsdotter c,a,
M. Kiss b, C. Marini Bettolo b, M. Arimoto d, C.-I. Björnsson a, P. Carlson b, Y. Fukazawa e,
T. Kamae f,g, Y. Kanai d, J. Kataoka d, N. Kawai d, W. Klamra b, G. Madejski f,g, T. Mizuno e, J. Ng f ,
H. Tajima f,g, T. Takahashi h, T. Tanaka e, M. Ueno d, G. Varner i, K. Yamamoto e
aStockholm Observatory, AlbaNova, SE-106 91 Stockholm, Sweden
bPhysics Department, Royal Institute of Technology, AlbaNova, SE-106 91 Stockholm, Sweden
cObservatory, PO Box 14, FIN-00014 University of Helsinki, Finland
dTokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan
eHiroshima University, Physics Department, Higashi-Hiroshima 739-8526, Japan
fStanford Linear Accelerator Center, 2575 Sand Hill Road, Menlo Park, CA 94025, USA
gKavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA
hInstitute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara 229-8510, Japan
iDepartment of Physics and Astronomy, University of Hawaii, 2505 Correa Road, Honolulu, HI 96822, USA
Abstract
Linear polarization in X- and gamma-rays is an important diagnostic of many astrophysical sources, foremost giving information
about their geometry, magnetic fields, and radiation mechanisms. However, very few X-ray polarization measurements have been
made, and then only mono-energetic detections, whilst several objects are assumed to have energy dependent polarization signatures.
In this paper we investigate whether detection of energy dependent polarization from cosmic sources is possible using the Compton
technique, in particular with the proposed PoGOLite balloon-experiment, in the 25–100 keV range. We use Geant4 simulations of a
PoGOLite model and input photon spectra based on Cygnus X-1 and accreting magnetic pulsars (100mCrab). Effective observing
times of 6 and 35 hours were simulated, corresponding to a standard and a long duration flight respectively. Both smooth and sharp
energy variations of the polarization are investigated and compared to constant polarization signals using chi-square statistics.
We can reject constant polarization, with energy, for the Cygnus X-1 spectrum (in the hard state), if the reflected component
is assumed to be completely polarized, whereas the distinction cannot be made for weaker polarization. For the accreting pulsar,
constant polarization can be rejected in the case of polarization in a narrow energy band with at least 50% polarization, and
similarly for a negative step distribution from 30% to 0% polarization.
Key words: Polarization, X-rays, Gamma-rays, Compton technique, PoGOLite, Geant4, Simulations
PACS: 95.55.Ka, 95.55.Qf, 95.75.Hi, 98.70.Qy
1. Introduction
In the areas of spectral and temporal studies, X-ray and
gamma-ray astronomers have been given a wealth of data
on a wide range of objects. Polarization has long been pre-
dicted to play a crucial role in determining physical and geo-
metrical parameters in many astrophysical sources, thereby
discriminating among current models. However, there have
so far been very few measurements of polarization at these
∗ Corresponding author.
E-mail: magnusa@astro.su.se
energies. In light of this, the possibility to detect energy
dependent polarization has hardly been discussed at all in
the literature. In this paper, we present the results from
simulations of a dedicated soft gamma-ray polarimeter us-
ing Compton scattering, and study the response when the
degree of polarization varies with the energy of the emit-
ted photons. While energy-dependent polarization is ex-
pected from many sources, its detection requires an instru-
ment of sufficiently good energy response. The Compton
polarimeter presented in this paper utilizes plastic scintilla-
tors, which are relatively inefficient for energy depositions
Preprint submitted to Elsevier 15 September 2021
http://arxiv.org/abs/0704.1603v2
below a few keV. Thus, simulations are necessary to deter-
mine how sensitive the instrument is and how large varia-
tions must be for detection.
We begin by describing the organisation of the pa-
per. The remainder of this section is devoted to giving
a background of polarimetry in the X/γ-ray regime, and
an overview of the scientific motivation for such measure-
ments. In Section 2 we focus on polarimetry using Comp-
ton scattering and describe an instrument design based on
this technique. We then present the set-up of our simula-
tion of the instrument in Section 3, and the results of the
simulations in Section 4. Finally, in Sections 5 and 6, we
discuss and summarise our results.
1.1. Measurement of polarization
The aim of any polarimetric measurement is to deter-
mine the degree and direction of polarization of incident
radiation. When combined with the traditionally measured
quantities of energy and time, polarimetry has the poten-
tial to double the parameter space available. As such, it can
be a powerful tool to discriminate between physical models
proposed for a given source.
