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arXiv:1303.5586v2 [astro-ph.CO] 18 Jun 2013
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 11 January 2014 (MN L
A
T
EX style file v2.2)
Star Formation and Metallicity Gradients in Semi-analytic
Models of Disk Galaxy Formation
Jian Fu1,2⋆, Guinevere Kauffmann1, Mei-ling Huang1, Robert M. Yates1,
Sean Moran3, Timothy M. Heckman4, Romeel Dav´e5, Qi Guo6,7,
Bruno M. B. Henriques1
1Max-Planck-Institut f¨ur Astrophysik, D-85741 Garching, Germany
2Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, CAS,
80 Nandan Rd., Shanghai, 200030, China
3Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
4Department of Physics and Astronomy, The Johns Hopkins University, MD21218, Baltimore, USA
5Astronomy Department, University of Arizona, AZ85721, Tucson, USA
6Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, UK
7National Astronomical Observatories, CAS, Beijing 100012, China
11 January 2014
ABSTRACT
We have updated our radially-resolved semi-analytic models of galaxy formation,
which track both the atomic and molecular gas phases of the interstellar medium. The
models are adapted from those of Guo et al. (2011) using similar methodology as in
Fu et al. (2010) and are run on halo merger trees from the Millennium and Millennium
II simulations with the following main changes: (1) We adopt a simple star formation
law ΣSFR ∝ΣH2. (2) We inject the heavy elements produced by supernovae directly
into the halo hot gas, instead of first mixing them with the cold gas in the disk. (3) We
include radial gas inflows in disks using a model of the form vin flow =αr. The models
are used to study the radial profiles of star formation rate and gas-phase metallicity in
present-day galaxies. The surface density profiles of molecular gas in L∗galaxies place
strong constraints on inflow velocities, favouring models where vinfl ow ∼7 km/s at a
galactocentric radius of 10 kpc. Radial gas inflow has little influence on gas-phase and
stellar metallicity gradients, which are affected much more strongly by the fraction of
metals that are directly injected into the halo gas, rather than mixed with the cold gas.
Metals ejected out of the galaxy in early epochs result in late infall of pre-enriched gas
and flatter present-day gas-phase metallicity gradients. A prescription in which 80%
of the metals are injected into the halo gas results in good fits to the flat observed
metallicity gradients in galaxies with stellar masses greater than 1010M⊙, as well
as the relations between gas-phase metallicity and specific star formation rate in the
outer parts of galactic disks. We examine the correlation between gas-phase metallicity
gradient and global galaxy properties, finding that it is most strongly correlated with
the bulge-to-total (B /T ) ratio of the galaxy. This is because gas is consumed when
the bulge forms during galaxy mergers, and the gas-phase metallicity gradient is then
set by newly-accreted gas.
Key words: galaxies: evolution - galaxies: formation - stars: formation - galaxies:
ISM - ISM: atoms - ISM: molecules
1 INTRODUCTION
In recent years, observations of radially resolved profiles of
atomic gas, molecular gas and star formation in represen-
⋆E-mail: fujian@mpa-garching.mpg.de
tative samples nearby galaxies (e.g THINGS for HI pro-
files, Walter et al. 2008; HERACLES for H2profiles, Leroy
et al. 2009) have motivated galaxy formation theorists to
include the physics of the atomic-to-molecular gas transi-
tion in their models. Some of these models make predic-
tions for the global atomic and molecular gas content of
2Fu et al.
galaxies (e.g. Obreschkow et al. 2009; Lagos et al. 2011),
while others attempt to model the radial structure of the
gas in more detail (e.g. Robertson & Kravtsov 2008; Fu et
al. 2010; Power, Baugh & Lacey 2010; Feldmann, Hernandez
& Gnedin 2012). Radial abundance gradients also place im-
portant constraints on disk formation models, particularly
on how supernova feedback (SN feedback) processes eject
metals into the surrounding gas and how these metals are
mixed throughout the disk as the galaxy evolves. A success-
ful model of disk galaxy formation should be able to repro-
duce the metallicity gradients in galaxies along with radial
density profiles of old stars, young stars and gas.
Observational data on metal abundance gradients in
galaxies remains rather sparse. Many studies focus only on
one galaxy or on a handful of galaxies (e.g Rudolph et
al. 2006 for the Milky Way; Magrini et al. 2007 for M33;
Bresolin et al. 2009 for M83). For many years, the Zaritsky,
Kennicutt & Huchra (1994) study of gas-phase metallicity
gradients derived for 159 HII regions in 14 spiral galaxies
in combination with published data for another 25 galax-
ies, has remained the standard reference in the field. More
recently, Moustakas et al. (2010) published metallicity gra-
dients for 65 galaxies from the SINGS survey and Moran
et al. (2012; hereafter Moran12) analyzed gradients for 174
galaxies with atomic and molecular gas mass measurements
from the GASS and COLD GASS surveys (Catinella et al.
2010; Saintonge et al. 2011a). These new observations make
it possible to carry out a statistical comparison with model
predictions for the first time.
There is a long history of models of the radial structure
and properties of disk galaxies in the literature, beginning
with Tinsley & Larson (1978) who implemented a model
for chemical evolution into the dynamical collapse calcula-
tions for gas clouds with rotation and axial symmetry in-
troduced by Larson (1976). Similar modelling efforts were
later undertaken by many others (e.g. Matteucci & Fran-
cois 1989; Kauffmann 1996; Chiappini, Matteucci & Grat-
ton 1997; Dalcanton, Spergel & Summers 1997; Avila-Reese,
Firmani, & Hern´andez 1998; van den Bosch 1998; Prantzos
1999; Dutton et al. 2007; Fu et al. 2009; Yin et al. 2009;
Cook et al. 2010; Fu et al. 2010). The models can be ar-
ranged in increasing order of complexity, from those that
use simple parameterized formulae to describe the infall of
gas onto the disk, to those that are embedded within high
resolution N-body simulations of structure formation in a
ΛCDM cosmology and where gas infall rates are determined
by explicitly following the growth of each dark matter halo
in the simulation and then calculating the rate at which gas
will cool as a function of time.
In order to model the radial distributions of stars and
gas in the disk, processes such as star formation, SN feedback
and chemical enrichment need to be treated in a radially re-
solved fashion. In previous work (Fu et al. 2010, hereafter
Fu10), we developed models in which each galaxy disk was
divided into a series of concentric rings. We adopted two
different prescriptions to partition the cold gas into atomic
and molecular components: one is adapted from the models
of Krumholz, McKee & Tumlinson (2009), in which the H2
fraction is parameterized as a function of local cold gas sur-
face density and gas-phase metallicity. The other prescrip-
tion is empirically-based and proposes that the H2fraction is
related to the interstellar pressure (Elmegreen 1989 & 1993;
Blitz & Rosolowsky 2004 & 2006). The Fu10 semi-analytic
model is based on the version of the L-Galaxies code de-
scribed in De Lucia & Blaizot (2007) (hereafter DLB07)
implemented on the halo merger trees of the Millennium
Simulation outputs (Springel et al. 2005). As discussed in
the paper, the model reproduces the radial profiles of stellar
mass, HI, H2, and star formation in L∗galaxies. However,
the model produces a stellar mass function that is too steep
at the faint end and the gas fractions of low mass galaxies
do not agree with observations, which is a problem inherit
from DLB07 model.
The analysis presented in this paper is based on the new
version of the L-Galaxies code described in Guo et al. (2011)
(hereafter Guo11), which runs on the halo merger trees of
both the Millennium Simulation (Springel et al. 2005) and
on the 125 times higher resolution Millennium II Simula-
tion (Boylan-Kolchin et al. 2009). This allows us to study
the formation and evolution of galaxies with masses rang-
ing from those of dwarfs to the most massive cD galaxies.
Based on the new models, the main purpose of this paper is
to study the radial profiles of the gas-phase metallicity and
star formation rate surface density of disk galaxies at z= 0
and compare them with the recent observational results, es-
pecially those from Moran12.
This paper is organized as follows. In Section 2, we
briefly describe the N-body simulation and the L-Galaxies
semi-analytic models, and then we will give the changes to
the semi-analytic model with respect to Fu10 and Guo11.
We discuss how we normalize the free parameters in our
models using gas profile data from Leroy et al. (2008) for
a small sample of nearby disks, along with the stellar and
gas mass functions of nearby galaxies at z= 0. In Section
3, we present stellar, HI and H2mass functions at z= 0
and compare them to observations. In Section 4, we ana-
lyze how the star formation rate surface density profiles of
galaxies depend on stellar mass. In Section 5, we study how
the radial profiles of gas-phase metallicity depend on stellar
mass, bulge-to-total ratio, stellar surface density and specific
star formation rate. We also look at the relations between
metallicity and star formation in the inner and outer disks of
galaxies. The model results presented in Sections 4 and 5 are
compared to observational results from Moran12. Finally, in
Section 6, we summarize our results and discuss avenues for
future work.
2 THE N-BODY SIMULATION AND
SEMI-ANALYTIC MODELS
In this section, we will briefly introduce the Millennium
and Millennium II simulations, the physical processes in L-
Galaxies semi-analytic galaxy formation models, the meth-
ods of treating the radial profiles and the conversion of
atomic to molecular gas in the semi-analytic model frame-
work.
2.1 The N-body simulations
The L-Galaxies semi-analytic models of galaxy formation
are run on two very large N-Body simulations: the Millen-
nium Simulation (hereafter MS, Springel et al. 2005) and
Millennium-II Simulation (hereafter MS-II, Boylan-Kolchin
Radial Gradients in SAMs of Disk Galaxy Formation 3
et al. 2009). Both simulations adopt a ΛCDM cosmogony
with parameters ΩΛ= 0.75,Ωm= 0.25,Ωbaryon =
0.045, σ8= 0.9 and h= 0.73, and they track 21603≈1010
particles from z= 127 to z= 0. The main difference between
the two simulations is the resolution. The particle mass in
MS is 8.6×108M⊙h−1and the periodic box is 500 Mpc h−1
on a side; the particle mass in MS-II is 6.8×106M⊙h−1and
the size of the periodic box is 100 Mpc h−1, which means
the resolution of MS-II is 125 times higher than MS. Since
the smallest halo traced in both simulations has 20 parti-
cles, MS can be used to study the formation and evolution
for Milky Way-sized or larger galaxies, while MS-II is more
appropriate to study dwarf galaxies. For our models of the
cold gas components in galaxies, the MS is suitable to study
atomic and molecular gas for disk galaxies with stellar mass
M∗&1010M⊙and MS-II is useful to study the gas compo-
nents in present-day dwarf galaxies and in the high-redshift
progenitors of present-day L∗galaxies. Another difference
between the two simulations is the number of output snap-
shots: the last 60 snapshots for MS and MS-II are identical,
but MS-II has four more snapshots at very high redshift to
increase the time resolution in early-forming first structures.
