Three-dimensional morphological asymmetries in the ejecta of Cassiopeia A using a component separation method in X-rays
Adrien Picquenot, Fabio Acero, Tyler Holland-Ashford, Laura A. Lopez, Jérôme Bobin
AAstronomy & Astrophysics manuscript no. main © ESO 2021Wednesday 3 rd February, 2021
Three-dimensional morphological asymmetries in the ejecta ofCassiopeia A using a component separation method in X-rays
A. Picquenot , F. Acero , T. Holland-Ashford , , L. A. Lopez , , and J. Bobin AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, F-91191 Gif-sur-Yvette, France Department of Astronomy, The Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 W. Woodru ff Ave., Columbus, OH 43210, USAWednesday 3 rd February, 2021
ABSTRACT
Recent simulations have shown that asymmetries in the ejecta distribution of supernova remnants can still reflect asymmetries fromthe initial supernova explosion. Thus, their study provides a great means to test and constrain model predictions in relation to thedistributions of heavy elements or the neutron star kicks, both of which are key to better understanding the explosion mechanisms incore-collapse supernovae.The use of a novel blind source separation method applied to the megasecond X-ray observations of the well-known Cassiopeia Asupernova remnant has revealed maps of the distribution of the ejecta endowed with an unprecedented level of detail and clearlyseparated from continuum emission. Our method also provides a three-dimensional view of the ejecta by disentangling the red-and blue-shifted spectral components and associated images of the Si, S, Ar, Ca and Fe, providing insights into the morphologyof the ejecta distribution in Cassiopeia A. These mappings allow us to thoroughly investigate the asymmetries in the heavy elementsdistribution and probe simulation predictions about the neutron star kicks and the relative asymmetries between the di ff erent elements.We find in our study that most of the ejecta X-ray flux stems from the red-shifted component, suggesting an asymmetry in theexplosion. In addition, the red-shifted ejecta can physically be described as a broad, relatively symmetric plume, whereas the blue-shifted ejecta is more similar to a dense knot. The neutron star also moves directly opposite to the red-shifted parts of the ejectasimilar to what is seen with Ti. Regarding the morphological asymmetries, it appears that heavier elements have more asymmetricaldistributions, which confirms predictions made by simulations. This study is a showcase of the capacities of new analysis methods torevisit archival observations to fully exploit their scientific content.
Key words.
ISM: supernova remnants – ISM: individual objects: Cassiopeia A – ISM: lines and bands – ISM: kinematics anddynamics – ISM: structure
1. Introduction
Cassiopeia A (hereafter, Cas A) is among the most studied as-tronomical objects in X-rays and is arguably the best-studied su-pernova remnant (SNR). Investigation of the distribution of met-als on sub-parsec scales is possible because it is the youngestcore-collapse (CC) SNR in the Milky Way (about 340 years old;Thorstensen et al. 2001), its X-ray emission is dominated by theejecta metals (Hwang & Laming 2012), and it is relatively close(3 . Chandra ),making it an ideal laboratory to probe simulation predictions re-garding the distribution of ejecta metals.In the last few years, three-dimensional simulations of CCsupernovae (SNe) have begun to produce testable predictions ofSNe explosion and compact object properties in models usingthe neutrino-driven mechanism (see reviews by Janka et al. 2016;Müller 2016). In particular, explosion-generated ejecta asymme-tries (Wongwathanarat et al. 2013; Summa et al. 2018; Janka2017) and neutron star (NS) kick velocities (DeLaney & Sat-terfield 2013) appear to be key elements in CC SN simulationsthat Cas A’s data can constrain. Although it is challenging to dis-entangle the asymmetries produced by the surrounding mediumfrom those inherent to the explosion, Orlando et al. (2016) haveexplored the evolution of the asymmetries in Cas A using sim- ulations beginning from the immediate aftermath of the SN andincluding the three-dimensional interactions of the remnant withthe interstellar medium. Similar simulations presenting the evo-lution of a Type Ia SNR over a period spanning from one yearafter the explosion to several centuries afterward have been madeby Ferrand et al. (2019), showing that asymmetries present in theoriginal SN can still be observed after centuries. The same maygo for the CC SNR Cas A, and a better knowledge of its three-dimensional morphology could lead to a better understanding ofthe explosion mechanisms by providing a way to test the simu-lations.An accurate mapping of the di ff erent elements’ distributions,the quantification of their relative asymmetries, and their rela-tion to the NS motion would, for example, allow us to probethe simulation predictions that heavier elements are ejected moreasymmetrically and more directly opposed to the NS motion thanlighter elements (Wongwathanarat et al. 2013; Janka 2017; Gess-ner & Janka 2018; Müller et al. 2019). On this topic, this papercan be viewed as a follow-up to Holland-Ashford et al. (2020), astudy that aimed to quantitatively compare the relative asymme-tries of di ff erent elements within Cas A, but which was hindereddue to di ffi culties in separating and limiting contamination in theelements’ distribution. Moreover, in that analysis, the separation Article number, page 1 of 14 a r X i v : . [ a s t r o - ph . H E ] F e b & A proofs: manuscript no. main of the blue- and red-shifted parts in these distributions was notpossible.Here, we intend to fix these issues by using a new methodto retrieve accurate maps for each element’s distribution, allow-ing us to further investigate their individual and relative physicalproperties. This method is based on the General MorphologicalComponents Analysis (GMCA, see Bobin et al. 2015), a blindsource separation (BSS) algorithm that was introduced for X-ray observations by Picquenot et al. (2019). It can disentangleboth spectrally and spatially mixed components from an X-raydata cube of the form ( x , y , E ) with a precision unprecedented inthis field. The new images thus obtained su ff er less contamina-tion by other components, including the synchrotron emission. Italso o ff ers the opportunity to separate the blue- and red-shiftedparts of the elements’ distribution, thereby facilitating a three-dimensional mapping of the X-ray emitting metals and a com-parison of their relative asymmetries. Specifically, the GMCAis able to disentangle detailed maps of a red- and a blue-shiftedparts in the distributions of Si, S, Ca, Ar, and Fe, thus providingnew and crucial information about the three-dimensional mor-phology of Cas A. This is a step forward as previous studiesintending to map the distribution of the individual elements andstudy their asymmetries in Cas A in X-rays (Hwang & Laming2012; Katsuda et al. 2018; Holland-Ashford et al. 2020) werenot able to separate red- and blue-shifted components.This paper is structured as follows. In Section 2, we will de-scribe the nature of the data we use (Section 2.1), our extractionmethod (Section 2.2), our way to quantify the asymmetries (Sec-tion 2.3), and our method to retrieve error bars (Section 2.4). InSection 2, we will present the images resulting from the appli-cation of our extraction method (Sections 3.1 and 3.2), and wewill discuss the interpretation of the retrieved images as blue- orred-shifted by looking at their associated spectra (Section 3.3),and will present the results of a spectral analysis on these samespectra (Section 3.4). Lastly, we will discuss in Section 4 thephysical information we can infer from our results. Section 4.1will be dedicated to the interpretation of the spatial asymmetriesof each line emission, while Sections 4.2 and 4.4 will focus re-spectively on the mean direction of each line’s emission and onthe NS velocity. A comparison with the NuSTAR data of Ti willfinally be presented in Section 4.5.