Historically, polarimetry has proven very successful at
optical and radio wavelengths. In these bands, it has been
extensively used to probe both radiation physics and ge-
ometry of sources (see, e.g., [1]). In the X-ray regime, how-
ever, the results are more meagre. Early rocket observa-
tions measured X-ray polarization from the Crab Nebula
[2]. This result was later confirmed by the Orbiting Solar
Observatory 8 (OSO-8, measuring a polarization degree of
19.2% ± 1.0%, [3,4]), the only satellite mission carrying a
dedicated polarimeter to date. As the design was based on
Bragg reflection on graphite crystals, the energies probed
were constrained to 2.6 keV and 5.2 keV.
A number of new polarimetric instruments, designed to
work in the X/γ-ray regime, have recently been proposed.
These include POLAR (10–300 keV, [5]), GRAPE (50–300
keV, [6]), PHENEX (40–300 keV, [7]), CIPHER (10 keV –
1MeV, [8]), and POLARIX (1.5–10 keV, [9]). In this paper
we present PoGOLite, a Compton polarimeter currently
under construction [10].
1.2. Expected objects of interest
The lack of polarimetric measurements in X-rays is not
due to a lack of potential targets. Indeed, from a theoreti-
cal point of view there are many sources that are expected
to display detectable degrees of polarization. Over the past
decades, there have been publications discussing the po-
tential for polarization in sources such as X-ray binary
(XRB) systems, active galactic nuclei (AGN), accretion
and rotation powered pulsars as well as cataclysmic vari-
ables (CVs); see e.g., [11,12,13,14,15]. Other work has fo-
cused on the processes producing polarized radiation, either
the radiative processes themselves (e.g., synchrotron and
Fig. 1. Likely geometry in the hard state of Cyg X-1. Mass being
accreted forms an accretion disc around the compact object. In the
inner regions, there is a hot inner flow/corona. Soft seed photons
from the disc may be Comptonized in the hot flow. A fraction of
the resulting hard photons can then be reflected off the disc, giving
a net polarization.
non-thermal bremsstrahlung, [16,17]), or processes such as
reflection/asymmetric scattering (e.g., [18,19,20]), strong-
field gravity [21,22] and vacuum birefringence in strong
magnetic fields [23].
In most sources, polarization is not expected to re-
main constant with energy. An example is radiation from
strongly magnetized plasmas where the polarization may
change dramatically near the cyclotron resonance energy.
It is therefore important to understand not only what
degree of polarization is needed for detection, but also
how sensitive a given instrument will be to the changes
of polarization with energy. To study such effects we have
chosen to simulate two example sources: Cygnus X-1 and
an accreting magnetic neutron star.
1.2.1. Cygnus X-1
Cygnus X-1 is a high-mass XRB where the compact ob-
ject is believed to be a black hole. The source exhibits two
main spectral states, commonly referred to as hard and soft.
Most of the time is spent in the hard state. Several mod-
els have been proposed to explain the observed states and
transitions. The two main components of such models are
usually a geometrically thin, optically thick accretion disc
and a hot inner flow or corona [24]. A schematic picture of
a likely geometry in the hard state is shown in Fig. 1. Soft
X-rays are produced in the accretion disc, and may then
be Comptonized in the hot inner flow/corona. A fraction
of the hard radiation can be reflected off the accretion disc
before reaching the observer. Polarization from this system
may arise through several processes. In this paper, we will
focus on the polarization introduced by the reflection (for
more details, see, e.g., [19,25,26]). In Cygnus X-1, this con-
tribution is strongest in the energy range of ∼ 20–100keV.
The polarization degree is expected to vary with energy,
following the relative strength of the reflection component.
1.2.2. Accreting magnetic neutron stars
In many high-mass XRBs the accreting object is a
highly magnetic neutron star. The strong magnetic field,
∼ 1013 gauss at the surface, directs the accretion flow to-
wards the magnetic poles of the star. Most of the accretion
energy is released just above the polar cap where the emis-
sion and propagation of radiation is directly connected to
the magnetic field as well as the local properties of the
plasma. For a number of sources cyclotron spectral features
have been observed in hard X-rays, and from these, mag-
netic field strengths have been deduced. The X-rays are
expected to be polarized and the degree, angle and energy
dependence of the polarization will depend on the physical
conditions in the emission region [27]. Measurements of the
detailed polarization properties would therefore provide a
new and very powerful probe of the radiating plasma near
the surface of the neutron star.
2. The Compton technique
Apart from the special case of Bragg reflection, all three
main physical processes of photon-matter interaction in
the X/γ-ray regime may be used in polarimetry: photoab-
sorption, Compton scattering and pair production. Each of
these preserves information on the polarization of the in-
coming radiation. For photon energies between ∼ 100 keV
and 1MeV, Compton scattering is the dominant process. In
this section, we will briefly outline the theoretical basis for
a polarimeter based on Compton scattering, and present a
design for a dedicated polarimeter based on this technique.