2.2 The galaxy formation models
The L-Galaxies is the semi-analytic model of galaxy for-
mation developed by Munich group (Kauffmann et al. 1999;
Springel et al. 2001; Croton et al. 2006; DLB07; Guo11; Guo
et al. 2013). The model tracks various physical processes on
the halo merger trees of MS and MS-II: re-ionization, gas
infall and cooling, star formation and metal production, SN
feedback, ram-pressure stripping of gas in satellite galaxies,
tidal disruption of satellites, galaxy mergers, bulge forma-
tion, black hole growth and AGN feedback.
The L-Galaxies model tracks four phases: stars, inter-
stellar cold gas, hot gaseous haloes (hot gas) and ejecta
reservoirs (ejected gas). In Fig. 1, we illustrate how mass
is exchanged between different phases when different phys-
ical processes operate. The exchange of metals follows the
exchange of the gas. All metals are fully mixed with the gas
in one timestep. In Appendix A, we briefly summarize the
physics and the equations of the processes in Fig. 1, and
the descriptions of bulge formation and mergers will be de-
scribed in Sec. 2.3.2. The reader is also referred to Section 3
of Guo11 for a more detailed discussion of how the physical
processes in the model are calculated.
The cosmological parameters for both MS and MS-II
and the semi-analytic models in Guo11 and previous papers
are from WMAP1 (Wilkinson Microwave Anisotropy Probe
first-year results, Spergel et al. 2003). To update the cosmol-
ogy, Guo et al. (2013, hereafter Guo13) adopt the technique
described in Angulo & White (2010) to rescale the growth
of structure to be appropriate for WMAP7 parameters
(ΩΛ= 0.728,Ωm= 0.272,Ωbaryon = 0.0454, σ8= 0.807
and h= 0.704; Komatsu et al. 2011). In this paper, we make
use of these rescaled models, but we note that differences for
the quantities explored in this paper are negligible compared
to systematic uncertainties in our treatment of the various
gas-physical processes.
The main changes made in this paper with respect to
Guo11 are the following: (1) We include the methodology to
track gas and stellar components in radial rings described in
Cold Gas
Stars
Hot Gas
Ejected Gas
Star
Formation Recycling
Cooling SN
Reheating
Re-
incorporation SN Ejection
Figure 1. Illustration of how mass is exchanged between different
phases in model galaxies in response to the physical processes
treated by the model.
Fu10. (2) We include the prescriptions for the transition be-
tween atomic gas and molecular gas described in Fu10. (3)
We abandon the two-regime star formation model in Fu10
and adopt a simpler prescription where the star formation
surface density is always proportional to the molecular gas
surface density (e.g. Leroy et al. 2008; Bigiel et al. 2008;
Schruba et al. 2011). (4) We allow a fraction of heavy ele-
ments to be mixed directly with the hot gas in the halo. We
describe these modifications in more detail below.
2.3 Treatment of the radial distribution of gas
and star in disks
In the “standard” L-Galaxies recipes (DLB07 & Guo11),
each galaxy disk is treated as a single zone. However, these
single zone recipes are not suitable to study the transition
of atomic gas to molecular gas, since molecular clouds form
where the local density is high enough for molecules to be
shielded from the surrounding ionizing radiation. Single zone
models are also unable to predict metallicity gradients. The
Fu10 models are based on the DLB07 models, except that
we divide each model galaxy disk into a set of radial con-
centric rings to study the radial distribution of star and gas
components on galaxy disk. We trace the physical processes
related to the galaxy disks in each ring in each timestep.
Thus, our model can predict the surface density profiles of
stars, gas and star formation rate, and also the radial profiles
of the stellar and gas-phase metallicity.
In Fu10, we divide each galaxy disk into 30 rings, and
the radius of the rings is given by a geometric series. In
this paper, we divide each disk into 12 rings instead of 30
rings. This is done to decrease the computational memory
requirements based on MS-II halo merger trees. The ring
radii are given by the geometric series
ri= 0.44 ×1.5i[h−1kpc] (i = 1,2...12) (1)
The radius for the innermost ring in Eq. (1) is about 0.9 kpc,
and the radius for the outermost ring is about 80 kpc. As we
mentioned in Fu10, the radial profiles are insensitive to the
precise scheme, if the adopted number of rings is sufficiently
large. This sub-division of the disk in Eq. (1) is sufficient to
study both small and large galaxies.
4Fu et al.
2.3.1 The radial distribution of gas cooling onto the
galaxy disks and inside-out disk growth
To model the radial distribution of stars and gas in the
disk of the galaxy, we adopt the same prescription de-
scribed in Fu10, in which the surface density profile of
newly infalling cold gas at each timestep has exponential
form with a uniform metallicity (see Fig. 13 and the discus-
sion in last section on the pre-enriched gas accretion). The
scale length of the exponential infalling profile is given by
rinfall =rd=λ/√2rvir, and this equation assumes that
angular momentum is conserved during gas cooling and in-
fall (Mo, Mao & White 1998). In our simple scheme, the
profile of the newly infalling gas is directly superposed onto
the pre-existing gas profile from the previous timestep.
Since the disk and halo size is smaller and compact at
high redshift, the scale length of the infalling gas is smaller
at higher redshift. The size of the galaxy disk grows with
time (see Figure 1 in Fu10). This is the so-called “inside-
out” disk growth paradigm, that has been incorporated in
many disk formation models (e.g Kauffmann et al. 1996;
Dalcanton, Spergel, & Summers 1997; Avila-Reese, Firmani,
& Hern´andez 1998; Dutton 2009; Fu et al. 2009; Pilkington
et al. 2012).
2.3.2 The radial distribution of bulge formation and
galaxy mergers
The DLB07 model does not include a prescription for bulge
sizes and the Fu10 model did not attempt to calculate the
radial distribution of bulge stars. The Guo11 models include
a method for calculating the sizes of bulges produced during
galaxy mergers and by disc instabilities.
The size of the bulge after a merger is calculated using
the following equation
Cm2
new
rnew
=Cm2
1
r1
+Cm2
2
r2
+αm1m2
r1+r2
(2)
where Cparametrizes the binding energy of the galaxy and
αparametrizes the effective interaction energy deposited in
the stellar components. In Guo11, C= 0.5 and α= 0.5 are
adopted for galaxy mergers. mnew and rnew in Eq. (2) are
the mass and half mass radius of the newly formed bulge.
For major mergers, in which the ratio of the total (stel-
lar+gas) mass of the satellite to the central galaxy is larger
than 0.3, both disks are destroyed. The stars in both pro-
genitor galaxies and the stars formed in the merger-induced
starburst become bulge stars. In this case, m1is sum of
the stellar mass of the two progenitor galaxies and m2is the
sum of the stellar mass converted from cold gas in the major
merger starburst (see later); r1and r2are the corresponding
half-mass radii.
For a minor merger (msat/mcen <0.3), the stellar com-
ponent of the smaller satellite galaxy is added to the bulge
of the central galaxy, while the cold gas component of the
satellite is added to the disk of the central galaxy. m1in
Eq. (2) is the bulge mass of the central galaxy before the
merger and m2is the stellar mass (bulge star+disk star)
of the satellite galaxy; r1and r2are the corresponding half
mass radii.
The main new prescription implemented in this paper
concerns the radial profile of gas added to the central galaxy
from the satellite before the starburst occurs. The Fu10 code
directly superposes the gas radial profiles of two gas disks
together and then processes the starburst. In this paper, we
treat the accreted gas from the satellite in the same way as
gas cooling from the halo; the scale length of the accreted
gas is determined from the spin parameter and virial radius
of the halo of the central galaxy using the equation rinfall =
(λ/√2)rvir.
The DLB07 and Guo11 codes use the “collisional star-
burst model” introduced by Somerville, Primack & Faber
(2001). In both major and minor mergers, the mass of stars
formed in the starburst is m∗=eburst mgas, with eburst given
by the equation
eburst =βburst (msat/mcen )αburst (3)
with αburst = 0.7 and βburst = 0.56.
In our model, we simply calculate the surface mass den-
sity of stars formed in the burst in ring number ias
Σ∗,i =eburstΣgas,i (4)
using Eq. (3) to define eburst . For starbursts occurring dur-
ing major mergers, the stars formed in all the rings become
bulge stars.
This is admittedly not a very realistic description of
what happens in an actual merger-induced starburst, but
since we mainly study quiescent disk galaxies in this paper,
this will not concern us. The other update we make with
respect to the Guo11 models during mergers is to calculate
the half mass radii directly from the stellar and gas profiles,
rather than to assume an exponential form.
Bulges also form through secular evolution. When
vmax <pGm∗,disk/r∗,disk , the disk is considered unstable.
A mass δm∗from the inner part of the disk is transferred to
the bulge
δm∗=m∗,disk −r∗,disk v2
max
G(5)
where m∗,disk and r∗,disk are the mass and scale length of the
stellar disk. If this material forms a new bulge, it is assumed
to have a half-mass radius δr equal to the outer radius of
the region. If there is already a bulge in the galaxy, then the
resulting bulge is calculated using
0.5m2
new
rnew
= 0.5m2
1
r1
+ 0.5m2
2
r2
+ 2.0m1m2
r1+r2
(6)
where m1and r1are the mass and half mass radius of the
existing bulge; m2=δm∗and r2=δr are the mass and
radius of the transferred region. Once again we have up-
dated this prescription by calculating the radius δr directly
from the stellar radial distribution of the disk rather than
by assuming an exponential stellar disk profile as in Guo11.
As in Guo11, we assume a Jaffe profile (Jaffe 1983) for
the bulge of the form
ρbulge (r) = mbulge
4πr3
br
rb−21 + r
rb−2
(7)
in which mbulge is the bulge mass and rbis the half mass
radius of the galaxy bulge. The stellar mass in a given radial
ring is the combined mass of disk and bulge stars in that
Radial Gradients in SAMs of Disk Galaxy Formation 5
ring:
m∗=m∗,disk + 4πZrout
rin
ρbulge (r)r2dr
=m∗,disk +mbulge "1 + rin
rb−1
−1 + rout
rb−1#(8)
where m∗,disk is the stellar mass of the disk component in
the ring, and rin ,rout are the inner and outer radii of the
radial ring. We note that the stellar surface density profiles
presented in the next sections include both bulge and disk
components.
