Formation and Evolution of the Dust in Galaxies. II. The Solar Neighbourhood

Abstract

Over the past decade a new generation of chemical models have included the
dust in the treatment of the ISM. This major accomplishment has been spurred by
the growing amounts of data on the highly obscured high-z Universe and the
intriguing local properties of the Solar Neighbourhood (SoNE). We present here
a new model able to simulate the formation and evolution of dust in the ISM.
The model follows the evolution of 16 elemental species, with particular
attention to those that are simultaneously present in form of gas and dust,
e.g. C, N, O, Mg, Si, S, Ca and Fe. In this study we focus on the SoNe and the
MW Disk as a whole which are considered as laboratories to test the physical
ingredients governing the dust evolution. Infall of primordial gas, birth and
death of stars, radial flows of matter between contiguous shells, presence of a
central bar, star-dust emission by SNae and AGB stars, dust destruction and
accretion are taken into account. The model reproduces the local depletion of
the elements in the gas, and simultaneously satisfies other constraints
obtained from the observations. The evolution of the element abundances in the
gas and dust has been well reproduced for plausible choices of the parameters.
The Mg/Si ratio, in particular, drives the formation of silicates. We show that
for most of the evolution of the MW, the main process for dust enrichment is
the accretion in the cold regions of the ISM. SNae dominate in the early phases
of the evolution. We have examined the main factors controlling the temporal
window in which SNae govern the dust budget both in low and high star forming
environments. The role played by AGB stars is discussed. We find that IMFs with
regular slope in the range of massive stars better reproduce the observed
depletions. The results obtained for the SoNe lead us to safely extend the
model.

