Particle acceleration controlled by ambient density in the southwestern rim of RCW 86

In order to perform the proper motion study with the best position accuracy available, we apply the aspect correction to the three observations in 2013. The six point-like sources indicated in Figure 4 are selected and used for the aspect correction because they are bright and close to the nominal points in all the observations. Before the aspect correction, the positions of these sources differ among the observations typically by $\sim 0\farcs 3$. We first run the wavdetect tool for each observation to determine the central positions of these sources, which are summarized in Table 2. Then we run the wcs_match tool to find the best-fit transformation matrices to correct the coordinate systems of the three observations in 2013 to match those of the observation in 2001¹¹1All the six point sources are used in all the calculations without being excluded in the wcs_match process.. The transformation matrix only uses two-dimensional translation without rotation and scaling because the number of available point-like sources is relatively small. Finally, the wcs_update tool applies these corrections to the observations in 2013.

After the correction, for each of the three observations in 2013, the resultant correction accuracy is evaluated based on the standard deviation of the position offsets of the six sources with respect to those in 2001. The correction accuracies are obtained as $0\farcs 18$, $0\farcs 24$, and $0\farcs 26$, for OBSIDs 13748, 15610, and 15611, respectively. We merge the three observations in 2013 for use in the following analysis. The aspect correction accuracy for these merged data is estimated by taking an exposure-weighted average of those for the three observations, to be $0\farcs 22$, which is converted to $0\farcs 018$ yr^-1. This is considered as the systematic uncertainty associated with the position accuracy.

Proper motion velocities of the filaments indicated in Figure 1 are measured as follows. Flux profiles are extracted from the vignetting-corrected images in 2001 and 2013, which are presented in Figures 17 and 18. To evaluate their proper motion velocities, a $\chi^{2}$-test is used as below. We artificially shift the profile in 2013 by $\Delta x$ and calculate $\chi^{2}(\Delta x)$, which is defined as

where $f_{i}$ and $\Delta f_{i}$ indicate the flux and error of the bin number $i$ in 2001, and $g_{i}$ and $\Delta g_{i}$ indicate those of the shifted profile in 2013. This calculation is repeated with various values of $\Delta x$ to plot $\chi^{2}$ as a function of $\Delta x$. The minimum $\chi^{2}$ value ($\chi^{2}_{\rm min}$) and corresponding profile shift ($\Delta x_{\rm min}$) are determined by fitting the $\chi^{2}$–$\Delta x$ plot with a parabola function. An example of the $\chi^{2}$–$\Delta x$ plot and parabola fitting is presented in Figure 5. The best-fit $\Delta x_{\rm min}$ is converted to the proper motion velocity. The profile shift is not limited to an integer multiple of the bin width. We re-bin the shifted profile $g(\Delta x)$ with the same bin arrangement as $f$ with an assumption of a uniform probability distribution inside each bin. Then, the profile-shift ranges which give $\chi^{2}(\Delta x)=\chi^{2}_{\rm min}+1$ are calculated from the best-fit parabola functions. These ranges are considered to be 1$\sigma$ confidence ranges of the proper motion velocities.

The resultant proper motion estimates are summarized in Table 3, and shown in Figures 17 and 18. Note that the filaments SW7 and SW10 are moving inward, toward the SNR center. These inward movements are confirmed by spectral softening toward presumable downstream in SW7 and SW10 as described in Section 3.3. We also note that the projection effect, i.e., the difference between the measured radial velocities and actual three-dimensional velocities, will be small ($\lesssim 15\%$) considering the positions of the filaments with respect to the apparent SNR radius.²²2We assume the SNR radius of $\approx 22.6^{\prime}$ and radial distances of the filaments from the explosion center of $\approx 19.5\text{--}22.6^{\prime}$.

