MNRAS 454, 1620–1633 (2015)

doi:10.1093/mnras/stv2095

Dust processing in elliptical galaxies
Hiroyuki Hirashita,1‹ Takaya Nozawa,2 Alexa Villaume3 and Sundar Srinivasan1
1 Institute of Astronomy and Astrophysics, Academia Sinica, PO Box 23-141, Taipei 10617, Taiwan
2 National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
3 Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA

Accepted 2015 September 7. Received 2015 September 3; in original form 2015 August 4

ABSTRACTWe reconsider the origin and processing of dust in elliptical galaxies. We theoretically formulate the evolution of grain size distribution, taking into account dust supply from asymptotic
giant branch (AGB) stars and dust destruction by sputtering in the hot interstellar medium(ISM), whose temperature evolution is treated by including two cooling paths: gas emission
and dust emission (i.e. gas cooling and dust cooling). With our new full treatment of grain
size distribution, we confirm that dust destruction by sputtering is too efficient to explain theobserved dust abundance even if AGB stars continue to supply dust grains, and that, except
for the case where the initial dust-to-gas ratio in the hot gas is as high as ∼0.01, dust coolingis negligible compared with gas cooling. However, we show that, contrary to previous expectations, cooling does not help to protect the dust; rather, the sputtering efficiency is raised by
the gas compression as a result of cooling. We additionally consider grain growth after the gas
cools down. Dust growth by the accretion of gas-phase metals in the cold medium increase
the dust-to-gas ratio up to ∼10−3 if this process lasts 10/(nH /103 cm−3 ) Myr, where nH is
the number density of hydrogen nuclei. We show that the accretion of gas-phase metals is a
viable mechanism of increasing the dust abundance in elliptical galaxies to a level consistent
with observations, and that the steepness of observed extinction curves is better explained withgrain growth by accretion.
Key words: methods: analytical – dust, extinction – galaxies: elliptical and lenticular,cD – galaxies: evolution – galaxies: ISM.

1 I N T RO D U C T I O N
In the nearby Universe, elliptical galaxies are known to have little
ongoing star formation activity. However, elliptical galaxies are notcompletely devoid of gas: they are also known to have hot X-rayemitting halo gas (e.g. O’Sullivan, Forbes & Ponman 2001) andcold gas (e.g. Wiklind, Combes & Henkel 1995). Moreover, dust
is detected for a large (>50 per cent) fraction of elliptical galaxies
by optical extinction (e.g. Goudfrooij et al. 1994a; van Dokkum &Franx 1995; Ferrari et al. 1999; Tran et al. 2001) or far-infrared(FIR) emission (e.g. Knapp et al. 1989). Dust mass is estimated tobe 104 –105 M from the reddening in the optical (Goudfrooij et al.
1994a). The dust mass estimated from FIR dust emission tends tobe even larger (∼105 –107 M ; e.g. Leeuw et al. 2004; Temi et al.
2004). The existence of dust is also important in determining thechemical properties of the cold gas through dust-surface reactions
and freeze-out of molecules on to the dust (Fabian, Johnstone &Daines 1994; Voit & Donahue 1995). In spite of such an important E-mail: hirashita@asiaa.sinica.edu.twrole of dust, the origin of dust in elliptical galaxies is still being
debated, and there is no clear theoretical explanation for the total
amount of dust there.
Because the stellar population is dominated by old stars whoseages are comparable to the cosmic age, the dust is predominantly
supplied by asymptotic giant branch (AGB) stars in elliptical galaxies. However, the supplied dust is quickly destroyed by sputteringin the X-ray-emitting hot gas. This destruction is so efficient thatthe observed dust mass cannot be explained by the dust abundanceachieved by the balance between the supply from AGB stars andthe destruction (e.g. Patil et al. 2007). Therefore, the existence ofsuch an ‘excessive’ amount of dust has long been a mystery.
Some authors argue that the dust existing in elliptical galaxies
is possibly injected from outside via the merging or accretion of
external galaxies (Forbes 1991; Temi et al. 2004; Fujita et al. 2013).
The lack of correlation between dust FIR luminosity and stellar luminosity is also taken as evidence of external origin of dust (Temi,
Brighenti & Mathews 2007); however, this argument implicitly assumes that the existing dust is tightly related to the stellar dust
production activity. If dust is processed by mechanisms not related
to stars, it may be natural that we do not find a correlation betweenC 2015 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society

Dust processing in elliptical galaxies
dust emission and stellar luminosity. In particular, dust may growthrough the accretion of gas-phase metals in the cold gas, and thisdust growth is suggested to be the most efficient mechanism of
dust mass increase in nearby galaxies (e.g. Dwek 1998; Hirashita
1999; Zhukovska, Gail & Trieloff 2008; Inoue 2011; Mattsson &Andersen 2012; Asano et al. 2013a; de Bennassuti et al. 2014;
Rémy-Ruyer et al. 2014). Indeed, Fabian et al. (1994) and Voit &Donahue (1995) pointed out a possibility of dust reformation in acooling flow. More recently, Martini, Dicken & Storchi-Bergmann(2013) also suggested the importance of accretion in maintaining
the dust abundance to a level consistent with observations by analogy with the Milky Way. Since the efficiency of dust growth is notnecessarily related to the stellar properties of elliptical galaxies,
the lack of correlation between dust and stellar emissions does notnecessarily mean the external origin of dust.
The interstellar dust is not only passively processed in the ISM,
but could also actively regulate the physical state of the ISM, sincedust is able to radiate away the energy obtained by collisions withgas particles. This is one of the most important cooling paths ofthe hot ISM, and referred to as dust cooling. Mathews & Brighenti(2003) argued that dust cooling could occur faster than the dynamical time in the galactic potential and thus possibly enable the dust
to survive against sputtering. However, they did not treat the dust
supply from AGB stars consistently, and they assumed a relatively
high initial dust-to-gas ratio (∼0.01, which is comparable to thedust-rich ISM seen in the Milky Way) in the hot gas. Since there
has been some advance in the theoretical understanding of dust production in AGB stars (Ferrarotti & Gail 2006; Ventura et al. 2012),
it is worth reconsidering their models using the new knowledge
about the dust yield. Moreover, since Mathews & Brighenti (2003)
assumed a single grain size for dust cooling, it would be interesting
to calculate the evolution of grain size distribution: indeed, the grain
size distribution is of fundamental importance in comparison withobserved extinction curves as we argue below. The rates of sputtering and dust cooling also depend on the grain size distribution(Dwek 1987; Nozawa, Kozasa & Habe 2006).
The evolution of dust is also linked to active galactic nucleus(AGN) activities in elliptical galaxies. Temi et al. (2007) proposed
the transportation of dust from dust reservoirs in the central regions
to explain the extended dust emission in elliptical galaxies (seealso Mathews et al. 2013). In this context, the origin, production,and survival of dust are important in clarifying the origin of thedust reservoirs. As mentioned above, dust may also contribute tocooling. The cool gas that falls into the centre may fuel the AGN(e.g. Werner et al. 2014), which in turn works as a heating source
and reforms the hot gas (Ciotti & Ostriker 2001). Recently, Valentini& Brighenti (2015) pointed out that the AGN feedback could alsoenhance cooling locally through compression.
In this paper, we focus on the processing of dust in ellipticalgalaxies, both in the hot and cold gas components. In order toavoid complexity arising from the AGN feedback, we do not treat
the supply of hot gas through the feedback but start the calculation given the pre-existing hot gas. Instead, we do treat coolingof the gas in a consistent manner with the dust supply and destruction in the hot gas. Since dust cooling and dust destruction
depend on the grain size distribution, we fully take the evolutionof grain size distribution into account. We additionally consider
the impact of dust growth in the cooled gas not only on the dust
amount, but also on the grain size distribution. Through the modelling of the above processes, we will be able to clarify the evolution and survival of dust in a cooling cycle of the ISM in ellipticalgalaxies.

