The IXPE View of GRB 221009A

We start with the analysis of the core, which arises from the burst afterglow. We select the region as a disk centered on the brightest pixel ⁶⁹ of the IXPE image and radius of 26'' (0.43:00). Beyond this radius, the radial profile of the emission deviates from the PSF of the instrument by more than 20%, as shown in Figure 1 (left panel). Such a deviation informs us on possible contamination from the emission of dust-scattered X-rays (from the GRB prompt and/or afterglow emission) that we cannot fully resolve. We verified that the bright core emission from the central point-like source dominates the final result as we find consistent numbers within the 1σ uncertainty when varying slightly the selecting radius. According to the IXPE PSF, cutting at a radius of 26'' eliminates less than ∼15% of the total source emission.

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Given the high photon statistics of the core (signal-to-noise ratio >100 for any background), the results of the analysis are not affected by the choice of the background spectrum. Here, we report the results for the background template BKG2. Through the PCUBE analysis, we find an unconstrained polarization in the 2–8 keV energy range and derive a 99% C.L. upper limit of 16.1%. No evolution with time or energy is observed. For the spectropolarimetric analysis, we model the observed spectrum with an absorbed power law decreasing in energy, with intrinsic parameters fixed to the values of the Swift/XRT automated online analysis (Evans et al. 2007), which are consistent with other reported values (Kennea et al. 2022). In particular, the intrinsic absorption parameters are fixed to 1.36 × 10²² cm⁻² (Evans et al. 2007) at a redshift of 0.151 (Ugarte Postigo et al. 2022), while the Galactic absorption value is fixed to 5.38 × 10²¹ cm⁻² (Willingale et al. 2013). To account for mismatches of inter-calibration among the different IXPE telescopes, a constant normalization is left free to vary for DU2 and DU3 with respect to DU1.

The best-fit values of the spectropolarimetric analysis are provided in Table 1. We find a best-fit power-law index of ${{\rm{\Gamma }}}_{\mathrm{core}}=1.98\pm 0.03$, in agreement with expectations from a late-afterglow emission (see Section 3.2). The polarization results are slightly more constraining than, but consistent with, the PCUBE analysis with a PD of 6.1% ± 3.0%. The right panel of Figure 1 shows the Q/I versus U/I distribution of the core emission.

Table 1. Spectropolarimetric Core Analysis

Parameter	Value
PD	(6.1 ± 3.0)%
PD U.L. 99% C.L.	<13.8%
PD U.L. 95% C.L.	<12.1%
${{\rm{\Gamma }}}_{\mathrm{core}}$	1.98 ± 0.03

Note. Summary table of the spectropolarimetric analysis of the core. The spectropolarimetric fit is performed in the 2–8 keV energy band, and it assumes Gaussian statistics. The PA is unconstrained. The best-fit Q/I and U/I constants are (5.6 ± 2.9) × 10⁻² and (2.8 ± 2.9) × 10⁻², respectively.

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The Stokes parameters Q/I and U/I are expected to be normally distributed with respective means $\overline{{\text{}}Q/I}$ and $\overline{{\text{}}U/I}$, and equal standard deviations. An error contour in (Q/I, U/I) space is a circle of radius centered on ($\overline{{\text{}}Q/I},\overline{{\text{}}U/I}$), where ${\left(\epsilon /\sigma \right)}^{2}$ is distributed as χ²(d.o.f=2). The probability that the observed polarization exceeds the measured value, under the null hypothesis of unpolarized emission, is 9.7%. This is inconsistent with a 0° of polarization at 90% C.L. We therefore set a upper limit to the PD (1D distribution) of 13.8% at 99% C.L. (12.1% at 95% C.L.). For completeness, the I, Q, U spectra are reported  in Appendix B.3.

According to current models, once beyond the early flaring stages, GRB afterglows arise via synchrotron processes from electrons accelerated through interactions of the GRB jet with the circumstellar material. This is consistent with observations from radio to high energies of GRB afterglows (Kumar & Zhang 2015). The physics is well understood and follows a set of closure relations (e.g., Sari & Meszaros 2000), which, when observations fit a self-consistent picture, can be used to infer properties of the underlying emitting region through observables such as the spectral indices and rate of temporal decay. The synchrotron spectrum is described by a set of power laws with different spectral indices, each with its own closure relation depending on the particle density distribution of the circumstellar environment. We model the core X-ray emission as observed by IXPE with this interpretation, in order to utilize the polarimetric observation to constrain intrinsic properties of the jet and our viewing angle.