2.3.3 Radial gas inflows in disks
In Fu10, we showed that a star formation law of form
ΣSFR ∝ΣH2was problematic in that it led to cold gas sur-
face density profiles that were too shallow to match obser-
vations (see Sec. 3.4 & Fig. 3 in Fu10). In that paper, we
introduced the following two tweaks to solve the problem:
(i) we adopted a star formation law in regions of the disk
with fH2<0.5 of the form ΣSFR ∝Σ2
gas. (ii) we assumed
that the SN reheating efficiency was inversely proportional
to gas surface density (∆mreheat ∝∆m∗/Σgas). The second
assumption increases the gas consumption timescale in the
inner disk.
In this paper, we will introduce a radial gas inflow pre-
scription as a more realistic solution, i.e the gas from the
outer disk flows inwards towards the inner disk. As we will
now show, a plausible inflow prescription combined with a
simpler star formation law in which the star formation rate
surface density is always proportional to the molecular gas
surface density, produces gas surface density profiles that
match observations without need for a radially-dependent
SN feedback efficiency.
There are a number of physical mechanisms that can
drive radial inflows of gas in the disk. The ones that are in-
voked most frequently involve gravitational interaction be-
tween gas in the disk and non-axisymmetric stellar struc-
tures such as bars and spiral structures (Kalnajs 1972). Sim-
ple physical considerations yield estimates of flow velocities
ranging from 0.1 to a few km/s (Lacey & Fall 1985; Bertin &
Lin 1996). Attempts to measure radial flow rates in galaxies
have been confined to very small samples and have yielded
inconclusive results (e.g. Wong, Blitz & Bosma 2004; Haan
et al. 2009; Zhang & Buta 2012). Difficulties arise from the
fact that flow patterns in individual galaxies are often irreg-
ular and disks are frequently not axisymmetric. In addition,
different methods for estimating mass inflow rates yield dis-
crepant results.
As a result, modellers generally resort to simple param-
eterized inflow prescriptions (e.g Lacey & Fall 1985; Porti-
nari & Chiosi 2000; Sch¨onrich & Binney 2009; Spitoni &
Matteucci 2011). In this work, we have tested a number of
inflow prescriptions. We first tried the simplest prescription
in which the radial gas inflow velocity is constant for the
whole disk (model (a) in Lacey & Fall 1985), but we found
that the inflow velocities required to move enough gas from
the outer disk to the inner disk to compensate the depletion
by star formation, lead to an unacceptably large pile-up of
gas in the very inner disk. A prescription in which the rate
of change of the angular momentum is proportional to the
0 5 10 15
0
2
4
6
8
10
12
r[kpc]
vinflow [km/s]
Figure 2. The relation between galactocentric radius rand radial
gas inflow velocity vinflow from Eq. (10) (αv= 0.7 km s−1kpc−1
is adopted).
angular momentum yields results that agree best with ob-
servational data, i.e
dLgas
dt =CLgas (9)
where Cis a constant for all galaxies. Because Lgas =
mgasrgas vcir, we have that
vinflow =αvr(10)
The radial inflow velocity of the gas is thus proportional
to the galactocentric radius r. This is equivalent to model
(b) in Lacey & Fall (1985). Spitoni & Matteucci (2011) also
invoke a radial flow prescription of this form to reproduce
the observed stellar metallicity gradient in the Milky Way.
The constant αvin Eq. (10) is an adjustable model pa-
rameter. As discussed later, αv= 0.7 km s−1kpc−1is re-
quired to fit observations. This results in negligible flow ve-
locities in the inner disk and flow speeds of several km s−1
in the outer disk (see Fig. 2).
In the models, we calculate the gas inflow velocity ac-
cording to Eq. (10) after processing gas cooling and infall in
each timestep and derive the new radius r′=r−vinflow ∆tfor
the gas in each ring, where ∆tis the length of the timestep.
Proceeding from the inner ring to the outer ring, we move
the gas to its new radius; metals associated with the cold
gas move inward together with the gas.
In Fig. 3, we show the best-fit radial surface density
profiles of stars, atomic gas, molecular gas, and total cold
(atomic+molecular) gas for disk galaxies in haloes with
circular velocities comparable to that of the Milky Way
(200 < vcirc <235 km/s) at z= 0. The four panels on
the left show the results if radial gas inflows are not in-
cluded, and the four right panels show the results of our
inflow model. Without inflows, both the molecular gas and
average total cold gas radial surface density profiles are too
flat in comparison to the data of Leroy et al. (2008). The
right panels show that the inclusion of radial flows leads to
gas profiles that are in much better agreement with observa-
tions. We note that the stellar mass profiles are insensitive
to the radial flow prescription; this is because the majority
of the mass in the inner galaxy forms at high redshifts from
gas cooling in smaller, denser haloes. The gas radial profiles
are set by gas that cools at late times. Because the timescale
over which H2is transformed into stars is ∼2 Gyr (see Sec.
2.5), the H2surface densities in the inner disk quickly drop
to low values as a result of star formation and SN feedback,
unless gas is transferred from the outer disk to the inner
regions of the galaxy.
6Fu et al.
101
102
103
Σstar[M⊙/pc2]
100
101
102
ΣHI[M⊙/pc2]
0 5 10 15
100
101
102
r[kpc]
ΣH2[M⊙/pc2]
0 5 10 15 20
100
101
102
r[kpc]
Σgas[M⊙/pc2]
101
102
103
Σstar[M⊙/pc2]
100
101
102
ΣHI[M⊙/pc2]
0 5 10 15
100
101
102
r[kpc]
ΣH2[M⊙/pc2]
0 5 10 15 20
100
101
102
r[kpc]
Σgas[M⊙/pc2]
Figure 3. The radial surface density profiles of stars, atomic gas, molecular gas, and total cold gas for disk galaxies similar to the Milky
Way. Results from the model with H2fraction prescription 1 (see Sec. 2.4) are shown at at z= 0. The two panels show the model results
with (right panel) and without (left panel) radial gas inflow. The light blue curves with error bars are taken from the data of Leroy et
al. (2008) for disk galaxies in the range of circular velocities 200kms−1< vcir <235kms−1(NGC 0628, NGC 3184, NGC 5194, NGC
3521). The red solid and dotted curves are the mean and the median values from model results, and the black dashed curves are the ±1σ
deviations about the mean values.
r[kpc]
log10[Zgas /Z⊙]
0 5 10 15
−0.5
0
0.5
0 5 10 15
−0.5
0
0.5
r[kpc]
log10[Zstar /Z⊙]
Figure 4. The mean radial metallicity profile of gas (left panel) and stars (right panel) for a Milky Way-sized galaxy at z= 0 from
models with (green curve) and without (blue curve) radial gas inflow. In the left panel, the black dots are the oxygen metallicity from HII
regions in the Milky Way (Rudolph et al. 2006) and black dashed line is the corresponding linear fit to the data between 0 and 15 kpc. The
gray area shows the ±1σdeviation around the median radial profile of oxygen metallicity for galaxies with 10.5<log10[M∗/M⊙]<11.0
from Moran12. In the right panel, the black dots represent the stellar oxygen gradients from Cepheids (Andrievsky et al. 2002a,b,c;2004;
Luck et al. 2003) and the red dots with error bars are corresponding mean values.
We note that we have tuned the parameters of the ra-
dial inflow model (in particular the constant αvin Eq. 10)
so that the gas radial profiles in Milky Way-type disks are
reproduced. One might ask whether metallicity profiles pro-
vide an additional check on the inflow model. In Fig. 4, we
show the mean radial stellar metallicity profile (right panel)
and gas-phase metallicity profile (left panel) for the same
set of Milky Way sized galaxies at z= 0 (disk galaxies with
200 < vcirc <235 km/s). Green curves show results with ra-
dial inflow and blue curves show models without radial gas
inflow. In the right panel, the black dots represent the stel-
lar oxygen measurements gradients derived from Cepheids
by Andrievsky et al. (2002a,b,c;2004) and Luck et al. (2003).
Red dots with error bars are the corresponding mean values.
In the left panel, the black dots represent oxygen metallic-
ity measurements from HII regions in the Milky Way by
Rudolph et al. (2006) and the black dashed line is the corre-
sponding linear fit to the data between 0 and 15 kpc, which
exhibits a -0.031 dex/kpc for gas-phase metallicity gradi-
ent between 0 and 15 kpc. The metallicity gradient of the
galaxies in our models is a bit shallower than found for the
data (-0.018 dex/kpc for models without radial gas inflow
and -0.02 dex/kpc for models with inflow), because the pa-
rameters of the models were tuned to fit the average gas-
phase metallicity profiles of Milky Way mass galaxies from
Moran12, which are flatter than the Milky Way disk (see
Sec. 5 for detail). The gray area in the left panel shows the
±1σdeviation around the median radial profiles of oxygen
metallicity for galaxies with 10.5<log10[M∗/M⊙]<11.0
from Moran12.
We see from Fig. 4 that inclusion of radial gas inflow
yields gas and stellar metallicity gradients that are slightly
Radial Gradients in SAMs of Disk Galaxy Formation 7
steeper in the region of the disk interior to 3 kpc, but oth-
erwise the predicted metallicity gradients are virtually in-
distinguishable in the two cases. In our model, infalling gas
from the halo has already been pre-enriched by the ejection
of metals during the early formation phase of the galaxy, and
the outer disk metallicity is mainly affected by the metal-
licity of the gas accreted recently because of inside-out disk
growth. On the other hand, very small inflow velocities in
the inner disk only slightly change the inner disk metallici-
ties. These two aspects mean that the radial gas inflow has
only slight influence on the radial metallicity gradients. As
we will discuss in Sec. 2.6, the fraction of the metals pro-
duced by SNe we choose to put into the hot gas rather than
the cold interstellar medium has a much stronger influence
on metallicity gradients than the radial gas inflow.
2.4 Prescriptions for the transition between
atomic and molecular gas in the ISM
In Fu10, we implemented two prescriptions for the transi-
tion between atomic and molecular gas in the interstellar
medium. One is from Krumholz et al. (2009) (H2fraction
prescription 1), who calculate an equilibrium H2fraction
(fH2= ΣH2/(ΣH2+ ΣHI)) for a spherical cloud with given
dust content surrounded by a photo-dissociating UV field.
In this prescription, the H2fraction is primarily a function
of local cold gas surface density and metallicity. The sec-
ond prescription (H2fraction prescription 2) originates from
Elmegreen (1989, 1993) and Blitz & Rosolowsky (2006), in
which the fH2is a function of the pressure in the ISM. We
adopt the approximation by Obreschkow & Rawlings (2009),
in which the local ISM pressure is expressed as a function
of gas and stellar surface density in the radial ring. More
details can be found in Section 3.2 of Fu10.
