© The Authors 2023

1 Introduction

Electrons accelerated in the forward shocks of young supernova remnants (SNRs) can emit synchrotron radiation. This emission is mostly seen at radio wavelengths, but can also be seen in X-rays thanks to the fast (≥3000 km s⁻¹; see Aharonian & Atoyan 1999) shocks of young SNRs. In some cases, the X-ray brightness fades downstream and produces a bright shell-like structure of rims and filaments.

A better understanding of these rims could help us to constrain the downstream magnetic field structure. Current models use three physical effects to account for the existence of the synchrotron filaments in X-ray: advection (bulk motion of the plasma), diffusion (random motion of electrons at small scales), and magnetic damping (where the X-ray emission reflects the magnetic field morphology). The relative importance of these effects has an influence on the evolution of the widths of the filaments with energy.

The models describing the formation of X-ray synchrotron rims can mostly be divided into two main categories depending on how they take magnetic damping effects into account. In the “loss-limited” models, the magnetic field is considered constant over the width of the rim. Electrons only travel a certain distance until they lose enough energy through diffusion and advection for their radiation to drop below the X-ray band (Vink & Laming 2003; Bamba et al. 2005; Parizot et al. 2006). If the magnetic field is constant downstream, most energetic electrons will radiate and cool more quickly through advection, which results in a narrowing of the synchrotron rims with energy. However, diffusion will dilute this effect: more energetic electrons may diffuse further than would be expected from pure advection, weakening the energy dependence of the widths at higher energies (Araya et al. 2010; Ressler et al. 2014). Therefore, loss-limited models predict a narrowing of the X-ray synchrotron filaments, with a weakening of the energy dependence with energy.

Loss-limited models require a strong magnetic field amplification to successfully account for the formation of thin rims. Pohl et al. (2005) proposed that the rim profiles could reflect the magnetic field morphology, adding a magnetic damping mechanism to the effects of diffusion and advection. The magnetic field could be damped downstream of the shock and prevent electrons from radiating efficiently, thus reducing synchrotron flux at all wavelengths. Therefore, a damping mechanism is supposed to produce relatively energy-independent rim widths below a threshold energy and may decrease or increase once advection and/or diffusion controls rim widths (Tran et al. 2015). Hence, the evolution of these widths with energy is a constraint on the magnetic field amplification and the amount of damping. However, there are various physically possible damping mechanisms, and so the relationship between the filament widths and the actual magnetic field variations is not straightforward.

Ressler et al. (2014) observed synchrotron narrowing in SN 1006 and concluded that it was too strongly energy dependent to be well described by the damping mechanism only: the damping lengths would need to be larger than the synchrotron-loss lengths. Tran et al. (2015) conducted a similar study of the Tycho SNR. These authors proposed a range of radio and X-ray rim profiles showing the influences of both the magnetic field and the damping length. Their work highlights the similarity between the effects of a strong magnetic field and a small damping length on lowering the dependence of rim width on energy, showing that it is difficult to probe a model with no further information on the magnetic field. In the same paper, a moderate narrowing of the rim widths was found in Tycho’s SNR in X-rays, but this was insufficient to constrain the model. However, loss-limited models alone cannot account for the formation of thin radio filaments: hydrodynamic models with diffusive shock acceleration cannot produce radio profiles with narrow rims in a purely advected magnetic field (Cassam-Chenai et al. 2007; Slane et al. 2014). The observation of thin radio filaments hence proved that there is a damping effect in Tycho, which is not necessarily sufficient to account for the narrowing in X-rays.

Cassiopeia A (hereafter, Cas A) is among the most studied astronomical objects at X-ray wavelengths. Its study benefits from extensive observations (about 3 Ms in total with Chandra), which reveal a surrounding synchrotron shell showing filamentary structures. This makes Cas A an ideal laboratory with which to investigate the potential narrowing of synchrotron rims with energy.

