The X-Ray Polarization View of Mrk 421 in an Average Flux State as Observed by the Imaging X-Ray Polarimetry Explorer

IXPE observed Mrk 421 from 10:00 UTC on 2022 May 4 to 11:10 UTC on May 6, with a live-time exposure of 97 ks, alongside a multiwavelength campaign. The latter included observations with the Swift X-Ray Telescope (XRT), XMM-Newton, and the Nuclear Spectroscopic Telescope Array (NuSTAR) at X-ray energies, the Nordic Optical Telescope (NOT) at the optical R band, Boston University's Perkins Telescope at IR wavelengths, and the Institut de Radioastronomie Millimétrique (IRAM) 30 m telescope at short millimeter wavelengths. The optical, IR, and radio flux densities and linear polarization measurements are summarized in Table 1. The data reduction procedure for all of the telescopes is described in Appendix. At the epoch of the IXPE observation, the 0.3–10 keV X-ray flux of Mrk 421 was ∼3.5 × 10⁻¹⁰ erg s⁻¹ cm⁻², which is near its typical flux state (Liodakis et al. 2019). As a comparison, during occasional flaring events reaching γ-ray energies, the X-ray flux of Mrk 421 can be 10 times higher (Donnarumma et al. 2009).

We have determined the 2–8 keV polarization of Mrk 421 using (1) the Kislat et al. (2015) method within the ixpeobssim simulation and analysis software (Baldini et al. 2022), (2) a spectropolarimetric fit of the IXPE data alone, (3) a spectropolarimetric fit of combined quasi-simultaneous data from IXPE, the XMM EPIC-pn camera, and NuSTAR, (4) a maximum-likelihood spectropolarimetric (MLS) fit of the IXPE data, implemented within the multinest algorithm, and (5) a maximum-likelihood calculation of the polarization properties based on the method of Marshall (2021). While the Kislat et al. (2015) approach and the maximum-likelihood calculation are model-independent, spectropolarimetric and MLS fits include modeling of the energy spectrum, allowing us to test whether the determination of the polarization properties is affected by the modeling of the spectral curvature.

For Mrk 421, we find a minimum detectable polarization at 99% significance (MDP99) of 6.4% in the 2–8 keV band. The measured polarization parameters from the three combined DUs are Π_x = 15% ± 2% and ψ_x = +35° ± 4°. Given the MDP99 level of the observation, the 2–8 keV polarization of Mrk 421 is detected at a confidence level of >99.99% (i.e., at a significance of 7σ). By computing the polarization properties in three energy bins, i.e., 2.0–3.0 keV, 3.0–4.0 keV, and 4.0–8.0 keV, we do not observe any significant variation of the polarization properties with the photon energy within maximum-error bars of ±5% and ±8° for the polarization degree and polarization angle, respectively.

We have also determined the degree Π_x and position angle ψ_x of polarization via a spectropolarimetric fit in Xspec (Arnaud et al. 1999), which implements the approach of Strohmayer (2017). In all of the fits, we included Galactic photoelectric absorption along the line of sight toward Mrk 421 (N_H = 1.34 × 10²⁰ cm⁻²; HI4PI Collaboration et al. 2016) using the TBABS model, with Fe abundances set according to Wilms et al. (2000). As a first test, we considered only the IXPE data (i.e., three Stokes parameters for the three DUs). In the fit we considered multiplicative constants, free to vary, to account for intercalibration differences among the three DUs. The shape of the I spectra is best fit ($\tfrac{{\chi }^{2}}{{\rm{d}}.{\rm{o}}.{\rm{f}}.}$ = 293/267) by a power law with a photon index Γ = 2.97 ± 0.01. By multiplying the model for the I spectra by a constant-polarization model (polconst model in Xspec), we obtained a statistically acceptable spectropolarimetric fit of the I, Q, and U spectra ($\tfrac{{\chi }^{2}}{{\rm{d}}.{\rm{o}}.{\rm{f}}.}$ = 841/795). The fluxes measured by the three IXPE DUs agree with each other within 12%, with the differences explained by uncertainties in the IXPE effective area at the time of observation. The estimated polarization degree and position angle, Π_x = 16% ± 2% and ψ_x = 34° ± 3°, are consistent with the ixpeobssim result.