We start by investigating the density of the interstellar matter around the GRB progenitor. The density profile in units of cm⁻³, n(R), where R is the distance from the central engine, is parameterized by the index k, such that n(R) ∝ R^−k . For example, k = 0 corresponds to a constant density medium, and k = 2 describes a wind medium, and in-between values are also possible. A wind medium may be expected around long GRBs because they arise in the deaths of massive stars. The density profile affects the time evolution of the synchrotron break frequencies. We assume that the IXPE energy range lies between the typical (ν_m ) and the cooling (ν_c ) synchrotron frequencies, and show that this assumption yields a consistent picture.

The time and energy evolution of GRB afterglow emission is described by (see, e.g., Granot & Sari 2002)

$$\begin{eqnarray}&&{F}_{\nu }\propto {t}^{-\alpha }{\nu }^{-\beta }\,.\end{eqnarray} \tag{ 1 }$$

The core spectrum is well fit by an absorbed power law with a photon index of 1.98 ± 0.03, which yields $\beta ={{\rm{\Gamma }}}_{\mathrm{core}}\,-1=0.98\pm 0.03$. The temporal evolution of GRB afterglow usually shows a break, which causes the index α to increase. This steepening is proportional to ${\left({{\rm{\Gamma }}}_{j}{\theta }_{j}\right)}^{2}$, where Γ_j is the jet Lorentz factor, and θ_j is the jet half opening angle. Taking into account the time evolution of the Lorentz factor, the increase in the temporal index will be Δα = (k − 3)/(4 − k). From the closure relations (e.g., Sari & Meszaros 2000) between the temporal and spectral indices, we can express the index of the density profile k as

$$\begin{eqnarray}&&k=\displaystyle \frac{2(4\alpha -6\beta +3)}{2\alpha -3\beta +3}.\end{eqnarray} \tag{ 2 }$$

For α = 1.634 ± 0.015, measured by Swift-XRT ⁷⁰ and β =0.98 ± 0.03, we get k = 2.20 ± 0.05. We note that using the Swift-XRT spectral index of 0.88 ± 0.15, we get k = 2.35 ± 0.21. In our model, we assume that the IXPE observation was preceded by an achromatic jet break at ∼1 day (D'Avanzo et al. 2022).

We will thus assume that the forward shock propagates in a wind medium with density n(R) = AR⁻², where A = 3.02 × 10³⁵ A_⋆ cm⁻¹. To estimate A_⋆, we introduce fiducial or base values for the energy density fraction in electrons and in magnetic fields:

$$\begin{eqnarray}&&{\epsilon }_{e}={10}^{-1}{\epsilon }_{e,-1}\,\,;\,\,\,\,\,\,\,\,\,\,{\epsilon }_{B}={10}^{-3}{\epsilon }_{B,-3}\,\,.\end{eqnarray} \tag{ 3 }$$

Furthermore, we set the kinetic energy of the outflow to E_k,iso ≈ 10⁵⁵ erg, and we use the Q = 10^x Q_x scaling convention for quantity Q in cgs units. With the above choice of parameters, and neglecting the effect of inverse Compton scattering on the cooling, we have the following (e.g., Granot & Sari 2002):

$$\begin{eqnarray}\begin{array}{ccc}{\nu }_{m} & = & 3.6\times {10}^{12}\,{E}_{k,55}^{1/2}\,{\epsilon }_{e,-1}^{2}\,{\epsilon }_{B,-3}^{1/2}\,{\left(t/3.5\,\mathrm{day}\right)}^{-3/2}\,\,\mathrm{Hz}\,;\\ {\nu }_{c} & = & 2\times {10}^{18}\,{E}_{k,55}^{1/2}\,{A}_{\star ,-1.5}^{-2}\,{\epsilon }_{B,-3}^{-3/2}\,{\left(t/3.5\,\mathrm{day}\right)}^{1/2}\,\,\mathrm{Hz}\,,\end{array}\end{eqnarray} \tag{ 4 }$$

confirming that indeed ν_m < ν_IXPE ≲ ν_c at the time of the IXPE observation, and this ordering persists at later times because ν_m ∝ t^−3/2 and ν_c ∝ t^1/2. In this spectral regime, the energy spectral index is given by β = (p − 1)/2, where p is the power-law index of the electron energy distribution (${{dN}}_{e}/d{\gamma }_{e}\propto {\gamma }_{e}^{-p}$, where γ_e is the electron's Lorentz factor). Using the β derived from IXPE observation, we find p = 2.96 ± 0.06.