2. Method
Spectro-imaging instruments, such as those aboard the currentgeneration of X-ray satellites
XMM-Newton and
Chandra , pro-vide data comprising spatial and spectral information: The de-tectors record the position ( x , y ) and energy E event by event,thereby producing a data cube with two spatial dimensions andone spectral dimension. For our study, we used Chandra obser-vations of the Cas A SNR, which was observed with the ACIS-Sinstrument in 2004 for a total of 980 ks (Hwang et al. 2004).We used only the 2004 data set to avoid the need to correct forproper motion across epochs. The event lists from all observa-tions were merged in a single data cube. The spatial (of 2 (cid:48)(cid:48) ) andspectral binning (of 14.6 eV) were adapted so as to obtain a suf-ficient number of counts in each cube element. No backgroundsubtraction or vignetting correction has been applied to the data.
The main concept of GMCA is to take into account the morpho-logical particularities of each component in the wavelet domainto disentangle them, without any prior instrumental or physicalinformation. Apart from the ( x , y , E ) data cube, the only inputneeded is the number n of components to retrieve, which is user-defined. The outputs are then a set of n images associated with n spectra. Each couple image-spectrum represents a component:The algorithm makes the assumption that every component canbe described as the product of an image with a spectrum. Thus,the retrieved components are approximations of the actual com-ponents with the same spectrum on each point of the image.Nevertheless, Picquenot et al. (2019) showed that when testedon Cas A-like toy models, the GMCA was able to extract mor-phologically and spectrally accurate results. The tested spectraltoy models included power-laws, thermal plasmas, and Gaussianlines. In particular, in one of these toy models, the method wasable to separate three components: two nearby partially overlap-ping Gaussian emission lines and power-law emission. The en-ergy centroids of both Gaussians were accurately retrieved, de-spite their closeness. Such a disentangling of mixed componentswith similar neighboring spectra cannot be obtained throughline-interpolation, and fitting of a two-Gaussian model region byregion is often time consuming, producing images contaminatedby other components with unstable fitting results.In the same paper, the first applications on real data of Cas Awere promising, in particular concerning asymmetries in the el-ements’ distribution. For Si, S, Ar, Fe and Ca, the GMCA wasable to retrieve two maps associated with spectra slightly blue-or red-shifted from their theoretical position. The existence ofblue- or red-shifted parts in these elements’ distribution was pre-viously known, and the Fe maps from Picquenot et al. (2019)were consistent with prior works but endowed with more details(see Willingale et al. 2002a; DeLaney et al. 2010). Thus, theyconstitute a great basis for an extensive study of the asymme-tries in the elements’ distribution in Cas A.In this paper, we will use a more recent version of theGMCA, the pGMCA, that was developed to take into accountdata of a Poissonian nature (Bobin et al. 2020). In the precedentversion of the algorithm, the noise was supposed to be Gaus-sian. Even with that biased assumption, the results were provento be reliable. However, a proper treatment of the noise is stillrelevant: It increases the consistency of the spectral morpholo-gies of the retrieved components and makes the algorithm ableto disentangle components with a fainter contrast.The mathematical formalism is similar to that of the GMCA,presented in Picquenot et al. (2019). The fundamental di ff erenceis that instead of a linear representation, the pGMCA uses thenotion of a Poisson-likelihood of a given sum of components tobe the origin of a certain observation. The problem solved bythe algorithm is thus essentially the same kind, the main di ff er-ence being a change in the nature of the norm that needs to beminimized. A more precise description of this new method isavailable in Bobin et al. (2020).The use of the pGMCA is also highly similar to that of theGMCA. One notable di ff erence is that the pGMCA is more sen-sitive to the initial conditions, so it needs a first guess for conver-gence purposes. The analysis therefore consists of two steps: afirst guess obtained with the GMCA and a refinement step usingthe Poissonian version pGMCA.The aforementioned workflow was applied to the Cas A Chandra observations by creating data cubes for each energyband shown in Fig. 1. These energy bands were chosen to be
Article number, page 2 of 14icquenot et al.: Morphological asymmetries in the ejecta of Cassiopeia A in X-rays
Energy (keV) C o un t s Fig. 1: Spectrum of Cas A obtained from the combination of thedeep
Chandra n was 3: the synchrotronemission and the blue- and red-shifted parts of the line emis-sion. We then tested using 4 and 5 components to ensure extracomponents were not merged into our components of interest.We also tested with 2 components to verify our assumption onthe presence of blue- and red-shifted parts was not imposing theapparition of a spurious component. For each emission line, wethen chose n as the best candidate to retrieve the most seeminglymeaningful components without spurious images.For each analysis, the algorithm was able to retrieve a com-ponent that we identify as the synchrotron emission (a power-lawspectrum and filamentary spatial distribution, not shown here)and multiple additional thermal components with strong linefeatures. We were able to identify two associated images withshifted spectra from the theoretical emission line energy for allthese line features except O, Fe L, and Mg. We use the power-ratio method (PRM) to quantitatively ana-lyze and compare the asymmetries of the images extracted bypGMCA. This method was developed by Buote & Tsai (1995)and previously employed for use on SNRs (Lopez et al. 2009a,b,2011). It consists of calculating multipole moments in a circularaperture positioned on the centroid of the image, with a radiusthat encloses the whole SNR. Powers of the multipole expansion P m are then obtained by integrating the m th term over the circle.To normalize the powers with respect to flux, they are dividedby P , thus forming the power ratios P m / P . For a more detaileddescription of the method, see Lopez et al. (2009b).The P / P and P / P terms convey complementary infor-mation about the asymmetries in an image. The first term is thequadrupole power-ratio and quantifies the ellipticity / elongationof an extended source, while the second term is the octupolepower-ratio and is a measure of mirror asymmetry. Hence, both are to be compared simultaneously to ascertain the asymmetriesin di ff erent images.Here, as we want to compare asymmetries in the blue- andred-shifted part of the elements’ distribution, the method isslightly modified. In a first step, we calculate the P / P and P / P ratios of each element’s total distribution by using the sumof