# Systematic Study of Acceleration Efficiency in Young Supernova Remnants with
Nonthermal X-ray Observations

Naomi Tsuji Interdisciplinary Theoretical & Mathematical Science Program
(iTHEMS), RIKEN, 2-1 Hirosawa, Saitama 351-0198, Japan  Department of Physics,
Rikkyo University, 3-34-1 Nishi Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan
Yasunobu Uchiyama  Department of Physics, Rikkyo University, 3-34-1 Nishi
Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan  Department of Artificial
Intelligence and Science, Rikkyo University, 3-34-1 Nishi Ikebukuro, Toshima-
ku, Tokyo 171-8501, Japan  Dmitry Khangulyan  Department of Physics, Rikkyo
University, 3-34-1 Nishi Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan  Felix
Aharonian  Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin
2, Ireland  Max-Planck-Institut für Kernphysik, P.O. Box 103980, D 69029
Heidelberg, Germany

###### Abstract

Cutoff energy in a synchrotron radiation spectrum of an suprenova remnant
(SNR) contains a key parameter of ongoing particle acceleration. We
systematically analyze 11 young SNRs, including all historical SNRs, to
measure the cutoff energy, thus shedding light on the nature of particle
acceleration at the early stage of SNR evolution. The nonthermal (syncrotron)
dominated spectra in filament-like outer rims are selectively extracted and
used for spectral fitting because our model assumes that accelerated electrons
are concentrated in the vicinity of the shock front due to synchrotron
cooling. The cutoff energy parameter ($\varepsilon_{0}$) and shock speed
($v_{\rm sh}$) are related as $\varepsilon_{0}\propto v_{\rm sh}^{2}\eta^{-1}$
with a Bohm factor of $\eta$. Five SNRs provide us with spatially resolved
$\varepsilon_{0}$–$v_{\rm sh}$ plots across the remnants, indicating a variety
of particle acceleration. With all SNRs considered together, the systematic
tendency of $\eta$ clarifies a correlation between $\eta$ and an age of $t$
(or an expansion parameter of $m$) as $\eta\propto t^{-0.4}$ ($\eta\propto
m^{4}$). This might be interpreted as the magnetic field becomes more
turbulent and self-generated, as particles are accelerated at a greater rate
with time. The maximum energy achieved in SNRs can be higher if we consider
the newly observed time dependence on $\eta$.

acceleration of particles — ISM: supernova remnants — radiation mechanisms:
non-thermal — X-rays: ISM

††software: NuSTARDAS (v1.4.1), HEAsoft (v6.19), nuskybgd
(https://github.com/NuSTAR/nuskybgd; Wik et al. (2014)), XSPEC (v12.9.0,
Arnaud (1996)), CIAO (v4.9, Fruscione et al. (2006))

ic short = IC , long = inverse Compton , first-style = default nustar short =
NuSTAR , long = Nuclear Spectroscopic Telescope Array , first-style = default
bi short = BI , long = backside illumination , first-style = default fi short
= FI , long = frontside illumination , first-style = default fov short = FoV ,
long = field of view , first-style = default sim short = SIM , long = Science
Instrument Module , first-style = default hetg short = HETG , long = High
Energy Transmission Grating , first-style = default letg short = LETG , long =
Low Energy Transmission Grating , first-style = default hrc short = HRC , long
= High Resolution Camera , first-style = default acis short = ACIS , long =
Advanced CCD Imaging Spectrometer , first-style = default hrma short = HRMA ,
long = High Resolution Mirror Assembly, first-style = default compton short =
Compton , long = Compton Gamma Ray Observatory, first-style = default hst
short = HST , long = Hubble Space Telescope , first-style = default iact short
= IACT , long = Imaging Atmospheric Cherenkov Telescope , first-style =
default wcd short = WCD , long = Water Cherenkov Detector , first-style =
default hawc short = HAWC , long = High-Altitude Water Cherenkov , first-style
= default cta short = CTA , long = Cherenkov Telescope Array , first-style =
default em short = EM , long = electromagnetic , first-style = default ism
short = ISM , long = interstellar medium , first-style = default csm short =
CSM , long = circumstellar medium , first-style = default sne short = SNe ,
long = supernovae , first-style = default iss short = ISS , long =
International Space Station , first-style = default uhecr short = UHECRs ,
long = ultra high energy cosmic rays , first-style = default ta short = TA ,
long = Telescope Array , first-style = default auger short = Auger , long =
Pierre Auger Observatory , first-style = default ams short = AMS , long =
Alpha Magnetic Spectrometer , first-style = default pamela short = PAMELA ,
long = Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics
, first-style = default cmb short = CMB , long = Cosmic Microwave Background ,
first-style = default sed short = SED , long = spectral energy distribution ,
first-style = default mhd short = MHD , long = magnetohydrodynamical , first-
style = default dof short = dof , long = degree of freedom , first-style =
default cco short = CCO , long = central compact object , first-style =
default lmc short = LMC , long = Large Magellanic Cloud , first-style =
default mpik short = MPIK , long = ããã¯ã¹ãã©ã³ã¯æ ¸ç©çç ç©¶æ
, first-style = default hess short = H.E.S.S. , long = High Energy
Spectroscopic System , first-style = default snr short = SNR , long =
supernova remnant , first-style = default sn short = SN , short-plural = e ,
long = supernova , long-plural = e , first-style = default nw short = NW ,
long = northwest , first-style = default hxc short = HXC , long = hard X-ray
component , first-style = default cr short = CR , long = cosmic ray , first-
style = default psf short = PSF , long = point spread function , first-style =
default hpd short = HPD , long = half power diameter , first-style = default
fwhm short = FWHM , long = full width of half maximum , first-style = default
pic short = PIC , long = particle-in-cell , first-style = default cxb short =
CXB , long = Cosmic X-ray Background , first-style = default grxe short = GRXE
, long = Galactic Ridge X-ray Emission , first-style = default pa short = PA ,
long = Positional Angle , first-style = default dsa short = DSA , long =
diffusive shock acceleration , first-style = default

##  1 Introduction

snr are believed to be prominent accelerators of the galactic cr with energies
up to a few PeV. Under the conditions typical for shocks in snr, dsa should be
capable of boosting protons to the PeV energies. The rate of sn in the Galaxy
and their typical energy output are high enough to explain the entire flux of
PeV cr particles at the Earth. It favors the scenario where the dominant
fraction of cr below the knee is produced in Galactic snr. However, this
scenario contains some problems, among which the absence of any empirical
evidence for existence of PeV particles in snr is the most critical. Nearly
all snr detected in gamma-ray show power-law energy spectra with cutoffs at
energies of at most tens of TeV. Thus, it is questionable whether snr are
indeed capable of producing PeV particles (i.e., whether they are “PeVatron”).
From the theoretical point of view, the maximum attainable energy for
particles accelerated by dsa in snr is not well constrained because of some
unknown physical parameters such as a diffusion coefficient of the non-thermal
particle and magnetic field turbulence.

In dsa, which is widely accepted as the main acceleration mechanism in snr,
particles gradually gain energy by crossing the shock wave forward and
backward. A particle changes its direction by being scattered by magnetic
field. Assuming Bohm diffusion, one can characterize the diffusion coefficient
using mean free path that is represented by the product of the particle
gyroradius and energy independent factor $\eta$ (the so-called Bohm factor).
In the case of $\eta=1$ (also known as the Bohm limit), particle mean free
path takes the minimum value, and the particles are accelerated most
efficiently. The diffusion coefficient (or the $\eta$ parameter) is strongly
related to the spectrum of the turbulent magnetic field that scatters
particles. Theoretical studies and numerical simulations propose that the
turbulence is generated by the instability resulting from the pressure
gradient derived from CR streaming (e.g., Bell (1978); Caprioli & Spitkovsky
(2014a)). However, many facets of the mechanism of turbulent production are
poorly understood.

Synchrotron radiation is a good channel to trace electron acceleration at
shocks in snr. The synchrotron X-ray, in particular, is emitted by multi-TeV
electrons. Therefore, an observation of synchrotron radiation in the X-ray
domain is a great means for diagnosing the nature of the acceleration and of
the maximum energy attainable in snr. In young snr where the shock speed is
high, the conditions for particle acceleration are especially favorable. Thus,
young snr, with ages less than a few thousand years, are expected to radiate
synchrotron X-rays. The detection of the synchrotron emission has been
reported in a few tens of galactic snr to date.

One can indirectly estimate $\eta$ from measuring the shock speed and cutoff
energy in the synchrotron X-ray spectrum (e.g., Reynolds (1998); Zirakashvili
& Aharonian (2007)). RX J1713.7$-$3946, an TeV emitting snr, is recognized as
an accelerator operating at the most efficient rate (i.e., the acceleration
proceeds in a regime close to the Bohm limit of $\eta=1$ (Tanaka et al., 2008;
Tsuji et al., 2019)). Since NuSTAR was launched in 2012, it has provided us
with spatially resolved hard X-ray spectra, thus enabling precise measurements
of the cutoff energy. Although young snr are believed to be efficient
accelerators, some recent observations with NuSTAR have in fact revealed that
the acceleration efficiency $\eta$ varies significantly and depends on the
acceleration site and/or the object. Based on NuSTAR observations, it has been
determined that $\eta$ is $\sim$1 in the forward shock and 3–8 in the reverse
shock or reflection shock in Cassiopeia A (Sato et al., 2018). In addition,
$\eta$ turned out to be $\sim 20$ in the youngest galactic snr, G1.9$+$0.3
(Aharonian et al., 2017). Toward a unified understanding of the particle
acceleration in young snr, gaps between G1.9$+$0.3 ($\sim$190 year),
Cassiopeia A ($\sim$330 year), and RX J1713.7$-$3946 ($\sim$1600 year) must be
filled.

In this paper, we present a systematic analysis and measurement of the cutoff
energy in the synchrotron radiation spectrum of young 11 snr: G1.9$+$0.3,
Cassiopeia A, Kepler’s snr (hereafter Kepler), Tycho’s snr (Tycho),
G330.2$+$1.0, SN 1006, RX J1713.7$-$3946, RCW 86, Vela Jr., HESS J1731$-$347,
and SN 1987A. Most of them are historical, and thus their ages are well
constrained. The dataset of observations analyzed here and the data reduction
are presented in Section 2. Section 3 presents imaging and spectral analyses.
The results are presented in Section 4. Our analysis for the first time
reveals a variety of instances of particle acceleration in the remnants and
clarifies a systematic tendency of acceleration efficiency. Discussion and
interpretation are given in Section 5, and conclusion are provided in Section
6.

##  2 Observations and data reduction

We systematically studied eleven young snr using X-ray observations. To reveal
a tendency of acceleration efficiency in the young snr, they should be strong
synchrotron emitters, and their physical parameters, such as age, distance,
and shock speed, should be well constrained. The following 11 snr were
selected based on the aforementioned criteria: G1.9$+$0.3, Cassiopeia A,
Kepler, Tycho, G330.2$+$1.0, SN 1006, RX J1713.7$-$3946, RCW 86, Vela Jr., and
HESS J1731$-$347 in our galaxy, and SN 1987A in the lmc. The other candidates
that are dominated by synchrotron radiation in the X-ray energy band (e.g.,
G32.45$+$0.1 and HESS J1640$-$465) were not included because there are large
uncertainties regarding their ages and distances. The known parameters of the
11 snr are listed in Table 1.

Table 1: Properties of SNRs analyzed in this paper

Name | Age |  $m$† | Distance | Shock speed | References
---|---|---|---|---|---
| (yr) |  | (kpc) | (${\rm km}~{}{\rm s}^{-1}$) |
G1.9$+$0.3 | 190$\pm$50 | 1.1$\pm$0.2 |  $\sim$8.5 | 3600–13000 | Borkowski et al. (2010, 2017)
Cassiopeia A (SN 1680) | 335$\pm$20 | 0.7$\pm$0.1 | 3.4 | 4400–5500 | Patnaude & Fesen (2009)
Kepler (SN 1604) | 415 | 0.47–0.82 | 4$\pm$1 | 3400–5700 | Cassam-Chenaï et al. (2004); Katsuda et al. (2008b)
Tycho (SN 1572) | 440 | 0.6$\pm$0.1 | 2.3 | 3200–4000 | Williams et al. (2013)
G330.2$+$1.0 | 700$\pm$300 | 0.9$\pm$0.3 |  $\sim$5 | 3700–9100 | Park et al. (2006); Borkowski et al. (2018)
SN 1006 | 1010 | 0.5$\pm$0.1 | 1.9$\pm$0.3 | 3000–7200 | Winkler et al. (2014)
RX J1713.7$-$3946 (SN 393) | 1600 $\pm$10 | 0.7$\pm$0.1 | 1 | 800–4000 | Tsuji & Uchiyama (2016); Acero et al. (2017)
RCW 86 (SN 185) | 1835 | 0.3$\pm$0.1 | 2.8 | 1800–3000 | Rosado et al. (1996); Yamaguchi et al. (2016)
Vela Jr. | 3000$\pm$1000 | 0.5$\pm$0.2 | 0.5–0.9 |  $\sim$2000 | Allen et al. (2015)
HESS J1731$-$347 | 2500–14000 | 0.5$\pm$0.1 | 3.6$\pm$0.4 |  $\sim$2500 | H. E. S. S. Collaboration et al. (2011)
SN 1987A | 30 | 0.9$\pm$0.1 | 51.4 |  $\sim$6700 | Frank et al. (2016)

Note. — † Expansion parameter (i.e., $R\propto t^{m}$ where $R$ and $t$ are
the radius of the SNR and time from the explosion, respectively).

We utilized archival X-ray observations using Chandra and NuSTAR. The dataset
used in the analysis is summarized in Table 4 and Table 5 in Section A.

We reduced all the observational data using the following software and
calibration. The Chandra data were processed using CALDB version 4.8.3 in CIAO
version 4.11. The NuSTAR data were calibrated and screened using nupipeline of
NuSTAR Data Analysis Software (NuSTARDAS version 1.4.1 with CALDB version
20180814) included in HEAsoft version 6.19. To screen the NuSTAR data, we used
the strictest mode (SAAMODE = STRICT and TENTACLE = YES cut). The effective
observational time reduced by these processes is shown in Table 4 and Table 5
(Section A).

##  3 Analysis

###  3.1 Image

Figure 1 illustrates flux images taken with Chandra. The energy band is set to
be 0.5–7 keV for synchrotron dominated snr (i.e., G1.9$+$0.3, G330.2$+$1.0, SN
1006, and HESS J1731$-$347) and RCW 86, while is set to be 4–6 keV for snr
which contain thermal emissions (i.e., Cassiopeia A, Kepler, and Tycho). All
the epochs were combined by using merge_obs, the exposure was corrected, and
the background was not subtracted. Note that we show the images of the entire
remnants taken with XMM-Newton and Suzaku for RX J1713.7$-$3946 and Vela Jr.,
respectively. We omitted the image of SN 1987A since it is treated as a point-
like source in this paper, although it shows spatial extent (e.g., McCray &
Fransson (2016)).

Some snr have strong thermal line emissions in addition to the synchrotron
component, making it difficult to extract the pure synchrotron spectrum. For
snr that have thermal components, the 4–6 keV channel can be dominated by
nonthermal radiation, while the other X-ray band can be contaminated by
thermal line emissions. The rims or the filament-like structures, which are
likely located in the outermost regions, appear bright in the 4–6 keV energy
range (Figure 1). These regions are expected to contain the synchrotron
emission from electrons accelerated at the forward shock, and are thus the
best targets for our analysis to study particle acceleration in snr. We
defined regions to extract the spectra along these outer rims (Figure 1). In
order to investigate the relation between the shock velocity and synchrotron
radiation, we also defined subregions along the regions where the proper
motions were measured in the previous works. Five snr were picked up for this
purpose: proper motions along the entire rims were already measured in
G1.9$+$0.3 (Borkowski et al., 2017), Cassiopeia A (Patnaude & Fesen, 2009),
Kepler (Katsuda et al., 2008b), Tycho (Williams et al., 2013), and SN 1006
(Winkler et al., 2014). It should be noted that the superb angular resolution
of Chandra enabled us to extract spectra from relatively small (a few arcsec
scale) subregions.

