The PoGO+ view on Crab off-pulse hard X-ray polarization

Abstract

The linear polarization fraction (PF) and angle of the hard X-ray emission from the Crab provide unique insight into high-energy radiation mechanisms, complementing the usual imaging, timing, and spectroscopic approaches. Results have recently been presented by two missions operating in partially overlapping energy bands, PoGO+ (18–160 keV) and AstroSat CZTI (100–380 keV). We previously reported PoGO+ results on the polarization parameters integrated across the light curve and for the entire nebula-dominated off-pulse region. We now introduce finer phase binning, in light of the AstroSat CZTI claim that the PF varies across the off-pulse region. Since both missions are operating in a regime where errors on the reconstructed polarization parameters are non-Gaussian, we adopt a Bayesian approach to compare results from each mission. We find no statistically significant variation in off-pulse polarization parameters, neither when considering the mission data separately nor when they are combined. This supports expectations from standard high-energy emission models.

instrumentation: polarimeters, methods: statistical

1 INTRODUCTION

The Crab pulsar and wind nebula are an archetypal multiwavelength laboratory for the study of high-energy astrophysics. As such, the system is one of the most studied celestial objects (Bühler & Blandford 2014). Additional data are required to fully understand high-energy emission processes for X-rays. Determining the linear polarization of the emission provides complementary information to observation methods which result in images, energy spectra, and temporal light curves (Krawczynski et al. 2011). Linear polarization is described using two parameters: (i) the polarization fraction (PF,  per cent) describing the degree of polarization and (ii) the polarization angle (PA, deg.) which describes the orientation of the electric field vector.

Polarized emission is a consequence of the synchrotron processes which are thought to dominate for the Crab (Hester 2008). The maximum allowed PF for synchrotron emission in a uniform magnetic field is high, ∼75 per cent (Lyutikov et al. 2003). The Crab is a complex system comprising a rotation-powered pulsar surrounded by a diffuse pulsar wind nebula which includes resolved structures in the inner nebula including a jet, toroidal structures, and synchrotron shock fronts, seen as knots and wisps (Moran et al. 2013). Measured PF values are integrated over these features, and so are expected to be significantly lower than the theoretical maximum, motivating the need for sensitive and well-calibrated instruments. The dependence of the measured polarization parameters on the rotational phase of the pulsar can be used to differentiate between emission models (Cheng et al. 2000; Dyks et al. 2004; Petri 2013; Harding&Kalapotharakos 2017).

More than 40 yr have passed since the OSO-8 mission made the first significant observations of linear polarization in X-ray emission from the Crab nebula. In Weisskopf et al. (1978), a relatively high PF is measured at 2.6 keV, (19.2±1.0) per cent for PA=(156.4±1.4)°, supporting the hypothesis that synchrotron processes dominate the emission. In the gamma-ray regime, inventive use of instruments onboard INTEGRAL has provided polarimetric data on the Crab (Dean et al. 2008; Forot et al. 2008; Chauvin et al. 2013; Moran et al. 2016). We note, however, that the INTEGRAL analyses are complicated by the lack of pre-launch studies of the polarimetric response of the instrument.

Recently, new results on the Crab in the hard X-ray regime (10–100 s of keV) have been presented by two complementary polarimetry missions – the Swedish–Japanese stratospheric ballooning platform, PoGO+ (Chauvin et al. 2017a), and the Indian earth-orbiting satellite, AstroSat (Vadawale et al. 2018). The PoGO+ mission is a development of the PoGOLite Pathfinder which detected polarization in hard X-ray emission from the Crab in 2013 (Chauvin et al. 2016). Polarimetry measurements obtained by the CZTI instrument of AstroSat allowed the phase dependence of the polarization parameters to be studied, whereas PoGO+ estimated polarization parameters in relatively wide phase windows. The team analysing the AstroSat CZTI data concluded that the polarization properties vary across the Crab off-pulse region. This implies that the pulsar contributes significantly to the off-pulse emission. Vadawale et al. (2018) point out that the viewing geometry may give rise to such effects (Takata et al. 2007; Bai & Spitkovsky 2010) and that a similar effect has been reported for radio pulsars (Basu et al. 2011). None the less, this is a surprising and intriguing result which warrants further study since it challenges prevailing high-energy emission models.

In this paper, we extend our previous analysis of PoGO+ Crab data by examining the phase dependence of the polarization parameters. We compare our results to those obtained by AstroSat CZTI, to elucidate the off-pulse behaviour of the polarization parameters in a lower, but partially overlapping, energy band.