We can now estimate A_⋆ by comparing the observed flux density at 10 keV, F_ν,obs ≈ 10⁻⁶ Jy, to the synchrotron model prediction (e.g., Granot & Sari 2002), valid after the jet break:

$$\begin{eqnarray}&&{A}_{\star }\approx 3\times {10}^{-1}\,{E}_{k,55}^{-(3+p)/2}{\epsilon }_{e,-1}^{2-2p}{\epsilon }_{B,-3}^{-(p+1)/2}\,{\mathrm{cm}}^{-1}.\end{eqnarray} \tag{ 5 }$$

We note that A_⋆ depends strongly on the _e parameter (${A}_{\star }\propto {\epsilon }_{e}^{-4}$ for p ≈ 3).

A separate constraint for our afterglow model comes from the measured jet break time, t_jet, which scales as ${t}_{\mathrm{jet}}\propto {E}_{k}{\theta }_{j}^{4}{A}_{\star }^{-1}$ if the ratio between the jet opening angle θ_j and viewing angle θ_v is known. This parameter, and its position in time with respect to the time of the observation, is relevant for polarization, as it can be broadly associated with the time when the PD lightcurve has a zero-point, and the PA rotates by 90°. In fact, for uniform (top-hat) jet structure with no sideways expansion, significant polarization arises from the break in symmetry of the visible surface. This surface is typically an annulus when projected to the plane of the sky. As the annulus grows, it encompasses a progressively larger fraction of the jet surface. Eventually, for an off-axis observer, the annulus will grow beyond the size of the jet on one side, while still collecting emission from the opposite side, resulting in net polarization. The polarization lightcurve exhibits the typical two-bump structure (Sari 1999; Ghisellini & Lazzati 1999), where the jet break time approximately corresponds to the minimum between the bumps. Our model is constructed so that the PD zero-point between the two bumps is at ≈1 day, to match the estimated jet break time (D'Avanzo et al. 2022). We derive the expected PD by integrating the intensity and polarization of the comoving volume elements of the jet over the equal arrival time surfaces (Sari 1999; Granot & Knigl 2003; Shimoda & Toma 2021; Pedreira et al. 2022). For each comoving volume element, the maximum PD of a synchrotron-emitting, shock-accelerated electron population with power-law distribution with index p will be as follows: PD = (p + 1)/(p + 7/3) ≲ 75% (Rybicki & Lightman 1979). The observed polarization will be reduced from this value by integrating over all the parts of the jet that contribute to the flux at a given observer time (see, e.g., Lyutikov et al. 2003).

The evolution of the polarization as a function of time depends strongly on a variety of parameters. We take a set of parameters (base values) that give a polarization consistent with the IXPE spectropolarimetric measurement: jet opening angle θ_j = 1.5° , viewing angle ${\theta }_{v}=\tfrac{2}{3}{\theta }_{j}$ (Ghisellini & Lazzati 1999), electron energy distribution index p = 2.96, kinetic energy E_k = 10⁵⁵ erg, density parameter A_⋆ = 3 × 10⁻¹ cm⁻¹, and ξ = 0. The parameter ξ is the ratio of the magnetic field strength in two directions defined as follows: ${\xi }^{2}=2\langle {B}_{| | }^{2}\rangle /\langle {B}_{\perp }^{2}\rangle$. Here, B_∣∣ and B_⊥ are the magnetic field parallel and perpendicular to the shock normal, respectively. The case ξ = 0 yields the maximum attainable polarization for any given set of afterglow parameters. Our model with base values and several additional realizations is presented in Figure 2. In general terms, all realizations have zero-points anchored at ≈1 day and yield increasing PD at the time of the IXPE observations. For a given θ_v /θ_j ratio, we can choose a set of parameters (E_k , A_⋆, and θ_j ) so that t_jet = 1 day is satisfied. A higher θ_v /θ_j ratio results in a higher peak polarization and earlier jet break time, due to the higher level of asymmetry as we move away from the jet axis.

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All presented models in Figure 2, except the low jet opening angle, are consistent with the upper limit. Taking the PD=6.1% ± 3.0% at face value, models with jet opening angle θ_j < 1.5 deg (while keeping all other base values fixed) are disfavored. Similarly, models with θ_v /θ_j > 2/3 tend to overpredict the IXPE measurement. Assuming a magnetic field ratio, ξ, closer to 1 simply scales down the PD. In principle, any model that overpredicts the observations can be made consistent by appropriate choice of ξ. The IXPE measurement, considering the base values, favors the cases where ξ² ≲ 1/2.