2.1. Basic principle
The differential cross section for Compton scattering is
given by:
− 2 sin2 θ cos2 φ
, (1)
where re is the classical electron radius, E0 and E are the
photon frequency before and after scattering, θ is the an-
gle between incident and scattered direction, and φ is the
azimuthal scattering angle relative to the plane of polar-
ization. When projected on a plane, the angle of scattering
will thus be modulated as cos2 φ.
To measure the scattering angles, it is necessary to de-
tect both the site of scattering and that of photoabsorp-
tion. If more than two scattering sites are identified, the
relative energy depositions can be used to help distinguish
between Compton scattering and photoelectric absorption
sites. Some form of segmentation of the detector is neces-
sary to provide spatial resolution, required to determine
the positions of the signals.
Fig. 2. The design of the PoGOLite instrument. The side anticoinci-
dence shield has been partially cut away for clarity. The total length
of the instrument will be ∼ 100 cm.
2.2. PoGOLite
The Polarized Gamma-ray Observer - Light weight ver-
sion (PoGOLite) is a balloon-borne polarimeter, planned
for launch with a stratospheric balloon in 2009. Figure 2
shows the design of the instrument. The instrument con-
sists of 217 phoswich detector cells (PDCs) arranged in
a hexagonal pattern. Each PDC is made up of a hollow
slow scintillator tube, a fast scintillator detector, a bottom
bismuth germanate (BGO) crystal, and a photomultiplier
tube (PMT). Signals from the different optical components
are distinguished using a pulse shape discrimination tech-
nique based on the different scintillation decay times of the
materials [28]. The configuration is surrounded by an an-
ticoincidence shield made of BGO crystals. Together with
the bottom BGO crystals, this allows side and back enter-
ing photons and cosmic rays to be rejected.
The hollow slow scintillator tube acts as an active col-
limator. Photons or charged particles entering the instru-
ment off-axis will be registered in the slow scintillator and
can be rejected. The desired events are from photons that
enter cleanly through the slow scintillator and scatter in
the fast scintillator. After scattering, the photon may be
absorbed in one of the neighbouring fast scintillator cells,
allowing the azimuthal scattering angle to be determined.
The well-type design of PoGOLite allows for efficient
background rejection [29,30], and gives a field of view of
∼ 5 deg2. This allows the instrument to be accurately
pointed at specific sources. As both the initial Compton
scattering and subsequent photon absorption occur in the
same material (the plastic fast scintillator), the effective
energy range is determined by the cross-sections for both
these processes, as well as the background. PoGOLite will
have an energy range of ∼ 25–100keV, which is lower than
the range where Compton scattering dominates. A more
detailed description of the instrument may be found in
[28,31].
The capability of PoGOLite to measure the energy de-
Azimuth angle
Fig. 3. Simplified sketch of the distribution of scattering angles,
used to determine the modulation factor. The maximum (Cmax),
minimum (Cmin) and average (T = [Cmax + Cmin]/2) values of the
distribution are indicated.
pendence of polarization is limited both by the signal-to-
background ratio and the energy resolution. Due to redis-
tribution, some of the higher energy photons will produce
events at lower energies. The flux and polarization in the
low energy band will therefore be affected by the spectrum
at higher energies but not vice-versa. The energy response
has been carefully simulated using Geant4 [32].
Figure 3 shows a hypothetical distribution of azimuthal
scattering angles. The maximum (Cmax) and minimum
(Cmin) values of the distribution and the average (T =
[Cmax+Cmin]/2) can be used to define a modulation factor:
Cmax − Cmin
Cmax + Cmin
Cmax − Cmin
. (2)
The modulation factor is determined by fitting the follow-
ing function to the distribution of azimuthal scattering an-
f(x) = T (1 +M cos(2x+ 2α)) , (3)
with angle x (a function variable, not a fitting parameter),
average T , modulation factorM , and polarization angle α.
In this work, the modulation factor is the discriminator
between different polarizationmodels. If the response of the
instrument to a 100% polarized source is known, the mod-
ulation factor can be used to determine the polarization of
the incoming photon beam [33].
3. Simulations
In this section we will describe the setup of our simula-
tions. The source models used as input are also presented,
as well as the background considered.
3.1. Geant4
Geant4 1 is a multi-purpose software package for sim-
ulating particles travelling through and interacting with
1 http://geant4.cern.ch
matter, using Monte Carlo techniques [34]. The standard
Geant4 package was earlier found [35] to have incorrect im-
plementations concerning photon polarization in Compton
and Rayleigh scattering; the Geant4 version used here is a
corrected version of 4.8.0.p01.
