Detection of polarized gamma-ray emission from the Crab nebula with Hitomi Soft Gamma-ray Detector

We present the results from the Hitomi Soft Gamma-ray Detector (SGD) observation of the Crab nebula. The main part of SGD is a Compton camera, which in addition to being a spectrometer, is capable of measuring polarization of gamma-ray photons. The Crab nebula is one of the brightest X-ray / gamma-ray sources on the sky, and, the only source from which polarized X-ray photons have been detected. SGD observed the Crab nebula during the initial test observation phase of Hitomi. We performed the data analysis of the SGD observation, the SGD background estimation and the SGD Monte Carlo simulations, and, successfully detected polarized gamma-ray emission from the Crab nebula with only about 5 ks exposure time. The obtained polarization fraction of the phase-integrated Crab emission (sum of pulsar and nebula emissions) is (22.1 $\pm$ 10.6)% and, the polarization angle is 110.7$^o$ + 13.2 / $-$13.0$^o$ in the energy range of 60--160 keV (The errors correspond to the 1 sigma deviation). The confidence level of the polarization detection was 99.3%. The polarization angle measured by SGD is about one sigma deviation with the projected spin axis of the pulsar, 124.0$^o$ $\pm$0.1$^o$.


Introduction
In addition to spectral, temporal, and imaging information gleaned from observations of any astrophysical sources, polarization of electromagnetic emission from those sources provides the fourth handle on understanding the operating radiative processes. Historically, measurement of high radio polarization from celestial sources implicated synchrotron radiation as such process, first suggested by Shklovsky (1970). Measurement of radio or optical polarization is relatively straightforward: first, it can be done from the Earth's surface, and second, the instruments are relatively simple. The measurements in the X-ray band are more complicated: those have to be conducted from space which constrains the instrument size, and, unlike e.g. radio waves, X-rays are usually detected as particles and require large statistics to measure the polarization.
One of the brightest X-ray sources on the sky, with appreciable polarization measured in the radio and optical bands is the Crab nebula. It was detected by (probably) every orbiting X-ray astronomy mission (for a recent summary, see Hester 2008). It was thus expected that X-ray polarization should be detected as well, and in fact, the first instrument sensitive to X-ray polarization, the OSO-8 mission, observed the Crab nebula, and detected X-ray polarization (Weisskopf et al. 1978). The measurement, performed at 2.6 keV, measured polarization at roughly ∼ 20 ± 1% level. It was some 30 years later that the INTEGRAL mission observed the Crab nebula and detected significant polarization of its hard X-ray / soft γ-ray emission (Chauvin et al. 2013;Forot et al. 2008). Moreover, INTEGRAL teams reported gammaray polarization measurements from the black hole binary system Cygnus X-1 (Laurent at al. 2011;Jourdain et al. 2012;Rodriguez et al. 2015). However, the interpretation of the measurements with INTEGRAL are not straightforward, because its instruments were not designed for, or calibrated for polarization measurements.
More recently, the Crab nebula was observed by the balloon-borne mission PoGOLite Pathfinder (Chauvin et al. 2016), and PoGO+ (Chauvin et al. 2017;Chauvin et al. 2018), with clear detection of soft γ-ray polarization in the ∼ 18−160 keV band, thus expanding the X-ray band where the Crab nebula emission shows polarization. The PoGO+ is an instrument employing a plastic scintillator, with an effective area of 378 cm 2 and optimized for polarization measurements of Compton scattering perpendicular to the incident direction where the modulation factor of the azimuth scattering angle is high; the PoGO+ team reports the polarization of the phase-integrated Crab emission of 20.9 ± 5.0% with a polarization angle of 131.3 • ± 6.8 • , while in the off-pulse phase, it is 17.4 +8.6 −9.3 % with a polarization angle of 137 • ± 15 • .
