# Properties of shocked dust grains in supernova remnants

F. D. Priestley1, H. Chawner1, M. J. Barlow2, I. De Looze2,3, H. L. Gomez1, M.
Matsuura1
1School of Physics and Astronomy, Cardiff University, Queen’s Buildings, The
Parade, Cardiff CF24 3AA, UK
2Department of Physics and Astronomy, University College London, Gower Street,
London WC1E 6BT, UK
3Sterrenkundig Observatorium, Ghent University, Krijgslaan 281 - S9, 9000
Gent, Belgium
Email<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Shockwaves driven by supernovae both destroy dust and reprocess the surviving
grains, greatly affecting the resulting dust properties of the interstellar
medium (ISM). While these processes have been extensively studied
theoretically, observational constraints are limited. We use physically-
motivated models of dust emission to fit the infrared (IR) spectral energy
distributions of seven Galactic supernova remnants, allowing us to determine
the distribution of dust mass between diffuse and dense gas phases, and
between large and small grain sizes. We find that the dense ($\sim
10^{3}\,{\rm cm}^{-3}$), relatively cool ($\sim 10^{3}\,{\rm K}$) gas phase
contains $>90\%$ of the dust mass, making the warm dust located in the X-ray
emitting plasma ($\sim 1\,{\rm cm}^{-3}$/$10^{6}\,{\rm K}$) a negligible
fraction of the total, despite dominating the mid-IR emission. The ratio of
small ($\lesssim 10\,{\rm nm}$) to large ($\gtrsim 0.1\,{\rm\mu m}$) grains in
the cold component is consistent with that in the ISM, and possibly even
higher, whereas the hot phase is almost entirely devoid of small grains. This
suggests that grain shattering, which processes large grains into smaller
ones, is ineffective in the low-density gas, contrary to model predictions.
Single-phase models of dust destruction in the ISM, which do not account for
the existence of the cold swept-up material containing most of the dust mass,
are likely to greatly overestimate the rate of dust destruction by supernovae.

###### keywords:

dust, extinction – ISM: supernova remnants – ISM: evolution

††pubyear: 2022††pagerange: Properties of shocked dust grains in supernova
remnants–D

##  1 Introduction

Core-collapse supernovae (CCSNe) both form (Dunne et al., 2009; Matsuura et
al., 2011; Gomez et al., 2012; De Looze et al., 2017; De Looze et al., 2019;
Chawner et al., 2019; Niculescu-Duvaz et al., 2021) and destroy (Jones et al.,
1996; Slavin et al., 2015) significant quantities of dust. The balance between
these two processes determines whether CCSNe are net dust sources or sinks,
which has important consequences for the evolution of the interstellar medium
(ISM) (Morgan & Edmunds, 2003; De Looze et al., 2020; Galliano et al., 2021).
With extensive observational evidence for efficient dust formation by CCSNe,
the key uncertainties in models of ISM evolution are now those related to dust
destruction in the shockwaves driven by these same objects.

Although theoretical predictions for the destruction of newly-formed dust in
CCSN ejecta span nearly the entire range from complete destruction to complete
survival (Kirchschlager et al. 2019 and references therein), models of dust
destruction in the ISM have settled on a typical gas mass ‘cleared’ of dust of
order $\sim 10^{3}\,{\rm M}_{\odot}$ per SN (Jones et al. 1996; Slavin et al.
2015, although Kirchschlager et al. 2022 report larger values). This
corresponds to $\sim 10\,{\rm M}_{\odot}$ of dust destroyed per SN for a
typical Galactic dust-to-gas (DTG) ratio of $\sim 0.01$, much larger than the
observed dust masses of $\lesssim 1\,{\rm M}_{\odot}$ found in supernova
remnants (SNRs), suggesting that CCSNe are net destroyers of dust under
present-day ISM conditions.

These models generally assume that shocks propagate into a spherically
symmetric, uniform density ISM, with properties appropriate for the warm
neutral medium ($n_{\rm H}\sim 1\,{\rm cm}^{-3}$, $T\sim 10^{4}\,{\rm K}$).
The real ISM is multi-phase, with most of the mass concentrated in colder,
denser gas ($\sim 30\,{\rm cm}^{-3}$/$100\,{\rm K}$; McKee & Ostriker 1977).
More realistic ISM structures can significantly reduce the quantity of dust
destroyed (Hu et al., 2019; Martínez-González et al., 2019), as denser regions
of the ISM experience less violent shock interactions. Models also typically
assume the standard Mathis et al. (1977) (MRN) grain size distribution for the
ISM dust, but this may vary depending on the ISM phase (Hirashita & Yan, 2009)
and can be altered by the SN itself (Hoang et al., 2019), with potentially
dramatic effects on the resulting mass of dust destroyed (Kirchschlager et
al., 2019).

Observationally, infrared (IR) studies of SNRs often find dust temperatures
lower than those predicted by models of grains in the high-temperature
($\gtrsim 10^{6}\,{\rm K}$) gas produced by SN shocks in low-density material
(Seok et al., 2015; Koo et al., 2016; Chawner et al., 2020a). Using data
extending into the far-IR, Chawner et al. (2020b) showed that the observed
spectral energy distribution (SED) of the Tornado SNR cannot be explained by
dust grains in the hot, X-ray emitting shocked material swept up by the
expanding SNR. The far-IR emission requires the presence of colder grains,
located within the SNR but not exposed to the high-temperature gas. This cold
dust component has a mass at least an order of magnitude larger than that of
the warmer ($\sim 100\,{\rm K}$) grains located in the hot gas. The same
phenomenon was found for the three SNRs investigated in Priestley et al.
(2021), suggesting that it is not uncommon. Most observational studies of dust
in SNRs (e.g. Borkowski et al., 2006; Williams et al., 2006; Temim et al.,
2012) have focused on the mid-IR emission, which is mainly produced by the
warm grains, and thus may represent a very small fraction of the total dust
mass within the SNR. The dust properties derived from the mid-IR data, and the
resulting destruction efficiencies, are potentially unrepresentative of most
of the swept-up material.

A significant issue in interpreting far-IR observations of SNRs is that the
dust properties are poorly constrained; in Chawner et al. (2020b) and
Priestley et al. (2021), we were forced to assume standard ISM properties for
both the cold dust, and for the radiation field presumably responsible for
heating it. The impact of the SNR blastwave can be expected to significantly
alter both of these properties. In this paper, we develop a comprehensive
model for the IR emission of shocked dust grains in SNRs, allowing us to
derive the properties of these grains from observational data, and thus
provide empirical constraints on models of dust destruction.