3.2. Simulation setup
The Geant4 implementation includes the essential parts
of PoGOLite: 217 PDCs with slow and fast plastic scintil-
lators and bottom BGO crystals together with a BGO side
shield. The model has no PMTs, and uses a solid BGO side
shield instead of discrete pentagonal bars (cf. Fig. 2). Lay-
ers of tin (50µm) and lead (50µm) surrounding each slow
scintillator and the BaSO4 coating (200µm) of the BGO
crystals are included. The mechanical support structure is
not represented.
During the simulation, separate photons are generated
with random energies from a spectral model. An event is
triggered by a hit in two or three of the fast plastic scintil-
lators. The following is saved as output data for each event:
information about the original gamma momentum, the ID-
number of the cells that had an interaction (ranging from
1 to 217) and the energy deposited in each cell. These data
are preprocessed to simulate the resolution of the PMTs,
as described in Sect. 4.1.
3.3. Source Models
As stated in Sect. 1.2, two sources were considered:
Cygnus X-1 (in the hard state) and an accreting neutron
star. Below we describe the model used for the incident
radiation and polarization in each case. As shown in [10],
PoGOLite is expected to detect polarization in both these
sources; what we are investigating is the sensitivity to
changes in polarization degree with energy.
3.3.1. Cygnus X-1
For our simulations of Cygnus X-1 we used an input spec-
trum of a power-law, with photon index α = −1.2 and an
exponential cutoff at energy Ecut = 120 keV. It was nor-
malised to match the observed spectrum of Cygnus X-1.
The spectrum of the reflection was approximated by the
logarithmic quadratic curve
EFE = 10
−c(logE−log a)(logE−log b), (4)
with a = 24, b = 98 and c = 1.89. Figure 4 shows the
observed radiation of Cygnus X-1, and our model of the
total spectrum as well as that assumed for the reflection
component.
In our simulations, the polarization is assumed to arise
due to the reflection component. Two scenarios were tested:
100% and 20% polarization for the reflection component,
with unpolarized direct emission. This corresponds to a to-
tal average polarization around 17% and 3% respectively.
Fig. 4. Observed radiation spectrum and input model used in the
simulations of observations of Cygnus X-1. Gray lines: Typical ra-
diation spectrum of Cygnus X-1 in the hard state. Black lines: As-
sumed input spectrum: a cut-off power law with index α = −1.2 and
cut-off Ecut = 120 keV, normalized to match the measured flux. The
reflection component is shown, and the energy range of PoGOLite
is indicated by vertical lines.
The energy dependent polarization Π(E) used as input was
set to the relative strength of the reflection component com-
pared to the total flux, scaled down in the case of 20% po-
larization. Simulations were performed for effective observ-
ing times of 6 hours and 35 hours. These times are chosen
as realistic estimates for short and long duration balloon
flights, respectively.
3.3.2. Accreting Magnetic Neutron Star
In the case of the neutron star, we study the observability
of energy dependent effects by simulations of three different
idealized polarized spectra:
– Polarization in a narrow band.
– Polarization only at low energies.
– Polarization only at high energies.
The neutron star spectrum was in all cases approxi-
mated with an exponentially cut-off power law, with index
α = −1.1 and energy cut-off at Ecut = 70keV. It was
normalized to correspond to a 100mCrab source.
Assuming a cyclotron energy Ec at 50 keV, we use three
toy models of the polarization energy dependence Π(E),
with Πmax ≡ p%:
– A Gaussian peak centred at 50 keV, Gp, modelling a
rise in polarization from 0% to maximum p%, using the
Gaussian curve
Π(E) = pe
−(E−50)2
2σ2 % (5)
with E measured in keV and σ = 5 keV.
– Two step functions, Sp and S−p, with polarization
0% if E < 50 keV
p% if E ≥ 50 keV
for Sp, and
10 20 30 40 50 60 70 80 90 100
E (keV)
Fig. 5. Example of assumed energy dependence of the polarization
fraction in the case of an accreting magnetic neutron star. The figure
shows a Gaussian curve with Πmax = 20% (G20), and a positive step
with Πmax = 10% (S10).
p% if E < 50 keV
0% if E ≥ 50 keV
for S−p. Simply put, p is the jump in polarization that
occurs at E = 50 keV.
Figure 5 illustrates examples of G20 and S10, the
Gaussian and positive steps with maxima 20% and
10% respectively. In the simulations, the values p =
{10, 20, 30, 40, 50}% were used, each assuming an observa-
tion time of 35 hours.
3.4. Background
Balloon-borne gamma-ray polarimetry measurements
are subject to several significant sources of background.
Through the use of the well-type phoswich detector tech-
nique, the PoGOLite instrument has been designed to
reduce these backgrounds, allowing 10% polarization of a
100 mCrab source to be measured in one 6 hour balloon
observation in the 25–100keV energy range. The basic
phoswhich design was used in the WELCOME series of
balloon-borne observations and allowed effective back-
ground suppression [36,37,38,39,40,41,42]. The concept
was subsequently improved and effectively used in a satel-
lite instrument, the Suzaku Hard X-ray Detector (HXD)
[29,43,44,45].
