The Gas Pixel Detectors
for the IXPE mission
Candidato Federico Pucci
Relatore Prof. Luca Baldini
i n t r o d u c t i o n 1
1 x-ray polarimetry 3
1.1 Basic principles of polarimeters 3
1.2 X-ray polarimetry techniques 7
1.2.1 Bragg diffraction 7
1.2.2 Thomson scattering 9
1.2.3 Photoelectric effect 11
1.3 History of X-ray polarization measurements 16
1.3.1 Sounding rocket experiments 16
1.3.2 Orbiting Solar Observatory 8 18
1.3.3 Electron tracking 20
2 t h e i m a g i n g x-ray polarimetry explorer 21
2.1 Scientific goals 21
2.1.1 Supernova remnants 21
2.1.2 Accretion disks 24
2.1.3 Black holes 31
2.1.4 Emission in strong magnetic fields 35
2.2 Technical overview 43
2.2.1 Launch 44
2.2.2 Mirror Module Assemblies 45
2.2.3 Deployable boom 47
2.2.4 Detector Units 49
3 t h e g a s p i x e l d e t e c t o r 52
3.1 The GPD for IXPE 52
3.1.1 Gas cell 54 3.1.2 GEM 56 3.1.3 Readout anode 58 3.1.4 DAQ 60 3.2 Reconstruction algorithm 61 3.2.1 Clustering stage 62 3.2.2 Reconstruction stage 63 3.2.3 Moment analysis 65 4 d e t e c t o r s i m u l at i o n 67 4.1 Detector construction 67
4.3 Simulation flow 69
4.3.1 Generation and propagation of the
photo-electron 69
4.3.2 Drift and multiplication 71
4.3.3 Digitization 72
4.4 Comparison of real and simulated data 73
4.4.1 Reconstruction of simulated data 77
4.5 Study on the gas mixture 78
4.5.1 Theoretical premises 79
4.5.2 Simulation results 81
5 p e r f o r m a n c e s t u d i e s 83
5.1 Study on the attachment 83
5.1.1 Introduction 83 5.1.2 Setup 84 5.1.3 Attachment simulation 86 5.1.4 Analysis 88 5.1.5 Test results 90 5.2 Thermal tests 91 5.2.1 Setup 91 5.2.2 Analysis 94 5.2.3 Test results 103 c o n c l u s i o n s 105
Astronomical polarimetry allows to investigate physical an-isotropies linked to astrophysical phenomena like non-spheric distributions of matter and ordered magnetic fields. X-ray as-tronomy has a vast history of successful missions and obser-vations that, nevertheless, include almost no polarimetric data: X-ray polarimetry counts just one target, the Crab nebula, with a highly significant, unambiguous result, and a handful of mar-ginal measurements.
The reason for such a gap is mainly found in the very na-ture of X-ray polarimetry itself: it requires significantly more data to produce meaningful results, when compared to spectral or photometric measurements. Moreover, polarimeters used in the 1960s and 1970s (the only period when polarimetry mis-sions have been performed) had low efficiencies, thus further worsening the problem. The physical processes exploited to build those types of polarimeters, as well as the missions that have carried one, are described in chapter 1.
The Imaging X-ray Polarimetry Explorer (IXPE) is a mission set to be launched in 2021 that will allow again, after a more than 40 years gap, to perform polarimetric measurements in the X band. IXPE will be able to investigate tens of astrophys-ical sources, giving a whole new set of data to investigate the physical processes happening inside and around them.
IXPE will comprise three telescopes, each one made of a mir-ror section to focus X-rays onto the detectors, called Gas Pixel Detectors (GPDs). The GPD is the heart of IXPE, the piece of technology that made it possible. A GPD is a new kind of po-larimeter that exploits photoelectric effect in a gas cell to mea-sure the direction of polarization of incoming radiation. With respect to the legacy detectors of the past, it also has imaging and spectral capabilities, that greatly increase the amount of information that can be inferred from the data.
Chapter 2 presents an overview of the physical phenomena that can be studied just with polarimetric data, as well as a de-scription of the IXPE spacecraft. Chapter 3 focuses on the GPD and describes its structure, how it works and the reconstruction algorithm needed to extract polarimetric results from the raw data.
As part of my personal contribution, I have actively taken part in the development of the detector simulator, a tool that has proven to be fundamental for the optimization of the de-tector concept. In chapter 4, I describe the libraries used in the simulator, the simulation flow and the results of some studies I have performed using it, concerning the optimization of the gas mixture.
Chapter 5 is about two performance studies for whom I have taken part in the data collection and I have carried out the analysis. The first study investigates the recombination rate of electrons, that should normally be detected as signal, inside the gas cell of the GPD. The test setup has involved a beam of X-rays not aligned with the normal to the plane of the GPD, to correlate the depth of absorption (not detected in normal cir-cumstances) to the detected image. The recombination effect and another uncorrelated mechanism yield a similar effect in the output data, but by using the simulator I have been able to separate the two contributions. With the results of the test, those effects can be appropriately described and effectively cor-rected.
The second study characterizes the response of the GPD with respect to temperature. The temperature stability of the detector will be at the level of ±5◦C. Knowing if the
perfor-mance is stable when subject to unavoidable temperature fluc-tuations is crucial to foresee possible issues that might arise
in orbit. In the investigated temperature range, from 15◦C to
40◦C, the GPD performance is remarkably stable on all
high-level quantities; in particular, polarimetry-related ones show just marginal fluctuations with temperature.
The tests described in this thesis work have contributed to get to the final design of the detector. The analysis scripts, in-cluding the ones I have developed, will evolve and be used in the upcoming calibration phase of the flight models of the De-tector Units.
1
X - R AY P O L A R I M E T R YThe detection of electromagnetic waves is the main way in-formation about astronomical sources can be obtained. When analyzing detected photons, there are four interesting quanti-ties to study.
• The direction the photons come from is a direct indicator of the position of the source. It can be inferred by know-ing how the instrument is positioned and, if the detector has imaging capabilities, which part of it has detected the signal.
• The time when a signal is detected is important to know if the emission of the source is constant and, if not, how it varies over time.
• The energy of photons can be used to build a spectrum of the source, from whom a wealth of information can be inferred.
• The distribution of the degree and angle of polarization of incoming photons is an indicator of the morphology of the source, as well as the physical processes happening inside it or along the line of sight.
Polarimetry concerning photons in the X band is an almost completely unexplored field, that might observationally con-firm — or disprove — a number of theoretical physical models.
1.1 basic principles of polarimeters
A traditional polarimeter is an object made of an analyzer and a detector, free to rotate around their common axis ([1]). During the rotation, a modulated signal can be detected, whose frequency is double the rotation frequency of the polarimeter:
N(φ) = A+B cos2(φ−φ0) , (1.1)
where φ0 is the angle of polarization of incoming radiation.