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We emphasize that the treatment of grain size distribution is oneof the most important features in our modelling. Indeed, there is anobservational clue to the grain size distribution, which enables us
to test our results. The wavelength dependence of extinction, theso-called extinction curve, is investigated for a statistical number(10) of elliptical galaxies: Goudfrooij et al. (1994b) showed thatthe ratio of total to selective extinction RV , which is a measure of theflatness of optical extinction curve, tends to be smaller (RV = 2.1–
3.3) than the mean value of the Milky Way extinction curves (3.1;
e.g. Pei 1992; Weingartner & Draine 2001; Draine 2003; Fitzpatrick& Massa 2007; Nozawa & Fukugita 2013). This implies that the
typical grain size is smaller in the elliptical galaxies than in the
Milky Way. Patil et al. (2007) obtained a similar range (RV = 2.03–
3.46) for 11 elliptical galaxies. These extinction curves can be used
to examine the evolution of grain size distributions through various
processes in the ISM.
The paper is organized as follows: we explain our models forthe basic processes of dust evolution in the hot ISM in Section 2.
We show the results in Section 3. We additionally consider dust
processing in the cold ISM, focusing on dust growth, in Section 4.
In Section 5, some predicted features of the models such as dust
abundance and extinction curves are compared with observations.
Finally, we conclude in Section 6.
2 P RO C E S S I N G I N T H E H OT I S M
We explain the method of calculating the evolution of dust grain
size distribution in the hot ISM in an elliptical galaxy. Our models
are composed of dust supply by AGB stars and dust destruction
by sputtering. Cooling of the hot gas by dust thermal emission andgas line emission is also calculated in a consistent manner with theevolution of dust grain size distribution. To avoid the complexity
arising from dynamical evolution determined by the gravitational
field and the AGN heating, we only consider the evolution of a local‘fluid element’ of the ISM without modelling the global dynamics.
The simplicity of this approach enables us to focus on dust processing which is determined by the local environment where the dust
resides.
2.1 Evolution of grain size distribution
The grain size distribution at time t is defined so that n(a, t) da is thenumber density of grains in the range of grain radii between a anda + da. The time t is measured from the onset of dust processingin the hot gas (t is not the age of the galaxy). As mentioned above,
we focus on a local region without considering the global gas anddust distributions within the galaxy. We also assume that the gas
and dust are well coupled (i.e. once dust is injected into a certain
gas element, that dust stays in the same element without any displacement; in other words, the Lagrangian trajectories of gas anddust are the same). Under these assumptions, the evolution of grain
size distribution in the hot gas is described as∂
d ln ρgas∂n(a, t)+[ȧn(a, t)] = S(a, t) + n(a, t)
,
(1)
∂t
∂a
dt
where ȧ = da/dt is the changing rate of grain radius by sputtering(ȧ is negative; see Section 2.3), S(a, t) is the source term due tothe dust production by AGB stars (Section 2.2), and ρ gas is the gas
mass density. The last term in equation (1) is the increase of thegrain number density by the change of the background gas density(Appendix A): as formulated in Section 2.4, we consider the changeof gas density by cooling. We give the number density of hydrogen
MNRAS 454, 1620–1633 (2015)

1622

H. Hirashita et al.

nuclei, nH , as an input parameter, and the relation between nH andρ gas is given byρgas = μmH nH ,(2)

where mH is the mass of hydrogen atom, and μ is the mean weightof the gas per hydrogen (we adopt μ = 1.4).
Since we are often interested in the grain mass, we introduce thegrain size distribution per unit grain mass by defining ñ(m, t) dm
as the number density of grains with mass between m and m +dm; i.e. ñ(m, t) dm = n(a, t) da. The grain mass is estimated as
m = 43 πa 3 s (i.e. grains are assumed to be spherical), where constants is the material density of the grain. For the calculation of grain size
distribution, we adopt s = 3.3 g cm−3 based on a silicate material.
Using the fact that ȧ is independent of a (Section 2.3), we obtain(see also Hirashita 2012)∂(mñ)
1d ln ρgas∂(mñ)+ ṁ= ṁñ + mS̃(m, t) + mñ
,
∂t
∂m
3
dt

(3)

where ṁ ≡ dm/dt = 4πa 2 s ȧ and S̃(m, t) dm = S(a, t) da.
The dust-to-gas ratio as a function of time, D(t), is calculated
by
 ∞
4 3
1
πa sn(a, t) da
D(t) =
μmH nH 0 3 ∞1
mñ(m, t) dm.
(4)=μmH nH 0
2.2 Supply from stars
2.2.1 Formulation
We consider the stellar population contributing to the dust supply inthe local gas element considered above. Since we are interested in asystem as old as the cosmic age, the current stellar population contributing to the dust formation is AGB stars, which originate fromstars with m < 8 M (m is the mass at the zero-age main sequence;e.g. Zhukovska et al. 2008). The dust mass density supplied from
AGB stars per unit time, ρ̇dust,AGB , is estimated as mupρ̇dust,AGB =md (m)φ(m)ρ̇∗ (tage − τm ) dm,
(5)
mtage

where mtage is the turn-off mass (i.e. the mass of the star whose
lifetime is equal to the galaxy age tage ), mup is the upper cut-off ofstellar mass, md (m) is the total dust mass produced by an AGB starwith mass m, φ(m) is the stellar initial mass function (IMF), ρ̇∗ (tage )is the star formation rate density as a function of galaxy age, and τ mis the lifetime of the star with mass m. We take the stellar lifetime
from Raiteri, Villata & Navarro (1996). We adopt a Salpeter IMF[φ(m) ∝ m−2.35 ] with mup = 100 M and mlow = 0.1 M , wheremlow is the lower cut-offof stellar mass, and the normalization of themup
mφ(m)dm = 1. The data of dust yield
IMF is determined by mlow
md (m) is described in Section 2.2.2. We assume that dust grainsejected from AGB stars are instantaneously (on a time-scale shorter
than 105 yr; Mathews & Brighenti 1999) injected into the hot gas(Temi et al. 2007). We approximate the star formation history withan instantaneous burst at tage = 0, which is a good approximation
for nearby elliptical galaxies:ρ̇∗ (tage ) = ρ∗,tot δ(tage ),(6)

where ρ ∗,tot is the total stellar mass density ever formed and δ(t)is Dirac’s δ function. Noting that only stars with mass mtage arecontributing to the current dust formation, we obtain from equations
MNRAS 454, 1620–1633 (2015)(5) and (6) dmt ρ̇dust,AGB = md (mtage )φ(mtage )ρ∗,tot .
dt t=tage(7)

Taking the mass loss of stars that have ended their lives into account,the current stellar mass, ρ ∗ , is estimated asρ∗ = (1 − R)ρ∗,tot ,(8)

where R is the returned fraction. We assume that R = 0.25 (Kuo,
Hirashita & Zafar 2013, this value is derived for mtage 1 M , but
is not sensitive to tage as long as tage > 109 yr; Hirashita & Kuo2011). Correspondingly, we adopt tage = 9.5 Gyr, the lifetime ofm = 1 M star. We fix tage , since we are interested in the time
evolution at t  108 yr, which is much shorter than tage .
Now, we formulate the source term in equation (1). This term
expresses the dust input from AGB stars, and we hereafter refer tothe dust supplied by AGB stars as AGB dust. We specify the grain
size distribution of AGB dust in such a way that the source term inequation (1) is written as
S(a, t) = ρ̇dust fdust (a),(9)

where fdust (a) is the grain size distribution function whose normalization is determined by ∞4 3
πa sf (a) da = 1.
(10)30Or equivalently, it is possible to use equation (3) with S̃(m, t) =of mass distribution functionρ̇dust f̃dust (m), where the normalization
∞f̃dust (m) is determined by 0 mf̃dust (m) dm = 1. For the grain size
distribution, we consider two cases: one is a power-law form withthe same power index as used in Tsai & Mathews (1995),
fdust (a) = CM a −3.5 (0.001 μm ≤ a ≤ 0.25 μm),
and the other is a lognormal form,Cl[ln(a/a0 )]2
fdust (a) =
,
exp −
a
2σ 2

