Systematic effects on a Compton polarimeter at the focus of an X-ray mirror

XL-Calibur is a balloon-borne Compton polarimeter for X-rays in the $\sim$15-80 keV range. Using an X-ray mirror with a 12 m focal length for collecting photons onto a beryllium scattering rod surrounded by CZT detectors, a minimum-detectable polarization as low as $\sim$3% is expected during a 24-hour on-target observation of a 1 Crab source at 45$^{\circ}$ elevation. Systematic effects alter the reconstructed polarization as the mirror focal spot moves across the beryllium scatterer, due to pointing offsets, mechanical misalignment or deformation of the carbon-fiber truss supporting the mirror and the polarimeter. Unaddressed, this can give rise to a spurious polarization signal for an unpolarized flux, or a change in reconstructed polarization fraction and angle for a polarized flux. Using bench-marked Monte-Carlo simulations and an accurate mirror point-spread function characterized at synchrotron beam-lines, systematic effects are quantified, and mitigation strategies discussed. By recalculating the scattering site for a shifted beam, systematic errors can be reduced from several tens of percent to the few-percent level for any shift within the scattering element. The treatment of these systematic effects will be important for any polarimetric instrument where a focused X-ray beam is impinging on a scattering element surrounded by counting detectors.


Introduction
X-ray polarimetry in the hard X-ray band [1] (∼10-100 keV) probes compact celestial sources, such as X-ray binary black holes and accretion-and rotation-powered neutron stars, which cannot be directly imaged [2,3].Polarized emission arises from source asymmetry caused by magnetic fields or the distribution of radiation/matter.The linear polarization of hard X-rays can be measured using Compton scattering, described by the Klein-Nishina differential scattering cross-section [4], where θ is the polar photon scattering angle and ϕ is the azimuthal angle between the scattering direction and the polarization vector of the incident photon.The distribution of azimuthal scattering angles is described by a modulation curve with 180 • periodicity: where µ and ξ are the amplitude and the phase of the modulation curve, respectively, and N is the mean.The polarization fraction, p (%), is defined as µ/µ 100 , where the denominator is the modulation amplitude for a 100% polarized beam.Since X-rays preferentially scatter perpendicular to the polarization (electric field) vector, see Eq. ( 1), the polarization angle, ψ ( • ), can be determined from the modulation phase as ψ = ξ + 90 • .
Scattering-polarimeter designs often combine a low-atomic-number scatterer symmetrically surrounded by high-atomic-number X-ray detectors, allowing the distribution of azimuthal scattering angles to be sampled over the range of polar angles subtended [1].In the balloon-borne polarimeter XL-Calibur [5], X-rays are focused over 12 m onto a 12 mm diameter beryllium (Be) rod, whence scattered X-rays are detected by pixelated CdZnTe (CZT) detectors arranged around the rod.The assembly is continuously rotated about the center-line of the Be rod to mitigate temporal and spatial variations in detector response.XL-Calibur is operating in the ∼15-80 keV energy range.The lower bound is due to the limited atmospheric transmission at ballooning altitudes of ∼40 km, while the upper limit is dictated by the effective area of the X-ray mirror [6].For typical source spectra, the ∼20-40 keV range will be most favorable for polarimetry.
The next XL-Calibur flight is scheduled for the spring/summer of 2024, from Esrange Space Center in northern Sweden.The science programme for the two primary northern-sky targets, the Crab pulsar and the black-hole binary Cygnus X-1, requires percent-level minimum detectable polarization (MDP) [5] (see Section 5 and Eq.(3) for a definition) in order to determine where in the magnetosphere high-energy emission originates, and to constrain the space-time geometry and coronal shapes in the vicinity of the black hole, respectively.
Equation (2) implicitly assumes that scatterings take place along the symmetry axis of the scatterer.For XL-Calibur, the beam from the X-ray mirror will impinge across the end of the Be rod, as characterized by the mirror point-spread function (PSF).
Photons that do not scatter from the symmetry axis will both alter the 180 • polarization signature and introduce additional harmonics in the modulation response if scattering along the symmetry axis is assumed.Additional components are dominated by, but not restricted to, 360 • periodicity.They are not canceled out by rotating the polarimeter, and their presence in the modulation response acts as a marker of systematic effects influencing the reconstructed polarization.
Systematic effects from an offset beam were first explored for the Stellar X-Ray Polarimeter instrument, which was developed for the (ultimately canceled) Spectrum X-Gamma mission [7].A lithium cylinder served as a Thomson-scattering target and xenon-filled imaging proportional counters were used to detect scattered X-rays in the 4-20 keV range.The effect of an offset beam on X-Calibur (the predecessor of XL-Calibur) was studied using a synchrotron beam and Monte-Carlo simulations [8].The scatterer comprised a 13 mm diameter plastic-scintillator rod and the configuration of CZT detectors was different from that in XL-Calibur.Insights into the response systematics inform the data analysis strategy for XL-Calibur, but are not fully representative for the realistic mirror PSF, since the beam used was circularly symmetric and the polarimeter was not rotating.
A subsequent iteration of X-Calibur (8 m focal-length telescope), which used a 12 mm diameter Be scattering rod and a CZT arrangement similar to XL-Calibur, observed the accretion-powered X-ray pulsar GX 301-2 during an Antarctic balloon-flight in December 2018 [9].The polarimeter rotated during observations, and the focal-point offset was corrected for when calculating scattering angles, but systematic effects could not be studied in detail as a balloon leak forced an early termination of the flight.
In this paper, the modulation response of XL-Calibur is characterized in detail, using Geant4 [10,11] Monte-Carlo simulations, for a realistic beam incident on the beryllium stick, including the source spectrum, atmospheric attenuation and the measured mirror PSF.The fidelity of the simulation is established by comparing the measured and simulated instrument response to beams of both polarized and unpolarized X-rays 1 .Systematic effects of shifting the PSF in arbitrary directions from the geometric center of the Be rod are studied, since the position of the mirror focal point will change during flight, e.g., due to pointing offsets, and thermal or gravitational forces on the structure.Methods to determine the beam offset and correct the modulation response are discussed, along with strategies to improve sensitivity.
Throughout this paper, polarization parameters are determined using modulation curves.Polarimeter data is commonly analyzed using Stokes parameters, as outlined e.g. in [13], due to attractive mathematical properties, such as the additive nature of the parameters allowing simple background subtraction, and their more straight-forward (Gaussian) statistical properties.By definition, the linear-polarization Stokes parameters are only sensitive to the 180 • harmonic.Asymmetries prominent in the modulation curve may remain undetected if Stokes parameters are directly applied, and the Stokes formalism should therefore only be used after inspecting the modulation curve for additional harmonics, or after applying suitable corrections to the scattering angles.