150 100 50 0 50 100 150 Polarization angle [deg]
0 1000 2000 3000 4000 5000 6000 7000 8000 Entries/bin
Figure 1.1: Modulation curve resulting from incoming polar-ized X-rays. The shape of the distribution is that of eq. 1.1. that a polarimeter allows to produce, because the polarimetric-relevant quantities are obtained by fitting it.
It is worth noting that a polarimetry measurement is ulti-mately sampling a cosine squared distribution. This means that the number of events needed for a meaningful result is much larger than with photometric or spectral measurements. If, as a bare minimum, 3 events can be called a detection and 100 can be used to build a decent spectrum, the order of magnitude for a polarimetric result is 100 000 events detected. That, of course, implies much larger observation times. To understand why that number is so high, it is first necessary to introduce some polarimetry-related quantities.
Back to eq. 1.1, if m= [N(φ)]minand M = [N(φ)]maxare the extrema of the distribution, then the quantity
V = M−m
M+m
is called visibility of the modulation. In the case of 100 % polar-ized incoming radiation, with no background, the visibility is referred to as the modulation factor µ. It represents the response of a polarimeter to a 100 % polarized source. The modulation factor spans from 0 (an insensitive detector) to 1 (maximum
150 100 50 0 50 100 150 Polarization angle [deg]
0 1000 2000 3000 4000 5000 6000 7000 8000 Entries/bin
Figure 1.2: Modulation curve resulting from incoming unpolar-ized X-rays. To obtain a plot like that, systematic effects must be considered and corrected, in order not to get a non-zero modulation.
sensitivity). Eq. 1.1 can be re-written in the following way, to highlight the modulation factor:
N(φ) = C1+µcos 2(φ−φ0) ; µ = B B
+2A . (1.2)
Since the degree of linear polarization is positive definite, the random noise will not average to zero and thus the output will show a certain degree of polarization even for an unpolar-ized source. Moreover, systematic effects can introduce more spurious modulation. A distribution like the one in fig. 1.2 is the ideal response of a polarimeter to unpolarized radiation. When systematics are under control, that is the result, but if there are effects that are not corrected, a certain degree of mod-ulation might show up even if the incoming radiation is exactly 0 % polarized.
One of the most important quantities to consider is the min-imum detectable polarization (MDP) at a certain confidence level: it is the degree of polarization that will not be exceeded by
chance with said confidence level. The general formula, at the level of nσ standard deviations, is ([2])
MDP(nσ) = nσ
µεs r
2εs+b
Σt , (1.3)
where s is the source flux, b is the background rate per unit area, ε and Σ are the efficiency and effective sensitive area of the detector and t is the observing time. In the limits of bright source, where the background in negligible, and faint source, where the background is dominant, the MDP is proportional to
MDP ∝ 1
µ√ε if s b; MDP ∝
1
µε if s b.
Because of that, the relevant figure of merit of the instrument is different depending on the brightness of the source:
FOM =µ√ε if s b; FOM =µε if s b.
Typically, the MDP at 99 % confidence level is used to de-termine how the instrument is performing. Using the effective counting rates for the source and the background S and B, in the absence of systematic effects eq. 1.3 becomes
MDP99 = 4.29
µ√t
√
S+B
S . (1.4)
In the limit of bright source, that becomes
MDP99 = 4.29
µ√St =
4.29
µ√N ,
where N is the number of detected photons. Assuming a mod-ulation factor µ =0.5 (a reasonable figure, as described in the next section) and solving for N to aim for a 2 % MDP, the result is N= 4.29 µMDP 2 ≈1.84×105 .
That is why, as stated earlier, the number of needed photons to perform meaningful polarization is of the order of 100 000.
It should be noted that the MDP does not represent the un-certainty of a measurement, but is a statistical value that yields an upper limit in the case of no polarization. A practical im-plication of this is that, if a given value P for the degree of polarization has to be detected, an observing time that gives an
MDP99 = P is not enough to have a meaningful measurement.
dsinθ θ
d
Figure 1.3: Visual representation of Bragg diffraction: photons scattered at the right angle, determined by eq. 1.5, interfere constructively after the reflection.
1.2 x-ray polarimetry techniques
The detection of linear polarization of X-rays is less straight-forward than, say, photometric and spectral measurements per-formed in the same energy band. Three physical phenomena have been exploited throughout the history of X-ray polarime-try: those are described in this section.
1.2.1 Bragg diffraction
Schnopper and Kalata first described Bragg diffraction at 45° in [4], in 1969. The principle it is based on is the Bragg condi-tion.
Bragg diffraction occurs when X-rays coherently scatter on a crystal lattice. If d is the separation between the lattice planes of the crystal and θ is the angle between the photon incoming direction and the planes, the Bragg condition is written as
2d sin θ =nλ ; (1.5)
this means that the difference in path length must be an integer number of wavelengths, so that the interference between the scattered photons is constructive. This can be seen in fig. 1.3.
If the polarization of the incident X-ray is considered to be split into two components, the corresponding intensities are I//
and I⊥, with respect to the scattering1
plane. If the incidence angle is θi =45°, then the intensities after reflections are I//0 =0 and I0
⊥ = I⊥. The variations from those values are small for
1The scattering plane is defined by the direction of the incident photon and
1.2 x-ray polarimetry techniques
polarimeter is ideal. Practical implementations
accept larger angles at the expense of the
modula-tion factor (
Fig. 1
) Thomson scattering competes at
at room temperature). It must be encapsulated
with beryllium or plastics and 4-keV is a lower
boundary for the energy. Multiple scattering and
Fig. 1. Concept of Bragg crystal dispersive polarimeter (top)and Thomson scattering non-dispersive polarimeter (bottom).
Rotation Angle° Count s s -1 * 1 0 0 0 10 20 30 40 50 0 90 180 270 360
Fig. 2. (Top) Polarimeter on-board of OSO-8. (Middle) Modulation curve from the Crab Nebula [22]. (Bottom) The focal plane of SXRP[30] with detectors and analyzers.
P. Soffitta et al. / Nuclear Instruments and Methods in Physics Research A 510 (2003) 170–175 172
Figure 1.4: Schematic view of an X-ray polarimeter exploiting Bragg diffraction; two configurations are shown ([5]).
incidence angles that are not far from 45°. That means that the modulation factor approaches 100 % for the right angles.
As just the radiation polarized perpendicularly to the scat-tering plane is reflected, a modulated signal as in eq. 1.1 can be detected by rotating the crystal around the direction of in-coming photons. The intensity of scattered radiation actually depends on the Fourier transform
I(q) ∝ Z ei(kf−ki)·rρ(r)d3r 2 ,
where kiand kf are the initial and final momenta of the photon.
In a perfect crystal, I(q) would be a series of delta functions, corresponding to the solutions of the Bragg condition 1.5. Real crystals, though, have imperfections and border effects, so the delta spikes become peaks with a finite width.