Figure 1:  Flux images in nonthermal dominated energy bands taken with
Chandra, except for RX J1713.7$-$3946 and Vela Jr. taken with XMM-Newton and
Suzaku, respectively. The color map is in units of photons cm-2 s-1. 1 pixel
corresponds to 1′′, except for 5′′ in RX J1713.7$-$3946 and 8′′ in Vela Jr.
The region highlighted by the thick line is used in the analysis in Section
4.2. The dashed boxes in Cassiopeia A, Kepler, and Tycho indicate the
reference regions used for determination of thermal components (see Section
3.2).

###  3.2 Spectrum

To extract spectra, we used specextract and nuproducts for Chandra and NuSTAR
data, respectively. The source regions are shown in Figure 1, while the
background spectra are extracted from nearby regions of the source otherwise
mentioned. The “extended=yes” set was applied to all the NuSTAR spectra except
for SN 1987A, which is spatially compatible with a point-like source. The
spectra of different epochs and different detector modules were combined using
addascaspec. Although most of the snr examined in this paper showed expanding
motions, we could safely combine the spectra of the different epochs because
the extracted regions were substantially larger than the shifts due to their
proper motions. It also should be noted that summing up the spectra of the
different epochs is not affected by time evolution and variations across the
fov of the response files. We checked this by simultaneously fitting the
spectra of the different epochs, showing the consistent result with when
fitting the combined spectrum. The spectra obtained from the representative
regions of each snr are shown in Figure 2.

In many cases NuSTAR spectra contain uncertainties related to stray light. We
therefore used NuSTAR data only for sources which it can be reliably dealt
with: namely, for G1.9$+$0.3, Cassiopeia A (the SE and NE regions), SN 1006
(the subregions 2–4 in NE and 11–13 in SW), RX J1713.7$-$3946, Vela Jr., and
SN 1987A. The background spectrum with NuSTAR here does include uncertainty of
non-uniform distribution due to stray light and instrumental components (Wik
et al., 2014), except for RX J1713.7$-$3946 which the background was
adequently subtracted as reported in Tsuji et al. (2019). As G1.9$+$0.3 and SN
1987A have small angular sizes, the background can be extracted from the
surrounding region of the source, and the non-uniform distribution of the
background should not noticeably affect the results. For largely extended
sources, such as SN 1006, RX J1713.7$-$3946, and Vela Jr., a careful treatment
with the non-uniform background is necessary. See Grefenstette et al. (2014),
Li et al. (2018) and Tsuji et al. (2019) for the detailed studies of the
background spectra in NuSTAR observations of Cassiopeia A, SN 1006, and RX
J1713.7$-$3946, respectively. In the case of the NuSTAR spectrum of Vela Jr.,
we simply checked that changing normalization of the present background by
$\pm$10% and 20% resulted in differences in the spectral parameters within 7%
and 15%, respectively.

Figure 2:  Spectrum of the representative region of each SNR with the best-fit
model. The spectra extracted from the Chandra and NuSTAR data are respectively
shown in black and red.

###  3.3 Model

We applied the model of synchrotron radiation from cooling-limited electrons,
as proposed in Zirakashvili & Aharonian (2007) (hereafter the ZA07 model),
given by

$\displaystyle\frac{dN_{X}}{d\varepsilon}$ $\displaystyle\propto$
$\displaystyle\left(\frac{\varepsilon}{\varepsilon_{0}}\right)^{-2}\Bigg{[}1+0.38\left(\frac{\varepsilon}{\varepsilon_{0}}\right)^{\nicefrac{{1}}{{2}}}\Bigg{]}^{\nicefrac{{11}}{{4}}}\exp\Bigg{[}-\left(\frac{\varepsilon}{\varepsilon_{0}}\right)^{\nicefrac{{1}}{{2}}}\Bigg{]},$
(1)

with $\varepsilon_{0}$ being a cutoff energy parameter. In this model, we
adopted the $\kappa=\sqrt{1/11}$ case in which $\kappa$ is the ratio of the
upstream magnetic field to the downstream magnetic field, $\kappa=B_{\rm
up}/B_{\rm down}$. This corresponds to an enhancement of random isotropic
magnetic field due to the standard shock compression ratio of $\sigma=4$.

The model is described with an absorbed ZA07, where the interstellar
absorption is considered by the TBabs model in XSEPC. The energy band of the
Chandra spectra is set to be 0.5–7 keV unless otherwise mentioned. For the
NuSTAR spectra, the energy band is set to be 3–20 keV for SN 1006, RX
J1713.7$-$3946, and Vela Jr.; 3–40 keV for G1.9$+$0.3 and SN 1987A; and 3–50
keV for Cassiopeia A. We performed spectral fitting of the broadband X-ray
observations (i.e., Chandra $+$ NuSTAR joint fitting) if NuSTAR data are
available. Although the combination of the Chandra and NuSTAR spectra gives us
the more precise measurement of $\varepsilon_{0}$, the fitting result with
solely the Chandra data would be less affected by systematic errors. We
confirmed this by measuring the $\varepsilon_{0}$ parameter separately using
only the Chandra and NuSTAR spectra, showing the consistent results with those
obtained by the joint fit. The fitting results are presented in Figure 2 and
Table 2. Spectral fitting was performed using XSPEC version 12.9.0.

####  3.3.1 Synchrotron dominated spectra

X-ray spectra can constrain the process of particle acceleration in snr where
the synchrotron emission dominates. Such completely featureless and nonthermal
spectra were obtained from G1.9$+$0.3, G330.2$+$1.0, RX J1713.7$-$3946, Vela
Jr., and HESS J1731$-$347111Note that these synchrotron dominated remnants
also have thermal components, particularly in the central regions.. They are
well reproduced by the absorbed ZA07 model.

On the other hand, some spectra contain non-negligible thermal emission in
Cassiopeia A, Kepler, Tycho, SN 1006, RCW 86, and SN 1987A. For these snr, we
added a thermal model, namely, TBabs $\times$ (ZA07 + thermal plasma). The
thermal model is described by VNEI for Cassiopeia A, Kepler, and Tycho, and by
Vpshock in SN 1006 and RCW 86. We added two-component thermal plasma, Vpshock
and Vequil, in SN 1987A (see, e.g, Frank et al., 2016). In the former three
remnants, a plasma temperature ($kT$) and an ionization parameter ($nt$) are
fixed to the values which are derived from the reference regions in Figure 1.
Thermal parameters are fixed to the values in the literature for SN 1006
(Miceli et al., 2009), RCW 86 (Tsubone et al., 2017), and SN 1987A (Frank et
al., 2016). Additional line emissions described by Gaussian are added when
necessary. The parameters of the thermal component are summarized in Table 6
(Section B). Although the spectra are well fitted with this model, X-rays in
the higher energy range might originate from thermal Bremsstrahlung and/or a
hot plasma instead of synchrotron radiation. Therefore, we defined synchrotron
dominated spectra as those which 1.) were featureless (i.e., line emissions
are week), and 2.) the normalization ratio of the synchrotron model to the
thermal model was relatively large. The threshold normalization ratio is
$\sim$10–20, but we cannot adopt a common value for all snr because the
thermal components differ. This selection resulted in that 5 regions
(subregions 4–8) in Kepler, 11 regions (subregions 3, 4, 6–9, 11–13, 17, and
20) in Tycho, and 12 region (subregions 1–6 and 10–15) in SN 1006 were
synchrotron dominated, and the others are referred to as thermal dominated. We
presumed that all the spectra in Cassiopeia A, RCW 86, and SN 1987A defined in
this paper belonged to the synchrotron dominated group. It should be noted
that the origin of a hard X-ray component in SN 1987A is unknown yet.

For those snr containing strong thermal components, we also fitted with only
the ZA07 model, but using the spectra above 2.5 keV where the thermal
components can be suppressed to some extent. We excluded the 6.4–6.8 keV
channel if the spectrum has a sign of iron line emission. The cutoff energy
parameter was roughly consistent between these two methods, although the one
including the thermal model yielded slightly higher cutoff energy parameter.
It should be noted that the thermal-dominated spectra defined above cannot be
fitted only with the synchrotron model, resulting in large reduced chi-squared
values of $\chi_{\nu}^{2}\geq 2$. We focus on the results of the synchrotron
dominated spectra in this paper and leave, to the future publication,
discussion of the relation between the thermal properties and the synchrotron
radiation.

####  3.3.2 Acceleration efficiency

In the present framework, the cutoff energy parameter $\varepsilon_{0}$ is a
key factor characterizing shock acceleration in snr as it is determined by the
balance between acceleration and synchrotron cooling. Zirakashvili & Aharonian
(2007) derived a relation between the cutoff energy parameter and the shock
velocity,

$\displaystyle\varepsilon_{0}=1.6\left(\frac{v_{\rm sh}}{4000~{}{\rm
km}~{}{\rm s}^{-1}}\right)^{2}\eta^{-1}~{}{\rm keV}.$ (2)

Equation 2 yields a relation determining the Bohm factor:

$\displaystyle\eta=1.6\left(\frac{v_{\rm sh}}{4000~{}{\rm km}~{}{\rm
s}^{-1}}\right)^{2}\left(\frac{\varepsilon_{0}}{\rm keV}\right)^{-1}.$ (3)

Combining the measured cutoff energy parameter and the known shock speed, we
can estimate the value of $\eta$ using Equation 3. Table 2 lists the obtained
$\eta$ parameter as well as the shock velocity in each region of snr analyzed
in Section 3.2.

####  3.3.3 Validity of cooling-limited model

We need to test the validity of the models (Equation 1 and Equation 2) which
are derived based on the assumption of the cooling-limited case, since
acceleration in young snr can be limited by an age instead. In the
synchrotron-cooling-limited case, the acceleration timescale ($\tau_{\rm
acc}$) is comparable to the synchrotron-cooling timescale ($\tau_{\rm synch}$;
defined as, e.g., Equation (23) in Zirakashvili & Aharonian (2007) for
$\kappa=\sqrt{11}^{-1}$) and is shorter than the characteristic dynamical
timescale of the source, such as the age of the accelerator ($\tau_{\rm
age}$). Therefore, to fulfill our assumption, the condition that follows
should be satisfied for the energy of synchrotron-emitting electrons:

$\displaystyle\tau_{\rm acc}\approx\tau_{\rm synch}\leq\tau_{\rm age}.$ (4)

We can place a lower limit value on the magnetic field by imposing the
condition of $\tau_{\rm synch}\leq\tau_{\rm age}$,

$\displaystyle B\geq B_{\rm low}=12\left(\frac{t_{\rm age}}{1~{}{\rm
kyr}}\right)^{-\frac{2}{3}}\left(\frac{u_{1}}{4000~{}\hbox{${\rm km}~{}{\rm
s}^{-1}$}}\right)^{-\frac{2}{3}}\eta^{\frac{1}{3}}\ \hbox{$\mu\mathrm{G}$}.$
(5)

The lower limit of $B$ calculated for each region is listed in Table 2.
$B_{\rm low}$ was estimated to be 10–20 $\mu\mathrm{G}$ in the galactic snr
older than G330.2$+$1.0 with the age of at most a thousand years. This can be
reasonably achieved given the magnetic field strength in the interstellar
medium of $\sim 4$ $\mu\mathrm{G}$ and the standard shock compression of 4.
For the younger snr, we obtained $B_{\rm low}$ of 30–40 $\mu\mathrm{G}$, which
is still acceptable when considering magnetic field amplification. Indeed, it
has been reported that the magnetic field strength is enhanced up to $\sim$100
$\mu\mathrm{G}$, inferred from the filament width (see, e.g., Bamba et al.,
2005; Berezhko & Völk, 2006). An exceptionally high value of $B_{\rm
low}=170~{}\hbox{$\mu\mathrm{G}$}$ was required for SN 1987A. It is worth
noting that, for gamma-ray emitting snr, $B_{\rm low}$ derived from Equation 5
is compatible with or smaller than that obtained by modelling the sed with the
leptonic (ic) models in the literature.

Table 2: Nonthermal parameters and properties of particle acceleration (see
also Table 6 for the parameters of the thermal model in Cassiopeia A, Kepler,
Tycho, SN 1006, RCW 86, and SN 1987A).