The background to PoGOLite measurements can arise
from charged cosmic rays, neutrons (atmospheric and in-
strumental) and gamma-rays (primary and atmospheric).
The background from charged cosmic rays (predominantly
protons, ∼90%, and helium nuclei, ∼10%) is rejected by
the BGO anticoincidence shields and slow plastic collima-
tors. Cosmic rays are minimum ionizing particles and can
be identified through their relatively large energy deposits.
The background presented by atmospheric neutrons and
neutrons produced in the PoGOLite instrument and sur-
rounding structures is currently being studied in detail [46].
Fig. 6. Estimated contribution to the gamma-ray background in the
PoGOLite energy range (∼ 25–100 keV). The radiation from the two
considered sources, Cygnus X-1 and an accreting neutron star, are
also shown.
For the purposes of the study presented in this paper, par-
ticular attention has been paid to what is expected to be the
dominant background: primary and atmospheric gamma-
rays. The gamma-ray background rate is estimated from a
model derived from measurements taken in Texas with the
GLAST Balloon Flight Engineering Model [47].
The primary gamma-ray component originates outside
the atmosphere, i.e., above PoGOLite. The angular distri-
bution of the radiation is uniform within the hemisphere
above PoGOLite. The energy spectrum is modeled by
a doubly-broken power-law with breaks at 50 keV and
1 MeV [48].
Secondary gamma-rays are created in the Earth’s at-
mosphere through bremsstrahlung interactions of charged
cosmic-rays. Two separate components are considered, one
directed upwards and one downwards. The upward flux is
dependent on the zenith angle [49], and the energy spec-
trum consists of a doubly-broken power-law with breaks
at 10 MeV and 1 GeV, and a 511 keV line from electron-
positron annihilation. The downward component is simi-
lar, but with breaks at 1 MeV and 1 GeV. Energies up to
100 GeV were generated for all components. These models
are based on data from satellite- and balloon-borne instru-
ments ([50] and [51], and references therein).
Figure 6 shows the estimated gamma-ray backgrounds
compared to the accreting pulsar and Cygnus X-1 models.
The total gamma-ray background is at the 10 mCrab level.
4. Analysis and Results
4.1. Data Processing
In the first data processing step, the resolution of the
scintillator-PMT assembly is simulated by fluctuating the
number of photo-electrons generated in the scintillating
materials. It is assumed that when the energyE is deposited
in a cell, the average number of photo-electrons generated
is En, with n set to 0.5 photo-electrons per keV. Now, we
fluctuate En to (En)fluct by applying a Gaussian spread.
If En ≤ 10, we do it in two steps: First we subject it to a
Poissonian spread, and thereafter a Gaussian spread with
variance Enσ2, with σ set to 0.4. If En > 10, only a Gaus-
sian spread with variance En is used. If (En)fluct < 0, then
it is set to 0. Finally, we take
Emes =
(En)fluct
as the energy actually measured by the PMT in the cell
of interest. We reject all fast scintillator interactions with
Emes below a certain measurement threshold (2 keV). For
the analysis described in this paper, only events with two
or three hits in the fast scintillators are retained (the veto
logic is not considered at this stage). At PoGOLite energies,
more than 80% of the events are from photons interacting
in no more than three detector cells [52].
For two-site events, the chronological order of the two
cells does not matter for angle calculation, as the distribu-
tion is periodic over the angle π. In the case of three hits,
we calculate the scattering angle by ignoring the hit with
the lowest energy measured, assuming that a low-energy
interaction does not affect direction much, and derive an
angle from the positions of the two cells with highest en-
ergy deposits.
Most photons do not scatter very far; about half will only
go from one cell to its neighbour. As the range of possible
scattering angles resulting in detection in a given adjacent
cell is large, this causes strong peaks in each of the six direc-
tions corresponding to the neighbouring cells. The PoGO-
Lite instrument will rotate about its axis, causing the range
for a given cell to smoothly vary and thereby creating a
continuous distribution over angles. In the simulations, the
uncertainty of the angle determination is instead approxi-
mated by introducing a Gaussian spread to the measured
scattering angles.
In the last steps of data processing we take into account
the mass of air in the atmosphere above the balloon, filter-
ing out roughly half of our incident source radiation, as-
suming the atmospheric overburden 4 g/cm2 at 40 km alti-
tude. We also apply the veto logic, rejecting all events with
detection in any slow plastic scintillator or BGO crystal.