Next, we used the simultaneous XMM-Newton and NuSTAR data to refine the constraints on the broadband spectral shape and check whether the derived polarization is sensitive to the addition of broadband spectral curvature in the model of the energy spectrum. We initially fit the data from XMM EPIC-pn, NuSTAR FPMA, and NuSTAR FPMB plus the three IXPE Stokes I spectra with a power-law model subject to photoelectric absorption (see above). We fixed the normalization and photon index Γ for all of the instruments, while we allowed five multiplicative constants to vary to account for cross-calibration between the instruments and source variability between the observations (see Table 1). This model does not provide a statistically acceptable fit to the broadband data ($\tfrac{{\chi }^{2}}{{\rm{d}}.{\rm{o}}.{\rm{f}}.}$ = 2764/611), which indicates that, when considering an energy range broader than that of IXPE, a more complex model of the spectral curvature is needed. The broadband spectrum of Mrk 421 is better fit by a log-parabolic model (Massaro et al. 2004), in which the photon index varies as a log-parabola in energy, i.e.,

$$\begin{eqnarray*}&&N{(E)=K(E/{E}_{{\rm{p}}})}^{(\alpha -\beta \mathrm{log}(E/{E}_{{\rm{p}}}))},\end{eqnarray*}$$

where the pivot energy E_p is a scaling factor, α describes the spectral slope at the pivot energy, β describes the spectral curvature, and K is a normalization constant. This spectral shape is typical for HSP blazars, including Mrk 421, both in quiescence and in flaring states (e.g., Baloković et al. 2016; Donnarumma et al. 2009). As the photon index varies with the energy in this model, the choice of reference energy E_p changes the determination of α. In our fits, we set the pivot energy E_p to 5.0 keV (e.g., Baloković et al. 2016), while the α and β parameters were allowed to vary, although they were coupled across the three telescopes. With our setup, α approximates the photon index in the 3.0–7.0 keV range (i.e., the core of the IXPE band).

Next, to determine the polarization properties from the polconst model, we added to the fit the IXPE Stokes Q and U spectra. This fitting exercise returned an estimation of the polarization parameters of Π_x = 16% ± 2% and ψ_x = 35° ± 3°, which are consistent with the ixpeobssim result and with the fit using only IXPE data. As a final check, we tested whether allowing the spectral shape to differ between NuSTAR, XMM EPIC-pn, and IXPE could further improve the fit and thereby affect the values of Π_x and ψ_x. Decoupling the α and β parameters among the three instruments resulted in a statistically significant improvement of the fit (Δχ² = −72 for 4 more degrees of freedom (d.o.f.), corresponding to a decrease in the Bayesian information criterion ΔBIC = −43). This discrepancy in spectral shape between the three telescopes is likely due to the nonsimultaneity of the observations as the source varies (see Section 2.2). We note, however, that this fitting exercise returned the same result, within the uncertainty, for the polarization properties: Π_x = 16% ± 2%, ψ_x = 34° ±4°. This indicates that the polarization determination is robust against differences in details of the spectral shape modeling. Our final spectropolarimetric fit, including XMM EPIC-pn, NuSTAR, and IXPE, is displayed in Figure 1. To highlight the significance of the polarization detection and the agreement among the determinations via different methods that use different models for the spectral curvature in Figure 2, we compared the two-dimensional confidence contours for the X-ray polarization obtained with ixpeobssim and with our spectropolarimetric fits.

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Finally, for the MLS method, we fit the IXPE data with a power-law model with Galactic photoelectric absorption (see above), while holding the intrinsic values of Q and U constant. In this way, we derived a photon index Γ = 2.97 ± 0.01 and polarization properties (Π_x = 16% ± 2% and ψ_x = 34° ± 2°) consistent with the values from the previously described methods. In addition, using a model-independent maximum-likelihood calculation, we obtained Π_x = 16% ± 2% and ψ_x =34° ± 2°. This is a further, independent confirmation of our polarization measurement of Mrk 421.

To obtain additional information about the physical conditions in the jet of Mrk 421, we have searched for variability of the X-ray polarization properties as a function of time within the total duration of the IXPE pointing. We first exploited the Swift-XRT monitoring of the source (nine separate pointings, each ∼1 ks) to sample the spectral evolution of the source before, during, and after the IXPE pointing. As a comparison, we created a light curve of the IXPE count rate with time bins of 10 ks (Figure 3). The flux of the source increased during the IXPE pointing in both the soft (0.3–2 keV) and hard (2–10 keV) XRT bands, while maintaining the same spectral slope, as witnessed by the hardness ratio light curve. In the IXPE data, we find an increase of the count rate by a factor of 2.2.