Regarding the background (sky and particle-induced background), we estimate the sky background from a region outside the SNR shell (shown in Figure 1), and model the particle-induced background using the tool mkacispback (Suzuki et al., 2021). We simultaneously model the spectra extracted from the background region for both 2001 and 2013 observations. The spectral model for the background region is FE + Abs. (MWH + CXB) + PB, where the FE, Abs., MWH, CXB, and PB indicate the foreground emission (Yoshino et al., 2009; Kuntz & Snowden, 2000), interstellar absorption, Milky Way halo (transabsorption emission) (Masui et al., 2009; Yoshino et al., 2009; Kuntz & Snowden, 2000), cosmic X-ray background (Snowden et al., 1992; Kushino et al., 2002; Hickox & Markevitch, 2006), and particle-induced background model suited for our background region.

The FE component is described by the apec model with the fixed temperature of 0.1 keV and metal abundances of solar values. The emission measure, which is defined as (4$\pi$D²)^-1$\int$n_en_HdV cm^-5, where D, n_e. and n_H stand for distance, electron and hydrogen number densities, respectively, is treated as a free parameter. The absorption column density of the Abs. component (tbabs model) is fixed to $N_{\rm H}=6.4\times 10^{21}$ cm^-2 (HI4PI Collaboration et al., 2016). The MWH component is also described with the apec model with the fixed metal abundances of solar values and free temperature and emission measure. The CXB component is described by the powerlaw model with the fixed spectral index of 1.4 and the normalization corresponding to the flux of $6.38\times 10^{-8}$ erg cm^-2 s^-1 str^-1 (Kushino et al., 2002). As for the particle-induced background, the acispback model predicts a lower flux than the observation in the $\sim 2$–7 keV band only in 2001 (Suzuki et al., 2021). Thus, we apply an additional powerlaw model to acispback only for the spectra in 2001.

The spectral fitting results are shown in Figure 10. The best-fit spectral parameters are summarized in Table 4. The spectral parameters are consistent with Kuntz & Snowden (2000) and Yoshino et al. (2009). Note that the visible flux decrease from 2001 to 2013 at the energies below 1 keV is due to the increased contamination on the sensor surface.

As the source emission, we apply the vpshock (non-equilibrium ionizing plasma) model to the thermal, and the powerlaw model to the non-thermal components following Tsubone et al. (2017). In addition, a Gaussian with the fixed energy centroid of 6.4 keV and width of zero is added for all the regions (Yamaguchi et al., 2011; Tsubone et al., 2017). The spectral model is described as Abs. (source emission) + (sky background) + PB. The source emission is powerlaw for SW5, SW6, SW7, and SW10 because thermal components are found to be unnecessary. The model vpshock + powerlaw is applied for the other regions. We note that the regions SW9 and SW12 are intermediate cases between the non-thermal- and thermal-dominated ones. The absorption column density of the Abs. component (tbabs model) is treated as a free parameter. For vpshock, the temperature, ionization timescale ($n_{\rm e}t$), emission measure, and the metal abundances of O, Ne, Mg, Si, and S are treated as free parameters. The other metal abundances are fixed to solar values. All the parameters of vpshock are tied between 2001 and 2013. The powerlaw normalization and index are treated as free parameters and are basically tied between 2001 and 2013. Only for SW5, SW6, SW7, and SW10, both parameters are untied between 2001 and 2013. The sky background component is the same as that determined in Section 3.4.1. Their overall normalizations are scaled by the area ratio between the source regions and background region. The PB component for the spectra in 2001 is composed of acipback plus powerlaw models, whereas that for 2013 is a simple acispback model, which is the same treatment as that for the background estimation.

The spectra and best-fit models are presented in Figure 11. The resultant spectral parameters are summarized in Table 5. The power-law fluxes of SW5, SW6, SW7, and SW10 all show significant decrease from 2001 to 2013. Similarly, the power-law indices increase (spectra soften) from 2001 to 2013. The absorption column densities, temperatures, metal abundances, and ionization timescales are roughly consistent with those presented in Tsubone et al. (2017). From the plasma densities inferred from the emission measures ($\sim 1$ cm^-3) and ionization timescales ($5\text{--}13\times 10^{10}$ s cm^-3), we confirm⁵⁵5With an assumption that our analysis regions are cuboids. that the elapsed times after the shock heating are comparable to the remnant age ($\sim 2000$ yr). The SW3 region may show smaller elapsed time of $\sim 600$ yr. This is consistent with the fact that this region is more distant from the remnant center than the other filaments and will be less affected by the projection effect, which mixes the newly heated and older regions. We also search for possible variations of the non-thermal parameters in the thermal-dominated cases where the spectral model fits the data relatively well (e.g., SW9 and SW12), and find consistent parameters in 2001 and 2013. ⁶⁶6For example, the power-law fluxes of SW9 in the 0.5–7.0 keV band in 2001 and 2013 are $3.6\text{--}4.5\times 10^{-13}$ erg and $4.4\text{--}5.5\times 10^{-13}$ erg, respectively.