4.2. A χ2 measure
Tomeasure the polarization energy dependence, one can-
not simply calculate the polarization at certain energies and
construct Π(E), since the photon energy Eγ always will be
unknown due to the response of the instrument. Instead we
calculate the modulation factor at different measured en-
ergies, obtaining a curve M(Emes). This curve can then be
compared with theoretic curves resulting from other mod-
els, possibly from the same family of curves, enabling us to
reject complete families of energy dependencies. One such
family, which we will be concerned with here, is the set of
constant polarizations.
Fig. 7. Example of the measured distribution of events as a function
of scattering angle (histogram). In this example, the simulated results
of a six hour observation of Cygnus X-1 are shown, assuming a
completely polarized reflection component. The data are from the
measured energy range of 30–35 keV. A sinusoidal function is fit to
the data (solid line, cf. Eq. 3) and a modulation factor is calculated.
For a given source model (polarization energy depen-
dence) A, the modulation factor M (Eq. 2) was fitted
at different measured energies, yielding a curve MA =
MA(Emes). Figure 7 shows an example of a modulation
curve in the 30–35keV band, generated for a six hour
observation of Cygnus X-1 in the case of a completely
polarized reflection component. The resulting modulation
factor is 2.69± 0.30. The modulation factor is in this way
calculated for each energy band. The result is a curve
showing how the modulation factor varies with energy,
which can then be compared to the corresponding curves
for various models of polarization.
To test the results against constant polarization, we also
generated curves MΠ with constant levels of polarization
Π. Since these latter curves should not be thought of as
measured, but fluctuation-free theoretical constructs, they
were generated by much longer simulations than the obser-
vational curves.
A measure of how much two curves differ is defined as
χ2Π =
(MA,Emes −MΠ,Emes)
σ2Emes
, (9)
with σi as the sum of the two errors in fitting MA,Emes and
MΠ,Emes . When this is calculated for all reasonable values
of Π, we can reject the hypothesis of constant polarization
if the minimum of χ2Π is high enough. For 16 degrees of
freedom, corresponding to data points up to 100 keV, the
95% certainty level requires χ2 > 26.3.
4.3. Cygnus X-1
Table 1 summarises the results for the simulations of
Cygnus X-1. For a 100% polarized reflection component,
the energy dependence is detected both after 6 hours of
observation and after 35 hours. Figure 8 shows the expected
modulation factors and χ2 values for a 35h observation. In
Reflection 100% Reflection 20%
Obs. time 6h 35h 6h 35h
Significance 99.4% >99.99% 19.1% 51.2%
Table 1
The significance in rejecting constant polarization models, shown for
different reflection polarization strengths and observation times in
the case of Cygnus X-1.
Gaussian peaks
Πmax (%) 10 20 30 40 50
Significance 24.3% 24.3% 35.3% 79.9% 99.7%
Positive steps
Πmax (%) 10 20 30 40 50
Significance 46.3% 70.0% 96.8% 91.8% 83.5%
Negative steps
Πmax (%) 10 20 30 40 50
Significance 51.0% 99.3% 95.8% 99.6% >99.99%
Table 2
The significance in rejecting constant polarization models, shown for
different polarization shapes and maxima in the case of a 100mCrab
neutron star.
the case of 20% polarization of the reflection component,
shown in Fig. 9, constant polarization cannot be ruled out
at any higher significance level.
4.4. Magnetic NS
The results for the simulations of the accreting neutron
star are summarised in Table 2. For a 100mCrab source,
the background will start to become significant already in
the higher end of the PoGOLite energy range. To be con-
servative, only M(E) data points up to 60 keV were used
for the χ2 analysis in order to limit the errors due to un-
certainties in the background flux. This is equivalent to 8
degrees of freedom, demanding χ2 > 15.5 for 95% certainty
in rejecting constant polarization.
4.4.1. Gaussian peaks
In Fig. 10 we seeM(E) curves forG50 and a few constant
polarizations (left panel), clearly illustrating their different
characteristics. The resulting χ2 curve (right panel) con-
firms that these models are significantly (99% level) differ-
ent. However, this was not the case for any lower value of
Πmax.
4.4.2. Positive steps
Figure 11 shows the result for S50 when compared to
cases of constant polarization. In this case, the difference in
characteristics between the constant models and the energy
dependent model is not large enough, and constant polar-
ization cannot be ruled out. The only value of Πmax yielding
a significant (95% level) difference was Πmax = 30%. What
is interesting to note is that a higher Πmax, which implies
20 30 40 50 60 70 80 90 100
Modulation factors, 35h
E (keV)
Cygx1 P(E), ref=100
P=15%
P=17%
Fig. 8. Results from simulations of a 35h observation of Cygnus X-1, with the reflection component assumed to be 100% polarized. Left panel:
Expected modulation factor M(E), together with models of constant polarization at 15% and 17%. The modulation is fitted in intervals
of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed line
marks the limit for 99.9% significance in rejecting constant polarization. Constant polarization is rejected with high significance.