Table 1: Distance, radius, estimated age, volume, the hot component density and temperature, the initial ISM density and shock velocity reproducing these hot component properties, and the scaling factor to reproduce observed X-ray luminosities, for each SNR in our sample. Note that for G11, G27 and G29, ‘volume’ refers to that of the shell of shocked material, rather than the spherical volume. References: (1) Priestley et al. (2021); (2) Chawner et al. (2020b); (3) Green (2004); (4) Ranasinghe & Leahy (2018); (5) Verbiest et al. (2012); (6) Sawada et al. (2011); (7) Frail et al. (1996); (8) Kilpatrick et al. (2016); (9) Chawner et al. (2020a); (10) Kothes & Dougherty (2007); (11) Tian & Leahy (2014); (12) Keohane et al. (2007); (13) Park et al. (2010); (14) Leahy et al. (2020); (15) Koo et al. (2016); (16) Chawner et al. (2019). SNR |  $D$/kpc |  $r$/pc | Age/$\,{\rm kyr}$ |  $V$/pc3 |  $n_{\rm H}$/cm-3 |  $T$/$10^{6}\,{\rm K}$ |  $n_{\rm ISM}$/cm-3 |  $v_{\rm sh}$/km s-1 | $f$ | Ref.
---|---|---|---|---|---|---|---|---|---|---
G11 | $4.4$ | $2.8$ | $1.4-2.4$ | $33.7$ | $6.8$ | $8.2$ | $1.7$ | $770$ | $0.028$ | 1, 3, 15, 16
G27 | $5.8$ | $2.9$ | $0.8-2.1$ | $37.1$ | $6.5$ | $9.1$ | $1.6$ | $810$ | $0.019$ | 1, 4, 15, 16
G29 | $5.8$ | $3.2$ | $<0.8$ | $59.4$ | $1.6$ | $26.0$ | $0.4$ | $1370$ | $0.010$ | 1, 5, 15, 16
Tornado | $11.8$ | $4.5$ | $2-8$ | $382$ | $0.5$ | $8.5$ | $0.13$ | $780$ | $0.091$ | 2, 6, 7
G43 | $11.3$ | $8.2$ | $1-4$ | $2310$ | $1.0$ | $18.6$ | $0.25$ | $1170$ | $0.121$ | 8, 9, 12, 15, 16
G340 | $15.0$ | $13.6$ | $2.6$ | $10530$ | $1.0$ | $11.6$ | $0.25$ | $920$ | $0.121$ | 9, 10, 13, 16
G349 | $11.5$ | $4.5$ | $1.8$ | $382$ | $3.4$ | $7.0$ | $0.85$ | $710$ | $0.091$ | 9, 11, 14, 15, 16

Figure 1: Far-IR Herschel three-colour images (red $250\,{\rm\mu m}$; green
$160\,{\rm\mu m}$; blue $70\,{\rm\mu m}$) of G11 (left) and G340 (right), with
X-ray flux overlaid as contours. Reproduced from Chawner et al. (2019).

Figure 2: Left: shock-generated radiation field for G11 (black) compared to
the Mathis et al. (1983) ISM radiation field (red). Right: flux per unit mass
for $0.1\,{\rm\mu m}$ (green) and $5\,{\rm nm}$ (purple) carbon grains, heated
by the G11 (solid lines) and ISM (dashed lines) radiation fields.

Figure 3: Best-fit dust SEDs for G11 - data (black crosses), total model SED
(black line), and individual component SEDs: hot/large (blue); hot/small
(orange); cold/large (green); cold/small (purple). Note that some component
SEDs may not be visible, due to their contributing a negligible amount to the
total SED. Left: carbon grains. Right: silicates. Figure 4: Flux per unit mass
for carbon grains in G11: hot/large (blue); hot/small (orange); cold/large
(green); cold/small (purple).

Figure 5: Best-fit dust SEDs for carbon grains - data (black crosses), total model SED (black line), and individual component SEDs: hot/large (blue); hot/small (orange); cold/large (green); cold/small (purple). Top left: G27. Top right: G29. Middle left: Tornado. Middle right: G43. Bottom left: G340. Bottom right: G349. Table 2: Median SNR dust masses returned by the MCMC for each component, with the 16th and 84th percentiles as uncertainties, and the estimated initial swept-up dust mass $M_{0}$, assuming an initial density $n_{\rm ISM}$ and volume $V$ from Table 1, and a DTG mass ratio of $0.01$. | Dust mass / $\,{\rm M}_{\odot}$ |
---|---|---
SNR | Hot/large | Hot/small | Cold/large | Cold/small | $M_{0}/\,{\rm M}_{\odot}$
Carbon
G11 | $3.0^{+0.7}_{-0.8}\times 10^{-3}$ | $<10^{-4}$ | $<0.2$ | $4.1^{+0.7}_{-0.8}$ | $0.020$
G27 | $4.6^{+0.1}_{-0.9}\times 10^{-3}$ | $<10^{-4}$ | $9.6^{+1.3}_{-1.7}$ | $<0.5$ | $0.021$
G29 | $0.017^{+0.003}_{-0.003}$ | $<2\times 10^{-4}$ | $0.004^{+0.396}_{-0.003}$ | $<0.3$ | $0.008$
Tornado | $0.86^{+0.05}_{-0.04}$ | $<10^{-4}$ | $<0.2$ | $12.5^{+0.6}_{-0.9}$ | $0.017$
G43 | $1.61^{+0.07}_{-0.09}$ | $<5\times 10^{-3}$ | $<2$ | $<1$ | $0.203$
G340 | $<0.7$ | $<4\times 10^{-3}$ | $167^{+98}_{-31}$ | $123^{+14}_{-99}$ | $0.928$
G349 | $0.30^{+0.03}_{-0.02}$ | $<10^{-4}$ | $<0.5$ | $38.2^{+1.2}_{-1.8}$ | $0.114$
Silicate
G11 | $5.3^{+1.3}_{-5.0}\times 10^{-3}$ | $<10^{-3}$ | $<4$ | $6.6^{+1.0}_{-1.2}$ | $0.020$
G27 | $6.4^{+0.1}_{-2.5}\times 10^{-3}$ | $<6\times 10^{-4}$ | $40.5^{+5.0}_{-6.4}$ | $<0.6$ | $0.021$
G29 | $0.014^{+0.003}_{-0.004}$ | $<1\times 10^{-3}$ | $<1$ | $<4$ | $0.008$
Tornado | $0.307^{+0.020}_{-0.019}$ | $<10^{-4}$ | $<0.5$ | $45.1^{+1.1}_{-1.3}$ | $0.017$
G43 | $0.47^{+0.10}_{-0.10}$ | $<10^{-3}$ | $<2$ | $142^{+20}_{-23}$ | $0.203$
G340 | $<0.5$ | $<0.03$ | $962^{+171}_{-117}$ | $130^{+25}_{-71}$ | $0.928$
G349 | $0.22^{+0.02}_{-0.02}$ | $<2\times 10^{-4}$ | $<0.5$ | $101.6^{+2.4}_{-2.8}$ | $0.114$
Figure 6: Ratio of the dust mass in the hot component to the total dust mass
versus hot component gas density. Crosses represent median values of the MCMC,
error bars the 16th and 84th percentiles. Values for carbon and silicate
grains are shown in black and red respectively. G43 has been plotted at
$n_{\rm H}=1.1\,{\rm cm}^{-3}$ to avoid confusion with G340, and the carbon
and silicate data slightly offset from each other.