The Japanese mission Hitomi (Takahashi et al. 2018), launched in 2016, included the Soft Gamma-ray Detector (SGD), an instrument sensitive in the 60-600 keV range, but also capable of measuring polarization (see Tajima et al. 2018) since it employs a Compton camera as a gammaray detector. The SGD was primarily designed as a spectrometer, but, it was also optimized for polarization measurements (see, e.g., Tajima et al. 2010). For example, the Compton camera of the SGD is highly efficient for Compton scattering perpendicular to the incident photon direction and is symmetric with 90 • rotation. The calibration and the performance verification as a polarimeter have already been performed by using polarized soft gamma-ray beam at SPring-8 (Katsuta et al. 2016). Hitomi did observe the Crab nebula in the early phase of the mission. Since the goal of the observation reported here was to verify the performance of Hitomi's instruments rather than to perform detailed scientific studies of the Crab nebula, the observation time was short. Even though this observation was conducted during orbits where the satellite passed through the high-background orbital regions including orbits crossing the South Atlantic Anomaly, the Crab nebula was still readily detected, as we report in subsequent sections. We discuss the data reduction and analysis in section 2 and section 3, the measurement of Crab's polarization in section 4, compare our measurement to previous measurements in section 5, and also discuss the implications on the modeling of the Crab nebula in section 5. We note that the Crab nebula observations with Hitomi's Soft X-ray Spectrometer were published recently (Hitomi Collaboration 2018a), and observations with the Hard Xray Imager are in preparation. Moreover, the data analysis of the Crab pulsar with Hitomi's instruments were also published (Hitomi Collaboration 2018b).

Instrument and Data Selection
The Soft Gamma-ray Detector (SGD) was one of the instruments deployed on the Hitomi satellite (see Takahashi et al. 2018 for the detailed description of the Hitomi mission). The instrument was a collimated Si/CdTe Compton camera with the field of view of 0.6 • × 0.6 • , sensitive in the 60-600 keV band; for details of the SGD, see Tajima et al. (2018). The SGD Compton camera consisted of 32 layers of Si pixel sensors where Compton scatterings take place primarily. Each layer of the Si sensor had a 16 × 16 array of 3.2 × 3.2 mm 2 pixels with a thickness of 0.6 mm. In order to efficiently detect photons scattered in the Si sensor stack, it was surrounded on 5 sides by 0.75 mm thick CdTe pixel sensors where photo-absorptions take place primarily. In the forward direction, 8 layers of CdTe sensors with a 16 × 16 array of 3.2 × 3.2 mm 2 pixels were placed, while 2 layers of CdTe sensors with 16 × 24 array of 3.2 × 3.2 mm 2 pixels were placed on four sides of the Si sensor stack. For details of SGD Compton camera, see Watanabe et al. (2014). The SGD consisted of two detector units, SGD1 and SGD2, each containing three Compton cameras, named as CC1, CC2, and CC3, respectively. Those detectors were surrounded on five sides by an anti-coincidence detector containing BGO scintillator. The observation of the Crab nebula with Hitomi was performed from 12:35 to 18:01 UT on March 25, 2016. This observation followed the start-up operations for the SGD, which were held from March 15 to March 24, and, all cameras of both SGD1 and SGD2 went into the nominal observation mode before the Crab nebula observation. However, just before the Crab nebula observation it was found that one channel in the CdTe detectors of SGD2 CC2 became noisy, and subsequently we set the voltage value of the high-voltage power supply for the CdTe sensors of SGD2 CC2 to 0 V during the Crab nebula observation. Since CC3 shares the same high-voltage power supply with CC2, the CdTe sensors in CC3 are also disabled. Therefore, four of six Compton cameras (SGD1 CC1, CC2, CC3 and SGD2 CC1) were operated in the nominal mode, which enabled the Compton event reconstruction.