Full text

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arXiv:1107.4561v1 [astro-ph.GA] 22 Jul 2011
Astronomy & Astrophysics manuscript no. PiovanII
July 25, 2011
c ? ESO 2011
Formation and Evolution of the Dust in Galaxies. II.
The Solar Neighbourhood
L. Piovan1,2, C. Chiosi1, E. Merlin1, T. Grassi1, R. Tantalo1, U. Buonomo1and L. P. Cassar` a1
1Department of Astronomy, Padova University, Vicolo dell’Osservatorio 3, I-35122, Padova, Italy
2Max-Planck-Institut f¨ ur Astrophysik, Karl-Schwarzschild-Str. 1, Garching bei M¨ unchen, Germany
e-mail: lorenzo.piovan@unipd.it
Received: July 2011; Revised: *** ****; Accepted: *** ***
ABSTRACT
Context. Over the past decade a new generation of chemical models, in addition to the gas, have included the dust in
the treatment of the interstellar medium (ISM). This major accomplishment has been spurred by the growing amounts
of data on the highly obscured high-z Universe and the intriguing local properties of the Solar Neighbourhood (SoNe)
of the Milky Way (MW) Disk.
Aims. We present here a new model able to simulate the formation and evolution of dust in the ISM of the MW. The
model follows the evolution of 16 elemental species, with particular attention to those that are simultaneously present
in form of gas and dust, e.g. C, N, O, Mg, Si, S, Ca and Fe. In this study we focus on the SoNe and the MW Disk as a
whole which are considered as laboratories to test the physical ingredients governing the dust evolution.
Methods. The MW is described as a set of concentric rings of which we follow the time evolution of gas and dust.
Infall of primordial gas, birth and death of stars, radial flows of matter between contiguous shells, presence of a central
bar, star-dust emission by SNæ and AGB stars, dust destruction and accretion are taken into account. The model
reproduces the local depletion of the elements in the gas, and simultaneously satisfies other constraints obtained from
the observations.
Results. The evolution of the element abundances in the gas and dust has been well reproduced for plausible choices of
the parameters. The Mg/Si ratio, in particular, drives the formation of silicates. We show that for most of the evolution
of the MW, the main process for dust enrichment is the accretion in the cold regions of the ISM. SNæ dominate in
the early phases of the evolution. We have also examined the main factors controlling the temporal window in which
SNæ govern the dust budget both in low and high star forming environments. The role played by AGB stars is also
discussed. We find that IMFs with regular slope in the range of massive stars better reproduce the observed depletions.
Conclusions. The classical chemical models nicely reproduce the abundances, depletion factors and dust properties of
the SoNe and the main ingredients of the models are tested against observational data. The results obtained for the
SoNe lead us to safely extend the model to the whole Galactic Disk or galaxies of different morphological types.
Key words. Galaxies - Dust; Galaxies – Spirals; Galaxies – Milky Way
1. Introduction
In the fascinating subject of the origin and evolution of
galaxies, the interstellar dust is acquiring a primary role
because of its growing importance in the observations of
the high-z Universe (Omont et al. 2001; Shapley et al.
2001; Bertoldi et al. 2003; Robson et al. 2004; Wang et al.
2008a,b; Gallerani et al. 2010; Micha? lowski et al. 2010b,a)
andthetheoretical spectro-photometric,
and chemical modeling of galaxies (Schurer et al. 2009;
Narayanan et al. 2010; Jonsson et al. 2010; Grassi et al.
2010; Pipino et al. 2011; Popescu et al. 2011).
Indeed, the evidence of highly obscured QSOs and
galaxies already in place at high-z leads necessarily to a
new generation of theoretical models where dust is a key
ingredient that cannot be neglected, if we want to obtain
precious clues on the fundamental question about when
and how galaxies formed and evolved. First of all, dust
absorbs the stellar radiation and re-emits it in the infrared
deeply changing the shape of the observed spectral energy
distributions (SEDs) of obscured galaxies (Silva et al.
1998; Piovan et al. 2006; Popescu et al. 2011); second, it
dynamical,
strongly affects the production of molecular hydrogen and
the local amount of UV radiation in galaxies thus playing
a strong role in the star formation process via the cooling
mechanisms (Yamasawa et al. 2011). The inclusion of dust
in the models leads to a growing complexity and typically
to a much larger set of parameters influencing the results
of the simulations to be then compared with the observa-
tions. Indeed several question must be addressed, each one
easily expanding the model: who are the main stardust
injectors in the interstellar medium (ISM) (Gail et al.
2009; Valiante et al. 2009; Gall et al. 2011a; Piovan et al.
2011a)? How much dust do they produce and on which
timescales (Draine 2009; Dwek et al. 2009)? What is the
contribution and the role played by the molecular clouds
(MCs)-grown dust that form in the cold dense regions of
the ISM (Zhukovska et al. 2008; Pipino et al. 2011)? How
much dust is destroyed by SNæ shocks (Nozawa et al. 2006,
2007; Bianchi & Schneider 2007; Jones & Nuth 2011)?
What is the typical minimal set of dust grains whose
evolution should be followed and what could be a minimal
set of dust grains to be used for a satisfactory description
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L. Piovan et al.: Formation and evolution of the dust in galaxies
of the chemical or spectrophotometric properties of the
galaxy?
To answer all these questions the theoretical models
must include (i) a set of grains with suitable composition
and properties and/or an ISM made of gas and dust in
which the abundances of the elements are followed, (ii)
a recipe for their formation/accretion and destruction
in the ISM and (iii) a prescription for the yields of dust
by the stellar sources. The duty cycle of the dust can
be schematically summarized as follows (Jones 2004).
Stars, mainly AGBs and SNæ, inject material in the ISM,
mainly in form of gas, but with a variable amount that
condenses into the so-called star-dust. Once injected into
the ISM, star-dust grains are subjected to destruction
processes that restitute the material to the gaseous phase.
However in cold and dense regions dust can accrete on the
so-called seeds: the competition between the accretion and
destruction processes, mainly via shocks, determines the
total budget of dust in the ISM and the observed depletion
of the elements that are involved in the formation of dust
grains (Dwek 1998). Dust accretion mainly occurs in the
very cold molecular clouds (MCs), where it induces strong
cooling thus leading to the formation of new stars. The
stellar winds from AGB stars SNa explosions more and
more enrich the ISM with new metals and star-dust grains
that are able to survive to the local shocks caused by SNa
explosions.
The Milky Way (MW) is the ideal laboratory to our dis-
posal to study the dust cycle (Zhukovska et al. 2008) and
its impact on the wider subject of galaxy formation. For
obvious reasons, the MW provides plenty of observational
data to which we can compare theoretical predictions,
thus setting useful constraints on theoretical simulations
and highlighting the role of the most important physical
quantities leading the whole problem. Once this important
step is accomplished, our modelling of the role played by
dust can be extended to other galaxies such as local disk
and spheroidal galaxies, high-z galaxies and QSOs.
Starting from these considerations, in this paper we
simulate the formation, evolution and composition of dust
in the MW, both locally in the Solar Neighbourhood
(SoNe) and radially along the Galactic Disk. We build up a
detailed chemical model (the theoretical simulation is still
the main tool to investigate the formation and evolution
of dust in galaxies) starting from the pioneering study
by Dwek (1998) and taking into account the more recent
ones by Zhukovska et al. (2008), Calura et al. (2008),
Valiante et al.(2009), Gall et al.
(2011b),Mattsson(2011),
Kemper et al. (2011) and Dwek & Cherchneff (2011).
The theoretical model we are building up stems from the
basic one with infall by Chiosi (1980), however updated
to the more recent version with radial flows of matter and
presence of a central bar developed by Portinari & Chiosi
(2000). The stellar yields of chemical elements in form
of gas are those calculated by Portinari et al. (1998).
The model follows the evolution of the abundances of a
number of elements composing the ISM gas, includes the
formation/destruction and evolution of dust and, finally,
follows in detail also the abundances of those elements that
are embedded in the dust grains. To this aims, the model
makes use of the best prescriptions available in literature
concerning dust accretion, destruction and condensation in
the AGB winds and SNæ explosions. These prescriptions
(2011a),
Valiante et al.
Gall et al.
(2011),
have been already presented by Piovan et al. (2011a) to
whom the reader should refer and will also be discussed in
some detail here.
The main test for any model of dust formation is
given by the data on element depletion provided by SoNe
of the MW to which we will compare our results. In a
forthcoming paper (Piovan et al. 2011c) we will investigate
the radial dependence of chemical abundances and dust
depletion across the Disk of the MW.
The plan of the paper is as follows. In Sect. 2 we
introduce the formalism and basic equations governing the
temporal evolution of the gas, stars, and dust in an open
model with radial flows of gas and dust for the galactic
Disk of the MW. In Sect. 3 we summarize the current
prescriptions for the star formation rate and initial mass
function. In Sect. 4 we introduce and describe in some
detail the various processes responsible for the formation
and growth of dust grains in the ISM, whereas in Sect. 5 we
present the accretion rates into dust grains for a number of
important elements we have considered. The yields of dust
from AGB stars and SNa explosions are adopted according
to Piovan et al. (2011a) to whom the reader should refer
for all the details. In Sect. 6 the problem of dust destruc-
tion by SNa shocks is faced. Then, in Sect. 7 we present
the observational data for the elemental abundances in
the Solar Neighborhood and define the reference set of
abundances we have adopted. In Sect. 8 after summarizing
the main ones between the many available parameters, we
discuss and compare the effect of them on the formation
and evolution of dust, at varying them between the many
possible choices. In particular we examine the influence on
dust of a different CO fraction in the ISM (Sect. 8.1), the
effect of the IMF (Sect. 8.2) and of the SF law (Sect. 8.3),
the choice between different models for the accretion of
dust in cold regions (Sect. 8.4) and, finally we discuss some
interesting parameters however not discussed in our model.
In Sect. 9 we present the results for our models of the Solar
Vicinity in presence of dust and as a function of three
important ingredients, namely, the initial mass function,
the efficiency of star formation, the accretion time scale of
primordial gas onto the system mimicking the evolution of
the MW Disk. The effect of radial flows and central bar are
always included according to the prescription developed
in previous studies (Portinari & Chiosi 2000) and also
adopted in Piovan et al. (2011c). The simulations are
compared with the depletion of the elements in the SoNe
under the constraints that we want that the local chemical
properties are satisfied, such as the time evolution of the
elemental abundances, the metallicity and iron enrichment.
In Sect. 10 we discuss the results we have obtained and
draw some general conclusions.
2. Chemical Evolution Model
In classical models of chemical evolution, the Disk of the
MW is subdivided in N concentric circular rings of a cer-
tain thickness ∆r, where r is the galacto-centric distance,
in the case of plane geometry or N concentric cylindrical
shells if the third dimension is considered. Each ring or
shell is identified by the mid radius rk with k = 1,.....,N.
In most cases, radial flows of interstellar gas and dust are
neglected, so that each ring / shell evolves independently
from the others. The physical quantity used to describe the
Disk is the surface mass density as a function of the radial
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L. Piovan et al.: Formation and evolution of the dust in galaxies
coordinate r and time t: σ(rk,t) is the mass surface den-
sity at radius rkand time t. Depending on the model, σ can
refer to the ISM (σM), in turn split into dust or gas (σD
or σGrespectively), to the stars (σ∗) or to the total mass
(simply σ). At every radius rk, the surface mass density is
supposed to slowly grow by infall of either primordial or
already enriched gas and to fetch at the present age tGthe
mass density profile across the Galactic Disk for which an
exponential profile is best suited to represent the surface
mass density distribution: σ(rk,tG) ∝ exp(−rk/rd), where
rdis the scale radius of the Galactic Disk, that is typically
estimated of the order of 4 − 5 kpc. Since the final density
profile is a priori known, one may normalize to it the cur-
rent total surface mass density of the ISM ”M” (sum of
gas and dust),
M(rk,t) =σM(rk,t)
σ(rk,tG). (1)
Introducing the fractionary mass of the generic i-th element
we have:
Mi(rk,t) =σM
i
(rk,t)
σ(rk,tG)= χi(rk,t)M(rk,t)(2)
and therefore the fractional mass abundance χi(rk,t) =
Mi(rk,t)/M(rk,t), with?
cated by ”D) and the gas (indicated by ”G”)
D(rk,t) =σD(rk,t)
iχi(rk,t) = 1.
Similar expressions can be derived for the dust (indi-
σ (rk,tG)
(3)
G(rk,t) =σG(rk,t)
σ (rk,tG)
(4)
with σM(rk,t) = σG(rk,t) + σD(rk,t) and σ(rk,t) =
σM(rk,t) + σ∗(rk,t), where σ∗(rk,t) is the surface mass
density of stars. For single chemical elements we may write:
Di(rk,t) =χD
i(rk,t)σD(rk,t)
σ(rk,tG)
= χD
i(rk,t)D(rk,t) (5)
Gi(rk,t) =χG
with?
The fundamental equation describing the evolution of
the ISM in absence of radial flows of matter between con-
tiguous shells (Portinari & Chiosi 2000) and processes of
dust accretion/destruction (Dwek 1998) is:
i(rk,t)σG(rk,t)
σ(rk,tG)
i(rk,t)?= 1, from which it follows
= χG
i(rk,t)G(rk,t) (6)
i
iχD
?χD
i(rk,t) + χG
i(rk,t) ?= 1 and?
that?
iχG
i(rk,t) ?= 1.
d
dtMi(rk,t) = −χi(rk,t)ψ (rk,t)
?Mu
?d
where φ(M) is the IMF and Ml and Mu are the lower
and upper limits for the stellar masses, ψ (rk,t − τM) is the
star formation rate (SFR) at the radius rkand at the time
t′= t−τM, Ri(M) = EiM/M (Portinari et al. 1998) is the
fraction of a star of initial mass M ejected back in form
of the chemical species i-th. The three terms at the r.h.s.
+
Ml
ψ (rk,t − τM)Ri(M)φ(M)dM
?
+
dtMi(rK,t)
inf
(7)
represent the depletion of the ISM due to star formation,
its increase by stellar ejecta, and the increase by infall of
external gas (either primordial or already enriched).
Adding supernovae and radial flows. Type Ia supernovae
originate in binary systems and have a fundamental role,
in particular concerning the iron enrichment. The super-
novae rate and the adopted formalism are the ones of
Greggio & Renzini (1983) and the formulation of equation
describing chemical evolution of the ISM is modified follow-
ing Matteucci & Greggio (1986). The contribution of single
stars, corresponding to a fraction (1 − A) of the total, is
separated from the contribution of binary system, a frac-
tion A of the total. Inserting the contribution of type Ia
supernovae and integrating in time, instead that in mass,
the equation for the evolution of the i-th component of the
ISM is,
d
dtMi(rk,t) = −χiψ+
?t−τMB,l
+
0
ψ
?
φ(M)Ri·
?
−dM
dτM
??
M(τ)
dt′+
+ (1 − A)
?t−τMB,u
?
t−τMB,l
ψ
?
−dM
dτM
φRi·
?
??
−dM
dτM
??
M(τ)
dt′+
+
?t−τMu
?t−τM1,max
+ RSNI · ESNI,i+
t−τMB,u
ψφRi·
?
M(τ)
dt′+
+ A
t−τM1,min
ψ
?
f(M1)Ri,1·
?
−dM1
dτM1
??
M1(τ)
dt′+
+
?d
?d
dtMi(rk,t)
?
?
inf
−
?d
dtMi(rk,t)
?
out
+
dtMi(rk,t)
rf
(8)
where ψ = ψ(rk,t′), φ = φ(M), χ = χi(rk,t), Ri =
Ri(M), Ri,1= Ri(M1) and M (t − t′) = M (τ). The first
term at the r.h.s. is as usual the one describing the de-
pletion of interstellar material because of the process of
star formation and it depends from the star formation rate
and from the abundance of the i-th element considered.
The next three terms represent the contribution of sin-
gle stars to the enrichment of the i-th element. The fifth
term is the contribution of the primary star in a binary
system (assumed to be independent from the secondary
star as far as it concerns the chemical yields). The sixth
term is the contribution of type Ia supernovae. Finally,
the last three terms are the infall rate of external gas,
the outflow rate of matter due for example to the onset
of galactic winds powered by supernovæ explosions, and
the radial flows of gas that determine the ISM exchange
between contiguous shells (Portinari & Chiosi 2000), re-
spectively. Furthermore, f(M1) is the distribution function
of the mass of the primary star M1 in a binary system,
between M1,min = MB,l/2 and M1,max = MB,u, where
MB,l and MB,u are the lower and upper limit of the bi-
nary systems assumed respectively 3M⊙and 12M⊙. RSNI
is the rate of type Ia SNæ and ESNI,i their ejecta of the
i-th chemical species. M (τ) = M (t − t′) is the mass of
a star of lifetime τ, born at t′. It is worth noticing that
various quantities depend on the metallicity Z (t) as well
as on M: M (τ) = M (t − t′) = M (t − t′,Z (t − t′)) and
Ri(M) = Ri(M,Z (t − t′)) as stellar lifetimes and ejecta
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L. Piovan et al.: Formation and evolution of the dust in galaxies
depend on metallicity. Ri(M) are calculated on the base of
the detailed stellar yields from Portinari et al. (1998) and
keep track of finite stellar lifetimes (no instantaneous recy-
cling approximation). Eqns. (8) govern the evolution of the
ISM.
Separating gas from dust. For our purposes we need
to formulate the equations governing the evolution of the
dust in the ISM. Separating the ISM in gas and dust, the
equations governing the evolution of the generic elemental
species i-th in the dust are
d
dtDi(rk,t) = −χD
?t−τMB,l
iψ+
+
0
ψ
?
φδw
c,iRi·
?
?
?
?
−dM
dτM
??
?
?
??
M(τ)
dt′+
+ (1 − A)
?t−τMSNæ
?t−τMB,u
?
t−τMB,l
ψ φδw
c,iRi·−dM
dτM
??
??
M(τ)
dt′+
+ (1 − A)
t−τMSNæ
ψ φδII
c,iRi·−dM
dτM
M(τ)
dt′+
+
?t−τMu
?t−τM1,max
?t−τMSNæ
+ RSNIESNI,iδI
t−τMB,u
ψ φδII
c,iRi·−dM
dτM
M(τ)
dt′+
+ A
t−τMSNæ
ψ
?
f(M1)δII
c,iRi,1·
?
−dM1
dτM1
−dM1
dτM1
??
??
M(τ)
dt′+
+ A
t−τM1,min
ψ
?
f(M1)δw
c,iRi,1·
?
M(τ)
dt′+
c,i+
?
?
−
?d
?d
dtDi(rk,t)
out
+
?d
?d
dtDi(rk,t)
?
rf
?
+
+
dtDi(rk,t)
accr
−
dtDi(rk,t)
SN
(9)
where χD
(9) is the depletion of dust because of star formation that
consumes both gas and dust (uniformly mixed in the ISM).
The second term is the contribution by stellar winds from
low mass stars to the enrichment of the i-th component of
the dust. Following Dwek (1998), we introduce the so-called
condensation coefficients δw
c,ithat determines the fraction
of material in stellar winds that goes into dust with respect
to that in gas (local condensation). The third term is the
contribution by stars not belonging to binary systems and
not going into type II SNæ (the same coefficients δw
used). The fourth term is the contribution by stars not
belonging to binary systems, but going into type II SNæ.
For the condensation efficiency in the ejecta of type II SNæ
we introduce the coefficients δII
possible choices for these coefficients are discussed in detail
in Piovan et al. (2011a). The fifth term is the contribution
of massive stars going into type II SNæ. The sixth and
seventh term represent the contribution by the primary star
of a binary system, distinguishing between those becoming
type II SNæ from those failing this stage and using in each
situation the correct coefficients. The eighth term is the
contribution of type Ia SNæ, where again we introduced the
condensation coefficients δI
c,ito describe the mass fraction
of the ejecta going into dust. The last four terms describe:
(1) the outflow of dust due to galactic winds (in the case
of disk galaxies this term can be set to zero); (2) the radial
flows of matter between contiguous shells; (3) the accretion
i= χD
i(rk,t). The first term at the r.h.s. of Eqn.
c,iare
c,i, the analog of δw
c,i. The
term describing the accretion of grain onto bigger particles
in cold clouds; (4) the destruction term taking into account
the effect of the shocks of SNæ on grains, obviously giving
a negative contribution. The infall term in the case of dust
can be neglected because we can safely assume that the
material entering the galaxy is made by gas only without a
solid dust component mixed to it.
Finally, from the equation for Di(rk,t) we can get the
equation describing the evolution of the gaseous component
Gi(rk,t), where Gi(rk,t) = Mi(rk,t) − Di(rk,t):
d
dtGi(rk,t) = −χG,iψ+
+ (1 − A)
?t−τMSNæ
?t−τMB,u
?
?
t−τMB,l
ψ
?
φ?1 − δw
φ
?
c,i
?Ri·
?
−dM
dτM
?
?
??
−dM
dτM
??
??
M(τ)
dt′+
+ (1 − A)
t−τMSNæ
ψ
?
1 − δII
c,i
Ri·−dM
dτM
M(τ)
dt′+
+
?t−τMB,l
?t−τMu
?t−τM1,max
?t−τMSNæ
0
ψφ?1 − δw
φ(M)
c,i
?Ri·
?
M(τ)
dt′+
+
t−τMB,u
ψ
?
1 − δII
c,i
?
Ri·
?
−dM
dτM
??
−dM1
dτM1
M(τ)
dt′+
+ A
t−τMSNæ
ψ
?
f(M1)
?
1 − δII
c,i
?
Ri,1·
?
−dM1
dτM1
??
M(τ)
dt′+
+ A
t−τM1,min
ψ
?
f(M1)?1 − δw
1 − δI
c,i
?
+
dtGi(rk,t)
c,i
?Ri,1·
?
?
??
M(τ)
dt′+
+ RSNIESNI,i
?
?
?
+
?d
dtGi(rk,t)
inf
+
−
?d
?d
dtGi(rk,t)
out
?d
?d
?
rf
?
dtGi(rk,t)?
?d
+
−
dtDi(rk,t)
accr
+
dtDi(rk,t)
SN
(10)
where again the outflow term −?d
spirals with continuous star formation. Since the primor-
dial material is likely dust-free we have
?d
els, upon which are based our yields, predict that stars
with mass higher than 6M⊙ go into SNæ, whereas those
with mass lower than 6M⊙ first become AGB stars and
later White Dwarfs. We must therefore split the third and
the fifth member of Eqn. (8) in two parts, both in Eqns.
(10) and (9), because the minimum mass dividing the in-
tervals of AGB and/or SNæ belongs to the mass interval
(3 − 12M⊙) describing binary systems going into type Ia
SNæ1.
outwill be
fixed to zero because we do not have galactic wind for
dtMi(r,t)?
in=
dtGi(r,t)?
in.
It is worth noticing the following point: the stellar mod-
1For example in Eqn. (9), using δc,i to indicate the generic
condensation coefficient we have the following split:
(1 − A)
?t−τMB,u
t−τMB,l
ψ
?
φδc,iRi(M)
?
−dM
dτM
??
M(τ)
dt′
is divided into:
(1 − A)
?t−τMSNæ
?t−τMB,u
t−τMB,l
ψ
?
φδw
c,iRi
?
−dM
dτM
??
M(τ)
dt′
+ (1 − A)
t−τMSNæ
ψ
?
φδII
c,iRi
?
−dM
dτM
??
M(τ)
dt′
4