Figure 7: Ratio of the dust mass in small and large grains versus hot
component gas density, in the hot (left) and cold (right) components. Crosses
represent median values of the MCMC, error bars the 16th and 84th percentiles,
and triangles the 84th percentile as an upper limit. Values for carbon and
silicate grains are shown in black and red respectively. G43 has been plotted
at $n_{\rm H}=1.1\,{\rm cm}^{-3}$ to avoid confusion with G340, and the carbon
and silicate data slightly offset from each other. An MRN small/large ratio of
$0.16$ is marked with a dashed line.

##  2 Method

###  2.1 Observational sample

We consider a sample of seven core-collapse SNRs: G$11.2$-$0.3$ (hereafter
G11), G$27.4$+$0.0$ (G27), G$29.7$-$0.3$ (G29), G$43.3$-$0.2$ (G43),
G$340.6$+$0.3$ (G340), G$349.7$+$0.2$ (G349), and G$357.7$-$0.1$ (the
Tornado). We have previously investigated the Tornado in Chawner et al.
(2020b), and G11, G27 and G29 in Priestley et al. (2021). In all four SNRs,
the IR SEDs were inconsistent with collisionally-heated dust grains, for gas
properties derived from X-ray data. We found that an additional, colder dust
component is required to reproduce the far-IR SEDs, but were unable to
constrain its properties beyond estimating the dust masses involved.

The other three SNRs (G43, G340 and G349) were selected from the Chawner et
al. (2019, 2020a) catalogue of Galactic plane SNRs due to having both clear
SNR-associated IR emission in at least one band (most often Spitzer MIPS
$24\,{\rm\mu m}$), and X-ray emission indicating ongoing shock interation with
the surrounding ISM. These were excluded from the analysis in Priestley et al.
(2021) due to their irregular morphologies; Figure 1 shows far-IR three-colour
images of G11 and G340, with X-ray contours overlaid. While G11 has a
coincident shell-like structure in both the far-IR and X-ray data, indicating
an interaction between the SNR and the surrounding ISM, G340 is much more
confused. Nonetheless, the combination of enhanced X-ray emission and dust
temperature (represented by the $70\,{\rm\mu m}$ flux) within the SNR strongly
suggests interaction with ambient material. Table 1 lists relevant physical
properties for the SNRs.

G11 and G29 both show evidence of newly-formed ejecta dust interior to the
region of interaction with the ISM, found in the pulsar wind nebulae located
at the centres of these SNRs111In both cases, the radiation field generated by
the pulsar wind nebula is insufficient to power the central dust emission
(Priestley et al., 2020). As the swept-up dust of interest is located at even
greater distances, we assume the impact of the central object is negligble.
(Chawner et al., 2019). For these objects, we take IR fluxes from Priestley et
al. (2021), extracted from annuli excluding the central regions, and thus
presumably dominated by the swept-up ISM. G27 has little central IR flux, but
does have a shell-like X-ray structure, so we again use the Priestley et al.
(2021) annulus fluxes. Fluxes for the Tornado are taken from Chawner et al.
(2020b), and those for G43, G340 and G349 from Chawner et al. (2020a), using
circular apertures. These last four SNRs have previously-derived dust masses
($\gg 1\,{\rm M}_{\odot}$) far in excess of what could be produced by a single
CCSN, so we assume the IR fluxes are primarily due to ISM dust, and that any
contribution from ejecta dust is negligble. The IR data for each SNR are
listed in Appendix A.

###  2.2 Dust SED model

For each SNR, we calculate dust emission models using dinamo (Priestley et
al., 2019), which determines the temperature distribution for grains heated by
the local radiation field and electron/ion collisions. We assume there are two
gas components in each SNR; a ‘hot’ component responsible for the X-ray
emission, with typical densities of $\sim 1\,{\rm cm}^{-3}$ and temperatures
$\gtrsim 10^{6}\,{\rm K}$, and an additional ‘cold’ component with a much
higher pre-shock density ($\gtrsim 100\,{\rm cm}^{-3}$) and thus a much lower
post-shock temperature ($\ll 10^{6}\,{\rm K}$). The two components are assumed
to be spatially well-mixed (e.g. cold clumps embedded in a hot diffuse medium)
, so that the local radiation field is the same for both.

The gas properties of the hot component are taken from analysis of the X-ray
data in the literature, listed in Table 1, and we assume $n_{\rm e}=n_{\rm H}$
for simplicity. Properties for G11, G27 and G29 were derived from modelling
X-ray data in Priestley et al. (2021), and those for the Tornado obtained
similarly by Sawada et al. (2011). The temperature for G43 is taken from
Keohane et al. (2007), and we use the lower end of their quoted range of
densities ($1-3.5\,{\rm cm}^{-3}$). Park et al. (2010) report a temperature
for G340 of $1-1.5\,{\rm keV}$; we again take the lower limit, and assume a
density of $1\,{\rm cm}^{-3}$, typical for the rest of our sample. Leahy et
al. (2020) provide the temperature and the emission measure for G349, again
from X-ray modelling, and we obtain the density from the SNR volume, the
emission measure, and the assumption $n_{\rm H}=n_{\rm e}={\rm constant}$
throughout the SNR.

For the cold component, we assume $n_{\rm H}=1000\,{\rm cm}^{-3}$, $n_{\rm
e}=0.1\,{\rm cm}^{-3}$ and $T=5000\,{\rm K}$ for all SNRs, as determined for
G11 from fits to the Andersen et al. (2011) H2 line observations in Priestley
et al. (2021). These properties may not be appropriate for the rest of our
sample, but they are fairly typical values for SNRs interacting with dense
ambient material, as derived from molecular line observations (e.g. Reach et
al., 2005; Zhu et al., 2014). We discuss the importance of this assumption in
Appendix B.

We assume the local radiation field in both the hot and cold component is
primarily due to emission from the shock-ISM interaction. We use mappings
(Sutherland & Dopita, 2017) to calculate plane-parallel radiative shock
models, with the initial gas density, $n_{\rm ISM}$, and shock velocity,
$v_{\rm sh}$, chosen to reproduce the post-shock density and temperature
reported by X-ray studies of each object (the initial temperature is fixed to
$10^{4}\,{\rm K}$). We then scale the resulting SED by a factor $f$, such that
the X-ray luminosity of a spherical shell of emitting material, with the same
radius as the SNR, matches the observed values given by Koo et al. (2016). The
Tornado and G340 are not included in Koo et al. (2016), so we adopt the values
of G349 and G43 respectively, being the best-matched SNRs in terms of X-ray
derived gas properties.