XL-Calibur design
The XL-Calibur telescope [5] comprises a 12 m long rigid truss, with carbon-fiber struts interconnected by aluminum joints.Suspended under a ∼10 6 m 3 stratospheric balloon, the truss can be aimed with arc-second precision using the Wallops Arc-Second Pointer (WASP) [14].A 45 cm diameter X-ray mirror is mounted at one end of the truss, with the polarimeter at the other end, as shown in Figure 1.

Simulation and validation
A Monte-Carlo simulation has been implemented within the Geant4 framework (version 10.02) to model the polarimeter response.This simulation was previously used for the mission design [5] and for optimizing the anticoincidence-shield configuration [15].For the current study, the mirror point-spread function was also implemented.Source photons are sampled from a given spectrum and propagated with a spatial distribution based on beam-line calibration data [6] for the three mirror sections.Each section was tested at energies 20, 30, 40, 50 and 70 keV, and the simulated PSF uses the measured mirror response at the closest energy.The PSF can be translated across the face of the scattering element to allow the study off-center pointings or misalignments.Figure 3 shows an example of the mirror PSF implemented in the simulation, and the resulting hit-distribution in detector 17, for a power-law spectrum with photon index 2.1.The twelve-cornered star-shape results from three images, recorded with a square CMOS camera during the beam test [6], rotated by 120 • relative to each other, with one image for each mirror section.The blue circle and blue square indicate the beryllium stick and the CZT walls, respectively.Right: the resulting simulated distribution of photon hits in detector 17, deconvolved for the detector roll.For each hit above the detection threshold, the coordinate is randomized within the 2.5 mm × 2.5 mm pixel size, to provide sub-pixel binning.The sharp boundary near the center of the image is from the central 2 × 2 pixels detecting most of the hits.Asymmetries arise from the true mirror behavior, with subtle differences between mirror sections.Even though the mirror is perfectly aligned in the simulation, the resulting beam distribution within the beryllium stick (dashed magenta line) is shifted, by ∼0.3 mm, approximately in the zenith direction.Dashed black lines represent the detector walls and the size of detector 17 (20 mm × 20 mm).As the detector rotates, events outside this region can become populated.The tungsten collimator is tapered to a minimum diameter of ∼18 mm (dashed gray line) to prevent direct illumination of the CZT walls.
To bench-mark the simulations, measurements with a ∼1.75 GBq 241 Am source were performed at Esrange, immediately prior to the 2022 XL-Calibur launch2 .In one configuration, source photons directly illuminated the beryllium rod, allowing the response to unpolarized radiation to be studied.In a second configuration, source photons were scattered through 90 • in a dedicated polarizer, resulting in a beam with close to 100% polarization.Emission lines from the radioactive decay of 241 Am were simulated with their relevant branching ratios, with the most prominent X-ray line at 59.5 keV.
Generated source photons were propagated through the simulated experimental setup, with no manual tuning of spectra or photon spatial distributions.
Polarized (unpolarized) measurement data was acquired over 36 hours (1 hour), yielding 954930 (1692873) valid events (hit in one CZT pixel only and no coincident hit in the anticoincidence shield above a 100 keV threshold), for a rate of ∼7 Hz (∼500 Hz).The contemporaneous background trigger rate was ∼1 Hz.Signal (source exposed) and background (source blocked) observations were interspersed, allowing accurate background subtraction under variable measurement conditions.Based on relative rates, the beam-on/beam-off ratio was taken to be 2:1 (10:1) for the polarized (unpolarized) beam, which ensures the minimization of the uncertainty incurred by the background subtraction [13].These rates and exposure times result in a statistical error on the reconstructed modulation amplitude of <0.2 percentage points.