Region | $N_{H}$ | $\varepsilon_{0}$ | Flux (a) |  $\chi^{2}$ (dof) |  $v_{\rm sh}$ (b) |  $m$ (c) | $\eta$ |  $B_{\rm low}$ (d)
---|---|---|---|---|---|---|---|---
| $\left(10^{22}~{}{\rm cm}^{-2}\right)$ | (keV) |  |  | (${\rm km}~{}{\rm s}^{-1}$) |  |  | ($\mu\mathrm{G}$)
G1.9$+$0.3 |  |  |  |  |  |  |  |
whole | 6 $\pm$ 0.04 | 1.2 $\pm$ 0.1 | 3.25${}^{+0.02}_{-0.04}$ | 902.2 (585) | 13000$\pm$1000 | 1.1$\pm$0.2 | 14.0$\pm$2.8 | 39$\pm$7
N | 5.4 $\pm$ 0.1 | 1.2 $\pm$ 0.1 | 0.99${}^{+0.02}_{-0.01}$ | 697.3 (477) | 3600$\pm$500 | 0.4$\pm$0.2 | 1.1$\pm$0.4 | 39$\pm$9
N-RS(e) | 5.4 $\pm$ 0.1 | 1.2 $\pm$ 0.1 | 0.99${}^{+0.02}_{-0.01}$ | 697.3 (477) | 5000$\pm$1000 | 0.4$\pm$0.2 | 2.2$\pm$1.0 | 39$\pm$10
E | 6 $\pm$ 0.1 | 1.4 $\pm$ 0.1 | 2.02${}^{+0.03}_{-0.01}$ | 654.2 (492) | 13000$\pm$1000 | 1.1$\pm$0.2 | 12.1$\pm$2.6 | 37$\pm$7
S | 5.7 $\pm$ 0.1 | 0.9 $\pm$ 0.1 | 0.77${}^{+0.02}_{-0.01}$ | 573.5 (479) | 3600$\pm$500 | 0.4$\pm$0.2 | 1.5$\pm$0.5 | 44$\pm$10
S-RS(e) | 5.7 $\pm$ 0.1 | 0.9 $\pm$ 0.1 | 0.77${}^{+0.02}_{-0.01}$ | 573.5 (479) | 5000$\pm$1000 | 0.4$\pm$0.2 | 2.9$\pm$1.4 | 44$\pm$11
W | 6.2 $\pm$ 0.1 | 1.1 $\pm$ 0.1 | 1.47${}^{+0.01}_{-0.02}$ | 641.5 (480) | 13000$\pm$1000 | 1.1$\pm$0.2 | 15.2$\pm$3.3 | 40$\pm$8
Cassiopeia A |  |  |  |  |  |  |  |
N1 | 1.16${}^{+0.02}_{-0.01}$ | 0.54${}^{+0.03}_{-0.02}$ | 27.1${}^{+0.3}_{-0.4}$ | 542 (398) | 4351 $\pm$ 322 | 0.60 $\pm$ 0.04 | 3.6 $\pm$ 0.6 | 36 $\pm$ 3
N2 | 1.06 $\pm$ 0.02 | 0.44 $\pm$ 0.03 | 10.4${}^{+0.1}_{-0.3}$ | 411 (338) | 4351 $\pm$ 322 | 0.60 $\pm$ 0.04 | 4.4 $\pm$ 0.7 | 39 $\pm$ 3
S | 1.50${}^{+0.03}_{-0.04}$ | 0.49 $\pm$ 0.03 | 19.3${}^{+0.2}_{-0.4}$ | 512 (369) | 5479 $\pm$ 483 | 0.75 $\pm$ 0.07 | 6.3 $\pm$ 1.2 | 37${}^{+4}_{-3}$
NW | 1.33 $\pm$ 0.03 | 1.59${}^{+0.26}_{-0.18}$ | 18.4${}^{+0.1}_{-1.0}$ | 375 (376) | 4512 $\pm$ 483 | 0.62 $\pm$ 0.07 | 1.3 $\pm$ 0.3 | 25 $\pm$ 3
SE | 1.98 $\pm$ 0.03 | 1.72 $\pm$ 0.09 | 10.0${}^{+0.1}_{-0.2}$ | 1249 (661) | 5157 $\pm$ 644 | 0.71 $\pm$ 0.09 | 1.6 $\pm$ 0.4 | 24 $\pm$ 3
E | 1.12 $\pm$ 0.04 | 0.89${}^{+0.21}_{-0.15}$ | 4.3${}^{+0.1}_{-0.4}$ | 310 (267) | 5157 $\pm$ 322 | 0.65 $\pm$ 0.04 | 3.1${}^{+0.8}_{-0.6}$ | 30 $\pm$ 3
NE | 1.45 $\pm$ 0.02 | 1.55 $\pm$ 0.09 | 16.2 $\pm$ 0.2 | 896 (689) | 4835 $\pm$ 483 | 0.67 $\pm$ 0.07 | 1.5 $\pm$ 0.3 | 25 $\pm$ 3
Kepler |  |  |  |  |  |  |  |
1 | 0.38${}^{+0.02}_{-0.01}$ | 1.54${}^{+1.73}_{-0.58}$ | 1.08${}^{+0.04}_{-0.54}$ | 334 (170) | 2067 $\pm$ 247 | 0.49 $\pm$ 0.09 | 0.3${}^{+0.3}_{-0.1}$ | 22${}^{+9}_{-4}$
2 | 0.50 $\pm$ 0.02 | 0.39${}^{+0.24}_{-0.12}$ | 0.40${}^{+0.02}_{-0.28}$ | 273 (150) | 1972 $\pm$ 284 | 0.47 $\pm$ 0.10 | 1.0${}^{+0.7}_{-0.4}$ | 35${}^{+9}_{-6}$
3 | 0.41 $\pm$ 0.02 | 0.76${}^{+0.19}_{-0.14}$ | 1.51${}^{+0.04}_{-0.21}$ | 217 (178) | 2825 $\pm$ 322 | 0.63 $\pm$ 0.10 | 1.1${}^{+0.4}_{-0.3}$ | 28${}^{+4}_{-3}$
4 | 0.61${}^{+0.05}_{-0.02}$ | 0.64${}^{+0.15}_{-0.09}$ | 1.26${}^{+0.03}_{-0.14}$ | 143 (149) | 3622 $\pm$ 284 | 0.68 $\pm$ 0.08 | 2.1${}^{+0.6}_{-0.4}$ | 29 $\pm$ 3
5 | 0.64${}^{+0.05}_{-0.04}$ | 0.49${}^{+0.08}_{-0.06}$ | 1.51${}^{+0.04}_{-0.15}$ | 148 (164) | 3906 $\pm$ 303 | 0.70 $\pm$ 0.07 | 3.2${}^{+0.7}_{-0.6}$ | 32 $\pm$ 3
6 | 0.60 $\pm$ 0.03 | 1.65${}^{+0.72}_{-0.41}$ | 1.63${}^{+0.02}_{-0.34}$ | 170 (179) | 5727 $\pm$ 910 | 0.98 $\pm$ 0.17 | 2.0${}^{+1.1}_{-0.8}$ | 21 $\pm$ 4
7 | 0.51 $\pm$ 0.03 | 0.53${}^{+0.12}_{-0.09}$ | 0.92${}^{+0.03}_{-0.12}$ | 177 (142) | 3375 $\pm$ 569 | 0.63 $\pm$ 0.13 | 2.2${}^{+0.9}_{-0.8}$ | 31${}^{+6}_{-5}$
8 | 0.51${}^{+0.08}_{-0.06}$ | 0.85${}^{+0.29}_{-0.19}$ | 1.19${}^{+0.04}_{-0.19}$ | 176 (147) | 4589 $\pm$ 474 | 0.82 $\pm$ 0.09 | 2.5${}^{+1.0}_{-0.8}$ | 27${}^{+4}_{-3}$
9 | 0.46${}^{+0.03}_{-0.02}$ | 0.80${}^{+0.24}_{-0.18}$ | 1.13${}^{+0.03}_{-0.22}$ | 227 (154) | 3906 $\pm$ 398 | 0.71 $\pm$ 0.09 | 1.9${}^{+0.7}_{-0.6}$ | 27${}^{+4}_{-3}$
10 | 0.36 $\pm$ 0.04 | 1.94${}^{+4.32}_{-0.88}$ | 0.56${}^{+0.01}_{-0.55}$ | 215 (126) | 3072 $\pm$ 417 | 0.55 $\pm$ 0.10 | 0.5${}^{+1.1}_{-0.3}$ | 20${}^{+15}_{-4}$
11 | 0.49${}^{+0.05}_{-0.04}$ | 1.14${}^{+1.62}_{-0.48}$ | 0.34${}^{+0.005}_{-0.281}$ | 96 (92) | 4020 $\pm$ 759 | 0.72 $\pm$ 0.15 | 1.4${}^{+2.1}_{-0.8}$ | 24${}^{+12}_{-6}$
12 | 0.65${}^{+0.09}_{-0.10}$ | 1.06${}^{+0.72}_{-0.33}$ | 0.59${}^{+0.01}_{-0.31}$ | 178 (119) | 3413 $\pm$ 512 | 0.62 $\pm$ 0.11 | 1.1${}^{+0.8}_{-0.5}$ | 25${}^{+7}_{-4}$
13 | 0.54 $\pm$ 0.02 | 1.53${}^{+2.61}_{-0.64}$ | 1.27${}^{+0.03}_{-1.18}$ | 456 (177) | 1441 $\pm$ 607 | 0.34 $\pm$ 0.16 | 0.1${}^{+0.3}_{-0.1}$ | 22${}^{+15}_{-9}$
14 | 0.12 $\pm$ 0.04 | 0.42${}^{+0.16}_{-0.10}$ | 1.00${}^{+0.03}_{-0.28}$ | 314 (171) | 2162 $\pm$ 266 | 0.50 $\pm$ 0.09 | 1.1${}^{+0.5}_{-0.4}$ | 34${}^{+6}_{-5}$
Tycho |  |  |  |  |  |  |  |
1 | 0.89 $\pm$ 0.04 | 0.24 $\pm$ 0.02 | 5.5${}^{+0.2}_{-0.3}$ | 484 (283) | 2380 $\pm$ 238 | 0.37 $\pm$ 0.04 | 2.5 $\pm$ 0.5 | 40 $\pm$ 4
2 | 0.77 $\pm$ 0.04 | 0.18 $\pm$ 0.01 | 9.8${}^{+0.2}_{-0.3}$ | 401 (301) | 3000 $\pm$ 500 | 0.51 $\pm$ 0.08 | 5.0 $\pm$ 1.7 | 43 $\pm$ 7
3 | 0.59 $\pm$ 0.03 | 0.33 $\pm$ 0.01 | 23.8 $\pm$ 0.4 | 518 (369) | 3200 $\pm$ 320 | 0.55 $\pm$ 0.05 | 3.2 $\pm$ 0.6 | 35 $\pm$ 3
4 | 0.66 $\pm$ 0.01 | 0.38 $\pm$ 0.01 | 29.8${}^{+0.3}_{-0.4}$ | 561 (399) | 3580 $\pm$ 358 | 0.56 $\pm$ 0.06 | 3.4 $\pm$ 0.7 | 34 $\pm$ 3
5 | 0.91 $\pm$ 0.04 | 0.40 $\pm$ 0.02 | 30.5${}^{+0.3}_{-0.6}$ | 717 (404) | 3700 $\pm$ 370 | 0.58 $\pm$ 0.06 | 3.5 $\pm$ 0.7 | 33 $\pm$ 3
6 | 0.76${}^{+0.05}_{-0.04}$ | 0.42${}^{+0.03}_{-0.02}$ | 26.3${}^{+0.3}_{-0.6}$ | 640 (396) | 3850 $\pm$ 385 | 0.60 $\pm$ 0.06 | 3.6 $\pm$ 0.8 | 33 $\pm$ 3
7 | 0.77 $\pm$ 0.01 | 0.36 $\pm$ 0.01 | 26.4${}^{+0.3}_{-0.5}$ | 605 (396) | 3920 $\pm$ 648 | 0.61 $\pm$ 0.10 | 4.3 $\pm$ 1.4 | 34 $\pm$ 5
8 | 0.78${}^{+0.02}_{-0.01}$ | 0.39 $\pm$ 0.02 | 20.4${}^{+0.3}_{-0.5}$ | 419 (363) | 3980 $\pm$ 418 | 0.62 $\pm$ 0.07 | 4.1 $\pm$ 0.9 | 33 $\pm$ 3
9 | 0.81 $\pm$ 0.02 | 0.55${}^{+0.04}_{-0.03}$ | 16.9${}^{+0.2}_{-0.4}$ | 486 (366) | 4060 $\pm$ 634 | 0.63 $\pm$ 0.10 | 3.1 $\pm$ 1.0 | 30 $\pm$ 4
10 | 1.16 $\pm$ 0.06 | 0.35 $\pm$ 0.03 | 11.8${}^{+0.2}_{-0.3}$ | 651 (335) | 3780 $\pm$ 378 | 0.59 $\pm$ 0.06 | 4.2 $\pm$ 0.9 | 35 $\pm$ 3
11 | 0.74 $\pm$ 0.03 | 0.41${}^{+0.03}_{-0.02}$ | 17.7${}^{+0.2}_{-0.4}$ | 427 (371) | 3240 $\pm$ 468 | 0.54 $\pm$ 0.07 | 2.6 $\pm$ 0.8 | 33 $\pm$ 5
12 | 0.64${}^{+0.05}_{-0.04}$ | 0.42 $\pm$ 0.03 | 14.9${}^{+0.2}_{-0.5}$ | 536 (360) | 3480 $\pm$ 538 | 0.59 $\pm$ 0.08 | 3.0 $\pm$ 0.9 | 33 $\pm$ 5
13 | 0.74 $\pm$ 0.03 | 0.40${}^{+0.04}_{-0.03}$ | 9.2${}^{+0.2}_{-0.5}$ | 452 (308) | 3330 $\pm$ 333 | 0.57 $\pm$ 0.05 | 2.8 $\pm$ 0.6 | 33 $\pm$ 3
14 | 1.09 $\pm$ 0.04 | 0.40 $\pm$ 0.04 | 12.8${}^{+0.2}_{-0.4}$ | 601 (351) | 3510 $\pm$ 351 | 0.55 $\pm$ 0.05 | 3.1 $\pm$ 0.7 | 33 $\pm$ 3
15 | 0.55 $\pm$ 0.03 | 0.44 $\pm$ 0.03 | 12.0${}^{+0.2}_{-0.5}$ | 458 (344) | 3110 $\pm$ 311 | 0.49 $\pm$ 0.05 | 2.2 $\pm$ 0.5 | 32 $\pm$ 3
16 | 0.73 $\pm$ 0.03 | 0.41${}^{+0.03}_{-0.02}$ | 11.9${}^{+0.2}_{-0.4}$ | 601 (329) | 2210 $\pm$ 378 | 0.38 $\pm$ 0.06 | 1.2 $\pm$ 0.4 | 33 $\pm$ 5
17 | 0.65 $\pm$ 0.02 | 0.57 $\pm$ 0.02 | 56.6${}^{+0.4}_{-0.6}$ | 810 (428) | 2000 $\pm$ 400 | 0.36 $\pm$ 0.06 | 0.7 $\pm$ 0.3 | 30 $\pm$ 6
18 | 1.10${}^{+0.14}_{-0.06}$ | 0.28 $\pm$ 0.02 | 12.5${}^{+0.2}_{-0.4}$ | 664 (341) | 2360 $\pm$ 236 | 0.39 $\pm$ 0.04 | 2.1 $\pm$ 0.4 | 37 $\pm$ 4
19 | 0.80 $\pm$ 0.02 | 0.41 $\pm$ 0.02 | 17.5${}^{+0.2}_{-0.4}$ | 583 (371) | 3310 $\pm$ 331 | 0.52 $\pm$ 0.05 | 2.7 $\pm$ 0.6 | 33 $\pm$ 3
20 | 0.58 $\pm$ 0.02 | 0.38 $\pm$ 0.02 | 17.1${}^{+0.2}_{-0.3}$ | 530 (372) | 3660 $\pm$ 366 | 0.57 $\pm$ 0.06 | 3.6 $\pm$ 0.7 | 34 $\pm$ 3
G330.2$+$1.0 |  |  |  |  |  |  |  |
whole | 2.5 (fixed) | 1.17 ${}^{+0.32}_{-0.22}$ | 2.52${}^{+0.39}_{-0.03}$ | 244.6 (269) | 7000$\pm$2000 | 0.9$\pm$0.3 | 4.3${}^{+2.7}_{-2.6}$ | 17$\pm$7
SN 1006 |  |  |  |  |  |  |  |
1 | 0.13 $\pm$ 0.02 | 0.15 $\pm$ 0.01 | 6.3${}^{+0.3}_{-0.6}$ | 292 (279) | 6500 $\pm$ 500 | 0.75 $\pm$ 0.06 | 28.6 $\pm$ 4.7 | 26 $\pm$ 2
2 | 0.11 $\pm$ 0.01 | 0.37 $\pm$ 0.01 | 75 $\pm$ 1 | 609 (505) | 5900 $\pm$ 500 | 0.68 $\pm$ 0.06 | 9.6 $\pm$ 1.6 | 20 $\pm$ 2
3 | 0.10 $\pm$ 0.01 | 0.50 $\pm$ 0.02 | 88 $\pm$ 1 | 566 (525) | 5500 $\pm$ 800 | 0.64 $\pm$ 0.09 | 6.2 $\pm$ 1.8 | 18 $\pm$ 2
4 | 0.09 $\pm$ 0.01 | 0.37${}^{+0.02}_{-0.01}$ | 65${}^{+1}_{-2}$ | 555 (431) | 5000 $\pm$ 500 | 0.64 $\pm$ 0.06 | 6.9 $\pm$ 1.4 | 20 $\pm$ 2
5 | 0.12 $\pm$ 0.01 | 0.31 $\pm$ 0.02 | 17${}^{+0}_{-1}$ | 290 (277) | 4900 $\pm$ 500 | 0.62 $\pm$ 0.06 | 8.0 $\pm$ 1.7 | 21 $\pm$ 2
6 | 0.09 $\pm$ 0.01 | 0.20 $\pm$ 0.01 | 21 $\pm$ 1 | 403 (318) | 5800 $\pm$ 500 | 0.74 $\pm$ 0.06 | 17.2 $\pm$ 3.1 | 24 $\pm$ 2
7 | 0.11 | 0.13${}^{+0.03}_{-0.04}$ | 1.3${}^{+0.3}_{-0.7}$ | 325 (171) | 7000 $\pm$ 1200 | 0.89 $\pm$ 0.15 | 37.4${}^{+16.1}_{-16.4}$ | 27 $\pm$ 5
8 | 0.11 | 0.56${}^{+0.56}_{-0.23}$ | 5.6${}^{+0.3}_{-4.2}$ | 233 (179) | 7200 $\pm$ 1000 | 0.92 $\pm$ 0.13 | 9.5${}^{+9.9}_{-4.7}$ | 17${}^{+6}_{-3}$
9 | 0.14${}^{+0.05}_{-0.04}$ | 0.09 $\pm$ 0.01 | 2.3${}^{+0.3}_{-0.4}$ | 295 (202) | 5000 $\pm$ 800 | 0.64 $\pm$ 0.10 | 27.4${}^{+9.2}_{-9.1}$ | 31 $\pm$ 5
10 | 0.15 $\pm$ 0.02 | 0.20 $\pm$ 0.01 | 27 $\pm$ 1 | 327 (319) | 5000 $\pm$ 800 | 0.64 $\pm$ 0.10 | 12.5 $\pm$ 4.1 | 24 $\pm$ 4
11 | 0.14 $\pm$ 0.01 | 0.30 $\pm$ 0.01 | 60 $\pm$ 1 | 574 (414) | 5500 $\pm$ 500 | 0.70 $\pm$ 0.06 | 10.4${}^{+2.0}_{-1.9}$ | 21 $\pm$ 2
12 | 0.11 $\pm$ 0.01 | 0.31 $\pm$ 0.01 | 48 $\pm$ 1 | 498 (457) | 5900 $\pm$ 800 | 0.75 $\pm$ 0.10 | 11.4 $\pm$ 3.1 | 21 $\pm$ 3
13 | 0.10${}^{+0.02}_{-0.01}$ | 0.32 $\pm$ 0.02 | 37 $\pm$ 1 | 439 (352) | 7500 $\pm$ 400 | 0.96 $\pm$ 0.05 | 18.0 $\pm$ 2.2 | 21 $\pm$ 1
14 | 0.12 $\pm$ 0.02 | 0.23 $\pm$ 0.02 | 10.3${}^{+0.4}_{-0.9}$ | 216 (228) | 5500 $\pm$ 500 | 0.70 $\pm$ 0.06 | 13.6 $\pm$ 2.7 | 23 $\pm$ 2
15 | 0.01${}^{+0.04}_{-0.01}$ | 0.17${}^{+0.01}_{-0.02}$ | 2.3${}^{+0.2}_{-0.5}$ | 223 (197) | 7000 $\pm$ 1000 | 0.89 $\pm$ 0.13 | 28.8${}^{+8.5}_{-9.1}$ | 25${}^{+3}_{-4}$
16 | 0.23${}^{+0.05}_{-0.03}$ | 0.03${}^{+0.02}_{-0.01}$ | 0.04${}^{+0.01}_{-0.03}$ | 283 (184) | 3000 $\pm$ 1200 | 0.44 $\pm$ 0.18 | 31.4${}^{+30.7}_{-28.2}$ | 46${}^{+19}_{-18}$
17 | 0.12 | 0.03${}^{+0.02}_{-0.01}$ | 0.53${}^{+0.06}_{-0.49}$ | 173 (122) | 3000 $\pm$ 1200 | 0.38 $\pm$ 0.15 | 27.8${}^{+27.3}_{-23.2}$ | 44${}^{+18}_{-17}$
RX J1713.7$-$3946 |  |  |  |  |  |  |  |
NW | 0.75 $\pm$ 0.01 | 1.14$\pm$0.06 | 11.5${}^{+0.3}_{-0.2}$ | 687.9 (587) | 3900$\pm$300 | 0.7$\pm$0.1 | 1.4$\pm$0.3 | 9.7$\pm$0.9
RCW 86 |  |  |  |  |  |  |  |
NE | 0.33 (fixed) | 0.32 $\pm$0.01 | 3.22${}^{+0.03}_{-0.05}$ | 527 (407) | 2500$\pm$700 | 0.3$\pm$0.1 | 2.0$\pm$1.1 | 14$\pm$4
Vela Jr. |  |  |  |  |  |  |  |
NW | 0.67 (fixed) | 0.54 $\pm$ 0.04 | 3.9$\pm$0.1 | 289.3 (258) | 2000$\pm$600 | 0.5$\pm$0.2 | 0.7$\pm$0.5 | 8${}^{+3}_{-4}$
HESS J1731$-$347 |  |  |  |  |  |  |  |
NE | 1 (fixed) | 0.97 ${}^{+0.46}_{-0.26}$ | 3.6${}^{+1.2}_{-0.1}$ | 102.7 (86) | 2500$\pm$1000 | 0.5$\pm$0.1 | 0.65${}^{+0.61}_{-0.55}$ | 6${}^{+3}_{-12}$
SN 1987A |  |  |  |  |  |  |  |
whole | 0.18$\pm$0.02 | 0.70${}^{+0.36}_{-0.08}$ | 2.49${}^{+0.07}_{-0.02}$ | 610.7 (497) | 6700$\pm$800 | 0.9$\pm$0.1 | 6.5$\pm$2.6 | 165${}^{+16}_{-17}$
whole-RS(e) | 0.18$\pm$0.02 | 0.70${}^{+0.36}_{-0.08}$ | 2.49${}^{+0.07}_{-0.02}$ | 610.7 (497) | 4800$\pm$300 | 0.9$\pm$0.1 | 3.4$\pm$1.0 | 165${}^{+11}_{-13}$