For a spherical shell of emitting material, the flux at any point within the
shell is the same as the flux at the centre by symmetry. We approximate the
local radiation field heating the dust using this central radiation field.
While swept-up grains located at the shock front are much closer to some of
the shocked ISM generating this radiation field, they are also much further
away from most of it. The $r^{-2}$ scaling of the received flux with distance
to the source should, roughly, cancel out the $\sim r^{2}$ growth in the
amount of material emitting at a given distance. The typical radiation field
experienced by a dust grain should therefore be comparable to that at the
centre. While several of the SNRs in our sample clearly deviate from spherical
symmetry, this represents a substantial upgrade on our previous work (Chawner
et al., 2020b; Priestley et al., 2021), where we assumed a Mathis et al.
(1983) ISM field scaled by an arbitrary constant.

Typical shock-generated radiation fields are very different from those in the
wider ISM, affecting the resulting dust SEDs. Figure 2 shows the radiation
field produced by this method for G11 compared to the Mathis et al. (1983) ISM
field, and corresponding SEDs for carbon grains of different sizes heated by
the two fields. The higher ultraviolet (UV) flux from the ISM field results in
$0.1\,{\rm\mu m}$ carbon grains being heated to higher temperatures than for
the G11 field, whereas the much greater flux of X-ray photons from the SNR
causes non-equilibrium effects in $5\,{\rm nm}$ grains to become more
important, reflected in the increased mid-IR grain emission. These effects are
not necessarily universal, even for different grain types in the same
radiation field (large silicate grains have similar temperatures for both the
G11 and ISM fields) - it is essential to consider the local radiation field on
a case-by-case basis. Shock parameters and scaling factors are listed in Table
1.

We calculate dust SEDs for $0.1\,{\rm\mu m}$ and $5\,{\rm nm}$ grains,
representing the largest and smallest sizes typically present in the ISM
(Mathis et al., 1977), for the hot and cold components of each SNR. We
demonstrate in Appendix C that our results are not sensitive to the specific
choice of ‘large’ and ‘small’ grain sizes. We then fit the observed IR
SEDs222Flux measurements shortwards of $24\,{\rm\mu m}$ are treated as upper
limits, due to the potential for significant non-dust contamination at these
wavelengths., after convolving with the appropriate filter response curves,
with the mass of each dust component (small/large, hot/cold) as the four free
parameters. In the following, we depict the hot/large dust component in blue,
hot/small in orange, cold/large in green, and cold/small in purple. Fitting is
done using emcee, a Monte Carlo Markov chain (MCMC) code (Foreman-Mackey et
al., 2013), with $500$ walkers, $5000$ steps per walker, and $500$ burn-in
steps, which is sufficient for convergence. For dust properties, we use either
carbon grains, with optical constants taken from Zubko et al. (1996) and a
bulk density of $1.6\,{\rm g\,cm}^{-3}$, or silicates, with MgSiO3 optical
constants from Dorschner et al. (1995) (extended to far-UV/X-ray wavelengths
with values from Laor & Draine 1993) and a bulk density of $2.5\,{\rm
g\,cm}^{-3}$. While ISM dust includes both species, the data are insufficient
to fit both simultaneously, and our main conclusions hold regardless of the
assumed grain composition.

##  3 Results

Figure 3 shows the results of our SED fitting for G11. The cold dust mass is
at least three orders of magnitude larger than that of the hot component,
consistent with the results from Priestley et al. (2021). We also find strict
limits on the mass of small grains that can be present in the hot component,
due to their high emissivity around $\sim 10\,{\rm\mu m}$ (Figure 4) combined
with strong upper limits on the observed flux in this wavelength region, again
consistent with our previous results. For either grain composition, we find
that a significant mass of small grains are required in the cold component, at
least comparable to that in large grains, and possibly much larger. This is in
contrast to the MRN size distribution we assumed for this component in
Priestley et al. (2021), where most of the mass is in the largest grain sizes,
and suggests significant dust processing in the shocked material.

Figure 5 shows the best-fit carbon grain SEDs for the remaining six SNRs in
our sample (results for silicate grains are shown in Appendix D). While the
total dust mass in each SNR ranges from $\sim 1-100\,{\rm M}_{\odot}$, in all
cases this mass is primarily in the cold component, with hot grains typically
making up a negligible fraction of the total. The hot component consists only
of large grains in all SNRs for which it has a non-negligible mass, and for
all but one (G27), small grains make up a substantial fraction of the dust
mass in the cold component. The parameter distributions returned by the MCMC
confirm these findings. Median masses for each dust component333These can
differ significantly from the best-fit masses in Figures 3 and 5, generally
indicating that the mass of that component is poorly constrained due to large
observational uncertainties (e.g. G29). and the 16th and 84th percentiles,
listed in Table 2, show that for all seven SNRs, the hot/small dust mass is
consistent with zero, and typically constrained to be much lower than the
other three dust components, regardless of the assumed grain composition.

The total SNR dust masses for G11, G27, G29 and the Tornado are all within a
factor of a few of our previous estimates for these objects (Chawner et al.,
2020b; Priestley et al., 2021). For G340 and G349, our masses for carbon
grains are in good agreement with those obtained from blackbody fits by
Chawner et al. (2020a), while with silicate optical properties the dust masses
are somewhat larger, due to the lower mass opacity. Our carbon mass for G43 is
significantly lower than the Chawner et al. (2020a) estimate because in this
case, the far-IR fluxes can be reproduced by grains in the hot component, with
high grain temperature and thus high emissivity. With silicate grains, for
which this is not the case, our estimated mass is a factor of a few larger, as
with G340 and G349. In general, our model results in dust masses basically
consistent with those from previous work. The large ($>100\,{\rm M}_{\odot}$)
dust masses found for some of the SNRs are required by the observed far-IR
fluxes, for the assumed distances in Table 1 and typical dust mass opacities
at these wavelengths.

We estimate the mass of swept-up ISM in the SNRs using the ambient densities
derived from the X-ray data, $n_{\rm ISM}$, and volumes corresponding to the
regions the IR SEDs were extracted from, $V$, as given in Table 1. Assuming an
ISM DTG ratio of $0.01$ and a hydrogen mass fraction of $0.7$, the total
swept-up dust masses for each SNR are given in Table 2, if the average density
of the ambient ISM is that of the (pre-shock) hot component. These values are
all $<0.1\,{\rm M}_{\odot}$. The total observed dust mass exceeds our estimate
for all seven SNRs, and by factors of $\gtrsim 100$ for all SNRs except G29.
In order to contain this much dust, the average ambient ISM density around the
SNRs would have to be larger than the $\sim 1\,{\rm cm}^{-3}$ values in Table
1 by a similar factor, i.e. $\left<n_{\rm ISM}\right>\gtrsim 100\,{\rm
cm}^{-3}$, comparable to the typical average densities of molecular clouds on
these scales (Larson, 1981).