Good time intervals (GTI) of SGD during the Crab observation are listed in Table 1. The intervals during the Earth occultation and South Atlantic Anomaly (SAA) passages are excluded. The total on-source duration was 8.6 ks. The exposure times of each Compton camera after dead-time corrections are listed in Table 2. In the SGD1 Compton cameras, the dead-time corrected exposure time can be derived from the number of "clean" pseudo events (Watanabe et al. 2014), which have no FBGO flag and no HITPATBGO flag. The pseudo events are events triggered by "pseudo triggers", which are generated randomly in the Compton camera FPGA based on the pseudorandom numbers calculated in the FPGA. The count rate of the pseudo triggers is set to be 2 Hz. FBGO and HITPATBGO flags indicate existence of anti-coincidence signals from the BGO shield. The pseudo events are processed in the same manner as usual triggers, and, are discarded if the pseudo trig-ger is generated while a "real event" is inhibiting other triggers. Therefore, the dead-time fraction can be estimated by counting a number of pseudo events, and, the deadtime by accidental hits in BGOs can be also estimated from the pseudo events with FBGO flags and HITPATBGO flags. However, it was found that there was an error in the on-board readout logic of adding the HITPAT BGO flags to pseudo events for the parameter setting of SGD2 CC1. Due to this error, dead-time fraction by the accidental hit in the BGOs cannot be derived from the number of pseudo events generated from SGD2 CC1. Therefore, for SGD2 CC1, the dead-time fraction due to accidental hit in BGOs was calculated from the fraction of "clean" pseudo events in the SGD2 CC2, allowing the determination of the dead-time corrected exposure time. For SGD2 CC2, a parameter setting to avoid the error has been used. And, the dead-time fraction by accidental hits in BGOs must be same among the Compton cameras in SGD2, because the BGO signals are common among all three Compton cameras in SGD2.
The attitude of the Hitomi satellite was stable throughout the Crab GTI. The nominal pointing position is (R.A., DEC.) = (83.6334 • , 22.0132 • ) and the nominal roll angle is 267.72 • that is measured from the north to the satellite Y axis counter-clockwise. The distance from the nominal pointing position is within 0.3 arcmin for the 98.7% of the observation time. The difference from the nominal roll angle is within 0.05 • for the 99.6% of the observation time. Therefore, these offsets from the true direction of Crab are negligible and we have not considered them in the analysis. Figure 1 shows the Hitomi satellite position during the Crab GTI and one day before the Crab GTI, when the satellite was pointing at RXJ 1856.5−3754, which is a very weak source in the hard X-ray/soft gamma-ray band, and thus such "one day earlier" observation is a good proxy to measure the background. The time interval information about observations performed one day earlier than the Crab GTI are listed in Table 3. Because the observations start soon after the SAA passages, the background rate during the Crab GTI was higher than the average due to short-lived activated materials produced in the SAA. Although the Crab nebula is one of the brightest sources in this energy region, the background events were not negligible for spectral analysis and polarization measurements. As shown in Figure 1, the satellite positions and the orbit conditions one day earlier than the Crab GTI are similar to those during the Crab GTI, which would imply background conditions could be similar.

Background Determination
In order to confirm that the satellite encountered the similar background environments during similar orbit conditions, we compare the SGD data between an epoch one day earlier and also two days earlier than the Crab observation GTIs. The single hit spectra obtained by the CdTe-side sensors are shown in Figure 2. The CdTe-side sensors are located on the four sides around the stack of Si/CdTe sensors inside the Compton camera, and, are not exposed to gamma rays from the field of view. Therefore, the influence of the background environment should be reflected strongly in the single hit event in the CdTe-side detectors. The red and the black points show the spectra for the epochs one day and two days earlier than the Crab GTI, respectively. These two spectra have the same spectral shape including various emission lines from activated materials. The flux levels were the same within 3%. On the other hand, the blue spectrum shows the single hit events of CdTe-side detectors on the orbit where the satellite does not pass the SAA region. Although the background environment varied during one day, it was found that the background estimation becomes possible by using the data from one day earlier.
In order to further verify the background subtraction using the data one day earlier, the count rates as a function of the time during the Crab GTI and one day earlier are compared in Figure 3. The red and the blue points show the count rates during the Crab GTI and one day earlier.
The black points show the count rates of the Crab GTI after subtracting the count rates one day earlier, which corresponds to the count rates of the Crab nebula. Since the black points do not show any visible systematic trend implying additional backgrounds, it implies this background subtraction is appropriate.   2. Spectra of CdTe side single hit events. The red and the black show the spectra for the one day and two days earlier than the Crab GTI, respectively. The blue spectrum shows the single hit events of CdTe-side sensors on the orbit that the satellite does not pass the SAA region.

Data Processing with Hitomi tools
The data processing and the event reconstruction are performed by the standard Hitomi pipeline using the Hitomi ftools (Angelini et al. 2018) 1 . In the pipeline process for SGD, the ftools used for the SGD are hxisgdsff, hxisgdpha and sgdevtid. The hxisgdsff converts the raw event data into the predefined data format. The hxisgdpha calibrates the event energy. The sgdevtid reconstruct each event. These tools are included in HEASoft after version is 6.19. The version of the calibration files used in these  process is 20140101v003.