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L. Piovan et al.: Formation and evolution of the dust in galaxies
To summarize. Indicating the contribution to the yields
by stellar winds and type Ia and II SNæ with the symbols
Wi,D(rk,t), Wi,G(rk,t) and Wi,M(rk,t) (they can easily
be reconstructed by comparison) and neglecting the outflow
term, Eqns. (8), (9) and (10) become:
d
dtMi(rk,t) = −χM
?d
i (rk,t)ψ (rk,t) + +Wi,M(rk,t)
?
+
dtMi(rk,t)
rf
(11)
d
dtDi(rk,t) = −χD
?d
?d
i(rk,t)ψ (rk,t) + +Wi,G(rk,t)
?
?
+
dtDi(rk,t)
accr
−
?d
dtDi(rk,t)
?
SN
+
dtDi(rk,t)
rf
(12)
d
dtGi(rk,t) = −χG,i(rk,t)ψ (rk,t) + Wi,G(rk,t)
?d
?d
−
dtDi(rk,t)
?
?
accr
+
?d
?d
dtDi(rk,t)
?
SN
+
dtGi(rk,t)
inf
+
dtGi(rk,t)
?
rf
.(13)
It is soon evident that the dust creation/destruction and
the radial flows make the system of differential equations
more complicated than the original one by Talbot & Arnett
(1975) for a one-zone closed-box model. As the ISM is given
by the sum of gas and dust, only two of these equations
are required, furthermore Eqn. (11) can be used only if
gas and dust flow with the same velocity. To proceed fur-
ther we must now specify the law of star formation, the
IMF, the stellar ejecta and the various rates describing gas
infall, dust accretion/destruction, and radial flows/bar ef-
fect. No details will be given about these ones. They are
included into the model and they are mainly useful in or-
der to reproduce the radial gradients of abundance in the
MW. The reader should refer to Portinari & Chiosi (2000)
and Piovan et al. (2011c).
where MSNæ is the separation mass that tell us if we must use
the condensation coefficients of stellar winds (condensation of
dust in the envelopes of AGB stars) in the mass interval between
MB,l and MSNæ or the condensation coefficients of supernovæ
between MSNæ e MB,u. In the same way:
A
?t−τM1,max
t−τM1,min
ψ
?
f(M1)δc,iRi,1
?
−dM1
dτM1
??
M1(τ)
dt′
splits itself into:
A
?t−τM1,max
?t−τMSNæ
t−τMSNæ
ψ
?
f(M1)δII
c,iRi,1
?
−dM1
dτM1
??
M(τ)
dt′
+ A
t−τM1,min
ψ
?
f(M1)δw
c,iRi,1
?
−dM1
dτM1
??
M(τ)
dt′.
3. The Star Formation Laws and Initial Mass
Functions
Star Formation. The law of Star Formation (SF) is a key
ingredient of any model of galaxy formation and evolution.
Unfortunately it is poorly known, so that many prescrip-
tions for the SF rate can be found in literature. In this study
we have considered several well known SF laws adopted for
the MW (see Portinari & Chiosi 1999, for details).
A very popular prescription is the Schmidt (1959) law.
In our formalism it becomes:
Ψ(rk,t) = −
?dG(rk,t)
dt
?
∗
= ν
?σ(rk,tG)
σ(r⊙,tG)
?κ−1
Gκ(rk,t) (14)
where the normalization factor is σ(r⊙,tG)−(κ−1)and ν
is in [t−1]. Following Portinari & Chiosi (1999) we adopt
κ = 1.5.
This simple dependence of the SFR can be complicated
by including other physical effects. For instance, the SF
suited to spiral galaxies such as the MW, may include the
effect of gas compression by density waves (Roberts 1969;
Shu et al. 1972; Wyse & Silk 1989; Prantzos & Silk 1998)
or gravitational instabilities (Wang & Silk 1994). We have:
Ψ(rk,t) = ν
?r
r⊙
?−1?σ(rk,tG)
σ(r⊙,tG)
?κ−1
Gκ(rk,t)(15)
where ν is always in [t−1] and k = 1 (Kennicutt 1998;
Portinari & Chiosi 1999).
Another possibility is to describe the SF as a balance
between cooling and heating processes, that is the grav-
itational settling of the gas onto the Disk and the energy
injection from massive stars (Talbot & Arnett 1975; Dopita
1985; Dopita & Ryder 1994). In our formalism, we have
Ψ(rk,t) = ν
?σn(rk,t)σm−1(rk,tG)
σ(r⊙,tG)n+m−1
?
Gm(rk,t) (16)
where n = 1/3, m = 5/3 (Portinari & Chiosi 1999) and
ν in [t−1]. This formulation is similar to the original one
by Talbot & Arnett (1975) thus leading to similar results
(Portinari et al. 1998).
Initial Mass Function. The initial mass function (IMF) is
perhaps the most important ingredient of chemical models
of any kind (see Kroupa 2002b, for a recent review of the
subject). Brown dwarfs and very low mass stars whose life-
times are longer than the age of the Universe, in practice
lock up forever the chemical elements present in the ISM at
the age of their birth, whereas intermediate and high mass
stars of short life continuously enrich the environment with
the products of thermonuclear reactions, thus driving the
chemical evolution of the host system. They are also the
factories of star-dust to be injected into the ISM by SNæ
explosions and strong stellar winds. The adoption of an IMF
has two effects worth being mentioned here. First of all, a
different slope of the IMF in the intermediate-high mass
range would imply a different relative population of the
stars contributing to the dust yields. Second, the net yield
of metals and dust per stellar generation varies. According
to its definition (see for instance Tinsley 1980; Pagel 1997;
Portinari et al. 2004a) the net yield is the the amount of
metals globally produced by a stellar generation over the
5

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L. Piovan et al.: Formation and evolution of the dust in galaxies
the fraction of mass locked up in living stars and remnants.
Therefore, efficiency of metal and dust enrichment depends
not just on the amount of metals produced per unit mass
involved in star formation but on the ratio between this and
the mass that remains locked in remnants or ever-lived low
mass stars. The locked-up fraction, is therefore as crucial to
the metal and dust enrichment as is the absolute number
of the high-mass stars directly responsible for the produc-
tion of dust itself. In a given model of fixed total mass, it
is clear that an IMF bending down steeply at low masses
will lead to a different locked up fraction with respect to
a power-law, low-mass oriented IMF. The metal and dust
production is accordingly affected.
In this section we shortly present the IMFs we have in-
cluded in our model of the Galactic Disk and SoNe. For the
purposes of our study the IMF is assumed to be constant
in time and space. All the IMFs are normalized assum-
ing that the total mass encompassed by the IMF from the
lower, ML, to the upper, MU, mass limit of stars is equal to
1 M⊙. To this aim, following Talbot & Arnett (1975) and
Bressan et al. (1994), we define the parameter ζ, which de-
scribes the fraction of total mass in form of stars stored in
the IMF above a given mass M∗. In other words, M∗is the
minimum mass contributing to the nucleo-synthetic enrich-
ment of the ISM over a timescale of the order of the galaxy
life
ζ =
?MU
M∗φ(M)dM
?MU
MLφ(M)dM
. (17)
This equation, at varying ζ and for fixed MU and M∗,
can be reversed numerically to determine the lower limit of
the distribution ML. The IMFs included in our model are:
- The Salpeter IMF. Salpeter-like IMFs are very popular.
These are an extension over the desired mass range of the
original Salpeter IMF (Salpeter 1955). This IMF is φ(M) =
CSM−1.35with CSdepending on the value of ζ. For a mass
range [0.1 − 100]M⊙(Portinari et al. 2004a) we have Cs=
0.1716 and a ζ = 0.3925.
- The Kroupa IMF. In a series of papers Kroupa revised
and updated the power-law IMF with a set of continuous
multi-slope power-laws (Kroupa et al. 1993; Kroupa 2001,
2002b,a, 2007, e.g.). In the following we consider two cases.
First, the IMF derived by Kroupa (1998) for field stars in
the SoNe and used by Portinari et al. (2004b). This IMF
is typical of models of chemical evolution of disk galaxies
(Boissier & Prantzos 1999, 2000; Prantzos & Boissier 2000;
Hou et al. 2008)2. For ML= 0.1M⊙and MU = 100M⊙,
we obtain ζ = 0.405. Second, the Kroupa (2007) IMF,
where taking ML = 0.01M⊙ and MU = 100M⊙ we get
ζ = 0.38, slightly lower than in the above Kroupa (1998)
because of the lower limit extended to brown dwarf regime.
- The Larson IMF. Larson (1998) proposed an IMF in
which the relative percentage of very low mass stars and
sub-stellar objects is decreased due to the presence of an
exponential cut-off. As a consequence of this there is a neg-
ligible contribution to the locked-up mass, and in contrast
2Along this line it worth recalling that IMF with slope
(1.6 ∼ 1.7) in the high mass range, i.e. steeper than the
Salpeter value, is from Scalo (1986) and it is widely used
in literature in chemical models of the MW even with dust
(Matteucci & Fran¸ cois 1989; Chiappini et al. 1997; Dwek 1998;
Romano et al. 2000; Fran¸ cois et al. 2004; Calura et al. 2008)
a very high net yield per stellar generation, and a high pro-
duction of metals and dust. The Larson (1986) IMF is
φ(M) = CLM−1.35exp
?
−ML
M
?
. (18)
In practice this IMF recovers the Salpeter IMF at high
masses, whereas at low masses the exponential cut-off de-
termines the steep downfall after the peak mass MP =
ML/1.35 = 0.25M⊙. For a typical mass range [0.01 −
100]M⊙we have ζ = 0.653 thus allowing for a high num-
ber of intermediate-high mass stars. For the present aims,
we will simply keep MLconstant with time or metallicity.
We also consider the possibility for a modified Larson IMF
adapted to the SoNe, in which the slope in the power-law
factor for the high mass range has the value M−1.7, ac-
cording to Scalo (1986) and in agreement with recent IMFs
proposed by Kroupa (see above). In this last case with the
same mass range [0.01 − 100], we have ζ = 0.5, lower than
the other case.
- The Chabrier IMF. Along the line of thought of Larson
(1998), Chabrier (2001) proposes:
φ(M) = CCM−2.3exp
??
−MC
M
?1/4?
.(19)
The two parameters in equation (19) are tuned on local field
low mass stars and the functional form is proved to be o
be valid down to the brown dwarfs regime (Chabrier 2002).
With a mass range [0.01−100]M⊙and MC= 716.4 we get
CC = 40.33 and ζ = 0.545, not as high as in the Larson
IMF, but still leading to high net yield and low locked up
mass fraction.
- The Kennicutt IMF. The Kennicutt (1983) IMF is used
in literature to describe the global properties of spiral galax-
ies and it is inspired by the observations of Hα luminosities
and equivalent width in external galaxies (Kennicutt et al.
1994; Portinari et al. 2004b; Sommer-Larsen 1996). With a
mass range [0.1 − 100]M⊙, we get ζ = 0.59.
- The Arimoto IMF. This top-heavy IMF has been sug-
gested by Arimoto & Yoshii (1987) to simulate elliptical
galaxies and is introduced just for the sake of comparison as
an extreme case: φ(M) = CAriM−1.0. With a mass range
[0.1 − 100]M⊙, we get ζ = 0.5.
4. Growth dust grains in the ISM
Dust grains form by a number of physical processes (see
below) and once in the ISM they may grow in mass by
accreting a number of atoms or molecules. In the following
we examine in some detail the accretion mechanisms for
which two different models are proposed.
4.1. The Dwek (1998) model or case A
Dwek (1998) view of grain growth can be summarized as
follows. Let us consider an ISM whose dust component is
formed by grains made of a single element i-th with mass
mgr,i. Let mibe the mass of an atom (or molecule) of the el-
ement i-th in the gaseous phase. Niis the number of atoms
(molecules) of the element i-th locked up into the mono-
composition grains that we are considering and ngr,i the
density of grains of i-th type in the ISM, considered in this
6