Fig. 1.4 represents a schematic Bragg polarimeter. If we con-sider just the first diffraction peak at the energy
E1 = 2d sin θhc ,
at a given incidence angle θ, the counting rate can be written as R= I(E)·ε(E)·S(E)·ctg θ·∆Θ(E),
where I(E) is the source intensity at the energy E, ε is the effi-ciency of the instrument, S is the effective collection area and ∆Θ is the integrated reflectivity, given by
∆Θ(E) =
Z
R(θ, E)dθ .
The last quantity is a fundamental property of the crystal, be-cause it determines the number of photons detected. Typical values of ∆Θ are 10−4– 10−3for a 45° incidence angle.
The extremely narrow bandwidth is the main issue of Bragg diffraction that, although with modulation factors near 100 %, has an overall low efficiency that limits its effectiveness.
1.2.2 Thomson scattering
Incoherent scattering from free electrons provides an alter-native way of detecting X-ray polarization over a broad energy range. That technique was described by Novick and Wolff in [6], in 1971.
In non-relativistic approximation, the scattering of linearly polarized X-rays on free electrons is described by
dσ
dΩ =
r2 e
2 cos2θcos2φ+sin2φ
, (1.6)
where θ is the scattering angle and φ is the angle between the polarization vector of incoming photons and the scattering plane. The radiation scattered at 90° is thus modulated as
I(φ) ∝ sin2φ, (1.7)
so only photons with electric field vectors normal to the plane of incidence are scattered at 90°. That anisotropy in the scatter-ing cross section can be exploited to measure the polarization of the incident flux.
Fig. 1.5 shows a schematic picture of a Thomson polarime-ter. Incident X-rays hit a target typically made of lithium and scatter on a proportional counter that surrounds the target.
The modulation factor of a Thomson polarimeter typically ranges between 30 % and 50 %, depending on the geometry of the instrument. The main factor is the cross section 1.6, that
1.2 x-ray polarimetry techniques
Figure 1.5: Schematic picture of an X-ray polarimeter exploiting Thomson scattering. The proportional counter is placed around the scattering material because radiation is nicely modulated as in eq. 1.7 just when it is scattered at 90° [7].
is 100 % modulated only if θ = 90°. Off-axis reflections and
multiple scattering further deteriorate information. The overall efficiency of the system can be written as
η ≈ε µs
µs+µa 1−e
−(µs+µa)z0 e−(µs+µa)r0 ,
where r0 and z0 are the radius and the height of the scattering
target, µs and µa are the scattering and absorption coefficients
and ε is the efficiency of the proportional counter. Apart from the last one, that is self-explanatory, there are three factors to consider.
The first factor, the term µs(µs+µa)−1, gets larger as the
scattering cross section rises and the absorption cross section gets smaller. The most efficient absorption process in the en-ergy range (2 – 8)keV is photoelectric effect, whose cross
sec-tion depends on Z5. Since Thomson scattering cross section
depends on Z, the scattering material must be chosen to have the lowest atomic number possible. Among materials that are
solid at room temperature, the lightest one is lithium (Z = 3),
that is actually the most used material for this application. The second factor means that the more the target is high (or long), the more photons will be scattered; that is obvious, but comes with a cost. The modulation in intensity described by eq. 1.7 happens only when the scattering angle is 90°; as the
detector lengthens, though, the possibility of detecting photons scattered at much lower angles rises, causing a loss of informa-tion.
The last term states that the scattering target should be as thin as possible, because multiple scattering deteriorates the in-formation. The target area is effectively limited by the factor rmax ≈ (µs+µa)−1: above that, the probability of multiple
in-teractions of a photon inside the material gets too high. In the case of lithium, rmax ≈7 cm.
Moreover, the efficiency poses a practical lower limit to the energies that can be detected. In the example given in [7] and schematized in fig. 1.5, if z0 = 5 cm the efficiency at 7 keV is
about 10 %, but at 4 keV it is just 0.4 %.
The mechanism that Thomson polarimetry exploits limits the efficiency of the technique and the constrains in size does not allow to scale up this kind of device. Those limits are funda-mental, so there seems to be little to no possibility of improving the performance of this type of detector.
1.2.3 Photoelectric effect
Photoelectric absorption is the dominant interaction process for X-rays in the(2 – 8)keV energy range. It is the absorption of a photon followed by the emission of an atomic electron, whose energy is the difference between the energy of the photon and the binding energy of the electron:
Epe=Eγ−Eb .
An exact treatment of the subject is complicated; suffice it to
say that the cross section dependence ranges between Z4 and
Z5, depending on the energy. The asymptotic behaviors, with
respect to the binding energy EK of electrons in the innermost
shell (the K-shell), are
σpe ∝ Z
5
hν if E EK; σpe∝
Z4
(hν)3 if E EK.
The typical trend of photoelectric cross section is shown in fig. 1.6. Since photoelectric absorption is more effective in inner shells, the cross section has abrupt rises as soon as the binding energy for a shell is overcome.
Figure 1.6: Photoelectric cross section with respect to energy for lead. The bumps correspond to the binding energies of more and more internal electrons, that are more likely to inter-act with photons but require higher energies to pass to a free state ([8, p. 51]).
Photoelectric effect in the K-shell
In non-relativistic approximation, K-shell photoelectric ef-fect has an analytical solution ([9, p. 122]). If E EK, the
interaction in the final state can be neglected, so that the photo-electron can be considered a free photo-electron. To solve the problem, one must compute the matrix element of the interaction Hami-latonian
HI =− e
mec A· p
between the electron initial and final wave functions. If p is the final momentum of the electron and ˆe is the polarization versor of the incoming photon, then
hf|HI|ii = −me e r h2c2 2πk Z ψ∗fp·ˆe ei(k·r)ψi,
where ψi is the wave function of an s electron and ψf is a plane wave: ψi =s Z3 πa30 exp −Zra 0 , ψf =exp 2πi (p·r) hc . If θ is the angle between the directions of the incoming photon and the emitted electron and φ is the azimuthal angle of the photoelectron with respect to the polarization direction of the photon, it is possible to obtain the cross section
dσ dΩ =r2eZ5α4 m ec2 hν 7/2 4√2 sin2 θcos2φ (1−βcos θ)4 , (1.8)
where β = ve/c and re = e2/(mec2) is the classical electron
radius.
The term sin2θ at the numerator means that the
photoelec-tron is emitted preferably in a plane orthogonal to the direction of propagation of the photon. The angle where the cross section is maximum is given by θmax=cos−1 1 2β 1−q1−8β2 ≈cos−1(2β) :
the term(1−βcos θ)4in eq. 1.8 causes a forward bending of the distribution. For example, a 5 keV photoelectron corresponds to
θmax≈73°.
The projection of the distribution on the plane orthogonal to the wave vector of the photon is obtained integrating eq. 1.8 in θ and is given by
dσ
dφ ∝ cos2φ.
The emission of the photoelectron leaves the ionized atom with a hole that typically is in the innermost shell. So, the atom is in a metastable configuration and must decay to the ground state; it can do so in two main ways.