Araya et al. (2010) investigated some properties of the synchrotron rims visible at the forward shock in the 1 Ms Chandra observation of Cas A, including a comparison between the linear profiles of some filaments in the 0.3–2.0 keV, 3.0–6.0 keV, and 6.0–10.0 keV energy bands. These authors only found a slight difference in the widths between the 0.3–2.0 keV and the 3.0–6.0 keV images, but no difference between the 3.0–6.0 keV and the 6.0–10.0 keV images, as shown in Table 1 of their paper. They also found that the shape of the decline in emission downstream of the shock was similar to that upstream, contrary to model predictions.

Here, we obtained more detailed images using a new method to retrieve accurate maps of the synchrotron emission around different energy bands. This method is based on the General Morphological Components Analysis (GMCA, see Bobin et al. 2015), a blind source-separation algorithm that was introduced for X-ray observations by Picquenot et al. (2019). It can disentangle spectrally and spatially mixed components from an X-ray data cube of the form (x, y, E). The new images thus obtained suffer less contamination by other components, such as thermal emission from lines or continuum. An updated version of this algorithm, the pGMCA (see Bobin et al. 2020), has been developed to take into account the Poissonian nature of X-ray data. It was first used on Cas A data to probe the three-dimensional morphological asymmetries in the ejecta distribution (Picquenot et al. 2021), and proved perfectly suited to producing clear, detailed, and unpolluted images of both the ejecta and the synchrotron at different energies. Thanks to these new images, we are able to study the profiles of some filamentary structures associated with the forward shock, and also find some associated with the reverse shock. We label the sides of the filament profiles according to their location relative to the shock they are associated with, therefore denoting them “upstream” and “downstream”. However, the widths of the profiles do not necessarily correspond to the actual width of the upstream and downstream shock as projection effects might have an effect.

This paper is structured as follows. In Sect. 2, we show the images of the synchrotron we used, and the way the filament profiles are defined. In Sect. 3, we present a way to quantify the narrowing of the filaments, and discuss our results.

2 Using pGMCA to probe the widths of the synchrotron rim in Cas A

Being one of the brightest sources in the X-ray sky, Cas A is an ideal extended source to showcase the capabilities of pGMCA. Cas A has been the target of years of extensive observations, meaning that large sample sizes are available for strongly overlapping components that the algorithm is well suited to disentangle. Here, we obtain three images of the synchrotron of Cas A at different wavelengths, from which we derived the linear profiles of some wisely chosen synchrotron rims.

Fig. 1 Synchrotron emission obtained with pGMCA in three energy bands, with square-root scaling. The boxes used to define the filament profiles are shown in the bottom right corner. The profiles at the forward shock, the reverse shock, and those that are unidentified are shown in black, green, and blue, respectively.

2.1 Image definitions

For our study, we used Chandra observations of the Cas A SNR, which was observed with the ACIS-S instrument in 2004 for a total of 980 ks (Hwang et al. 2004, ObsId 4634, 4635, 4636, 4637, 4638, 4639, 5196, 5319 and 5320). We used only the 2004 data set in order to avoid the need to correct for proper motion across epochs. The event lists from all observations were merged in a single data cube. The spatial bin size is the native Chandra bin size of 0.5 arcsec, which was chosen in order to produce images that are as detailed as possible. As we are not interested in the spectral lines, we chose a spectral bin size of 58.4 eV to keep a good number of counts in every pixel.

We applied the pGMCA algorithm on three bandwidths: between 0.4 and 1.7 keV, between 2.5 and 4 keV, and between 5 and 8 keV. We chose these bandwidths on different criteria: large enough for pGMCA to work properly and not too large in order to avoid a possible dependency of the filament width on energy. We also tried to avoid, as best we could, any line emission in the same bandwidths. This last criterion could not be fulfilled at lower energies, and some pollution from other emission can be seen in the first image displayed in Fig. 1, particularly in the southwest. The two other images seem clear and unpolluted, even with a square root scaling. They probably constitute the most detailed and accurate maps of the synchrotron in X-rays in Cas A to this day, especially at low energy.