In this paper, we make a number of adaptations to the
Krumholz et al. prescription. We update the prescription
using the recent fitting equations in McKee & Krumholz
(2010) with the molecular gas fraction fH2given by
fH2= 1 −3
4
s
1 + 0.25s(11)
for s < 2 and fH2= 0 for s>2. The sin Eq. (11) is defined
as
s=ln 1 + 0.6χ+ 0.01χ2
0.6τc
(12)
in which χ= 3.11 + 3.1Z′0.365/4.1 and τc=
0.66 Σcomp/M⊙pc−2Z′. Note that Z′=Zgas /Z⊙is the
gas-phase metallicity relative to the solar value, and Σcomp
is the gas surface density of the giant gas cloud. Since the
gas surface density in the model is the azimuthally averaged
value in each concentric ring, a clumping factor cfis intro-
duced to consider the difference between Σcomp and Σgas
Σcomp =cfΣgas (13)
One problem with the Krumholz et al. prescription is
that it can easily yield non-convergent results. The reason
for this is that the value of fH2is very sensitive to the ex-
act value of the gas-phase metallicity when the metallic-
ity is well below the solar value (see Figure 5 in McKee &
Krumholz 2010). Molecular clouds do not form until metals
have been produced, and metal production is dependent on
the formation of stars, which can only take place in molec-
ular clouds, so the conditions for star formation to occur in
the first generation of haloes to form in the simulation are
subject to considerable uncertainty. Moreover, at the resolu-
tion limit of the simulation, the assembly histories of haloes,
and hence the star formation histories of galaxies embedded
within them, cannot be tracked accurately. It thus becomes
very difficult to get results that are consistent for galaxies
forming in haloes of the same mass in simulations of differ-
ent resolutions (e.g. MS and MS-II). Similar problems were
reported by Kuhlen et al. (2012) in their attempts to model
dwarf galaxies with Krumholz et al. prescription.
In the top panels of Fig. 5, we plot the relation between
the gas-phase metallicity of a central galaxy and the mass
of its parent halo at different redshifts. The red curves and
green curves show results for the MS and MS-II simulations,
respectively. We have adopted cf= 1.5 for all galaxies at all
redshifts, as in Fu10. As can be seen, at fixed halo mass, the
gas-phase metallicity is higher in the MS-II run than in the
MS run, particularly for low mass haloes at high redshifts.
Since the halo resolution of MS-II is 125 times higher than
MS, a larger fraction of the formation and chemical enrich-
ment history of low mass haloes lies below the resolution
limit for MS than MS-II. Because the Krumholz et al. pre-
scription is so sensitive at low metallicities, the resolution
discrepancies are enhanced for high redshift galaxies in low
mass haloes.
The question arises as to why stars are able to form at
all in low-metallicity galaxies. For example, 1ZW18 is the
most metal-poor galaxy known, yet it is a starburst system
(e.g. Martin 1996). The main reason is that the gas in low-
metallicity dwarf galaxies is clumpy on scales of ∼100 pc
(Lo, Sargent & Young 1993; Stil & Israel 2002). To account
for this, Kuhlen et al. (2012) adopt a clumping factor cf= 30
for their models of dwarf galaxies at high redshift; Luo et
al. (2011) adopt cf= 20 to model high-redshift damped
Lyman-alpha systems.
In this paper, we adopt a variable clumping factor that
depends on gas-phase metallicity. We use the following equa-
tion to describe the relation between gas-phase metallicity
and clumping factor
cf=Z′−0.7(14)
for 0.01 < Z′<1 and cf= 1 for Z′>1. The clumping
factor for log10 Z′>−1 is between 1 and 5, which agrees
with the values suggested for normal galaxies in KMT09.
In the four bottom panels of Fig. 5, we plot central
galaxy gas-phase metallicity as a function of halo mass for
models that adopt a variable cf. Interestingly, the differ-
ence between MS-I and MS-II results is largely eliminated
for galaxies in haloes with larger than 1011M⊙, because the
higher clumping factor compensates the low molecular gas
surface densities in dwarf galaxies and high redshift galax-
ies with low gas-phase metallicities. Of course, this does not
mean that our clumping factor prescription is correct! It just
means that we are able to match together the results from
the two simulations with greater ease.
2.5 Star formation prescription
DLB07 and Guo11 both adopt the following star forma-
tion law which relates the star formation rate surface den-
8Fu et al.
10 11 12 13
−2
−1
0
log10[Zgas /Z⊙]
z~3
10 11 12 13
z~2
10 11 12 13
z~1
10 11 12 13
z~0
10 11 12 13
−2
−1
0
log10[Mhalo /M⊙]
log10[Zgas /Z⊙]
10 11 12 13
log10[Mhalo /M⊙]
10 11 12 13
log10[Mhalo /M⊙]
10 11 12 13
log10[Mhalo /M⊙]
Figure 5. The relation between gas-phase metallicity and galaxy halo mass at redshifts 3, 2, 1, and 0. The red curves and green curves
are the mean values derived for the MS and the MS-II simulations, respectively. The four top panels are from the Krumholz et al. H2
prescription with fixed clumping factor cf= 1.5, and the four bottom panels are from the Krumholz et al. H2prescription with variable
clumping factor dependent on gas-phase metallicity (see text).
sity to the cold gas surface density of the ISM, ˙m∗=
α(mgas −mcrit)/tdyn (see Eq. (A5) in Appendix). In Fu10,
we adopt a two-regime star formation law related to both
molecular and atomic gas phases
ΣSFR =(αΣH2(fH2>0.5)
α′Σ2
gas (fH2<0.5) (15)
The star formation law in atomic gas dominant regions helps
solve the problem of gas consumption times being too slow
in outer disks.
Since we now include radial gas inflow process in Sec.
2.3.3 to solve the problem of gas consumption timescales in
both inner and outer disks, we now abandon the complex
two-regime star formation form Eq. (15) in the model. We
adopt a simple law in which the star formation surface den-
sity is proportional to the H2surface density: ΣSFR ∝ΣH2
(e.g Leroy et al. 2008; Bigiel, Leroy & Walter 2011; Schruba
et al. 2011; Leroy et al. 2013). We adopt a constant molec-
ular gas consumption timescale for all galaxies at all red-
shifts. There is evidence that molecular gas consumption
times may be longer in massive, early-type galaxies than
in normal, present-day star-forming spirals (Saintonge et al.
2011b) and shorter in starburst galaxies (Genzel et al. 2010),
but we will neglect these complications, because it is still un-
clear what the underlying physical cause of these effects are.
The star formation law adopted in the model can be
simply written as
ΣSFR =αH2ΣH2(16)
in which the H2star formation efficiency αH2= 5.3×
10−10yr−1is slightly adapted to yield a better fit to the
results stellar mass function at z= 0. We note that the
molecular gas surface density ΣH2in Eq. (16) does not in-
clude the helium component.
2.6 The gas-phase radial abundance gradients and
the mixing of metals
In the models of DLB07, Guo11 and Fu10, metals produced
by star formation in the disk are mixed instantaneously with
the cold gas in the disk (see the sixth paragraph in Ap-
pendix). SN reheating results in part of the cold gas (along
with the metals that have been mixed into it) being trans-
ferred to the hot gas in the halo. If the SN explosion energy
ESN is larger than the energy required to reheat the cold
gas Ereheated, the remaining energy ESN −Ereheated will eject
part of the halo hot gas out of the halo: this corresponds to
the so-called “ejected component” (see Fig. 1, and Eq. (A7),
(A8), (A9)).
In the models, the gas-phase metallicity radial gradient
is mainly determined by two processes. One is the fraction
of metals from star formation and SN explosions that re-
main in the interstellar cold gas, which mainly determines
the gas-phase metallicity in the inner disk. The other one
is the metallicity of recent infalling gas pre-enriched by the
SN ejecta in early epochs, which mainly determines the gas-
phase metallicity in outer disk because of the inside-out disk
growth (e.g Pilkington et al. 2012). This is why the fraction
of metals produced by young stars that is injected into the
hot gaseous halo can affect the slope of the metallicity gra-
dient.
On the other hand, we note that there is direct observa-
tional evidence from deep X-ray spectral imaging of nearby
star-forming disk galaxies that the hot gas produced by su-
pernovae type II (SNe-II) is highly metal-enriched (Martin,
Kobulnicky & Heckman 2002) and also enhanced in alpha
elements (Strickland et al. 2004). Even in “normal” spirals,
the gas is directly observed at scale heights 4-8 kpc above the
galactic disk, before its emissivity decreases below the detec-
tion threshold (Strickland et al. 2004). The scenario favoured
by current observations is that SN feedback in the disks of
Radial Gradients in SAMs of Disk Galaxy Formation 9
star-forming galaxies create exponential atmospheres of hot
gas via blow-out and venting of hot gas from the disk.
We have experimented with mixing a fraction fz,hot of
the metals produced by star formation directly with the hot
gas in the halo, i.e a fraction of the metal from the star is
directly recycled to the hot gas in Fig. 1. This turns out to
have very strong impact on the predicted gas-phase metal-
licity gradients in the galaxy. In Fig. 6, we plot the mean
radial gas-phase metallicity gradients for disk galaxies with
stellar mass 1010.0< M∗/M⊙<1010.5at z= 0. The blue
curve shows results for fz,hot = 0.0, which is effectively the
DLB07 and Guo11 mixing prescription, in which none of
the metals are directly mixed into the hot gas. This yields a
steep gradient, with more than 0.6 dex change in metallicity
in the inner 15 kpc of the galaxy. The green curve shows
what happens if we adopt fz,hot = 1.0, i.e. all metals are
mixed with the hot gas. In this case, the metallicity gra-
dient nearly disappears, because we assume the hot gas in
the halo is fully mixed and the infalling gas onto the disk
has a uniform metallicity at different radii. In this paper,
we assume that metals from AGB star winds are retained
in the interstellar cold gas and metals from core-collapse SN
explosions are mixed into hot gaseous halo. According to the
yield data compiled by Marigo et al. (2001) for AGB stars
and Portinari, Chiosi & Bressan (1998) for SNe-II, about
20 percent of the total metal yield comes from AGB stars
of low and intermediate masses when we adopt a Kroupa
IMF between 0.8M⊙and 120 M⊙.1The red curve in Fig.
6 shows the case where metals from AGB star outflows are
retained in the cold ISM and metals from core-collapse SN
explosions are mixed into hot gas (fz,hot = 1 −0.2 = 0.8).
In Sec. 5, we will compare the model gas-phase metal-
licity gradients to observations and also explore how the pre-
dicted metallicity profiles depend on galaxy properties such
as mass and gas mass fraction. We will show that fz,hot = 0.8
gives reasonably good agreement with observations.
2.7 The parameters in the models
As discussed in Fu10, we tune some of the model parameters
to fit the observed stellar, HI and H2mass functions at z= 0.