The sgdevtid is one of key tools for the SGD event reconstruction, which determines whether the sequence of interactions is valid and computes the event energy and the 3-dimensional coordinates of its first interaction. The event reconstruction procedure of the sgdevtid is described in Ichinohe et al. 2016. The first step of the process is to merge signals that are consistent with fluorescence Xrays with the original interaction sites according to their locations and energies. The merging process combines the separated signals into a hit for each interaction. The second step is to analyze the reconstructed hits and determine whether the sequence is consistent with an event. This step depends on the number of reconstructed hits. If there is only one hit, the process is done, and, the energy information and the hit position information are recorded on the output event file as a "single hit" event. In the case of the event which has 2 to 4 hits, the process determines whether the event is a valid gamma-ray event and whether the first interaction is Compton scattering by applying the Compton kinematics equation: where θK is scattering angle defined by Compton kinematics, mec 2 is the rest energy of an electron, E1 is the first hit energy corresponding to the recoil energy of the scattered electron and Eγ is the reconstructed energy of the incoming gamma-ray photon. All possible permutations for the sequence of hits are tried and all sequences with non-physical Compton scattering angle (| cos θK| > 1) are rejected. Besides the kinematic scattering angle θK, the geometrical scattering angles θgeometry can be derived from the directions of the incident gamma ray and the scattered gamma ray. The incident gamma ray is assumed to be aligned with the line of sight. The direction of the scattered gamma ray is reconstructed from the positions of the first and the second hits. The difference of them is called angular resolution measure (ARM): If more than one sequence remains, the order of hits with the smallest ARM value is selected as the most likely sequence. Moreover, in the case of 3-hit events, the second interaction is assumed to be the Compton scattering, and, in the case of 4-hit events, the second and the third interactions are assumed to be the Compton scatterings. For these interactions, the tests of Compton kinematics and differences between kinematic scattering angles and geometrical scattering angles are performed. If the sequences have any non-physical Compton scatterings or any inconsistent kinematic angles to the geometric scattering angles, the sequences are rejected. In the first calculation, the reconstructed energy of the incoming gamma-ray photon Eγ is set to be where Ei is the energy information of the i-th hit. For 3-hit and 4-hit events, if all sequences are rejected in this calculation, the sgdevtid calculates the escape energy, the unabsorbed part of the energy of a photon that is able to exit the camera after detections, and executes the previous tests again. Finally, for such good "Compton event" after the process, the information for the first interaction such as cos θK, the azimuthal angle φ of scattered gammarays, and the ARM value as 'OFFAXIS' are recorded on the output event file in addition to the reconstructed energy information and the first hit position information 2 . Figure 4 shows a relation between OFFAXIS and energy spectrum for the "Compton-reconstructed" events where sgdevtid finds the position of the first Compton scattering with physical cosθK in Si sensors. The histogram in the left-hand panel is made from the events during the Crab GTI, and, that in the right-hand panel is made from the events collected one day earlier than the Crab GTI. An excess at around OFFAXIS ∼ 0 • can be seen in the histogram of the Crab GTI corresponding to the gamma rays from the Crab nebula.

Processing of Crab Observation Data
In order to obtain good signal to noise ratio, selections of 60 keV < Energy < 160 keV, −30 • < OFFAXIS < +30 • , 50 • < θgeometry < 150 • are applied. The histograms of Energy, OFFAXIS, θgeometry are shown in Figure 5. The selections of Energy, OFFAXIS, and θgeometry are not applied in the histograms of Energy, OFFAXIS, and θgeometry, respectively. The red histograms are made from the events during the Crab GTI, and, the events collected during the period one day earlier than the Crab GTI are shown in black as a reference.
We measure the gamma-ray polarization by investigating the azimuth angle distribution in the Compton camera since gamma rays tend to be scattered perpendicular to the Energy [keV] counts/bin Fig. 4. Two-dimensional histograms of Compton-reconstructed events. The relation between OFFAXIS and energy is shown. The left-hand panel is the histogram made from the events during the Crab GTI, and, the right-hand panel is prepared from the events collected one day earlier than the Crab GTI.