Optical polarization observations occurred during the IXPE observation window at the Nordic Optical Telescope (Lindfors et al. 2022). The sky conditions allowed an estimation of an upper limit to the optical PD of 8.3% at 99% C.L. (5.1% at 95% C.L., Lindfors et al. 2022). In Appendix B.1, we provide more details about the optical data reduction. The optical band falls in the same spectral regime as the X-rays (ν_m < ν_optical < ν_c ) for most choices of parameters around the base values. Thus the optical upper limit can be used to constrain the models. The optical limit is slightly stronger, but gives qualitatively the same constraints as the X-ray limit. As the above analysis assumes a homogeneous jet (top-hat profile), more elaborate profiles —e.g., Gaussian jet or structured jet (Rossi et al. 2004)— could give different results.

As mentioned in the introduction, the observed rings are the result of a known effect involving Galactic dust along the line of sight of a bright transient event. A fraction of the photons emitted in the prompt phase of the GRB is scattered by dust clouds in the Milky Way. Those scattered inwards toward the line of sight arrive at Earth after traveling a longer path length with respect to the unscattered ones. This results in a later arrival time of the scattered photons, with a delay that depends on the distance of the dust cloud to Earth and the scattering angle θ_s . The angular size of the halos θ_h is related to the scattering angle, as θ_s (1 − x), where x is the ratio between the distance of the cloud and the distance of the source. Because we are dealing with a transient event at cosmological distance (x ≪ 1), the approximation θ_h ∼ θ_s applies (Miralda-Escudé 1999; Draine 2003a).

Being produced by a short transient event, the rings expand radially in time. This arises as photons with different scattering angles travel different path lengths. In order to study the radial evolution of the rings and correctly select prompt, scattered photons as the rings expand, we devise a method inspired by the procedure described in Tiengo & Mereghetti (2006). For each photon i detected at a time T_i and at a sky coordinate (a_i , δ_i ), we define the following variables

$$\begin{eqnarray}&&{t}_{i}={T}_{i}-{T}_{0};\,\mathrm{and}\,{K}_{i}=[{\left({a}_{i}-{a}_{B}\right)}^{2}+{\left({\delta }_{i}-{\delta }_{B}\right)}^{2}]/2{{ct}}_{i}={\theta }_{i}^{2}/2{{ct}}_{i}\,,\end{eqnarray} \tag{ 6 }$$

where (a_B , δ_B ) are the coordinates of the unscattered emission (the center of the core in the IXPE image), and θ_i is the angular distance (in arcsec) of the photon i from the point (a_B , δ_B ). The trigger time of the prompt emission is taken from the Fermi/GBM ⁷¹ (Veres et al. 2022). The advantage of this approach is that, in the plane K_i versus t_i , shown in the left panel of Figure 3, the expanding rings appear as horizontal bands, facilitating the event selection. We remove the dominant emission from the core by removing events within 0.85' from the center to avoid contamination from the bright core (afterglow) emission. We estimate that the contamination from the core emission at radial distances larger than 0.85' is less than 4%. The distribution n(K_i ), after subtracting the simulated background events, is shown in the right panel of Figure 3: the contribution of the two rings is prominent. We fit the distribution around the peaks with the sum of two Lorentzian functions, which approximate well the observed distribution.

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We define the event selection cut on the K_i distribution as illustrated in Figure 3 (right panel). The area under each best-fit Lorentzian between ${R}_{\min }^{i}$ and ${R}_{\max }^{i}$ (orange areas in the plot), where i = 1, 2 denotes r ₁ and r ₂ respectively, is at least a factor of 20 larger than the area under the other Lorentzian in the same range (gray areas in the plot). This ensures a negligible contamination from the emission of one ring onto the other. The edges of the selection for the wider ring are symmetric with respect to the peak, while the innermost edge of the smaller ring is naturally defined by our region cutoff at 0.85' to exclude the core emission.