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L. Piovan et al.: Formation and evolution of the dust in galaxies
model as a single phase, without distinction between dif-
fuse ISM and cold molecular clouds. Let now αi be the
sticking coefficient, telling us the probability that an atom
(molecule) of the element i in the gaseous phase binds to
the grains increasing the number of atoms/molecules on it.
Finally ngas,iis the number density the gas made of atoms
(molecules) of type i-th and v =
?8KBT
πmi
?1
2
is the mean
thermal velocity of the particles of the element i in the
gaseous phase with respect to the dust grains. We get:
dNi
dt
Multiplying both members of the equation by the constant
mass mi, multiplying and dividing the second member by
the mass of one grain mgr,i, introducing the mean thermal
velocity v and using the following relation3(Dwek 1998):
= αiπa2ngr,ingas,iv .(20)
mingas,i= 2mHnH2
?
1 −σD
i
σM
i
?
(21)
where σD
time, we obtain:
i(r,t) and σi(r,t) are functions of position and
dσD
dt
i
=σD
τi,0
i
?
1 −σD
i
σM
i
?
. (22)
For the dimensional consistency of the equation, the multi-
plying quantity has the dimension of the inverse of a time
indicated by τi,0
= 2nH2αiπa2
Ai
It may be worth of interest to give an estimate of the in-
volved timescale: by means of typical values of the involved
quantities, Dwek (1998) obtains ∼ 3×104yrs. It is possible
now to define the accretion time scale of our i-th element
onto the grains as:
1
τi,accr
τ0,i
1
τi,0
?8KBT
πmi
?1
2
. (23)
=
1
·
?
1 −σD
i
σM
i
?
(24)
that is the inverse of τ0,imultiplied by the fraction of the
i-th element in the gaseous phase. Again, we drop the de-
pendence from r and t. Dwek (1998) adopting the ratio
σD
i
≈ 0.7 gets τi,accr≈ 105yrs which turns out to be
significantly shorter than the lifetime of molecular clouds
where accretion takes place. This lifetime is estimated to
be of the order of tMC ∼ 2 − 3 × 106− 107yrs. The life-
time of a molecular cloud is comparable to the lifetime of
the most massive stars born in it. These stars indeed in-
jecting great amounts of energy by stellar winds and SNæ
explosions eventually disrupt the cloud. Similar estimates
have been made in studies on the temporal evolution of
dust in the ISM by Calura et al. (2008), ∼ 5 · 107yrs,
and Zhukovska et al. (2008), ∼ 107yrs, and are consistent
with observational and theoretical estimates of the lifetimes
of molecular clouds (Matzner 2002; Krumholz et al. 2006;
Blitz et al. 2007, see e.g.). One may argue that we are com-
paring two different timescales, i.e. with ti,accr ≪ tMC.
i/σM
3The above relation says that the mass of the element i-th
in the gaseous phase can be expressed by means of the density
of the molecular clouds, i.e. the site in which the grains grow.
2mHnH2is the mass of the molecular cloud,
the fraction of the element i-th in the gaseous phase as a function
of the abundance of the same element in the dust and ISM.
?
1 − σD
i/σM
i
?
is
However, inside molecular clouds there are many physical
processes continuously stripping atoms and and molecules
from the grains, like UV radiation and cosmic rays. The net
effect of it could be that τi,accris considerably lengthened
(Dwek 1998), likely up to 6· 107yrs, so that ti,accr≃ tMC.
Furthermore, all the dust grains are in cold molecular
clouds where the growing occurs. The simplest way to take
this into account is to divide the above estimate for ti,accr
by the fraction of dust in molecular clouds. The resulting
accretion timescale would be ti,accr∼ 1 − 2 × 108yrs.
There is another important point to consider on which
indeed several studies of the ISM evolution are based: both
the destruction timescale (driven by SNæ explosions) and
the accretion timescale ti,accr have the same dependence
on the star formation rate ψ and surface mass density σi.
SNæ rate can be expressed as τi,snr∝ σ/ψ. The lower the
star formation rate, the lower is the number of supernovae
explosions and finally the longer the destruction time of
grains due to shocks. We clearly expect τi,accrto vary over
the evolutionary history of the Galaxy: Dwek (1998) divides
the accretion timescale by σMC/σ and reasonably assumes
that ψ ∝ σMC. In this way, τi,accr∝ σψ. Since both τi,accr
and τi,snr have the same dependence from σISM/ψ Dwek
(1998) suggests that the ratio between the two timescales
is constant: τi,accr/τi,snr= 1/2. If now we consider the ac-
cretion/destruction terms in eqn. (12) we have:
?dσD
i
dt
?
accr
+
?dσD
i
dt
?
snr
=
σD
i
τi,accr
σD
i
τi,snr
−
σD
i
τi,snr
σD
i
τi,snr
=
= 2 ·
σD
i
τi,snr
−=.(25)
The contribution of the creation/destruction terms is
greatly simplified and the numerical solution of the dif-
ferential equations describing the dust evolution becomes
very simple. More refined is the solution adopted by
Calura et al. (2008). In brief, (i) only τ0,iis fixed and τi,accr
remains an explicit function of the ratio σD
ori correlation between τi,accrand τi,snris supposed to ex-
ist, the two timescales are defined independently, and both
plays a role in the σD
ievolution. In our model we adopt the
same strategy.
i/σi; (ii) no a pri-
4.2. The Zhukovska et al. (2008) model or case B
A step forward in modelling grain accretion has been
made by Zhukovska et al. (2008). In brief, Zhukovska et al.
(2008)’s picture can be summarized as follows: (i) grain ac-
cretion is tightly related to molecular clouds and a multi-
phase description of the ISM would be required; (ii) the
evolution of chemical elements is calculated in presence of
some typical dust compounds that are representative of the
grain families growing inside the cold regions of the ISM or
ejected by the stars, namely silicates, carbonaceous grains,
iron grains, and silicon carbide. This model has two impor-
tant advantages. First, it allows for different physical situa-
tions each of which implying different timescales for each el-
ements and different relationships between τi,accrand τi,snr
which are independent; second, accretion in cold regions is
described in a realistic way; third, the model can be eas-
ily incorporated in a multi-phase description of the ISM to
estimate the amounts of gas in the warm and cold phases.
7