(11)

(12)

where CM and Cl are the normalizing constants determined byequation (10), and a0 and σ are the central grain radius and thestandard deviation of the lognormal distribution, respectively. The
first one (equation 11) is based on the one derived for the Milky
Way ISM (Mathis, Rumpl & Nordsieck 1977), and is referred toas the MRN grain size distribution. The second one in equation(12) is based on the argument that AGB stars produce large (0.1–1 μm) grains (Asano et al. 2013b, and references therein). We
adopt the same parameters as in Asano et al. (2013b): a0 = 0.1 μm
and σ = 0.47, which are based on theoretical calculations of dust
condensation in AGB star winds by Yasuda & Kozasa (2012).
2.2.2 Dust yield of AGB stars
We take md (m) (the ejected dust mass by an AGB star for different progenitor mass) from theoretical calculations by Zhukovska
et al. (2008), who adopted Ferrarotti & Gail (2006)’s framework to
calculate the dust condensation in AGB star winds. Since we areinterested in nearby elliptical galaxies, it is reasonable to consider
the solar metallicity case (Z = 0.02). The stellar mass around 1 M ,which is contributing to the current dust production, is relevant tothis paper. Their calculations indicate that around 4 × 10−4 –10−3
M of dust is produced by an m = 1–2 M star. If we adopt
Z = 0.015 (0.03) instead, the dust mass becomes three times asDust processing in elliptical galaxies
small (large) as the case above. Therefore, we should keep a factorof 3 uncertainty in mind. Ventura et al. (2014) also calculated theoretical dust yield for AGB stars, focusing on lower metallicities.
Their highest metallicity case (Z = 8 × 10−3 ) shows that the dust
mass produced by an AGB star with m = 1–2 M is 2 × 10−4 –10−3
M , similar to that adopted above.
2.3 Sputtering
The changing rate of grain radius by sputtering, ȧ, in equation (1) isestimated by a fitting formula derived by Tsai & Mathews (1995):ȧ = −h̃

ρgas
mH




1+

T̃
T

2.5 −1

Table 1. Model parameters adopted from Tsai & Mathews (1995).
Model
a
b
c

LB
(1010 L )

ρ∗(g cm−3 )ρ gas a(g cm−3 )T(107 K)

10.6
3.31
0.976

1.19 × 10−21
1.36 × 10−20
1.83 × 10−19

1.43 × 10−25
3.01 × 10−25
8.74 × 10−25

0.922
0.468
0.252

Notes. The densities at the core radius are adopted.
a The corresponding hydrogen number densities are n = 0.0610, 0.128, andH
0.373 cm−3 for Models a, b, and c, respectively.

simultaneously varied as,(13)

where the coefficients of this fitting function, h̃ = 3.2 ×
10−18 cm4 s−1 and T̃ = 2 × 106 K, are determined to give a good
fit to the sputtering rate of both silicate and graphite in Draine &Salpeter (1979). This formula correctly reproduces the temperaturedependence of sputtering in that it is efficient at T  2 × 106 K
and steeply declines at T  106 K (Nozawa et al. 2006). We also
define the sputtering time-scale, τsput ≡ |a/ȧ|, which is estimatedat T  T̃ using equation (2) as


−1
a
nH
a
7.1 × 105
yr.
(14)
τsput
−3
1 μm
1 cm
h̃μnH
This is consistent with the values derived by a detailed treatment of
sputtering yield (Nozawa et al. 2006, see their fig. 2).
2.4 Cooling of the hot ISM
Following Mathews & Brighenti (2003), we assume that the gaspressure in the local region of interest is constant. The time evolutionof the gas temperature T through isobaric cooling is described by(Mathews & Brighenti 2003)
2tot (T )
dT
=−
,
dt
5nk

1623

(15)

where tot (T) is the cooling rate (energy lost per volume per time)
including both gas and dust cooling, n is the number density of gas
particles, and k is the Boltzmann constant. Assuming that H and He
are fully ionized, we estimate that n = 2.3nH .
The cooling rate by gas emission is taken from Sutherland & Dopita (1993). Specifically, we adopt nt ne N (T) [N (T) is the cooling
function for gas cooling] for the cooling rate per volume with electron number density ne = 1.2nH and total number density of ionsnt = 1.1nH . We stop the calculation when T reaches 105 K (this timeis denoted as t5 ), below which the temperature evolution occurs
much faster and the effect of recombination would change ne significantly. Since sputtering does not work at such a low temperature,
the resulting grain abundance and size distribution are insensitive
to the detailed treatment of thermal evolution at  105 K.
The gas also cools through gas–grain collisions (e.g. Dalgarno& McCray 1972; Dwek 1987; Seok, Koo & Hirashita 2015). The
rate of this dust cooling is estimated as ne nH d (T), where d (T) is
the cooling function for dust cooling. We compute d (T) based on
Dwek (1987) under the dust abundance and grain size distribution
calculated in this paper. We again adopt ne = 1.2nH and assume silicate as dust species (adopting graphite instead does not significantly
change the cooling rate; Dwek 1987).
After all, we evaluate the total cooling rate as tot = nt ne N+ ne nH d . We assume a constant pressure, so that the density isd ln Td ln ρgas
=−
.
dt
dt

(16)

2.5 Models and ICs
For simplicity, we only focus on a ‘typical’ single region in an
elliptical galaxy by adopting representative gas density and temperature. Tsai & Mathews (1995) considered three models of ellipticalgalaxies which are classified by B-band luminosities, LB . The models are referred to as Models a, b, and c depending on the B-band
luminosity. Although we do not directly use B-band luminosity, thequantities used in the models (ρ ∗ , ρ gas , T) are related to it. In Table 1,
we list the adopted parameters for each model. The values at thecore radius are adopted for the star and gas densities. (We convert
the central densities given in table 1 of Tsai & Mathews 1995 to thevalues at the core radius by assuming the radial profiles given inthat paper.) The stellar mass density ρ ∗ is used to estimate the dust
supply rate from AGB stars (Section 2.2.1), and the gas density ρ gas
and gas temperature T are used for the initial conditions (ICs).
We start with no dust (D0 = 0, where D0 is the initial dust-togas ratio), referred to as the dust-free IC, or D0 = 0.0075, referred
to as the dusty IC. The latter value of D0 is typical of the solarmetallicity ISM and the same functional form for the grain size
distribution as equation (11) [i.e. n(a, t = 0) ∝ a−3.5 ] is assumed
with the normalization determined by equation (4). We expect thatthe realistic situation lies between these two extreme conditions
for the IC, but that the dust-free IC is more realistic considering thatthe dust is rapidly destroyed by sputtering as shown below. Thus,
we focus on the dust-free IC as a ‘fiducial’ model. As we explained
above, we investigate two cases for the grain size distribution of
AGB dust: MRN and lognormal.
3 R E S U LT S
3.1 Thermal evolution and dust-to-gas ratio
The thermal evolution of the hot gas is of fundamental importancein determining the fate of dust. On the other hand, the temperatureof hot gas could also be affected by dust cooling. For dust cooling,
the dust abundance is the key factor. Thus, we first show the time
evolutions of gas temperature (T) and dust-to-gas ratio (D) in Figs 1
and 2, respectively.
We compare the results with the different two ICs for the dust-togas ratio, D0 = 0 and 0.0075 (dust-free and dusty ICs, respectively).
We adopt the MRN grain size distribution for AGB dust. As shownin Fig. 1, the temperature drops more rapidly for the dusty IC thanfor the dust-free IC because of more efficient dust cooling. The timeat which the temperature drops down to 105 K, t5 , is by a factor of∼1.5 shorter for the dusty IC than for the dust-free IC. In Table 2,
MNRAS 454, 1620–1633 (2015)

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H. Hirashita et al.