Similar to the core analysis, we proceed with the PCUBE polarization analysis in the 2–8 keV energy band. The observed spectra of the two rings are expected to be different because they are generated from the same prompt emission scattered at different angles. As discussed later on, given a scattering angle, the scattering efficiency of X-rays by dust grains is energy dependent (Draine 2003b). This leads to the realization that combining the two ring selections into one single PCUBE analysis would be inaccurate. Furthermore, the low statistics of the individual ring selections prevents a proper background template subtraction for the PCUBE analysis. This implies that the estimated uncertainties are not accurate because they are computed on a boosted statistic that includes background events. The results of the PCUBE analysis for the individual rings are reported in Appendix B.3. We find a PD${}_{{r}_{1}}=19.6 \% \pm 8.7$%, and PD${}_{{r}_{2}}=17.2 \% \pm 8.8$%, in agreement with each other. These values indicate a higher PD than that observed in the core, although never exceeding the 99% C.L. required to claim a detection. ⁷²

The spectropolarimetric fit, allowing a proper combination of the rings selections, can give a more accurate estimation of the underlying polarization. The phenomenological model we define to describe the rings' emission allows for the spectral parameters of the rings to be different while sharing common polarization parameters. The spectra of both rings are modeled as absorbed simple power laws, while we assume constant polarization parameters. The intrinsic and Galactic absorption parameters are kept fixed to the same values adopted for the core analysis. As opposed to the PCUBE analysis, we perform the subtraction of the background spectrum. We test different background assumptions, subtracting the spectra derived from BKG1, BKG2, and BKG3 templates, to which we applied the analogous event selection as for r ₁ and r ₂.

The results are summarized in Table 2. We find that r ₁ has a best-fit photon index, averaged over the values obtained assuming different backgrounds, ${{\rm{\Gamma }}}_{{{\rm{r}}}_{1}}=2.89$, with a relative statistical error of about 7% and negligible systematic uncertainty due to the choice of the background spectrum. r ₂ has a steeper spectrum, with photon index ${{\rm{\Gamma }}}_{{{\boldsymbol{r}}}_{1}}=3.98$ with a relative statistical error of about 6% as well as a 7% relative systematic error associated with different background assumptions. This is also illustrated in Appendix B.3. As we will discuss in the next section, such a difference in spectral index between the two rings is expected owing to the energy dependence of the X-ray scattering cross section. We note that, for the case of r ₂, the estimated photon index found assuming BKG1 shows a larger statistical error: the softer spectrum of the emission from this ring with respect to r ₁ makes the measurement more sensitive to the spectral characteristics of the subtracted background (Appendix A shows that BKG1 has the hardest spectrum).

Table 2. Summary Table of the Spectropolarimetric Analysis for the Rings' Emission

Summary Table of the Rings Analysis
PD (BKG1)	(32.1 ± 19.2)%
PD (BKG2)	(22.0 ± 12.6)%
PD (BKG3)	(25.1 ± 13.5)%
PDave	${27.2}_{\pm 4.0\,(\mathrm{sys}.)}^{\pm 15.1\,(\mathrm{sta}.)}$ %
PD U.L. 99% C.L.	<[54.6% − 81.5%]
PD U.L. 95% C.L.	<[47.1% − 71.4%]
${{\rm{\Gamma }}}_{{{\boldsymbol{r}}}_{1}}$	${2.89}_{\pm 0.07\,(\mathrm{sys}.)}^{\pm 0.20\,(\mathrm{sta}.)}$
${{\rm{\Gamma }}}_{{{\boldsymbol{r}}}_{2}}$	${3.98}_{\pm 0.30\,(\mathrm{sys}.)}^{\pm 0.25\,(\mathrm{sta}.)}$

Note. The spectropolarimetric fit is performed assuming Poissonian statistics of the background-rejected data with background subtraction applied. We report the best-fit PD value for each of the three assumed background models, as well as the weighted average with associated statistical and systematic errors. The upper limits are strongly dependent on the assumed subtracted background: we report the range of values defined by the minimum and maximum value obtained. The PA is unconstrained.

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As for the polarization, we find that the PD value and uncertainty depend upon the assumed background. The statistical-error-weighted average is (27.2 ± 15.1 (sta.)± 4.0(sys.))%, where the statistical error is the average among the statistical uncertainties obtained assuming different backgrounds, and the systematic uncertainty is given by the variation of the best-fit value assuming different backgrounds. Figure 4 shows the results for the different background subtractions. The significance of this result, tested against the null hypothesis of unpolarized emission, is about 81% C.L. The 1D 99% C.L. upper limit on the PD varies between 54.6% and 81.5%, depending on the assumed background. Such a difference is due to the low-statistic regime we have for the rings data selection, which causes the statistical uncertainty to be strongly affected by small changes of the subtracted background.

