Dust Formation and Survival in Supernova Ejecta
Mon. Not. R. Astron. Soc. 000, 1–11 (2007) Printed 26 October 2018 (MN LATEX style file v2.2)
Dust Formation and Survival in Supernova Ejecta
Simone Bianchi
and Raffaella Schneider
1 INAF - Istituto di Radioastronomia, Sezione di Firenze, Largo Enrico Fermi 5, 50125 Firenze, Italy
2 INAF - Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy
3 April 2007
ABSTRACT
The presence of dust at high redshift requires efficient condensation of grains in SN
ejecta, in accordance with current theoretical models. Yet, observations of the few
well studied SNe and SN remnants imply condensation efficiencies which are about
two orders of magnitude smaller. Motivated by this tension, we have (i) revisited the
model of Todini & Ferrara (2001) for dust formation in the ejecta of core collapse SNe
and (ii) followed, for the first time, the evolution of newly condensed grains from the
time of formation to their survival - through the passage of the reverse shock - in the
SN remnant. We find that 0.1 - 0.6 M⊙ of dust form in the ejecta of 12 - 40 M⊙ stellar
progenitors. Depending on the density of the surrounding ISM, between 2-20% of
the initial dust mass survives the passage of the reverse shock, on time-scales of about
4−8×104 yr from the stellar explosion. Sputtering by the hot gas induces a shift of the
dust size distribution towards smaller grains. The resulting dust extinction curve shows
a good agreement with that derived by observations of a reddened QSO at z = 6.2.
Stochastic heating of small grains leads to a wide distribution of dust temperatures.
This supports the idea that large amounts (∼ 0.1M⊙) of cold dust (T ∼ 40K) can be
present in SN remnants, without being in conflict with the observed IR emission.
Key words: dust, extinction - shock waves - supernova remnants - supernovae:
individual (Cassiopeia A)
1 INTRODUCTION
In the last few years, mm and submm observations of sam-
ples of 5 < z < 6.4 quasars have provided a powerful way of
probing the very existence and properties of dust within 1
Gyr of the Big Bang. The inferred far-IR luminosities are
consistent with thermal emission from warm dust (T <
100 K), with dust masses > 108M⊙ (Bertoldi et al. 2003;
Robson et al. 2004; Beelen et al. 2006; Hines et al. 2006).
Despite the uncertainties due to the poorly constrained dust
temperatures and absorption coefficients, the estimated dust
masses are huge, implying a high abundance of heavy ele-
ments at z ≈ 6, consistent with the super-solar metallicities
inferred from the optical emission-line ratios for many of
these systems (Pentericci et al. 2002; Freudling et al. 2003;
Maiolino et al. 2003).
Although high redshift quasars are extreme and rare
objects, hardly representative of the dominant star forming
galaxies, the above observations show that early star forma-
tion leads to rapid enrichment of the Interstellar Medium
(ISM) with metals and dust.
It is difficult for the dust to have originated from low-
mass evolved stars at z > 5 as their evolutionary timescales
(108 to 109 yr) are comparable to the age of the Universe
at that time (Morgan & Edmunds 2003; Marchenko 2006).
Thus, if the observed dust at z > 5 is the product of stellar
processes, grain condensation in supernova (SN) ejecta pro-
vides the only viable explanation for its existence. This sce-
nario has recently been tested through the observation of the
reddened quasar SDSSJ1048+46 at z = 6.2 (Maiolino et al.
2004). The inferred extinction curve of the dust responsible
for the reddening is different with respect to that observed
at z < 4 (Small Magellanic Cloud-like, Hopkins et al. 2004),
and it shows a very good agreement with the extinction
curve predicted for dust formed in SN ejecta.
Theoretical models, based on classical nucleation the-
ory, predict that a few hundred days after the explo-
sions silicate and carbon grains can form in expanding SN
ejecta, with condensation efficiencies in the range 0.1-0.3
(Kozasa et al. 1991; Todini & Ferrara 2001; Clayton et al.
2001).
Direct observational evidences for dust production have
been collected only for a limited number of SNe, such
as 1987A (Wooden et al. 1993) 1999em (Elmhamdi et al.
2003), and 2003gd (Sugerman et al. 2006). With the excep-
tion of 2003gd, the dust masses derived from the IR emission
are ≈ 10−3M⊙, corresponding to condensation efficiencies
which are two orders of magnitude smaller than what the-
ory predicts. A fraction of dust could escape detection if
it is cold and concentrated in clumps. This has been con-
c© 2007 RAS
http://arxiv.org/abs/0704.0586v1
2 S. Bianchi & R. Schneider
firmed to be the case for SN 2003gd where a radiative trans-
fer code has been used to simultaneosly fit the optical ex-
tinction and IR emission, leading to an estimated dust mass
of 2× 10−2M⊙ (Sugerman et al. 2006). However, when ap-
plied to SN 1987A, the same numerical model gives dust
mass estimates which do not differ significantly from previ-
ous analytic results (Ercolano et al. 2007).
Similar low dust masses have been inferred from in-
frared observations of galactic SN remnants with Spitzer
and ISO satellites (Hines et al. 2004; Krause et al. 2004;
Green et al. 2004). The consistent picture that emerges is
that the mid- and far-IR excess observed is due to emission
from small amounts of warm dust, with indicative tempera-
tures T ∼ 80−270 K and masses 3×10−3−10−5M⊙ for Cas
A, and temperatures T ∼ 50 K and masses 3×10−3−0.02M⊙
for the Crab nebula. Cold dust has also been detected
through far-IR and submm observations of these remnants
(Dunne et al. 2003; Krause et al. 2004). However, the inter-
pretation of these data is complicated by the strong contam-
ination from cold dust along the line of sight, providing so
far only upper limits of 0.2M⊙ on the amount of cold dust
associated to the SN remnants.