Figure 6 shows the ratio of the hot component dust mass to the total dust mass
for each SNR, plotted against the density of the hot component. With a few
exceptions, this value is well-constrained to be $<0.1$, and in most cases
$<0.01$ \- the dust located in the high-temperature shocked material makes up
a percent-level fraction of the total swept-up dust mass. Due to the higher
grain temperatures, this dust component typically contributes most of the mid-
IR flux, but it is unrepresentative of the bulk of the swept-up dust mass.
There is a suggestion of a negative correlation between the hot component dust
mass fraction and the gas density. This could indicate either an intrinsically
lower mass fraction of this component in denser regions, or more efficient
sputtering in denser gas destroying a larger proportion of the initial dust
mass.

Figure 7 shows the mass ratio of small to large grains in both the hot and the
cold components, again plotted against density. The mass ratio of grains with
radii $\leq 10\,{\rm nm}$ to those $\geq 0.1\,{\rm\mu m}$ for an MRN size
distribution ($\sim 16\%$) is indicated for comparison. In general, it appears
that the hot component is substantially depleted in small grains compared to
the undisturbed ISM, whereas the cold component either has a more typical size
distribution, or is enhanced in small grains, to the extent that the mass of
large grains is negligible in some SNRs (note that because grain mass is
proportional to $a^{3}$, the enhancement in grain number is even larger than
indicated by Figure 7).

##  4 Discussion

Figure 8: Best-fit dust SEDs for G11, with radiative heating by both the
shock-generated radiation field, and a Mathis et al. (1983) ISM field enhanced
by a factor of five - data (black crosses), total model SED (black line), and
individual component SEDs: hot/large (blue); hot/small (orange); cold/large
(green); cold/small (purple). Note that some component SEDs may not be
visible, due to their contributing a negligible amount to the total SED. Left:
carbon grains. Right: silicates.

###  4.1 Caveats

The flux measurements we model (Appendix A) are not the raw observational
fluxes, but the estimated contribution from the SNRs at each wavelength, after
the subtraction of background flux from unrelated material. There is no clear
consensus on the best way to estimate these background contributions, and even
if they are explicitly included as an additional model component, some
assumptions have to be made about its properties (such as SED shape; see De
Looze et al. 2017; De Looze et al. 2019). Given this freedom of choice, it is
impossible to rigorously account for the possible impact on our results. We
have thus refrained from drawing any conclusions based on single objects in
our sample. The consistent qualitative results seen in Figures 6 and 7 across
the sample of SNRs suggest that these are real physical phenomena, even if the
numerical values for any individual SNR should be viewed with some caution.

Our sample was selected from the Chawner et al. (2019, 2020a) SNR catalogue
based on the presence of co-spatial X-ray and warm dust ($24/70\,{\rm\mu m}$)
emission, taken as a sign of interaction. We are therefore biased towards SNRs
which are detectable in both these shock tracers, which likely corresponds to
high ambient densities. In fact, all seven of our SNRs show signs of
interaction with molecular material (G11, G27, G29, G43 - Kilpatrick et al.
2016; G340 - Green et al. 1997; G349 - Lazendic et al. 2010; Tornado - Hewitt
et al. 2008). These interactions are not rare; $\sim 20\%$ ($88/383$) of the
SNRs in the Ferrand & Safi-Harb (2012) catalogue are listed as interacting
with molecular clouds, and this seems likely to be an underestimate given the
requirement for targeted molecular line observations. While our sample may
well be biased, we do not appear to be selecting for a particularly uncommon
class of object.

The best-fit models for several SNRs (Figures 3, 5 and 12) have either large
$\chi_{\rm red.}^{2}$s, indicating a poor correspondance between model and
data, or $\chi_{\rm red.}^{2}<1$, which suggests that the model has overfit
the data. In the latter case, it is clear that the SEDs of some objects can be
fit with only two, or even one, dust component, rather than the four we use.
However, the MCMC fitting approach accounts for degeneracies between model
parameters. Figures 6 and 7 include all the information from the chain, so we
are confident that the conclusions drawn from them are robust against
overfitting issues.

For those SNRs with high $\chi_{\rm red.}^{2}$ values, we note that these are
typically driven by relatively small uncertainties on the far-IR fluxes. This
is the wavelength range where background subtraction is most uncertain, as it
contains the emission peak of cold ($\sim 20\,{\rm K}$) ISM dust, so there are
likely to be significant additional systematic uncertainties not accounted for
by our model. This wavelength range is also where the least well-constrained
model parameters, such as the cold gas properties (Appendix B), have the most
impact on the model dust SEDs. Modest variations in the assumed cold gas
properties would almost certainly be able to obtain statistically-good fits to
the data, without substantially altering our main conclusions.

We have assumed that the radiation field in both components is solely due to
shock interactions, but Figure 2 suggests that heating by the ISM radiation
field could also be relevant, particularly for those SNRs located at small
Galactocentric distances where the typical radiation field strength is higher
(Mathis et al., 1983). If we include an additional Mathis et al. (1983)
radiation field in the dust heating model, multiplied by a factor of five to
approximate the stronger $5\,{\rm kpc}$ field, the best-fit models (shown in
Figure 8) have much lower small/large mass ratios in the cold component than
those in Figure 3 without the additional ISM field. However, the upper limit
on this quantity (as given by the 84th percentile of the MCMC chain) is $0.36$
for carbon grains and $2.2$ for silicates, still consistent with values above
the $0.16$ of the MRN size distribution (and the silicate median of $0.25$ is
also higher than the ISM value). Depending on the details of the dust heating
model, significant masses of small grains in the cold component may not be
necessary to fit the observed SEDs, but they do not seem to be ruled out
either, in contrast to the situation in the hot component.

Another implicit assumption in our models is that all IR flux comes from dust
which has already been swept-up by the SNR. An alternative possibility is that
the far-IR emission comes from dust grains ahead of the shock, which would
explain the large dust masses relative to the inferred swept-up gas masses,
and the grain size distributions being consistent with that in the ISM. We
disfavour this explanation, as the shock radiation field should only penetrate
a short distance into the ambient medium (Docenko & Sunyaev, 2010), and so the
volume of dust preheated above ambient temperatures is quite small. The
average density in this layer would then have to be orders of magnitude larger
than our estimate above, to a somewhat implausible degree (see discussion in
Priestley et al. 2021). In any case, this possibility does not alter our
conclusions about the grain sizes in the hot component, or that the mass of
this component - in both gas and dust - is a negligible fraction of the total
in the immediate surroundings of the SNR.