2 NEW RESULTS IN HARD X-RAY POLARIMETRY

PoGO+ (18–160 keV) is specifically designed for polarimetry, while AstroSat CZTI (100–380 keV) is a coded aperture spectrometer for general hard X-ray observations, including polarimetry. The response of both instruments was determined for both polarized and unpolarized radiation before launch (Vadawale et al. 2015; Chauvin et al. 2017b). Results are summarized in Table 1 and relevant characteristics of the two missions are summarized in Table 2. In both cases, the reported values are corrected for the bias due to the positive definite nature of the measurement, e.g. Maier et al. (2014), as discussed in Section 3. The PF values reported are statistically compatible, although the AstroSat CZTI values are consistently higher than PoGO+ which may indicate an energy dependence of PF when considered together with the OSO-8 results. The PoGO+ PA is compatible with the pulsar spin axis, (124.0±0.1)°, as determined from high-resolution Chandra X-ray images (Ng&Romani 2004). The AstroSat CZTI angle is further rotated. When compared to OSO-8 results for the off-pulse (nebula-dominated) emission, the PA does not appear to exhibit a simple evolution with energy.

Table 1.

Comparison of bias-corrected Crab hard X-ray polarization parameters. Errors correspond to 1σ. PA is defined relative to celestial North in the Easterly direction.

	Phase range	PF ( per cent)	PA (°)
PoGO+	All	20.9±5.0	131.3±6.8
	Off-pulse	17.4|$^{+8.6}_{-9.3}$|	137±15
AstroSat CZTI	All	32.1|$^{+5.7}_{-6.0}$|	143.5±4.9
	Off-pulse	37.7|$^{+10.0}_{-11.1}$|	141.0±7.6

Table 2.

Comparison of the PoGO+ and AstroSat CZTI missions (polarimetric mode).

	PoGO+	AstroSat CZTI
Platform	stratospheric balloon	satellite
Overburden	5.8 g cm−2 average	0 g cm−2
Detector	plastic scintillator	CZT
Pixels, geometry	61, hexagonal	16 384, square
Geometrical area	378 cm2	976 cm2
Field of view	∼2°	∼90°
Energy band	18–160 keV	100–380 keV
Observation	Jul.’16	Sep.’15–Mar.’17
t  source	92 ks	800 ks
t  bkgnd	79 ks	180 ks
Signal/Bkgnd	0.14	0.05

3 OBSERVATION METHODOLOGY

Both PoGO+ and AstroSat CZTI utilize Compton scattering interactions in a segmented detector to determine the polarization of incident X-rays. According to the Klein–Nishina scattering cross-section, X-rays will preferentially scatter in a direction perpendicular to the polarization vector. This implies that the azimuthal scattering angle, ϕ (defined relative to the polarization vector), is modulated for a given range of polar scattering angles, θ.

For PoGO+, the azimuthal scattering angle is determined from events with exactly two interactions at any location in the scintillator array. The distribution of such angles is a harmonic function (‘modulation curve’), where the phase defines PA, and the modulation amplitude defines PF. In order to separate instrumental effects from source polarization, the polarimeter is rotated around the viewing axis during observations. The symmetric geometry of the instrument pixels allows the scattering angle distribution to be determined independent of computer simulations. The background is dominated by albedo atmospheric neutrons. Since a fake polarization signal can be generated by such an anisotropic background, 79 ks of interspersed observations were conducted on fields 5° to the East and West of the Crab. Temporal behaviour of the background was tracked by transitioning between the fields every 15 min. Unbinned and background-subtracted Stokes parameters were used to determine polarization parameters.

Above ∼100 keV, Compton scattering dominates in the AstroSat CZTI Cadmium-Zinc-Telluride (CZT) detector, and polarization events are identified through coincident interactions in adjacent pixels. The telescope structure becomes increasingly transparent across the energy range which results in a large field of view where, like PoGO+, observations are spatially averaged over the entire nebula. Polarimetric data were gathered for 21 Crab observations (totalling 800 ks) after the launch on 2015 September 28. The low-inclination (6°) orbit provides a low background environment for measurements, with the Cosmic X-ray Background dominating. The background response is determined through observations (180 ks) of fields with a declination close to that of the Crab, with bright X-ray sources such as the Crab and Cygnus X-1 outside the field of view. In contrast to PoGO+, the detector pixels have a square geometry which yields a non-symmetric scattering geometry. The CZTI instrument is not rotated during observations and uniformity corrections are derived from computer simulations. Resulting background-subtracted and geometry-corrected modulation curves are fitted with a harmonic function to determine PA and PF.