Figure 1. Time evolution of gas temperature T. Note that the gas density
is always inversely proportional to T during the entire evolution. The upper
thicker (lower thinner) solid, dotted, and dashed lines present the results for
Models a, b, and c, respectively, with the dust-free IC, D0 = 0 (with the
dusty IC, D0 = 0.0075). The MRN grain size distribution is adopted for
AGB dust. The evolutionary tracks of the cases without cooling and withthe lognormal size distribution of AGB dust for D0 both overlap with the
thick lines (thus, they are not shown in the figure), which means that dust
cooling does not affect the thermal evolution of the hot gas for the dust-freeIC.

we list t5 for each galaxy model with different D0 and grain size
distributions of AGB dust.
In Fig. 2, we show the evolution of dust-to-gas ratio, D, calculated
by equation (4). First we explain the evolution of D for the dust-free
IC. At the beginning of the evolution, the dust-to-gas ratio increases
because of the dust supplied from the AGB stars. The increase of
D is saturated because of the destruction by sputtering. At thisstage, the dust-to-gas ratio approaches to the value determined by
the balance between dust supply and destruction as we show later.
Indeed, the sputtering time-scale given by equation (14) (note thata < 0.25 μm for the MRN grain size distribution) is much shorter
than t5 , which allows the grain size distribution to approach theequilibrium between supply and destruction. At the end, the dustto-gas ratio decreases because the compression of the gas due to thetemperature drop by cooling enhances the sputtering rate (but thetemperature is still higher than ∼2 × 106 K). Indeed, the decreaseof dust-to-gas ratio coincides with the temperature drop (Fig. 1). As
shown in equation (13), the sputtering efficiency scales as ∝ ρ gas at
any gas temperature, while it is not sensitive to the gas temperature
as long as T  T̃ = 2 × 106 K. Because the effect of density islarger than that of temperature, cooling eventually enhances the dust
destruction, contrary to the expectation by Mathews & Brighenti(2003).
In order to show the level of D achieved by the equilibrium between dust supply and destruction under the initial gas density, we
also show the cases where we neglect cooling (i.e. the gas temperature and density are constant) in Fig. 2. In these cases without
cooling, the dust-to-gas ratio is the same as the above case at the
beginning of evolution before the gas cools significantly, and it
approaches to the value determined by the balance between dust
supply and sputtering. The dust-to-gas ratio achieved by the equiMNRAS 454, 1620–1633 (2015)

Figure 2. Time evolution of dust-to-gas ratio D. (a) The MRN grain size
distribution is adopted for AGB dust. The solid, dotted, and dashed lines
present the results for Models a, b, and c, respectively, with the dust-free anddust-rich ICs (lines with enhanced and medium thickness, respectively) withcooling included, and with the dust-free IC without cooling (thin lines). In
the cases with cooling, the calculation is stopped when the gas temperaturereaches 105 K (i.e. at time t5 ). (b) Comparison between the two different
grain size distributions of AGB dust: lognormal (thick lines) and MRN (thinlines). The solid, dotted, and dashed lines present the results for Models a,b, and c, respectively.

librium is the highest in Model c, primarily because of the highest
stellar mass per gas mass (i.e. the highest dust supply rate per gasmass).
We also present the results for the dust-rich ICs (D0 = 0.0075) inFig. 2a. In Models a and b, the final dust-to-gas ratio at t5 is almost
the same regardless of D0 (note that t5 is different, though, between
the different ICs; Table 2). On the other hand, the effect of IC on
the final dust-to-gas ratio remains in Model c, because the evolution
starts with lower temperature so the effect of sputtering is smaller

Dust processing in elliptical galaxies

1625

Table 2. Time (t5 ) at which the temperature drops down to 105 K.
ICa

AGB dustb

Model at5 (yr)Model bt5 (yr)Model ct5 (yr)Dust-free
Dusty
Dust-free
Dust-free

MRN
MRN
lognormal
no dustc

4.9 × 107
3.3 × 107
4.9 × 107
5.0 × 107

8.3 × 106
5.3 × 106
8.3 × 106
8.6 × 106

7.3 × 105
4.5 × 105
7.4 × 105
7.4 × 105

Notes. a The IC is chosen from the dust-free and the dusty cases with theinitial dust-to-gas ratio D = 0 and 0.0075, respectively. For the dusty initialcondition, the grain size distribution is assumed be the MRN (i.e. power law
with an index of −3.5; see the text for details).
b The grain size distribution of AGB dust is chosen from MRN or lognormal.
c This case is for no dust cooling (i.e. only gas cooling).

than Models a and b. Yet, even in the dusty IC of Model c, thedust-to-gas ratio at t = t5 is much lower than the initial dust-to-gas
ratio (D0 = 0.0075).
Although the large fraction of the initial dust is destroyed in any
case, the IC has an imprint on the temperature evolution as shownin Fig. 1. Therefore, if the initial dust-to-gas ratio is as large as thevalue in the Milky Way ISM, dust cooling has a significant influenceon the thermal evolution of the gas. However, such a high dust-togas ratio is never achieved if the dust enrichment and destruction
take place simultaneously.
In order to show the effect of AGB dust, we also examine thecases with and without dust enrichment for the dust-free IC, findingthat dust cooling by AGB dust has little influence on the thermal
evolution of the hot ISM. Indeed, the evolutionary track is indistinguishable between the cases with and without dust enrichment(thus, the case without dust enrichment is not shown in Fig. 1). Thisis also clear from Table 2 (compare the first and last rows): t5 is only
slightly longer with no dust cooling than with dust cooling. We also
confirmed that change of the grain size distribution of AGB dust to
the lognormal size distribution does not alter the thermal history.
Indeed as shown in Table 2, t5 is almost identical between the MRN
and lognormal AGB dust.
In Fig. 2b, we examine the effects of the grain size distributionof AGB dust on the evolution of D. Since the lognormal grain size
distribution is more biased to large sizes than the MRN grain size
distribution, it has a longer sputtering time-scale τ sput (equation 14).
The final dust-to-gas ratio at t ∼ t5 in the lognormal case is aboutfour times as large as in the MRN case. Nevertheless, since the dust
abundance is still kept much lower than 0.01 by dust destruction,
the temperature evolution is not affected by dust cooling even in thelognormal case as mentioned above.

3.2 Grain size distribution
Now we examine the evolution of grain size distribution in detail. In
Fig. 3, we show the evolution of grain size distribution for the dustfree IC. The MRN grain size distribution is adopted for AGB dust.
In the initial stage at t t5 , when the change of gas density due tocooling is negligible, the grain abundance gradually increases, converging to the equilibrium grain size distribution determined by thebalance between the supply from AGB stars and the destruction by
sputtering, and the slope is described by n(a) ∝ a−2.5 (Appendix B).
We have already seen above that the dust-to-gas ratio approaches tothe equilibrium value. As is also observed above in Fig. 2, the grainabundance decreases at the final stage of evolution around t ∼ t5 because the density increase by cooling enhances the sputtering rate.

Figure 3. Grain size distribution per hydrogen atom for different times. The
dust-free IC is adopted. Grain size distributions are multiplied by a4 to show
the grain mass distribution per logarithmic grain radius. Panels (a), (b), and(c) show the results for Models a, b, and c, respectively. The correspondenceof the line species to the time is shown in each panel. The dotted line marked
as ‘MRN’ is the MRN grain size distribution with a dust-to-gas ratio of
0.0075, shown as a reference for the grain size distribution appropriate for
the Milky Way ISM. Note that the final time (t5 ) in each model is determinedby the time when the gas temperature reaches T = 105 K. The dot–dashedline shows the equilibrium grain size distribution without dust cooling.