20 30 40 50 60 70 80 90 100
Modulation factors, 35h
E (keV)
Cygx1 P(E), ref=20
Fig. 9. Same as Fig. 8, but with the reflection component assumed to be 20% polarized. Left panel: Expected modulation factor M(E),
together with models of constant polarization at 2% and 5%. Right panel: χ2 values when comparing M(E) for the simulation with different
degrees of constant polarization. The dashed line marks the limit for 95% significance in rejecting constant polarization. Constant polarization
cannot be rejected with high significance.
20 30 40 50 60 70 80
Modulation factors, 35h
E (keV)
NS, G50
Const. 0%
Const. 10%
Const. 20%
Fig. 10. Results from simulations of a 35h observation of an accreting magnetic neutron star, assuming a Gaussian polarization curve. Left
panel: Expected modulation factor M(E), together with models of constant polarization at 0%, 10% and 20%. The modulation is fitted in
intervals of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The
dashed line marks the limit for 95% significance in rejecting constant polarization. Constant polarization is rejected with high significance.
20 30 40 50 60 70 80
Modulation factors, 35h
E (keV)
NS, S50
Const. 20%
Const. 30%
Const. 40%
Fig. 11. Results from simulations of a 35h observation of an accreting magnetic neutron star, assuming a positive step with Πmax = 50%. Left
panel: Expected modulation factor M(E), together with models of constant polarization at 20%, 30% and 40%. The modulation is fitted in
intervals of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed
line marks the limit for 95% significance in rejecting constant polarization. Constant polarization cannot be rejected with high significance.
a higher total polarization, does not necessarily make the
energy dependence easier to measure.
4.4.3. Negative steps
The result of the S−50 model is shown in Fig. 12, where
the χ2 plot (right panel) allows us to reject constant polar-
ization with a certainty much higher than 99.99%. Other
models yielding significant detections were S−20 (99%),
S−30 (95%), and S−40 (99%). One reason these models are
so easily measurable stems from the fact that, due to the
shape of the neutron star spectrum, data and statistics are
much poorer at higher energies. If the polarized part only
lies at high energies, the total polarization will be much
lower or even undetectable.
5. Discussion
Although no measurements have been made of polariza-
tion in the energy range covered by PoGOLite, some theo-
retical models predict changes in polarization with energy.
The results of the simulations clearly show that it is possible
for the PoGOLite instrument to detect the energy depen-
dence of polarization for several of the investigated cases.
The highest significance is found for Cygnus X-1 assuming
a fully polarized reflection component and a neutron star
in the case of a negative step; in the remaining cases, con-
stant polarization can not be rejected. Our results there-
fore show that PoGOLite has the potential to discriminate
among these models.
As described above, in Cygnus X-1 the hard X-rays are
believed to originate from Comptonization of soft seed pho-
tons in a predominantly thermal electron distribution. Al-
though this process involves Compton scattering – which
could introduce a net polarization – multiple scatterings
are required, making our assumption of the direct compo-
nent being unpolarized a reasonable one. The degree of po-
larization of the reflected component is however more un-
certain. Our idealized case of 100% polarization is certainly
an overestimation. Calculations [19] show that the degree
of polarization in the reflected component varies with in-
clination, with a maximum of ∼ 30% expected at high in-
clination. The inclination of the Cygnus X-1 system is not
well known, but estimates put it at 30◦–50◦ [53], making
our assumption of 20% polarization in the reflected compo-
nent reasonable. The relative size of the reflected compo-
nent compared to the direct emission is in turn dependent
on both inclination, where the dependence is the opposite
one, and system geometry. We note that in other sources
the reflected component may be much stronger, or even
dominate the radiation spectrum (e.g., Cygnus X-3, [54]).
Another issue which may complicate measurements of
energy dependent polarization is the behaviour of the angle
of polarization. In our simulations, we have implicitly as-
sumed that the angle does not vary with energy. However,
for both black hole and neutron star systems, this assump-
tion may be an oversimplification. It is certainly true that
emission originating from the region close to a black hole
will be affected by the strong gravity, affecting the polar-
ization angle [21]. It is not clear how large this effect would
be on the reflected component in, for instance, Cygnus X-1,
but results from both temporal and spectral analysis show
that the accretion disc – assumed to be the reflector – is pre-
sumably truncated at a large distance (Rin & 30Rg, [55,56])
from the black hole in the hard state. We therefore do not
expect this effect to alter the outcome of our simulations.