The aim of the present paper is to critically assess the
model developed by Todini & Ferrara (2001) exploring a
wider range of initial conditions and model assumptions. We
then follow the evolution of dust condensed in SN ejecta on
longer timescales with respect to previous theoretical mod-
els. In particular, we are interested in understanding how the
passage of the reverse shock affects the newly formed grain
size distributions and masses, so as to make predictions for
the expected dust properties from the time of formation in
the ejecta (a few hundred days after the explosion) to its
survival in the SN remnant, hundreds of years later. So far
this process has received little attention, most of the stud-
ies being dedicated to the destruction of ISM dust grains
caused by the SN forward shock (Draine & Salpeter 1979;
Jones et al. 1994; Nozawa et al. 2006), with the notable ex-
ception of Dwek (2005), who, on the basis of timescale con-
siderations, finds that the reverse shock is able to destroy
much of the initially formed dust.
The paper is organised as follows: Sect. 2 revisits the
dust formation models based on the nucleation theory;
Sect. 3 describes the model adopted for the propagation of
the reverse shock into the ejecta and shows the effect of
sputtering on the grain size distribution and total mass; in
Sect. 4 we compare the extinction and emission properties of
the surviving SN dust with observations. Finally, the results
are summarised in Sect. 5.
2 SN DUST FORMATION REVISITED
Models of dust formation in the ejecta of core collapse SNe
typically predict that large masses of dust (0.1−1.0M⊙) are
formed within 1000 days from the onset of the explosion,
when the ejecta are still compact (radius of order 1016cm;
Kozasa et al. 1991; Todini & Ferrara 2001; Nozawa et al.
2003). If these freshly formed dust grains were distributed
homogeneously within the ejecta, their opacity would be
very high, with center-to-edge optical depths of order 102 −
104 in optical wavelengths, depending on the grain material
and size distribution. The ejecta would thus be opaque to
radiation produced within it (Kozasa et al. 1991).
Observations of recent SNe, instead, reveal extinc-
tions smaller than a couple of magnitudes, which imply
dust masses of only 10−4 − 10−2M⊙ (Sugerman et al. 2006;
Ercolano et al. 2007). The dust mass derived from extinc-
tion measures could be underestimated if grains are dis-
tributed in clumps with a small volume filling factor: for
a given amount of grains, a clumpy distribution would pro-
duce a lower effective extinction. However, the comparison
between observations of dust extinction/emission and radia-
tive transfer models shows that the neglect of clumping can
only produce a moderate underestimation of the dust mass
in the ejecta (Ercolano et al. 2007).
To check whether the dust production in SNe is overesti-
mated, we have reconsidered the model of Todini & Ferrara
(2001). In the model, dust formation is investigated in the
framework of standard nucleation theory: when a gas be-
comes supersaturated, particles (monomers) aggregate in
a seed cluster which subsequently grows by accretion of
other monomers (Feder et al. 1966). For grain materials
whose molecules are not present in the gas phase, the key
species approach is adopted (Kozasa & Hasegawa 1987).
Six materials where considered in the original work: amor-
phous carbon (AC), iron, corundum (Al2O3), magnetite
(Fe3O4), enstatite (MgSiO3) and forsterite (Mg2SiO4). Fol-
lowing Schneider et al. (2004), we have added the formation
of SiO2 grains. SiC grains, found in meteorites and consid-
ered to be of SN origin from their anomalous isotopic ratios
(Clayton & Nittler 2004), are not considered since their for-
mation is impeded by the formation of AC and Si-bearing
grains (Nozawa et al. 2003). The model of Todini & Ferrara
also considers the formation and destruction of SiO and CO
molecules: while the first is necessary to study the formation
of Si-bearing grains, the second may be a sink for carbon
atoms that otherwise would accrete on grains.
The ejecta are taken to have a uniform composition
and density, with initial temperature and density chosen to
match the observations of SN1987A. The initial composition
depends on the metallicity and mass of the progenitor star,
Mstar, while the dynamic is given by the mass of the ejecta
Meje and the kinetic energy of the explosion Ekin: the models
of Woosley & Weaver (1995) were used.
In the models of Todini & Ferrara, the gas becomes su-
persaturated after a few hundred days from the explosion.
The nucleation process starts at temperature between 1800K
(for AC) and 1200K (for Si-bearing materials). At the be-
ginning the gas is moderately supersaturated and large seed
clusters, made of N monomers, tend to form. However, their
formation rate per unit volume (the nucleation current) is
small. As the volume of the ejecta increases, the supersat-
uration rate grows and smaller clusters aggregate with a
larger formation rate. This occurs until the gas becomes
sufficiently rarified (because of expansion and/or exhaus-
tion of monomers in the gas phase) and the formation rate
drops. The nucleation process, together with accretion, re-
sults in a typical log-normal grain size distributions (see,
e.g., Todini & Ferrara 2001; Nozawa et al. 2003).
For materials apart from AC, the supersaturation rate
increases quickly during the ejecta expansion, and the seed
clusters can become very small. In Todini & Ferrara (2001)
seed clusters were allowed to be of any size. In this paper we
c© 2007 RAS, MNRAS 000, 1–11
Dust Formation and Survival in SN ejecta 3
Figure 1. Size distribution for grains formed in the ejecta of a
SN with a progenitor star of solar metallicity and mass Mstar =
20M⊙ (N = 2). The distributions of Al2O3 and Mg2SiO4 are
shown with dashed lines for ease of identification.
consider only clusters with N > 2, and introduce discrete
accretion of monomers. While these two (more physical) re-
quirements have a limited effect on AC grains, they alter
the size distributions and masses of grains composed by the
other materials. In Fig. 1 we show the size distribution of
grains formed in the ejecta of a SN with a progenitor star of
solar metallicity and Mstar = 20M⊙. Only AC grains retain
the usual log-normal distribution. Instead, the size distribu-
tion of grains of other materials lacks the low-radius tail.