As shown by the statistical uncertainty of the results presented in Section 2, the two missions have comparable polarimetric performance for Crab observations. The shorter observation time for PoGO+ is compensated¹ by the lower energy range (higher photon flux) and the larger Compton scattering cross-section offered by plastic scintillators. The modulation response is simpler for PoGO+ due to the symmetric scattering geometry. Both missions report a Crab polarization sensitivity [Minimum Detectable Polarization, MDP (Weisskopf et al. 2010)] before background subtraction of ∼10  per cent. Unpolarized radiation has a 1 per cent probability of exhibiting PF>MDP. Polarization measurements are positive definite, with the PF following a Rice distribution. This results in a bias to positive values unless the reconstructed PF≫MDP. This is not the case for either mission, so bias corrections are applied as described in the Supplementary Information of Chauvin et al. (2017a) and Vadawale et al. (2018).

4 PHASE DEPENDENT ANALYSIS OF PoGO+ DATA

In Chauvin et al. (2017a), we determined polarization parameters integrated over the off-pulse region. We now extend this work by following the ‘dynamic binning’ approach detailed in Vadawale et al. (2018) in order to directly compare with AstroSat CZTI. Polarization parameters are determined across the full phase range, 0 ≤ η ≤ 1, in bins of width 0.1, spaced with a phase interval of 0.01. Results are shown in Fig. 1. The error bars shown for the AstroSat CZTI data are different to those presented in Vadawale et al. (2018) (Fig. 7, Supplementary Information) since confidence levels were inadvertently shown instead of credibility intervals (Vadawale et al. 2018). For the majority of the phase bins, the difference is negligible but low significance bins were reported with overestimated errors, e.g. for η = 0.83 the credibility interval should be (0 per cent, 23.6 per cent) not (0 per cent, 29.8 per cent). The interpretation of polarization parameter trends from this binning approach is complicated by the presence of correlations – these figures therefore also show parameters derived for 10 independent phase bins, selected to follow the AstroSat CZTI convention. For the PF data, PoGO+ indicates more variation in the pulsar peaks (where pulsar and nebula emission are mixed), while AstroSat CZTI data vary more in the off-pulse region. This off-pulse variation is stated as a main result of the AstroSat CZTI analysis. In the PA data, no significant variation is seen in either sets of data. In the remainder of this paper, we present a statistical analysis on the three independent phase bins in the off-pulse region where variation in PF is reported by AstroSat CZTI.

Figure 1.

Upper two panels: Bias-corrected PF values, binned dynamically (blue points), and independently (red points), from PoGO+ and AstroSat CZTI data. Middle panel: The background-subtracted PoGO+ Crab light curve (shown for reference) with the off-pulse region bounded by dashed lines. Lower two panels: Bias-corrected PF values, binned dynamically (blue points), and independently (red points), from PoGO+ and AstroSat CZTI data. The binning convention follows that presented in Vadawale et al. (2018). The AstroSat CZTI error estimates are revised, as described in Section 4.

5 STATISTICAL COMPARISON TO ASTROSAT CZTI RESULTS

We have studied the statistical significance of variations in polarization parameters across the Crab off-pulse phase region, using the independently binned PF results shown in Fig. 2. A standard χ² approach is not suitable since PF∼MDP, which yields non-Gaussian errors. We instead use a Bayesian approach in our analysis.

Figure 2.

PF in the off-pulse region displayed in three independent phase bins, as selected by Vadawale et al. (2018). PoGO+ (AstroSat CZTI) data are shown as red (black) data points. The light curve reconstructed by PoGO+ is shown in blue. Compared to Fig. 1, the light curve has been shifted by 0.02 in phase in order to accommodate the width of the data points. The AstroSat CZTI error estimates are revised, as described in Section 4.

5.1 Bayesian model comparison

The Bayesian methodology provides an automatic³³ way of applying Occam’s razor to model selection (Trotta 2007). An important aspect is that under-utilized data set space, which favours simple models, is automatically balanced against differences between predicted and observed data, which favours complex models.