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L. Piovan et al.: Formation and evolution of the dust in galaxies
Let us now examine some of the key points and basic
equations of the model and how they are adapted to our de-
scription, where contrary to what made in Zhukovska et al.
(2008) we track the evolution abundances of single elements
in dust instead of ad-hoc families of dust grains.
The growth of the generic type of grains j-th is driven
by the least abundant element indicated as element i-th
(otherwise called the key element) among those forming
the grain. All the other elements concurring to form the
j-th compound adapt their abundances to that of the key
element. For a mixture of accreting grains made of silicates
(pyroxenes and olivines, i.e. magnesium-iron sulfates), car-
bonaceous and iron grains, the key elements are Mg, Si or
Fe for silicates, depending on which has the lowest abun-
dance, C for carbon grains, and Fe for iron grains. Most
of these elements accrete some specific atomic or molecular
species called growth species. The growth in mass of a single
j-th type grain is:
dmj
dt
= Sjαjvgw
jAD
jmH
νgw
i,j
νD
i,j
ngw
j
(26)
where mj is the mass of one grain of the j-th type, Sj is
the surface area, αj is the sticking coefficient, vgw
relative velocity of the accreting species with respect to the
grain, AD
jmH is the mass of one atom (or molecule) that
is accreting on the grain of type of type j-th, ngw
number density of the growth species j, and finally νgw
is the ratio between the number of atoms of the key element
i-th in the growth species for the dust grain of type j-th
and the number of atoms of i-th element we need to build
one grain of the j-th type.
After some substitutions and simplifications for which
all the details can be found in Zhukovska et al. (2008), it
comes out:
dfi,j
dtτgr
j
where fi,j∝ ρD
that is already condensed onto the dust type j-th, called
degree of condensation. 1 − fi,j is therefore the fraction of
the key element i-th still in the gaseous phase and available
to form growth species for the j-th dust component. An
explicit formula for τgr
j
as a function of all the relevant
quantities (Zhukovska et al. 2008, see Eqn. 31) is:
j
is the
j
is the
i,j/νD
i,j
=
1
fi,j(1 − fi,j)(27)
jis the fraction of the key element i-th
τgr
j
= 46(Agw
j)1/2νD
AD
j
i,j
?
ρC
j
3
??103
nH
??3.5 · 10−5
ǫi,j
?
Myr. (28)
Integrating eqn. (27) with the initial conditions fi,j =
f0,i,j for t = 0 we get the final equation describing the
evolution of the condensed fraction:
f0,i,jet/τgr
1 − f0,i,j+ f0,i,jet/τgr
As in real conditions the grain accretion takes place in cold
MCs with a finite lifetime τMC, the above accretion time
scale must be compared with τMC. As already discussed for
model A of accretion, the molecular lifetime constrains the
real time interval during which the exchange of matter be-
tween cold clouds and surrounding medium can happen. It
is the interplay between the accretion/disruption timescales
to determine if and how much material can be condensed
inside the cloud before it dissolves into the surrounding
fi,j(t) =
j
j
. (29)
ISM. If f0,i,jis the initial degree of condensation of the key
element i-th of the grains j-th and the molecular cloud will
be destroyed after a lifetime t we have that the effective
amount of material that is given back to the ISM after the
cloud dissolution is:
MMC
j
(t) = (fi,j(t) − f0,i,j)χD
j,maxMMC
(30)
where fi,j(t)−f0,i,jis the fraction of key element condensed
with respect to the initial condensation fraction, χD
the maximum possible fraction of dust of the kind j-th
that can be formed in the molecular cloud and that would
exist if all the material is condensed in dust of type j-th.
Multiplying by the mass of the cloud we get the mass of
newly formed material MMC
j
fraction χD
j,maxis given by
j,maxis
returned to the ISM. The
χD
j,max=
AD
jǫi,j
(1 + 4ǫHe)νD
i,j
(31)
as shown in the footnote below4.
Not all the MCs have the same lifetime: therefore there
is a certain probability P (t) that a molecular cloud is
destroyed in the time interval t – t + dt. According to
Zhukovska et al. (2008) the probability is well represented
by an exponential law with time scale τMC. The instanta-
neous condensation fraction at the time t of the key element
i-th relative to the dust compound j-th is derived from inte-
grating over time and considering for each cloud of lifetime
x, born at t − x the condensation fraction holding at the
time of birth f0,i,j(t − x). Since the lifetime distribution
function quickly decays over time, only the MCs born some
τMCbefore t will contribute in practice. The instantaneous
condensation fraction is
fi,j(t) =
1
τMC
?t
0
f0,i,j(t − x)ex/τgr
1 − f0,i,j(t − x)(1 − ex/τgr
je−x/τMC
j)dx.(32)
This procedure is however time consuming when applied
in practice. A much simpler approach is provided by using
the average restitution mass RMC
densation fraction fi,jassuming f0,i,jas a constant. Using
Eqn. (30) and integrating between 0 and t = ∞ we obtain
the mean value of fi,j:
j
and the average con-
fi,j=
1
τMC
?∞
0
f0,i,jet/τgr
j
1 − f0,i,j+ f0,i,jet/τgr
j
e−t/τMCdt.(33)
4The number density nH of hydrogen atoms is given by
nH = ρMC/[(1 + 4ǫHe)mH] which in turn follows from the series
of equalities: ρMC = mHnH+4mHnHe = mHnH+4mHnHǫHe =
mHnH(1 + 4ǫHe), where ǫHe = nHe/nH. The fraction χj,d,max
ρD
j,max
ρMC
is given by χD
j,max =
=
AD
jmH(ǫi,jnH)
i,j· mHnH(1 + 4ǫHe)νD
=
AD
jǫi,j
(1 + 4ǫHe)νD
number of atoms ǫi,jnHof the key element available in the molec-
ular cloud by the number of atoms νD
that are tied up when we form one dust grain, we get the max-
imum number of dust grains that we can form per unit volume.
Knowing the mass of a single dust grain, AD
density ρD
i,j
. The explanation is quite simple: dividing the
i,jof the same key element
jmH, the the mass
j,maximmediately follows.
8

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L. Piovan et al.: Formation and evolution of the dust in galaxies
Inserting Eqn. (33) into Eqn. (30) we get Mj,MC, that is
the time averaged mass of dust of type j-th returned to
the ISM. However, this is only part of the story, because
the molecular clouds have also different mass, so we need
to evaluate their mean mass MMC and use this instead of
MMC.
Fractionary mass of dust grains of j-type. With aid of
Eqn. (33) we can derive the fractionary mass of dust of type
j-th produced by the accretion of grains in cold clouds per
unit area and unit time:
?dσD
Looking at Eqn. (34), it requires to know the amount
of molecular gas of the galaxy we are going to model.
To do this, a multi-phase description of the ISM taking
into account the exchange of matter between cold and
warm phases would be required. This is not possible with
the present model because Eqns. (8), (9), and (10) have
been formulated for the single phase description and with-
out a description of the gas exchange between cold and
warm phases. To overcome this limitation some adjustment
of the model are needed. Introducing χMC = σMC/σM
as the fraction of molecular gas and with some passages
(Zhukovska et al. 2008) the final equation for grain growth
in MCs in the frame of a single-phase ISM.
i
dt
?
accr
=
1
τMC
?fi,j− f0,i,j
?· χD
j,maxσMC.(34)
?dσD
We need however to fix χMC: in Zhukovska et al. (2008) is
fixed to 0.2 in agreement to the observations for the SoNe
at the current time. However this approximation could not
probably hold for other phases of the evolution of the SoNe
or other regions of the MW. In Piovan et al. (2011c), by
means of the data available for the MW disk about molec-
ular hydrogen, neutral hydrogen, total amount of gas and
star formation we suitably extended χMC by means of
Artificial Neural Networks (ANNs) in such a way to have
the amount of MCs as a function of the current local vari-
ables describing the system. This recipe will be adopted in
this work. For more details see Piovan et al. (2011c).
Degree of condensation. Another critical parameter to
be examined with accuracy is the condensation degree fi,j.
It depends on the comparison between τMC, the average
lifetime of the MC, and the typical accretion time of grains
τgr
j
of Eqn. (28). If τgr
j
≫ τMC then we expect that only
small amounts of dust are produced because the cold clouds
will be destroyed before that dust has enough time to grow.
On the contrary, if τgr
j
≪ τMC, we expect the condensation
of dust grains to occur almost completely and consequently
dust to increase. Zhukovska et al. (2008) propose an ana-
lytical expression for the condensation degree as a function
of the ratio τgr
j/τMC(assuming a constant f0,i,j):


According to Zhukovska et al. (2008), this expression is no
longer valid only for very small values of f0,i,j. The situation
for our models is more complicated because we are dealing
i
dt
?
accr
=
χMC
(1 − χMC)τMC
?fi,jσD
j,max− σD
j
?. (35)
fi,j=

1
f2
0,i,j
?
1 +τMC
τgr
j
?2+ 1



−1
2
.(36)
01
3
5
7
10
0
0.2
0.4
f0,i,j
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
|fi,j(INT)−fi,j(AN)|
τj,gr [Gyr]
Fig.1. Absolute value of the differences between the fi,j
calculated integrating and averaging on the MCs lifetimes
as in Eqn. (33) and the same quantity derived from the an-
alytical formula of Eqn. (36) suggested by Zhukovska et al.
(2008). A grid of 200×200 suitably discrete values of f0,i,j
and τgr
j
has been considered.
with large variations of f0,i,j and τgr
we checked in advance whether or nor the above relation can
be still used. To this aim we have calculated the differences
between fi,jfrom t Eqn. (36) and the relation of Eqn. (33).
This comparison is made for τgr
τMC/50. If τgr
j
> 1000τMC or τgr
Taylor expansion suggested by Zhukovska et al. (2008) deal
with the extreme cases of very slow or very fast accretion
time.
The results are presented in Fig. 4.2 for a 200×200 grid
of models with uniform spacing of f0,i,j and τgr
linearly from low to high values. The error can be signifi-
cant, in particular for very low f0,i,jand very short τgr
high τgr
j
and average f0,i,j.
As we discussed in Sect. 4.1, the same accretion term in
Dwek (1998) and Calura et al. (2008) is
j. To clarify the issue,
j
< 1000τMC and τgr
< τMC/50 we use the
j
>
j
j
growing
j
or
?dσD
i
dt
?
accr
=
σD
i
τi,accr
=σD
τ0,i
i
·
?
1 −σD
i
σM
i
?
.(37)
The basic difference between the two approaches is that
while Dwek (1998) and Calura et al. (2008) with describes
the evolution of the abundance of the element i-th in the
dust, Zhukovska et al. (2008) follow the evolution of the
abundance of the j-th type dust in a given sample molecules
considered to represent the situation of the ISM. To adapt
the equations to our formulation for single elements (see
Sect. 2), we start from:
σG
i
i= σM
− σD
i
(38)
which summing up all the types of dust j-th in which a
given element is locked up the element i-th is locked (in-
dependently on whether or not it is a key element for that
kind of molecule) becomes:
9

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L. Piovan et al.: Formation and evolution of the dust in galaxies
σG
i= σM
i
−
?
νD
j
νD
i,j
Ai,j
AD
j
σD
j
(39)
with
σD
i=
?
j
i,j
Ai,j
AD
j
σD
j
(40)
where as usual σD
element in dust, while σD
j-th type dust. The above relations says that the amount
of element i-th contained in the dust is given by the sum
of the contributions by all the j-th kinds of dust in which
i is involved. For each one of the j-th types, we divide σD
by the unit mass AD
jmH of one species j-th and find the
number density of dust of type j-th. Then, for each one
of these molecules j-th we have νD
i-th. Finally, we multiply by the atomic mass Ai,jmH of
the element i-th to get the mass of the i-th element coming
from the j-th type dust. If we derive respect to time we
obtain
?dσD
j
where the total evolution of dust is given by the sum of the
variations due to the star formation, the stardust injected
in the ISM, the destruction of the grains by SNa shocks,
and the accretion process in the cold regions. Taking into
account only this last one:
?dσD
j
The amount of element i-th contained in dust is given by
the sum of the contributions by all the jth kinds of dust in
which i is involved. For each j-th type, we divide σD
unit mass AD
jmHof one species j-th of dust and find the
number density of the j-th type dust. For each j-th dust
we have νD
i,jatoms of the element i-th. Finally, we multiply
by the atomic mass Ai,jmHof the element i-th to get the
mass of this coming from the j-th type dust. Inserting the
expression for?dσD
i
dt
accr
j
j
?
Let’s now introduce σD
j,max, which is related to the amount
of key-element k-th in the dust of type j-th present in the
ISM, i.e. σM
element k-th of the j-th dust that is locked up into the
j-th component itself. This quantity is not known and not
tracked by our model. All we know is how much of a given
element is locked up into dust, but we do not know how
much of that element is locked in every kind of dust. For
example we are able to follow how much silicon is locked
up into dust, but not how much silicon is stored in pyroxene,
olivine and quartz. Doing the substitution we have:
?dσD
j
?
Ak,jνD
k,j
i
is the surface mass density of the i-th
jis the surface mass density of the
j
i,jatoms of the element
i
dt
?
TOT
=
?
νD
i,j
Ai,j
AD
j
?
dσD
dt
j
?
TOT
(41)
i
dt
?
accr
=
?
νD
i,j
Ai,j
AD
j
?
dσD
dt
j
?
accr
(42)
jby the
j/dt?
?
accrwe have:
?
?dσD
?
νD
i,jAi,j
AD
=
νD
i,j
Ai,j
AD
dσD
dt
j
?
gg
=
=
j
j
χMC
(1 − χMC)τMC
?fk,jσD
j,max− σD
j
?. (43)
k. We can express σD
j
by means of the key-
i
dt
?
accr
=
?
νD
i,jAi,j
AD
jτMC
χMC
(1 − χMC)·
·
fk,j
AD
j
σM
k,j−
AD
j
Ak,jνD
k,j
σD
k,j
?
(44)
?dσD
i
dt
?
accr
=
χMC
(1 − χMC)τMC
·?fk,jσM
?
j
Ai,jνD
Ak,jνD
i,j
k,j
·
k,j− σD
k,j
?
(45)
Posing K = χMC/(τMC(1 − χMC)), we have:

j,i=k
?dσD
i
dt
?
accr
= K
?
?
?fi,jσM
Ai,jνD
i,j
Ak,jνD
k,j
i,j− σD
i,j
?+
+
j,i?=k
?fk,jσM
k,j− σD
k,j
?