Energy[keV]
counts/sec/keV direction of the polarization vector of the incident gamma ray in Compton scatterings. Figure 6 shows the azimuth angle distribution of Compton events obtained with the SGD Compton cameras. The red and the black points show the distribution during the Crab GTI and that from one day earlier than the Crab GTI, respectively. The azimuthal angle Φ is defined as the angle from the satellite +X-axis to the satellite +Y-axis. The average count rate during the Crab GTI is 0.808 count s −1 .
counts/sec/bin  . 6. The azimuth angle distributions obtained with the SGD Compton cameras. The red and the black show the distribution during the Crab GTI and that from an epoch one day earlier than the Crab GTI, respectively. The definition of Φ is also shown. SATX and SATY mean the satellite +X-axis and the satellite +Y-axis, respectively.

Background Estimation for Polarization Analysis
Before the Crab observations, Hitomi also observed RXJ 1856.5−3754 which is fairly faint in the energy band of the SGD (Hitomi Soft X-ray Imager results were reported in Nakajima et al. 2018). The GTIs of RXJ 1856.5−3754 and the exposure times are listed in Table 4 and Table 5, respectively. The total exposure time of the all RXJ 1856.5−3754 observation is about 85.6 ksec and the number of the Compton-reconstructed events is about 24400. More than ten times larger number of events are available by using this observation than the observation of the Crab nebula. In order to obtain the azimuth angle distribution of the background events with better statics, the SGD data during the RXJ 1856.5−3754 GTI were investigated.
Comparisons of the incident energy, OFFAXIS, θgeometry and the azimuth angle Φ between the RXJ 1856.5−3754 GTI and one day earlier than the Crab GTI are shown in Figure 7. Since orbits with no SAA passage are included in the all RXJ 1856.5−3754 observation, the flux level was lower than that obtained one day earlier than the Crab GTI. The count rate of the events during the RXJ 1856.5−3754 GTI is 0.285 count s −1 , and that during the one day earlier than Crab GTI is 0.404 count s −1 . Therefore, the scale of the histograms for the RXJ 1856.5−3754 GTI are normalized to match those for one day earlier than the Crab GTI. The distributions of OFFAXIS, θgeometry and the azimuth angle Φ are similar. Since the incident energy spectrum of the RXJ 1856.5−3754 GTI looks slightly different from that observed one day earlier than the Crab GTI, we further investigated the effect on the Φ distrbution. We divided the data in five energy bands, 60-80 keV, 80-100 keV, 100-120 keV, 120-140 keV and 140-160 keV, and the number of events in each energy band is normalized to match those for one day earlier than the Crab GTI. The resulting Φ distribution for the RXJ 1856.5−3754 GTI is shown as the magenta points in the lower right panel of Figure 7. We do not observe any significant trend from the original distribution for the RXJ 1856.5−3754 GTI, which implies that the difference in the energy spectrum does not have significant effect on the Φ distribution. From above investigations, we conclude that the Compton reconstructed events during the RXJ 1856.5−3754 GTI can be utilized for the background estimation of the polarization analysis.

Monte Carlo simulation
Monte Carlo simulations of SGD are essential to derive physical parameters including gamma-ray polarization from the observation data. For the Monte Carlo simulations, we used ComptonSoft 3 in combination with a mass model of the SGD and databases describing detector parameters that affect the detector response to polarized gamma rays. ComptonSoft is a general-purpose simulation and analysis software suite for semiconductor radiation detectors including Compton cameras (Odaka et al. 2010), and depends on the Geant4 toolkit library (Agostinelli et al. 2003;Allison et al. 2006;Allison et al. 2016) for the Monte Carlo simulation of gamma rays and their associated particles. We chose Geant4 version 10.03.p03 and G4EmLivermorePolarizedPhysics as the physics model of electromagnetic processes. The mass model describes the entire structure of one SGD unit including the surrounding BGO shields. The databases of the detector parameters contain configuration of readout electrodes, charge collection efficiencies, energy resolutions, trigger properties, and data readout thresholds in order to obtain accurate detector responses of the semiconductor detectors and scintillators composing the SGD unit.