Compared to the results of Todini & Ferrara (2001), their
total number is reduced (since larger seed clusters have a
smaller formation rate) and their mean size is larger (since
the monomers not allowed to form the smaller clusters are
now available to accrete on the larger).
It is to be noted, however, that the use of the standard
nucleation theory is questionable when clusters are made
by N . 10 monomers (Draine 1979; Gail et al. 1984). To
check what influence this limit has on the results, we have
run models in which the formation of clusters with N < 10
is suppressed. The resulting size distributions confirm the
same trend: less non-AC grains form, and of larger mean
size. Again AC is unaffected.
In Fig. 2 we show Mdust, the mass of dust formed in the
ejecta of SNe of solar metallicity, as a function of Mstar. The
solid line refer to the models with N > 2. Though reduced
with respect to Todini & Ferrara, still considerable masses
of dust are formed, predominantly of AC and Fe3O4 grains.
If Mstar 6 25M⊙, all the available carbon condenses in dust
grains. In the more massive models, roughly equal amounts
of carbon goes in grains and in CO, since the molecule
destruction mechanism provided by 56Co decay is reduced
because of its low yield in the ejecta. Results are similar
(within a factor of two) if the metallicity of the progenitor
stars is below solar. The only distinction is the model with
zero metallicity, where stars with Mstar > 35M⊙ produce no
dust (Schneider et al. 2004). No substantial differences are
found if a different thermal history of the ejecta is assumed:
Mdust is still of the same order of magnitude if densities and
temperatures follow the evolution adopted by Nozawa et al.
(2003). As already seen in Fig. 1, imposing N > 10 results
Figure 2. Mass of dust formed in the ejecta of a SN as a func-
tion of the mass of the progenitor star, for models with different
minimum cluster size and sticking coefficient. The metallicity of
the progenitors is solar.
in a great reduction of the number of non-AC grains: the
dust mass in these models is entirely due to AC, which is
unaffected by the limit (Fig. 2, long-dashed line).
Dust formation models depend strongly on the sticking
coefficient α. In most of the published models, and in the
results presented so far, it is assumed that all gas particles
colliding on a grain will stick to it (α = 1). However, theory
predicts that α depends on the impact energy, on the grain
internal energy, and on the material involved: for the gas
temperature at which most grains form α is significantly re-
duced (Leitch-Devlin & Williams 1985). Indeed, laboratory
experiments on the formation of cosmic dust analogs shows
that α ≈ 0.1 for Si-bearing grains (Gail 2003). Thus, we
have also run models assuming α = 0.1 for all the species
considered. By reducing α, monomers stay in the gas phase
longer and dust formation is delayed to times when super-
saturation is larger: typically, smaller seed clusters form. For
N > 2, the number of non-AC grains is further reduced
and their mass becomes negligible compared to that of AC.
Again all available carbon is locked in AC grains, but their
size distribution is shifted towards lower radii and seed clus-
ters form with N < 10. For N > 10 (Fig. 2, dashed line)
their mass reduces and the size distribution becomes similar
to those of non-AC grains in Fig. 1. At least for low mass
progenitors, the predicted Mdust are closer to the values in-
ferred by observations.
Clearly, the thermodynamic properties of the ejecta are
at the limits of applicability of classical nucleation theory.
A different approach may be needed, expecially if realistic
α values are taken into account. Unless otherwise stated,
in the following we will study the evolution of dust grains
resulting from models with solar metallicity for the SN pro-
genitors and N > 2, α = 1 (so as to conform to most works
in literature). However, we will also discuss the results for
models with different assuptions on N and α.
c© 2007 RAS, MNRAS 000, 1–11
4 S. Bianchi & R. Schneider
3 SURVIVAL IN THE REVERSE SHOCK
As the ejecta expands, a forward shock is driven into the
ISM, which compresses and heats the ambient gas. The
ISM becomes an hostile environment for the survival of dust
grains preexisting the SN event, mainly because of sputter-
ing by collisions with gas particles (Draine & Salpeter 1979;
Jones et al. 1994; Nozawa et al. 2006). In turn, the shocked
ambient gas drives a reverse shock in the ejecta, which, by
about 1000 years, has swept over a considerable fraction of
its volume. The dust within the SNe, then, has to face hos-
tile conditions inside what had previously been its cradle.
We study the process in this Section.
3.1 Dynamics of the reverse shock
Truelove & McKee (1999) have studied the dynamics of a
SN remnant through its nonradiative stages, the ejecta dom-
inated and the Sedov-Taylor. They provide analytic approxi-
mations for velocity and position of the reverse and forward
shocks, as a function of the kinetic energy Ekin and mass
Meje of the ejecta, and of the ISM density ρISM. We use
here their solution for a uniform density distribution inside
the ejecta. The values for Ekin and Meje are the same that
were used in the dust formation models: Ekin = 1.2 × 10
erg and 10M⊙ . Meje . 30M⊙ for stellar progenitor masses
in the range 12-40 M⊙ and metallicities between zero and
solar (Woosley & Weaver 1995). We study the effect of three
different ISM environments, with ρISM = 10
−25, 10−24 and
10−23 g cm−3.