The modulation curve for the polarized beam is not well-described by Eq. ( 2): additional harmonics, such as a 360 • and a 120 • component, must be introduced to fit the angular distribution.These components arise from an offset of the beam impinging on the scatterer.To accurately replicate the offset present in the alignment of the experimental setup, the simulated position of the radioactive source geometry was moved iteratively in x and y, minimizing the difference between measured/simulated beam center-point positions, as reconstructed in detector 17 by the arithmetic (mean x, mean y).The result is shown in Figure 4, where the central positions agree within ∼0.1 mm.
Resulting measured/simulated scattering-angle distributions for the polarized and the unpolarized beams are presented in Figure 5.With only the simulated source position (x, y) as a free parameter, very good agreement is achieved in both cases, and no further fine-tuning is required, demonstrating the fidelity of the XL-Calibur Geant4 simulation.

Determining the beam offset
The simulations can be used to evaluate the effects of the PSF migrating across the beryllium stick.Several independent effects contribute to such excursions: (1) pointing offset, (2) a possible (static) misalignment of the mirror, and (3) a (time-dependent) bending of the truss.These effects relate to the position of the focal point of the mirror and can be corrected for.Individual photons are further dispersed across the Be rod based on the mirror PSF and can only be corrected for on average.
The pointing offset (1) has an approximately linear relationship with the shift in the focal spot, with a ∼1.7 arcmin offset resulting in a 6 mm shift (the radius of the beryllium stick) in the focal plane of the detector 3 .Since each recorded event is tagged with the pointing feedback from the Wallops Arc-Second Pointer [14], individual events can be corrected for the pointing offset.In the 2022 flight, a pointing stability of 10 arcsec (3σ) was achieved during observations of the Crab pulsar, which corresponds to a ∼0.6 mm jitter of the focal point.
A static misalignment (2) of the mirror can occur, e.g., due to mechanical alignment errors on ground.Although the mirror has no adjustable degrees of freedom during the flight, a systematic misalignment can partly be counteracted by adjusting the pointing in the opposite direction.The back-looking camera can be used to estimate the mirror alignment, under the assumption that the camera axis coincides with the mirror axis.
To confirm the X-ray alignment, images from detector 17 are required.As shown in Section 3, these can be sensitive to sub-mm offsets of the focal spot.A representative detector-17 image from the flight of X-Calibur can be found in [9].
A bending of the truss (3) can arise, e.g., due to pointing-dependent effects of gravity on the 12-m long structure.To address this, an elevation-dependent sag correction (pointing offset) is implemented in the pointing system.Prior to flight, the shift in the optical axis of the back-looking camera is mapped out over the full range of elevations (up to ∼80 • ). Figure 6 shows the truss deflection during the 2022 XL-Calibur flight, as determined from the back-looking camera.
Although the mean horizontal and vertical offsets are centered within ∼0.6 mm, excursions of up to ∼3-4 mm also occur.Integrating the image from detector 17 over a long time interval 4 would destroy any temporal information.Since the back-looking camera can afford a high rate of offset data, scattering angles can be corrected essen- tially on an event-by-event basis.
The instantaneous offset of the optical axis (back-looking camera) must be related to the most probable scattering site.This is taken as (mean x, mean y) in detector 17, calculated for events within the radius of the beryllium stick 5 .A systematic simulation study of the offset in detector 17 (mean x, mean y) as a function of the true offset (x, y) has been conducted for a grid of irradiation points across the beryllium stick.Both directions have a linear relation between the detector-17 offset and the true offset, with both regression coefficients exceeding 0.998.This is true for both the polarized and the unpolarized beam, indicating that (mean x, mean y) in detector 17 is a good proxy for the focal spot impinging on the top of the beryllium stick.Focal-spot excursions can thus be corrected on the time scale of the back-looking camera, instead of on the time scale of the integrated image in detector 17.