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To investigate whether there was also a variation in the polarization properties on different timescales within the IXPE pointing, we proceeded as follows. Using time bins in the range 5–50 ks, we created light curves of the normalized Stokes parameters. For each case, we fit the light curve with a constant model and recorded the probability of the null hypothesis of obtaining a value of χ² at least as large as that of the constant model. For all of the time bins tested, we find that this probability is always above 1%, which indicates that the Stokes parameters are consistent with no variation during the IXPE pointing. We therefore conclude that, despite the increase in flux during the observation, the polarization properties did not vary substantially. As a useful quantitative constraint, we computed the uncertainty in Π_x and ψ_x allowed by the statistic in each time bin. This is an indication of the amount of variation that can be excluded, on different timescales, by the present analysis. On a timescale of 50 ks, similar to the 1 day timescales sometimes used in studies of optical polarization variability (e.g., Blinov et al. 2015), variations in Π_x above ∼±4% for a constant value of ψ_x, or variations in ψ_x above ∼±7° for constant Π_x, are excluded by the present analysis.

IXPE data are processed using a pipeline ⁵⁴ that estimates the photoelectron emission direction, correcting for charging effects, detector temperature, gas electron multiplier gain nonuniformity, and polarization artifacts induced by spurious instrumental modulation (Rankin et al. 2022). The level-2 event files (one for each of the three DUs) produced by the pipeline contain all of the information typical of an imaging X-ray astronomy mission (e.g., photon arrival time, detector coordinates, and energy), with the addition of polarization information in the form of event-by-event Stokes parameters. Before proceeding with the science analysis, we corrected the data for small time-dependent changes to the gain correction obtained from data taken with the onboard calibration sources (Ferrazzoli et al. 2020) close to the actual time of observation.

We performed scientific analysis of the IXPE data using the ixpeobssim software (Pesce-Rollins et al. 2019; Baldini et al. 2022). At the angular resolution of IXPE (∼30''), blazars like Mrk 421 are pointlike sources. The selection of IXPE events was performed with the xpselect tool, which applies to the photon lists a user-defined selection on sky position, time, phase, and energy. In the case of Mrk 421, we extracted the source events using circular regions 60'' in radius, while we used an annulus centered on the source, with an inner and outer radius of 2' and 5', respectively, to estimate the background. We found that the background accounts for less than 2% of the total counts in the source region, and hence background subtraction is not critical for the polarization measurement. The Kislat et al. (2015) method for estimating the polarization of a user-defined set of events is implemented in the PCUBE algorithm. We created the Stokes parameter spectra of the source using the PHA1, PHAU, and PHAQ algorithms, which map the I, Q, and U Stokes parameters of the photons into OGIP-compliant pulse height analyzer (PHA) spectral files (three Stokes parameter spectra per three DUs). We prepared the spectra for the spectropolarimetric fit by binning the I spectrum, requiring that a minimum of 30 counts be reached in each energy bin. This is needed for correctly using the χ² statistics in the fit. Hence, as a last step, we applied the same binning to the Q and U spectra.

In order to catch any possible X-ray flux variations of the source contemporaneous with the IXPE pointing, and to accurately constrain the slope of the X-ray spectrum, we observed Mrk 421 in the 0.3–10 keV energy band with Swift-XRT (nine pointings, each with exposure time ∼1 ks, of which one was simultaneous with IXPE), and in the 3–70 keV energy range with NuSTAR (see Table 1). In addition, an XMM-Newton (5 ks) observation was carried out as a part of our program on 2022 May 3, 1 day before the IXPE pointing.

The Swift-XRT observations were performed in windowed timing (WT) mode, and the data were processed with the X-Ray Telescope Data Analysis Software (XRTDAS, v3.6.1). In the analysis, we used the latest calibration files available in the Swift-XRT CALDB (version 20210915). The X-ray source spectrum was extracted from the cleaned event file using a circular region with a radius of 47''. The background was extracted using a circular region with the same radius from a blank-sky WT observation available in the Swift archive. As a final step, we binned the 0.3–10 keV channels to achieve at least 25 counts in each energy bin.

We calibrated and cleaned the NuSTAR data using the NuSTAR Data Analysis Software (NuSTARDAS, v2.1.1) and the latest calibration data files in the NuSTAR CALDB (version 20220301). The net exposure time after cleaning the data set was 25 ks. For both the FPMA and FBMB detectors, we created level-2 cleaned event files with the nupipeline task, while high-level scientific data products were extracted using the nuproducts tool. We extracted the source spectrum from a circular region with a radius of 70'', while we used an annular region to estimate the background. Finally, we binned the spectra to achieve at least 50 counts in each energy bin.