Our results from simulating Cygnus X-1 indicate that
long observations with PoGOLite are required to search for
energy dependence of polarization. The first flights of the
instrument will likely be shorter flights, covering several
targets. While these observations should be long enough to
detect polarization down to the level of a few per cent, we
do not expect to detect any changes in polarizationwith en-
ergy. However, long duration flights spanning several days
are also planned, and such flights would provide the obser-
20 30 40 50 60 70 80
Modulation factors, 35h
E (keV)
NS, S−50
Const. 20%
Const. 30%
Const. 40%
Fig. 12. Same as in Fig. 11, but using the negative step. In this case, constant polarization is rejected with high significance.
vation time needed to search for variations of polarization
degree with energy.
A point to note from the neutron star simulations is the
result that it is easier to rule out constant polarization in
the case of a negative step than for a positive step. As noted
in Sect. 4, the energy response of the instrument is such
that the energy of the incoming photon cannot be uniquely
determined. The result will be a redistribution of energy,
with a possibility for higher energy photons to be detected
with lower energies. The reverse is however not true – a
low energy photon will not be detected as having a higher
energy. In the case of a positive step, some polarized high
energy photons will be detected at lower energies. This will
give a false polarization signal at lower energies, and act to
“smooth” the detected energy dependance of polarization.
For a negative step, the polarization is introduced at lower
energies and will not “spread” to higher energies. The low
energy polarization will be somewhat diluted by redistribu-
tion of high energy photons but the polarization contrast
will still be higher than in the positive case.
By excluding energies above 60 keV in the χ2 analysis of
accreting X-ray pulsars we have been fairly conservative in
our estimate of significances. The restriction of the energy
range was motivated by a potentially high sensitivity to
systematic errors in the background level at high energies.
In this analysis however, we have not taken advantage of
the fact that these sources are pulsating. By analysing the
polarization of the pulsed flux, rather than the total flux,
it should be possible to include all points up to 100 keV
and thereby increase the sensitivity. On the other hand the
polarization direction will probably change over the pulsa-
tion period which will have the opposite effect of reducing
the sensitivity. How important this effect is depends on the
precise source geometry, radiation beaming and our view-
ing angle.
As PoGOLite’s field-of-view is rather large (∼ 5 deg2),
the pointing errors with respect to the axis of rotation must
be small to avoid introducing systematic errors in the po-
larization measurements. The attitude control system used
for PoGOLite will assure accurate pointing to within a few
arcminutes, keeping the systematic error below 1% [10]. Al-
though this figure refers to the whole energy band, we do
not expect any such effect to change the results presented
here. A comprehensive study of systematic effects is beyond
the scope of this paper, but will be crucial once PoGOLite
is in operation.
The performance of the PoGOLite instrument has been
extensively evaluated, both with laboratory-based tests
[57], accelerator-based tests [28], and simulations [32].
These tests show that it will be able to detect low (∼ 10%)
degrees of polarization even for 100 mCrab sources. What
has not previously been tested is its sensitivity to a polar-
ization degree that varies with energy. Despite the rela-
tively modest inherent energy resolution, our results show
that PoGOLite has the capacity to detect changes in po-
larization degree with energy. The simulations show that
significant results can be obtained in a 35h observation,
attainable in the long duration flights already planned for
PoGOLite. We stress that the design is not optimized for
such detections, and future instruments will in all likeli-
hood develop this technique further.
6. Conclusions
The Compton technique applied to an array of plastic
scintillators is an effective method to measure broad en-
ergy band polarization, which is demonstrated with the
proposed PoGOLite mission, using Geant4 simulations. In
particular, energy dependence can be detected. However, in
our model of polarization from X-ray binaries, we require
the reflection to contribute a large fraction of the observed
flux and/or have high degree of polarization for energy de-
pendence to be detected. Similarly, for accreting magnetic
neutron stars, sharp energy variations in the polarization
are needed for a clear detection. This is made easier if the
lower energies contain most of the polarization.
Acknowledgments
The authors gratefully acknowledge support from the
Knut and Alice Wallenberg Foundation, the Swedish Na-
tional Space Board, the Swedish Research Council, the
Kavli Institute for Particle Astrophysics and Cosmology
(KIPAC) at Stanford University through an Enterprise
Fund, and the Ministry of Education, Science, Sports and
Culture (Japan) Grant-in-Aid in Science No.18340052.
J.K. and N.K. acknowledge support by JSPS.KAKENHI
(16340055).J.K. was also supported by a grant for the in-
ternational mission research, which was provided by the In-
stitute for Space and Astronautical Science (ISAS/JAXA).
T.M. acknowledges support by Grants-in-Aid for Young
Scientists (B) from Japan Society for the Promotion of
Science (No. 18740154).
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http://www.particle.kth.se/pogolite
	Introduction
	Measurement of polarization
	Expected objects of interest
	The Compton technique
	Basic principle
	PoGOLite
	Simulations
	Geant4
	Simulation setup
	Source Models
	Background
	Analysis and Results
	Data Processing
	A 2 measure
	Cygnus X-1
	Magnetic NS
	Discussion
	Conclusions
	References