the blue- and red-shifted maps as an image. Its centroid is thenan approximation of the center-of-emission of the consideredelement. Then, we calculate the power ratios of the blue- andred-shifted images separately using the same center-of-emission.Ultimately, we normalize the power ratios thus obtained by thepower ratios of the total element’s distribution: P i / P (shifted / total) = P i / P (red or blue image) P i / P (total image) , (1)where i = P i / P (red or blue image) is calculated us-ing the centroid of the total image. That way, we can compare therelative asymmetries of the blue- and red-shifted parts of di ff er-ent elements, without the comparison being biased by the origi-nal asymmetries of the whole distribution. As explained in Picquenot et al. (2019), error bars can be ob-tained by applying this method on every image retrieved by theGMCA applied on a block bootstrap resampling. However, aswas shown in that paper, this method introduces a bias in theresults of the GMCA. We show in Appendix B that the blockbootstrap method modifies the Poissonian nature of the data,thus impacting the results of the algorithm. Since the pGMCAis more dependent than GMCA on the initial conditions, the biasin the outputs is even greater with this newer version of the al-gorithm (see Fig. B.3). For that reason, we developed a new re-sampling method we named "constrained bootstrap," presentedin Appendix B.4.Thus, we applied pGMCA on a hundred resampled datacubes obtained thanks to the constrained bootstrap and plottedthe di ff erent spectra we retrieved around the ones obtained onreal data. As stated in Appendix B.4, the spread between the re-samplings has no physical significance but helps in evaluatingthe robustness of the algorithm around a given set of originalconditions. The blue-shifted part of the Ca line emission, a veryweak component, was not retrieved for every resampling. In thiscase, we created more resamplings in order to obtain a hundredcorrectly retrieved components. The faintest components are theones with the largest relative error bars, as can be seen in Fig. 3and Fig. 4, highlighting the di ffi culty for the algorithm to re-trieve them in a consistent way on a hundred slightly di ff erentresamplings.To obtain the error bars for the PRM plot of the asymme-tries, we applied the PRM to the hundred images retrieved by thepGMCA on the resamplings. Then, in each direction we plottederror bars representing the interval between the 10 th and the 90 th percentile and crossing at the median. We also plotted the PRMapplied on real data. Although our new constrained bootstrapmethod ensures the Poissonian nature of the data to be preservedin the resampled data sets, we see that the results of the pGMCAon real data are sometimes not in the 10 th -90 th percentile zone,thus suggesting there may still be some biases. It happens mostlywith the weakest components, showing once more the di ffi cultyfor the pGMCA to retrieve them consistently out of di ff erent Article number, page 3 of 14 & A proofs: manuscript no. main data sets presenting slightly di ff erent initial conditions. How-ever, even when the results on real data are not exactly in the10 th -90 th percentile zone, the adequation between the results onreal and resampled data sets is still good, and the relative posi-tioning for each line is the same, whether we consider the resultson the original data or on the resampled data sets.
3. Results
By applying the pGMCA algorithm on the energy bands sur-rounding the eight emission lines shown in Fig. 1, we were ableto retrieve maps of their spatial distribution associated with spec-tra, successfully disentangling them from the synchrotron emis-sion or other unwanted components. The O, Mg, and Fe L lineswere only retrieved as single features, each associated with aspectrum, whereas Si, S, Ar, Ca, and Fe-K were retrieved as twodi ff erent images associated with spectra that we interpret as be-ing the same emission lines slightly red- or blue-shifted. Fig. 2shows the total images for all eight line emissions, obtainedby summing the blue- and red-shifted parts when necessary. Italso indicates the centroid of each image that is adopted in thePRM. Fig. 3 shows the red- and blue-shifted parts of five lineemissions, together with their associated spectra, while Fig. 4presents the images of O, Mg, and Fe L together with their re-spective spectra. Fig. 2 is similar to Fig. 10 from Picquenot et al.(2019), but the images here are more accurate and less contam-inated by other components thanks to a proper treatment of thePoisson noise, and the associated spectra not shown in our firstpaper are presented here in Fig. 3 and Fig. 4. The fact that our algorithm fails to separate a blue-shifted from ared-shifted part in the O, Mg, and Fe L images is not surprising.At 1 keV, we infer that a radial speed of 4000 km s − would leadto a ∆ E of about 13 eV, which is below the spectral bin size ofour data. We see in Fig. 2 that while the O and the Mg imagesare highly similar, they are both noticeably di ff erent from theimages of the other line emissions. Both the O and Mg imagesexhibit similar morphology to the optical images of O ii and O iii from Hubble (Fesen et al. 2001; Patnaude & Fesen 2014). Theintermediate mass elements share interesting properties: Theirspatial distributions appears similar in Fig. 2, and the divisioninto a red- and a blue-shifted part (as found by the pGMCA)allows us to investigate their three-dimensional morphology. Wealso notice that the maps of Si and Ar are similar to that of theAr ii in infrared (DeLaney et al. 2010).As the reverse shock has not fully propagated to the inte-rior of Cas A (Gotthelf et al. 2001; DeLaney et al. 2010), ourimages may not reflect the full distribution of the ejecta. How-ever, Hwang & Laming (2012) estimates that most of the ejectamass has already been shocked: we can thus conclude that ourimages capture the bulk of the ejecta and our element images arelikely similar to the true ejecta distributions. In addition, our Fered-shifted image matches well with the Ti image, producedby radioactive decay instead of reverse-shock heating (see Sec-tion 4.5).Hence, we can quantify the asymmetries in the ejecta dis-tribution by using the PRM method described in Sect. 2.3 onour images. Fig. 5 presents the quadrupole power-ratios P / P versus the octupole power-ratios P / P of the total images fromFig. 2. Fig. 6 shows the quadrupole power-ratios versus the oc- SiFe KMg Fe LArCaSO