###  4.2 Implications

The overall picture we find for our sample of SNRs is that the high-
temperature, diffuse material contains a relatively small mass of large
($\gtrsim 0.1\,{\rm\mu m}$) grains, while dust in the denser, cooler shocked
gas makes up virtually all the total swept-up dust mass, and may include a
substantial mass of small ($\lesssim 10\,{\rm nm}$) grains. This is similar to
the situation in the Cassiopeia A reverse shock (Priestley et al., 2022), and
suggests that the physical processes affecting dust grains in shocks do not
differ between the ISM and metal-enriched CCSN ejecta. The observed
distributions of grain sizes in the two gas components are suggestive of the
two main processes responsible for processing dust grains in shocks: small
grains in the hot component are rapidly destroyed by sputtering; large grains
in the cold component are efficiently converted into smaller grains via
shattering in grain-grain collisions (Kirchschlager et al., 2019).

Although there is some evidence for shattering (large small-grain mass
fractions) in the dense, cool ejecta (subject to the caveats discussed above),
we find strict upper limits on the mass of small grains in the hot component
for all SNRs. The hot component small/large mass ratios in Figure 7 suggest
that either shattering is inefficient in this phase, or is at least
subdominant to sputtering (i.e. newly-produced small grains are destroyed by
sputtering on shorter timescales than they are produced by shattering of large
grains). Most theoretical studies of dust destruction (Jones et al., 1996;
Slavin et al., 2015; Kirchschlager et al., 2022) assume ambient ISM densities
of $\sim 0.1-1\,{\rm cm}^{-3}$, similar to the hot component densities in
Table 1, but find much more efficient shattering than appears to be the case
in our SNRs (see e.g. the post-shock grain size distributions from Slavin et
al. 2015). These models may be overestimating the importance of grain-grain
collisions in the processing of shocked ISM dust.

The severe mismatch between the estimated total swept-up dust masses, if the
ambient ISM densities are those derived from X-ray measurements, and the
observed present-day dust masses in Table 2, effectively requires that the
total (gas plus dust) mass of the hot component is a small fraction of that
swept up by the SNRs. Making the conservative assumption that no dust has been
destroyed, the implied swept-up (or soon to be swept-up) gas masses for a DTG
ratio of $0.01$ are (with the exception of G29) hundreds of times larger than
those of the SNR hot components (estimated using the gas densities and volumes
in Table 1). Any post-shock reduction in dust mass, or a hot component filling
factor lower than unity, will make this discrepancy larger. When considering
the overall effect of the SNR on the surrounding ISM, what occurs in (or to)
the hot component - the only phase typically modelled by theoretical work - is
effectively negligible.

As the gas temperatures in the cold component are unlikely to be high enough
for thermal sputtering444Kinetic sputtering may be effective regardless of gas
temperature for a sufficiently strong shock, although in higher-density gas
the shock strength is also necessarily reduced, while the coupling between gas
and grain motions is increased. We would thus expect the kinetic sputtering
rate to also be lower in the cold component. to be effective ($>10^{5}\,{\rm
K}$; Biscaro & Cherchneff 2016), and grain collisions themselves mostly
reprocess rather than destroy dust (Kirchschlager et al., 2019), it is
conceivable that a significant fraction of the dust in this component has
survived being shocked. It is generally thought that the majority of the mass
in the ISM is contained in much denser, colder phases than those typically
investigated by models of dust destruction (McKee & Ostriker, 1977; Jones &
Nuth, 2011). There is observational evidence, in some cases, for a large
fraction of the total SN momentum and energy going into these phases (e.g.
Cosentino et al., 2022), rather than the more diffuse ISM. If the destruction
efficiency in the dense ISM is in fact lower, it seems likely that models
assuming a uniform, low-density ISM are overestimating the rate of dust
destruction, particularly if grain shattering in the low-density ISM is also
being overestimated (which may have a huge impact on the destruction
efficiency; Kirchschlager et al. 2022).

##  5 Conclusions

We have modelled the IR dust emission from the shocked ISM for a sample of
seven SNRs, taking into account the multiphase nature of the shocked material,
and the uncertain (and likely non-standard) grain properties. We find
consistent results across the sample: grains located in the cooler ($\sim
1000\,{\rm K}$) shocked gas make up $>90\%$ of the total surviving swept-up
dust mass; only large ($\gtrsim 0.1\,{\rm\mu m}$) grains have survived in the
hot ($>10^{6}\,{\rm K}$) phase of the shocked ISM; grain size distributions in
the colder phase are consistent with those in the ISM, or possibly even biased
towards small ($\lesssim 10\,{\rm nm}$) grains. We suggest that this indicates
efficient sputtering in the hot phase, and either efficient shattering or
generally inefficient dust processing in the cold phase.

The lack of evidence for grain shattering in the hot, diffuse swept-up ISM is
contrary to models of dust destruction in shocks. Most theoretical predictions
of the dust destruction efficiency in SNRs assume a uniform, low-density
ambient medium. The multi-phase nature of the observed SNRs, with a very small
fraction of the total dust mass in the low-density material, suggests that
these predicted values may be significantly overestimating the dust
destruction efficiency of SNe.

## Acknowledgements

We are grateful to Florian Kirchschlager for comments on a draft version of
this paper. FDP is supported by a consolidated grant (ST/K00926/1) from the
Science and Technology Facilities Council. HC and HLG acknowledge support from
the European Research Council (ERC) grant COSMICDUST ERC-2014-CoG-647939. IDL
acknowledges support from ERC starting grant 851622 DustOrigin. MJB
acknowledges support from the ERC grant SNDUST ERC-2015-AdG-694520.

## Data Availability

The data underlying this article will be made available upon request.