For each model, we have divided the ejecta into Ns
spherical shells. We have assumed that all shells have the
same initial width ∆R = Reje/Ns, with Reje the initial radius
of the ejecta. The mass of each shell is conserved throughout
the evolution. For the j shell (counting shells outwards), the
initial velocity of the gas at its inner boundary is given by
homologous expansion,
vj = veje
, veje =
, (1)
where Rj is the initial radius of the inner shell boundary and
veje is the velocity of the external boundary for ejecta of uni-
form density. For practical purposes, we start our simulation
at a time t0 (ideally, t0 → 0), and we set Reje = vejet0. The
results do not depend on the exact value of t0, provided it
is taken small enough (we use a value of order a few tens of
years).
After setting the initial conditions, we study the evo-
lution of the ejecta with time. At each time step, the re-
verse shock goes inward through a single shell. Thus, at time
ti, the reverse shock has travelled inward through i shells,
and lies at the inner boundary of shell jrs = Ns − i − 1.
Shells that have not been visited by the reverse shock (for
0 6 j 6 jrs − 1) continue to follow homologous expansion,
i.e. the inner and outer radii grow linearly with time, with
velocity given by Eq. 1. Following the shell expansion (in-
crease in the shell volume Vj), the shell gas density decreases
as ρj ∝ V
j . For an adiabatic expansion, the shell tempera-
ture scales as Tj ∝ V
j , with γ = 5/3 (Truelove & McKee
1999). Since the shock is strong, the results are independent
on the initial choice for the gas temperature in the ejecta.
For the shell j = jrs that has been swept over by the
Figure 3. Velocity, density and temperature of the ejecta at
t = tch as a function of the radius, for the model with progen-
itor star of solar metallicity and mass Mstar = 20M⊙ (Meje =
18M⊙) expanding in a ISM with density ρISM = 10
−24 g
cm−3. All quantities are normalised to their characteristic val-
ues (Truelove & McKee 1999). For the model shown here tch =
5800 yr, Rch = 10.7 pc, Tch = 5.7 × 10
7 K, vch = 1800 km s
It is also ρch = ρISM. The contact discontinuity marks the border
between the shocked ejecta and the ISM swept by the forward
shock.
shock at time ti, we apply the standard Rankine-Hugoniot
jump conditions for a strong adiabatic shock. The density,
velocity and temperature change as
γ + 1
γ − 1
vj = v
γ + 1
ṽrs,
Tj = 2
γ − 1
(γ + 1)2
where ρ′j and v
j are the density and velocity before the shock
(i.e. following the same evolution as for shells with j < jrs),
ṽrs is the velocity of the reverse shock in the reference frame
of the unshocked ejecta (provided by Truelove & McKee
1999), m is the mean particle mass and k the Boltzmann’s
constant. To ensure mass conservation, the volume of shell
j = jrs is reduced by a factor (γ − 1)/(γ + 1).
For the shells jrs < j < Nrs shocked at earlier times
t < ti, we compute the velocity vj by interpolating between
the velocity of the j = jrs shell and the velocity of the for-
ward shock (in the ambient rest frame), as a function of the
logarithm of the shell inner radius. Velocity and position
of the forward shock are also given by Truelove & McKee
(1999). As for shells with j < jrs, the evolution of density
and temperature is derived from the condition of adiabatic
expansion and conservation of the shell mass. The typical
trends for velocity, density and temperature around the re-
verse shock are shown in Fig. 3.
We checked the results (in particular the assumption
for the evolution of j > jrs shells) with the 1-D hydrody-
namical models of SN blast waves of van der Swaluw et al.
(2001) and with simulations kindly provided by L. Del Zanna
c© 2007 RAS, MNRAS 000, 1–11
Dust Formation and Survival in SN ejecta 5
(based on the code described in Del Zanna et al. 2003).
However crude, our approximation provide a simple and fast
solution for the density and temperature evolution of the
ejecta during the passage of the reverse shock. Chosing an
adequate number of shells (we use Ns = 400), it agrees with
the complete hydrodynamical solution within a factor of 2.
3.2 Dust grain survival
We assume that dust grains are distributed uniformly within
the ejecta, and that the size distribution is the same every-
where. In the shells that have been visited by the reverse
shock, dust grains are bathed in a gas heated to high tem-
perature (of order 107 − 108 K for the cases studied here).
Also, the gas is slowed down and dust grains decouple from
it, attaining a velocity relative to the gas
vdj =
γ + 1
ṽrs.
Gas particles thus impact on dust grains transferring ther-
mal and kinetic energy, which are of the same order of mag-
nitude (both depending on the reverse shock velocity ṽrs,
which is of order 103 km s−1). Thermal and non-thermal
sputtering result, which erode the dust grain, reducing its
size. Eventually, the gas drag due to direct and Coulomb
collisions slows the grain and non-thermal sputtering weak-
ens. In this work we consider both thermal and non-thermal
sputtering, but we neglect the gas drag and the grain charge:
once passed through the reverse shock, the grain retains its
velocity relative to the gas. We can thus provide upper limits
on the influence of non-thermal sputtering.
The number of atoms that are sputtered off a dust
grain per unit time is given by the sputtering rate dN/dt,
a complex function of the gas density, temperature and of
the nature of the dust/gas (target/projectiles) interaction
(full expressions for dN/dt can be found elsewere, see e.g.
Bianchi & Ferrara 2005). The sputtering rate depends on
the sputtering yield, Y , the fraction of atoms that leave the
target per projectile collision, which is a function of the en-
ergy of the impact. We use here the Y functions described
in Nozawa et al. (2006), and we consider collisions of dust
grains with H, He and O atoms in the ejecta. The grain
radius decreases with sputtering as
, (2)
where, q is the number of atoms in a molecule of the grain
material, and am is the molecule radius, computed from
the material density and the molecule mass. The values for
am can be derived easily from the a0j values of Table 2 in
Nozawa et al. (2003). At each time step, we reduce the grain
size according to Eq. 2 in all shells that have been swept by
the reverse shock. We follow the evolution until the reverse
shock arrives near the center of the ejecta: this is the limit of
validity of the approximations in Truelove & McKee (1999).