Fig. 2: Total images of the di ff erent line emission spatial struc-ture as retrieved by the pGMCA. The blue symbol represents theimage centroid adopted in the PRM analysis. The color-scale isin square root.tupole power-ratios of the red- and blue-shifted images presentedin Fig. 6 normalized with the quadrupole and octupole power-ratios of the total images (Fig. 2) as defined in Eq. 2.3. As stated before, it is the spectra retrieved together with theaforementioned images that allow us to identify them as "blue-"or "red-shifted" components. Here we will expand on our rea-sons for supporting these assertions.The spectra in Fig. 3 are superimposed with the theoreticalpositions of the main emission lines in the energy range. In thecase of Si, the retrieved features are shifted to lower or higherenergy with respect to the rest-energy positions of the Si xiii andSi xiv lines. Appendix A shows that this shifting is not primarilydue to an ionization e ff ect as the ratio Si xiii / Si xiv is roughlyequal in both cases. The same goes for S, where two lines corre-sponding to S xv and S xvi are shifted together while keeping asimilar ratio.A word on the Ca blue-shifted emission: This component isvery weak and in a region where there is a lot of spatial overlap,making it di ffi cult for the algorithm to retrieve. For that reason,the retrieved spectrum has a poorer quality than the others, and Article number, page 4 of 14icquenot et al.: Morphological asymmetries in the ejecta of Cassiopeia A in X-rays
Red-shift Blue-shiftSiSArCaFe K
Fig. 3: Red- and blue-shifted parts of the Si, S, Ar, Ca, and Fe line emission spatial distribution and their associated spectrum asfound by pGMCA. The spectra in red correspond to the application of the algorithm on real data, while the dotted gray spectracorrespond to the application on a hundred constrained bootstrap resamplings illustrating statistical uncertainties. The x-axis is inkeV and the y-axis in counts. The dotted vertical lines represent the energy of the brightest emission lines for a non-equilibriumionization plasma at a temperature of 1.5 keV and ionization timescale of log ( τ ) = . − s produced using the AtomDB (Fosteret al. 2012). These parameters are the mean value of the distribution shown in Fig.2 of Hwang & Laming (2012).it was imperfectly found on some of our constrained bootstrapresamplings. Consequently, we were compelled to run the algo-rithm on more than a hundred resamplings and to select the ac-curate ones to obtain a significant envelop around the spectrumobtained on the original data. Using the spectral components retrieved for each data subsetshown in Fig. 3, we carried out a spectral fitting assuming aresidual continuum plus line emission in
XSPEC (power-law + Article number, page 5 of 14 & A proofs: manuscript no. main
OMgFe L
Fig. 4: Images of the O, Mg, and Fe L line emission spatialstructures and their associated spectra as found by pGMCA. Thespectra in red correspond to the application of the algorithm onreal data, while the dotted gray spectra correspond to the applica-tion on a hundred constrained bootstrap resamplings. The x-axisis in keV and the y-axis in counts.Line E rest E red E blue ∆ V V red V blue keV keV keV km / s km / s km / sSi xiii xiii ∗ xv xvii xix ∗ uses a di ff erent rest energy, the one neededto match the ACIS and HETG Si velocities discussed in DeLaneyet al. (2010), to illustrate possible ACIS calibration issues.Gauss model). In this analysis, the errors for each spectral datapoint are derived from the constraint bootstrap method presentedin Appendix B. This constrained bootstrap eliminates a bias in-troduced by classical bootstrap methods and that is critical topGMCA, but underestimates the true statistical error. Therefore,no statistical errors on the line centroids are listed in Table 1 as,in addition, systematic errors associated with ACIS energy cali-bration are likely to be the dominant source of uncertainty.The resulting line centroid and equivalent velocity shifts areshown in Table 1. To transform the shift in energy into a velocityshift, a rest energy is needed. The ACIS CCD spectral resolu- Fe KCaO Fe L ArSMgSi
Mirror Asymmetry E lli p t i ca l A sy mm e t r y Fig. 5: Quadrupole power-ratios P / P versus the octupolepower-ratios P / P of the total images of the di ff erent line emis-sions shown in Fig. 2. The dots represent the values measured forthe pGMCA images obtained from the real data, and the crossesthe 10 th and 90 th percentiles obtained with pGMCA on a hun-dred constrained bootstrap resamplings, with the center of thecross being the median. SiSArCa Fe KSiSAr CaFe K
Mirror Asymmetry E lli p t i ca l A sy mm e t r y Fig. 6: Quadrupole power-ratios P / P versus the octupolepower-ratios P / P of the red- and blue-shifted images of thedi ff erent line emissions shown in Fig. 3, normalized with thequadrupole and octupole power-ratios of the total images. Thedots and error bars are obtained in the same way as in Fig. 5tion does not resolve the line complex and cannot easily disen-tangle velocity and ionization e ff ects. However, given the rangeof ionization state observed in Cas A (with ionization ages of tau ∼ − cm − s, see Fig. 2 of Hwang & Laming 2012),there is little e ff ect of ionization on the dominant line for Si, S,Ar, and Ca, as discussed in more details in Appendix A. The linerest energy was chosen as the brightest line for a non-equilibriumionization plasma with a temperature of 1.5 keV temperature and log ( τ ) = . − s, the mean values from Fig. 2 of Hwang &Laming (2012).For the specific case of the Si xiii line, a very large asym-metry in the red / blue-shifted velocities is observed. This couldbe due to possible energy calibration issues near the Si line asshown by DeLaney et al. (2010) in a comparison of ACIS andHETG line centroid, resulting in a systematic blue-shift e ff ect Article number, page 6 of 14icquenot et al.: Morphological asymmetries in the ejecta of Cassiopeia A in X-rays
Si S ArCaFe K Si S ArCa Fe KCenter of explosion 20 arcsecNeutron star Ti Fig. 7: Centroids of the blue- and red-shifted parts of each lineemission and their distance from the center of explosion ofCas A. For reference, we added the direction of motion of the Ti in black, as shown in Fig. 13 of Grefenstette et al. (2017).Only the direction is relevant as the norm of this specific vectoris arbitrary.in ACIS data. The Si xiii ∗ line in Table 1 uses a corrected restline energy to illustrate systematic uncertainties associated withcalibration issues.For the Fe-K complex of lines, we rely on the analysis ofDeLaney et al. (2010) who derived an average rest line energyof 6.6605 keV (1.8615 Å) by fitting a spherical expansion modelto their three-dimensional ejecta model. We can note that withthis spectral analysis, what we measure here is the radial veloc-ity that is flux weighted over the entire image of the associatedcomponent. Therefore, we are not probing the velocity at smallangular scale but the bulk velocity of the entire component.With the caveats listed above, we notice an asymmetry in thevelocities where ejecta seem to have a higher velocity toward us(blue-shifted) than away from us, even in the case of Si xiii aftercalibration corrections. A comparison of those results with pre-vious studies and possibles biases are discussed in Sect. 4.3. Thelarge uncertainties associated with the energy calibration and thechoice of rest energy has little impact on the delta between thered- and blue shifted centroids and hence on the ∆ V. We notethat all elements show a consistent ∆ V of ∼ − .