In the top part of Tab. 1, we list the model parameters which
are changed in this paper with respect to those listed in
previous papers. The value of H2star formation efficiency
αH2, the amplitude of SN reheating efficiency ǫ, and the
quiescent hot gas black hole accretion rate κAGN (see Eq.
(A2)) are tuned to fit the amplitudes of both the stellar
and the gas mass functions. The normalization of the radial
1In our current model, we adopt instantaneous recycling approx-
imation for the metal production and chemical evolution, and we
only consider the metal yield from AGB stars and SNe-II. The
mass of metal elements ejected from AGB stars is about 0.1M⊙
per star and about 10M⊙per star from SNe-II (see Figure 2 and 4
in Yates et al. 2013 for the yield data from Marigo 2001 and Porti-
nari et al. 1998). We adopt a Kroupa IMF and assume that stars
with masses in the range 0.8< M/M⊙<8 end their lives as AGB
stars, and that stars with masses in the range 8 < M/M⊙<120
explode as SNe-II. This results in an estimate of 20% of the metal
yield from AGB stars and 80% from SNe-II. Our future work will
include more detailed and accurate prescriptions for chemical evo-
lution, similar to Yates et al. (2013).
0 5 10 15
−0.2
−0.1
0
0.1
0.2
0.3
0.4
r [kpc]
log10[Zgas/Z⊙]
fz,hot = 1.0
fz,hot = 0.8
fz,hot = 0.0
Figure 6. The radial gas-phase metallicity gradient for disk
galaxies with stellar mass 1010.0< M∗/M⊙<1010.5from the
model results at z= 0. The 3 different colour curves represent
the model results with fz,hot = 0.0,0.8,1.0, respectively.
gas inflow rate αvin Sec. 2.3.3 is tuned to reproduce the
gas surface density profiles of Milky Way-type galaxies and
to fit the observed HI and H2mass functions. The fraction
of metals directly mixed with halo hot gas is chosen to be
0.8. Other parameters are not changed in previous papers
(bottom section of Tab. 1): fb,fBH,Tmerger,Yare the same
as in Table 1 in Croton et al. (2006) (they remain unchanged
in DLB07, Fu10, Guo11 and Guo13); Ris same as in DLB07
(it remains unchanged in Guo11 & Guo13); β1,Vreheat ,η,
β2,Veject are same as in Table 2 in Guo13; γis the same
as in Table 1 in Guo11 (it remains unchanged in Guo13); ξ,
P0,αPare the same as in Table 1 of Fu10.
3 STELLAR AND GAS MASS FUNCTIONS AT
Z= 0
The Fu10 models use the SN feedback prescriptions in
DLB07, resulting in stellar, HI and H2mass functions that
are too steep at low masses (see Figure 1 in Fu et al. 2012).
In this paper, we use the prescriptions in Guo11, which pro-
vide a better fit to the low-mass end of the stellar mass
function.
In Fig. 7, we show the mass functions of stars, molecular
gas and atomic gas for our revised models. Red solid curves
show results for H2fraction prescription 1 and green solid
curves for H2fraction prescription 2. We have computed the
mass functions using the combined results from running the
code on the MS and MS-II simulations. Blue circles show
the stellar mass function derived from the Data Release 7
of SDSS by Li & White (2009); black diamonds show SDSS
Data Release 4 results from Baldry et al. (2008). The ob-
served H2mass function was derived from the FCRAO sur-
vey (Young et al. 1995) assuming a constant CO-H2con-
version factor by Keres, Yun & Young (2003). The HI mass
functions are from Zwaan et al. (2005) (blue circles) and
Martin et al. (2010) (black diamonds).
As mentioned in Sec. 2.7, we tune the model parame-
ters to fit the HI, H2and stellar mass functions, so it is no
surprise that the agreement with models is very good.
4 STAR FORMATION GRADIENTS
In this section, we compare the star formation rate gradients
in our model galaxies with observations. This provides an
10 Fu et al.
Table 1. The model parameters introduced/changed in this paper (top section) and other parameters whose values are the same as in
previous papers (bottom section). Note that the parameters marked as “Croton et al. (2006)” remain unchanged in DLB07, Fu10, Guo11
and Guo13.
Parameter Value Description Remark
αv0.7 km s−1kpc−1the ratio of radial gas inflow and gas radius Eq. (10)
αH25.3×10−10yr−1H2star formation efficiency Eq. (16)
ǫ5.0 Amplitude of SN reheating efficiency Tab. 1 in Guo11 & Tab. 2 in Guo13
κAGN 1.5×10−5M⊙yr−1Quiescent hot gas black hole accretion rate Eq. (A2)
fz,hot 0.8 Fraction of metal elements from quiescent star Sec. 2.6
formation directly mixed with halo hot gas
fb0.17 Cosmic baryon fraction Croton et al. (2006)
fBH 0.03 Merger cold gas BH accretion fraction Croton et al. (2006)
Tmerger 0.3 Major merger mass ratio threshold Croton et al. (2006)
Y0.03 Yield of metals produced per unit star formation Croton et al. (2006)
R0.43 Instantaneous recycled fraction of star DLB07, Guo11 & Guo13
formation to the cold gas
β13.2 Slope of SN reheating efficiency Guo13
Vreheat 80 km s−1Normalization of SN reheating efficiency Guo13
η0.18 Amplitude of SN ejection efficiency Guo13
β23.2 Slope of SN ejection efficiency Guo13
Veject 90 km s−1Normalization of SN ejection efficiency Guo13
γ0.3 Ejecta reincorporation efficiency Guo11 & Guo13
ξ1.3 Warm-phase correction factor Fu10
P0,αP5.93 ×10−13 Pa, 0.92 Constant and index of the relation Fu10
between molecular ratio and ISM pressure
7 8 9 10 11
−6
−5
−4
−3
−2
−1
0
log10[M∗/M⊙h−2]
log10[Φ∗/Mp c3h−3]
7 8 9 10
−6
−5
−4
−3
−2
−1
0
log10[MH2/M⊙h−2]
log10[ΦH2/Mp c3h−3]
Keres et al. 2003
8 9 10 11
−5
−4
−3
−2
−1
0
log10[MHI /M⊙h−2]
log10[ΦHI /Mpc3h−3]
Martin et al. 2010
Zwaan et al. 2005
Li et al. 2009
Baldry et al. 2008
Figure 7. The stellar, H2and HI mass functions from model galaxies at z= 0 compared with the observations. The red solid curves
and green solid curves in each panel are from the models with H2fraction prescription 1 and 2, respectively. The observed stellar mass
functions are from Li & White (2009) and from Baldry et al. (2008). The observed H2mass function is from the FCRAO CO survey
by Keres et al. (2003). The observed HI mass function is from Zwaan et al. (2005) using HIPASS data, and Martin et al. (2010) using
ALFALFA data.
important test of whether the present-day growth of disks
predicted by the models agrees with the data.
We have adopted two observational samples for the
analysis. Leroy et al. (2008) analyzed star formation gra-
dients in a sample of 23 nearby spiral galaxies from the
THINGS/HERACLES survey, finding that the scale length
of the star formation rate surface density profile lSFR is
roughly proportional to the scale length of the stellar disk l∗
(lSFR = (1±0.2)l∗). We also make use of star formation rate
surface density measurements from the MMT long-slit ob-
servations of 174 galaxies from Moran12. 119 of these galax-
ies (i.e. around 70% of the total sample) have emission lines
strong enough to measure both SFR and gas-phase metal-
licity gradients out to 2r50.
In Fig. 8, we plot the star formation rate surface den-
sity as a function of scaled radius for model galaxies in
four different stellar mass bins: 9.0<log10 [M∗/M⊙]<9.5,
9.5<log10[M∗/M⊙]<10.0, 10.0<log10[M∗/M⊙]<10.5,
and 10.5<log10[M∗/M⊙]<11.0. For the model sample, rd
is calculated by fitting exponentials to the stellar profiles of
the disk. The red solid and green dashed curves show results
of mean radial profiles for H2fraction prescriptions 1 and
2, respectively, and the error bars on the red solid curves
represent the ±1σscatter between different model galaxies.
In each panel, black curves with error bars are star for-
mation surface density profiles from Leroy et al. (2008); rd
for each galaxy is taken from Table 4 of that paper. The light
blue dots are from Moran12, who obtained long-slit spectra
Radial Gradients in SAMs of Disk Galaxy Formation 11
012345
100
10−1
10−2
10−3
10−4
10−5
ΣSFR[M⊙]yr−1kpc−2]
9.0<log10[M/M⊙]<9.5
0 2 4 6
9.5<log10[M/M⊙]<10.0
r/rd
ΣSFR[M⊙]yr−1kpc−2]
10.0<log10[M/M⊙]<10.5
012345
100
10−1
10−2
10−3
10−4
10−4
10−5
r/rd
10.5<log10[M/M⊙]<11.0
0 2 4 6
Figure 8. The radial profiles of star formation surface density for model galaxies at z= 0 compared with observations from Leroy
et al. (2008) (black curves with error bars) and Moran12 (blue dots). The gray areas represent the ±1σdeviations around the median
values for the Moran12 data. The red solid curves show model results of mean radial profiles for H2prescription 1 and the green dashed
curves are for H2fraction prescription 2. The error bars on the red solid curves represent ±1σscatter about the mean values for the
model. The panels show results in 4 stellar mass bins: 109.0< M∗/M⊙<109.5, 109.5< M∗/M⊙<1010.0, 1010.0< M∗/M⊙<1010.5,
1010.5< M∗/M⊙<1011.0. The radii plotted on the x-axis are scaled by dividing by the disk scale length rd.
of 174 star-forming galaxies with stellar masses greater than
1010M⊙from the GALEX Arecibo Sloan Digital Sky Sur-
vey (GASS) survey (Catinella et al. 2010). In this case, rd
is estimated from r90 assuming an exponential profile (i.e.
r90 = 3.9rd). Note that we have simply plotted the star for-
mation rate surface density measured for each spectral bin
along the slit, i.e. these data points are not averaged in ra-
dial bins as for the Leroy et al. data, so the scatter from
one galaxy to another will be substantially larger. The grey
shaded region shows the 1σscatter around the median for
the Moran12 sample. As can be seen, the agreement between
the Moran et al. and the Leroy et al. data is quite good.
Our results for H2fraction prescription 1 agree well with
the data, particularly for the 3 highest stellar mass bins. H2
fraction prescription 2 yields higher star formation rate sur-
face densities in the outer disks. This is because the pressure-
based prescription yields higher H2fractions in regions of
low gas surface density with low gas-phase metallicities.