Effects of a beam offset
From any point along the central axis of the beryllium stick, the path length through the scatterer is the same for all azimuthal angles.Intrinsic differences between pixels, such as angular coverage, detection efficiency, threshold levels and noise characteristics are canceled out by the polarimeter rotation.The result for a fully centered beam is then a modulation curve as described by Eq. ( 2), having only a 180 • harmonic.
The assumption of an isotropic response ceases to be valid when the beam is offset from the center of the scattering element.Even for a fully centered beam, asymmetries can arise (see mirror PSF in Figure 3).A schematic view of how the offset alters the modulation curve is shown in Figure 7.  Once the focal-point offset is known in the plane of the polarimeter, scattering angles can be calculated assuming this origin instead of the center of the scatterer, while the absorption site is assigned a randomized position within the triggered pixel 6 .This correction only takes the azimuthal-angle acceptance (in the plane of the detector) into account.The change in polar-angle acceptance (axial direction) cannot be accounted for, as the height of the interaction along the axis of the beryllium stick is not known.
For the same reason, the difference in path length through the stick cannot be treated.
This effect is small compared to the change in angular acceptance caused by the beam offset: for a 30 keV photon, shifting the interaction site by 3 mm from the center (half of the stick radius) changes the attenuation in the perpendicular direction to the nearest wall by less than 10%, while the azimuthal-angle acceptance in the same direction changes by more than 40%.For higher energies, the relative difference becomes even more pronounced, as the absorption probability decreases while the solid angle remains unchanged.