4. Physical interpretation
Fig. 5 shows that the distribution of heavier elements is generallymore elliptical and more mirror asymmetric than that of lighterelements in Cas A: O, Si, S, Ar, Ca, and Fe emission all exhibitsuccessively higher levels of both measures of asymmetry. Thisresult is consistent with the recent observational study of Cas Aby Holland-Ashford et al. (2020), suggesting that the pGMCAmethod accurately extracts information from X-ray data cubeswithout the complicated and time-consuming step of extractingspectra from hundreds or thousands of small regions and analyz-ing them individually.Similar to the results of Holland-Ashford et al. (2020) andHwang & Laming (2012), Mg emission does not follow the ex-act same trend as the other elements : it has roughly an order ofmagnitude lower elliptical asymmetry ( P / P ) than the other el- ements. In contrast to Holland-Ashford et al. (2020) and Hwang& Laming (2012), our Mg image (as shown in Fig. 4) presentsa morphology highly di ff erent from that of the Fe L; we believethat the pGMCA was able to retrieve the Mg spatial distributionwith little continuum or Fe contamination.Fig. 6 presents the relative ellipticity / elongation and mirrorasymmetries of the blue- and red-shifted ejecta emission com-pared to the total ejecta images (Fig. 2). A value of “1” in-dicates that the velocity-shifted ejecta has equivalent levels ofasymmetry as the full bandpass emission. In the cases wherewe can clearly disentangle the red- and blue-shifted emission(i.e.Si, S, Ar, Ca, and Fe-K, described in previous paragraphs),we see that the red-shifted ejecta emission is less asymmetricthan the blue-shifted emission. This holds true both for ellipticalasymmetry P / P and mirror asymmetry P / P . Thus, we couldphysically describe the red-shifted ejecta distribution as a broad,relatively symmetric plume, whereas the blue-shifted ejecta isconcentrated into dense knots. This interpretation matches withthe observation that most of the X-ray emission is from the red-shifted ejecta, as we can also see in the flux ratios shown in Ta-ble 2 and in the images of Fig. 3, suggesting that there was moremass ejected away from the observer, NS, and blue-shifted ejectaknot. We note that there is not a direct correlation between ejectamass and X-ray emission due to the position of the reverse shock,the plasma temperature and ionization timescale, but the indica-tion that most of the X-ray emission is red-shifted is consistentwith our knowledge of the Ti distribution (see Sect. 4.5 for amore detailed discussion).Furthermore, in all cases, the red-shifted ejecta emission ismore circularly symmetric than the total images, and the blue-shifted ejecta is more elliptical and elongated than the total im-ages. Moreover, the red-shifted ejecta is more mirror symmet-ric than the blue-shifted ejecta, though both the red-shifted andblue-shifted Si are more mirror asymmetric than the total image.The latter result may suggest that the red-shifted and blue-shiftedSi images’ asymmetries sum together such that the total Si im-age appears more mirror symmetric than the actual distributionof the Si.
Fig. 7 shows the centroids of the blue- and red-shifted parts ofeach emission line relative to the center-of-explosion of Cas A,revealing the bulk three-dimensional distribution of each compo-nent. We note that this figure was only made using the centroidsof the red- and blue-shifted images retrieved by pGMCA, with-out using the PRM method. We can see the red-shifted ejectais mainly moving in a similar direction (toward the northwest),while all the blue-shifted ejecta is moving toward the east. Asdiscussed in Section 4.5, this result is consistent with previousworks on Cas A investigating the Ti distribution with
NuSTAR data (Grefenstette et al. 2017).The blue-shifted ejecta is clearly moving in a di ff erent direc-tion than the red-shifted ejecta, but not directly opposite to it.The angles between the blue- and red-shifted components are allbetween 90 ◦ and 140 ◦ . This finding provides evidence against ajet and counter-jet explosion mechanism being responsible forthe explosion and resulting in the expansion of ejecta in Cas A(e.g., Fesen 2001; Hines et al. 2004; Schure et al. 2008). Wecan also note a trend where heavier elements exhibit increas-ingly larger opening angles than lighter elements, from Si show-ing a 90 ◦ angle to Ca and Fe that show opening angles of about130 − ◦ . Article number, page 7 of 14 & A proofs: manuscript no. main
By fitting the line centroids, we obtained the velocities discussedin Sect. 3.4. As stated before, the e ff ects of ionization on possible"imposter velocities" are discussed in Appendix A. Our derivedvelocities showed higher values for the blue-shifted componentthan for the red-shifted one for all elements. Those results are indisagreement with spectroscopic studies and in agreement withsome others. On the one hand, the X-ray studies of individualregions Willingale et al. (2002b) (Fig. 8, XMM-Newton EPICcameras) and DeLaney et al. (2010) (Fig. 10 and 11, ChandraACIS and HETG instruments) indicate higher velocities for thered-shifted component. But on the other hand, the highest veloc-ity measured in the Ti NuSTAR analysis is for the blue-shiftedcomponent (Table 3 of Grefenstette et al. 2017). We note that thecomparison is not straightforward as the methods being used aredi ff erent. Our method measures a flux weighted average velocityfor each well separated component whereas in the X-ray studiespreviously mentioned, a single gaussian model is fitted to thespectrum extracted in each small-scale region. In regions whereboth red- and blue-shifted ejecta co-exist (see Fig. 3), the Gaus-sian fit will provide a flux weighted average velocity value of thetwo components as they are not resolved with ACIS. As the red-shifted component is brighter in average, a systematic bias thatwould reduce the blue velocities could exist. This could be thecase in the southeastern region where most of the blue-shiftedemission is observed and where a significant level of red-shiftedemission is also seen. Besides this, calibration issues may alsoplay an important role. Although the GMCA method was suc-cessful in retrieving the centroid energy of nearby emission linesusing a simple toy model (Picquenot et al. 2019, Fig. 7 in ), wedo not rule out that the higher velocity of the blue-shifted com-ponent is an artifact of the method. Further tests of the methodwith the help of synthetic X-ray observations using numericalsimulations could shed light on this issue. The NS in Cas A is located southeast of the explosion site, mov-ing at a velocity of ∼
340 km s − southeast in the plane of thesky (Thorstensen et al. 2001; Fesen et al. 2006). In Hwang &Laming (2012) it was stated that, contrary to expectations, theFe structure was not observed to recoil in the opposite direc-tion to the NS. Here, thanks to our ability to disentangle red-and blue-shifted structures, we find that the red-shifted ejectais moving nearly opposite the NS : The angles between thered-shifted structures and the NS tangential motion range be-tween 154 ◦ (Fe) and 180 ◦ (Si). Table. 2 also shows that the bulkemission is from red-shifted ejecta (consistent with Milisavlje-vic & Fesen 2013). This correlation is consistent with theoreti-cal predictions that NSs are kicked opposite to the direction ofbulk ejecta motion, in adequation with conservation of momen-tum with the ejecta (Wongwathanarat et al. 2013; Müller 2016;Bruenn et al. 2016; Janka 2017). Specifically, observations haveprovided evidence for the "gravitational tugboat mechanism" ofgenerating NS kicks asymmetries proposed by Wongwathanaratet al. (2013); Janka (2017), where the NS is gravitationally ac-celerated by the slower moving ejecta clumps, opposite to thebulk ejecta motion.It is impossible to calculate the NS line-of-sight motion byexamining the NS alone as its spectra contains no lines to beDoppler-shifted. However, limits on its three-dimensional mo-tion can be placed by assuming it moves opposite the bulkof ejecta and examining the bulk three-dimensional motion of Fig. 8: Counts image of the Fe-K red-shifted component over-laid with the extraction regions used for the Ti NuSTAR studyof Grefenstette et al. (2017). The regions 19 and 20, which dom-inate our image in terms of flux, have respective velocities mov-ing away from the observer of 2300 ± ±