We note that we tuned the radial inflow prescriptions
to match the H2surface density profiles of the galaxies in
the Leroy et al. sample for stellar masses log10[M∗/M⊙]∼
10.6. Our star formation prescription is also motivated
by results obtained for this sample. This means that star
formation rate surface density profiles for galaxies with
log10[M∗/M⊙]∼10.6 will match the data, essentially by
construction. The main check in this section is to test
whether we can match to SFR profiles over a large range
in stellar mass. One significant conclusion that we reach is
that H2fraction prescription 1 does a better job than H2
prescription 2 at matching the outer gas profiles of galaxies
with stellar masses less than 1010M⊙.
5 GAS-PHASE METALLICITY GRADIENTS
IN GALAXIES
In this section, we will compare the gas-phase metallicity
gradients predicted by the model with data from Moran12.
We will first study how the gradient depends on the stel-
lar mass of the galaxy, and then examine its dependence
on the global gas fraction. When we compare the model
with data from Moran12, we choose galaxies that have HI
and H2gas mass fractions greater than ∼1−3%, such
that HI and CO lines would be detected in the GASS
and COLD GASS surveys: i.e. log10 [MHI /M∗]>−1.82 for
galaxies with log10[M∗/M⊙]>10.3, and log10[MHI /M∗]>
−1.066 log10[M∗/M⊙] + 9.16 for galaxies with 10.0<
log10[M∗/M⊙]<10.3 (Kauffmann et al. 2012). This is a
slightly more stringent cut than adopted by Moran12 for
his emission-line analysis, but we have verified that the pre-
cise location of the cut makes negligible difference to all the
results presented in this paper.
We adopt the gas-phase oxygen abundance based on
the O3N2 empirical index described by Pettini & Pagel
(2004), which is derived from the N [II] λ6583/Hαand
12 Fu et al.
O [III] λ5007/Hβemission line ratios (see Section 3.1 of
Moran12 for more details).
In the models, we only track total metallicity; the value
of Zgas/Z⊙represents the cold gas-phase metallicity in units
of the solar value. To make a comparison to oxygen abun-
dance from observations, we adopt 12 + log10(O/H) = 8.69
as the solar value (Asplund et al. 2009). We convert Zgas to
(O/H)gas using the equation
12 + log10(O/H)gas = log10 [Zgas /Z⊙] + 8.69 (17)
5.1 Gas-phase metallicity gradients as a function
of stellar mass
In Fig. 9, we plot mean gas-phase metallicity profiles for
model galaxies in 3 stellar mass bins: 109.5< M∗/M⊙<
1010.0, 1010.0< M∗/M⊙<1010.5, and 1010.5< M∗/M⊙<
1011.0. The top two panels show results for H2fraction pre-
scription 1 and the bottom two panels are for H2fraction
prescription 2. In the left panels, metallicity is plotted as a
function of radius in kpc, and in the right panels it is plot-
ted as a function of radius scaled by r90, the radius enclosing
90% of the total stellar mass of the galaxy. The main result
is that the gas radial metallicity profiles are flatter in higher
mass galaxies; this is seen both in the left and in the right
panels, where the radius has been scaled by the size of the
galaxy.
We have investigated the cause of varying metallicity
gradients in the models by examining how the metallicity
gradient in a central galaxy of a given halo evolves over
time. We have found that the evolution of the gas-phase
metallicity gradient is most closely tied to the merger his-
tory of the galaxy. The starbursts induced by the mergers
consume all the cold gas (in the case of major mergers) or
a large fraction of it (in the case of minor mergers) (see
Sec. 2.3.2), which destroy the radial gas-phase metallicity
gradient formed in early epochs. The metallicity gradient is
re-established by the new gas accreted after mergers, and as
a result, the gas-phase radial metallicity gradient in the new
disk will be weaker. The models thus predict that gas-phase
metallicity gradients correlate more strongly with the bulge
mass fraction of the galaxy than with its stellar mass. The
correlation with stellar mass arises because massive galaxies
have experienced more mergers and have larger bulge mass
fractions than less massive galaxies.
This is illustrated in detail in Fig. 10, which shows re-
sults from models with H2fraction prescription 1. In the top
left panel, the black solid curve shows the mean gas-phase
metallicity gradient (in units of dex/r90) as a function of
galaxy stellar mass. The black dashed curves indicate the
1σscatter around the mean. As can be seen, the mean gra-
dient increases from -0.35 dex/r90 for galaxies with stellar
masses M∗∼109.5M⊙to values near zero for galaxies with
M∗>1010.5M⊙. In the top middle panel, the gas-phase
metallicity gradient is plotted as function of bulge-to-total
(B/T ) mass fraction of the galaxy. A slightly stronger, and
more linear correlation is seen with B/T than with stellar
mass. In the top right panel, we plot the average bulge mass
fraction as a function of stellar mass, which shows that more
massive galaxies tend to have higher bulge mass fractions
(and hence weaker metallicity gradients).
In the bottom right panel, the gas-phase metallicity gra-
dient is plotted as a function of B/T for model galaxies in a
narrow stellar mass interval (10.0<log10[M∗/M⊙]<10.2)
and in the bottom left panel metallicity gradient is plotted
against M∗in a narrow interval of B/T (0.25 < B/T < 0.3).
These two panels demonstrate that the gas-phase metallicity
gradient correlates primarily with bulge mass fraction.
The purple dots superposed on the top three panels of
Fig. 10 show data from Moran12. We convert the concen-
tration index r90/r50 measured for galaxies in this sample
to a rough estimate of B/T using the fitting equation in
Gadotti (2009) r90/r50 = 1.93 + 2.02 B /T. Comparing the
purple dots with black curves, the qualitative trends are in
line with the model predictions, but the scatter is very large
– once again, this is because the data points represent mea-
surements along a 1-dimensional slit, rather than averages
in radial bins as in the models. The sample is too small to
investigate trends in metallicity gradient as a function of
bulge fraction in a narrow stellar mass bin or as a function
of stellar mass in a narrow bin of bulge-to-total ratio. It is
thus not possible to assess whether the observed gas-phase
metallicity gradient is more intrinsically correlated with stel-
lar mass of B/T . In addition, we note that there are some
galaxies in the observational data set with positively sloped
radial metallicity gradients. This is not seen in the models.
Integral field spectroscopic observations will be required in
order to ascertain whether the outliers with positive gradi-
ents are still present in similar numbers once metallicity is
averaged radially.
5.2 Relations between Zgas, sSFR and µ∗in the
inner and the outer regions of galaxies
In this section, we will examine correlations between gas-
phase metallicity, specific star formation rate and stellar
mass surface density. In particular, we will ask whether these
relations are the same or are different in the inner and the
outer regions of galaxies. In a simple “closed-box” model,
where gas is transformed into stars at a rate regulated by
its density, chemical enrichment proceeds in the same way
in the inner and outer regions of the galaxy. The main dif-
ference is that less gas is consumed into stars in the outer,
lower density regions, resulting in lower gas-phase metallici-
ties. In our semi-analytic models, where the inner regions of
disks assemble before the outer regions, and where gas in-
flows and outflows regulate gas content and metallicity, star
formation/metallicity correlations should be very different
in the inner and outer regions of galaxies.
Moran12 examined relations between gas-phase metal-
licity versus specific star formation rate and stellar sur-
face mass density for spectral bins located in the inner
(r < 0.7r90 ) and the outer (r > 0.7r90 ) regions of galaxies. It
is difficult to draw clear conclusions from their analysis, be-
cause the individual spectral bin measurements in the outer
regions exhibit so much scatter. In this section, we work with
SFR-weighted gas-phase metallicities averaged over the in-
ner and the outer disk, which yield considerably less scat-
tered results. We note that the individual gas-phase metal-
licity measurements are well-constrained, so the scatter is a
real physical effect, likely to do with longer timescales for
mixing of metals in the outer disk. In our models, heavy ele-
ments injected into one radial bin are instantaneously mixed
throughout that radial bin, so working with quantities that
Radial Gradients in SAMs of Disk Galaxy Formation 13
0 5 10
8
8.2
8.4
8.6
8.8
9
12 + log10 (O/H)gas
10−1 100
8
8.2
8.4
8.6
8.8
9
9.5<log10[M∗/M⊙]<10.0 10.0<log10[M∗/M⊙]<10.5 10.5<log10[M∗/M⊙]<11.0
0 5 10
8
8.2
8.4
8.6
8.8
9
r[kpc]
12 + log10 (O/H)gas
10−1 100
8
8.2
8.4
8.6
8.8
9
r/r90
Figure 9. The mean radial profiles of gas-phase metallicity from model galaxies at z= 0 in 3 mass bins: 109.5< M∗/M⊙<1010.0(blue),
1010.0< M∗/M⊙<1010.5(green), 1010.5< M∗/M⊙<1011.0(red). The top two panels are the results from H2fraction prescription
1, and bottom two panels are from H2fraction prescription 2. The radii plotted on the x-axis are scaled by divising by the disk scale
length rd.
are averaged over a larger region of the disk should yield a
fairer comparison. We restrict our analysis to models that
use H2fraction prescription 1 in this section (very similar
results are obtained for the other prescription). The inner
disk region is defined as r < 0.7r90 , and the outer disk region
as r > 0.7r90.
In the top panels of Fig. 11, we plot the relation between
SFR-weighted gas-phase metallicity and mean specific star
formation rate (sSFR=SFR/M∗) in the inner disk (left) and
in the outer disk (right). The blue dots show results for
the Moran12 sample and the red and yellow contours show
results for the models. We have taken care to compute the
average inner and outer metallicities and sSFR in the same
way in the models as in the data.
It is now clear that in the Moran12 sample, the
metallicity-sSFR relations are quite different in the inner
and the outer regions of the galaxy. At fixed sSFR, the gas-
phase metallicity is systematically higher in the inner disk
compared to the outer disk. The systematic offset increases
towards higher values of SFR/M∗. In the inner region of
the galaxy, gas-phase metallicity and SFR/M∗are weakly
correlated. Regions with the highest specific star formation
rates have slightly higher gas-phase metallicities. However,
the opposite is true in the outer regions of the galaxy: gas-
phase metallicity decreases at higher specific star formation
rates. The data and the models agree quite well. 2
2We note that the sSFR can no longer be estimated at all accu-
The reason for the “inverted” metallicity-sSFR relation
in the outer disk can be clarified by examining the bottom
two panels of Fig. 11, where we plot gas-phase metallicity
as a function of mean stellar mass-weighted stellar surface
mass density in the inner (left) and outer (right) regions of
galaxies for both models and data. Because stellar surface
density is a very steeply declining function of radius in disks
(top left panels in Fig. 3), the range of stellar surface den-
sities in inner and outer disks is almost completely disjoint.