The format of the simulation output file is same as the SGD observation data. The simulation data can be processed with sgdevtid, and as a result, it is guaranteed that the same event reconstructions are performed for both observation data and simulation data.
For the Compton camera part, the accuracy of the simulation response to the gamma-ray photons and the gammaray polarization was confirmed through polarized gammaray beam experiments performed at SPring-8 (Katsuta et al. 2016). The better than 3% systematic uncertainty was validated in the polarized gamma-ray beam experiments. On the other hand, the effective area losses due to the distortions and misalignments of the fine collimators (FCs) are not implemented in the SGD simulator (Tajima et al. 2018). We have not obtained measurements of the FC distortions and misalignments with the calibration observations: this is because the satellite operation was terminated before we had opportunities to make such measurements. In the simulator, the ideal shape FCs with no distortion and no misalignment are implemented. Since the losses due to the distortions and misalignments of the fine collimators does not affect the azimuthal angle distribution of the Compton scattering, the effects on the polarization measurements are negligible.
In the simulation of the Crab nebula emission, we assumed a power-law spectrum, N · (E/1 keV) −Γ , with a  photon index (Γ) of 2.1. In the first step, unpolarized gamma-ray photons are assumed. And, the normalization of the simulation model (N ) is derived from the Si single-hit events. Figure 8 shows the Si single hit spectrum obtained with the 4 Compton cameras. The background spectrum is estimated from the observations taken one day earlier than those for the Crab GTI, and the background subtracted spectrum is shown in Figure 8, together with the simulated spectrum. By scaling the integrated rate of the simulation spectrum in the 20-70 keV range to match the observed rate, we obtained N = 8.23 which corresponds to a flux of 1.89 × 10 −8 erg/s/cm 2 in the 2-10 keV energy range.
We compare the Compton reconstructed events between the observation and the simulation. In Figure 9, the distributions of OFFAXIS for the observation and the simulation are shown. The distribution of OFFAXIS for the simulation is slightly narrower than that for the observation. If the same selection of −30 • < OFFAXIS < +30 • is applied for the both events, the observation count rate becomes 8.6% smaller than the simulation count rate. We think that one of causes of this discrepancy is in modeling the Doppler broadening profile of Compton scattering for electrons in silicon crystals. However, at this time, we have not found a solution to eliminate the discrepancy from first principles. Therefore, by adjusting the OFFAXIS selection value of the simulation, we decided to match the count rate of the simulation to the observed count rate of 0.40 count sec −1 . The relation between the count rate and the OFFAXIS selection for the simulation events is shown in Figure 10. From the relation, we obtained 22.13 degree as the OFFAXIS selection value of the simulation. The effect of adjusting the OFFAXIS selection for the simulation is discussed later in this section.
The observational data, the background data, and the simulation data are plotted in Figure 11. The simulation data with the selection of −22.13 • < OFFAXIS < +22.13 • is shown in black, the background data derived from the entire RXJ 1856.5−3754 observation is shown in green. Sum of the simulation data and the background data is plotted in blue, and, is comparable with the observation data shown in red. The θgeometry distribution is well reproduced by the Monte Carlo simulation while the energy spectrum shows small discrepancy due to the background data as shown in section 3.3.
The azimuth angle distributions of the simulated data are shown in Figure 12. The left-hand side panel shows the azimuth angle distributions of the simulated data with the OFFAXIS selection of −22.13 • < OFFAXIS < +22.13 • and −30 • < OFFAXIS < +30 • . The normalization for the −30 • < OFFAXIS < +30 • selection is scaled. There is little difference in the azimuth angle distribution between these two selections. The right-hand side panel of Figure 12 shows how the azimuth angle distribution depends on the OFFAXIS selection. It is found that the azimuth angle distribution changes by less than 1% when the angle selection range is changed from 15 • to 45 • .

Parameter search for the polarization measurement
We obtained the azimuth angle distributions for the Crab observation, the background, and the unpolarized gamma-ray simulation, respectively. In order to derive the polarization parameters of the Crab nebula from these data, we adopt a binned likelihood fit. Although the bin width of the histograms for the azimuth angle distributions was 20 • for the figures in the previous subsections, 1 • per bin histograms are prepared for the binned likelihood fit. The histograms are shown in Figure 13. For the simulation data, the OFFAXIS selection of −22.13 • < OFFAXIS < +22.13 • is adopted.