Note. — (a) Flux is calculated for the 4–6 keV band in Cassiopeia A, Kepler,
Tycho, and SN 1006 in units of $10^{-14}$ ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm
s}^{-1}$, while is for the 2–10 keV band in the other SNRs in units of
$10^{-12}$ ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$.

(b) References (see also Table 1). G1.9$+$0.3: Borkowski et al. (2017).
Cassiopeia A: Patnaude & Fesen (2009). Kepler: Vink (2008); Katsuda et al.
(2008b). Tycho: Katsuda et al. (2010a); Williams et al. (2013). G330.2$+$1.0:
Borkowski et al. (2018). SN 1006: Winkler et al. (2014). RX J1713.7$-$3946:
Tsuji & Uchiyama (2016); Acero et al. (2017). RCW 86: Yamaguchi et al. (2016).
Vela Jr.: Katsuda et al. (2008a); Allen et al. (2015). HESS J1731$-$347: H. E.
S. S. Collaboration et al. (2011). SN 1987A: Frank et al. (2016).

(c) Expansion parameter.

(d) Lower limit of magnetic field required for cooling-limited assumption
(Equation 5).

(e) RS: Assuming the reverse shock or reflection shock.

##  4 Results

###  4.1 Relation between shock speed and cutoff energy

In this section we study the possible correlation between the cutoff energy
from our analysis in Section 3.3 and the shock speed taken from the literature
(see references in Tables 1 and 2). Our results show that there is a
significant variance of both the cutoff energy parameter and the shock
velocity in the considered SNR (see in Figure 3). Below we discuss the result
for each SNR, and in Section 4.2 we consider the source averaged acceleration
efficiencies. We also show in Figure 3 (top left panel) a compilation of the
largest-$\varepsilon_{0}$ region of each SNR, which is highlighted by the
thick line in Figure 1 and nearly corresponds to the maximum shock-speed
region. In this maximum-$\varepsilon_{0}$ region, the acceleration is the most
efficient, and the result is discussed in Section 4.2 and Section 5.2.

In Figure 3, we also show the obtained $v_{\rm sh}$–$\varepsilon_{0}$ relation
of G1.9$+$0.3, Cassiopeia A, Kepler, Tycho, and SN 1006. For Kepler, Tycho,
and SN 1006, we separately showed the thermal dominated and nonthermal
(synchrotron) dominated spectra (see Section 3.3.1). As mentioned in Section
3.3.1, in the thermal dominated spectrum the cutoff energy might not be
appropriate because the spectrum in the higher energy range, fitted by the
ZA07 model, may not be of synchrotron radiation origin, but Bremsstrahlung or
hotter thermal plasma. Furthermore, the proper motion in the thermal-dominated
regions might represent the speed of ejecta components rather than blast
waves. Hence, our objective to examine acceleration efficiency cannot be
achieved in these thermal regions (i.e., both cutoff energy and shock speed
would not be robust in terms of particle acceleration). There exist variations
of the shock speed and cutoff energy across the remnants in Figure 3, and the
summary will be given in Section 5.1. Note that the spatially resolved $v_{\rm
sh}$–$\varepsilon_{0}$ plots can be obtained only in these five snr, since the
shock velocity was not measured across the remnant in the other snr222The
proper motion was measured to be different from region to region in
G330.2$+$1.0, as reported in Williams et al. (2018), but due to its faint
radiation we could not obtain the spatially resolved cutoff energy. In RCW 86,
proper motion measurement with X-ray observations was conducted in the
northeast rims (Yamaguchi et al., 2016), while proper motion with optical
observations was measured in the other regions. However X-ray and optical
measurements were different by a factor of about 2. We made use of the shock
speed measured with X-rays since accelerated electrons are traced by the
synchrotron X-ray emission.. In addition to them, the $v_{\rm
sh}$–$\varepsilon_{0}$ relation in the north west rim of RX J1713.7$-$3946 was
already provided in details in Tsuji et al. (2019). We make a brief
description of the nature of particle acceleration of individual snr in the
following and present a systematical tendency of particle acceleration of
young snr in Section 4.2.

Figure 3:  $\varepsilon_{0}$–$v_{\rm sh}$ diagram. Green solid, dashed, dash-
dotted, and dotted lines indicate $\eta$ of 1, 3, 10, and 20, respectively.
Top left: Each plot is taken from the region having the largest
$\varepsilon_{0}$ (or the maximum $v_{\rm sh}$) in each SNR, where the region
is highlighted in Figure 1. For Kepler, Tycho, and SN 1006, results of the
synchrotron-dominated regions are shown in black, while those of the thermal-
dominated regions are shown in grey (see Section 3.3.1). In G1.9$+$0.3, NR and
SR indicate the results assuming that the emission originates from the reverse
shock in the north and south regions, respectively.

####  4.1.1 G1.9$+$0.3

G1.9$+$0.3, known as the youngest SNR in our galaxy, showed that the cutoff
energy parameters were slightly variable inside the remnant: $\varepsilon_{0}$
was obtained to be 1.1–1.2 keV in the entire remnant, the western and northern
regions, whereas it was slightly higher in the eastern rim
($\varepsilon_{0}\approx 1.4$ keV) and lower in the southern part
($\varepsilon_{0}\approx 0.9$ keV). It should be noted that these differences
in the measured cutoff energy parameters may have been underestimated because
of the overlapped subregions and the limited angular resolution of NuSTAR. In
the case of forward shock, the Bohm factor was estimated to be 12–15 in the
bright rim of E and W with the faster shock velocity of $\sim$13000 ${\rm
km}~{}{\rm s}^{-1}$. This was roughly consistent with $\eta\sim 20$ obtained
in Aharonian et al. (2017). However, $\eta$ was 1–2 in the fainter area of N
and S, with the slower shock speed of $\sim$4000 ${\rm km}~{}{\rm s}^{-1}$.
The $\eta$ parameter averaged over the entire remnant (the N, E, S, and W
regions) was $7.5\pm 0.8$. If we assume the outward proper motions in the N
and S rims, which were measured in Borkowski et al. (2017), correspond to the
reverse shock, the upstream speed in the rest frame of the reverse shock is
given by

$\displaystyle u_{1}=\frac{R_{\rm ref}}{t_{\rm age}}-v_{\rm obs}$ (6)

(Sato et al., 2018; Tsuji et al., 2019). Indeed the reverse shock might play a
major role in the northern region where the radio synchrotron emission is
strong, and the proper motion of the shock is reduced, as proposed in (Brose
et al., 2019). Using Equation 6, $u_{1}$ in N and S is approximately
calculated to be 5000 ${\rm km}~{}{\rm s}^{-1}$, resulting in $\eta\sim$2–3.
We apparently see a significant difference in the acceleration efficiency
across the remnant: the Bohm factor in the bright E–W rim is approximately one
order of magnitude higher than that in the N–S part in the case of either the
forward shock or reverse shock.

####  4.1.2 Cassiopeia A

The seven sectors (N1–2, NE, E, SE, S, and NW) are roughly compatible with the
regions which Patnaude & Fesen (2009) measured proper motions and showed that
the speeds are almost constant of $\sim$5000 ${\rm km}~{}{\rm s}^{-1}$,
assuming a distance of 3.4 kpc. The cutoff energy parameters indicate
significant varieties, showing the highest value in the SE rim and the lowest
value in the N2 region. Because of the constant shock speeds and different
cutoff energy parameters, the $\eta$ values also varied from 1.5 in SE and 6.3
in S. This trend was consistent with the previous result in Stage et al.
(2006). In the case of the forward shock in Cassiopeia A, the variation of
$\eta$ is attributed to the different values of $\varepsilon_{0}$, which is
opposite to G1.9$+$0.3 showing the variable shock speed and roughly constant
$\varepsilon_{0}$. If we averaged the $\eta$ parameters obtained in the seven
small regions, the averaged $\eta$ was derived to be $3.1\pm 0.3$.

####  4.1.3 Kepler’s SNR

We defined 14 subregions that are almost comparable to those defined by
Katsuda et al. (2008b), who performed proper motion measurements. Five of
these subregions (labeled as 4–8) turned out to be synchrotron dominated and
highlighted in Figure 3. The observed $v_{\rm sh}$–$\varepsilon_{0}$ relation
of these five regions indicated a clear correlation of $\varepsilon_{0}$ and
$v_{\rm sh}$ (i.e., the higher the shock speed is, the larger the cutoff
energy parameter is). In addition, the correlation is well described with the
theoretical curve with $\eta$ of 2–3. This clarifies that the acceleration
efficiency is constant ($\eta=$2–3) and independent of the sites within the
forward shock located in the outermost eastern and southern rims of Kepler.
The averaged $\eta$ over the 5 synchrotron dominated regions was
$2.4^{+0.3}_{-0.4}$. Our analysis, for the first time, found that the particle
acceleration in Kepler at the evolutional age of $\sim$400 years significantly
deviates from the maximum rate of $\eta=1$.

####  4.1.4 Tycho’s SNR

Figure 3 presents the $v_{\rm sh}$–$\varepsilon_{0}$ scatter plot measured in
Tycho, in which 11 regions (subregions 3, 4, 6–9, 11–13, 17, and 20) are
synchrotron dominated. Tycho shows slightly variable but theoretical predicted
plots with $\varepsilon_{0}=0.3$–0.6 keV and $v_{\rm sh}=3000$–4000 ${\rm
km}~{}{\rm s}^{-1}$ (Katsuda et al., 2010a; Williams et al., 2013). This
corresponds to nearly constant acceleration efficiency with $\eta$ of 3–4 (and
the average value of $3.1\pm 0.3$) in the forward shock of Tycho. One
exceptional region is subregion 17 located in the eastern rim. The $\eta$
value should be $\sim$1 on the subregion 17, inferred from a slow shock speed
of 2000 ${\rm km}~{}{\rm s}^{-1}$ and a higher cutoff energy parameter of 0.57
keV. This nonthermal radiation in the E rim might be unique and behave
differently from the other parts because a precursor has been detected (Lee et
al., 2010), and a dense clump known as knot g is present (see, for example,
Ghavamian et al. (2000)). The dense density might affect the lower shock
velocity of knot g. Note that our result was roughly consistent with that of
Lopez et al. (2015), although the model and region definition were different
(i.e., they also showed the greater roll-off energies in the higher shock
speeds and the exception of knot g).

####  4.1.5 SN 1006

The filaments in the NE and SW limbs were dominated by synchrotron emission.
Thus, the spectra of the subregions 2–4 (NE) and 11–13 (SW), extracted both
with Chandra and NuSTAR, represent the typical properties of the synchrotron
radiation from this remnant. They showed somewhat small cutoff energy
parameters ($\varepsilon_{0}=0.3$–0.4 keV) and relatively high shock speeds
($v_{\rm sh}=5000$–8000 ${\rm km}~{}{\rm s}^{-1}$), suggesting inefficient
particle acceleration ($\eta\sim 10$) in SN 1006. Note that the cutoff energy
parameters obtained in this paper were roughly consistent with those of Li et
al. (2018). The other subregions (1, 5–10, and 14–17) were analyzed using only
Chandra, as NuSTAR covered only two limbs in the NE and SW parts. The
synchrotron dominated regions (1–6 and 10–15) showed a clear variation in
$\varepsilon_{0}$ but a nearly constant value in $v_{\rm sh}$. The averaged
$\eta$ parameter over the synchrotron dominated spectra was $14\pm 1$.

The validity of the loss-limited assumption is controversial in SN 1006.
Katsuda et al. (2010b) suggested that the accelerated electrons may not be
loss-limited, while Miceli et al. (2013) found evidence supporting the loss
limited scenario. In this work, we obtained $B$ should be larger than 20–30
$\mu\mathrm{G}$ for the ZA07 model to be applicable to SN 1006. This condition
can be sufficiently fulfilled, adopting $B\sim 300$ $\mu\mathrm{G}$ that was
estimated by the filament width (Bamba et al., 2003; Völk et al., 2005).