• Emission of an X fluorescence photon: an electron in the outermost shell fills the hole left by the photoelectron, thus creating a photon because of the de-excitation. • Auger emission: the energy released in the de-excitation
process is captured by another electron that is expelled if the energy it receives is higher than its binding energy.
Figure 1.7: Probability of the two possible types of emission due to de-excitation after a K-shell photoelectric absorption. The de-exciting atom either emits a photon (fluorescence emis-sion) or an electron (Auger emisemis-sion). The latter is by far the most likely to happen in materials with low atomic num-ber ([10]).
There is also the possibility of intermediate steps, where the hole is filled by an electron that is not in the outermost shell. In that case, the atom is still excited, so another de-excitation must occur; the overall emitted energy is the same, but it is split among the steps.
Fig. 1.7 shows the probabilities of the two processes, depend-ing on the atomic number Z. When considerdepend-ing light atoms, with Z<10, the probability of Auger emission is near 100 %. Photoelectric polarimetry
Photoelectric absorption is 100 % modulated by the direc-tion of polarizadirec-tion of incoming radiadirec-tion, so in principle it is an ideal way to make a polarimeter. The main problem con-nected to this technique is that, in the energy range where it is efficient, the mean free path of electrons is far lower than that of photons. Because of that, they are scattered after a short dis-tance, so that the initial direction (that is the actual information that one would like to measure) tends to be randomized.
Figure 1.8: Rate of energy loss of a particle in a medium. The highest rate is near the end of the track, at the Bragg peak. The data refers to 5.5 MeV α particles in air; few keV electrons would have been more relevant to the subject, but the result-ing curve, while maintainresult-ing a similar shape, would have been compressed and thus just less clear to understand.
So, even if the process is ideal, the method of detection is crucial when determining the modulation factor. One of the approaches that has been tried ([11]) is using a solid state de-tector, specifically a Charge Coupled Device (CCD). The reason is that pixels can be small enough that a photoelectron spans over more than one pixel, thus giving a hint of what the initial direction was. The modulation factors, though, are very small, in the range 3 % – 10 % in the hard X-rays. The main factor for that is the high sensitivity of CCDs to systematic effects when they are unevenly illuminated.
A relevant improvement to that idea is to use a gas as the material where photoelectrons propagate, because the range of an electron in a gas is far higher, so the track can go over mul-tiple pixels and it can be easier to infer the emission direction. In that case, an algorithm that reconstructs the track of the pho-toelectron from the pixelated image is necessary.
An extended track exhibits features, such as the Bragg peak, that can be used to reconstruct its path. The rate of energy loss of a particle in a medium is at its highest near the end of the track, when it is about to stop. Fig. 1.8 shows a typical curve, where the peak near the end is evident. The plot refers to
5.5 MeV α particles in air; the curve for electrons of a few keV has a similar trend, but the peak is much more compressed, thus being less clear to read. In a gas polarimeter, the presence of the Bragg peak translates to the highest signal being located at the end of the track. Unfortunately, the part of the track that contains the relevant information is the start.
1.3 history of x-ray polarization measurements
The literature on astronomical X-ray polarimetry measure-ments is extremely poor. Due to the technological challenges that that type of detection demands to overcome, only a hand-ful of missions has ever housed an X-ray polarimeter within its instruments. To date, X-ray polarimetry counts just one astro-physical target with a high significance measurement.
That is due in part to the fact that meaningful X-ray po-larimetry is difficult, for a number of reasons ([12]).
• As already said in sec. 1.1, the random noise in polarime-try measurements does not average to zero, because the degree of linear polarization is positive definite.
• Typical modulation factor values for most instrument are 20 % – 40 %. Those two effects raise the noise and lower the signal, ultimately reducing the quality of detected in-formation.
• We do not expect most sources to be strongly polarized,
with that meaning P 10 %. Because of that, most of the
incoming information will carry no polarization and thus further increase the background.
In this section, a brief history of missions housing an X-ray polarimeter will be given, as well as the achieved results ([13]). 1.3.1 Sounding rocket experiments
The first mission flown with an on-board polarimeter in an attempt to detect X-ray sources beyond the Solar System was a sounding rocket launched in July of 1968; the chosen target was Sco X-1 ([14]). The polarimeter exploited the polarization dependence of Thomson scattering using lithium as a scatter-ing material (the technique was described in subsec. 1.2.2); a schematic view of the instrument can be seen in fig. 1.9. The
1.3 history of x-ray polarization measurements
SEARCH
FOR X-RAY
POLARIZATION
INSco
X-1*
J.
R.
P.
Angel,
R.
Novick,
f
P.
Vanden
Bout,
and
R.
Wolff
Columbia
Laboratory
for
Astrophysics
andSpace
Physics,
Columbia University, NewYork,
New York10027
(Received
1 April1969)
An
x-ray
polarimeter
sensitive to
x rays
in the energy rangefrom
6to
18
keV, flownabove the atmosphere on 27 July
1968,
was used bothto
set
an upper limit on thepolar-ization
ofSco
X-1
andto check
for
spurious indications ofpolarization
which mightre-sult
from
the anisotropyof
thecosmic
rays.
%'ithin thestatistical
limitations of theda-ta,
no evidence was foundfor
spurious backgroundpolarization.
A
Thomson-scattering
x-ray
polarimeter
sensi-tive
in
the
spectral
range
of
about
6
to 18
keV
was
flown
from
the
White
Sands
Missle
Range
at
0356
UT
on
27 July
1968
in an
Aerobee-150
sounding
rocket.
Initially
the
instrument
was
pointed
at
Sco
X-1;
later
it
was pointed
to
a
re-gion
free
from
knownpoint
x-ray
sources
to
check
for
apparent
polarization
that
might
arise
from
the
anisotropy of
primary
cosmic-ray
pro-tons
and
terrestrial-albedo
gamma
rays.
Within
the
statistical
limitations
of the
experiment,
we
found
no
evidence
either
for
polarization
of
the
x
rays from
Sco
X-1
or
for
spurious
instrumental
polarization
from
the
cosmic-ray
background.
The
polarimeter
provides
a direct
measure
of the
normalized
Stokes
parameters
and
the
compo-nents
of the
polarization
vector
Q/I=
p =pcos28
and
U/I=
p =psin28,
where
p
is
the
magnitude
of
the
polarization
and
6Iis
the
position angle.
For
Sco
X-1
we
obtain
Q/I=p„=
(-4.
9+6.
0)%
and
U/I=p&=(5.
4+6.
0)
/o.The
6.
0/ouncertainty
given
for
each of
these
quantities
includes the
un-certainties
in
both
the background
and
source
measurements.
The
interest
in
stellar
x-ray
polarization
is
stimulated
bythe
hope
of
obtaining
information
about
source
mechanisms.
Polarization
could
be
expected if
the
emission
process
were
synchro-tron radiation.