Systematic error with and without offset correction
The detector-17 offset (mean x, mean y), calculated within the 6 mm radius of the scatterer, is assumed as the scattering vertex for the offset correction 7 .Although the correction is not perfect on an event-by-event basis -due to incomplete information about the scattering kinematics of individual events -it is highly successful in reducing the systematic error on the reconstructed polarization resulting from a beam offset.
Figure 8 shows the effect on the reconstructed polarization fraction for the polarized case, with and without offset correction based on the image from detector 17, for a grid of hypothetical beam offsets.Effects on the reconstructed polarization angle are shown in Figure 9. Figure 8: Reconstructed polarization fraction for an offset mirror beam with 100% polarization (horizontal direction), in the absence of offset correction (left) and with a correction based on calculating scattering angles using detector-17 (mean x, mean y) (right).The color scale is normalized relative to the central point (zero beam offset, corresponding to the ∼0.3 mm shift shown in Figure 3).Irradiation points where the PSF is centered outside the beryllium stick (magenta circle) can also be kept, and benefit even more from the offset correction.Such points have been excluded here, as the flux drops to below 50% once the focal point exceeds the boundary of the beryllium stick.During the 2022 flight, 90% of the observation time was spent within a total offset of less than 2.65 mm (black dashed circle).The error remaining after correction (right) is slightly shifted upwards, likely resulting from the mirror asymmetry which shows a bias in the zenith direction (see Figure 3).Figure 9: Shift in reconstructed polarization angle resulting from an offset of the incident mirror beam with 100% polarization (horizontal direction), in the absence of offset correction (left) and with a correction based on calculating scattering angles using detector-17 (mean x, mean y) (right).The color scale is defined relative to the central point (zero beam offset, corresponding to the ∼0.3 mm shift shown in Figure 3. Diagonal offsets are more affected than those perpendicular to or parallel with the polarization vector.After correction, systematic effects are small, revealing fluctuations due to the structure of the mirror PSF.During the 2022 flight, 90% of the observation time was spent within a total offset of less than 2.65 mm (black dashed circle).
In the absence of a correction, systematic errors of up to several tens of percentage points in the reconstructed polarization fraction can arise for offsets within the radius of the scattering element.Even modest offsets of 1 mm (2 mm) result in absolute errors of >2 percentage points (>6 percentage points).For a polarized beam, shifts along the polarization vector result in a reduction in apparent polarization, while perpendicular shifts increase the reconstructed polarization relative to the central point.Nonphysical values with apparent polarization fraction in excess of 100% can thus occur.After correction, systematic errors are significantly reduced, e.g., even a 2 mm offset now gives <2 percentage points of error, for any offset direction.The knowledge of the location of the focal spot is thus essential for high-accuracy polarization measurements.
The reconstructed polarization angle is similarly altered in the absence of a correction, and both the magnitude and the sign of the systematic error depend on the direction of the beam offset.The largest error is incurred when the offset is diagonal relative to the polarization direction, reaching almost a ±10 • shift for a 6 mm offset.
Offsets parallel with or perpendicular to the polarization vector are shifted by less than ±2 • for the same offset.With offset correction, the true polarization direction is recovered within ∼1.5 • for any systematic offset on the beryllium stick.
Results for an unpolarized beam are shown in Figure 10.In the absence of an offset correction, even a 2-mm offset can cause spurious polarization as high as 5%, and the effect increases drastically for larger offsets.This can immediately preclude the sought percent-level MDP, and underlines the importance of applying an offset correction.A systematic error from offset in one direction is not obviously counteracted by an error caused by an offset in the opposite direction: instead, these errors may compound.Once a correction is applied, systematic errors become significantly lower 8 .
The remaining asymmetry in the corrected grid results from the anisotropy in the beam PSF, seen in Figure 3. Figure 10: Spurious polarization resulting from an offset of the incident mirror beam in the unpolarized case, in the absence of offset correction (left) and with a correction based on calculating scattering angles using detector-17 (mean x, mean y) (right).The color scale is normalized relative to the central point (zero beam offset, corresponding to the ∼0.3 mm shift shown in Figure 3) for the 100% polarized beam.During the 2022 flight, 90% of the observation time was spent within a total offset of less than 2.65 mm (black dashed circle).

Observation-specific modulation response and spectro-polarimetry
For accurate polarization measurements, the modulation response (µ 100 ) must be determined for a given observation.In the simulation, factors such as float altitude, source spectrum, elevation-dependent atmospheric attenuation and systematic pointing offset can be taken into account.Additionally, for spectro-polarimetry [17], the energy dependence of the modulation response must be considered.This is shown in Figure 11, comparing results for an infinitesimally narrow pencil beam and for the simulated mirror PSF.Within the XL-Calibur energy range, µ 100 can vary by as much as ∼7 percentage points.The mirror PSF reduces the value by an additional ∼5 percentage points as compared to the pencil beam.
When populating the modulation curve, individual event weights can provide observation-specific corrections such as for a systematic offset.The weighting strategy is based on the expression for the minimum detectable polarization [18], which can be