We also investigated a shock-obliquity dependency on particle acceleration in
SN 1006. SN 1006 is a unique remnant of which the ambient magnetic field was
reported to be along the galactic plane that is approximately 60∘
counterclockwise inclined from the north (see, e.g., Reynoso et al., 2013).
This oriented magnetic field, combined with the Type Ia explosion in a high
Galactic latitude of 14.6∘, makes SN 1006 an ideal laboratory for studying the
dependence of particle acceleration on magnetic field configurations (i.e.,
parallel or perpendicular shocks). The azimuthal variations of roll-off
frequencies (with the SRCUT model in XSPEC) and shock velocities were studied
in Rothenflug et al. (2004); Miceli et al. (2009); Winkler et al. (2014). Our
measurements of $\varepsilon_{0}$ indicated the larger cutoff energy
parameters near the polar regions (NE and SW), which is consistent with the
previous studies. The observed variations of $\varepsilon_{0}$ and $v_{\rm
sh}$ enabled us to reproduce a clear correlation between the shock obliquity
$\theta_{\rm Bn}$ and acceleration efficiency $\eta$ in the synchrotron
dominated regions, as illustrated in Figure 4. It should be noted that in NW
and SE (subregions 7–9 and 16–17), where $\theta_{\rm Bn}$ is close to 90∘,
the thermal emission dominates and the proper motion there is representative
of X-ray emitting ejecta knots, not shock waves. The shock-obliquity
dependence on particle acceleration is discussed in Section 5.

####  4.1.6 SN 1987A

Although the soft X-ray spectrum of SN 1987A below $\sim 7$ keV is well-fitted
by a thermal model of two-temperature plasmas (i.e., Vpshock and Vequil in
XSPEC), there exists an apparent excess in the NuSTAR spectrum above 10 keV,
which cannot not described by the thermal model. The origin of the hard X-ray
emission with NuSTAR remains unclear (Boggs et al., 2015; Reynolds et al.,
2015). Malyshev et al. (2019) recently reported GeV emission detected with
Fermi likely from SN 1987A. Presuming that this GeV radiation originates from
SN 1987A, there should be a population of nonthermal particles, and thus its
synchrotron emission could be considered as a plausible interpretation of the
hard X-ray excess. In this paper, we simply assumed that the hard component of
NuSTAR is described by the nonthermal (synchrotron) radiation, resulting in
$\varepsilon_{0}\approx 0.7$ keV.

The expansion velocity of SN 1987A was reduced from $\sim$7000 ${\rm
km}~{}{\rm s}^{-1}$ to $\sim$1000 ${\rm km}~{}{\rm s}^{-1}$ when impacting
with the equatorial ring around 2003. We are interested in the shock velocity
which is relevant for the acceleration of the X-ray emitting particles, thus
adopted the former value (before the impact) for the shock speed of SN 1987A.
The following two factors should be addressed: (i) X-ray emitting particles
cannot be accelerated if the shock speed is as slow as 1000 ${\rm km}~{}{\rm
s}^{-1}$. (ii) The cooling time of TeV electrons even in a strong magnetic
field of $B\sim 300$ $\mu\mathrm{G}$ exceeds significantly 10 years, $t_{\rm
syn}=140~{}{\rm yr}~{}(E/1~{}{\rm
TeV})^{-1}~{}(B/300~{}{\rm\hbox{$\mu\mathrm{G}$}})^{-2}$. Therefore, in the
framework on the adopted stationary approach model by Zirakashvili & Aharonian
(2007), we should use the shock velocity relevant to the moment of TeV
particle acceleration (i.e., we chose $v_{\rm sh}\approx 7000$ ${\rm
km}~{}{\rm s}^{-1}$). Combined with the cutoff energy parameter, the Bohm
factor was estimated to be $\sim$7\. The detailed discussion of particle
acceleration in SN 1987A will be given in the future publication.

Figure 4:  The $\eta$ variation in SN 1006 as a function of shock obliquity,
assuming the shock inclination is 60∘ from the north. The black and grey plots
show the synchrotron dominated and thermal dominated regions, respectively.

####  4.1.7 Others

We could not investigate the spatially resolved $\varepsilon_{0}$–$v_{\rm sh}$
relation in the other SNRs (i.e., G330.2$+$1.0, RX J1713.7$-$3946, RCW 86,
Vela Jr., HESS J1731$-$347, and SN 1987A) due to lacking observational data.
The proper motions were measured along the rim of G330.2$+$1.0 (Borkowski et
al., 2018), but due to the large uncertainty of the shock speed and the cutoff
energy parameter the spatially resolved $\varepsilon_{0}$–$v_{\rm sh}$ plot
was not presented in this paper. The $\varepsilon_{0}$–$v_{\rm sh}$ diagram of
the NW rim in RX J1713.7$-$3946 was presented in Tsuji et al. (2019). To date
the shock speeds were obtained only in specific regions, the NE rim in RCW 86
and the NW rim in Vela Jr., since these two SNRs have relatively large angular
sizes of $\sim$0.7∘ and $\sim$2∘, respectively.

###  4.2 Systematic tendency of $\eta$

We obtained the $\eta$ values that are indicative of acceleration efficiency
(Bohm factor) in the 11 individual snr. This subsection presents the
systematic trend of $\eta$ when all these snr are considered together. Because
five snr showed significantly different properties of particle acceleration
depending on the sites (Section 4.1), we investigated the systematical $\eta$
values estimated from the selected regions based on the
largest-$\varepsilon_{0}$ regions in Section 4.2.1. We also present the
results using the $\eta$ values averaged over the remnants. Section 4.2.2
presents the result of different supernova explosion types.

####  4.2.1 Largest-$\varepsilon_{0}$ (maximum-$v_{\rm sh}$) region

Figure 5:  $\eta$ as a function of the age (left) and expansion parameter $m$
(right). The $\eta$ parameters averaged over the synchrotron dominated regions
are shown in red, and the model fitted with the averaged values is shown with
the red dashed line.

We investigated acceleration efficiency ($\eta$) obtained in the region with
the largest $\varepsilon_{0}$ of each snr. Note that the
largest-$\varepsilon_{0}$ region nearly corresponded to the fastest-$v_{\rm
sh}$ region, except for SN 1006, which showed a slightly inverse correlation
between $\varepsilon_{0}$ and $v_{\rm sh}$. Figure 5 (left) illustrates the
relation between the estimated $\eta$ value and the age of each snr. We
confirmed a tendency in which $\eta$ decreased as the age of snr increased. In
snr younger than a few 100 years, the acceleration efficiency substantially
deviated from the Bohm limit (i.e., $\eta>1$). In the later evolutional stage
older than a few 1,000 years, the acceleration proceeded at the most efficient
rate (i.e., Bohm limit with $\eta\approx 1$).

To quantify the relation between $\eta$ and $t_{\rm age}$, we fit the observed
diagram using an empirical equation,

$\displaystyle\eta=C_{\rm age}\left(\frac{t_{\rm age}}{1\ {\rm
kyr}}\right)^{-\delta_{\rm age}},$ (7)

where $C_{\rm age}$ and $\delta_{\rm age}$ indicate a constant value of $\eta$
at the age of 1 kyr and a slope of the age–$\eta$ relation, respectively. With
chi-squared fitting, we obtained $C_{\rm age}=1.5\pm 0.2$ and $\delta_{\rm
age}=0.41\pm 0.08$. A correlation coefficient of the logarithm-scale plots in
Figure 5 (left) was approximately $-$0.70 with a significance of 2.4$\sigma$.
The best-fit model is indicated by the grey line, and the 90% uncertainty is
indicated by the light grey region in Figure 5. The best-fit parameters are
summarized in Table 3. When we considered the averaged $\eta$ instead of the
largest-$\varepsilon_{0}$ region for G1.9$+$0.3, Cassiopeia A, Kelper, Tycho,
and SN 1006, $C_{\rm age}$ and $\delta_{\rm age}$ were $2.2\pm 0.2$ and
$0.37\pm 0.06$, respectively, and a correlation coefficient was $-0.65$ with a
significance of 2.2$\sigma$. Equation 7 nicely reproduces the observation,
although G1.9$+$0.3 and SN 1006 significantly deviate from the best-fit model.
The deviations in the two snr might arise from the fact that they are limited
by some different factors, such as magnetic field or age, instead of cooling.
If we omitted the results of G1.9$+$0.3 and SN 1006, the correlation
coefficient was obtained to be $-$0.81 with 2.7$\sigma$ ($-$0.89 with
3.0$\sigma$ using the averaged $\eta$).

Figure 5 (right) indicates the observed relation between the $\eta$ value and
an expansion parameter $m$, which is an alternative parameter indicating the
evolutional phase of snr. Note that $m$ refers to the evolutional stage at
which the snr currently resides, and it depends not only on the snr age, but
also on other physical parameters (e.g., ambient density, progenitor mass, and
total energy of sne). The radius of the snr, $R$, and $m$ are described
respectively with

$\displaystyle R\propto t^{m},$ (8)

and

$\displaystyle m\equiv\frac{v_{\rm obs}}{R/t_{\rm age}},$ (9)

where $v_{\rm obs}$ is the observed expansion velocity. $m=1$ indicates the
earliest evolutional stage (also known as the free expansion phase or the
ejecta dominated (ED) phase). $m=0.4$ corresponds to the Sedov stage at which
the snr continues to expand self-similarly. As the snr evolve from $m=1$ to
0.4, $\eta$ decreases to reach $\sim$1 in Figure 5 (right). We also fit the
observed $m$–$\eta$ diagram using an experimental model given by

$\displaystyle\eta=C_{m}m^{\delta_{m}}.$ (10)

The best-fit values were $C_{\rm m}=5.2\pm 1.0$ and $\delta_{\rm m}=4.0\pm
0.8$. The correlation coefficient of the logarithm plots in Figure 5 (right)
was approximately 0.60 with a significance of 1.9$\sigma$. Using the average
value of the $\eta$ parameter, we obtained $C_{\rm m}=3.3\pm 0.4$ and
$\delta_{\rm m}=0.67\pm 0.34$ and a correlation coefficient of 0.52 with a
significance of 1.6$\sigma$.

Here we summarize the results of 6 SNRs (G1.9$+$0.3, Cassiopeia A (the SE and
NE regions), SN 1006 (the NE and SW regions), RX J1713.7$-$3946, Vela Jr., and
SN 1987A) which were analyzed with the available NuSTAR data. Although the
analysis using only the Chandra is reliable to some extent, the NuSTAR
observation in the higher X-ray domain enables us to derive the more robust
result. When we reduce to those 6 SNRs, the best-fit parameters are $(C_{\rm
age},\ \delta_{\rm age})=(1.5\pm 0.3,0.41\pm 0.09)$ with $\chi^{2}$ of 16 (dof
of 3) and $(C_{m},\ \delta_{m})=(5.3\pm 0.7,7.7\pm 1.2)$ with $\chi^{2}$ of
6.1, which are roughly consistent with those derived by using all SNRs.

####  4.2.2 Supernova explosion type

In this subsection, we present the $\eta$ values from two types of sne: Type
Ia SN which is driven by a thermonuclear explosion of a white dwarf star and
Type II which is a core-collapse explosion. In general, the ambient
interstellar medium of a Type Ia snr is rarefied, whereas the circumstellar
medium surrounding a core-collapse snr is clumpy and complex due to stellar
wind from the massive progenitor star.

In this paper, G1.9$+$0.3, Kepler, Tycho, SN 1006, and RCW 86 are classified
as Type Ia, and Cassiopeia A, G330.2$+$1.0, RX J1713.7$-$3946, Vela Jr., HESS
J1731$-$347, and SN 1987A are classified as core-collapse snr. We group snr
with cco as Type II. It should be noted that this might not be a precise
classification. Figure 6 shows age–$\eta$ and $m$–$\eta$ diagrams of Type Ia
(top) and Type II (bottom). The best-fit parameters are $(C_{\rm age},\
\delta_{\rm age})=(0.85\pm 0.55,1.5\pm 0.5)$ for Type Ia and $(C_{\rm age},\
\delta_{\rm age})=(1.4\pm 0.3,\ 0.41\pm 0.09)$ for Type II. If we took the
averaged $\eta$, $(C_{\rm age},\ \delta_{\rm age})=(4.4\pm 0.6,\ -0.29\pm
0.17)$ for Type Ia and $(C_{\rm age},\ \delta_{\rm age})=(1.8\pm 0.2,0.40\pm
0.07)$ for Type II. As seen in Figure 6 and Table 3, the observed $\eta$ value
of Type Ia SNRs does not have a significant correlation with the age nor the
expansion parameter, while Type II SNRs show the relatively strong
correlation. Overall, $\eta$ of core-collapse snr appears smaller than that of
Type Ia snr at $t_{\rm age}\lesssim 10000$ years. We should be careful
regarding the small number of data samples and the fact that Type II snr
presented here consist of relatively older snr.

Figure 6:  Same as Figure 5 for Type I (top) and Type II (bottom).  Table 3:
Parameters of evolution of $\eta$ (Equation 7 and Equation 10)

| $C_{\rm age}$ | $\delta_{\rm age}$ |  $\chi^{2}$ (dof) | $\rho^{({\rm a})}$ | $C_{m}$ | $\delta_{m}$ |  $\chi^{2}$ (dof) | $\rho^{({\rm a})}$
---|---|---|---|---|---|---|---|---
All SNRs | 1.5$\pm$0.2 | 0.41$\pm$0.08 | 17 (8) |  $-$0.70 (2.4$\sigma$) | 5.2$\pm$1.0 | 4.0$\pm$0.8 | 19 (8) | 0.60 (1.9$\sigma$)
All SNRs with averaged $\eta$ | 2.2$\pm$0.2 | 0.37$\pm$0.06 | 89 (8) |  $-$0.65 (2.2$\sigma$) | 3.3$\pm$0.4 | 0.67$\pm$0.34 | 118 (8) | 0.52 (1.6$\sigma$)
Type Ia | 0.85$\pm$0.55 | 1.5$\pm$0.5 | 6.8 (2) |  $-$0.53 (0.9$\sigma$) | 4.4$\pm$1.2 | 0.69$\pm$0.63 | 11 (2) | 0.50 (0.86$\sigma$)
Type Ia with averaged $\eta$ | 4.4$\pm$0.6 | -0.29$\pm$0.17 | 75 (2) |  $-$0.17 (0.27$\sigma$) | 3.5$\pm$0.4 | 0.023$\pm$0.30 | 77 (2) | 0.32 (0.5$\sigma$)
Type II | 1.4$\pm$0.3 | 0.41$\pm$0.09 | 2.5 (3) |  $-$0.86 (2.2$\sigma$) | 9.6$\pm$3.2 | 6.2$\pm$1.3 | 1.6 (3) | 0.95 (2.9$\sigma$)
Type II with averaged $\eta$ | 1.8$\pm$0.2 | 0.40$\pm$0.07 | 1.9 (3) |  $-$0.90 (2.4$\sigma$) | 5.8$\pm$2.2 | 3.1$\pm$1.2 | 19 (3) | 0.94 (2.8$\sigma$)

Note. — (a) Pearson’s correlation coefficient.

##  5 Discussion

###  5.1 Variety in $v_{\rm sh}$–$\varepsilon_{0}$

First, we summarize the properties of particle acceleration in the individual
sources. Significant variations across the remnants were confirmed in
G1.9$+$0.3, Cassiopeia A, Kepler, Tycho, and SN 1006 (see also Tsuji et al.,
2019, for the NW of RX J1713.7$-$3946). Spatially resolved studies could not
be conducted in the other snr due to lacking the observational data. Figure 7
shows the $v_{\rm sh}$–$\varepsilon_{0}$ scatter plots of the former five snr,
which showed significant varieties, indicating four types of $v_{\rm
sh}$–$\varepsilon_{0}$ dependencies. In G1.9$+$0.3, the cutoff energy
parameters showed small variations, but the shock speeds were different from
region to region. This was opposite to the case of Cassiopeia A, which the
cutoff energy parameters were variable, whereas the shock speeds were nearly
constant. Only Kepler and Tycho (expect for knot g) presented the $v_{\rm
sh}$–$\varepsilon_{0}$ scatter plots nicely described with the theoretical
prediction of $\varepsilon_{0}\propto v_{\rm sh}^{2}$. Finally, SN 1006 showed
a slightly inverse trend, in which the cutoff energy parameters tended to be
smaller in the higher shock-speed regions.