In the data, gas-phase metallicity correlates with stellar sur-
face mass density only in the outer regions of galaxies, where
stellar surface densities are low. In the models, there is a cor-
relation between metallicity and µ∗in both the inner and
outer regions, but it is nevertheless very much stronger in
the outer disk. Our “inside-out” disk formation models pre-
dict that most of the stellar mass in the inner disk formed at
high redshifts by cooling and collapse of gas within a denser
progenitor halo. The outer disks are still in the process of
formation at present as the halo continues to accrete dark
matter and higher angular momentum gas is able to cool.
The outer disks with the lowest stellar surface mass densi-
ties are found in galaxies where gas has been accreted, but
has not yet reached high enough densities to form molecules
and stars. The fact that the outer gas-phase metallicities in
rately below a value of ∼ −12, so the extension of the data points
to sSFR values of ∼ −13 should not be regarded as a significant
discrepancy.
14 Fu et al.
9.5 10 10.5 11
−0.5
0
0.5
log10[M∗/M⊙]
∆Zgas(r < r90)(dex/r90)
0 0.2 0.4 0.6 0.8
−0.5
0
0.5
B/T
∆Zgas(r < r90)(dex/r90)
9.5 10 10.5 11
0
0.2
0.4
0.6
0.8
log10[M∗/M⊙]
Mbulge/M∗
9.5 10 10.5 11
−0.5
0
0.5
log10[M∗/M⊙]
∆Zgas(r < r90)(dex/r90)
0 0.2 0.4 0.6 0.8
−0.5
0
0.5
B/T
∆Zgas(r < r90)(dex/r90)
0.25 <B/T<0.310 <log10 [M/M⊙]<10.2
Figure 10. Top left panel: gas-phase metallicity (in units of dex/r90 ) vs. stellar mass; top middle panel: gas-phase metallicity vs. bulge-
to-total (B/T ) ratio; top right panel: the B/T ratio vs. stellar mass; bottom left panel: gas-phase metallicity vs. stellar mass for model
galaxies with B/T ratio in the range 0.25 < B /T < 0.3; bottom right panel: gas-phase metallicity vs. B/T ratio for model galaxies in the
stellar mass range 10.0<log10[M∗/M⊙]<10.2. In each panel, the black solid curves show the mean values from the model sample. The
black dashed curves indicate the ±1σscatter around the mean. The purple dots show the results of individual spectral bin measurements
from the Moran12 data set.
models do not reach metallicities much below 0.4 solar arises
because gas that cools from the surrounding halo has been
significantly pre-enriched with heavy elements (see discus-
sion in Sec. 6).
Finally, we would like to comment on the result in
Moran12 that around 10% of disk galaxies exhibit outer
“metallicity drops”, i.e. they exhibit a sharp turndown in
metallicity beyond ∼r90. The magnitude of this drop was
found to be strongly correlated with total MHI /M∗ratio, but
not with total MH2/M∗ratio. We have already commented
on the large scatter in metallicity from one spectral bin to
another in the outer regions of galaxies in the Moran12 sam-
ple. Indeed, close examination of individual profiles in Figure
10 of Moran12 reveals that there is significant diversity in
profile shape, even among the “metal-drop” objects.
In the top two panels Fig. 12, we examine how the in-
ner/outer disk metallicity difference [O/H]in −[O/H]out cor-
relates with total MHI/M∗and MH2/M∗ratios. Interest-
ingly, for both the Moran12 data and for our model galax-
ies the metallicity difference does not correlate with either
MHI/M∗or MH2/M∗. This suggests that outer metallic-
ity drops reflect the presence of isolated spectral bins with
low metallicity. Once again integral field spectroscopy data
would be extremely useful to test this conjecture in more de-
tail. Inspired by the relation between gas-phase metallicity
radial gradient vs. B/T fraction in Sec. 5.1, we plot metallic-
ity difference as a function of B/T in the bottom left panel
of Fig. 12. This anti-correlation is by far the strongest one
in both models and data.
6 SUMMARY AND DISCUSSION
In this paper, we describe how we have transplanted the
Fu10 prescriptions for modelling the gas and stellar profiles
of disk galaxies and for tracking the conversion of atomic
to molecular gas to the semi-analytic model framework of
Guo11. Our models are run on dark matter halo merging
trees constructed from both the MS and MS-II. Each galaxy
disk is divided into a series of radial concentric rings. This
allows us to track the radial distribution of gas, stars and
metals in each galaxy.
The main changes with respect to the Fu10 recipes are:
(i) We adopt a simple star formation law in which the star
formation rate surface density is proportional to the molec-
ular gas surface density ΣSFR ∝ΣH2, rather than the two
regime star formation model in Fu10.
(ii) In Krumholz et al. H2prescription, we adopt a
metallicity-dependent gas clumping factor so that the gas
is assumed to be more clumpy in low metallicity galaxy re-
gions. This accelerates molecule formation, star formation
and metal production in low mass galaxies and in low metal-
licity regions and allows us to obtain convergent results in
simulations with different resolutions.
(iii) SNe feedback processes are no longer assumed to be less
Radial Gradients in SAMs of Disk Galaxy Formation 15
log10[SFR/M∗][yr−1]
12 + log10 [O/H]in
−13 −12 −11 −10 −9
8.3
8.4
8.5
8.6
8.7
8.8
8.9
log10[SFR/M∗][yr−1]
12 + log10 [O/H]out
−13 −12 −11 −10 −9
8.3
8.4
8.5
8.6
8.7
8.8
8.9
log10 µ∗[M⊙pc−2]
12 + log10 [O/H]in
7 8 9 10 11
8.3
8.4
8.5
8.6
8.7
8.8
8.9
0.6 0.8 0.9 0.95
log10 µ∗[M⊙pc−2]
12 + log10 [O/H]out
6 7 8 9
8.3
8.4
8.5
8.6
8.7
8.8
8.9
galaxy number fraction
inner disks outer disks
Figure 11. The relation of SFR-weighted mean metallicity vs. mean stellar mass-weighted stellar surface density µ∗and mean specific
star formation rate SFR/M∗for inner and outer disks. The left two panels are the results for inner disks and right two panels are for
outer disks. The blue dots are from Moran12 data and the contours indicate the fraction of model galaxies located in a given region of
parameter space, as given by the colour key at the top of the plot.
efficient in the regions with higher gas surface density. In-
stead, we include radial gas inflow so that cold gas from the
outer disk moves inwards to compensate the gas consump-
tion in the inner disk.
(iv) The prescription for the mixing of heavy elements pro-
duced by star formation has been modified. Instead of mix-
ing all the metals directly with the cold gas in the disk, we
mix 80% of the metals with the hot gas in the halo, and 20%
of the metals with the cold gas in the disk. This partition
corresponds roughly to the fraction of total metals produced
by SNe as compared to AGB stars.
Based on the model results, we study the radial profiles
of gas-phase metallicity and star formation rate surface den-
sity and we also examine the correlation between gas-phase
metallicity gradient and some global galaxy properties. The
main conclusions are:
(i) The radial gas inflow prevents too fast gas consumption
in the inner region of the galaxy disk, because gas flowing in
from the outer disk compensates for the consumption. The
surface density profiles of molecular gas in L∗galaxies can
constrain the inflow velocities.
(ii) The radial gas inflow has only weak influence on the gas
and stellar metallicity profiles, especially in the outer regions
of galaxies. Because of inside-out disk growth, the outer disk
metallicity is mainly affected by the gas accreted recently,
which has been enriched by star formation and SN feedback
in early epochs. On the other hand, the small inflow velocity
in inner disk leads to weak influence of inner disk metallicity.
(iii) The gas-phase metallicity gradient is strongly affected
by the fraction of metals directly injected into the halo gas
from dying stars, rather than the interstellar cold gas of the
galaxy. Metals ejected from a galaxy in early epochs are later
re-accreted and this leads to flatter present-day gas-phase
metallicity gradients. We demonstrate that a prescription
in which 80% of all the metals are injected into the halo
gas provides the best fit to the relatively shallow observed
metallicity gradients of galaxies with stellar masses greater
than 1010M⊙(Kewley et al. 2010; Moran12). We also show
that such a prescription results in a good fit to the relation
between gas-phase metallicity and specific star formation
rate in the outer parts of galactic disks, which are still being
built by gas accretion at the present day.
(iv) Bulge formation through galaxy mergers is the other
main process that determines the strength of the gas-
phase metallicity gradient. This is because most of the gas
in the galaxy is consumed when the bulge is formed in
merger induced starburst, and the metallicity gradient is
re-established once new gas is able to accrete. In the mod-
els, galaxies with the strongest gas-phase metallicity gra-
dients are those that have accreted gas in an undisturbed
way over the age of the Universe and that have low bulge
mass fractions. We have re-examined metallicity gradient
16 Fu et al.
log10[MHI /M∗]
[O/H]in −[O/H]out
−2 −1.5 −1 −0.5 0
−0.2
−0.1
0
0.1
0.2
0.3
0.4
log10[MH2/M∗]
[O/H]in −[O/H]out
−3 −2 −1 0
−0.2
−0.1
0
0.1
0.2
0.3
0.4
B/T
[O/H]in −[O/H]out
0 0.2 0.4 0.6 0.8
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.6 0.8 0.9 0.95
galaxy number fraction
Figure 12. The relation of inner and outer disk mean gas-phase metallicity difference vs. total MH2/M∗,MHI/M∗and B/T ratios. The
meaning of blue points and colour contour are same to those in Fig. 11.
trends in the Moran12 sample and we do indeed find that
the gas-phase metallicity gradient correlates more strongly
with bulge-to-total ratio than with any other property.
It is worth comparing our results with other recent
work on modelling metallicity gradients. Pilkington et al.
(2012) use 25 galaxies from four previous hydrodynamical
simulation samples (Stinson et al. 2010; Rahimi et al. 2011;
Kobayashi & Nakasato 2011; Few et al. 2012) together with
two simple chemical evolution models (Chiappini et al. 2001;
Moll´a & D´ıaz 2005) to study the metallicity gradients in
disks in cosmological context. They study the radial abun-
dance gradient of stars at different age and find that the
stellar radial metallicity gradient tends to be flatter at lower
redshifts. They conclude that the stellar abundance radial
gradient originates from an inside-out disk formation, and
the merger histories have less effect on the metallicity gra-
dient than the recipes that treat the sub-grid physics. Gas-
phase metallicity gradients are not directly considered in
this paper.
Spitoni & Matteucci (2011) use chemical evolution mod-
els that include gas radial inflow processes to study the ra-
dial metallicity gradient for Milky Way disk. They test con-
stant inflow and inflow velocity proportional to the galacto-
centric radius with constant and inside-out infall prescrip-
tions in the models, and they conclude that the most im-
portant factor in reproducing the abundance gradient is the
radial inflow with a variable speed. They claim that radial
inflows are required to explain the observed metallicity gra-
dients in the Milky Way.