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The comparison of the best-fit PDs found assuming different backgrounds is illustrated in the right plot of Figure 4. For completeness, the Q/I versus U/I distributions obtained for the different assumed background are provided in Appendix B.3. Note that, given the different approaches and handling of the background, the PCUBE and spectropolarimetric analyses are not directly comparable in this case.

Appendix B.3 reports the Q/I versus U/I plots for the different background assumptions. Additionally, we show in the same figure the equivalent plots for the spectropolarimetric fit of the individual rings. Furthermore, Appendix B.3 reports the results of the PCUBE analysis of the individual rings resolved in two logarithmic energy bins between 2 and 8 keV. We refer the reader to Appendix B for further discussion on this matter.

As discussed in Draine (2003b), the effect of the dust scattering at such a small angles is not expected to alter the intrinsic polarization of the incoming radiation (see their Figure 5). However, an explicit demonstration of this statement in the X-ray band is not directly discussed in the literature. Therefore we investigated the effect on polarization from reflection, scattering, and transmission considering the common dust compounds. All of the above processes lead to a negligible effect on the polarization of the X-ray radiation, as discussed in Appendix B.2. Therefore, we can reasonably assume that any polarization observed from the X-ray scattering halos is attributable to the original emission.

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A high PD (PD ≳20%) in the prompt phase, when viewing the jet at angles smaller than the opening angle, θ_v < θ_j , can be achieved by synchrotron emission in an ordered, toroidal magnetic field configuration (Toma et al. 2009). Alternatively, high polarization can be achieved by random magnetic fields or Compton drag models, in a geometry where we are viewing the jet close to its edge, θ_j ≲ θ_v < θ_j + 1/Γ_j (Granot 2003). This scenario will result in a very early jet break and potentially high PD in the afterglow, which is disfavored by the observations. In what follows, therefore, we will focus on the ordered synchrotron scenario.

We estimate the PD integrated over the duration of the prompt emission. The PD mainly depends on the photon index, the viewing angle, and the product of the jet opening angle and the Lorentz factor, ${y}_{j}={\left({\theta }_{j}{{\rm{\Gamma }}}_{j}\right)}^{2}$. For the IXPE observation, only the low-energy photon index is relevant. We consider the two extreme values 0.62 and 1.25, which correspond to the minimum and maximum best-fit values of the prompt GRB intrinsic spectral index given by the assumed background bracketing (see Section 4.3). For Γ_j = 700 (Liu et al. 2023), θ_j = 1.5°, and ${\theta }_{v}=\tfrac{2}{3}{\theta }_{j}$, we obtain 16% and 36%, respectively. This range is consistent with the measured upper limits.

In this section, we derive some constraints on the dust clouds' distance and optical depth. We attempt to derive the intrinsic spectrum of the GRB prompt emission from the observed rings spectra. However, such considerations are limited by the imaging capabilities of our instrument with respect to other missions that were observing the burst at the same time. We therefore anticipate that the higher angular resolution and wider field of view of XMM-Newton and Swift/XRT data, possibly resolving the presence of more than 2 rings, will better determine the characteristics of the dust clouds visible at the time of the IXPE observation. Constraining the spectral parameters of the rings from independent observations might help constrain the IXPE polarization parameters. This will be explored in a follow-up paper.

The dust cloud distance D_dust is given by

$$\begin{eqnarray}&&{D}_{\mathrm{dust}}=2c\,\displaystyle \frac{{\rm{\Delta }}t}{{\theta }_{h}^{2}},\end{eqnarray} \tag{ 7 }$$

where c is the speed of light, Δt is the difference between the time of the burst T₀ and the time of the observation of the rings. Hence, from the fit of the distribution K_i defined in Equation (6), we can easily derive the distance of the clouds. In fact, the best-fit center of each Lorentzian is the inverse of the distance in kiloparsecs of the related dust clouds. We find the dust cloud associated with r ₁ to be at a distance of 14.41 kpc with a relative statistical error of 6%. Considering the Galactic latitude of the GRB, a cloud at such distance would be located at about 1.1 kpc above the Galactic plane. The second cloud, responsible for r ₂ emission, is estimated to be at a distance of 3.75 kpc with a relative statistical error of 1%. ⁷³ The variance of this measurement, given by the different assumed backgrounds, is negligible with respect to the statistical error, as shown in Figure 5 (left panel). The half width at half maximum of the two Lorentzian curves correspond to ∼(9.5 ± 1.1) kpc and ∼(0.7 ± 0.1) kpc for r ₁ and r ₂, respectively. Such values are larger than the values expected from the effect of the IXPE PSF ⁷⁴ by a factor of ∼3 (for r ₁) and ∼2 (for r ₂). This could be symptomatic of a nonnegligible thickness of the dust clouds, or, more likely, the presence of several rings within r ₁ and r ₂, which we do not resolve.