These features were interpreted as follows. Particle acceleration in Kepler
and Tycho exhibited the theoretically predicted $v_{\rm sh}$–$\varepsilon_{0}$
relation. Therefore, the standard framework was applicable in these two snr.
However, in the other snr in which the observed $v_{\rm sh}$–$\varepsilon_{0}$
plots were not described by the theoretical curve, particle acceleration may
have been strongly affected by the surrounding environment. In the case of SN
1006, particle acceleration was probably determined by the ambient magnetic
field, because the resulting Bohm factors were correlated with the shock
obliquities (see Figure 4 and also Miceli et al., 2009). For G1.9$+$0.3, the
different shock speeds may have been caused by differences in the ambient
density, resulting in variable Bohm factors across the remnant. It is more
likely that G1.9$+$0.3, at the age of $\sim$190 years, is too young to
accelerate sufficiently energetic particles, and our framework of the cooling-
limited electron is not appropriate. Instead, the particle acceleration in
G1.9$+$0.3 might be limited by the age. Finally, in the case of Cassiopeia A,
the acceleration efficiency might be related to thermal emission (Tsuji,
2020). The investigation of the nonthermal and thermal components will be
discussed in the separated paper.

Figure 7:  $v_{\rm sh}$–$\varepsilon_{0}$ scatter plots of the synchrotron-
dominated regions in G1.9$+$0.3, Cassiopeia A, Kepler, Tycho (expect for knot
g), and SN 1006. Green solid, dashed, dash-dotted, and dotted lines indicate
$\eta$ of 1, 3, 10, and 20, respectively.

###  5.2 Evolution of $\eta$

The evolution of acceleration efficiency, as previously mentioned and shown in
Figure 5, was observationally revealed for the first time in this study. The
Bohm factor $\eta$ characterizes the diffusion coefficient and is explicitly
related to the magnetic turbulence. $\eta$ can also be described as:

$\displaystyle\eta\equiv\left(\frac{B_{0}}{\delta B}\right)^{2},$ (11)

where $B_{0}$ and $\delta B$ are the initial background magnetic field and the
turbulent magnetic field, respectively. A higher value of $\eta$ (i.e., a
larger diffusion coefficient) implies that insufficient turbulent magnetic
field to scatter the particles exists. A lower value of $\eta$ (i.e., a
smaller diffusion coefficient) implies that the magnetic field is sufficiently
turbulent. The observed age–$\eta$ plots may suggest generation of the
turbulent magnetic field is related to the evolution of snr: Figure 5 implies
the turbulence of magnetic field growing with time. Here we address some open
issues, such as a quantitative explanation and a physical meaning of the
growth index of $-\delta$.

The acceleration efficiency depends on a Mach number of the shock wave, as
studied in Caprioli & Spitkovsky (2014a, b, c) using numerical simulations.
Caprioli & Spitkovsky (2014c) derived the relation (Eq. (16) therein),

$\displaystyle\eta\propto{\cal M}^{-1/2},$ (12)

where ${\cal M}$ is the Mach number of the snr shock. However, the Mach number
decreases with the evolution of snr because the shock is decelerated, and
$\eta$ increases following Equation 12. Thus, the theoretical ${\cal
M}$–$\eta$ relation predicts the opposite tendency of the observation.

The shock obliquity ($\theta_{\rm Bn}$) also plays a critical role in particle
acceleration in snr (Petruk et al., 2011; Caprioli & Spitkovsky, 2014a).
Whether the acceleration occurs at parallel ($\theta_{\rm Bn}\sim 0$∘) or
perpendicular ($\theta_{\rm Bn}\sim 90$∘) shocks remains controversial. Some
numerical calculations have suggested a quasi-parallel shock produces particle
injection more efficiently and generates a more amplified magnetic field
(e.g., Caprioli & Spitkovsky, 2014a). They found that acceleration efficiency
($\varepsilon_{\rm cr}$)333$\varepsilon_{\rm cr}$ is defined as the fraction
of the postshock energy density of particles with energies greater than a
certain threshold energy. This is different from our definition of
acceleration efficiency with the Bohm factor, $\eta$. drastically drops off
for $\theta_{\rm Bn}\geq$45∘. SN 1006 is the only target to investigate the
dependence on shock obliquity, as we have known the direction of the field.
Although one should be cautious about the different definitions of
acceleration efficiency, our result also indicates inefficiency (larger
$\eta$) above $\sim$50∘ for shock obliquity in SN 1006, as shown in Figure 4.
The observed gradual decrease of acceleration efficiency for increasing
$\theta_{\rm Bn}$ might be consistent with the simulation result with a higher
Mach number, for example ${\cal M}=50$. From a morphological point of view,
G1.9$+$0.3 and Vela Jr. resemble bilateral structures similar to SN 1006, but
shock obliquities are unknown. If the initial ambient field is aligned, the
acceleration efficiency could depend on the shock obliquity in the earlier
phase, as confirmed in SN 1006. In the later evolution stage or in the
preexisting randomly turbulent field, the shock obliquity is averaged over the
outer rim, resulting in efficient acceleration.

Another possible explanation for the mismatch between the measured and theory
predicted values of $\eta$ is a nonlinear modification of the shock profile.
In the test-particle limit, the shock front is presumed to be a sharp jump.
Considering the backreation of accelerated particles, the jump profile is
modified, and this may have a considerable impact on the acceleration regime
(see, for example, Reynolds, 2008, for a review of snr and references
therein). For fast shock waves in young snr, this nonlinear effect is not
important. However, for slow shocks in older snr, the nonlinear effect becomes
non-negligible, which might change the acceleration efficiency.

It should be noted that the observed age–$\eta$ plot is somewhat biased
because of a selection problem in systematical analyses. There might exist snr
with larger Bohm factors at the older ages (a few 1,000 years), but their
synchrotron emission cannot be detected due to the inefficient acceleration.
At that point, the age–$\eta$ plot here is considered as the lower limit of
$\eta$ as a function of age. Our analysis disfavours efficient acceleration
with $\eta\sim 1$ in young snr (of a few 100 years).

####  5.2.1 Maximum attainable energy

Evolving acceleration efficiency, as confirmed in Figure 5, would modify a
picture of the maximum attainable energy of accelerated particles in snr.
Because the energy loss timescale for protons is significantly longer than the
age of snr at the earlier evolutional phase, the maximum energy of the proton
is expected to be limited by its age. With the governing equation of
$\tau_{\rm acc}=\tau_{\rm age}$, the maximum energy of the accelerated proton
is derived as

$\displaystyle E_{\rm max,age}=\frac{3}{20}\frac{q}{c}tv_{\rm
sh}^{2}B\eta^{-1}.$ (13)

We simply assume that the shock velocity is constant in the ED phase and is
given by a self-similar solution in the Sedov-Taylor (ST) phase:

$\displaystyle v_{\rm
sh}(t)\propto\left\\{\begin{array}[]{ll}t^{0}&{\rm(ED)}\\\
t^{-3/5}&{\rm(ST)}.\end{array}\right.$ (16)

Note that sophisticated analytical solutions to express a smooth connection
between the ED and ST phases have been well studied (e.g., Truelove & McKee,
1999; Tang & Chevalier, 2017). Since the evolution of the magnetic field is
not well understood, we presume that $B$ is dependent on the evolutional age
as

$\displaystyle B(t)\propto t^{-\mu}.$ (17)

Our analysis (see in Figure 5) suggests that the $\eta$ value also depends on
the time evolution of the snr. Thus, we apply

$\displaystyle\eta(t)\propto t^{-\delta}.$ (18)

We assume that this empirical relation holds until $\eta$ becomes unity with
$\delta$ being $\sim 0.4$ (Equation 7). Substituting into Equation 13, we
obtain time dependence on the maximum energy of the proton:

$\displaystyle E_{\rm
max,age}\propto\left\\{\begin{array}[]{ll}t^{1-\mu+\delta}&{\rm(ED)}\\\
t^{-1/5-\mu+\delta}&{\rm(ST)}\end{array}\right..$ (21)

Equation 21 insists that $E_{\rm max,age}$ can be greater than expected before
because of the newly added term of $\delta$. Our result, $\delta\sim 0.4$,
suggests that $E_{\rm max,age}$ increases as $t^{0.2}$ even in the ST stage on
the assumption of $\mu=0$.

Let us demonstrate the prediction of Equation 21 for the case of Cassiopeia A.
The GeV gamma-ray spectrum from this snr seems to show a $\pi^{0}$ bump that
is a characteristic feature of the hadronic scenario. The gamma-ray spectrum
shows exponential cutoff at $\sim$2 TeV, which roughly translates into the
proton spectrum cutoff energy of $20$ TeV (Abeysekara et al., 2020).
Cassiopeia A is obviously not a PeVatron in the current stage with its age of
$\sim$300 years. Our measurements showed that in Cassiopeia A $\eta$ varies
from 1 to 6 depending on the site, and the averaged $\eta$ is about 3. We
adopt the averaged value here since the gamma-ray observation could not
spatially resolve the exact location of the gamma-ray emission. If we apply
$\delta=0.4$ and $\mu=0$ for the ED (ST) phase of Equation 21, the maximum
energy of the proton becomes $\sim$400 TeV ($\sim$30 TeV) at $\sim$3000 years
when the $\eta$ value reaches 1 according to Figure 5 and Equation 7. This
maximum energy is approximately twice as large as that in the same later stage
with $\delta=0$, which has been sometimes assumed. The time evolution of
$\eta$ may suggest that there would be a possibility for snr to become more
efficient accelerators of multi-TeV particles at the late stage of $\gtrsim$1
kyr. This can be consistent with the evolution of the gamma-ray radiation,
which the $\sim$kyr snr appeared to be strong TeV gamma-ray emitters (Funk,
2015). However, it should be emphasised that the assumption above is the most
optimistic, and that the maximum energy would be predominantly dependent on
other parameters, such as the shock velocity, the magnetic field, and the
condition of escape.

Observational studies of $\eta$ in the later stage of snr remain as a future
work. The snr we analyzed in this study included those in both ED and ST
phases, but most them were relatively young and might be in the very
transition stage from ED to ST. Altogether, they indicate the Bohm factor
decreases until it becomes close to unity. Our work could not fully determine
the behavior of $\eta$ in the later phase, particularly the phase that has
significantly entered into the ST stage. It is also interesting but
challenging to measure the $\eta$ value of escaping particles from the SNRs.

####  5.2.2 Supernova explosion type

Figure 6 showed that Type II snr tend to have the lower value of $\eta$. Since
a core-collapse (Type II) snr has exploded in the surroundings produced by
stellar wind of the massive progenitor, the ambient medium is more irregular
and complex than that of Type I. If the obtained evolution of $\eta$
represents the growth of turbulence, the complexity of the circumstellar
medium of a Type II snr is expected to make the turbulent production faster,
which might be observed as smaller $\eta$ value of Type II. This would be
investigated in the future by measurements of the cutoff energy and shock
speed with higher accuracy, which can help us reduce the uncertainty of $\eta$
and distinguish the difference between Type I and II snr.

##  6 Conclusions

We analyzed X-ray observations of 11 young snr to determine the cutoff energy
parameter in the synchrotron spectrum and constrain the corresponding Bohm
factor of each snr. Our model of synchrotron radiation is based on the
framework that the accelerated electron is limited by synchrotron cooling and
Bohm diffusion. This assumption is reasonable for snr older than 1,000 years
if $B\geq 10$–$20~{}\hbox{$\mu\mathrm{G}$}$ and for snr as young as a few 100
years if $B\geq 30$–$40~{}\hbox{$\mu\mathrm{G}$}$. We should be cautious about
G1.9$+$0.3 (150–190 years) and SN 1987A (30 years) because $B$ should be much
larger for the assumption to be valid in these sources, and it is questionable
for such young snr to be capable of amplifying magnetic fields in their short
lifetimes.

The $v_{\rm sh}$–$\varepsilon_{0}$ relations obtained for the five individual
snr are interpreted as the following cases:

  1. 1.

The $v_{\rm sh}$–$\varepsilon_{0}$ scatter plot is nicely represented by the
theoretical curve, $\varepsilon_{0}\propto v_{\rm sh}^{2}\eta^{-1}$, with
constant $\eta$ throughout the remnant in the cases of Kepler and Tycho.

  2. 2.

Cassiopeia A showed variable cutoff energy parameters but nearly constant
shock velocities, resulting in different acceleration efficiency depending on
the sites.

  3. 3.

The acceleration is affected by a surrounding magnetic-field configuration in
the case of SN 1006: the $\eta$ parameters is small near the polar limbs where
quasi-parallel shocks are expected to form.

  4. 4.

The different $\eta$ values are attributed to the different shock speeds in
the case of the youngest snr in our Galaxy, G1.9$+$0.3.

With all 11 snr together, including G330.2$+$1.0, RX J1713.7$-$3946, RCW 86,
Vela Jr., HESS J1731$-$347, and SN 1987A, the systematic tendency of the Bohm
factor has been unveiled for the first time. The $\eta$ in the
maximum-$\varepsilon_{0}$ (or maximum-$v_{\rm sh}$) region of each snr depends
on the evolutional age as $\eta=1.5(t_{\rm age}/1~{}{\rm kyr})^{-0.41}$ or on
the expansion parameter as $\eta=5.2m^{4.0}$. This can be related to the
turbulent generation: the turbulence becomes more self-generated as particles
become more accelerated with time. Comparing the time dependence on $\eta$
between Types I and II supernova explosions, Type II shows a relatively lower
$\eta$ value and a flatter rate of the growth, which might suggest that Type
II snr exploded in the more complex surroundings of the csm and could
facilitate the turbulence to grow faster. Finally, if we consider the time
dependence on $\eta$ as $\eta\propto t^{-\delta}$ with $\delta\approx 0.4$,
which has not been expected before, and assume this condition holds until
$\eta$ reaches unity even in the Sedov-Taylor phase, the attainable maximum
energy appears greater by the term of $\delta$, possibly in the multi-TeV
range.

The scientific results reported in this paper are based on observations made
by the Chandra X-ray Observatory. This work was also made use of data from the
NuSTAR mission, a project led by the California Institute of Technology,
managed by the Jet Propulsion Laboratory, and funded by the National
Aeronautics and Space Administration. We appreciate the NuSTAR Operations,
Software, and Calibration teams for support with the execution and analysis of
these observations. N.T. is supported by the Japan Society for the Promotion
of Science (JSPS) KAKENHI grant No. JP17J06025. This work was partially
supported by JSPS KAKENHI grant Nos. JP18H03722 and JP18H05463.

##  Appendix A Dataset

The dataset of archival observations of Chandra and NuSTAR are listed in Table
4 and Table 5, respectively.