In the binned likelihood fit, we scaled the background data and unpolarized simulation with the exposure time of the Crab GTI. Expected counts nexp(φi) in each bin are expressed by the following equation using the background n bkg (φi) and unpolarized simulation data nsim(φi) in count space: nexp(φi) = nsim (φi) (1 − Q cos (2 (φi − φ0))) + n bkg (φi), (4) where Q is a modulation amplitude due to a polarization, φ0 is a polarization angle in the coordinate of the Compton camera, i is a bin number (i >= 1), and φi is the azimuthal angle at i-th bin center. We assume that the Crab observation counts n obs is given by Poisson distributions which can be expressed as Poisson (n obs (φi)|nexp(φi)) = n n obs exp e −nexp n obs ! .
Best fit parameters of Q and φ0, can be obtained by searching a combination of the parameters that yields the mini- The errors of estimated value are evaluated from the confidence level. In the large data sample limit, the difference of the log likelihood L from the minimum L0, ∆L = L − L0, follows χ 2 . Since we have two free parameters, ∆Ls of 2. 30, 5.99, 9.21 correspond to the coverage probabilities of 68.3%, 95.0% and 99.0%, respectively.

Polarization results and validation
The dependence of L on Q and φ0 is shown in Figure 14.
The modulation amplitude for the 100% polarized gamma-ray photons (Q100) is slightly dependent on φ0 and is estimated to be Q100 = 0.6534 with the Monte Carlo simulation for φ0 = 67 • and a power-law spectrum with a photon index (Γ) of 2.1 As the result, the polarization fraction (Π) of the Crab nebula is calculated as Π = 0.1441/0.6534 = 22.1%, and, the error is also calculated as 0.0688/0.6534 = 10.5%.
In order to validate the statistical confidence, we made 1000 simulated Crab observation data sets and derived the parameters with the binned likelihood fits for each data set. Because the exposure time of the Crab observation was about 5 ksec, the exposure time of the simulated Crab observation data is also set to be 5 ksec. In the Monte Carlo simulations, the polarization fraction Π = 0.22 and the polarization angle φ0 = 67 • are assumed. The background data is prepared using the azimuth angle distribution of the background data shown in Figure 13. By using the random number according to the azimuth angle distribution of the background data, 1000 sets of 5 ksec background data are obtained. The 1000 sets of the simulated Crab observation data are prepared by summing each Monte Carlo data and background data.
The distribution of the best combinations of Q and Φ0 counts/sec/keV Fig. 11. The distribution of the θgeometry (left) and the energy spectrum (right). The observational data are plotted in red. The simulation data with the selection of −22.13 • < OFFAXIS < +22.13 • are shown in black, and, the background data derived from the all RXJ 1856.5−3754 observation are shown in green. Sum of the simulation data and the background data are plotted in blue. The red data points are identical to the red ones in Figure 5, and, the green data points are identical to the green ones in Figure 7.   from the fits for the 1000 sets of the Crab simulation data are shown as the red points of Figure 15. The numbers of the data sets inside the contours of ∆Ls of 2. 30, 5.99, 9.21 are 668, 945 and 984, respectively. These numbers match the coverage probabilities in the case of two parameters.
In order to validate the confidence level for the detection of the polarized gamma-rays, we also prepared 1000 sets of unpolarized simulation data. The results of the binned likelihood fits for the data sets are shown in the blue points of Figure 15. The distribution of the difference between the minimum of the log likelihood (L0) and the log likelihood of Q = 0 (LQ=0) is shown in Figure 16. It is confirmed that the value of the difference corresponds to the coverage probabilities in the case of two parameters. Therefore, the ∆L against the case of Q = 0 of 10.03 derived from the Crab observation corresponds to the confidence level of 99.3%. Q Fig. 15. The results of Likelihood estimations for 1000 sets of simulation data. The red points show the best-fit parameters for the Crab simulation data with the polarization parameters (Π = 0.22 and φ0 = 67 • ) derived from the observation data, and, the blue points show the best-fit parameters for the unpolarized simulation data. The contours are same as in Figure 14. Figure 17 shows the phi distribution of the gamma rays from the Crab nebula with the parameters determined in this analysis. Figure 18 shows the relation between the satellite coordinate and the sky coordinate. The roll angle during the Crab observation was 267.72 • , and then, φ0 = 67.02 • corresponds the polarization angle of 110.70 • .