These images present filamentary structures throughout the ejecta, including some following the known layout of the reverse shock. In order to assess the nonthermal nature of these filaments, we extracted the spectrum of one and fitted a phabs*powerlaw model in Xspec. The results are shown in Fig. 2, together with a spectrum extracted from a bright region with low synchrotron level fitted with the same model. The filament presents a spectrum that can be well fit with a simple nonthermal power-law model, while the other region obviously cannot. This is consistent with the results from Helder & Vink (2008), who attributed synchrotron emission from filaments inside of the remnant to the reverse shock.

Fig. 2 Assessment of the non-thermal nature of the filamentary structures found by pGMCA. The northeast region of Cas A from our merged observations is shown in the top left panel. The top right panel shows an image of the synchrotron retrieved by our method on the 2.5–4 keV band. The blue and green contours in these latter two panels show the regions of extraction, and the white rectangle shows the fourth reverse shock box from Fig. 1. The extracted spectra from these two regions are shown in the panels below, both fitted with a simple phabs*powerlaw model in Xspec in red.

2.2 The filaments linear profiles

Thanks to the highly detailed images of the synchrotron emission, we were able to find that using pGMCA we could investigate the narrowing with energy of the filaments both at the forward shock and, for the first time, at the reverse shock.

In order to compare the widths of the rims, we defined boxes surrounding small regions crossing a filament at the forward shock, and likewise for the reverse shock. The boxes are shown in Fig. 1; the normalized linear profiles obtained perpendicular to the rims are presented in Fig. 3 for the forward shock, and in Fig. 4 for the reverse shock.

We also looked at some filaments whose positions prevented us from labeling them clearly. A line-of-sight effect is likely the cause of the problem, and these filaments could either be attached to the forward or to the reverse shock. The boxes are also shown in Fig. 1, in blue, and the linear profiles obtained perpendicular to the rim are presented in Fig. 5.

3 Quantifying the narrowing of the synchrotron rims

3.1 Modeling the linear profiles of the filaments

In order to measure the widths of the filament profiles we extracted, we fitted them with a piecewise two-exponential model, as in Tran et al. (2015): (1)

All parameters are free, except for A_d, which is fixed to ensure continuity at r = r₀, with A_d = A_u + (C_u − C_d).

This model is well adapted to describing the sharp peak displayed by most profiles, but some filaments (such as the sixth or seventh of the forward shock) present plateaus or Gaussian-like features that are not accounted for in this model. However, this model remains a good way to estimate the widths of the profiles we extracted overall, without neglecting the asymmetry around the peak between the upstream and downstream media. The resulting fitted models are displayed in Fig. 3 (forward shock), Fig. 4 (reverse shock), and Fig. 5 (other profiles). In some cases, we had to remove secondary peaks in the profiles to focus on the main one when fitting our model.

From the parameters of our model, we can derive two full widths at half maximum (FWHM) for each profile, FWHM_u = 2 ln(2)w_u and FWHM_d = 2 ln(2)w_d, describing the “sharpness” of the profile on both sides, with a larger FWHM meaning a wider profile. The mean between these FWHMs gives an estimation of the actual width of the profile. We also define m_E: (2)

where FWHM₁ and FWHM₂ are the mean FWHMs of the same filament in two energy bands, and E₁ and E₂ are the lower energy values for each energy band. For each filament, we calculate the m_E between the 0.4–1.7 keV and 2.5–4.0 keV energy bands (E₁ = 0.4, E₂ = 2.5), and between the 2.5–4.0 keV and 5.0–8.0 keV energy bands (E₁ = 2.5, E₂ = 5.0). This parameter is designed to evaluate the narrowing of the widths of the filaments and to quantify the dependence of this narrowing on energy: positive values mean widening, and negative values mean narrowing; higher values mean higher energy dependence, while lower values mean weaker energy dependence.