500 kmsec − . Red-shifted part Blue-shifted partSi 0.60 0.40S 0.61 0.39Ar 0.63 0.37Ca 0.80 0.20Fe-K 0.70 0.30Table 2: Fractions of the counts in the total image that belong tothe red-shifted or the blue-shifted parts, for each line.ejecta. Grefenstette et al. (2017) studied Ti emission in Cas Aand found that the bulk Ti emission was tilted 58 ◦ into the planeof the sky away from the observer, implying that the NS is mov-ing 58 ◦ out of the plane of the sky toward the observer. Thisfinding is supported by three-dimensional simulations of a TypeIIb progenitor by Wongwathanarat et al. (2017) and Jerkstrandet al. (2020), which suggested that the NS is moving out of theplane of the sky with an angle of ∼ ◦ .The results of this paper support the hypothesis that, if theNS is moving away from the bulk of ejecta motion, the NS ismoving toward us. Furthermore, we could tentatively concludethat the NS was accelerated toward the more slowly movingblue-shifted ejecta, which would further support the gravitationaltugboat mechanism. The strong levels of asymmetry exhibitedby the blue-shifted emission combined with the lower flux wouldimply that the blue-shifted ejecta is split into relatively smallejecta clumps, one of which would possibly be the source ofthe neutron star’s gravitational acceleration. However, the ve-locities determined in Table 1 contradict this hypothesis as theblue-shifted clumps seem to move faster. Ti Ti is a product of Si burning and is thought to be synthesizedin close proximity with iron. The Ti spatial distribution hasbeen studied via its radioactive decay with the
NuSTAR tele-scope and revealed that most of the material is red-shifted and
Article number, page 8 of 14icquenot et al.: Morphological asymmetries in the ejecta of Cassiopeia A in X-rays does not seem to follow the Fe-K X-ray emission (Grefenstetteet al. 2014, 2017). In our study, we have found that 70% of theFe-K X-ray emission (see Table 2) is red-shifted and that themean direction of the Fe-K red-shifted emission shown in Fig. 7is compatible with that of the Ti as determined in Fig. 13 ofGrefenstette et al. (2017). Yet, we can see the mean Ti direc-tion is not perfectly aligned with the mean red-shifted Fe-K di-rection. This may be caused by the fact that the Fe-K emissionis tracing only the reverse shock-heated material and may notreflect the true distribution of Fe, whereas Ti emission is fromradioactive decay and thus reflects the true distribution of Ti.In Fig. 8, we overlay the ten regions where Grefenstette et al.(2017) detected Ti with our red-shifted component image. Theregions 19 and 20 (which dominate our Fe-K red-shifted compo-nent image) have respective Ti velocities of 2300 ± ±
500 km s − , values that are compatible with our mea-sured value of ∼ − shown in Table 1.Concerning our Fe-K blue-shifted component map, its X-rayemission is fainter and located mostly in the southeast of thesource (see Fig 3). This southeastern X-ray emission is spatiallycoincident with region 46 in the Fig. 2 NuSTAR map of Grefen-stette et al. (2017), not plotted in our Fig 8 as the Ti emissionwas found to be below the detection threshold.We note that blue-shifted Ti emission is harder to detectfor
NuSTAR than a red-shifted one as it is intrinsically fainter.In addition, any blue-shifted emission of the 78.32 keV Ti lineplaces it outside the
NuSTAR bandpass, precluding detection ofone of the two radioactive decay lines in this case.
5. Conclusions
By using a new methodology and applying it to Cas A
Chan-dra
X-ray data, we were able to revisit the mapping of the heavyelements and separate them into a red- and a blue-shifted parts,allowing us to investigate the three-dimensional morphology ofthe SNR. These new maps and the associated spectra could thenbe used to quantify the asymmetries of each component, theirmean direction and their velocity. The main findings of the pa-per are consistent with the general results found in the previousstudies cited in Part 4, and are summarized below: – Morphological asymmetries:
An extensive study of theasymmetries shows the distribution of heavier elements isgenerally more elliptical and mirror asymmetric in Cas A,which is consistent with simulation predictions. For the ele-ments we were able to separate into a red- and a blue-shiftedparts (Si, S, Ar, Ca, Fe), it appears that the red-shifted ejectais less asymmetric than the blue-shifted one. The red-shiftedejecta can then be described as a broad, relatively symmetricplume, while the blue-shifted ejecta can be seen as concen-trated into dense knots. Most of the emission from each el-ement is red-shifted, implying there was more mass ejectedaway from the observer, which agrees with past studies. – Three-dimensional distribution:
The mean directions ofthe red- and blue- shifted parts of each element areclearly not diametrically opposed, disfavouring the idea ofa jet / counter-jet explosion mechanism. – NS velocity: We find that the NS is moving most opposite tothe direction of the red-shifted ejecta that forms the bulk ofthe ejecta emission. This supports the idea of a "GravitationalTugboat Mechanism" of generating NS kicks through a pro-cess consistent with conservation of momentum between NSand ejecta. This result implies that the NS is moving toward us, which is consistent with the findings of past studies. How-ever, we find the blue-shifted clumps to be faster than the red-shifted ones, which is not consistent with the gravitationaltug-boat mechanism’s prediction that the NS is moving op-posite to the faster ejecta. – Comparison with Ti:
Our finding that the bulk of ejecta isred-shifted and moving NW is consistent with the Ti distri-bution from NuSTAR observations. Its direction is similar tothat of the red-shifted Fe-K emission, but a slight di ff erencecould be explained by the fact that the Fe-K only traces thereverse shock-heated ejecta and not the full distribution ofthe Fe ejecta.The component separation method presented here enabled athree-dimensional view of the Cas A ejecta despite the low en-ergy resolution of the Chandra
CCDs by separating entangledcomponents all at once, without the need of a detailed spec-tral analysis on hundreds of regions. In the future, X-ray mi-crocalorimeters will enable kinematic measurements of X-rayemitting ejecta in many more SNRs. In its short operations, the
Hitomi mission demonstrated these powerful capabilities. In par-ticular, in a brief 3.7-ks observation, it revealed that the SNRN132D had highly red-shifted Fe emission with a velocity of ∼
800 km s − without any blue-shifted component, suggestingthe Fe-rich ejecta was ejected asymmetrically (Hitomi Collab-oration et al. 2018). The upcoming replacement X-ray Imagingand Spectroscopy Mission XRISM will o ff er 5–7 eV energy res-olution with 30 (cid:48)(cid:48) pixels over a 3 (cid:48) field of view (Tashiro et al.2018). In the longer term, Athena and
Lynx will combine this su-perb spectral resolution with high angular resolution, fosteringa detailed, three-dimensional view of SNRs that will revolution-ize our understanding of explosions (Lopez et al. 2019; Williamset al. 2019). While the new instruments will provide a giant leapforward in terms of data quality, development of new analysismethods are needed in order to maximize the scientific return ofnext generation telescopes.
Acknowledgements.