The XMM-Newton observation of Mrk 421 was obtained in timing mode, which limits photon pileup. We performed the data reduction using the XMM-Newton Science Analysis Software (SAS), version 20, and the latest calibration products. Starting from the raw observation data files, we created the calibrated EPIC-pn event list using the SAS task epchain. To determine good time intervals free from background flares, we cut the time intervals where the light curve of the background in the 10–13 keV band was above a fixed threshold of 0.4 counts s⁻¹ (SAS task tabgtigen). In the clean event files, we selected the source and the background using boxed regions (with a width of 27 pixels) that maximize the signal-to-noise ratio. We created the source and background spectra using only single and double events, and we created the spectral response matrices using the SAS tasks rmfgen and arfgen. Finally, we prepared the spectrum for analysis by binning the energy channels to achieve at least 30 counts in each spectral bin. Model fitting to the spectrum was carried out with Xspec version 12.12.1, as described in the main text. To allow simultaneous spectropolarimetric fitting with EPIC-pn and NuSTAR, we added the keyword "XFLT0001 Stokes:0" to the header of the non-IXPE PHA files. This allowed Xspec to identify the spectra as I Stokes spectra.

Optical polarization observations were performed using ALFOSC mounted at the 2.5 m NOT on the night of 2022 May 4 (MJD 59,703) in the R band (λ = 6500 Å/E = 1.9 eV, FWHM = 1300 Å/0.4 eV). The data were obtained in the standard polarimetric mode and analyzed using a 25 aperture radius and standard photometric procedures included in the pipeline developed at Tuorla Observatory (Hovatta et al. 2016; Nilsson et al. 2018). Unpolarized and polarized standard stars were also used for calibration. The R magnitude of the source was 12.98 ± 0.02, which corresponds to I = 19.8 ± 0.3 mJy, while Π_O,obs = 2.42% ± 0.06% and ψ_O = 21 ± 1. For the 25 aperture used in the analysis, the host galaxy contributes I_host = 3.2 ± 0.4 mJy to the total emission (Nilsson et al. 2007). We corrected for host galaxy contamination of the polarization degree by subtracting the host galaxy flux density as follows: ${{\rm{\Pi }}}_{\mathrm{intr}}={{\rm{\Pi }}}_{\mathrm{obs}}\times I/(I-{I}_{\mathrm{host}})$ (Hovatta et al. 2016). The intrinsic polarization is therefore Π_O,int = 2.9% ± 0.1%.

IR observations in the H band were obtained at the Boston University 1.8 m Perkins telescope located in Flagstaff, Arizona (Perkins Telescope Observatory), using the IR camera Mimir ⁵⁵ on 2022 May 5, 6, and 9. Each observation consisted of six dithering exposures of 5 s each at 16 positions of a half-wave plate, rotated in steps of 225 from 0° to 360°. The camera and data reduction procedures are described in detail by Clemens et al. (2012). The H-band degree of polarization changed from 1% to 2.2%, with an average 1σ uncertainty of 0.5%. However, the Π measurements in the H band are lower limits, since the level of contamination from the host galaxy is unknown. The position angle of polarization in the H band agrees within the uncertainty with that of the IXPE observation.

Radio polarization at 3.5 mm (86.24 GHz) and 1.3 mm (230 GHz) was measured with the 30 m telescope of IRAM, located at the Pico Veleta Observatory (Sierra Nevada, Granada, Spain), on 2022 May 5 and 7 (MJD 59,704 and 59,706), within the POLAMI program (Agudo et al. 2018b, 2018a). Under the POLAMI ⁵⁶ observing setup, the four Stokes parameters (I, Q, U, and V) were recorded simultaneously using the XPOL procedure (Thum et al. 2008). We reduced, calibrated, and managed the data following the POLAMI procedure described in detail in Agudo et al. (2018b). In the two measurements at 3.5 mm, the source flux density increased from 0.278 ± 0.014 Jy to 0.384 ± 0.019 Jy, while at the same time the polarization properties remained relatively stable (see Table 1). Only one measurement was taken at 1.3 mm, on 2022 May 7, providing an upper limit at a 95% confidence level of 6% for the degree of polarization.