The main difference between our model and these mod-
els for the origin of radial abundance gradient is the metal-
licity of the cooling gas. Most of other models assume the
cooling gas always has the initial metallicity, while our model
assumes that the infalling gas is pre-enriched by star forma-
tion and SN ejecta in early epochs, which leads to different
conclusions regarding outer disk metallicities and radial gra-
dients.
In Fig. 13, we plot the metallicity of the hot gas sur-
rounding present-day central galaxies as a function of their
stellar mass. This metallicity will be the same as that of the
recently accreted gas. As can be seen, the hot gas metallic-
ity is predicted to increase from around 0.1 solar for central
galaxies with stellar masses of ∼109M⊙to around 0.4 so-
lar for galaxies with masses comparable to or greater than
that of the Milky Way. We note that significant fraction of
the hot gas around central galaxies originates from material
that has been ejected out of galaxies by supernovae in early
epochs. In low mass haloes, SNe eject most of the gas out
of the halo into the so-called “ejected” component (see Fig.
1). Re-incorporation of this ejected gas into the halo occurs
after a few dynamical times for a Milky Way type galaxy,
but substantially longer in dwarf systems (see Eq. A12).
The value of accreted gas metallicity in Fig. 13 is consis-
tent with the recent observations by Bresolin et al. (2012),
who find that the gas-phase metallicities in the outer disks
Radial Gradients in SAMs of Disk Galaxy Formation 17
9 9.5 10 10.5 11
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
log10[M∗/M⊙]
log10[Zhot /Z⊙]
Figure 13. The mean hot gas metallicity relative to the solar
value as a function of the stellar mass of the central galaxy.
of some nearby spiral galaxies (NGC 1512, NGC 3621, M83,
NGC 4625) are about 0.35Z⊙. This could reflect the accre-
tion of pre-enriched gas from the intergalactic medium. Our
results also agree with earlier work by Tosi (1988), who in-
cludes the infall of pre-enriched gas in galaxy chemical evo-
lution models and conclude that the metallicity of infalling
gas cannot be higher than 0.4Z⊙for Milky Way sized galax-
ies, otherwise the metallicity gradient in the model will be
too flat compared to the Milky Way observations. We note
that the gas-phase metallicity gradient of the Milky Way
is slightly steeper than typical galaxies of the same stellar
mass (left panel of Fig. 4). In our model, the relatively high
metallicity of infalling gas for Milky Way sized galaxy leads
to relatively flatter metallicity gradient compared to the ob-
servations in Milky Way.
The effect of galaxy merger on the metallicity gradient
is also interesting. The observations in galaxy close pairs
presented by Kewley et al (2010) show that there is strong
relationship between metallicity gradient and galaxy inter-
actions and mergers. Some hydrodynamic simulation works
(e.g Rupke, Kewley & Barnes 2010; Torrey et al. 2012) con-
clude that the gas inflows by galaxy tidal interaction during
galaxy mergers disrupt and flatten the gas-phase metallic-
ity gradient, which is similar to what is predicted by our
semi-analytic model. Although we do not trace the detail
of gas flows during the galaxy mergers, gas-phase metallic-
ity gradient is disrupted by the merger-induced starburst in
our model. As a result, galaxies that have experienced more
mergers in their history (or higher B/T ratio) tend to have
flatter metallicity gradients.
Finally, we should caution that we have used a very
simple model Eq. (10) to describe radial gas inflows and our
models also neglect radial transport of gas due to bar insta-
bilities in disks. Some observational constraints on inflows
do exist (e.g Levine et al. 2006; Zhang & Buta 2012), but the
measurements are often complicated by non-radial motions
in individual galaxies, and average inflow measurements for
galaxy samples selected by stellar mass, size, presence or
absence of bars etc. are still lacking. In principle, the stel-
lar and gas-phase metallicity profiles in the inner regions of
galaxies do provide additional constraints on the inflow pre-
scriptions (see Fig. 4) and the hope is that future data sets
will motivate more detailed work on this topic.
In this paper, we note that the evolution of radial gas,
SFR and metallicity gradients to higher redshifts is a topic
that is not addressed and that may be of interest to explore
in future work.
ACKNOWLEDGMENTS
We are grateful to the comments from the anonymous ref-
eree.
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Radial Gradients in SAMs of Disk Galaxy Formation 19
APPENDIX A: THE PHYSICAL PROCESSES
In this appendix, we will give very brief description for the
physical processes related to this paper in the L-Galaxies
semi-analytic models of galaxy formation, especially the pro-
cesses in Fig. 1. The following introduction is mainly from
model by Guo11 and Guo13, which is based on MS and MS-
II halos. As our previous work Fu10 is based DLB07 model,
we also mention the equations in DLB07, which is a previous
version of the model based on MS haloes.
In both MS and MS-II, the time interval between two
successive output snapshots is about 200 Myr. To model the
physical processes affecting the baryonic matter, each snap-
shot is divided into 20 timesteps. Each timestep is about 10
Myr, which is approximately the timescale for massive stars
to evolve off the main sequence. All the physcial processes in
L-Galaxies are updated in each timestep, but the dark mat-
ter halo properties from MS and MS-II are only updated
when one snapshot begins.
As discussed in White & Frenk (1991), the hot gas dis-
tributes isothermally in the dark matter halo. When the
cooling radius rcool (defined as the radius where the lo-
cal cooling time equals to the halo dynamical timescale) is
smaller than the halo virial radius rvir, the halo is in cooling
flow regime. The halo hot gas cools to the central galaxy
quasi-statically with cooling rate
˙mcool =mhotrco ol
rvirthalo
dyn
(A1)
in which thalo
dyn = 0.1H(z)−1is the dynamical timescale of
the halo. When rcool > rvir , the halo is in the rapid infall
regime and all the halo hot gas will accrete on to the central
galaxy through a “cold flow” in one timestep. The hot gas
in the halo is assumed to have specific angular momentum
identical to the dark matter halo and the spin parameter of
the cooling gas is defined as λ=J|E|1/2G−1M−5/2
vir accord-
ing to Mo, Mao & White (1998), in which Jand Eare the
angular momentum and energy of the dark matter halo.
AGN feedback can suppress the cold gas accretion pro-
cess. The black hole in the central galaxy can accrete some
hot halo gas through the so-called “radio mode” and the
energy is fed back to the hot halo. Thus it can decrease or
even stop the cooling flow to galaxy disks and quench the
supply of cold gas for star formation in high mass galaxies.
The radio mode accretion rate is
˙mBH,R=κAGN mBH
108M⊙fhot
0.1vvir
200km s−13(A2)
in which mBH is the black hole mass, fhot is the ratio of hot
gas to halo mass, and vvir is the virial velocity of the halo.
The power of energy supplied by the black hole accretion is
˙
EBH = 0.1 ˙mBH,Rc2(A3)
in which ˙mBH,Ris from Eq. (A1) and cis the speed of light.
The energy is used to suppress the amount of cooling gas,
and thus the cooling rate is
˙m′
cool = ˙mcool −˙
EBH
0.5v2
vir
(A4)
for ˙
EBH <0.5v2
vir ˙mcool and 0 for ˙
EBH >0.5v2
vir ˙mcool.
Stars form from the cold gas in ISM, and the star for-
mation model in Guo11 & DLB07 is
˙m∗=α(mgas −mcrit)/tdyn (A5)
for cold gas mass mgas larger than critical mass mcrit, which
is a simplified version of star formation law by Kennicutt
et al. (1998) including disk instabilities (Toomre 1964). The
star formation efficiency αis a constant and the disk dy-
namical timescale is defined tdyn = 3rgas /vmax as in Guo11
(rgas is the scale length of gas disk and vmax is the maximum
circular velocity of the subhalo) and tdyn = 3rd/vvir as in
DLB07 (rdis the scale length of the galaxy disk and vvir is
the virial velocity of the subhalo).
As the stars evolve, a fraction Rof the newly formed
stars in each timestep is returned instantaneously to the
interstellar gas. According to the Chabrier 2003 IMF, R=
0.43 is adopted. The metal elements from the star formation
are all injected into the interstellar cold gas and mixed im-
mediately, and the mass of newly produced metal elements
is
∆mZ=Y∆m∗(A6)
where Y= 0.03 is the yield of all metal elements and ∆m∗
is the mass of newly formed stars in a given timestep.
The energy from supernova explosion is
∆ESN = 0.5ǫhalo∆m∗v2
SN (A7)
where 0.5v2
SN is the energy of supernova ejecta per unit mass
of newly formed stars, and vSN = 630 km s−1is adopted.
ǫhalo in Eq. (A7) is a halo-dependent efficiency. The super-
nova energy can reheat part of the disk cold gas into halo
hot gas, and the mass of reheated cold is
∆mreheat =ǫdisk∆m∗(A8)
where ǫdisk is a disk-dependent efficiency. If the supernova
energy is large enough, the excess energy of supernova
∆ESN −∆Ereheat will eject part of the hot gas out of the
halo and be placed in the ejected component. The mass of
ejected hot is
∆meject = 2 (∆ESN −∆Ereheat )/v2
max (A9)
In the previous version of L-Galaxies code DLB07, ǫhalo and
ǫdisk in supernova feedback are both constants and ǫdisk =
3.5, ǫhalo = 0.35 are adopted for all galaxies. In Guo11, both
ǫhalo and ǫdisk are related to the galaxy halo
ǫdisk =ǫ0.5 + vmax
vreheat −β1
ǫhalo =η0.5 + vmax
veject −β2(A10)
in which ǫ= 4, η= 0.18, vreheat = 80km s−1,veject =
90km s−1and β1=β2= 3.2 are adopted in Guo13. This
kind of supernova reheating and ejection efficiency that
scales primarily with the potential well depth of the host
halo results in much higher feedback efficiencies in low mass
galaxies, which suppresses the star formation in small galax-
ies and decreases the number of small galaxies so that it fits
the observed stellar mass functions at the low mass end.
With the growth of dark matter halos through dark
matter particle accretion and halo mergers, the ejected gas
will be reincorporated into the halo of the central galaxy
and become the hot gas again. The mass of reincorporated
gas at each timestep (∆t) is
∆mrein =γreinmejected
∆t
thalo
dyn
(A11)
20 Fu et al.
in which meject is the mass of the ejecta reservoir and thalo
dyn is
the halo dynamical timescale. The reincorporation efficiency
γrein in DLB07 is a constant γrein = 0.5, and γrein in Guo11
is related to the halo virial velocity
γrein = 0.3vvir
220km s−1(A12)
In Eq. (A12), smaller haloes tend to reincorporate the ejecta
reservoir more slowly, which also helps to decrease the num-
ber of low mass galaxies in the Guo11 models.