The posterior distribution over the models |$\mathcal {M}_i$| for all data |$\mathcal {D}$| is given by

\begin{equation} P(\mathcal {M}_i|\mathcal {D})\propto P(\mathcal {D}|\mathcal {M}_i)P(\mathcal {M}_i), \end{equation}

(1)

where |$P(\mathcal {M}_i)$| is a prior over the models and |$P(\mathcal {D}|\mathcal {M}_i)$| is the evidence (the marginal likelihood),

\begin{equation} P(\mathcal {D}|\mathcal {M}_i)=\int P(\mathcal {D}|\mathbf {w}, \mathcal {M}_i)P(\mathbf {w}|\mathcal {M}_i)\mathrm{d}\mathbf {w}. \end{equation}

(2)

Here, |$\mathbf {w}$| is a set of parameters, |$P(\mathbf {w}|\mathcal {M}_i)$| is the prior over parameters given in a model, and |$P(\mathcal {D}|\mathbf {w}, \mathcal {M}_i)$| is the likelihood (explicit form given in Appendix A). When comparing two models, the Bayes factor is a ratio of the evidences,

\begin{equation} B_{01}=\frac{P(\mathcal {D}|\mathcal {M}_0)}{P(\mathcal {D}|\mathcal {M}_1)} \end{equation}

(3)

and can be interpreted as an odds ratio between models when their priors are equal, i.e. |$P(\mathcal {M}_0)=P(\mathcal {M}_1)=0.5$|⁠. In that case, the probability for |$\mathcal {M}_1$|⁠, given that there are only two possible models, is

\begin{equation} P(\mathcal {M}_1|D)=\frac{1}{1+B_{01}}. \end{equation}

(4)

5.2 Models and priors

We follow a parametric approach to quantify statistically the claim by Vadawale et al. (2018) that PF varies in the off-pulse region. The function |$f(\eta ;\mathbf {w})$| (see Appendix A) should contain few parameters since there are only three data points in Fig. 2. A Bayesian model comparison where there are more parameters than data points is possible; however, such models will be penalized since there is only unit probability density to distribute among all possible data sets, |$\mathcal {D}$|⁠.

We consider the models in Table 3, where |$\mathcal {M}_0$| corresponds to no change in PF across the off-pulse region. The other models, |$\mathcal {M}_{1\text{--}4}$|⁠, are V-shaped functions with different parameter constraints. The parameters, w₀, w₁, and w₂, correspond to an offset from zero, the phase of the extreme point and the gradient, respectively. Model |$\mathcal {M}_4$| is, arguably, the most physical since it requires the change to occur at the same phase and in the same direction for the partially overlapping energy bands of PoGO+ and AstroSat CZTI. However, a general approach is followed and different weights are allowed under the same model unless specified in the ‘shared traits’ column. A simple first-order polynomial is not considered since it has significantly lower evidence for both missions.

Table 3.

The models and functions used in the likelihood |$P(\mathcal {D}|\mathbf {w}, \mathcal {M}_i)$| as specified by equation (A1), where η is the pulsar phase. The ‘shared traits’ column indicates which parameters are forced to be the same for PoGO+ and AstroSat CZTI data.

Model	Function fi(η)	Shared traits
|$\mathcal {M}_0$|	w  0	None
|$\mathcal {M}_1$|	w  2|η − w1| + w0	None
|$\mathcal {M}_2$|	w  2|η − w1| + w0	w  1
|$\mathcal {M}_3$|	w  2|η − w1| + w0	sign(w2)
|$\mathcal {M}_4$|	w  2|η − w1| + w0	w  1 and sign(w2)

The parameters, w₀, w₁, and w₂, are chosen from uniform distributions but not all combinations are possible as they would yield unphysical results, e.g. PF < 0 or PF > 1. Instead of determining the inter-dependence of parameters, the parameters are sampled randomly, discarding combinations that are non-physical. After sufficiently many iterations, the entire valid parameter space is sampled. This results in the priors shown in Fig. 3.

Figure 3.

Prior distributions for the weights |$\mathbf {w}$|⁠. The priors are constructed by drawing uniform random numbers from a large range and then excluding |$\mathbf {w}$| where |$f_i(\eta ;\mathbf {w})<0$| or |$f_i(\eta ;\mathbf {w})>1$| for 0.70 ≤ η ≤ 0.94. The number of samples is shown on z-axis.