'
The
fact
that
Sco
X-1
shows
spec-tral characteristics
similar
to those
of
an
old
no-va,
'
the
absence of appreciable polarization
in
the
visible region of
the
spectrum,
'
and
the
exponen-tial
form
of
the
x-ray
spectrum
are
generally
tak-en
as
evidence that the
x rays
are
produced
by
thermal bremsstrahlung
and
not
bysynchrotron
emission.
'
It
has
recently
been
shown
that
even
with
a
thermal-bremsstrahlung
model
we might
expect polarization
of
a
few
percent
if
the
source
is
not
spherically
symmetric
and
if
the
electron
density
is great
enough
to
give
a
highprobability
for electron scattering.
'
Finally,
it
is
important
to
note
that both
optical
and
x-ray flares
have
been
observed
in
Sco
X-1
and
that
these
might
be
POL IN CID PHOT COUNTERS~ LITHIUM SCATTERING BLOCK COUNTERS ELECTRONICS CAMERA (b) (DEPLOYED) ANTICO NCIDENCE DETECTOR SCATTERING DISTRIBUTION SCATTERED PHOTONS PREAMPLIFIER DETECTOR / ~COLLIMATOR TYPICAL LITHIUM~~i ii ROCHE~ BLOCK ~r '
'
SKIN'
.'I 8 N ii:,' M3 ii~TYPICALOFFAXIS ' ~COUNTER
LITHIUM
X-RAY
~COUNTER SOURCE ROCKET
DETE('TOR ROTATION
Fig.
l.
(a)Schematic representation
cf
thepolarime-ter
concept.
(b) Mounting of thepolarimeter
and ancil.—lary
equipment in therocket.
polarized.
'~'
'
Clearly,
a
study
of
both
the
spec-tral
and
temporal
dependence
of
polarization
in
Sco
X-1
and in
other
stellar x-ray
sources
will be
essential
to
a
full
elucidation
of
their
structure.
The instrument
used
in
the
present
work
ex-ploits
the
polarization
dependence
of Thomson
scattering.
The
probability
of
scattering at
an
an-gle
IIIto
the
electric
vector
of the incident
radia-tion
is
proportional
to
sin g. Metallic-lithium
scattering
blocks
are
used,
with
3-atm
xenon-methane
proportional
counters arranged
to detect
the
radiation
scattered
out through
the
sides
of
the
blocks.
This
is
shown
schematically
in
Fig.
1(a);
the
mounting
of the
polarimeter
in
the
rocket
is
shown
inFig.
1(b).
In
use,
the
polarim-eter is
pointed
toward
the
source
and
rotated
about
the line
of sight. If
the
incident
radiation
is
polarized,
the
counting
rate
in
each of
the
counters
will
be
modulated
at
a
frequency
equal
to
twice the
rotation
frequency of the
polarimeter;
the
depth and
phase of
the
modulation
provide
a
direct
measure
of the
magnitude
and
position
an-gle of the
polarization
vector
[see
Fig.
2(a)]
.
This
mode
of operation
avoids
false
indications
of
polarization
that
wouldotherwise
arise
from
861
Figure 1.9: Schematic view of the first sounding rocket experi-ment with an X-ray polarimeter ([14]).
sounding rocket made no detection, providing the upper limit PSco X-1 . 20 %. More sensitive detectors were needed,
espe-cially within the observing time limitations of a sounding rock-et flight.
The limitations of Thomson scattering polarimetry led the same team to launch another sounding rocket ([15]), this time adding a Bragg crystal polarimeter (described in subsec. 1.2.1). That second rocket successfully provided the first X-ray polar-ization data from the selected target, the Crab nebula and its
pulsar; the measurement confirmed the synchrotron2
origin of the X-rays. The degree of polarization was PCrab = (15±5)%
and the position angle φCrab = 156°±10°. The modulation
curves of the two instruments are superimposed in fig. 1.10; it is striking how much the background noise is reduced with the crystal polarimeter that uses Bragg diffraction. Recalling how the modulation factor is linked to the shape of the modulation curve, in eq. 1.2, it is clear that the crystal polarimeter achieves a much larger one.
2Synchrotron emission is one of the key physical phenomena that can be
detected just with polarimetric measurements; it is discussed in more detail in sec. 2.1.
1.3 history of x-ray polarization measurements
197
2ApJ.
.
.17
4L.
.
No. 1, 1972
X-RAY POLARIZATION OF CRAB NEBULA
L3
TABLE 1
Best Fit of the Raw Data to = S
0+ Si cos 2<f> + S
2sin 2<f>
So Si S
2M <f>'*
Experiment (counts s
-1) (counts s
_1) (counts s
-1) (%) (°)
Crystal Door 2 6.99 + 0.18 +0.37 + 0.24 -0.52 + 0.26 9.1+3.6 -27 + 11
Crystal Door 3 8.08 + 0.17 -0.18 + 0.24 +0.18 + 0.24 3.1 + 3.1 +67 + 28
Lithium-Xt 99.23 + 0.59 -0.40 + 0.84 -1.64 + 0.80 1.7 + 0.8 -51 + 14
Lithium-Ff 128.70 + 0.74 -1.49 + 0.99 +0.38+1.11 1.2 + 0.8 +83 + 21
* The phase angle 0' is the position of a fiducial on the rocket in celestial coordinates when the
counting rates in the detectors associated with the experiments listed in the first column are a maximum.
f The two orthogonal sets of lithium polarimeter detectors are labeled X and Y. The lithium polar-
imeters differed in azimuthal orientation from the crystal polarimeters by 9°.
unequal number of detectors. It is encouraging to see, particularly in the case of the
crystal polarimeter, that the modulation measured by orthogonal detectors is consistent
with a 90° phase difference, which is what one would expect in the observation of a po-
larized source. We must emphasize, however, that the statistical uncertainties are too
large to draw any definite conclusions regarding evidence for polarization if the four in-
dependent experiments are treated separately. Accordingly, the data from each crystal
polarimeter were weighted and combined. The data from the lithium polarimeters were
similarly treated. These results are plotted in figure 1 and listed in table 2. The solid
lines in figure 1 are the best fit to equation (1) with co = 2co
0. If the modulations are pro-
duced by polarization, then the corresponding position angle 6 for the electric vectors
are those listed in table 2. The position angles are related to the phase angles <t>
fby the
relative orientation of the instruments in the rocket.
As the modulation is positive-definite, there is always a finite probability of obtaining
a nonzero measurement of this quantity in the observation of an unpolarized source. The
Fig. 1.—Flight data from the lithium and crystal polarimeters versus position angle 0 modulo 180°.
The solid lines are the best fits to a function periodic at twice the rotation frequency of the rocket.
© American Astronomical Society • Provided by the NASA Astrophysics Data System
Figure 1.10: Modulation curves of the Thomson (lithium) and Bragg (crystal) polarimeters on board the sounding rocket that made the first positive detection of the polarization degree of the Crab nebula. The background noise of the crystal detector is vastly smaller, thus resulting in a larger modulation factor, according to how it is defined in eq. 1.2 ([15]).