We also use the spectral model by Zirakashvili & Aharonian (2007) (hereafter the ZA07 model) for the non-thermal component to discuss the acceleration efficiency. The ZA07 model is described as

where $E$ and $E_{0}$ indicate photon energy and spectral turnover energy, respectively. This model describes the synchrotron emission spectrum with several assumptions: loss-limited maximum energy, arbitrary energy dependence of the diffusion coefficient, shock compression ratio of four, and upstream-to-downstream magnetic field ratio of $\sqrt{11}$. We replace the powerlaw model in the spectral model set used above with the ZA07 model. The free parameters for spectral modeling are the same as those in the case with the powerlaw model except for the normalization and turnover $E_{0}$ of the ZA07 model. The results are summarized in Table 6. One can see that the non-thermal-dominated filaments, SW5, SW6, SW7, and SW10, exhibit decreasing $E_{0}$ in time, which is the same tendency as our results with the powerlaw model.

Based on the proper motions and spectral parameters we have obtained, we discuss here their parameter correlations to see how the environmental parameters affect the acceleration processes. Figure 14 shows the power-law parameters versus shock velocity, thermal emission measure, and electron temperature. Focusing on the filaments with both thermal and non-thermal parameters obtained (black crosses), the power-law parameters clearly depend on the emission measure ($\propto n_{\rm e}^{2}V$, where $n_{\rm e}$ and $V$ are plasma density and volume, respectively) and do not or only weakly on the shock velocity and post-shock electron temperature. If we evaluate the correlation between the downstream plasma density and power-law parameters with a power-law function, we obtain $\text{(power-law flux)}\propto n_{\rm e}^{1.0\pm 0.2}$ and $\text{(power-law index)}\propto n_{\rm e}^{0.38\pm 0.10}$. Note that the non-thermal-dominated filaments, SW5, SW6, SW7, and SW10 are not included in these evaluations. Such correlations were indeed implied by Tsubone et al. (2017) using Suzaku by investigating the whole remnant, but are more clarified in this work with Chandra. An increase in the non-thermal flux associated with high-density regions was also observed in Cassiopeia A (Sato et al. 2018; Fraschetti et al. 2018)

As seen in Figure 15, the emission measure is negatively correlated with the shock velocity $v_{\rm sh}$. This correlation appears to follow the function $n_{\rm e}v_{\rm sh}=\text{const.}$ A simple assumption of the constant ram pressure $n_{\rm e}v_{\rm sh}^{2}=\text{const.}$ seems to be inapplicable. This may reflect the inhomogeneity in the ambient density suggested by the complicated structure of the remnant and discussed in previous works (Williams et al., 2011; Tsubone et al., 2017). We note that additional uncertainties of the shock velocity are expected in the moving direction and projection effect, which would be less than a factor of $\sim$ two.

Certain correlations among magnetic field strength $B$, density $n_{\rm e}$, and shock velocity $v_{\rm sh}$ are expected due to the Bell instability (Bell, 2004), and were in fact confirmed from observations of several young SNRs (Völk et al., 2005; Vink, 2006; Helder et al., 2012; Vink, 2017): $B^{2}\propto n_{\rm e}v_{\rm sh}^{3}$ or $B^{2}\propto n_{\rm e}v_{\rm sh}^{2}$. If we plot the filament width ($\propto B^{-3/2}$ by Eq. 5) over the shock velocity for SW5, SW6, SW7, and SW10, we obtain Figure 16. As can be seen, the data do not follow the relations expected above.