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H. Hirashita et al.

Figure 4. Grain size distributions at t = t5 (after cooling) per hydrogen
atom (multiplied by a4 ). Solid and dashed lines show the results for the
dust-free (D0 = 0) and dust-rich (D0 = 0.0075) ICs (IC), respectively. The
dotted line marked as ‘MRN’ is the MRN grain size distribution with a dustto-gas ratio of 0.0075, shown as a reference for the grain size distribution
appropriate for the Milky Way ISM.

This is the reason why the grain size distributions shift downwards
at the final stage in all the three models.
In Fig. 4, we compare the grain size distributions at t = t5 forthe three models with the dust-free and dust-rich ICs (the MRN
grain size distribution is adopted for AGB dust). As is consistent
with Fig. 2, the dust abundance per hydrogen atom (or the dustto-gas ratio) is the largest in Model c: as explained above, this isdue to the highest stellar mass per gas mass (Table 1), that is, thehighest dust supply rate per gas mass. Moreover, the comparison
between the cases of different D0 in Models a and b shows almost
no difference, which indicates that almost all the dust that existed
initially in the hot gas has been destroyed. In Model c, the effect of
IC remains as explained in Section 3.1 (Fig. 2): this is because of
the low temperature, i.e. low sputtering rate. The difference in the
IC affects at the largest grain sizes because the largest grains havethe longest sputtering time-scale (equation 14). However, even in
Model c, the grain abundance decreases far below the initial MRN
size distribution (shown by the dotted line in Fig. 4). Therefore, inall cases, the final grain size distribution deviates significantly fromthe initial grain size distribution.
We also examine the case of the lognormal size distribution for
AGB dust (equation 12). We adopt the dust-free IC. The resulting
grain size distributions are shown in Fig. 5, and their evolutionary
behaviour is similar to the case of the MRN grain size distribution
in the following aspects: the grain abundance increases before significant cooling occurs (i.e. at t t5 ), converging to the equilibrium
grain size distribution (Appendix B), while it decreases after cooling (i.e. around t ∼ t5 ). Compared with the lognormal function, thetail towards small grain sizes becomes prominent in the final grain
size distributions at t = t5 . This tail is due to the continuous destruction of large grains by sputtering and described by n(a) = constant(Appendix B). The grain abundance is reduced around t = t5 because the increase of gas density by cooling enhances the sputtering
rate, as explained above.
MNRAS 454, 1620–1633 (2015)

Figure 5. Same as Fig. 3 but for the lognormal size distribution of AGB dust.
Panels (a), (b), and (c) show the results for Models a, b, and c, respectively.
The correspondence of the line species to the time is shown in each panel.
Note that the final time (t5 ) in each model is determined by the time when
the gas temperature reaches T = 105 K. The dot–dashed line shows theequilibrium grain size distribution without dust cooling.

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1627

4 P RO C E S S I N G A F T E R C O O L I N G
In the above, we have traced the dust evolution in the hot gas up to
the point where the gas cools down to 105 K. After that, sputtering
does not act as a dust processing mechanism any more, but the
dust may still be processed by other mechanisms than sputtering.
In particular, many authors have argued the importance of dust
growth by the accretion of gas-phase metals for the total dust budget
in star-forming galaxies (e.g. Dwek 1998). Since this accretionmechanism works in cold and dense clouds, it is worth investigating
a possibility of dust growth also in the cold gas in elliptical galaxies.
As mentioned in the Introduction (and shown in Section 3), it has
been argued that the observed dust abundance in elliptical galaxies
cannot be explained by the internal dust supply from AGB stars
because of rapid destruction by sputtering. Thus, we examine dust
growth as a potential additional source of dust in elliptical galaxies(see also Fabian et al. 1994; Voit & Donahue 1995; Martini, Dicken& Storchi-Bergmann 2013).
Under a high pressure in the inner part of elliptical galaxy, a cold
phase with T ∼ 102 K is preferred rather than a warm phase withT ∼ 104 K (Wolfire et al. 1995). Such a cold dense medium is indeed
suitable for the site of dust growth. Since accretion and coagulation(grain–grain sticking) both occur in such a cold dense medium,
we treat them simultaneously (Hirashita 2012); however, as shown
later, coagulation only plays a minor role. Below we calculate thedust evolution driven by accretion and coagulation in the cold gas,
starting from the grain size distributions achieved after cooling of
the hot gas.

4.1 Formulation of accretion and coagulation
We calculate the evolution of grain size distribution by accretionand coagulation using the formulation in Hirashita (2012), which
has a wide application to, for example, polarization predictions(Voshchinnikov & Hirashita 2014). Again, the silicate dust properties are assumed, but adopting carbonaceous dust instead does
not change the evolution of grain size distribution significantly(Hirashita 2012). If we consider the pressure equilibrium with the
hot gas, nH > 103 cm−3 is achieved for all models at a temperatureappropriate for the cold ISM (T ∼ 102 K). Thus, we conservatively
assume that nH = 103 cm−3 , but the results for other densities can beeasily obtained by noting that the time-scale of grain processing is
simply proportional to n−1
H . The gas temperature, which determines
the thermal speed of the accreting material is assume to be 102 K.
The available gas-phase metals (in our case, silicon) is estimatedbased on the solar abundance of Si, (Si/H) = 3.55 × 10−5 (Däppen2000). We also assume that the mass fraction of Si in silicate is
0.166. The grain velocity, used to estimate the grain–grain collision
rate for coagulation, is treated as a function of grain radius, andis taken from the molecular cloud case in Yan, Lazarian & Draine(2004), which has a similar gas density to the one considered here.
We also impose the same coagulation threshold velocity as adopted
in Hirashita & Yan (2009, originally based on Chokshi, Tielens &Hollenbach 1993; Dominik & Tielens 1997): only if the relative
velocity is below this threshold, which varies with the sizes of colliding grains, the grains coagulate. However, detailed treatment of
coagulation does not affect the results, because, as shown later, theabundant gas-phase metals make the role of accretion stronger thanthat of coagulation in changing the grain size distribution. For the
IC of each model, we use the grain size distribution achieved in
the hot gas after cooling. More specifically, the initial grain size
distribution is produced by multiplying the grain size distributions

Figure 6. Time evolution of dust-to-gas ratio D by dust growth. The time
t is the elapsed time from the onset of dust growth in cooled gas. The thicksolid, dotted, and dashed lines present the results for Models a, b, and c in
the case where we use the result of the MRN AGB dust calculation at t5 for
the IC (i.e. the final state of each model in Fig. 3 is used for the IC). The
thin lines with the same line species represent the results for the lognormal
AGB dust calculation at t5 (i.e. the final state of each model in Fig. 5 is used
for the IC).