(46)
where k indicates the key element of the dust of type j-th,
because it may occur that a dust molecule contains the el-
ement i-th of which we tracking the evolution, but not the
key element that could be of another type. For example,
we could be following the evolution of carbon and consid-
ering the silicon carbide that contains carbon, but not as
key-element. As a consequence of this, the above has been
split in two parts: in the first one we include the part of the
j-th type dust grain in which the element i-th is locked as
key-element; in the second part we have the dust molecules
containing the i-th element but not as key element, which
instead corresponds to the element k-th. Therefore the sum-
mation goes for all the k ?= i. The notation σD
indicate the surface mass density of the k-th element locked
up into the dust compounds of type j-th.
k,jis meant to
5. Dust accretion rates for a few important
elements
At this stage we need to specify the dust accretion rate
given by Eqn. (46) for some specific elements whose evo-
lution we intend to study. The model follows 16 elements
(Portinari et al. 1998): H,4He,12C,13C,14N,15N,16O,
17O,18O,22Ne,20Ne,24Mg,28Si,32S,40Ca,56Fe.
Four families of dust can be identified and tracked by
the model: they are the silicates, carbonaceous and iron
grains, and silicon carbide (it is worth clarifying that SiC
does not form by accretion in MCs). We must distribute
the 16 elements followed by the model in the four families
of dust. First of all, H,4He,20Ne,22Ne are noble gases
not involved into the dust formation process by accretion.
Second, since the hydrogen is by far the most abundant el-
ement in the Universe, even if it may be contained in many
dust molecules (such as for instance the PAHs), its abun-
dance in the ISM gas will not be significantly affected by
dust. For what concern16O,17O,18O,24Mg,28Si,56Fe,
according to the simplified Zhukovska et al. (2008) scheme,
they are involved in silicates (all of them) and iron grains
(just some of them). For14N,15N,32S and40Ca there is no
specific dust family to associate them. However, the obser-
vations tell us that they are or could be depleted –see for
instance the case of Calcium (Whittet 2003; Tielens 2005)–
, so we must set a plausible yet simple scheme to describe
them. Finally,12C and13C are both involved into the for-
mation of carbonaceous grains and silicon carbide. In the
following we will examine the case of12C,13C,16O,17O,
18O,24Mg,28Si and56Fe in some detail.
10

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L. Piovan et al.: Formation and evolution of the dust in galaxies
5.1. Carbon
Carbon is present in dust as key constituent of many
molecules containing carbon (C-molecules), otherwise
named carbonaceous grains. It is also part of important
dust molecules like silicon carbide (SiC) where, however,
the key element ruling the process is silicon. SiC grains are
mainly related to AGB carbon stars and for them there is
no process of accretion in cold clouds of the ISM. Finally,
the problem of the formation of SiC in the ejecta of super-
novæ is still debated.
Nozawa et al. (2003), studying dust formation in pop-
ulation III supernovæ argues that they cannot form SiC
grains with radii comparable to the pre-solar SiC grains
identified as supernova condensates, because silicon and
carbon are firstly locked into other carbonaceous grains or
silicates before the formation of SiC grains can take place.
However, we observe a small but significant percentage of
X-type SiC grains (one of the minor types A, B, X, Y and
Z, that is believed to be formed in the ejecta of type II
SNæ because of the isotopic signature) with respect to the
mainstream SiC (about 90% of the total and whose ori-
gin is attributed to AGB C-stars, because of their similar
12C/13C ratio, signatures of s-processes and the 11.3 SiC
µm feature in C-stars). The ratio between X-type and main-
stream type can be quantified in about 0.01 (Hoppe et al.
2000). Therefore, as proposed by Deneault et al. (2003),
special conditions and/or chemical pathways might have
to be considered in order to realize the formation in type
II SNæ of SiC grains. To reproduce the observed X-type
over mainstream ratio, Zhukovska et al. (2008) considers a
small formation of SiC in Type II SNæ, however this is just
an ad-hoc recipe based on meteorites and does not throw
light over the uncertain amount of dust, and X-type SiC,
produced by SNæ.
The equation describing the accretion of carbonaceous
grains of the ISM is:
?dσD
C
dt
?
accr
= K?fC,C(1 − ξCO)σM
C,Cis the amount of the available carbon, ξCO a
factor accounting for the carbon embedded in molecules
like CO that do not take part to the accretion process,
σD
C,Cis the carbon already condensed into dust, and fC,C
is the average condensation factor (see Sect. 4 model B).
The accretion of C-molecules in molecular clouds happens
according to a time scale defined by Eqn. 28 which for the
specific case of carbon is:
C,C− σD
C,C
?
(47)
where σM
τgr
C= 46(Agw
C)
1
2νD
AD
C
C,C
?ρD
C
3
??103
nH
??3.5 · 10−5
ǫC,C
?
Myr. (48)
This relation deserves some comments. First of all νD
because carbon is present in the cold regions of molecular
clouds as single atoms or molecules with typically one
atom of carbon. Second, ρD
2001). To determine AD
Cwe should average the masses
of the various type of molecules present in the molecular
clouds. Carbon is embedded in CO (with a percentage eas-
ily up to 20 or 40%). There are also lots of C-molecules like:
CO2,H2CO,HCN,CS,CH4,C2H2,C2H6,HCOOH,OCS,....
We consider AD
C= 28 corresponding to CO as a sort of
mean value fairly representing the above group. Agw
C,C= 1
C= 2.26 g cm−3(Li & Draine
C= 12
assuming single atoms as the typical accreting species.
Finally, we need to specify ǫC,Cthe abundance by number
with respect to H of the key-species C driving the process
of accretion of dusty C-molecules. We have:
ǫC,C= (1 − ξCO)σM
C− σD
12σM
C
H
.(49)
In this relation, the factor (1 − ξCO) takes into account that
fact that the carbon already in CO is not available to the
accretion process. This correction is not negligible. Second,
we subtract from the carbon atoms available for accretion,
those ones already embedded in dust or belonging to un-
reactive species like SiC. Finally, we obtain for the typical
accretion time:
1.80
(1 − ξCO)nH
A similar relation is assumed for13C, in which the only dif-
ference is the 2.03 instead of 1.80. This assumption stands
on the notion that13C, much less abundant than12C, un-
dergoes the same kind of accretion processes and forms
similar C-molecules. Clearly this is a simplified view of the
subject, however sustained by the fact that, on average, the
mean N(12CO)/N(13CO) ratio in diffuse clouds is close or
compatible with the local interstellar isotope ratio12C/13C,
even if the N(12CO)/N(13CO) can be easily enhanced or re-
duced by a factor of two with a large spread (Liszt 2007;
Visser et al. 2009). This allows to think that the relative
percentage of carbon isotopes in molecules and dust should
not be on average too different.
To practically use Eqn. (47), we need f0,C,C to get the
average condensation factor fC,C: f0,C,Cis the initial con-
densation of the key-element C of the carbonaceous dust
compound. If ? χD
f0,C,C≈
χD
C,max
where to derive it, we simply replaced the mass fraction of
dust in the ISM not belonging to cold regions with the mass
fraction embedded in the cold ones: ? χD
ISM dust content always mirrors the initial mass-fraction
already condensed at the formation of the MC.
τgr
C=
?
σM
H
C− σD
σM
C
?
Myr.(50)
jis the mass fraction of dust j-th contained
in the part of the ISM that is not stored in cold clouds, we
have:
? χD
C
≈
χD
C
χD
C,max
≈σD
σM
C,C
C,C
(51)
C≈ χD
C. This simply
means that after every cycle of MCs there is complete resti-
tution and mixing with the ISM, so that on the average the
5.2. Silicon
Silicon is present in many types of grains generically re-
ferred to as silicates, one of the most complicated fami-
lies of minerals. We consider only some kinds of silicates,
commonly used in models of dust formation. First of all,
we list the pyroxenes that are inosilicates (i.e. a family
containing single chain and double chain silicates) in gen-
eral indicated by XY(Si,Al)2O6, where X represents ions
such as calcium, sodium, iron2+and magnesium, whereas
Y represents ions of smaller size such as chromium, alu-
minum, iron+3, magnesium. Aluminum, while commonly
replacing in other silicates, does not often do it in pyrox-
enes. Typical pyroxenes used in dust models are the end
members of the enstatite-hypersthene-ferrosilite series de-
scribed by MgxFe1−xSiO3 with 0 < x < 1 determining
the partition between the two possibilities. Enstatite is the
11