## References

  * Andersen et al. (2011) Andersen M., Rho J., Reach W. T., Hewitt J. W., Bernard J. P., 2011, ApJ, 742, 7
  * Biscaro & Cherchneff (2016) Biscaro C., Cherchneff I., 2016, A&A, 589, A132
  * Borkowski et al. (2006) Borkowski K. J., et al., 2006, ApJ, 642, L141
  * Chawner et al. (2019) Chawner H., et al., 2019, MNRAS, 483, 70
  * Chawner et al. (2020a) Chawner H., et al., 2020a, MNRAS, 493, 2706
  * Chawner et al. (2020b) Chawner H., et al., 2020b, MNRAS, 499, 5665
  * Cosentino et al. (2022) Cosentino G., et al., 2022, MNRAS, 511, 953
  * De Looze et al. (2017) De Looze I., Barlow M. J., Swinyard B. M., Rho J., Gomez H. L., Matsuura M., Wesson R., 2017, MNRAS, 465, 3309
  * De Looze et al. (2019) De Looze I., et al., 2019, MNRAS, 488, 164
  * De Looze et al. (2020) De Looze I., et al., 2020, MNRAS, 496, 3668
  * Docenko & Sunyaev (2010) Docenko D., Sunyaev R. A., 2010, A&A, 509, A59
  * Dorschner et al. (1995) Dorschner J., Begemann B., Henning T., Jaeger C., Mutschke H., 1995, A&A, 300, 503
  * Dunne et al. (2009) Dunne L., et al., 2009, MNRAS, 394, 1307
  * Ferrand & Safi-Harb (2012) Ferrand G., Safi-Harb S., 2012, Advances in Space Research, 49, 1313
  * Foreman-Mackey et al. (2013) Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306
  * Frail et al. (1996) Frail D. A., Goss W. M., Reynoso E. M., Giacani E. B., Green A. J., Otrupcek R., 1996, AJ, 111, 1651
  * Galliano et al. (2021) Galliano F., et al., 2021, A&A, 649, A18
  * Gomez et al. (2012) Gomez H. L., et al., 2012, ApJ, 760, 96
  * Green (2004) Green D. A., 2004, Bulletin of the Astronomical Society of India, 32, 335
  * Green et al. (1997) Green A. J., Frail D. A., Goss W. M., Otrupcek R., 1997, AJ, 114, 2058
  * Hewitt et al. (2008) Hewitt J. W., Yusef-Zadeh F., Wardle M., 2008, ApJ, 683, 189
  * Hirashita & Yan (2009) Hirashita H., Yan H., 2009, MNRAS, 394, 1061
  * Hoang et al. (2019) Hoang T., Tram L. N., Lee H., Ahn S.-H., 2019, Nature Astronomy, 3, 766
  * Hu et al. (2019) Hu C.-Y., Zhukovska S., Somerville R. S., Naab T., 2019, MNRAS, 487, 3252
  * Jones & Nuth (2011) Jones A. P., Nuth J. A., 2011, A&A, 530, A44
  * Jones et al. (1996) Jones A. P., Tielens A. G. G. M., Hollenbach D. J., 1996, ApJ, 469, 740
  * Keohane et al. (2007) Keohane J. W., Reach W. T., Rho J., Jarrett T. H., 2007, ApJ, 654, 938
  * Kilpatrick et al. (2016) Kilpatrick C. D., Bieging J. H., Rieke G. H., 2016, ApJ, 816, 1
  * Kirchschlager et al. (2019) Kirchschlager F., Schmidt F. D., Barlow M. J., Fogerty E. L., Bevan A., Priestley F. D., 2019, MNRAS, 489, 4465
  * Kirchschlager et al. (2022) Kirchschlager F., Mattsson L., Gent F. A., 2022, MNRAS, 509, 3218
  * Koo et al. (2016) Koo B.-C., Lee J.-J., Jeong I.-G., Seok J. Y., Kim H.-J., 2016, ApJ, 821, 20
  * Kothes & Dougherty (2007) Kothes R., Dougherty S. M., 2007, A&A, 468, 993
  * Laor & Draine (1993) Laor A., Draine B. T., 1993, ApJ, 402, 441
  * Larson (1981) Larson R. B., 1981, MNRAS, 194, 809
  * Lazendic et al. (2010) Lazendic J. S., Wardle M., Whiteoak J. B., Burton M. G., Green A. J., 2010, MNRAS, 409, 371
  * Leahy et al. (2020) Leahy D. A., Ranasinghe S., Gelowitz M., 2020, ApJS, 248, 16
  * Martínez-González et al. (2019) Martínez-González S., Wünsch R., Silich S., Tenorio-Tagle G., Palouš J., Ferrara A., 2019, ApJ, 887, 198
  * Mathis et al. (1977) Mathis J. S., Rumpl W., Nordsieck K. H., 1977, ApJ, 217, 425
  * Mathis et al. (1983) Mathis J. S., Mezger P. G., Panagia N., 1983, A&A, 128, 212
  * Matsuura et al. (2011) Matsuura M., et al., 2011, Science, 333, 1258
  * McKee & Ostriker (1977) McKee C. F., Ostriker J. P., 1977, ApJ, 218, 148
  * Morgan & Edmunds (2003) Morgan H. L., Edmunds M. G., 2003, MNRAS, 343, 427
  * Niculescu-Duvaz et al. (2021) Niculescu-Duvaz M., Barlow M. J., Bevan A., Milisavljevic D., De Looze I., 2021, MNRAS, 504, 2133
  * Park et al. (2010) Park S., Lee J., Hughes J. P., Slane P. O., Mori K., Burrows D., Garmire G. P., 2010, in American Astronomical Society Meeting Abstracts #215. p. 454.13
  * Priestley et al. (2019) Priestley F. D., Barlow M. J., De Looze I., 2019, MNRAS, 485, 440
  * Priestley et al. (2020) Priestley F. D., Barlow M. J., De Looze I., Chawner H., 2020, MNRAS, 491, 6020
  * Priestley et al. (2021) Priestley F. D., Chawner H., Matsuura M., De Looze I., Barlow M. J., Gomez H. L., 2021, MNRAS, 500, 2543
  * Priestley et al. (2022) Priestley F. D., Arias M., Barlow M. J., De Looze I., 2022, MNRAS, 509, 3163
  * Ranasinghe & Leahy (2018) Ranasinghe S., Leahy D. A., 2018, AJ, 155, 204
  * Reach et al. (2005) Reach W. T., Rho J., Jarrett T. H., 2005, ApJ, 618, 297
  * Sawada et al. (2011) Sawada M., Tsuru T. G., Koyama K., Oka T., 2011, PASJ, 63, S849
  * Seok et al. (2015) Seok J. Y., Koo B.-C., Hirashita H., 2015, ApJ, 807, 100
  * Slavin et al. (2015) Slavin J. D., Dwek E., Jones A. P., 2015, ApJ, 803, 7
  * Sutherland & Dopita (2017) Sutherland R. S., Dopita M. A., 2017, ApJS, 229, 34
  * Temim et al. (2012) Temim T., Sonneborn G., Dwek E., Arendt R. G., Gehrz R. D., Slane P., Roellig T. L., 2012, ApJ, 753, 72
  * Tian & Leahy (2014) Tian W. W., Leahy D. A., 2014, ApJ, 783, L2
  * Verbiest et al. (2012) Verbiest J. P. W., Weisberg J. M., Chael A. A., Lee K. J., Lorimer D. R., 2012, ApJ, 755, 39
  * Williams et al. (2006) Williams B. J., et al., 2006, ApJ, 652, L33
  * Zhu et al. (2014) Zhu H., Tian W. W., Zuo P., 2014, ApJ, 793, 95
  * Zubko et al. (1996) Zubko V. G., Mennella V., Colangeli L., Bussoletti E., 1996, MNRAS, 282, 1321

##  Appendix A SNR dust fluxes

Table 3 lists the IR fluxes used in the dust modelling for each SNR in our
sample.