6 RESULTS AND DISCUSSION

The Bayes factors and evidence values (Appendix B) are shown in Table 4 for each mission. The Bayes factors are close to unity, so the data provides little information about the phase evolution of PF. As expected, PoGO+ tends to favour the constant model |$\mathcal {M}_0$|⁠, while AstroSat CZTI favours the V-shape |$\mathcal {M}_1$|⁠. The Bayes factors and evidence values when combining data from PoGO+ and AstroSat CZTI are shown in Table 5. The Bayes factors are very close to unity and there is no clear separation between the models – independent of model choice. Consequently, we do not support the claim that there is a variation in polarization properties within the off-pulse region. While a re-flight of the PoGO+ mission is not foreseen, additional data from AstroSat CZTI may help to clarify the situation.

Table 4.

Mission-wise model comparison.

	|$P(\mathcal {D}|\mathcal {M}_0)$|	|$P(\mathcal {D}|\mathcal {M}_1)$|	B  01	|$P(\mathcal {M}_1|D)$|
PoGO+	3.88	2.39	1.62	0.38
AstroSat CZTI	0.34	0.70	0.48	0.68

Table 5.

Model comparison for combined data from PoGO+ and AstroSat CZTI.

Model	|$P(\mathcal {D}|\mathcal {M}_i)$|	B  0i	|$P(\mathcal {M}_i|D)$|
|$\mathcal {M}_0$|	1.31	–	–
|$\mathcal {M}_1$|	1.69	0.78	0.56
|$\mathcal {M}_2$|	1.83	0.72	0.58
|$\mathcal {M}_3$|	1.46	0.90	0.53
|$\mathcal {M}_4$|	1.50	0.88	0.53

ACKNOWLEDGEMENTS

We are grateful to S. Vadawale and the AstroSat CZTI Collaboration for generously making their data available to us and for valuable discussions. This research was supported by The Swedish National Space Board, The Knut and Alice Wallenberg Foundation, The Swedish Research Council, The Japan Society for Promotion of Science, and ISAS/JAXA.

APPENDIX A: LIKELIHOOD

PF is a positive definite quantity – an unpolarized source will always have a non-zero reconstructed polarization p_r. The likelihood for observing N data points with PF values |$p_{r_j}$|⁠, after marginalizing over the PA (which is assumed constant), is a product of Rice distributions,

\begin{eqnarray} P(\mathcal {D}|\mathbf {w}, \mathcal {M}_i)&=&\prod _{j=1}^{N}\frac{p_{r_j}}{\sigma ^2_j}\exp {\bigg (-\frac{p_{r_j}^2+f_i(\eta \,\mathbf {w})^2}{2\sigma ^2_j}\bigg )} \nonumber\\ \times I_0\bigg (\frac{p_{r_j}\times f_i(\eta ;\mathbf {w})}{\sigma ^2_j}\bigg ), \end{eqnarray}

(A1)

where I₀ is a modified Bessel function of the zeroth order, η is the pulsar phase, |$p_0=f(\eta ;\mathbf {w})$| is the true PF value parametrized with weights |$\mathbf {w}$|⁠, and σ is the effective uncertainty on PF. Note that σ does not correspond to a Gaussian sigma. Since the signal-to-background ratio |$\mathcal {R}=S/B$| is low for both PoGO+ and AstroSat measurements and the calibration parameters have insignificant uncertainties, i.e. μ_r = μ₀, it is possible to write

\begin{eqnarray} \sigma =\frac{2}{\mu _r}\sqrt{\frac{1}{S}\bigg (\frac{S+B}{2S}-\frac{\mu _0^2p_0^2}{4}\bigg )}=\frac{\sqrt{2(S+B)}}{\mu _r S}, \end{eqnarray}

(A2)

removing the dependency on p₀. This makes σ a fixed parameter for every data point since it does not depend on the polarization properties but only on the signal and background counts which have insignificant uncertainties; thus, |$\mathcal {D}=\lbrace p_{r_1},...,p_{r_N};\sigma _1,... \sigma _N\rbrace$|⁠.

APPENDIX B: COMPUTING THE EVIDENCE

We compute the evidences |$P(\mathcal {D}|\mathcal {M}_i)$| using the Monte Carlo integration

\begin{equation} P(\mathcal {D}|\mathcal {M}_i)=\lim _{S\rightarrow \infty } \frac{1}{S}\sum _{j=1}^S P(\mathcal {D}|\mathbf {w}_j, \mathcal {M}_i), \end{equation}

(B1)

where |$\mathbf {w}_j$| is randomly sampled from the prior |$P(\mathbf {w}|\mathcal {M}_i)$| and S is the number of samples. We use S = 10⁷ which is sufficient for a good approximation of the evidence.