The FWHM_u, FWHM_d, mean FWHM, and m_E values derived from our fitted models are shown in Table 1 (forward shock), Table 2 (reverse shock), and Table 3 (other profiles). As the “upstream” and “downstream” labels do not make sense for the profiles that were not clearly identified, we named the two sides “right” and “left,” the orientations corresponding to the plots of Fig. 5. As there is no straightforward way to estimate the pGMCA algorithm errors, the errors shown in our tables are the model-fitting errors and their propagation in the calculation of the mean FWHM and m_E values.

3.2 Discussion

A quick look at the mean FWHMs or at the m_E signs shows that there is indeed a narrowing of the filament profiles with energy. In all of the seven forward-shock profiles, five reverse-shock profiles, and five unidentified profiles, only two present a widening between 0.4–1.7 keV and 5.0–8.0 keV, namely FS 7 and unidentified profile 2. In both cases, this widening is so low that the fitting errors allow the possibility of a narrowing as well. Hence, we can reasonably conclude that the narrowing of the filaments with energy in Cas A is a global effect that can be observed on several filaments both at the forward and at the reverse shock. In Sect. 3.3, we see that this narrowing is not due to the evolution of Chandra’s point spread function (PSF) with energy. Future studies could also take into account the possible effects of dust scattering on the observed widths of the filaments. However, in a first approximation, scattering effects have an energy dependence in E⁻² (Costantini & Corrales 2022; Corrales & Paerels 2015, for example), which is not consistent with our observations. Hence, the narrowing we observe is likely not primarily due to dust scattering.

A common trend can be seen in our results regarding the evolution of m_E values: the mean m_E between 2.5–4.0 keV and 5.0–8.0 keV are significantly larger in absolute value than the mean m_E between 0.4–1.7 keV and 2.5–4.0 keV. While this result is to be taken with caution, given the importance of the errors, the low statistics on which the means are calculated, and the non-negligible number of outliers, it appears that the observed narrowing of the synchrotron rims has a stronger energy dependence at higher energies than at lower energies. Following Tran et al. (2015), this result would be characteristic of a damping mechanism: the rim widths are relatively energy-independent below a threshold energy and increase once advection controls rim width. This is in apparent contradiction with Araya et al. (2010), where a moderate narrowing was observed between 0.3–2.0 keV and 3.0–6.0 keV, but not between 3.0–6.0 keV and 6.0–10.0 keV. However, the energy bands we compared are not the same, and our “high energy m_E” is defined between 2.5 and 5.0 keV, while that of these latter authors is between 3.0 and 6.0 keV. The highly detailed synchrotron images we obtained with pGMCA may also have an influence, allowing for a more precise profile definition and width measurement with less thermal emission contamination.

In Araya et al. (2010), the model used to describe the forward shock filament profiles predicts a sharp decline upstream of the shock, that they did not observe. We can see from the FWHM_u and FWHM_d values from Table 1 that we did not observe a decline upstream either. The FWHM_u values even tend to be larger than FWHM_d values, meaning that the profiles are sharper downstream from the forward shock than upstream. On the contrary, we can see in Table 2 that at the reverse shock, FWHM_u values are mainly smaller than FWHM_d values, meaning that the profiles are sharper upstream (even though this is not apparent in the mean FWHM_u and FWHM_d values, which are driven by a few outliers). However, we see in Sect. 3.3 that this could be linked to the PSF.

There is little evidence with which to identify our “other” profiles as belonging to the forward or to the reverse shock. Although it might be tempting to base an identification on their FWHM on each side, line-of-sight effects are likely involved, and for a filament facing the observer, “left” and “right” of the peak are not trivially equivalent to “upstream” or “downstream”. Nonetheless, Table 1 mostly follows the same trend of narrowing – with stronger energy-dependence at high energies – that we observed on both forward- and reverse-shock rim profiles.

Fig. 3 Linear profiles at the forward shock along the black boxes presented in Fig. 1, and the models fitted to describe their widths, offset for clarity. The profiles and the models are normalized so that the peaks coincide, and the radius is given in arcseconds. The arrows show the direction of the forward shock (upstream is right of the plot).