Table 3: IR SEDs for our SNR sample, with fluxes given in Jy and filter effective wavelength in $\,{\rm\mu m}$. Data for G11, G27 and G29 are taken from Priestley et al. (2021); G43, G340 and G349 from Chawner et al. (2020a); and the Tornado from Chawner et al. (2020b). SNR | IRAC $8$ | WISE $12$ | MIPS $24$ | PACS $70$ | PACS $160$ | SPIRE $250$ | SPIRE $350$ | SPIRE $500$
---|---|---|---|---|---|---|---|---
G11 | - | $<6.2$ | $26.0\pm 3.1$ | $124.8\pm 22.4$ | $176.8\pm 85.6$ | $65.0\pm 52.1$ | $33.4\pm 21.9$ | $15.3\pm 7.7$
G27 | - | $<1.2$ | $13.0\pm 0.2$ | $25.6\pm 21.6$ | $124.7\pm 70.0$ | $94.4\pm 29.5$ | $47.1\pm 13.2$ | $16.6\pm 4.5$
G29 | - | $<1.9$ | $10.0\pm 2.1$ | $101.5\pm 39.1$ | $16.1\pm 112.4$ | $11.4\pm 47.4$ | $1.3\pm 17.3$ | $2.2\pm 8.2$
Tornado | $<1.66$ | - | $4.3\pm 0.2$ | $164.5\pm 11.5$ | $151.2\pm 10.6$ | $63.2\pm 3.5$ | $25.9\pm 1.4$ | $8.2\pm 0.5$
G43 | $<61.0$ | - | $80.1\pm 6.0$ | $744.5\pm 61.2$ | $283.6\pm 135.2$ | $74.4\pm 113.7$ | $20.9\pm 53.8$ | $5.3\pm 23.1$
G340 | $<24.5$ | - | $23.2\pm 7.8$ | $220.6\pm 29.4$ | $755.0\pm 117.8$ | $506.5\pm 85.1$ | $207.5\pm 36.7$ | $80.2\pm 14.1$
G349 | $<12.6$ | - | $47.8\pm 3.5$ | $588.4\pm 41.5$ | $456.5\pm 32.8$ | $158.8\pm 9.1$ | $60.1\pm 3.5$ | $19.6\pm 1.2$

##  Appendix B Cold gas properties

For collisional heating in the cold component, we have assumed gas properties
derived from H2 line observations by Priestley et al. (2021) for G11. There is
no guarantee that these properties are appropriate for the other SNRs in our
sample. Figure 9 shows the influence of the assumed cold component properties
on the resulting grain fluxes. We consider three cases: the G11 properties
($n_{\rm H}=1000\,{\rm cm}^{-3}$, $n_{\rm e}=0.1\,{\rm cm}^{-3}$,
$T=5000\,{\rm K}$); radiative heating only, with collisional heating turned
off; and $n_{\rm H}=n_{\rm e}=10\,{\rm cm}^{-3}$ with $T=10^{4}\,{\rm K}$,
representing more diffuse, fully-ionised gas. The emissivity of $0.1\,{\rm\mu
m}$ grains is almost completely unaffected by the assumed cold component gas
properties, as the grain heating is dominated by the radiation field in all
cases. For $5\,{\rm nm}$ grains, there is a significant difference between the
case with no collisional heating and the two cases where it is included, but
the grain SEDs are very similar for the two sets of gas properties we
consider.

While investigating the full three-dimensional collisional heating parameter
space ($n_{\rm H},n_{\rm e},T$) is beyond the scope of this paper, it appears
that for a range of post-shock dense gas properties, dust SEDs are similar
enough to be identical for our purposes. Even if we neglect collisional
heating entirely, our main conclusions are qualitatively unchanged. Figure 10
shows best-fit dust masses for G11, with the cold component only heated by the
radiation field. While the dust masses differ from those in Figure 3,
regardless of grain composition we still find that the majority of the dust
mass is in the cold component, that the mass of small grains is comparable to
that of large grains in this component, and that grains in the hot component
(if present) must be large.

Figure 9: Flux per unit mass for cold/large (green) and cold/small (purple)
carbon grains in G11. Solid lines have gas properties derived from H2
observations ($n_{\rm H}=1000\,{\rm cm}^{-3}$, $n_{\rm e}=0.1\,{\rm cm}^{-3}$,
$T=5000\,{\rm K}$), dashed lines are heated solely by the radiation field, and
dotted lines assume $n_{\rm H}=n_{\rm e}=10\,{\rm cm}^{-3}$, $T=10^{4}\,{\rm
K}$.

Figure 10: Best-fit dust SEDs for G11 with no collisional heating by the cold
component - data (black crosses), total model SED (black line), and individual
component SEDs: hot/large (blue); hot/small (orange); cold/large (green);
cold/small (purple). Left: carbon grains. Right: silicates.

##  Appendix C Grain size distributions

Figure 11: Left: flux per unit mass for carbon grains in G11: hot/large
(blue); hot/small (orange); cold/large (green); cold/small (purple). Solid
lines are for single grain sizes, dashed lines for size distributions. Right:
best-fit dust SED for G11 with a distribution of grain sizes - data (black
crosses), total model SED (black line), and individual component SEDs:
hot/large (blue); hot/small (orange); cold/large (green); cold/small (purple).

Our choice of $0.1\,{\rm\mu m}$ to represent large grains, and $5\,{\rm nm}$
to represent small ones, is somewhat arbitrary, although informed by the sizes
typically thought to be present in the ISM. Figure 11 shows the impact of
using a size distribution for each component, rather than a single
representative grain size, using carbon grains in G11 as an example. We
replace small and large grains with power-law distributions, with an MRN
exponent of $-3.5$, spanning the ranges $5-10\,{\rm nm}$ and $0.1-0.3\,{\rm\mu
m}$ respectively. Both the SEDs of the individual dust components, and the
results of the MCMC fit to the G11 data, are affected by at most a factor of a
few. This is a minor source of uncertainty compared to others in our modelling
procedure, and is not sufficient to qualitatively change our conclusions to
any significant extent.

##  Appendix D Silicate grain SED fits

Figure 12: Best-fit dust SEDs for silicate grains - data (black crosses),
total model SED (black line), and individual component SEDs: hot/large (blue);
hot/small (orange); cold/large (green); cold/small (purple). Top left: G27.
Top right: G29. Middle left: Tornado. Middle right: G43. Bottom left: G340.
Bottom right: G349.

Figure 12 shows the best-fit silicate grain SEDs for our sample of SNRs,
excluding G11, which is presented in Figure 3. While the different grain
composition results in non-trivial changes to both the total dust mass and its
distribution between components, we find identical qualitative results across
the sample of SNRs as for carbon grains: the hot component is typically a
small fraction of the total dust mass, and rarely (if ever) contains any
substantial mass of small grains, whereas the cold component requires a non-
negligible mass of small grains.