This research made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration et al.2013; Price-Whelan et al. 2018) and of gammapy, a community-developed corePython package for TeV gamma-ray astronomy (Deil et al. 2017; Nigro et al.2019). We also acknowledge the use of Numpy (Oliphant 2006) and Matplotlib(Hunter 2007). References
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Article number, page 10 of 14icquenot et al.: Morphological asymmetries in the ejecta of Cassiopeia A in X-rays
Appendix A: Ionization impact on line centroid
At the spectral resolution of CCD type instruments, most emis-sion lines are not resolved and the observed emission is a blurredcomplex of lines. The centroid energy of emission lines can shifteither via Doppler e ff ect or when the ionization timescale in-creases and the ions distribution in a given line complex evolves (Dewey 2010; Greco, Emanuele et al. 2020) . In Fig. A.1we compare the spectral model pshock at di ff erent ionizationtimescales with the spectra that we labeled red- and blue-shiftedin Fig. 3. The temperature of the model was fixed to 1.5 keVbased on the temperature histogram of Fig. 2 of Hwang & Lam-ing (2012). The e ff ective area and redistribution matrix from ob-servation ObsID: 4634 were used. We can see that as the ioniza-tion timescale τ increases the line centroid, which is a blend ofmultiple lines, shifts to higher energies. This is most visible inthe Fe-K region where a large number of lines exists. We notethat the spectral component that we labeled as blue-shifted iswell beyond any ionization state shown here and reinforces theidea that this component is dominated by velocity e ff ect. Thesituation is less clear for the red-shifted component where theshift in energy is not as strong. We also do not precisely knowwhich reference line it can be compared to. It is interesting tonote that for the purpose of measuring a velocity e ff ect whileminimizing the confusion with ionization e ff ects, the Ar and Calines provide the best probe. Indeed for τ > cm − s, the cen-troid of the main Ar and Ca lines shows no evolution given theCCD energy resolution. The Fe K centroid strongly varies with τ and the choice of a reference ionization state and reference en-ergy limits the reliability of this line for velocity measurementsin non-equilibrium ionization plasma. Appendix B: Retrieving error bars for a nonlinearestimator applied on a Poissonian data set
Appendix B.1: Introduction
The BSS method we used in this paper, the pGMCA, is one ofthe numerous advanced data analysis methods that have recentlybeen introduced for a use in astrophysics, among which we canalso find other BSS methods, classification, PSF deconvolution,denoising or dimensionality reduction. We can formalize the ap-plication of these data analysis methods by writing
Θ = A ( X ) ,where X is the original data, A is the nonlinear analysis operatorused to process the signal and Θ is the estimator for which wewant to find errors (in this paper for example, X is the originalX-ray data from Cas A, A is the pGMCA algorithm and Θ rep-resents the retrieved spectra and images). Most of these methodsbeing nonlinear, there is no easy way to retrieve error bars or aconfidence interval associated with the estimator Θ . Estimatingerrors accurately in a nonlinear problem is still an open ques-tion that goes far beyond the scope of astrophysical applicationsas there is no general method to get error bars from a nonlin-ear data-driven method such as the pGMCA. This is a hot topicwhose study would be essential for an appropriate use of com-plex data analysis methods in retrieving physical parameters, andfor allowing the user to estimate the accuracy of the results. Appendix B.2: Existing methods to retrieve error bars onPoissonian data sets
Our aim, when searching for error bars associated with a certainestimator Θ on an analyzed data set, is to obtain the variance of Θ = A ( X ) , where the original data X is composed of N elements. Fig. A.1: Comparison of our red and blue spectra (dotted curves)presented in Fig. 3 versus pshock Xspec models with di ff erentionization timescales for kT = Θ is to apply the considered dataanalysis method A on a certain number of Monte-Carlo (MC)realizations X i and look at the standard deviation of the results Article number, page 11 of 14 & A proofs: manuscript no. main
Original data First bootstrap resampling Second bootstrap resampling
Fig. B.1: Example of bootstrap resampling. Each square repre-sents a di ff erent event, each color a di ff erent value. N events aretaken randomly with replacement from the original data to createeach of the two bootstrap resamplings. Θ i = A ( X i ). The variance of the Θ i provides a good estimationof the errors. Yet, this cannot be done with real data as only oneobservation is available: the observed one. Thus, a resamplingmethod such as the jackknife, the bootstrap (see Efron 1979)or its derivatives, able to simulate several realizations out of asingle one, is necessary. Ideally, the aim is to obtain through thisresampling method a number of ”fake” MC realizations centeredon the original data: new data sets variating spatially and repro-ducing the spread of MC drawings with a mean equal or close tothe mean of the original data.The mechanisms at stake in jacknife or bootstrap resam-plings are similar. Jacknife and bootstrap resampling methodsproduce n resampled sets ˜ X i by rearranging the elements of X ,and allow us to consider the variance of Θ i = A ( ˜ X i ) for i in (cid:126) , n (cid:127) as an approximation for the variance of Θ . As jacknife and boot-strap methods are close to each other, and the bootstrap and someof its derivatives are more adapted to handle correlated data sets,we will in this Appendix focus on a particular method, repre-sentative of other resampling methods and theoretically suitedfor astrophysical applications: the block bootstrap, which is asimple bootstrap applied on randomly formed groups of eventsrather than on the individual events.In the case of a Poisson process, the discrete nature of theelements composing the data set can easily be resampled witha block bootstrap method. The N discrete elements composing aPoissonian data set X will be called "events." In X-rays for exam-ple, the events are the photons detected by the spectro-imaginginstrument. The bootstrap consists in a random sampling withreplacement from the current set events X . The resampling ob-tained through bootstrapping is a set ˜ X boot of N events taken ran-domly with replacement amid the initial ones (see Fig. B.1). Thismethod can be repeated in order to simulate as many realiza-tions ˜ X booti as needed to estimate standard errors or confidenceintervals. In order to save calculation time, we can choose to re-sample blocks of data of a fixed size instead of single events:This method is named block bootstrap. The block bootstrap isalso supposed to conserve correlations more accurately, makingit more appropriate for a use on astrophysical signals. The datacan be of any dimension but for clarity, we will only show in thisAppendix bi-dimensional data sets, that is images. Appendix B.3: Biases in classical bootstrap applied onPoissonian data sets