Comparison with other measurements
The detection of polarization, and the measurement of its angle indicates the direction of an electric field vector of radiation. In our analysis, the polarization angle is derived to be PA = 110.7 • +13.2 • −13.0 • . The energy range of gammarays contributing most significantly to this measurement is ∼ 60-160 keV. All pulse phases of the Crab nebula emission are integrated. The spin axis of the Crab pulsar is estimated 124.0 • ± 0.1 • from X-ray imaging (Ng & Romani 2004). Therefore, the direction of the electric vector of radiation as measured by the SGD is about one standard Roll angle =267.72º Projected spin axis: 124º (Ng & Romani 2004)  deviation with the spin axis.
The Crab polarization observation results from other instruments are listed in Table 6. These instruments can be divided into three types based on the material of the scatterer. The PoGO+ and the SGD employ carbon and silicon for as scatterer, respectively, while remaining instruments employ CZT or germanium. Since the cross section of the Compton scattering exceeds that of the photo absorption at around 20 keV for carbon, around 60 keV for silicon and above 150 keV for germanium and CZT, which constrain the minimum energy range for each instrument. Since the flux decreases with E −2 , the effective maximum energy for polarization measurements will be less than four times of the minimum energy. Therefore, the PoGO+, the SGD and the other instruments have more or less non-overlapping energy range and are complimentary. The PoGO+ team has reported the polarization angle PA = 131.3 • ± 6.8 • and the polarization fraction PF = 20.9% ± 5.0% for the pulse-integrated, and PA = 137 • ± 15 • and PF = 17.4% +8.6% −9.3% for off-pulse period (Chauvin et al. 2017). Our results are consistent with the PoGO+ results. On the other hand, for the higher energy range, the INTEGRAL IBIS, SPI and the AstroSat CZTI have performed the polarization observation of the Crab nebula in recent years, and, reported the slightly higher polarization fractions than our results. Furthermore, the AstroSat CZTI reported varying polarization fraction during the off-peak period (Vadawale et al. 2017). However, we have not been able to verify those results because of extremely short observation time, which was less than 1/18th of the PoGO+, and less than 1/100th of the higher energy instrument. Despite such short observation time, the errors of our measurements are within a factor of two of other instruments. This result demonstrate the effectiveness of the SGD design such as high modulation factor of the azimuthal angle dependence, highly efficient instrument design and low backgrounds. Extrapolating from this result, we expect that the 20 ks SGD observation can achieve statistical error equivalent with the PoGO+ and the AstroSAT CZTI, and the 80 ks SGD observation can perform phase resolved polarization measurements with similar errors.

.(8)
For the measured parameters, α = 1.1, θ = 60 • , this evaluates to Π = 0.37, The measured mean polarization is comfortably below this value suggesting that the magnetic field is moderately disordered relative to our simple model and the particle distribution function may be anisotropic. MHD and PIC simulations can be used to investigate this further.

Conclusions
The Soft Gamma-ray Detector (SGD) on board the Hitomi satellite observed the Crab nebula during the initial test observation period of Hitomi. Even though this observation was not intended for the scientific analyses, the gamma-ray radiation from the Crab nebula was detected by combining the careful data analysis, the background estimation, and the SGD Monte Carlo simulations. Moreover, polarization measurements were performed for the data obtained with SGD Compton cameras, and, po- larization of soft gamma-ray emission was successfully detected. The obtained polarization fraction of the phaseintegrated Crab emission (sum of pulsar and nebula emissions) was 22.1% ± 10.6% and, the polarization angle was 110.7 • +13.2 • /−13.0 • (The errors correspond to the 1 sigma deviation) despite extremely short observation time of 5 ks. The confidence level of the polarization detection was 99.3%. This is well-described as the soft gamma-ray emission arising predominantly from energetic particles radiating via the synchrotron process in the toroidal magnetic field in the Crab nebula, roughly symmetric around the rotation axis of the Crab pulsar. This result demonstrates that the SGD design is highly optimized for polarization measurements.