After that, we simply assume that the ejecta expands adi-
abatically, and we end the simulations when the sputtering
rate becomes negligible. Since we do not include gas drag
and grain charge, grains do not attain differential veloci-
ties for different sizes. Thus, we have neglected destruction
due to grain-grain collisions. However, sputtering dominates
Figure 4. Changes in the size distribution of AC and Fe3O4
grains. For each material, the thick line is the initial size distribu-
tion (the same as in Fig. 1). The thin line is the size distribution
after the passage of the reverse shock through the ejecta.
over this process for the high shock velocities considered here
(Jones et al. 1994).
Dust grains in the ionized shocked gas are heated mainly
by collisions with electrons. If the grains are small, heating
is stochastic and an equilibrium temperature does not exist.
Instead, a broad temperature distribution P (Td) establishes,
peaking at low temperature but extending also to high val-
ues (Dwek 1986). The temperature may be so high that
dust grains sublimate (Guhathakurta & Draine 1989). For
the cases studied here, however, sublimation is negligible.
Details on the calculation are presented in Appendix A.
In Fig. 4 we show the initial (thick lines) and final
(thin lines) size distributions for AC and Fe3O4 grains in the
ejecta of a star with Mstar = 20M⊙ expanding in a medium
with ρISM = 10
−24 g cm−3. As it is evident for the (initially)
more peaked size distributions of magnetite, sputtering pro-
duces a leaking towards smaller sizes. The evolution of the
size distribution is analogous to that of ISM grains destroyed
by the forward shock (Nozawa et al. 2006).
In Fig. 5 we show the mass of dust that survives the pas-
sage of the reverse shock. For the reference model (dashed
line), the erosion caused by sputtering reduces the dust mass
to about 7% of its initial value, almost independently of the
stellar progenitor model. Most of the dust (about 70% in
mass) is consumed within one characteristic time tch from
the explosion, when 95% of the original volume of the ejecta
has been reached by the reverse shock (tch = 4−8×10
4 yr for
the ejecta discussed here; Truelove & McKee 1999). Dust in
the inner shells is less affected by erosion, because the sput-
tering rate is lower. A minor fraction of the dust mass, less
than 10%, is consumed after the reverse shock bounces at
the center of the ejecta (for t & 2.6tch; Truelove & McKee
1999).
If the SN explodes in a denser ISM, the reverse shock
would travel faster inside the ejecta and would encounter a
gas at higher density. This increases the effect of sputtering.
In Fig. 5 (long-dashed line) we see the fraction of dust mass
that survives when ρISM = 10
−23 g cm−3: only about 2% of
the dust mass survives. Conversely, for a lower density ISM,
a larger fraction is left: for ρISM = 10
−25 g cm−3, it is 20%
c© 2007 RAS, MNRAS 000, 1–11
6 S. Bianchi & R. Schneider
Figure 5. Mass of dust that survives the passage of the reverse
shock in the ejecta, as a function of the mass of the progenitor
star and of the density of the surrounding ISM. The solid line
shows the initial dust mass (same as the solid line in Fig. 2).
(dotted line). While the number of surviving grains changes
with the ISM density, the shape of the size distributions
remain similar in all cases, with the typical patterns shown
in Fig. 4.
No substancial change is observed in models where the
dust was produced by progenitors of metallicity different
from solar. Dust destruction is instead enhanced in models
where a smaller sticking coefficient is adopted. If α = 0.1
(Sect. 2), only 10, 3 and 1% of the original dust mass sur-
vives, respectively, for ρISM = 10
−25, 10−24 and 10−23 g
cm−3 (compared to 20, 7 and 2% for α = 1.0). This is be-
cause for smaller values of α, the dust distribution is made
by grains of smaller radii, which are more easily destroyed.
4 EXTINCTION AND EMISSION FROM SN
Maiolino et al. (2004) measured the reddening in the rest-
frame UV spectrum of a z = 6.2 QSO and found it to be
different from that of the SMC, typically used to dered-
den the spectra of lower redshift QSOs. The measured red-
dening is instead compatible with the extinction law from
the Todini & Ferrara (2001) SN dust model. We repeat here
the same analysis using the updated dust formation mod-
els of Sect. 2 and the final distributions after the reverse
shock passage of Sect. 3. As in Maiolino et al. (2004), we
derive the extinction properties from the grain sizes using
the Mie’s (1908) theory for spherical dust grains and refrac-
tive indexes for dust materials from the literature (references
are provided in Table A1). The procedure is analogous to
that adopted by Hirashita et al. (2005) when modelling the
dust extinction from the SNe dust models of Nozawa et al.
(2003).
In Fig. 6 we show the results for dust formed in SNe
from progenitors of solar metallicity. The grain size distri-
butions from progenitors of different masses have been aver-
Figure 6. Extinction law for SN dust. The thick solid line is
the extinction law for dust freshly formed in the ejecta. The thin
solid line is the extinction law from dust processed by the re-
verse shock. The curves are computed from the IMF-averaged
size distributions of grains formed in SNe from solar metallicity
progenitors (see text for details). The gray line and shaded area
are the extinction law measured on a z = 6.2 QSO and its un-
certainty (Maiolino et al. 2004). The dashed line is the extinction
law of the SMC (Pei 1992).
aged over a stellar Initial Mass Function (IMF): we adopted
the Salpeter IMF, but the results do not depend heavily on
this choice (Maiolino et al. 2004). The thick solid line repre-
sents the extinction law of dust as formed within the ejecta,
without taking into account the grain processing caused by
the reverse shock. The SN dust extinction law is still flatter
than the SMC extinction law, but the agreement with the
observations (shaded area) is worse than in Maiolino et al.