1.3.2 Orbiting Solar Observatory 8
The next major step forward for X-ray polarimetry was the inclusion of two graphite crystal polarimeters ([16]) aboard the Orbiting Solar Observatory 8 (OSO-8). A schematic view of the instrument is shown in fig. 1.11a: there are an exploded view of the polarimeter assemblies and their positioning on the satel-lite, where it is clear how the polarimeters were rotated around the axis of the satellite, pointed at the target. The observation of the Crab nebula, with the contamination of the pulsar removed,
yielded the high-significance measurement PCrab = (19±1)%
1.3 history of x-ray polarization measurements
197
6ApJ.
.
.208L.125W
quick-look data gives the pulse-height spectra shown in Figures 2a and 2b .The first- and second-order Bragg reflections are clearly visible above the background, indicated by the dashed lines. The background spec- trum is based on 6 X 104 s of data obtained when the
field of view of the polarimeter was occulted by the Earth. The slightly broader pulse-height profiles in Figure 2b as compared to Figure 2a are due to the
poorer energy resolution of counter 2. The signal-to- background ratios listed in Table 1 illustrate the effectiveness of focusing, pulse-shape discrimination, and anticoincidence techniques to achieve background suppression.
The data from each detector in two energy band- widths, corresponding to the first- and second-order Bragg reflections (channels 6-13, 20-28 of counter 1;
COLLIMATOR (a) INCIDENT X-RAY
(b) PREAMP AND NETWORK UNIT COUNTER 1 GRAPHITE CRYSTAL REFLECTOR -COUNTER 2 HIGH-VOLTAGE POWER SUPPLY SOLAR PANEL ALWAYS FACING SUN
POLARIMETERS ORIENTED PARALLEL TO SPIN AXIS SPINNING WHEEL SPIN AXIS POINTED TOWARDS SOURCE OF INTEREST
STELLAR X-RAY SOURCE
Fig. 1.—(a) Exploded view of the OSO-8 polarimeter assemblies. (6) Location of the polarimeters in the satellite.
© American Astronomical Society • Provided by the NASA Astrophysics Data System (a) Exploded view of the polarimeter assem-blies and their positioning on the satellite.
(b) Modulation curves of the polarimeters. The two curves are shifted by 128° because of the relative orientation of the detectors.
Figure 1.11: Schematic view of OSO-8 and modulation curves resulting from the observation of the Crab nebula ([16]).
the two detectors are shown in fig. 1.11b; they are shifted by 128° because of the relative orientations of the polarimeters. That confirmed the previous result and left no doubt on the synchrotron origin of the Crab X-rays.
The OSO-8 satellite featured a number of different instru-ments; that caused a crowded observing program, with allo-cated time divided among all of them. That was a problem especially for polarimetry, that requires very long integration times. Because of that, there has been a limited number of
mea-surements performed with the OSO-8 polarimeter, that yielded minor3results.
1.3.3 Electron tracking
The next step in the development of detectors for X-ray po-larimetry came in the early 1990s, when an optical imaging chamber was presented by Austin and Ramsey ([21]); its goal was measuring the details of the tracks produced by the pho-toelectron in the gas of a proportional counter. The direction of the primary photoelectron released in the photoelectric ab-sorption of the incident photon depends on the polarization of the incident beam (as described in subsec. 1.2.3), so imaging the track provides information on the degree of polarization and the position angle of the incident flux. Also, the imaging capability is crucial because a large set of astronomical X-ray sources, such as supernova remnants, are extended.
A major advance in electron tracking was made with the introduction of a Gas Electron Multiplier and an ASIC for the readout. The subsequent refinement and development of this detector forms the basis for IXPE; more details on it are in sec. 3.1.
3The minor results of OSO-8 are upper limits on Sco X-1, Cen X-3 and
Her X-1, and very marginal detection of polarization from Cyg X-2 and Cyg X-1. For the relevant papers, see [17], [18], [19] and [20].
2
T H E I M A G I N G X - R AY P O L A R I M E T R YE X P L O R E R
The Imaging X-ray Polarimetry Explorer (IXPE) is a part-nership mission between NASA and ASI, selected by NASA as part of its Small Explorer program (SMEX) and currently set to be launched in April 2021. The goal of IXPE is measuring the polarization of X-rays coming from astrophysical sources.
Polarimetric data allows to observationally verify a number of theoretical models and predictions. Those, followed by the technical overview of the IXPE mission, designed to investigate such phenomena, are discussed in this chapter.
2.1 scientific goals
According to theoretical models of astrophysical objects, a large number of those should emit radiation with a significant degree of polarization ([1], [22]). As outlined in sec. 1.3, though, almost all the information available on X-ray-emitting bodies comes from spectral and temporal characteristics of radiation.
Linear polarization in radiation coming from astrophysical sources could arise due to the production mechanism of pho-tons (such as synchrotron radiation) or the interaction of emit-ted photons with the surrounding material. Because of that, acquiring information on polarization has the potential to be crucial in determining the physical processes that happen in-side the source of the radiation.
2.1.1 Supernova remnants
A supernova remnant (SNR) is the material expelled by an exploding supernova, a spectacular phenomenon that can emit energies of the order of 1051erg and typically produces a shock
wave that propagates for tens of parsecs, at speeds of thou-sands of km/s. SNRs have a crucial role in cosmic rays accel-eration and in the distribution of the heavy elements formed with the explosion. A simulation of an observation of the su-pernova remnant Cassiopeia A is shown in fig. 2.1. The
simula-Figure 2.1: Observation simulation of the supernova remnant Cas A for the IXPE optics, with 1.5 Ms of observing time. The ‘HPD’ circle is, roughly speaking, the imaging resolution of the
instrument.
tion is done considering the IXPE optics and an observing time of 1.5 Ms, corresponding to about 17 days. The circle labeled ‘HPD’ is the Half-Power Diameter, useful to see at a glance how the instrument spreads photons: it is the area enclosing half of the X-rays coming from a point source.
SNR spectra feature high energies excesses, clearly marking the presence of non-thermal mechanisms in the emission; in some cases, the non-thermal component of the spectrum can even be the dominant one ([23]). The synchrotron radiation is the natural explanation of this phenomenon.
The total power irradiated by an ultra-relativistic electron with energy E=γmec2 in a magnetic field B is
P= 2 3 e4B2 m2 ec3γ 2,
with a spectral distribution that has a maximum in νs = 12meB
ec2γ 2.
The synchrotron spectrum of an ensemble of particles can be obtained by taking the average of their energy distributions.
In a supernova shock wave, charged particles can be con-fined by the turbulence induced near the shock front and be forced to cross it many times ([24]). Such a mechanism can ac-celerate electrons to energies over the TeV, with a power-law distribution:
N(γ)∝ γ−s .
The synchrotron radiation emitted by an electron distribution like that has a spectrum with a similar distribution:
I(ν) ∝ ν−α , with a spectral index
α= s−1
2 .