Fig. 4 Linear profiles at the reverse shock along the green boxes presented in Fig. 1, and the models fitted to describe their widths, offset for clarity. The profiles and the models are normalized so that the peaks coincide, and the radius is given in arcseconds. The arrows show the direction of the reverse shock (upstream is left of the plot).

Fig. 5 Unidentified linear profiles along the blue boxes presented in Fig. 1, and the models fitted to describe their widths, offset for clarity. The profiles and the models are normalized so that the peaks coincide, and the radius is given in arcseconds.

3.3 Influence of the PSF

Previous studies by Araya et al. (2010), Ressler et al. (2014), and Tran et al. (2015) did not take into account the PSF in measurements of filament widths. Nevertheless, the PSF evolves with energy, and could have an impact on our width comparison between energy ranges. In a first attempt to take this into account, we generated the 1σ PSF maps of the merged 2004 observations of Cas A using the merge_obs routine from CIAO around the 0.4–1.7 keV, 2.5–4.0 keV, and 5.0–8.0 keV energy bands. We then intended to convolve our piecewise two-exponential model to the PSF profiles along each box for each energy range while fitting. However, the results we obtained were endowed with disproportionate errors due to the additional uncertainty brought by the spreading. We therefore renounced taking the PSF directly into account in our fitting, and decided to present a “worst-case scenario” to give an idea of the possible effects of the PSF on the results shown in Tables 1, 2, and 3.

To do so, we searched the box along which the PSF evolved the most, both spatially and with energy, which was the third box of our FS profiles, the furthest from the optical center. We then convolved the profiles of the PSF maps for the three energy ranges along this box with an infinitely thin filament, that is, a Dirac function. The results were then fitted with our model, and the differences between the retrieved FWHMs for each energy range give an idea of the influence of the PSF in the worst-case scenario, with infinitely thin filaments in our worst box as regards to the PSF.

The results are shown in Fig. 6, and it appears that the PSF can potentially have a significant influence. The FWHMs are consistently larger upstream than downstream, and the FWHMs on both sides increase with energy. It is important to note that the PSF profiles all behave in a similar way: the spreading radius values increase both with energy and with distance from the center (downstream to upstream for the forward shock, upstream to downstream for the reverse shock). Therefore, the narrowing of the filaments with energy we observed might be underestimated because of the PSF, and our observations regarding the sharper decline downstream than upstream of the forward shock might be due to the PSF. The m_E between the 0.4–1.7 keV and 2.5–4.0 keV profiles is 0.05 ± 0.03 and the m_E between the 2.5–4.0 keV and 5.0–8.0 keV profiles is 0.22 ± 0.08, indicating that the PSF would tend to widen the filaments more at higher energies.

Fig. 6 A Dirac delta function is convolved with the PSF profiles along the third FS box for our three energy ranges and normalized. This box was chosen because it was the one along which the PSF varied the most, both spatially and with energy. The FWHM values downstream and upstream are obtained by fitting our piecewise two-exponential model; values are given in arcsec.

3.4 Interpretation

The calculated m_E values tend to indicate a stronger energy dependence of the narrowing at high energies, which is likely underestimated by the PSF. According to Tran et al. (2015), this would be characteristic of a damping mechanism, where the rim widths are supposed to be energy independent below a threshold energy, and decrease or increase once advection and/or diffusion controls the widths. Following the same paper, the action of a damping mechanism can be more confidently assessed by the observation of thin synchrotron filaments in radio. In the radio VLA observations of Cas A, such filaments can be seen (see e.g., DeLaney et al. 2014), which also suggests that damping effects are influencing the synchrotron emission of this SNR. A thorough joint study of both radio and X-ray filaments could allow an estimation of the damping lengths.