(2004). This is mainly due to a change in the grain materials
that contribute to extinction: apart from AC, present in both
the old and new model, the rise at λ < 2000 Å was due to
Mg2SiO4 grains, with a minor contribution from Fe3O4. In
the new model, Mg2SiO4 contribution is insignificant, while
Fe3O4 grains (larger than in the original model) cause the
far UV rise. The bump at λ ≈ 2500 Å is due to AC grains
and it is typical of the optical properties derived from amor-
phous carbon formed in an inert athmosphere (the ACAR
sample of Zubko et al. (1996)).
During the passage of the reverse shock, Fe3O4 grains
are consumed more effectively than AC grains. The resulting
extinction law (thin solid line in Fig. 6) becomes flatter, lead-
ing to an excellent agreement with observations at λ 6 1600
Å. These results apply for ejecta expanding in a medium
with ρISM = 10
−24 g cm−3. There is no significant change in
the extinction law if different ISM densities are considered,
since the size distributions are similar in all cases (though
the extinction at any given wavelength is smaller for higher
ρISM, because less grains survive). It is worth noting that
grains with a . 20 Å though as abundant as larger grains,
c© 2007 RAS, MNRAS 000, 1–11
Dust Formation and Survival in SN ejecta 7
do not contribute to the extinction law because of their re-
duced extinction cross section. As in Maiolino et al. (2004),
we find that if progenitors of metallicity lower than solar are
considered, the difference in the resulting extinction laws are
small and lie within 0.1 y-axis units from the lines plotted
in Fig. 6.
Extending calculations to the infrared, we have derived
the dust emissivity. For all the IMF averaged size distribu-
tions, the emissivity in the wavelength range 10 6 λ/µm 6
1000 is rather featureless, and can be well described by a
power law in wavelength of index -1.4 with κ(100µm) = 40
cm2 g−1 for models where all dust has been processed by
the reverse shock. Emissivities for dust formed from progen-
itors of a given mass are within 10% of the IMF averaged
value, while the emissivity at the end of dust condensation,
before any significant sputtering has occurred, is found to
be about 20% lower. No significant dependence is found on
the metallicity of the progenitor and on ρISM. In all cases,
the emissivity is almost entirely due to the large AC grains1.
The amount of shock-heated dust in the ejecta can be
derived from infrared observations of SN remnants. A no-
table (and debated) case is that of Cas A, the remnant from
an historical SN which shows infrared emission from the re-
gion between the forward and reverse shocks. The identity of
Cas A’s progenitor is still highly debated. A star of 15-25M⊙
that loses its hydrogen envelope through winds (Chevalier
2006) or binary interactions (Young et al. 2006) and then
undergoes an energetic explosion can match all the avail-
able observational constraints. In particular, the age and
dynamics suggest a mass for the ejecta of 3 M⊙, with about
the same amount of gas reached by the reverse shock in the
ejecta and swept by the forward shock in the surrounding
ISM (Truelove & McKee 1999). Given these uncertainties,
and the dependence of the predicted dust masses on the
stellar progenitor (see Fig. 5), we can only give a tentative
estimate of the amount of dust predicted for Cas A by our
model. An ejecta evolution compatible with observations can
be obtained for a 12 M⊙ progenitor, provided we neglect the
hydrogen mass. In such a model, ≈0.1 M⊙ of dust forms.
By the age of the remnant (∼325 yr), ≈0.05 M⊙ survives in
the region reached by the reverse shock, where it is heated
by the hot gas. We also need to consider the contribution
to emission from dust in the ISM reached by the forward
shock. Typically, dust in the shocked ISM is exposed to a
gas of similar density and temperature to those in the reverse
shock (van der Swaluw et al. 2001). For a standard value of
the ISM gas-to-dust mass ratio, one would roughly expect a
similar mass of emitting dust in the ISM. Thus, a model for
Cas A remnant would have about 0.1 M⊙ of emitting dust.
This mass appears to be more than an order of magni-
tude larger than what could be derived fitting the observed
Spectral Energy distribution (SED) of Cas A (Hines et al.
2004). Using the emissivity predicted for SN dust, the flux in
the wavelength range 10 6 λ/µm 6 100 can be reasonably
well reproduced with a single modified blackbody with tem-
1 For the same reason, increasing the minimum cluster size N
and/or decreasing the sticking coefficient α does not affect the
predicted extinction laws and emissivities, which are similar to
those found for our reference model after the passage of the re-
verse shock.
Figure 7. Synchrotron-subtracted SED of dust emission in CasA.
Data points are from Hines et al. (2004). The solid line is a one-
component modified blackbody fit to the data for λ 6 100µm
(T ≈ 100K, Md ≈ 4.0 × 10
−3M⊙). The dashed line is a two-
component fit with T ≈ 110K, Md ≈ 3.0×10
−3M⊙ and T ≈ 35K,
Md ≈ 0.1M⊙. The dotted line is the spectrum from stochastically
heated dust in our model. See text for details.
perature T = 100K, and a dust mass of 4×10−3 M⊙ (Fig. 7,
solid line). Hines et al. (2004) obtain for the cold, more mas-
sive component a similar dust mass with T = 80K. However,
the large uncertainties and the limited FIR coverage allow to
fit, equally well, a two-component model with temperatures
110 and 35K and masses, respectively, of 3 × 10−3 and 0.1
M⊙ (Fig. 7, dashed line). Unfortunately Cas A lies on the
line of sight of dense molecular clouds which do not allow
a reliable estimate of the cold dust mass from observations
at longer wavelengths in the FIR and sub-mm. Still, upper
limits on the dust mass in the remnant are compatible with
our model predictions (Krause et al. 2004).