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L. Piovan et al.: Formation and evolution of the dust in galaxies
magnesium end-member MgSiO3of the series, while fer-
rosilite is the iron end-member FeSiO3: both are found in
iron and stony meteorites. Second, we have the olivines.
They are a series of minerals falling in between two end-
members, fayalite and forsterite. They can be described by
[MgxFe1−x]2SiO4with 0 < x < 1 determining the partition.
Fayalite is the iron-rich member with Fe2SiO4. Forsterite is
the magnesium-rich member with Mg2SiO4. The two min-
erals form a series where the iron and magnesium can be
interchanged without much effect, also difficult to detect,
on the crystal structure. Olivine can be found for example
in many meteorites like the iron-nickel ones, in the tails of
comets, and in the disks of dust around young stars. Both
pyroxenes and olivines are expected to exist in the interstel-
lar dust because they form from magnesium and iron that
are abundant. In addition to this and limited to the calcu-
lation of stellar dust yields, we included quartz, the most
common mineral on the surface of the Earth, composed by
SiO2 and silicon carbide, SiC. Silicon carbide in expected
to form in the ejecta of carbon-rich stars and type II SNæ,
but not by accretion in cold clouds.
Two parameters control the pyroxene-olivine mixture:
the Mg fraction x, assumed to be the same for pyroxenes
an olivines, and the abundance ratio between the amount
of magnesium and silicon bound in dust AD
latter determines the fraction of silicates with olivine sto-
ichiometry FOl = AD
we can define FPyr = 1 − FOl. The two parameters are
fixed and chosen according to present day observations of
IR emission and abundances in the diffuse ISM (Dwek et al.
1997; Whittet 2003; Dwek 2005; Zhukovska et al. 2008), i.e.
x = 0.8 and AD
Si= 1.06−1.07. Varying these parame-
ters does not significantly affect the total efficiency of dust
production by the MCs (Zhukovska et al. 2008). For the
purposes of this study, they can be kept fixed. The equa-
tion describing the evolution of silicon is nearly the same
no matter whether or not silicon is the key-element for py-
roxenes and olivines.
The Mg/Fe case. Let us first suppose that the key-
element is magnesium/iron and not silicon. Using Eqn. (46)
we get:
Mg/AD
Si. This
Mg/(AD
Si· x) − 1, while for pyroxenes
Mg/AD
?dσD
Si
dt
?
accr
= KASi,PyrνD
Ai,PyrνD
Si,Pyr
i,Pyr
·?fi,PyrσM
Si,Ol
Ai,OlνD
i,Ol
i,Pyr−
− σD
i,Pyr
?+ KASi,OlνD
i,Ol− σD
i,Ol
?fi,Ol·
· σM
?
(52)
where fi,Pyrand fi,Olare the average condensation frac-
tions of pyroxenes and olivines, obtained according to the
procedure described in Sect. 4. Furthermore, νD
number of silicon atoms needed for one molecule of py-
roxene, i.e. νD
Si,Pyr= 1. Since for each molecule we have
just one Si atom, νD
Si,Pyr· ASi,Pyr = 28. The quantity
Ai,Pyr= 24 or 56 (magnesium or iron). We must also de-
fine νD
i,Pyr(a number between 0 and 1). Assuming a fixed
partition x: νD
we have the physical constants for olivines: νD
ASi,Ol= 28, AMg,Ol= 24, AFe,Ol= 56, νD
nally νD
Fe,Ol= 0.4. Since FPyrand FOlare the fractions of
Si,Pyris the
Mg,Pyr= 0.8 and νD
Fe,Pyr= 0.2. Finally,
Si,Ol= 1,
Mg,Ol= 1.6, fi-
pyroxenes and olivines in which silicates are subdivided, we
have the following partition of magnesium in dust σD
σD
(2x·FOl)/(x·FPyr+2x·FOl)σD
and of iron in silicates (once subtracted the iron embedded
in iron grains) σD
(1 − x) · FPyr/((1 − x) · FPyr+ 2(1 − x) · FOl)σD
2(1 − x) · FOl/((1 − x) · FPyr+ 2(1 − x) · FOl)σD
FePyrσD
It is worth also noting that the dust compounds of which
we follow the accretion, i.e. pyroxenes/olivines for silicates,
carbonaceous grains and iron grains (the same considered
by Zhukovska et al. (2008)), are fewer in number than the
dust molecules considered by the stellar yields of dust in
the most detailed models calculated by CNT theory (see
Piovan et al. 2011a, for more details). The number and va-
riety of the dust molecules injected into the ISM are wider
than the number of accreting species. For this reason, before
splitting σD
Si(see below) between pyrox-
enes/olivines, we need to keep memory of the evolution of
the other dust species injected by AGB/SNæ (also eventu-
ally swept up and destroyed by shocks in the ISM), which
are not included in the accretion scheme.
The Si case. In alternative, silicon can be the key-
element for olivines and pyroxenes:
?dσD
+ K?fSi,OlσM
where again fSi,Pyrand fSi,Olare the average condensa-
tion fractions of silicon in pyroxenes and olivines. The cor-
responding partition of silicon in dust is σD
σD
Si. We need to calculate the accre-
tion time scales of pyroxenes and olivines, to be compared
with the lifetime of molecular clouds and included into
the integrations for fMg,Pyr, fMg,Ol, fSi,Pyrand fSi,Ol.
Applying Eqn. (28) we get:
Mg=
Mg+
Mg,Ol+ σD
Mg,Pyr= (x · FPyr)/(x · FPyr+ 2x · FOl)σD
Mg= MgPyrσD
Mg+MgOlσD
Mg
Fe,Sil= σD
Fe−σD
Fe,Fe= σD
Fe,Ol+σD
Fe,Pyr=
Fe,Sil+
Fe,Sil=
Fe,Sil+ FeOlσD
Fe,Sil.
Mg, σD
Fe,Siland σD
Si
dt
?
accr
= K?fSi,PyrσM
Si,Pyr− σD
Si,Pyr
?
?
Si,Ol− σD
Si,Ol
(53)
Si= σD
Si,Pyr+
Si,Ol= FPyrσD
Si+FOlσD
τgr
Pyr= 46(Agw
Pyr)1/2νD
AD
Pyr
i,Pyr
?
ρD
Pyr
3
??103
nH
??3.5 · 10−5
ǫi,Pyr
?
=
=
τgr
0,Pyr,i
σM
i
σM
H
nH
(54)
τgr
Ol= 46(Agw
Ol)1/2νi,d,Ol
Ad,Ol
σM
H
nH
0,Pyr,iand τgr
the key-element. τgr
the key-elements Si, Mg and Fe respectively, while τgr
equal to 2.05, 2.60 and 2.32 again for Si, Mg and Fe. Agw
and Agw
Pyrare the atomic weight of the growing species.
Finally, Ad,Ol= Ad,Pyr = 121.41. It is important to note
that in the calculation of the abundances of species avail-
able for dust accretion, we do not subtract the amount of
dust already formed σD
this is equivalent to assume that the grains of silicates al-
ready formed act as reactive species in the accretion pro-
cess. Dropping this assumption (and simplification), the
?ρd,Ol
3
??103
nH
??3.5 · 10−5
ǫi,Ol
?
=
=
τgr
0,Ol,i
σM
i
(55)
where τgr
0,Ol,iare two constants that depend on
0,Pyr,iis equal to 2.05, 1.30 and 1.16 for
0,Ol,iis
Ol
ifrom the total ISM abundance σM
i;
12

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L. Piovan et al.: Formation and evolution of the dust in galaxies
accretion process would be too slow. There is one excep-
tion though in the case of iron as the key-element for the
mixture of silicates mixture. In such a case we subtract
from σM
i
the amount of iron already embedded in the iron
dust grains (that do not react with silicates), thus obtaining
σM
i
= σM
The initial condensation fractions f0,i,Pyr, f0,i,Olneeded
to derive fi,Pyr, fi,Olcan be obtained as in Eqn. (51).
Fe− σD
Fe,Fe.
5.3. Oxygen
Oxygen is much more abundant than the refractory ele-
ments Si, Fe, Mg, Ca and S. Therefore, oxygen (at least
the most abundant isotope16O) will never become a key-
element determining the growth of the typical dust grains
in the ISM. The equation describing the evolution of the
mass abundance of the16O is obtained from Eqn. (46),
dropping the terms giving the contribution of the O as key-
element. Therefore, considering the accretion of pyroxenes
and olivines, the temporal evolution of the16O abundance
(we drop the mass number of the isotope) is:
?dσD
O
dt
?
accr
= KAO,PyrνD
Ai,PyrνD
O,Pyr
i,Pyr
?fi,PyrσM
O,Ol
Ai,OlνD
i,Ol
i,Pyr−
− σD
i,Pyr
?+ KAO,OlνD
?
?fi,OlσM
i,Ol−
− σD
i,Ol
(56)
where once again the key-element could be Mg, Si or Fe,
depending on which has the lowest abundance. The sub-
script i indicates a generic key-element. As usual, fi,Pyr
and fi,Olare the average condensation fractions when
the MC is dispersed, and σD
in Sect. 5.2. Introducing the Dirac delta function nota-
tion we have: Ai,Pyr = 28δ(Ai− 28) + 24δ(Ai− 24) +
56δ(Ai− 56) where Ai is the mass number of the key-
element i. Furthermore, AO,Pyr= 16 and, since in pyrox-
enes we have sulfite SiO3with 3 oxygen atoms, νD
νD
i,Pyr= 1δ(Ai− 28)+0.8δ(Ai− 24)+0.2δ(Ai− 56), and
AO,Ol= 16. Similarly, νD
have sulfate SiO4, Ai,Ol= 28δ(Ai− 28) + 24δ(Ai− 24) +
56δ(Ai− 56), and νD
0.4δ(Ai− 56). The accretion time scales τgr
we need for fi,Pyrand fi,Ol, the elemental abundances of
the key-elements and the initial condensation fractions are
the same as in Sect. 5.2.
In this context, we need also a simple description for
17O and18O. First of all let us examine how they behave
in the inert CO molecule. The way in which the isotopes of
O and C combine to form many isotopologues other than
12C16O is very complicated and thoroughly studied, be-
cause of the importance of CO molecule, its easy detection
and chemical stability (van Dishoeck & Black 1988; Liszt
2007; Visser et al. 2009). The photo-dissociation of CO is
a line process and consequently subject to self-shielding
that in turn depends on the column density. Therefore,
isotopologues other than C16O are not self-shielded un-
less located very deeply into the molecular clouds due to
the much lower abundance of17O and18O (Clayton 2003;
Lee et al. 2008; Visser et al. 2009). Looking at the case of
i,Pyrand σD
i,Olare defined as
O,Pyr= 3,
O,Ol= 4 because in the olivines we
i,Ol= 1δ (Ai− 28) + 1.6δ(Ai− 24) +
Pyrand τgr
Olthat
the Sun, where the isotopic ratios are about16O/18O ≈ 500
and16O/17O ≈ 2600 (Lodders et al. 2009; Asplund et al.
2009), the regions with abundances of the isotopologues
different than C16O significantly reduced with respect to
other isotopologues are located very deeply in the atmo-
sphere. Basing on these considerations, we set the reduction
factor of17O and18O, ξ′
CO, equal to 1/3 of the reduction
factor assumed for CO in Eqn. (49). All this agrees with
the observational ratios C16O/C17O and C16O/C18O in ζ
Oph, > 5900 and ≈ 1550 respectively (Savage & Sembach
1996)).
Another point to note is that the ratios16O to17O
and18O in the interstellar dust could be different from the
same ratios in the ISM (Clayton 1988; Leshin et al. 1997).
The subject is a matter of debate, because the above ratios
strongly depend on the site where those isotopes condense
into dust (Meyer 2009). Consequently, we should not simply
scale to17O and18O the results obtained for16O. However,
a detailed analysis of this issue is far beyond the purposes
of this paper and we leave it to future improvements of our
model. For the time being, basing on the following consid-
erations we adopt a simple recipe. As in the dust grains,
17O and18O are much less abundant than other elements,
consider them as the key element of the dust accretion pro-
cess. Therefore, Eqn. (46) applied to17O and18O (briefly
indicated asnO) becomes:
?dσD
(nO)
dt
?
accr
= K
?
f(nO),Pyr(1 − ξ′
CO)σM
(nO),Pyr−
σD
(nO),Pyr
?
?
+ K
?
f(nO),Ol(1 − ξ′
CO)σM
(nO),Ol
−σD
(nO),Ol
(57)
where ξ′
the ISM can be used to form new grains because part of
them are locked in the CO.
The relative amounts of17O and18O available to form
silicates are ǫ(nO),Sil= (1 − ξ′
the accretion timescales for17O and18O are:
COtakes into account that not all17O and18O in
CO)
?
σM
(nO)
?
/(A(nO)σM
H) and
τgr
17O=
0.97
(1 − ξ′
CO)nH
1.00
?σM
?σM
H
σM
17O
?
?
Myr(58)
τgr
18O=
(1 − ξ′
CO)nH
H
σM
18O
Myr (59)
where we have considered a typical silicate in which just
one atom of17O and/or18O is involved in the accretion
process.
5.4. Magnesium
In our simple picture of the the grain accretion process,
magnesium intervenes in pyroxenes and olivines, often as
the key-element. Furthermore, magnesium is present also
in other dust compounds such as MgO contained in the
SNæ yields of dust. In analogy to the case of Si, the set
of equations governing the temporal evolution of the Mg
abundance is different depending on whether or not Mg is
the key, similarly to silicon. If Mg is a key element:
13