Table 4: Log of Chandra observations  Name | Obs ID | Effective time | Date | RA | Dec | Roll
---|---|---|---|---|---|---
|  | (ks) | (yyyy-mm-dd) | (deg) | (deg) | (deg)
G1.9$+$0.3 | 6708 | 23.9 | 2007-02-10 | 267.2 | -27.2 | 91.8
G1.9$+$0.3 | 8521 | 25.7 | 2007-03-03 | 267.2 | -27.2 | 91.8
G1.9$+$0.3 | 10111 | 68.3 | 2009-07-23 | 267.2 | -27.2 | 270.0
G1.9$+$0.3 | 10112 | 50.8 | 2009-07-18 | 267.2 | -27.2 | 283.2
G1.9$+$0.3 | 10928 | 35.4 | 2009-07-13 | 267.2 | -27.2 | 270.2
G1.9$+$0.3 | 10930 | 82.1 | 2009-07-26 | 267.2 | -27.2 | 270.0
G1.9$+$0.3 | 12689 | 155.6 | 2011-07-14 | 267.2 | -27.2 | 277.3
G1.9$+$0.3 | 12690 | 48.2 | 2011-05-16 | 267.2 | -27.2 | 79.2
G1.9$+$0.3 | 12691 | 184.0 | 2011-05-09 | 267.2 | -27.2 | 79.2
G1.9$+$0.3 | 12692 | 162.6 | 2011-05-12 | 267.2 | -27.2 | 79.2
G1.9$+$0.3 | 12693 | 127.5 | 2011-05-18 | 267.2 | -27.2 | 79.2
G1.9$+$0.3 | 12694 | 159.3 | 2011-05-20 | 267.2 | -27.2 | 79.2
G1.9$+$0.3 | 12695 | 39.5 | 2011-05-23 | 267.2 | -27.2 | 79.2
G1.9$+$0.3 | 13407 | 48.4 | 2011-07-18 | 267.2 | -27.2 | 277.3
G1.9$+$0.3 | 13509 | 55.3 | 2011-07-22 | 267.2 | -27.2 | 277.3
G1.9$+$0.3 | 16947 | 38.8 | 2015-05-04 | 267.2 | -27.2 | 86.7
G1.9$+$0.3 | 16948 | 39.6 | 2015-07-14 | 267.2 | -27.2 | 271.6
G1.9$+$0.3 | 16949 | 9.1 | 2015-05-20 | 267.2 | -27.2 | 75.2
G1.9$+$0.3 | 17651 | 111.6 | 2015-05-05 | 267.2 | -27.2 | 86.7
G1.9$+$0.3 | 17652 | 26.2 | 2015-05-09 | 267.2 | -27.2 | 86.7
G1.9$+$0.3 | 17663 | 56.5 | 2015-07-24 | 267.2 | -27.2 | 271.6
G1.9$+$0.3 | 17699 | 19.8 | 2015-07-17 | 267.2 | -27.2 | 271.6
G1.9$+$0.3 | 17700 | 14.9 | 2015-08-31 | 267.2 | -27.2 | 260.2
G1.9$+$0.3 | 17702 | 36.9 | 2015-07-15 | 267.2 | -27.2 | 271.6
G1.9$+$0.3 | 17705 | 9.9 | 2015-07-25 | 267.2 | -27.2 | 271.6
G1.9$+$0.3 | 18354 | 29.7 | 2015-09-01 | 267.2 | -27.2 | 260.2
(total) |  | 1659.6 |  |  |  |
Cassiopeia A | 4634 | 148.6 | 2004-04-28 | 350.9 | 58.8 | 59.2
Cassiopeia A | 4635 | 135.0 | 2004-05-01 | 350.9 | 58.8 | 59.2
Cassiopeia A | 4636 | 143.5 | 2004-04-20 | 350.9 | 58.8 | 49.8
Cassiopeia A | 4637 | 163.5 | 2004-04-22 | 350.9 | 58.8 | 49.8
Cassiopeia A | 4638 | 164.5 | 2004-04-14 | 350.9 | 58.8 | 40.3
Cassiopeia A | 4639 | 79.0 | 2004-04-25 | 350.9 | 58.8 | 49.8
Cassiopeia A | 5196 | 49.5 | 2004-02-08 | 350.9 | 58.8 | 325.5
Cassiopeia A | 5319 | 42.3 | 2004-04-18 | 350.9 | 58.8 | 49.8
Cassiopeia A | 5320 | 54.4 | 2004-05-05 | 350.9 | 58.9 | 65.1
(total) |  | 980.3 |  |  |  |
Kepler | 116 | 48.8 | 2000-06-30 | 262.7 | -21.5 | 261.1
Kepler | 4650 | 46.2 | 2004-10-26 | 262.7 | -21.5 | 268.8
Kepler | 6714 | 157.8 | 2006-04-27 | 262.7 | -21.4 | 89.0
Kepler | 6715 | 159.1 | 2006-08-03 | 262.7 | -21.5 | 265.7
Kepler | 6716 | 158.0 | 2006-05-05 | 262.7 | -21.4 | 89.5
Kepler | 6717 | 106.8 | 2006-07-13 | 262.7 | -21.5 | 264.2
Kepler | 6718 | 107.8 | 2006-07-21 | 262.7 | -21.5 | 264.8
Kepler | 7366 | 51.5 | 2006-07-16 | 262.7 | -21.5 | 264.2
Kepler | 16004 | 102.7 | 2014-05-13 | 262.7 | -21.5 | 92.7
Kepler | 16614 | 36.4 | 2014-05-16 | 262.7 | -21.5 | 92.7
(total) |  | 975.1 |  |  |  |
Tycho | 7639 | 108.9 | 2007-04-23 | 6.3 | 64.1 | 29.2
Tycho | 8551 | 33.3 | 2007-04-26 | 6.3 | 64.1 | 29.2
Tycho | 10093 | 118.4 | 2009-04-13 | 6.3 | 64.1 | 29.2
Tycho | 10094 | 90.0 | 2009-04-18 | 6.3 | 64.1 | 29.2
Tycho | 10095 | 173.4 | 2009-04-23 | 6.3 | 64.1 | 29.2
Tycho | 10096 | 105.7 | 2009-04-27 | 6.3 | 64.1 | 29.2
Tycho | 10097 | 107.4 | 2009-04-11 | 6.3 | 64.1 | 26.3
(total) |  | 737.1 |  |  |  |
G330.2$+$1.0 | 6687 | 50.0 | 2006-05-21 | 240.2 | -51.6 | 3.3
G330.2$+$1.0 | 19163 | 74.1 | 2017-05-02 | 240.2 | -51.6 | 30.2
G330.2$+$1.0 | 20068 | 74.1 | 2017-05-05 | 240.2 | -51.5 | 30.2
(total) |  | 198.2 |  |  |  |
SN 1006 N | 13743 | 92.6 | 2012-04-28 | 225.8 | -41.7 | 19.9
SN 1006 NE | 9107 | 68.9 | 2008-06-24 | 226.0 | -41.9 | 280.4
SN 1006 NE | 732 | 68.1 | 2000-07-10 | 226.0 | -41.9 | 280.2
SN 1006 SW | 13739 | 100.1 | 2012-05-04 | 225.6 | -42.1 | 9.1
SN 1006 NW | 1959 | 89.0 | 2001-04-26 | 225.6 | -41.8 | 30.2
SN 1006 NW | 13737 | 87.1 | 2012-04-20 | 225.6 | -41.8 | 31.7
SN 1006 W | 13738 | 73.5 | 2012-04-23 | 225.4 | -42.0 | 25.3
SN 1006 W | 14424 | 25.4 | 2012-04-27 | 225.4 | -42.0 | 25.3
SN 1006 SE | 13741 | 98.5 | 2012-04-25 | 226.0 | -42.0 | 24.6
SN 1006 S | 13742 | 79.0 | 2012-06-15 | 225.8 | -42.1 | 289.1
(total) |  | 782.2 |  |  |  |
RX J1713.7$-$3946 NW | 736 | 29.6 | 2000-07-25 | 258.0 | -39.6 | 282.5
RX J1713.7$-$3946 NW | 6370 | 29.8 | 2006-05-03 | 257.9 | -39.6 | 64.8
RX J1713.7$-$3946 NW | 10090 | 28.4 | 2009-01-30 | 257.9 | -39.5 | 98.6
RX J1713.7$-$3946 NW | 10091 | 29.6 | 2009-05-16 | 257.9 | -39.5 | 53.8
RX J1713.7$-$3946 NW | 10092 | 29.2 | 2009-09-10 | 257.9 | -39.6 | 266.1
RX J1713.7$-$3946 NW | 12671 | 89.9 | 2011-07-01 | 257.9 | -39.6 | 304.5
(total) |  | 236.5 |  |  |  |
RCW 86 | 1993 | 92.0 | 2001-02-01 | 220.2 | -62.7 | 80.2
RCW 86 NE | 4611 | 71.7 | 2004-06-15 | 221.3 | -62.4 | 295.2
RCW 86 NE | 7642 | 69.2 | 2007-06-20 | 221.3 | -62.3 | 299.0
RCW 86 | 10699 | 2.0 | 2009-06-14 | 220.5 | -62.6 | 304.4
RCW86 | 13748 | 36.1 | 2013-02-14 | 220.1 | -62.7 | 70.7
RCW 86 | 14890 | 26.7 | 2013-02-03 | 220.4 | -62.2 | 75.2
RCW 86 | 15608 | 29.2 | 2013-02-05 | 220.4 | -62.2 | 75.2
RCW 86 | 15609 | 37.6 | 2013-02-10 | 220.4 | -62.2 | 75.2
RCW86 | 15610 | 23.1 | 2013-02-17 | 220.1 | -62.7 | 70.7
RCW86 | 15611 | 25.9 | 2013-02-12 | 220.1 | -62.7 | 70.7
RCW 86 NE | 16952 | 67.2 | 2015-06-25 | 221.3 | -62.4 | 293.6
(total) |  | 480.7 |  |  |  |
Vela Jr. NW | 3846 | 39.5 | 2003-01-05 | 132.3 | -45.6 | 30.2
Vela Jr. NW | 4414 | 34.5 | 2003-01-06 | 132.3 | -45.6 | 30.2
Vela Jr. NORTH | 9123 | 39.7 | 2008-08-31 | 132.3 | -45.7 | 146.2
(total) |  | 113.7 |  |  |  |
HESS J1731$-$347 | 9139 | 29.2 | 2008-04-28 | 263.0 | -34.7 | 81.2
SN 1987A | 14697 | 67.6 | 2013-03-21 | 83.9 | -69.3 | 264.2
SN 1987A | 14698 | 68.5 | 2013-09-28 | 83.9 | -69.3 | 79.4
SN 1987A | 15809 | 70.5 | 2014-03-19 | 83.9 | -69.3 | 266.1
SN 1987A | 15810 | 48.3 | 2014-09-20 | 83.9 | -69.3 | 84.2
SN 1987A | 17415 | 19.4 | 2014-09-17 | 83.9 | -69.3 | 84.2
(total) |  | 274.2 |  |  |  |

Table 5: Log of NuSTAR observations  Name | Obs ID | Effective time | Date | RA | Dec | Roll
---|---|---|---|---|---|---
|  | (ks) | (yyyy-mm-dd) | (deg) | (deg) | (deg)
G1.9$+$0.3 | 40001015003 | 85.4 | 2013-07-08 | 267.2 |  $-$27.2 | 327.3
G1.9$+$0.3 | 40001015005 | 121.6 | 2013-07-14 | 267.2 |  $-$27.2 | 327.3
G1.9$+$0.3 | 40001015007 | 144.7 | 2013-07-27 | 267.2 |  $-$27.2 | 327.3
(total) |  | 351.7 |  |  |  |
Cassiopeia A | 40021002002 | 270.9 | 2012-11-23 | 350.8 | 58.8 | 338.3
Cassiopeia A | 40021002006 | 135.6 | 2013-03-02 | 350.9 | 58.8 | 248.7
Cassiopeia A | 40021002008 | 189.3 | 2013-03-05 | 350.9 | 58.8 | 248.7
Cassiopeia A | 40021003003 | 197.8 | 2013-05-28 | 350.9 | 58.8 | 151.2
Cassiopeia A | 40021001002 | 170.1 | 2012-08-27 | 350.8 | 58.8 | 75.7
Cassiopeia A | 40021001004 | 25.7 | 2012-10-07 | 350.7 | 58.8 | 33.0
Cassiopeia A | 40021001005 | 184.5 | 2012-10-07 | 350.8 | 58.8 | 33.0
Cassiopeia A | 40021002010 | 12.4 | 2013-03-09 | 350.9 | 58.8 | 248.7
Cassiopeia A | 40021003002 | 12.4 | 2013-05-28 | 350.9 | 58.8 | 151.2
Cassiopeia A | 40021011002 | 235.1 | 2013-10-30 | 350.9 | 58.8 | 6.8
Cassiopeia A | 40021012002 | 205.8 | 2013-11-27 | 350.8 | 58.8 | 335.2
Cassiopeia A | 40021015002 | 74.4 | 2013-12-21 | 350.9 | 58.8 | 312.3
Cassiopeia A | 40021015003 | 136.9 | 2013-12-23 | 350.9 | 58.8 | 312.2
(total) |  | 1850.9 |  |  |  |
SN1006 NE | 40110001002 | 198.5 | 2016-03-02 | 225.9 |  $-$41.8 | 180.0
SN1006 SW | 40110002002 | 204.8 | 2016-03-08 | 225.5 |  $-$42.0 | 180.0
RX J1713.7$-$3946 NW | 40111001002 | 43 | 2015-09-27 | 257.86 |  $-$39.52 | 343.3
RX J1713.7$-$3946 NW | 40111002002 | 49 | 2016-03-30 | 257.93 |  $-$39.58 | 165.6
(total) |  | 92.0 |  |  |  |
Vela Jr. NW | 40101011002 | 69.0 | 2015-07-07 | 132.2 |  $-$45.7 | 40.4
Vela Jr. NW | 40101011004 | 102.1 | 2015-07-16 | 132.2 |  $-$45.7 | 44.4
(total) |  | 142.2 |  |  |  |
SN1987A | 40001014002 | 57.5 | 2012-09-07 | 84.0 |  $-$69.2 | 149.1
SN1987A | 40001014003 | 113.3 | 2012-09-08 | 84.0 |  $-$69.2 | 149.1
SN1987A | 40001014004 | 198.2 | 2012-09-11 | 83.9 |  $-$69.2 | 149.2
SN1987A | 40001014006 | 45.2 | 2012-10-20 | 84.0 |  $-$69.3 | 190.5
SN1987A | 40001014007 | 173.2 | 2012-10-21 | 83.9 |  $-$69.3 | 190.5
SN1987A | 40001014010 | 160.0 | 2012-12-12 | 83.9 |  $-$69.3 | 242.9
SN1987A | 40001014013 | 403.1 | 2013-06-29 | 83.8 |  $-$69.2 | 80.1
SN1987A | 40001014015 | 83.5 | 2014-04-21 | 83.8 |  $-$69.3 | 13.1
SN1987A | 40001014016 | 379.4 | 2014-04-22 | 83.8 |  $-$69.3 | 13.1
SN1987A | 40001014018 | 170.5 | 2014-06-15 | 83.8 |  $-$69.2 | 65.0
SN1987A | 40001014020 | 237.4 | 2014-06-19 | 83.8 |  $-$69.2 | 70.2
SN1987A | 40001014023 | 397.7 | 2014-08-01 | 83.9 |  $-$69.2 | 111.4
(total) |  | 2419.0 |  |  |  |

##  Appendix B Thermal parameters

Table 6 presents parameters of the thermal model which is used for fitting the
spectra of Cassiopeia A, Kepler, Tycho, SN 1006, RCW 86, and SN 1987A in
addition to the synchrotron radiation model (see Table 2 for the nonthermal
parameters).

Table 6: Thermal parameters.