4 Conclusions

From our study of the filament profiles in Cas A, we can conclude the following:

1. Our blind source-separation method was able to produce clear, detailed, and unpolluted maps of the synchrotron emission around three energy bands: 0.4–1.7 keV, 2.5–4.0 keV, and 5.0–8.0 keV. These images clearly show filamentary structures all around the remnant, some associated to the forward shock, and some to the reverse shock. Some profiles can also be seen that cannot be clearly attributed to the forward or to the reverse shock, possibly because of line of sight effects;

2. There is a noticeable narrowing of the synchrotron rims from lower to higher energies. Contrary to a previous study by Araya et al. (2010), we find that this effect is also visible between 2.5–4.0 keV and 5.0–8.0 keV, not only at lower energies. This narrowing is observed for filaments both at the forward and at the reverse shock, and for nonidentified filaments. The evolution of the PSF with energy cannot account for this narrowing; on the contrary, the PSF would tend to make filaments appear wider with energy;

3. There seems to be a stronger energy dependence of this narrowing at higher energies, which would be a sign that there is a damping mechanism in play. The observation of thin synchrotron profiles in the radio 6cm VLA observation of Cas A is another indicator of this damping mechanism. However, this does not exclude the possible simultaneous influence of the loss-limited effect;

4. The filament profiles at the forward shock tend to be sharper downstream than upstream. At the reverse shock, the profiles mainly tend to be sharper upstream. However, this could be due to the radial degradation of the PSF away from the aim point. It could also be caused by projection effects that do not have the same impact on the forward and the reverse shock as they propagate in opposite directions.

Our key finding is that there is indeed a narrowing with energy of the synchrotron filaments both at the forward and at the reverse shocks in Cassiopeia A. The energy dependency of this narrowing seems stronger at high energy, which is indicative of a damping effect that is also suggested by radio observations. A thorough joint study of both radio and X-ray filaments could allow an estimation of the damping lengths.

Acknowledgements

The material is based upon work supported by NASA under award number 80GSFC21M0002. We thank D. Castro for the useful discussion about the PSF potential influence, and E. Costantini for the interesting discussion about potential dust scattering effects.

References

All Tables

All Figures

	Fig. 1 Synchrotron emission obtained with pGMCA in three energy bands, with square-root scaling. The boxes used to define the filament profiles are shown in the bottom right corner. The profiles at the forward shock, the reverse shock, and those that are unidentified are shown in black, green, and blue, respectively.
In the text

Fig. 2 Assessment of the non-thermal nature of the filamentary structures found by pGMCA. The northeast region of Cas A from our merged observations is shown in the top left panel. The top right panel shows an image of the synchrotron retrieved by our method on the 2.5–4 keV band. The blue and green contours in these latter two panels show the regions of extraction, and the white rectangle shows the fourth reverse shock box from Fig. 1. The extracted spectra from these two regions are shown in the panels below, both fitted with a simple phabs*powerlaw model in Xspec in red.

In the text

	Fig. 3 Linear profiles at the forward shock along the black boxes presented in Fig. 1, and the models fitted to describe their widths, offset for clarity. The profiles and the models are normalized so that the peaks coincide, and the radius is given in arcseconds. The arrows show the direction of the forward shock (upstream is right of the plot).
In the text

	Fig. 4 Linear profiles at the reverse shock along the green boxes presented in Fig. 1, and the models fitted to describe their widths, offset for clarity. The profiles and the models are normalized so that the peaks coincide, and the radius is given in arcseconds. The arrows show the direction of the reverse shock (upstream is left of the plot).
In the text

	Fig. 5 Unidentified linear profiles along the blue boxes presented in Fig. 1, and the models fitted to describe their widths, offset for clarity. The profiles and the models are normalized so that the peaks coincide, and the radius is given in arcseconds.
In the text

	Fig. 6 A Dirac delta function is convolved with the PSF profiles along the third FS box for our three energy ranges and normalized. This box was chosen because it was the one along which the PSF varied the most, both spatially and with energy. The FWHM values downstream and upstream are obtained by fitting our piecewise two-exponential model; values are given in arcsec.
In the text