A broad span of temperatures is clearly needed for
a reliable estimate of the dust mass in the remnant. In
Fig. 7 we also show the SED of the shock heated dust in
the CasA model (dotted line). Because of stochastic heat-
ing (Appendix A), grains have temperatures mainly ranging
from 10 to 100K. The SED cannot be easily modelled using
a 2-component modified blackbody: the longer wavelength
side could be described with a cold component of T ≈ 60K,
which would underestimate the dust mass by about a factor
5; instead, a hot component at T ≈ 150K would leave a sub-
stantial residual in the fit at λ . 10µm. When comparing
to the data for CasA, it appears that dust temperature in
our models is overestimated. This could be due to an over-
estimate of the dust stochastic heating, to a reduction of
smaller grains with respect to the dust formation model, or
to differences between the emission properties of true and
modelled materials. However, the uncertainties in the ther-
mal/dynamical history of the ejecta of CasA and the im-
possibility of discriminating between ISM and ejecta dust
emission in the spectrum prevent a more detailed analysis.
c© 2007 RAS, MNRAS 000, 1–11
8 S. Bianchi & R. Schneider
5 SUMMARY
In the present work we have revisited the model of Todini
& Ferrara (2001) for dust formation in the ejecta of core
collapse SNe and followed the evolution of newly condensed
grains from the time of formation to their survival through
the passage of the reverse shock.
The main results can be summarized as follows:
(i) The new features introduced in the dust formation
model have only a minor impact on AC grains but signif-
icantly affect other species (Si-bearing grains, Al2O3, and
Fe3O4). For 12 - 40 M⊙ stellar progenitors with Z = Z⊙, the
predicted Mdust ranges between 0.1 - 0.6 M⊙; comparable
values (within a factor 2) are found if the progenitors have
Z < Z⊙. The dominant grain species are AC and Fe2O3.
(ii) We identify the most critical parameters to be the
minimum number of monomers, N , which define a critical
seed cluster, and the value of the sticking coefficient, α. As-
suming N > 10 (below which the application of standard
nucleation theory is questionable) results in a great reduc-
tion of non-AC grains because these species nucleate when
the gas in the ejecta is highly super-saturated and smaller
seed clusters form. This effect is further enhanced if α < 1:
for α = 0.1 and stellar progenitor masses Mstar < 20M⊙, the
total mass of dust is reduced to values in the range 0.001-
0.1 M⊙, comparable to those inferred from the IR emission
at 400-700 days after the explosion for 1987A and 2003gd,
the only two core-collapse SNe for which these data were
available.
(iii) Using a semi-analytical model to describe the dy-
namics of the reverse shock, we have found that thermal
and non-thermal sputtering produce a shift of the size dis-
tribution function towards smaller grains; the resulting dust
mass reduction depends on the density of the surrounding
ISM: for ρISM = 10
−25, 10−24, 10−23 g cm−3, about 20%,
7%, and 2% (respectively) of the initial dust mass survives.
Most of dust consumption occurs within one characteristic
time from the explosion, about 4−8×104 yr for core-collapse
SNe. Thus, the impact of the reverse shock needs to be taken
into account when comparing model predictions with obser-
vations of young SN remnants.
(iv) Averaging over a Salpeter IMF, we have derived dust
extinction and emissivity. We find that the extinction curve
is dominated by AC and Fe3O4 grains with radii larger than
20 Å. As a result, it is relatively flat in the range 1500-2500Å
and then rises in the far UV. Thus, the peculiar behaviour
of the extinction produced by SN dust, which has been suc-
cessfully used to interpret observations of a reddened QSO
at z = 6.2 (Maiolino et al. 2004), is preserved in the present
model, and it is further amplified by the modifications in-
duced by the passage of the reverse shock.
(v) Using dust emissivity predicted by the model, we can
reproduce the observed IR flux from the young SN rem-
nant CasA adopting a single modified black-body of tem-
perature T = 100 K, which implies a mass of warm dust
of 4 × 10−3M⊙, consistent with Hines et al. (2004). How-
ever, the limited observational coverage in the FIR allows
to equally well reproduce the data adding a cold component
with temperature T = 35 K and dust mass of 0.1M⊙. Ac-
cording to our model, such a mass of dust is what would
be produced by a single 12 M⊙ star that has exploded after
losing its hydrogen envelope, a plausible candidate for the
highly debated CasA’s progenitor. Because of the stochastic
heating of small grains by collisions with hot gas electrons,
dust in the shocked gas is predicted to have temperatures
ranging from 10 to 100K.
We conclude that our study supports the idea that
core-collapse SNe can be major dust factories. At the same
time, it shows that our knowledge of dust condensation and
its survival in SN ejecta still lacks to control some critical
parameters, which prevent reliable estimates of condensa-
tion efficiencies, especially for the less massive progenitors.
Within these uncertainties, the model can accomodate the
still sparse observational probes of the presence of dust in
SN and SN remnants.
ACKNOWLEDGMENTS
We are grateful to A. Ferrara for profitable discussions and
suggestions, and to L. Del Zanna for kindly providing us the
results of 1-D hydrodynamical simulations. We also acknowl-
edge DAVID members2 for fruitful comments and Cristiano
Porciani for precious help.
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APPENDIX A: STOCHASTIC HEATING FROM
ELECTRON COLLISIONS
We have derived the temperature distribution P (Td) follow-
ing the method of Guhathakurta & Draine (1989), to which
we refer for a more detailed description.