Pencil beam
Mirror PSF Figure 11: XL-Calibur modulation response to a 100% polarized flux (µ 100 ) for a pencil beam (blue) and the simulated mirror PSF (black).The dip around 30-40 keV is caused by characteristic (K α ) X-ray emission from the cadmium in the CZT detectors.Such secondary photons carry no polarimetric information pertaining to the original flux, and if they interact in a second pixel while the primary interaction is below the detection threshold, they will appear as a primary event and cannot be rejected, thus resulting in a suppression of the polarization detected for a given flux.
expressed at 99% confidence level as where N = S + B is the total number of signal (S ) and background (B) counts.Following Eq. ( 3), two types of weights will be considered: the modulation factor, µ, and the number of signal events, incorporated in the factor N. Maximizing µ reduces the uncertainty on the reconstructed polarization fraction (see, e.g, [12] for a discussion), while the benefit in weighting for N comes from optimizing the signal-to-background ratio as illustrated in Figure 12.
The pixel hit-map for a source observation is quantitatively different from a background observation, which is expected to be uniformly populated.Applying event weights based on the relative number of counts in a given pixel can thus favor signal events from this topology, while suppressing background.For example, events near the bottom of the CZT walls have a low probability of being signal and should be assigned along the length of the beryllium stick.Lobes with high count intensity arise from the solid-angle effect: as seen from the scatterer, pixels near the center of a wall subtend a larger solid angle than corner pixels.The beam from the mirror is incident "from the top" of the image.The vertical position of the lobes correlates to the beam energy (higher beam energies penetrate deeper into the beryllium stick), and the map thus provides indirect information on the polar scattering angles.
low weights.Based on Eq. ( 3), these weights are applied as √ N.
Following the Klein-Nishina formula, Eq. ( 1), the amplitude of the azimuthal modulation for a polarized flux depends on the polar scattering angle.This is encoded in the modulation factor, and photons scattering at polar angle ∼90 • are most favorable for polarimetry.Figure 13 shows this effect for XL-Calibur, through the modulation response for each row of CZT pixels in the detector.
For some pixel rows, modulation factors as high as ∼0.67 (∼0.55) are seen for the pencil (mirror PSF) beam.Folding this modulation response with the pixel-hit distribution (Figure 12) gives the effective modulation factor for a given observation.Instead of using the overall modulation factor for determining the polarization fraction of the flux, individual scattering events can be weighted with their corresponding modulation

Pencil beam
Mirror PSF Figure 13: Simulated modulation factor for each pixel row of the XL-Calibur detector for a Crab spectrum, for a pencil beam (red) and the mirror PSF (green).Vertical lines indicate the four CZT rings, and dips at rows 17 and 25 result from the gaps between adjacent rings.The top of the beryllium stick is approximately centered on the first ring, where the highest values of µ 100 are seen, resulting from scattering at close to 90 • in polar angle.The increase in error-bar size at high row numbers results from the reduced number of polarization events reaching the bottom of the detector (see Figure 12).factor, following Figure 13.Following Eq. ( 3), the µ weights apply linearly.
An optimized analysis [17] of flight data can benefit from event weights based on the product µ √ N, through the corresponding reduction in MDP.Within the established simulation framework, both µ and √ N weighting schemes (Figure 12, Figure 13, respectively) can be generated for any systematic offset of the mirror PSF.Thus, even if a systematic offset is present in a measurement, the analysis can benefit from the correction described here.

Summary and conclusions
For a polarimeter at the focus of an X-ray mirror, shifts in the focal point can occur due to pointing offsets, possible mirror misalignment and/or elevation-dependent structural deflection.In the absence of an offset correction, large systematic errors are incurred the reconstructed polarization parameters, both for polarized and for unpolarized beams.Effects for a mirror PSF, as described here, are not as severe as for a highly collimated synchrotron beam [8].This is because the extended PSF causes some photons to scatter at lower offsets, reducing the systematic error, while photons incident at larger offsets (leading to high systematic error) instead have a probability of missing the stick.Corrections can be applied by calculating the Compton-scattering angle not from the symmetry axis of the stick, but from the most probable scattering location at any given time.This is allowed by combining information from an imaging CZT, located underneath the beryllium scatterer, and a back-looking camera mounted on the optical axis of the mirror, to offer offset corrections on an event-by-event level.
Based on a simulation bench-marked with laboratory testing using radioactive sources, the polarimeter response for flight can be studied.Estimated values of µ 100 for the two primary targets for the upcoming flight from Esrange, taking into account source spectra, atmospheric attenuation, mirror response [6], energy dependence (Fig- ure 11) and detector response, are shown in Table 1.Assuming a modulation factor µ 100 ≈ 0.5 [5], derived from earlier simulations based on synchrotron beams [8], would then systematically under-estimate reconstructed polarization fractions.
Table 1: Representative µ 100 values for primary northern-hemisphere targets characterized by photon index Γ.Final values will be calculated accounting for the float altitude achieved and source variability.
Source µ 100 Crab (Γ = 2.11) (42.6 ± 0.2)% Cygnus X-1 low-hard state (Γ = 1.70) (43.1 ± 0.2)% Cygnus X-1 high-soft state (Γ = 2.47) (43.0 ± 0.2)% The minimum-detectable polarization can be improved through event weighting, using a factor µ √ N.These weights address the fact that detector rows subtend different polar-angle ranges.The benefit of the weighting depends on the source spectrum (through µ) as well as on the signal-to-background scenario in flight (through Based on simulations with a conservative 1:1 ratio of signal and background 9 , the MDP for a 1 Crab source at ∼45 • elevation is expected to improve from ∼4% to ∼3% for an on-source integration time of 24 hours.
XL-Calibur is scheduled for a flight from Esrange Space Center, Sweden, in 2024.
With the percent-level MDP achievable, the instrument is expected to provide strong polarimetric constraints for astrophysical sources in the hard X-ray range, ∼15-80 keV range, complimentary to the soft X-ray regime, 2-8 keV, accessible to the Imaging X-Ray Polarimetry Explorer (IXPE) [19].Joint observations are planned for the upcoming flight, to best utilize synergies between the measurements for polarimetry and spectro-polarimetry.