The properties of the data resampled strongly depends on the na-ture of the original data. Biases may appear in the resampled datasets, proving a block bootstrap can fail to reproduce consistent data that could be successfully used to evaluate the accuracy ofcertain estimators.In particular, Poissonian data sets, including our X-rays dataof Cas A, are not consistently resampled by current resamplingmethods such as the block bootstrap. A Poissonian data set X canbe defined as a Poisson realization of an underlying theoreticalmodel X ∗ , which can be written: X = P ( X ∗ ) , where P ( . ) is an operator giving as an output a Poisson realiza-tion of a set.A look on the histogram of a data set resampled from a Pois-sonian signal shows the block bootstrap fails to reproduce ac-curately the characteristics of the original data. Fig. B.2, top,compares the histogram of the real data X , a simple image of asquare with Poisson noise, with the histograms of the resampleddata sets ˜ X booti , and highlights the fact that the latter are moresimilar to the histogram of a Poisson realization of the origi-nal data P ( X ) = P ( P ( X ∗ )) than to the actual histogram of theoriginal data X = P ( X ∗ ), where X ∗ is the underlying model ofa square before adding Poisson noise. This is consistent withthe fact that the block bootstrap is a random sampling with re-placement, which introduces uncertainties of the same nature asa Poisson drawing.Fig. B.2, bottom, shows the comparison between the his-togram of the toy model Cas A image and the histograms of thedata sets resampled with a block bootstrap. We can see the re-sampling is, in this case too, adding Poissonian noise and giveshistograms resembling P ( P ( X ∗ )) rather than P ( X ∗ ). The samegoes with our real data cube of Cas A: Fig. B.3 shows an obvi-ous instance of this bias being transferred to the results of thepGMCA, thus proving the block bootstrap cannot be used assuch to retrieve error bars for this algorithm. Appendix B.4: A new constrained bootstrap method
Bootstrap resamplings consisting in random drawings with re-placement, it is natural that they fail to reproduce some charac-teristics of the data, among which the histogram that gets closerto the histogram of a Poisson realization of the original data thanto the histogram of the actual data. The block bootstrap methodis therefore unable to simulate a MC centered on the originaldata: the alteration of the histogram strongly impacts the natureof the data, hence the di ff erences in the morphologies observedby looking at the wavelet coe ffi cients. It is then necessary to finda new method in which we could force the histogram of the re-sampled data sets to be similar to that of the original data.A natural way to do so would be to impose the histogram wewant the resampled data to have before actually resampling thedata. To allow this constraint to be made on the pixel distribu-tion, we can no longer consider our events to be the individualelements of X or a block assembling a random sample of them.We should directly work on the pixels and their values, the pix-els here being the basic bricks constituting our data. Just as theblock bootstrap, our new method can work with data of any di-mension. In the case of images, the "basic bricks" correspond toactual pixels values. In the case of X-rays data cubes, they aretiny cubes of the size of a pixel along the spatial dimensions,and the size of an energy bin along the spectral dimension. Thesame goes for any dimension of our original data. The methodcan also be adapted for uni-dimensional data sets. The key ofour new method is then to work on the histogram of the datapresenting the pixels’ values rather than on the data itself,event by event.
Article number, page 12 of 14icquenot et al.: Morphological asymmetries in the ejecta of Cassiopeia A in X-rays Pixel values N u m b e r o f p i x e l s X = ( X * ) X booti ( ( X * )) Pixel values N u m b e r o f p i x e l s X = ( X * ) X booti ( ( X * )) Fig. B.2: Data sets and their associated histogram in two cases:on top, the very simple case of a Poisson realization of the imageof a square with uniform value 10; on the bottom, a toy modelCas A image obtained by taking a Poisson realization of a high-statistics denoised image of Cas A (hereafter called toy model).On the right, the black histogram correspond to the original data X = P ( X ∗ ). The red histograms are those of the data sets ˜ X booti obtained through resampling of the original data and the blueones are the histograms of a Poisson realization of the originaldata P ( X ) = P ( P ( X ∗ )). It appears that the resampled data setshave histograms highly similar to that of the original data withadditional Poisson noise. E (keV) T o t a l c o un t s Real dataResampled data
Fig. B.3: Spectrum of the synchrotron component retrieved bypGMCA on the 5 . . Step 1 :Step 2 :
Original imageDefining a new histogramImposing the new histogram on the original image
Fig. B.4: Scheme resuming the two steps of our new constrainedbootstrap method. Pixel values N u m b e r o f p i x e l s Constrained bootstrapMonte-CarloOriginal data Pixel values S t a n d a r d d e v i a t i o n Constrained bootstrapMonte-Carlo
Fig. B.5: Histograms and standard deviations of the original andresampled data sets. On the left, histograms of the original data,the resampled data sets and the MC realizations of the toy modelCas A image. On the right, the standard deviations of the resam-pled data sets and MC realizations bin by bin of the histogramon the left. We can notice the great adequation between the stan-dard deviations of the resampled data sets and that of the MCrealizations.law to select the pixels to exchange would introduce some spatialvariations, in order to reproduce what a MC would do.Our new constrained bootstrap method is thus composed oftwo steps, that are described below and illustrated in Fig. B.4:Firstly, obtaining the probability density function of the ran-dom variable underlying the observed data histogram using theKernel Density Estimation (KDE), and randomly generating n histograms from this density function with a spread around thedata mimicking that of a MC, with a constraint enforcing a Pois-sonian distribution of the total sums of pixel values of the n his-tograms.Secondly, producing resampled data sets associated with thenew histograms by changing the values of wisely chosen pixelsin the original image.During these steps, the pixels equal to zero remain equal tozero, and the nonzero pixels keep a strictly positive value. Thisconstraint enforces the number of nonzero pixels to be constantand avoids the creation of random emergence of nonzero pixelsin the empty area of the original data. While this is not com-pletely realistic we prefer constraining the resampled data sets inthis way than getting spurious features. We could explore waysto release this constraint in the future.Fig. B.5 highlights the similarities between the original his-togram and those obtained through MC realizations and our newconstrained bootstrap resamplings, while Fig. B.6 and the spec-tra in Fig. 3 and Fig. 4 show that even after being processed by Article number, page 13 of 14 & A proofs: manuscript no. main
E (keV) T o t a l c o un t s Resampled dataReal data
Fig. B.6: Spectrum of the synchrotron component retrieved bypGMCA on the 5 . . ff erent resamplings explore initial condi-tions slightly di ff erent from the original data, thus evaluating thedependence of our results on the initial conditions. Fig. 3 andFig. 4 indeed show that for some line emissions, the dispersionbetween the results on di ff erent resampled data sets is far greaterthan for others.This new constrained bootstrap method is a first and promis-ing attempt at retrieving error bars for nonlinear estimators onPoissonian data sets, a problem that is often not trivial. In nonlin-ear processes, errors frequently cannot be propagated correctly,so the calculation of sensitive parameters and the estimation oferrors after an extensive use of an advanced data analysis couldbenefit from this method. We will work in the future on a way toconstraint the variance of the results to be more closely relatedto that of a set of MC realizations in order to ensure the physicalsignification of the obtained error bars.erent resampled data sets is far greaterthan for others.This new constrained bootstrap method is a first and promis-ing attempt at retrieving error bars for nonlinear estimators onPoissonian data sets, a problem that is often not trivial. In nonlin-ear processes, errors frequently cannot be propagated correctly,so the calculation of sensitive parameters and the estimation oferrors after an extensive use of an advanced data analysis couldbenefit from this method. We will work in the future on a way toconstraint the variance of the results to be more closely relatedto that of a set of MC realizations in order to ensure the physicalsignification of the obtained error bars.