We have divided the range of possible dust tempera-
tures into Nb bins. For the case studied here, we found suf-
ficient to define Nb = 500 bins, logarithmically spaced in
the range 2 < Td/K< 2000. The i-bin has temperature Td,i,
energy Ei and energy width ∆Ei. The energy corresponding
to each value of Td is defined as
E(Td) =
C(T )dT,
where C is the specific heat, which is derived by fitting ex-
perimental data. We have adopted a piecewise power-law of
the form
C(T ) = A T
The fitted values for A and B are given in Table A1.
c© 2007 RAS, MNRAS 000, 1–11
10 S. Bianchi & R. Schneider
Table A1. Thermal and optical properties of SNe dust materials
Materials Specific heat Refractive index
A (erg cm3 K−1) B range in T (K) Refs. Refs.
Al2O3 4.44 3 (0, 110] Ditmars et al. (1982) Koike et al. (1995, ISAS)
1.22 102 2.29 (110, 200] Chase et al. (1985) Begemann et al. (1997, compact)
7.00 105 0.66 (200, 500]
1.45 107 0.17 (500, 2000]
5.44 107 0 (2000,∞)
Fe3O4 22.5 3 (0, 10] Chase et al. (1985) Mukai (1989)
7.65 3.47 (10, 30] Shepherd et al. (1985)
3.74 102 2.32 (30, 80] Koenitzer et al. (1989)
1.04 105 1.04 (80, 300]
3.95 107 0 (300,∞)
MgSiO3 9.22 3 (0, 20] Kelley (1943) Jäger et al. (2003)
2.01 3.51 (20, 50] Chase et al. (1985)
7.94 102 1.98 (50, 130] Krupka et al. (1985)
1.32 105 0.93 (130, 400]
3.47 107 0 (400,∞)
Mg2SiO4 22.7 3 (0, 60] Kelley (1943) Jäger et al. (2003)
6.45 102 2.18 (60, 120] Chase et al. (1985)
1.40 105 1.06 (120, 300]
1.42 107 0.25 (300, 2000]
9.44 107 0 (2000,∞)
AC 3.82 102 2 (0, 70] Chase et al. (1985) Zubko et al. (1996, ACAR)
3.27 103 1.50 (70, 300] Draine & Li (2001)
5.58 104 1.00 (300, 700]
1.09 107 0.19 (700, 3000]
5.09 107 0 (3000,∞)
SiO2 9.95 10
2 2 (0, 60] Chase et al. (1985) Philipp (1985)
5.50 104 1.02 (60, 500] Leger et al. (1985) Henning & Mutschke (1997)
3.11 107 0 (500,∞)
The function P (Td) is computed from the transition
matrix between the initial and final internal energy states,
Af,i. The probability per unit time that a dust grain in the
energy state Ei is heated to the energy state Ef (with f > i)
by collisions with electrons is given by
Af,i =
n πa2∆Ef×
2(Ef − Ei)
f(Ef − Ei) + if Ef − Ei < E⋆
2Eeff
f(Eeff)
1− (Ef − Ei)/Eeff
1− (Ef − Ei)/Eeff
0 otherwise.
In the above equation, n andm are the electron number den-
sity and mass, respectively, and f(E) is the Maxwell distri-
bution function defined such that nf(E)∆E is the number
of electrons of energy E per unit volume. E⋆ is the maximum
energy that an electron can transfer to a dust grain (Dwek
1986). If Ef −Ei > E⋆ it is not possible to jump from stage
i to stage f through electron collisions; if Ef − Ei < E⋆
the jump is allowed for all electrons with energy Ef − Ei
(which will transfer their entire energy to the grain) and for
the electrons with energy Eeff (which will transfer an energy
Eeffζ(Eeff) = Ef − Ei; the function ζ(E) is the fraction of
the electron energy which is deposited on a dust grain when
E > E⋆, as defined in Dwek 1986).
As in Guhathakurta & Draine (1989) only cooling
terms from level f+1 to level f are considered:
Af,f+1 =
Ef+1 − Ef
Qabs(a, λ) Bλ(Td,f+1)dλ,
where Bλ is the Planck function and Qabs(a, λ) the absorp-
tion efficiency of a grain of radius a. Values for Qabs(a, λ)
were derived using the Mie’s (1908) theory for spherical dust
grains, adopting the refractive index from the references in
Table A1.
In Fig. A1 we show the temperature distributions for
AC and Fe3O4 dust grains of radii a=10, 50 and 200Å.
Grains are exposed to a gas with T = 108K and n = 10
electrons cm−3, typical conditions encountered in the ejecta
swept by the reverse shock. Smaller grains have broader tem-
perature distributions, with a high temperature tail.
Using the sublimation rate of Draine & Hao (2002), we
have derived a sublimation temperature Ts, defined as the
temperature necessary to completely consume a grain in
about 20 yr, the typical time step of our simulation. For the
grain sizes encountered in our work, we have Td & 1200K. At
c© 2007 RAS, MNRAS 000, 1–11
Dust Formation and Survival in SN ejecta 11
Figure A1. Temperature distributions P (Td) for AC and Fe3O4
dust grains in a hot gas with T = 108K and n = 10 electrons
cm−3. P (Td) is shown for radii a = 10 Å (broader distributions),
50 Å and 200 Å (distributions spanning a smaller range of Td).
any time in the ejecta evolution, only a negligible fraction of
all grains would exceed that temperature. As a result, dust
sublimation is insignificant in our models.
c© 2007 RAS, MNRAS 000, 1–11
	Introduction
	SN dust formation revisited
	Survival in the reverse shock
	Dynamics of the reverse shock
	Dust grain survival
	Extinction and emission from SN dust
	Summary
	Stochastic heating from electron collisions