As mentioned in the previous section, the X-ray emission of the ring is the echo of the prompt GRB emission scattered by Galactic dust clouds. Therefore, assuming the characteristics of the dust to be known, it is possible to infer the intrinsic spectrum of the prompt soft X-ray emission of the GRB from the dust-scattered spectra. The relation between the scattered spectra and the intrinsic one reads as follows (Tiengo & Mereghetti 2006):

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{r}(E,{\theta }_{1},{\theta }_{2})={\rm{\Phi }}(E){\tau }_{d}(E)(g({\theta }_{1},E)-g({\theta }_{2},E))\,,\end{eqnarray} \tag{ 8 }$$

where the intrinsic spectrum is modeled with an absorbed power law decreasing in energy, θ₁ and θ₂ are the rings' extents at the beginning and at the end of the observations, and the functions

$$\begin{eqnarray}&&g(\theta ,E)=\frac{{\left(\theta /{\theta }_{s}(E)\right)}^{2}}{1+{\left(\theta /{\theta }_{s}(E)\right)}^{2}},<mml:mpadded>\,\mathrm{and}\,\,\,\,\,\,\,\,\,{\theta }_{s}(E)</mml:mpadded>=360^{\prime\prime} {\left(\frac{E}{\mathrm{keV}}\right)}^{-1}\end{eqnarray} \tag{ 9 }$$

account for both the fraction of halo we do not observe (because it lies outside the IXPE observing window) and the dependence of the scattering angle on the energy. ⁷⁵ In fact, larger (smaller) scattering angles correspond to a higher probability of scattering lower (higher) energy X-ray photons, which results in a steeper (harder) spectrum (Draine 2003b).

According to Draine & Bond (2004), the total scattering optical depth of the dust for photons between 0.8 and 10 keV can be estimated as

$$\begin{eqnarray}&&{\tau }_{d}={\tau }_{{d}_{1}}+{\tau }_{{d}_{2}}+{\tau }_{\mathrm{foreground}}=0.15{A}_{v}{\left(E/\mathrm{keV}\right)}^{-1.8}\end{eqnarray} \tag{ 10 }$$

with ${\tau }_{{d}_{1}}$ and ${\tau }_{{d}_{2}}$ being the optical depths associated with the two dust clouds that produce the echo rings r ₁ and r ₂, respectively, and τ_foreground is the total optical depth between us and the first dust cloud we detect. A_v is the V-band total Galactic extinction in magnitudes in the direction of the GRB. We assume A_v = 4.2 mag as reported in the circulars by the VLT and JWST groups (Izzo et al. 2022; Levan et al. 2022).

According to the measurements of Neckel & Klare (1980), the total A_v up to 3 kpc in the direction of GRB 221009A is about 3.3 mag. As the first cloud is at 3.75 kpc, this sum of three terms should be reasonably valid. Note that variations of the assumed value of A_v affect the normalization of the intrinsic GRB prompt spectrum, but do not affect the slope of the power law.

We perform a maximum likelihood analysis by simultaneously fitting both rings spectra with the model in Equation (8). As for the spectropolarimetric analysis, we have repeated the analysis three times for the different background models. Figure 5 (right panel) illustrates the results of this fit.

The fit is performed between 2 and 5 keV to avoid the low-count statistics part of the ring spectra at high energy. The best-fit estimate of the fraction of the total optical depth associated with the farther and closer dust clouds is n ∼ 0.36 and (1 − n) ∼ 0.64 respectively, and is not affected by the choice of the background. This translates into extinction values of ${A}_{v,{d}_{1}}\sim 0.33$ and ${A}_{v,{d}_{2}}\sim 0.57$.