Region | $kT$ | O | Ne | Mg | Si | S | Ar | Ca | Fe | $nt$ | Norm
---|---|---|---|---|---|---|---|---|---|---|---
| (keV) | (O⊙) | (Ne⊙) | (Mg⊙) | (Si⊙) | (S⊙) | (Ar⊙) | (Ca⊙) | (Fe⊙) | ($10^{10}$ s ${\rm cm}^{-3}$) | ($10^{-5}~{}{\rm cm}^{-5}$)
Cassiopeia A |  |  |  |  |  |  |  |  |  |  |
N1 | 1.70 | 1.0 | 1.0 | 1.0 $\pm$ 0.6 | 11 $\pm$ 2 | 10 $\pm$ 2 | 12${}^{+7}_{-6}$ | 12${}^{+16}_{-12}$ | 1.0 | 6.9 | 5.0 $\pm$ 0.9
N2 | 1.70 | 1.0 | 1.0 | 1.8${}^{+0.5}_{-0.4}$ | 7.9${}^{+1.1}_{-0.9}$ | 6.9${}^{+1.2}_{-1.0}$ | 9.5${}^{+4.2}_{-3.9}$ | 8.0${}^{+10.2}_{-8.0}$ | 1.0 | 6.9 | 5.0 $\pm$ 0.6
S | 0.69 | 1.0 | 1.0 | 1.3 $\pm$ 0.1 | 1.1 $\pm$ 0.1 | 1.4 $\pm$ 0.2 | 3.0${}^{+1.3}_{-1.2}$ | 1.0 | 1.0 | 22.6 | 118${}^{+14}_{-13}$
NW | 1.61 | 1.0 | 1.0 | 0.1${}^{+2.3}_{-0.1}$ | 20${}^{+36}_{-8}$ | 12${}^{+21}_{-5}$ | 42${}^{+82}_{-23}$ | 60${}^{+129}_{-49}$ | 1.0 | 9.6 | 1.2${}^{+0.9}_{-0.8}$
SE | 1.77 | 1.0 | 1.0 | 1.4 | 6.0 | 3.3 | 1.0 | 1.0 | 1.0 | 10.4 | 1.4 $\pm$ 0.4
E | 1.84 | 1.0 | 1.0 | 0.5 $\pm$ 0.5 | 4.4${}^{+1.0}_{-0.8}$ | 2.5${}^{+0.9}_{-0.8}$ | 1.0 | 1.0 | 1.0 | 14.3 | 3.6${}^{+0.7}_{-0.6}$
NE | 1.76 | 1.0 | 1.0 | 0.8 | 11.0 | 8.8 | 10.2 | 23.1 | 1.0 | 14.0 | 1.3 $\pm$ 0.2
Kepler |  |  |  |  |  |  |  |  |  |  |
1 | 0.80 | 0.42 | 1.0 | 1.1 | 11 $\pm$ 1 | 19 $\pm$ 1 | 1.0 | 1.0 | 1.0 | 5.9 | 5.8 $\pm$ 0.3
2 | 0.58 | 0.21 | 1.0 | 0.4 | 15 $\pm$ 1 | 27 $\pm$ 3 | 1.0 | 1.0 | 1.0 | 9.5 | 12 $\pm$ 1
3 | 0.60 | 0.98 | 1.0 | 0.4 | 21 $\pm$ 2 | 26 $\pm$ 4 | 1.0 | 1.0 | 1.0 | 82.1 | 3.9${}^{+0.4}_{-0.3}$
4 | 0.34 | 1.00 | 1.0 | 1.0 | 1 | 1.0 | 1.0 | 1.0 | 1.0 | 100.0 | ¡0.35
5 | 0.50 | 1.00 | 1.0 | 1.0 | 1 | 1.0 | 1.0 | 1.0 | 1.0 | 60.0 | 0.28${}^{+0.43}_{-0.28}$
6 | 0.53 | 1.00 | 1.0 | 1.0 | 44 | 28 | 130.8 | 1.0 | 1.0 | 69.8 | 0.15 $\pm$ 0.06
7 | 0.52 | 1.00 | 1.0 | 1.0 | 39 | 29 | 1.0 | 1.0 | 1.0 | 60.0 | 0.29 $\pm$ 0.07
8 | 0.45 | 2.21${}^{+3.37}_{-2.21}$ | 1.0 | 2.7${}^{+4.2}_{-2.7}$ | 67${}^{+77}_{-29}$ | 105${}^{+130}_{-54}$ | 1.0 | 1.0 | 1.0 | 42.1 | 0.47${}^{+0.36}_{-0.25}$
9 | 0.57 | 0.18 | 1.0 | 0.4 $\pm$ 0.3 | 18 $\pm$ 2 | 22 $\pm$ 4 | 1.0 | 1.0 | 1.0 | 20.5 | 3.4${}^{+0.4}_{-0.3}$
10 | 0.57 | 0.61 | 1.0 | 0.4 | 39${}^{+6}_{-5}$ | 42${}^{+9}_{-7}$ | 15${}^{+28}_{-15}$ | 1.0 | 1.0 | 87.1 | 1.8 $\pm$ 0.3
11 | 0.51 | 0.18 | 1.0 | 0.4 | 34${}^{+6}_{-5}$ | 43${}^{+11}_{-10}$ | 1.0 | 1.0 | 1.0 | 12.7 | 1.2 $\pm$ 0.2
12 | 0.50 | 0.72 | 1.0 | 0.7 | 21${}^{+7}_{-5}$ | 34${}^{+15}_{-10}$ | 62${}^{+87}_{-62}$ | 1.0 | 1.0 | 9.7 | 2.5${}^{+0.8}_{-0.7}$
13 | 0.69 | 0.59${}^{+0.08}_{-0.07}$ | 1.0 | 1.0 $\pm$ 0.1 | 7.5${}^{+0.5}_{-0.4}$ | 15 $\pm$ 1 | 34 $\pm$ 5 | 1.0 | 1.2 $\pm$ 0.1 | 8.2 | 26 $\pm$ 1
14 | 2.11 | 0.24${}^{+0.14}_{-0.11}$ | 1.0 | 2.8 $\pm$ 0.3 | 17${}^{+2}_{-1}$ | 17 $\pm$ 2 | 28 $\pm$ 5 | 1.0 | 1.0 | 3.0 | 2.5${}^{+0.3}_{-0.2}$
Tycho |  |  |  |  |  |  |  |  |  |  |
1 | 0.78 | 1.0 | 1.0 | 1.0 | 33${}^{+4}_{-3}$ | 45${}^{+6}_{-5}$ | 129${}^{+20}_{-18}$ | 316${}^{+67}_{-65}$ | 1.0 | 6.8 | 20 $\pm$ 2
2 | 2.17 | 1.9${}^{+0.3}_{-0.2}$ | 1.0 | 0.8 | 12 $\pm$ 1 | 11 $\pm$ 1 | 13 | 18.8 | 1.0 | 2.3 | 30 $\pm$ 3
3 | 3.38 | 2.3 $\pm$ 0.5 | 1.0 | 1.0 | 10${}^{+3}_{-2}$ | 7.2${}^{+2.3}_{-1.7}$ | 1 | 1.0 | 1.0 | 2.3 | 5.6${}^{+1.4}_{-1.2}$
4 | 5.46 | 10.9 | 1.0 | 1.0 | 107 | 68 | 74 | 1.0 | 1.0 | 2.6 | 0.94 $\pm$ 0.03
5 | 1.30 | 2.7 $\pm$ 0.4 | 1.0 | 1.0 | 32 $\pm$ 4 | 39${}^{+6}_{-5}$ | 55${}^{+10}_{-9}$ | 129${}^{+28}_{-25}$ | 1.0 | 3.5 | 30${}^{+5}_{-4}$
6 | 1.81 | 6.1${}^{+2.5}_{-1.2}$ | 1.0 | 1.8${}^{+1.4}_{-0.7}$ | 57${}^{+50}_{-20}$ | 61${}^{+54}_{-22}$ | 1 | 1.0 | 1.0 | 2.9 | 3.5${}^{+2.0}_{-1.7}$
7 | 2.64 | 4.0 | 1.0 | 3.5 | 31 | 23 | 1 | 1.0 | 1.0 | 3.0 | 3.6 $\pm$ 0.1
8 | 1.79 | 1.0 | 1.0 | 4.2 | 107 | 102 | 121 | 276.1 | 1.0 | 4.3 | 0.93 $\pm$ 0.03
9 | 2.52 | 5.5 | 1.0 | 5.1 | 68${}^{+21}_{-13}$ | 54${}^{+18}_{-11}$ | 65 | 1.0 | 1.0 | 3.2 | 1.2 $\pm$ 0.3
10 | 1.22 | 7.1 $\pm$ 0.7 | 1.0 | 1.2 $\pm$ 0.1 | 20${}^{+3}_{-2}$ | 22${}^{+4}_{-3}$ | 27${}^{+5}_{-4}$ | 60${}^{+17}_{-14}$ | 1.0 | 3.9 | 43${}^{+7}_{-6}$
11 | 1.25 | 5.1${}^{+1.2}_{-1.0}$ | 1.0 | 1.3${}^{+0.7}_{-0.6}$ | 40${}^{+12}_{-8}$ | 45${}^{+15}_{-10}$ | 87 | 1.0 | 1.0 | 3.6 | 3.6 $\pm$ 0.9
12 | 1.73 | 3.2 $\pm$ 0.6 | 1.0 | 1.4${}^{+0.6}_{-0.5}$ | 32${}^{+13}_{-8}$ | 32${}^{+14}_{-9}$ | 1 | 1.0 | 1.0 | 2.8 | 3.9${}^{+1.4}_{-1.1}$
13 | 1.64 | 3.4${}^{+0.6}_{-0.5}$ | 1.0 | 1.9 $\pm$ 0.3 | 24 $\pm$ 2 | 22 | 1 | 1.0 | 1.0 | 2.6 | 5.7 $\pm$ 0.4
14 | 1.95 | 6.4 $\pm$ 0.5 | 1.0 | 1.1 $\pm$ 0.1 | 17${}^{+2}_{-1}$ | 13 $\pm$ 1 | 10 | 13.5 | 1.0 | 2.3 | 32 $\pm$ 3
15 | 0.73${}^{+0.05}_{-0.02}$ | 1.0 | 1.0 | 1.0 | 30${}^{+6}_{-4}$ | 37${}^{+7}_{-6}$ | 36${}^{+11}_{-5}$ | 1.0 | 1.0 | 29${}^{+13}_{-12}$ | 22 $\pm$ 5
16 | 0.61 | 0.8 $\pm$ 0.1 | 1.0 | 1.9${}^{+0.2}_{-0.1}$ | 18 $\pm$ 2 | 40 $\pm$ 4 | 124 $\pm$ 18 | 1.0 | 1.0 | 7.1 | 37 $\pm$ 3
17 | 1.15${}^{+0.23}_{-0.11}$ | 1.1${}^{+0.2}_{-0.1}$ | 1.0 | 1.0 | 23 $\pm$ 2 | 26${}^{+3}_{-4}$ | 46${}^{+12}_{-13}$ | 1.0 | 1.0 | 3.0${}^{+0.5}_{-0.6}$ | 32${}^{+4}_{-5}$
18 | 0.77 | 1.0 | 1.0 | 1.3${}^{+0.4}_{-0.3}$ | 46${}^{+18}_{-15}$ | 55${}^{+22}_{-18}$ | 96${}^{+40}_{-34}$ | 329${}^{+141}_{-115}$ | 1.0 | 5.8 | 53${}^{+29}_{-15}$
19 | 1.15 | 1.7 | 1.0 | 1.4 | 38${}^{+4}_{-3}$ | 40${}^{+4}_{-3}$ | 61 $\pm$ 9 | 196${}^{+30}_{-29}$ | 1.0 | 3.8 | 18 $\pm$ 2
20 | 1.76 | 1.6 | 1.0 | 2.0 | 56${}^{+32}_{-15}$ | 49${}^{+29}_{-14}$ | 31 | 45.6 | 1.0 | 3.6 | 1.9 $\pm$ 0.7
SN 1006 |  |  |  |  |  |  |  |  |  |  |
1 | 0.41 | 5.0 | 4.7 | 12.0 | 29 | 1.0 | 1.0 | 1.0 | 1.0 | 1.6 | 1.1${}^{+0.4}_{-0.3}$
2 | 0.10${}^{+0.06}_{-0.10}$ | 5.0 | 4.7 | 12.0 | 29 | 1.0 | 1.0 | 1.0 | 1.0 | 1.6 | 63${}^{+166}_{-53}$
3 | 0.38${}^{+0.08}_{-0.09}$ | 4.4 | 1.5 | 15.0 | 50 | 1.0 | 1.0 | 1.0 | 1.0 | 1.3 | 1.5${}^{+0.5}_{-0.4}$
4 | 0.41${}^{+0.04}_{-0.05}$ | 4.4 | 1.5 | 15.0 | 50 | 1.0 | 1.0 | 1.0 | 1.0 | 1.3 | 2.9${}^{+0.5}_{-0.4}$
5 | 0.29${}^{+0.05}_{-0.06}$ | 4.4 | 1.5 | 15.0 | 50 | 1.0 | 1.0 | 1.0 | 1.0 | 1.3 | 2.8${}^{+1.2}_{-0.7}$
6 | 0.37 $\pm$ 0.02 | 4.4 | 1.5 | 15.0 | 50 | 1.0 | 1.0 | 1.0 | 1.0 | 1.3 | 6.1 $\pm$ 0.9
7 | 2.1${}^{+0.9}_{-0.5}$ | 9.0${}^{+1.4}_{-0.9}$ | 1.5 | 11${}^{+2}_{-1}$ | 44${}^{+11}_{-6}$ | 1.0 | 1.0 | 1.0 | 1.0 | 0.35${}^{+0.05}_{-0.03}$ | 3.3${}^{+0.4}_{-0.5}$
8 | 1.4${}^{+0.8}_{-0.4}$ | 6.7${}^{+0.9}_{-0.7}$ | 1.5 | 8.8${}^{+1.4}_{-1.2}$ | 48${}^{+15}_{-12}$ | 1.0 | 1.0 | 1.0 | 1.0 | 0.47${}^{+0.11}_{-0.07}$ | 10 $\pm$ 2
9 | 0.39${}^{+0.02}_{-0.03}$ | 4.4 | 1.5 | 15.0 | 241${}^{+63}_{-45}$ | 1.0 | 1.0 | 1.0 | 1.0 | 1.3 | 13${}^{+5}_{-4}$
10 | 0.40${}^{+0.04}_{-0.03}$ | 4.4 | 1.5 | 15.0 | 50 | 1.0 | 1.0 | 1.0 | 1.0 | 1.3 | 10${}^{+3}_{-2}$
11 | 0.39 $\pm$ 0.06 | 4.4 | 1.5 | 15.0 | 50 | 1.0 | 1.0 | 1.0 | 1.0 | 1.3 | 7.4${}^{+2.4}_{-2.0}$
12 | 0.38 $\pm$ 0.06 | 4.4 | 1.5 | 15.0 | 50 | 1.0 | 1.0 | 1.0 | 1.0 | 1.3 | 4.5${}^{+1.4}_{-1.2}$
13 | 0.12${}^{+0.16}_{-0.12}$ | 5.0 | 4.7 | 12.0 | 29 | 1.0 | 1.0 | 1.0 | 1.0 | 1.6 | 25${}^{+225}_{-22}$
14 | 0.09${}^{+0.19}_{-0.09}$ | 5.0 | 4.7 | 12.0 | 29 | 1.0 | 1.0 | 1.0 | 1.0 | 1.6 | 27${}^{+62}_{-26}$
15 | 0.14${}^{+0.15}_{-0.14}$ | 5.0 | 4.7 | 12.0 | 29 | 1.0 | 1.0 | 1.0 | 1.0 | 1.6 | 2.6${}^{+50.4}_{-2.2}$
16 | 0.27${}^{+0.02}_{-0.03}$ | 6.3${}^{+1.3}_{-0.7}$ | 4.3${}^{+0.9}_{-1.0}$ | 12.0 | 342${}^{+108}_{-88}$ | 1.0 | 1.0 | 1.0 | 1.0 | 1.6 | 41${}^{+33}_{-12}$
17 | 5.5${}^{+-5.5}_{-3.2}$ | 20${}^{+24}_{-9}$ | 13${}^{+10}_{-4}$ | 9.0${}^{+5.1}_{-3.1}$ | 29 | 1.0 | 1.0 | 1.0 | 1.0 | 0.12${}^{+0.24}_{-0.02}$ | 1.6${}^{+1.5}_{-0.7}$
RCW 86 |  |  |  |  |  |  |  |  |  |  |
NE | 0.44 | 1 | 1.9 | 1.9 | 1.9 | 1 | 1 | 1.2 | 1 | 2.7 | 17 $\pm$1
SN 1987A |  |  |  |  |  |  |  |  |  |  |
whole | 0.68$\pm$0.03/2.5${}^{+0.3}_{-0.2}$ | 0.34 | 3.0 | 1.0 | 1.4 | 2.5 | 2.7 | 1 | 0.61 | 8.5${}^{+2.3}_{-1.7}$ | 290$\pm$12/50$\pm$4

Note. — The parameter values without uncertainties are fixed. For the thermal
model, VNEI is used for Cassiopeia A, Kepler, and Tycho, while Vpshock is used
for SN 1006 and RCW 86 (see also the text). In SN 1987A, the first and second
terms of $kT$ and Norm indicate the best-fit parameters of the Vequil and
Vpshock models, respectively.

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