Page 14

L. Piovan et al.: Formation and evolution of the dust in galaxies
?
dσD
dt
Mg
?
accr
= K ·
?
fMg,PyrσM
Mg,Pyr− σD
Mg,Pyr
?
+
K ·
?
fMg,OlσM
Mg,Ol− σD
Mg,Ol
?
(60)
with the usual meaning of the symbols. The accretion time
scales and the initial condensation fractions of Mg can be
obtained in the same way as in Sect. 5.2, following Eqns.
(54), (55) and (51). If Mg is not the key-element, but Si or
Fe are playing the role, we have the following equation for
the Mg accretion:
?
dt
accr
?+ KAMg,OlνD
·?fi,OlσM
in which AMg,Pyr= 24, νD
(silicon/iron), νD
νD
The time scales for accretion of pyroxenes/olivines and the
initial condensation fractions are again from Eqns. (54),
(55) and (51). For σD
dσD
Mg
?
= K ·AMg,PyrνD
Ai,PyrνD
Mg,Pyr
i,Pyr
?fi,PyrσM
Mg,Ol
·
i,Pyr
−σD
i,Pyr
Ai,OlνD
?
i,Ol
i,Ol− σD
i,Ol
(61)
Mg,Pyr= 0.8, Ai,Pyr= 28 or 56
Si,Pyr= 1, νD
Mg,Ol= 1.6, ASi,Ol = 28, νD
Fe,Pyr= 0.2, AMg,Ol = 48,
Si,Ol= 1 and νD
Fe,Ol= 0.4.
i,Pyrand σD
i,Olsee Sect. 5.2.
5.5. Iron
Even considering the small number of accreting compounds
included in our model, iron can be locked up in grains of
various type thanks to accretion processes in cold regions
of the ISM. First of all, iron is locked up in iron grains
that act as the key element. Second, iron is also present in
pyroxene and olivine grains. In such a case, iron may or may
not be the key element. In any case it is removed from the
gaseous phase and stored in grains. The existence of two
channels for locking up iron into grains leads to a rather
complicated equation for the evolution of the iron. There
are the main terms to consider: in the first and second,
iron participates to the formation of silicates (pyroxenes
and olivines) as or not a key-element, and in the third one
iron plays the role of a key-element for the formation of iron
grains. Furthermore, the equation will be slightly different
depending on whether magnesium, silicon or iron is the key-
element for the formation of pyroxenes/olivines. The final
equation is
?dσD
Fe
dt
?
accr
= KAFe,PyrνD
Ai,PyrνD
?+ KAFe,OlνD
− σD
i,Ol
−?σD
Fe,Pyr
i,Pyr
?fi,PyrσM
Fe,Ol
Ai,OlνD
i,Ol
?σM
Fe,Sil
i,Pyr−
−σD
i,Pyr
?fi,OlσM
Fe− σD
i,Ol−
?+ K?fFe,Fe
Fe,Sil
?−
Fe− σD
??
(62)
where the subscript i stands for Si, Mg or Fe itself.
Using the Dirac delta function notation, we have Ai,Pyr=
28·δ (Ai− 28)+24·δ (Ai− 24)+56·δ (Ai− 56). The same
expression holds good for Ai,Ol. We have also: AFe,Pyr =
AFe,Ol = 56, νD
with x = 0.8), νD
i,Pyr= 1·δ (Ai− 28)+0.8·δ(Ai− 24)+0.2·
δ (Ai− 56), νD
with x = 0.8) and finally νD
i,Ol= 1 · δ(Ai,Ol− 28) + 1.6 ·
δ (Ai,Ol− 24) + 0.4 · δ (Ai,Ol− 56). Also, with the usual
meaning, fi,Pyr, fi,Oland fFe,Feare the average conden-
sations before cloud dispersion.
For internal consistency, the system of equations must
take into account that the iron abundance used to describe
the evolution of the iron grains should be corrected for the
iron already condensed into pyroxenes and olivines, sub-
tracting it from σM
Fe. If the key-element is magne-
sium, the amount of iron σD
Fe,Silembedded into pyroxenes
and olivines is given by:
Fe,Pyr= 0.2 (according to MgxFe1−xSiO3
Fe,Ol= 0.4 (according to [MgxFe1−x]2SiO4
Feand σD
σD
Fe,Sil=σD
MgAFe
24
?
νD
Fe,Ol
νD
Mg,Ol
MgOl+νD
Fe,Pyr
νD
Mg,Pyr
MgPyr
?
(63)
where AFeis the atomic weight of the iron and MgOland
MgPyrare defined in Sect. 5.2. Doing the correct substitu-
tion we get
σD
Fe,Sil= σD
Mg(0.583· MgOl+ 0.583 · MgPyr)(64)
In a similar way, if the role of key element is played by
silicon we have:
σD
Fe,Sil= σD
Si(0.8 · FOl+ 0.4 · FPyr)(65)
where FOland FPyrhave been defined in Sect. 5.2. If iron
is the key element for silicates, we simply have:
σD
Fe,Sil= σD
Fe− σD
Fe,Fe
(66)
This clearly requires to keep track of the accre-
tion/destruction and injection of the iron grains by SNæ
and AGB stars. The accretion time scales for pyrox-
enes/olivines are the same as in Eqns. (54) and (55),
whereas for iron grains:
τgr
Fe= 46(Agw
?3.5 · 10−5
Fe)1/2νD
AD
Fe,Fe
Fe
?
?ρFe
D
3
??103
σH
nH
?
·
·
ǫFe,Fe
=31.56
nH
?
σM
Fe− σD
Fe,Sil
?Myr.(67)
where νD
the accreting elements we consider the simplest case in
which they are in form of atoms. The initial condensation
fractions for magnesium/silicon as key-elements in pyrox-
enes/olivines follow Eqns. (??) and (??), while for iron we
have:
Fe,Fe= 1 and ρD
Fe= 7.86gcm−3. Concerning
f0,Fe,Fe=
χD
Fe
χD
Fe,max
=σD
σM
Fe− σD
Fe− σD
Fe,Sil
Fe,Sil
. (68)
where we subtract the amount of iron already locked up in
silicates and therefore not available to for iron grains.
14

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L. Piovan et al.: Formation and evolution of the dust in galaxies
5.6. Calcium
This element has a complicate behaviour difficult to fol-
low. First of all it is usually heavily depleted in the ISM
(Whittet 2003; Tielens 2005), and its abundance in the solar
system is quite low. The difficult arises from the total con-
densation efficiency and the big fluctuations generated by
the low abundance, compared to other refractory elements.
Second there is not an average molecule that could be used
to represent typical calcium grains in a simple theoretical
description. Furthermore, the measurements of depletion
cannot be easily derived from observational data: in prin-
ciple the ionization equilibrium equation should be solved
to derive the Ca abundance from the observations of Ca II.
Possible estimates of Ca abundances by electron densities
and strengths of the ionizing radiation fields are not easy
and relying on ratios of ionization and recombination rates
between different elements (like CaII to those of NaI or KI
for instance) is a cumbersome affair (Weingartner & Draine
2001). For all these reasons, Jenkins (2009, and 2011, pri-
vate communication) leaved Ca (and also Na and K) aside.
Trying to overcome this difficulty, we simplify the problem
as follows. Thanks to its low abundance, we consider Ca as
the key-element of the associated grains of dust; the equa-
tion for the evolution of Ca is:
?dσD
where all the symbols have their usual meaning. To derive
the accretion time scale τgr
Cawe do not refer to a typical Ca
dust grain but simply take the shortest timescale among
those of refractory elements that are most depleted, i.e.
Mg, Si and Fe (τgr
the formation of dust silicates-like molecules with one cal-
cium atom as key atom. This last-named timescale is mul-
tiplied for a correction factor CaX, eventually allowing for
a fast accretion. Therefore τgr
τgr
Ca,Ca}. The initial condensation fraction at the MC for-
mation is as usual fCa,Ca= σD
Ca
dt
?
accr
= K?fCa,CaσM
Ca(Ca) − σD
Ca(Ca)?
(69)
Si,τgr
Mg,τgr
Fe), and a timescale τgr
Ca,Cafor
Ca= min{τgr
Si,τgr
Mg,τgr
Fe,CaX·
Ca/σM
Ca.
5.7. Sulfur
Sulfur is a very important element: it is often used as a
reference case of nearly zero depletion in studies of local
and distant objects (Jenkins 2009). However the real de-
pletion efficiency of this element is a matter of debate and
the assumption of nearly zero depletion cannot be safe.
Calura et al. (2009) reviews data and theoretical interpre-
tations gathered over the past years to convincingly show
that S can be depleted in considerable amounts. Jenkins
(2009) points out the depletion of sulfur can be significant
along some lines of sight. We take Jenkins (2009) results
into account to set upper and lower limits to the sulfur
depletion. Considering sulfur as a refractory element, its
evolution in dust is given by
?dσD
where all the symbols have their usual meaning. Current
observations do allow us to choose a grain as representative
of the accretion process. In analogy to what made for cal-
cium, the accretion time τS,gris supposed to be the longest
between the timescales of the refractory elements, i.e. Mg,
S
dt
?
accr
= K?fS,SσM
S(S) − σD
S(S)?
(70)
Si and Fe, and a timescale τgr
silicates-like molecules with one sulfur atom as key atom,
allowing therefore for a slow accretion. We also introduce
a multiplying scaling factor SX to eventually correct this
timescale. Therefore τgr
S
= SX· max{τgr
The initial condensation fraction for fS,Sis fS,S= σD
S,Sfor the formation of dust
Si,τgr
Mg,τgr
Fe,τgr
S/σM
S,S}.
S.
5.8. Nitrogen
Nitrogen is known to be poorly depleted and if not depleted
at all (Whittet 2003; Tielens 2005). Jenkins (2009) suggests
that depletion is independent from the line of sight deple-
tion strength factor and that in any case depletion is very
low, thus confirming the poor ability of nitrogen to con-
dense into dust grains. As nitrogen is included in our list of
elements (both14N and15N), the evolution of both isotopes
in dust is governed by:
?dσD
N
dt
?
accr
= K
?
fN,NσM
N (N) − σD
N(N)
?
(71)
where for the accretion time we simply take the longest
between C and O and the accretion timescale with one ni-
trogen atom as growing species for a nitrogen molecule.
Carbon and oxygen are the nearest elements, both do not
show a strong depletion and, finally, similarly to nitrogen
both indicate (at least along some lines of sight) low values
of depletion. Taking the longest timescale, we implicitly as-
sume that nitrogen very slowly accretes onto dust. As usual
fN,N= σD
N/σM
N.
6. Supernovae: destruction time scales
Dust grains in the ISM can be destroyed by other physical
process such as the passage of shock waves by supernovæ
explosions. The destruction time scale of the element i-h in
dust grains in the ISM because of the shocks by local SNæ
is defined as the ratio between the available amount of that
element locked up into dust (the surface mass density of the
element i-th σD
i(r,t), at a given radius and evolutionary
time) and the rate at which grains containing that element
are destroyed refueling the gaseous phase :
τSNR,i= σD
i/
?dσD
i
dt
?
SNR
(72)
as usual, we drop the dependence from r and t. The de-
struction time scale can be expressed as follows:
?dσD
where Mdestr
i
= Mdestr
i
(r,t) is the amount of mass de-
stroyed by a single SNa event, while RSN = RSN(r,t) is
the global rate of SNæ, obtained by adding together the
rate of Type I and II SNæ. Multiplying numerator and de-
nominator by σD
i
and combining together Eqns. (72) and
(73) it follows:
i
dt
?
SNR
= Mdestr
i
· RSN
(73)
τSNR,i= σD
i/
?dσD
i
dt
?
SNR
=
1
RSN
σD
i
Mdestr
i
. (74)
The amount of mass destroyed by the interstellar shocks
can be defined as the amount of mass swept by the SNa
shock, multiplied by the fraction of dust mixed with the
ISM (in this way we get the total swept up mass of dust)
15