per hydrogen at t = t5 , n(a, t5 )/nH , in Section 3 by nH = 103 cm−3 .
The time is newly measured from the onset of grain growth in thecold gas; to avoid the confusion, we use a different notation for thetime, t , measured from the onset of grain growth.
4.2 Results
First, we show in Fig. 6 the evolution of dust-to-gas ratio D estimated by equation (4) at t ≤ 108 yr (as shown later, the observed
extinction curves are well covered by the grain size distributions
at t ≤ 108 yr). Note that accretion changes D while coagulation
does not. If we start with the grain size distribution at t5 for the
MRN AGB dust, D begins to increase drastically from t ∼ afew ×
106 yr and reaches 10−3 in a few × 107 yr. Therefore, if the lifetimeof the cold gas is longer than 107 yr, there is a possibility of dust
‘reformation’ using the remnants of sputtered dust grains, which
explains the observed level of dust abundance as we discuss in thenext section. The increase of D saturates most quickly in Modelc simply because the initial dust abundance is the largest, which
also means that the abundance of available gas-phase metals is thelowest.
For the calculation adopting the lognormal AGB dust case for
the IC, as shown in Fig. 6, the increase of dust-to-gas ratio is not sodrastic as in the case of MRN AGB dust models because the grain
size distribution is biased to large sizes (the accretion time-scale
is proportional to the grain radius; Hirashita 2012). This indicatesthat the initial grain size distribution in the cold gas, which is the
final grain size distribution of the cooled hot gas, is of fundamental
importance in determining the efficiency of dust mass increase in
the cold gas.
Next, we investigate the evolution of grain size distribution byaccretion and coagulation. In Fig. 7, we show the evolution of grain
size distribution, adopting the MRN AGB dust case for the IC. We
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H. Hirashita et al.
observe that accretion continue to increase the dust abundance over∼108 yr, which is consistent with the evolution of D shown above.
Because accretion saturates most quickly in Model c as explained
above, the grain radii achieved by the growth are the smallest in
Model c.
The most important feature of grain growth by accretion is thatthe effect appears most significantly at the smallest grain sizes. As
explained in Hirashita (2012), accretion enhances the dust abundance at the smallest grain sizes, which have the shortest accretiontime-scale because of the largest grain surface-to-volume ratio. Inorder to isolate the effect of accretion, we also examined the grain
size distribution with only accretion (i.e. without coagulation). We
confirmed that there is almost no difference between the cases withand without coagulation (we do not plot the results because thegrain size distributions with and without coagulation are almost
indistinguishable). This is partly because the effect of accretion isenhanced by the large amount of available gas-phase metals, partly
because coagulation does not proceed beyond a ∼ 0.1 μm at which
grains have larger velocities than the coagulation threshold (thismeans that the creation of larger grains than ∼0.1 μm only occurs
by accretion).
In Fig. 8, we show the same plot as above but we adopt the lognormal AGB dust case (i.e. the final grain size distribution in eachpanel of Fig. 5) for the IC. We observe that the change of the grain
size distribution occurs more slowly than the MRN AGB dust casebecause the grains are biased to larger sizes (recall that the accretion time-scale is proportional to the grain radius). Therefore, grain
growth is very sensitive to the size distribution of the grains incorporated in the cold medium. However, as shown later, extinction
curves calculated for the lognormal AGB dust cases are too flat andare inconsistent with the observed extinction curves (Section 5.2).
5 O B S E RVAT I O N A L P RO P E RT I E S A N D
DISCUSSION
5.1 Dust abundance

Figure 7. Evolution of grain size distribution by grain growth, starting fromthe final grain size distributions in Fig. 3. Panels (a), (b), and (c) show theresults for Models a, b, and c, respectively. The solid, dotted, dashed, dot–dashed, triple-dot–dashed lines present the grain size distributions at t = 0,
106 , 107 , 3 × 107 , and 108 yr, respectively. We assume nH = 103 cm−3 , but
the time-scale just scales as ∝ n−1
H .

MNRAS 454, 1620–1633 (2015)

For comparison of dust abundance, we adopt elliptical galaxy samples observed by Herschel from Smith et al. (2012) and di Serego
Alighieri et al. (2013). The dust-to-gas ratio is estimated for eachgalaxy based on the dust mass derived from the Herschel FIR emission and the gas mass estimated by the sum of H I and H2 masses(the latter is from CO luminosity). The sample galaxies adopted arelisted in Table 3. For the sample in Smith et al. (2012), we selectfour galaxies with (i) a morphological type of elliptical galaxies(E); (ii) detection of FIR emission (i.e. a successful estimate of dustmass); and (iii) detection of H I mass. We use the upper limits forH2 mass, but the treatment of H2 mass does not affect the results
since these upper limits of H2 mass are significantly smaller thanthe H I mass. For the sample in di Serego Alighieri et al. (2013), we
exclude galaxies with morphological type of S0 and Sa and only
select the objects for which both H I and H2 masses are constrained(i.e. detected or with an upper limit obtained). As a consequence,
we adopt four objects from their sample. NGC 4374 is common toboth samples. The B-band and X-ray luminosities are taken from
O’Sullivan et al. (2001).
As seen in Table 3, the observed dust-to-gas ratio is larger than
10−4 . However, as shown in Fig. 2, this level of dust-to-gas ratio
cannot be achieved in the hot ISM, especially in Models a andb, which are appropriate for most of the B-band luminosities ofthe sample. The situation does not change even if we change thegrain size distribution of AGB dust to the lognormal distribution

Dust processing in elliptical galaxies

1629(Fig. 2b). This confirms previous conclusions derived by various
authors that the dust destruction in the hot gas phase prevents thedust-to-gas ratio (or the total dust mass) from reaching the observed
level (Goudfrooij & de Jong 1995; Patil et al. 2007), and showsthat this is also true even if we consider the grain size distribution.
We have also shown that cooling does not help the dust to survive, contrary to Mathews & Brighenti (2003)’s expectation: gas
compression caused by cooling enhances the sputtering efficiency,
leading to an additional decrease of the dust abundance. In otherwords, the effect of cooling further emphasizes the discrepancy
between the expected and observed dust abundances in ellipticalgalaxies.
Some authors argue an external origin of the gas and dust basedon the evidence that the kinematic properties of the gas are decoupled from those of the major stellar component in some elliptical galaxies (Forbes 1991; Caon, Macchetto & Pastoriza 2000).
Since not all the galaxies fit this external-origin scenario (Forbes
1991), it is still worth considering how much dust could be explained by the ‘internal-origin’ scenario in which the dust is produced by their own stellar populations and/or inherent dust growthmechanisms.
Regarding the internal origin of dust, we have shown that theaccretion of gas-phase metals in cold clouds increases the dust-togas ratio to a level of ∼10−3 in ∼107 –108 yr. In this sense, coolingis important in the dust abundance, not through survival in the hotgas (as in Mathews & Brighenti 2003’s original idea) but throughthe formation of cold clouds hosting dust growth. The time-scaleof the dust abundance increase by dust growth is shorter than the
feedback time-scale (108 yr; Kaviraj et al. 2011; Pellegrini, Ciotti& Ostriker 2012), on which AGN activity heats the cooled gas.
Therefore, the ‘recovery’ of dust by accretion is a plausible scenario
of explaining the observed excessive dust abundance in ellipticalgalaxies.
5.2 Extinction curves
The grain size distributions obtained by our calculations are tested
against the observed extinction curves in Patil et al. (2007). Based on
the calculated grain size distributions above, we theoretically predictthe corresponding extinction curves. The extinction at wavelength λin units of magnitude (Aλ ) is calculated by weighting the extinction
efficiency factor Qext (a, λ) with the grain size distribution n(a) as ∞n(a)π a 2 Qext (a, λ)da,(17)
Aλ = C0

Figure 8. Evolution of grain size distribution by grain growth, starting fromthe final grain size distributions in Fig. 5. Panels (a), (b), and (c) show theresults for Models a, b, and c, respectively. The solid, dashed, dot–dashed,
triple-dot–dashed lines present the grain size distributions at t = 0, 107 , 3 ×
107 , and 108 yr, respectively. We assume nH = 103 cm−3 , but the time-scale
just scales as ∝ n−1
H .

where C is the normalizing constant, which cancels out when we
consider the ratio of Aλ at two wavelengths later. Although we assumed silicate for the grain property, the grain growth time-scaleof carbonaceous dust is similar (Hirashita 2012). The steepness of
extinction curve is often discussed in comparison with the Milky
Way extinction curve. Therefore, we simply assume a mass fractionof 0.54:0.46 for silicate and carbonaceous dust (graphite), which fits
the Milky Way extinction curve under the MRN grain size distribution (Hirashita & Yan 2009). We refer to Hirashita & Yan (2009)
for the adopted parameters for silicate and graphite. However, since
we are interested in wavelengths longer than 0.4 μm, the extinction
curve slope is broadly determined by the grain size distribution witha minor effect of the material.
Now we examine the extinction curves in the MRN AGB dustcase. In Fig. 9, we show the extinction curves for the grain size
distributions at t5 in Fig. 3 (i.e. the initial grain size distributions
before grain growth) and the variation of extinction curves by grain
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H. Hirashita et al.
Table 3. Elliptical galaxy sample adopted from Smith et al. (2012) and di Serego Alighieri et al. (2013).
Name