In order to compare the parameter correlations of the non-thermal-dominated filaments to those of the thermal-dominated ones, we evaluate the upper limits of the EMs for the non-thermal-dominated filaments (SW5, SW6, SW7, SW9, and SW10). We repeat the spectral modeling described in Section 3.4.2 with an additional ionizing plasma model with various electron temperatures (0.3–0.8 keV) and ionization timescales ($10^{10\text{--}11}$ s cm^-3) with the metal abundances fixed to solar, and derive the upper limit of the EM. ⁹⁹9The two parameter ranges are selected based on the values determined for the thermal-dominated filaments (Table 5). The resulting upper limits of the EMs versus the power-law parameters are shown in Figure 14 (b). The non-thermal-dominated filaments located at the inner regions, SW6, SW7, and SW10, do not follow the same tendency as that of the other filaments (shown with the red solid lines).

We have also investigated the dependence of the non-thermal parameters on the shock obliquity (angle between the shock normal and background magnetic field). The background field direction is estimated from the starlight polarization at similar distances to RCW 86 (see Appendix B for more details). The field direction around RCW 86 is found to be nearly parallel to the Galactic plane. We find no correlation between the non-thermal parameters and the shock obliquity as can be seen in Figure 20.

We propose here a shock-cloud interaction scenario to explain the parameter correlations we have obtained above. As the RCW 86 SW region is interacting with dense atomic and molecular clouds (Sano et al., 2017, 2019), X-ray emission is expected to trace such an interaction. The shock-cloud interaction will slow down the shocks and will damp the magnetic turbulence inside dense clumps but amplify the magnetic turbulence around them (Inoue et al., 2012; Fraschetti, 2013). If our analysis has resolved such clumpy structures, we expect harder synchrotron X-rays for lower-density regions due to higher maximum acceleration energies, which matches the measured trend (Figure 14 (b)). We note that our spectral extraction regions have a size of $\sim 0.2$–0.4 pc at a distance of 2.8 kpc, and the shock crossing time is estimated as $\sim 100$–1000 yr. Since the cooling timescale derived for the non-thermal-dominated filaments of $\sim 50$–200 yr is similar to the shock crossing timescale, it will be reasonable to assume that the spatial extent of clumps are similar to our region sizes. The fact that the power-law flux is proportional to the density (Figure 14 (b-1)) can be understood as more enhanced non-thermal particle density in higher-density regions. The shock-cloud interaction leads highly spatially inhomogeneous magnetic fields, and so the power-law parameters show large scatters even at similar shock velocities (Figure 14 (a)). However, interestingly, the plasma density seems to remain as a good tracer of the modification of the acceleration conditions even after the interaction (Figure 14 (b)).

The reason for the apparent inconsistency of our estimates of the magnetic field strengths and shock velocities with the Bell instability (Figure 16) is understandable if the magnetic turbulence induced by the shock-cloud interaction determines local acceleration activity. Note that, according to a comparison with other SNRs, the base magnetic field amplification level over the whole remnant seems to be controlled by the Bell instability (Vink, 2006; Helder et al., 2012). Thus, it is suggested that the local magnetic turbulence levels $\delta B/B$ are of great importance to understand the local acceleration conditions.

In Figure 14 (b), the tendency of the inner non-thermal-dominated filaments (SW6, SW7, and SW10) differs from that of the outer filaments. If we assume that the outer filaments are newly interacting with dense gas and the inner ones are still in the wind-blown bubble, our results may suggest that the cavity region has different acceleration conditions from those of the interacting regions, which is naturally expected.

As a conclusion, our findings suggest that the acceleration physics at the outer filaments of the RCW 86 SW region is governed by the ambient density, not or only weakly by the shock velocity and shock obliquity. We find that the shock-cloud interaction scenario can explain the measurements consistently, although not yet in a quantitative manner. Radio observations with high angular resolutions (e.g., Atacama Large Millimeter/submillimeter Array: ALMA) will greatly help test our scenario.