In that situation, the expected degree of linear polarization is Pl = s+1
s+7/3 .
Synchrotron radiation describes well the observed spectra. By measuring the degree of polarization, a direct check of this model can be performed. The successful polarization detection of the Crab nebula (subsecs. 1.3.1 and 1.3.2) confirmed the syn-chrotron origin of the X-rays.
With sources like Cas A, the imaging capabilities of IXPE are useful to detect where the regions of shock acceleration are and to measure the intensity and orientation of the magnetic field in those zones. Moreover, the fraction of non-thermal over thermal emission, and hence the degree of polarization, is ex-pected to be much higher in the filaments located at the shell boundaries than near the center ([25]).
Pulsar Wind Nebulae
Pulsar Wind Nebulae (PWNe) are a class of supernova rem-nants with a characteristic non-thermal continuum emission, ranging from the radio up to the soft gamma ray band, mainly due to synchrotron emission ([26]). PWNe are found inside
Figure 2.2: Observation simulation of the Crab nebula for the IXPE optics, with 30 ks of observing time.
SNRs that are powered by winds generated by their central pulsars. Optical and X-ray observations show that the nebular continuum emission features a jet-torus morphology. An exam-ple of that is the Crab nebula, a simulation of whom is shown in fig. 2.2. The simulated observing time is 30 ks, that is about 8 h.
Spatially resolved polarimetry allows to determine the mag-netic field orientation in the torus, in the jets and at various distances from the pulsar. With that information, the level of turbulence and instabilities can be evaluated, and consequently the acceleration mechanism responsible for the observed parti-cle distribution can be tested.
2.1.2 Accretion disks
Accretion disks are planar structures that form when a mass of gas with non-zero angular momentum is gravitationally at-tracted near a massive objects. Accretion disks provide ways to detect black holes, otherwise undetectable, and binary
sys-tems, allowing to discover additional information. If the main opacity source of the disk is scattering, the angle and degree of polarization give information about its inclination and the geometry of the system ([27]).
In the inner regions of accretion disks, scattering on elec-trons is typically the main contributor to its opacity. In non-relativistic approximation, the shift in the photon frequency is negligible and the Thomson differential cross section is
dσ dΩ = 1 2r2e 1+cos2θ . That formula gives the classical result
σT = 8π
3 r2e
when integrated on the scattering angle θ. The fraction of emerging radiation linearly polarized in a direction perpendic-ular to the scattering plane varies between 0, if θ =0, and 1, if
θ =90°.
If the energy of the photon is not negligible with respect to the rest mass of the electron, the shift in the photon frequency is given by
ν0 = ν
1+ hν
mec2(1−cos θ)
in the system of reference where the electron is initially still. In this case, the Klein-Nishina cross section for the Compton scattering must be used. In the X band, the polarizing process is essentially as the classic Thomson scattering. The mean energy gained by the electron in the reference where it is initially still is h∆Ei = (hν) 2 mec2 , if hν mec2 is small.
In the stationary system of reference, photons are Doppler shifted and, if the energy of electrons is high enough, on aver-age gain energy:
ν0
ν =
4 3γ2 .
This process is called inverse Compton scattering, because in that case photons gain energy, not electrons. The properties of out-going radiation is mainly dependent on the number of scatter-ing events inside the disk. If Thomson scatterscatter-ing is the domi-nant process, the mean free path of photons is
` = 1
where neis the electron density. The mean number of scattering
events depends on the optical depth of the system, defined by
τ =
Z
neσTds .
Diffusion by dust particles, that can be present in the outer, colder regions of the disk, is a much more complicated process that depends on the wavelength λ of radiation, the dimension
d of dust particles and their chemical composition. If λ d,
the cross section depends on the angle just as the Thomson scattering does, but scales as λ−4. In that case, the effects on the
degree of polarization are around 1 %, from the infrared to the ultraviolet. There is no general formula in more complicated situations.
The presence of magnetic fields, generated by turbulent mo-tion inside the disk, greatly complicates the matter. Normal propagation modes in magnetized plasma are, at the lowest or-der, circularly polarized with phase velocities that differ by
∆vp =2c
ω2pωc
ω3 cos ϕ , (2.1)
where ϕ is the angle between the direction of propagation and
the magnetic field lines and ωp and ωc are the plasma and
cyclotron frequencies, defined by
ωp = s 4πnee2 me , ωc = eB mec .
For a linearly polarized electromagnetic wave, the difference 2.1 causes a rotation in the polarization plane, called Faraday rota-tion: dΦ dτ ≈0.1 λ 500 nm 2B // 1 G . (2.2)
In the case of a magnetic field in equipartition with the thermal energy of the disk, the following approximation is valid:
B2 ≈aT4 .
The characteristic emission wavelength of the system is
λ ≈ hc
kT
so, with proportionality factors back in, the intensity of the magnetic field can be estimated as
B≈100 λ 500 nm −2 G .
ˆz
ˆk
θ
Figure 2.3: Chandrasekhar model of semi-infinite atmosphere. This is the simplest way of describing an accretion disk.
By substituting this value in eq. 2.2, the result is dΦ
dτ ≈10 ,
independently of λ, and thus of T. That happens because photons with different energies pass through different optical depths, so Faraday rotation cancels all the linear polarization in the thermal component of the emission.
That shows that in the optic range, the radiation emitted by accretion disks in active galactic nuclei has a low degree of po-larization. That is not valid, though, for the non-thermal com-ponent, that can be the main part of a spectrum in the X range. So, X polarimetry could be more useful than optical polarime-try in such cases. See the next subsection for a study case with active galactic nuclei.
Model of semi-infinite atmosphere
The simplest model that describes an accretion disk is the Chandrasekhar model, a semi-infinite atmosphere where the only interaction process is scattering of unpolarized primary pho-tons on matter ([28]); a representation of that is in fig. 2.3. The distribution of the intensity of outgoing radiation depends on the directional cosine µ =cos θ as
I(µ)∝ 1+2.06µ .
The system is rotationally invariant around the ˆz axis, thus
ˆz
ˆk θ
τ0
τ0
Figure 2.4: Model of finite thickness accretion disk. since all directions in the xy plane are the same. If θ 6=0, there are two directions orthogonal to ˆk that can be unambiguously found:
ˆe1 = ˆk׈z , ˆe2 = ˆe1× ˆk .
Those represent the versors of the electric field, corresponding
to the two normal propagation modes; ˆe1 lies on the plane of
the disk, ˆe2 on the plane kz. If I1 and I2 are the intensities of
the corresponding modes, the degree of linear polarization is P= I1−I2
I1+I2
and the direction of polarization will be ˆe1or ˆe2; by convention,
P is considered to be positive if the direction of polarization lies on the plane of the disk, negative otherwise. The degree of polarization ranges from Pmin = 0, for θ = 0, to Pmax ≈ 11.7 %,
for θ =90°, that is known as Chandrasekhar limit. The radiation is always polarized in the disk plane. The top curve in fig. 2.5, labeled with ∞, shows the dependence of linear polarization on the angle; the meaning of the other curves will be explained shortly.