Other namelog LB(L )log LX(erg s−1 )log Mdust(M )log MHI(M )log MH2
(M )

log D

Ref.a

VCC 763
HRS 150
HRS 186
HRS 241
VCC 345
VCC 1226
VCC 1619

NGC 4374, M84
NGC 4406, M86
NGC 4494
NGC 4636
NGC 4261
NGC 4472, M49
NGC 4550

10.57
10.66
10.62
10.51
10.70
10.90
9.72

40.83
42.05
<40.10
41.59
41.21
41.43
39.78

5.05b
6.63
5.08
5.06
5.81
5.49
5.41

8.96
7.95
8.26
9.0
<8.45
<7.90
<7.90

<7.23
<7.4
<7.35
<7.02
<7.70
<7.26
7.20

− 3.91c− 1.43c− 3.23c− 3.94c> − 2.71d> − 2.50d> − 2.57d

1, 2, 3
1, 3
1, 3
1, 3
2, 3
2, 3
2, 3

Notes. a References: (1) Smith et al. (2012); (2) di Serego Alighieri et al. (2013); (3) O’Sullivan et al. (2001).
b Adopted from Ref. 1. Ref. 2 gives 5.30 with the same mass absorption coefficient.
c The upper limit of H mass is used, but this does not affect the estimated dust-to-gas ratio since the gas mass is2
dominated by H I gas mass.
d We use the upper limits of gas mass to derive the lower limits of dust-to-gas ratio.

growth (time is measured from the onset of grain growth). We show
the total to selective extinction Rλ ≡ Aλ /E(B − V) = Aλ /(AB −AV ) (the wavelengths at the B and V bands are 0.44 and 0.55 μm,
respectively), which is sensitive to the extinction curve slope aroundthese two optical bands. The observational extinction curves for asample of elliptical galaxies are taken from Patil et al. (2007): weonly chose those galaxies that are classified as elliptical galaxies
and have extinction data at all bands (B, V, R, and I).
We observe in Fig. 9 that the initial extinction curves before grain
growth tend to overproduce Rλ since they are too flat [i.e. the colour
excess E(B − V) is small] compared with the observed extinction
curves. They are also flatter than the Milky Way extinction curve,
which has RV 3.1 (e.g. Draine 2003), because of the deficiencyof small grains (note that the grain size distribution after sputteringin Fig. 3 is ∝ a−2.5 while the Milky Way extinction curve is fitted
by a grain size distribution ∝ a−3.5 ; MRN). After grain growth byaccretion takes place, the extinction curve becomes steeper (or Rλbecomes smaller), which is consistent with the previous conclusion that grain growth by accretion steepens the extinction curve(Hirashita 2012). This is because, as explained in Section 4.2, accretion is much more efficient for smaller grains owing to theirlarger surface-to-volume ratio. Since Rλ is sensitive to the slope at
B and V bands, the decrease of Rλ is significant if the abundanceof grains with a  λ/2π 0.07 μm (for λ = 0.44 μm) increases(Bohren & Huffman 1983). This occurs up to 3 × 107 yr in Model aand 108 yr in Models b and c (Fig. 7). At 108 yr in Model a, almost
all grains exceed 0.07 μm, leading to a significant increase of Rλ (orsignificant flattening of optical extinction curve), as seen in Fig. 9a.
It is not likely that we observe the extinction of single-aged
clouds, but it is probable that we see a mixture of clouds withdifferent ages at the same time. Considering this, it is interesting topoint out that the observed extinction curves are just in the range
explained by the grain growth. Sputtering in the hot gas tends tomake the extinction curve too flat; the observed extinction curves
favour subsequent grain processing that could form grains witha  0.07 μm. Grain growth by accretion enhances the abundanceof such ‘small’ grains.
The lognormal grain size distribution of AGB dust produces completely flat extinction curve at λ  1 μm, even if we consider accretion. Since E(B − V) is slightly negative (i.e. Rλ is negative and|Rλ | is extremely large), the results cannot be displayed. Therefore,
the lognormal model does not fit the observed extinction curve unless there is an extra mechanism of small grain production (seeSection 5.3).

MNRAS 454, 1620–1633 (2015)

5.3 Shattering?
Hirashita & Yan (2009) also showed that the dust grains may be
shattered in the warm (∼104 K) diffuse medium in the Galactic
environment (see also Yan et al. 2004). If some fraction of the gas
remains to be diffuse after cooling of the hot gas and turbulent
velocity is as high as 1 km s−1 , grains may be shattered. This
would enhance the abundance of small grains and would steepen theextinction curve. If the dust is processed by shattering before being
included in the cold dense medium, the total surface area of the dust
is enhanced, resulting in more efficient dust growth by accretion andfurther steepening of the extinction curve. Therefore, the presenceof shattering would enhance our conclusion that dust processing
after cooling is a key to understanding the dust abundance and thesteepness of extinction curves in elliptical galaxies. However, in ahigh-pressure environment in the central part of elliptical galaxies,
such a diffuse warm medium is not realized as an equilibrium state(Wolfire et al. 1995). Therefore, a dynamical model is necessary totreat this intermediate temperature range ∼104 K, which is left for
the future work.

5.4 Implication of the internal origin of dust
We have shown that accretion in the cold medium is a viable mechanism of solving the discrepancy between the expected dust abundance after sputtering and the observed one in elliptical galaxies.
Moreover, the steepening of extinction curves by accretion in thecooled medium is favoured in terms of the steepness of observed
extinction curves. Although we do not intend to argue that all thedust in elliptical galaxies are of internal origin, it is worth notingthat more careful consideration is necessary before resorting to theexternal origin of elliptical galaxy dust.
Even if accretion is the most dominant dust source, the dust supplied by AGB stars works as seeds for accretion. Therefore, thedust supply by AGB stars is still important in obtaining the thorough understanding of the life-cycle of dust in elliptical galaxies.
The present-day mass loss rate of AGB stars in elliptical galaxies
can be constrained by the mid-infrared emission emitted by AGB
dust (e.g. Athey et al. 2002). Villaume, Conroy & Johnson (2015)
demonstrated that circumstellar dust around AGB stars can accountfor the mid-infrared (8–24 μm) flux in some early-type galaxies.
Spectral synthesis models including mid-infrared emission of AGB
dust such as those presented in Villaume et al. (2015) can be used
to constrain the dust yield of a population of AGB stars in a galaxy

Dust processing in elliptical galaxies

1631

using observed dust emission. This is a crucial component for understanding whether an internal origin is a plausible explanation forthe observed dust in early-type galaxies.
Theoretical dust yields of AGB stars are also tested by observations of nearby galaxies. Because of their proximity, the Magellanic
Clouds provide us with an opportunity of detailed observational
constraint on the dust formation in individual AGB stars (Srinivasan et al. 2009; Boyer et al. 2011; Srinivasan, Sargent & Meixner
2011; Riebel et al. 2012; Kemper 2013). According to Zhukovska& Henning (2013) and Schneider et al. (2014), the current modelsof dust production in AGB stars used or discussed in this paper (Ferrarotti & Gail 2006; Zhukovska et al. 2008; Ventura et al. 2014) are
broadly consistent with the observed dust production rate by AGB
stars in the Large Magellanic Cloud. However, Schneider et al.
(2014) also mentioned that there is still a discrepancy between themodel and observation for the Small Magellanic Cloud and that themetallicity dependence of the AGB dust yield is large. Since dust
yields of solar-metallicity AGB stars have not been tested against
observational data, observational studies on the dust formation by
AGB stars in elliptical galaxies, which are, broadly speaking, solarmetallicity objects, give an important constraint on theoretical dust
production models of AGB stars. The understanding of AGB dust
yield is also crucial even for high-redshift galaxies, in which AGB
stars may contribute to the dust enrichment (Valiante et al. 2009;
Gall, Hjorth & Andersen 2011; Pipino et al. 2011).
6 CONCLUSION