As for the intrinsic parameters of the GRB, we find that the prompt GRB power law has a photon index between 0.62 and 1.25 with a relative statistical uncertainty of the order of 10%. This spectral index could be directly compared to the index inferred by the STIX observation (Xiao & Krucker 2022) and to the extrapolation of the lower end of the energy spectrum measured by Fermi/GBM. Konus-Wind, which detects photons down to 20 keV in energy, has released a preliminary analysis of the prompt emission of this burst (Frederiks et al. 2022). These authors find a time-averaged spectrum at the onset of the brightest phase of the event with a low-energy photon index of 1.09 ± 0.01. Such value lies within range defined by the best-fit values we find assuming different background models (see Figure 5, right panel). Once confirmed, the direct observations of the prompt emission spectrum can provide a potential way to determine the best IXPE background model to use. In fact, the most representative background model could be selected based on agreement of these IXPE-inferred values with an externally measured index value. At the time of this work, however, such information is not publicly available, so we leave these considerations to a future work. Considering the bracketing given by the intrinsic spectra derived assuming different backgrounds, we infer a total fluence in the 1–10 keV band of F = [1.6 − 6.1] × 10⁻⁴ erg cm⁻². The fluence is obtained from the integrated flux between 1 and 10 keV and multiplied by the IXPE total time of the observation, to account for the missing fluence due to Earth occultation time. The range of values we report for the fluence is based on the best-fit parameters of the intrinsic power-law model, ignoring statistical uncertainties, which are of the order of 10%–15%.

During the IXPE pointing, we also performed optical polarization observations in the R band at the Nordic Optical Telescope (Lindfors et al. 2022). The observations were obtained using the Alhambra Faint Object Spectrograph and Camera (ALFOSC) in the standard linear polarimetric mode that includes a λ/2 retarder followed by calcite. At the time of the observations (2022 October 12 at 20:15UT), the sky conditions were clear with 12 seeing. However, GRB221009A is located in a crowded Galactic field. This resulted in the extraordinary beam of a nearby bright star to overlap with the ordinary beam of the source. As such, the standard polarimetric analysis was not possible (Hovatta et al. 2016; Nilsson et al. 2018, e.g.). Instead, we performed careful modeling of the PSF. We used the second brightest star within the ALFOSC field of view to create a model of the PSF, which was then subtracted from each image separately. This process, however, can result in background artifacts. To mitigate the effect of any artifact, we used a small aperture of 1.5'' radius to perform the measurements using standard formulas.

We investigated the effect on polarization from reflection, scattering, and transmission considering the dominant dust compounds, Carbon, and silicates (see, e.g., Costantini & Corrales 2022, for a recent discussion of the topic). At the small angles that we observe, even assuming a coherent reflection angle, any polarization induced by reflection of X-rays would result in a negligible modulation of less than 10⁻⁵, or a PD ∼0.001%. These values were obtained using the Center for X-Ray Optics database and online tools. ⁷⁷ Polarization from transmission is expected to be negligible as well for X-rays of energies at the peak of IXPE sensitivity, given that the common dust compounds do not show K or L shell edges there. A fraction of the scattered light might have a polarization status affected by big spheroidal dust grains via Mie scattering. We checked this by using the Python package Miepython, ⁷⁸ which calculates light scattering according to the Mie theory and Rayleigh–Gans approximation, and adopting the X-ray refraction index for silicates provided by Draine & Lee (1984), Laor & Draine (1993). We find that, at the scattering angles we are considering, the PD due to refraction is less than 5.5 × 10⁻⁵ at 2 keV for a binary population of grains (e.g., perfectly aligned, elongated and not aligned, spherical grains) with a power-law size distribution with an index of −3.5 (Costantini & Corrales 2022). Therefore, we can reasonably assume that any polarization observed from the X-ray scattering halos is attributable to the original emission.

In this appendix we report additional plots in support of the analysis described in the main text. Figure 9 is the time-integrated IXPE counts map obtained combining the three DUs. Figure 10 shows the time evolution of the dust-scattering halos in three time bins. Figure 11 illustrates the results, reported in Table 4, of the PCUBE analysis on the individual rings. Figure 12 shows the I, Q, and U Stokes parameters spectra for core, r ₁, and r ₂ regions. Figure 13 complements the right panels of Figure 3 showing the analog results subtracting a different background. Figure 14 provides further information on the spectral properties of the rings emission and how they are related to the dust-scattering process. Figure 15 reports the U/I vs Q/I best fit distributions for the combined fit of the two rings, as well as for the individual rings. Lastly, Figure 16 shows the corner plots obtained by fitting the coreemission and the combined rings emission using a Bayesian approach.

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