It should be noted that those results are valid in Newtonian approximation; systems where relativistic effects cannot be ne-glected, such as accretion disks around black holes, may have characteristics that greatly vary from the results presented here. Model with finite optical depth
If the model is modified to have a finite thickness, like in fig. 2.4, outgoing radiation has interesting characteristics. The
Figure 2.5: Degree of linear polarization for different values of optical depths τ0, defined as in fig. 2.4. When P becomes
nega-tive, it means that the direction of polarization is orthogonal to the disk ([28]).
main difference is that the degree of polarization greatly de-pends on the total optical depth.
Fig. 2.5 shows the degree of linear polarization of outgoing radiation, for different values of optical depths, if sources are
on the symmetry plane (z =0) and the electron density is
uni-form in the whole disk. The disk temperature is supposed to be uniform and, as in the semi-infinite model, the only process possible is scattering. The plot clearly shows that, for large opti-cal depths, the curve approaches the Chandrasekhar limit: the curve with τ =4 is very close.
If τ is small enough, the degree of polarization becomes always negative, meaning that the radiation is polarized in a plane orthogonal to the disk, and gets larger as the optical depths gets smaller. In that case, the polarization angle is ro-tated by 90° because with a small optical depth, the direction of
1993MNRAS.264..839M
Figure 2.6: Degree of polarization of radiation emerging from an optically thick accretion disk for different values of Λ (as defined in eq. 2.3) and for two different models of the internal source function B. The top curve refers to a pure-scattering
atmosphere (Λ = 1); the others, from top to bottom, refer to
values of Λ of 0.995, 0.9 and 0.5. If the sources are concentrated on the edges of the disk (the model shown with dashed lines), the resulting degree of polarization is higher ([29]).
photons before they get scattered out of the disk is most likely to be on the plane of the disk.
Absorption processes
The characteristics of outgoing radiation can change dramat-ically if scattering is not the only interaction process in the disk. If σscatand σabs are the cross sections for scattering and
absorp-tion processes, the relevant parameter is
Λ = σscat
σscat+σabs . (2.3)
The effect heavily depends on the distribution of the primary sources: if radiation is mainly emitted in the external strata of the disk, the degree of polarization of outgoing radiation is greater than if the sources are uniformly distributed in the disk. Fig. 2.6 shows the degree of linear polarization depending
on the direction of outgoing radiation in both cases: the solid lines, corresponding to a uniform source, clearly show a smaller degree of polarization.
The polarization degree becomes negative at low Λ because, in that case, absorption is the most efficient process, so photons can escape from the disk only if they are scattered near the edges. That scenario is analogous to the case of an optically thin disk that, as seen in fig. 2.5, behaves in a similar way.
X polarimetry might measure the fraction and angle of po-larization and hence give important information on the orien-tation of the disk with respect to the line of sight, its optical depth and the interaction mechanisms inside it.
2.1.3 Black holes
Black holes are objects that are intrinsically hard to study, because they do not emit radiation. The detection of radiation emitted by its accreting matter is a precious tool in that sense; polarimetric characteristics, in particular, should be strongly af-fected by relativistic effects.
Space-time around a rotating black hole is described by the Kerr metric. In natural units (G =c =1), it can be written as
ds2=− 1−2mr ρ2 dt2− 4amr sin2θ ρ2 ! dt dφ+ ρ 2 ∆ dr2+ +ρ2dθ2+ r2+a2+ 2a 2mr sin2θ ρ2 ! sin2θ2dφ2, where ρ2 =r2+a2cos2θ ; ∆ =r2−2mr+a2.
The geometry is determined by two parameters, the mass m
and the specific angular momentum a/m, and is symmetric for
rotations around the axis of the angular momentum. For a =0,
it becomes the Schwarzschild metric ds2=− 1−2m r dt2+ 1−2m r −1 dr2+ +r2dθ2+sin2θdφ2 , that describes the space-time with spherical symmetry around a non-rotating black hole.
Figure 2.7: Geodesics are significantly curved near black holes, so the angles of polarization of photons can be rotated if they come from a region near the event horizon.
The element grr of the Kerr metric is divergent for ∆ = 0,
that is for
r =rh =m+pm2−a2;
the surface defined by r =rhis called event horizon. A particle on a time- or light-type geodesic with r < rh cannot get out of the event horizon.
Around a Schwarzschild black hole, Keplerian circular
or-bits can be stable down to a minimum radius rmin = 6m.
Pres-sure gradients can allow stable orbits even for r <rmin. In the
Kerr metric,
lim
a→mrmin =rh .
This means that accreting matter around a black hole can create disks that extend down to the event horizon. Gas can stay in that configuration long enough that a non-negligible fraction of the radiation it emits comes from that region. Relativistic line broadening in some sources suggests that such a phenomenon happens, so the trajectory and polarization of radiation coming from the vicinity of a black hole can be highly affected by the curvature of spacetime. Fig. 2.7 schematically shows that effect: photons emitted far from the event horizon travel along a prac-tically straight geodesic, but near the black hole the geodesic curvature evidently changes the direction of polarization.
Radiation propagates in a vacuum along time-type geodes-ics, with the equation of motion
kβkα
A similar relation is valid for the polarization versor eα:
kβeα
;β =0 .
Moreover, according to Maxwell equations, e and k must be orthogonal:
eαk
α =0 .
Those three equations can be solved consistently and describe the trajectory of the photon and the propagation of its plane of polarization.
Accretion disks around black holes
The relativistic bending of the radiation along its trajectory causes additional effects with respect to the classical Newtonian case. In particular, the angle of linear polarization can either be 0 or 90° in the classical case, but can assume arbitrary values if relativistic effects are taken into account.
The degree of linear polarization P is actually a Lorentz-invariant quantity, so it does not change in the propagation along a relativistic trajectory. Gravitational bending, though, mixes radiation coming from different zones of the disk, so the overall effect is a partial loss of polarization, depending on frequency.
Active galactic nuclei
An Active Galactic Nucleus (AGN) is a galactic center that has a much higher than normal luminosity; the center of an AGN is believed to be a supermassive black hole, with an ex-tended accretion disk around it.
The primary source of X-ray emission of an AGN is due to inverse Compton scattering on electrons in a hot, tenuous corona, surrounding the main body of the accretion disk ([31]). The corona was assumed to be a slab, but in the past few years many theoretical models have proposed different shapes. Re-cent results seem to narrow the possible shapes to two: a slab or a sphere ([30]). The two shapes are shown in fig. 2.8.
The X-ray emission is visible both directly or with an inter-mediate reflection, as shown in fig. 2.9. The shape of the hot corona in that scenario is crucial, because a spherical corona would cause no polarization, because of its almost-isotropy, but if the corona is a slab, the anisotropy would produce a polar-ization in reflected radiation. This is what is shown in fig. 2.10,