Figure 9. Extinction curves normalized to E(B − V). Panels (a), (b), and(c) show the results for Models a, b, and c, respectively. The solid curveshows the extinction curves at the final stage of gas cooling in Fig. 3, whichcorresponds to the extinction curve at the onset of grain growth in the cold
medium. The dotted, dashed, dot–dashed, triple-dot–dashed lines present
the extinction curves at 106 , 107 , 3 × 107 , and 108 yr after the onset of grain
growth. The squares connected by the solid line are observational data for
an elliptical galaxy sample in Patil et al. (2007). The crosses show the Milky
Way extinction curve as a reference.

We have reconsidered the origin of dust in elliptical galaxies, focusing on the internal origin; that is, the dust originates from theproduction by AGB stars within the galaxy. The evolution of grain
size distribution is consistently solved along with the evolution ofgrain abundance by dust supply from AGB stars and dust destruction (sputtering) in the hot interstellar medium (ISM). We haveconfirmed that sputtering is so efficient that the dust abundance observed in the FIR cannot be explained. In addition, we have shownthat cooling does not help to protect the dust from sputtering; rather,gas compression induced by cooling raises the sputtering rate so thatthe dust-to-gas ratio is decreased by cooling. Because of the rapid
dust destruction, dust cooling has little influence on the thermal
history of the cooling hot gas, unless the hot gas initially has adust-to-gas ratio as high as ∼0.01 (i.e. comparable to the value in
the Galactic cold ISM).
In the latter part of this paper, we have considered grain growthby the accretion of gas-phase metals in the cold clouds formed asa result of cooling of the hot gas, in order to examine a possibility
that the dust abundance is increased or ‘recovered’ by this process.
We find that, if accretion lasts longer than 107 /(nH /103 cm−3 ) yr,
which is shorter than the heating time-scale of AGN feedback, thedust-to-gas ratio can increase up to ∼10−3 . Therefore, we do notnecessarily need to resort to the external origin for the observed
excessive dust abundance in elliptical galaxies, and we suggest dust
growth by accretion in the cooled gas as a plausible explanation forthe dust abundance. Since sputtering in the hot gas destroys small
grains more efficiently than large grains, extinction curves after
sputtering are too flat to explain the observed curves. The observed
extinction curves are better explained by considering the effect ofaccretion, which enhances the abundance of small (a  0.07 μm)
grains.
In summary, we conclude (i) that cooling does not protect the
dust grains against sputtering, (ii) that dust cooling does not change
the thermal history of the hot gas except for the case in which
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1632

H. Hirashita et al.

the hot gas initially has a dust-to-gas ratio as large as ∼ 0.01, (iii)that the formation of cold dense gas as a result of cooling helps dust
mass to increase through dust growth by the accretion of gas-phasemetals, and (iv) that the observed extinction curves are consistent
with this accretion scenario.
AC K N OW L E D G E M E N T S
We thank C. Conroy and F. Kemper for valuable discussions. We
also thank the anonymous referee for useful comments. HH is supported by the Ministry of Science and Technology (MoST) grant102-2119-M-001-006-MY3. TN is supported in part by the JSPS
Grant-in-Aid for Scientific Research (26400223).
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A P P E N D I X A : E VO L U T I O N O F G R A I N S I Z E
D I S T R I B U T I O N U N D E R A VA RY I N G
BAC K G RO U N D G A S D E N S I T Y
In the main text, we only concentrate on the single fluid element.
If we explicitly treat the spatial variation of grain size distribution
by denoting it as n(a, x, t), where x is the spatial coordinate, thecontinuity equation corresponding to equation (1) is written as (seeequation 37 in Tsai & Mathews 1995) ∂∂n(ȧn) + ∇ · (nu) = S,(A1)+
∂t a,x ∂awhere u is the velocity of the fluid, and (∂/∂t)a,x is the partial
derivative for time with fixed a and x. The gas density follows thecontinuity equation written in Lagrangian (comoving) form as
dρgas+ ρgas ∇ · u = 0.
dt
Eliminating ∇ · u from equations (A1) and (A2), we obtain ∂n
1 dρgas∂+ (u · ∇)n − n+(ȧn) = S.
∂t a,xρgas dt
∂a

(A2)

(A3)

Dust processing in elliptical galaxies

1633

The first two terms are combined to obtain the Lagrangian derivative(with a fixed): ∂n∂
1 dρgas+(ȧn) = S.
(A4)
−n
∂t a
ρgas dt∂aThus, we obtain equation (1).
APPENDIX B: EQUILIBRIUM GRAIN SIZE
DISTRIBUTIONAs seen in the text, the sputtering time-scale is often much shorter
than the cooling time-scale. Therefore, it is useful to discuss theequilibrium grain size distribution achieved by the balance between
sputtering and supply from AGB stars (we neglect cooling here).
Under this equilibrium condition, equation (1) is reduced to∂[ȧn(a)] = S(a),(B1)∂awhere we set all the time derivative terms as zero. Consideringthat lima → ∞ n(a) = 0, the equilibrium grain size distribution isanalytically written as
1 aS(a) da.
(B2)n(a) =ȧ ∞Figure B1. Equilibrium grain size distributions for Models a, b, and c(lower, middle and upper lines, respectively). We adopt the MRN grain
size distribution for AGB dust and neglect cooling (i.e. the gas temperatureand density are constant). The solid and dashed lines show the analytic
and numerical solutions, respectively. For the numerical solution, we plot
the grain size distributions at t = 107 yr, which is much longer than the
sputtering time-scale.

Figure B2. Same as Fig. B1 but for the lognormal grain size distribution
for AGB dust.

If we adopt the MRN grain size distribution (equation 11), we
obtainC ρ̇dust −2.5
−2.5
a
,
(B3)
− amaxn(a) =−2.5ȧ
where amax = 0.25 μm. Here, we have used equation (9) for S,
and we can use equation (13) for ȧ (note that ȧ is negative). If
a is significantly smaller than amax , n(a) ∝ a−2.5 . The grain size
distribution is less steep than that of AGB dust, since smaller grains
are more easily destroyed than larger grains.
In Fig. B1, we show the above analytic solution. For comparison,
we also show the numerical solution of equation (1) without coolingat t = 107 yr. Since the time-scales of sputtering and dust supply
are much shorter than 107 yr, we expect that the grain size distribution has achieved the equilibrium at t = 107 yr. The analytic andnumerical solutions are almost identical, which confirms that ournumerical scheme correctly solves the evolution of grain size distribution by dust supply and sputtering. Moreover, as expected above,the slope of the equilibrium grain size distribution is described bya power law with an index of −2.5.
We also show the equilibrium solution for the lognormal grain
size distribution of AGB dust in Fig. B2. The numerical solution
is produced in the same way as above; that is, we used the resultat t = 107 yr with cooling neglected. The ‘analytical’ solution
is obtained by numerically integrate equation (B2). We confirm
that the numerical results reproduce the analytical results well. Ataa0 = 0.1 μm (the central radius of the lognormal distribution),n(a) = constant since S(a) drops exponentially in that grain radius
range. We can confirm this in Fig. B2.
This paper has been typeset from a TEX/LATEX file prepared by the author.

MNRAS 454, 1620–1633 (2015)