Figure 2 :
Figure 2: Overview of the XL-Calibur shield and polarimeter.The focused X-ray beam from the mirror impinges on a beryllium rod and scattered photons are registered in the surrounding CZT detectors.

Figure 3 :
Figure3: Left: point-spread function (PSF) of the XL-Calibur mirror, measured at the SPring-8 synchrotron facility (logarithmic color scale).The twelve-cornered star-shape results from three images, recorded with a square CMOS camera during the beam test[6], rotated by 120 • relative to each other, with one image for each mirror section.The blue circle and blue square indicate the beryllium stick and the CZT walls, respectively.Right: the resulting simulated distribution of photon hits in detector 17, deconvolved for the detector roll.For each hit above the detection threshold, the coordinate is randomized within the 2.5 mm × 2.5 mm pixel size, to provide sub-pixel binning.The sharp boundary near the center of the image is from the central 2 × 2 pixels detecting most of the hits.Asymmetries arise from the true mirror behavior, with subtle differences between mirror sections.Even though the mirror is perfectly aligned in the simulation, the resulting beam distribution within the beryllium stick (dashed magenta line) is shifted, by ∼0.3 mm, approximately in the zenith direction.Dashed black lines represent the detector walls and the size of detector 17 (20 mm × 20 mm).As the detector rotates, events outside this region can become populated.The tungsten collimator is tapered to a minimum diameter of ∼18 mm (dashed gray line) to prevent direct illumination of the CZT walls.

Figure 4 :Figure 5 :
Figure4: Detector-17 hit-maps from the measurement (left) and simulation (right).Mean x and mean y values indicate that the beam is hitting the polarimeter off the optical/X-ray/rotation axis, due to a shift and/or a tilt in the radioactive-source alignment.The size of the beryllium scatterer is shown by the magenta circle.Mean x and mean y values for events hitting within the radius of the rod have been separately indicated.

Figure 6 :
Figure6: XL-Calibur truss alignment during the 2022 flight.Each data point corresponds to the center-point offset of the polarimeter, as reconstructed from the back-looking camera, for horizontal (red) and vertical (green) directions, and in total (magenta).The projections (right panel) reveal mean offsets of ∼0.3 mm and ∼0.6 mm in horizontal and vertical directions, respectively.Time-stamped data points from the back-looking camera allow event-by-event correction of the offset.

Figure 7 :
Figure7: Projected lobes (not to scale) of most probable scattering directions relative to the polarization angle ("PA") for scattering locations as indicated by the red stars (top), and simulated XL-Calibur modulation curves resulting from the corresponding beam offsets (bottom).The incident beam is infinitesimally narrow and has energy 53.3 keV.Red/green/blue arrows indicate phase for the 180 • , 360 • and 120 • components, respectively.Although the 180 • phase is consistently reproduced (perpendicular to the polarization vector), the presence of other harmonics indicates that the 180 • amplitude (giving the polarization fraction) is subject to systematic effects from an offset.

Figure 12 :
Figure12: Pixel hit-map of the 16 CZTs for a simulated Crab observation.The four vertical regions (delimited by thick black lines) correspond to the detector walls.Horizontal regions indicate the four CZT "rings" along the length of the beryllium stick.Lobes with high count intensity arise from the solid-angle effect: as seen from the scatterer, pixels near the center of a wall subtend a larger solid angle than corner pixels.The beam from the mirror is incident "from the top" of the image.The vertical position of the lobes correlates to the beam energy (higher beam energies penetrate deeper into the beryllium stick), and the map thus provides indirect information on the polar scattering angles.