The Astrophysical Journal, 707:1466–1484, 2009 December 20

doi:10.1088/0004-637X/707/2/1466


C 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

THE ROLE OF TYPE Ia SUPERNOVAE IN CHEMICAL EVOLUTION. I. LIFETIME OF TYPE Ia SUPERNOVAE
AND METALLICITY EFFECT
Chiaki Kobayashi1 and Ken’ichi Nomoto2,3
1 Research School of Astronomy & Astrophysics, The Australian National University, Cotter Road, Weston ACT 2611, Australia;chiaki@mso.anu.edu.au
2 Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8568, Japan
3 Department of Astronomy, School of Science, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan; nomoto@astron.s.u-tokyo.ac.jp

Received 2007 December 31; accepted 2009 November 4; published 2009 December 4

ABSTRACT
We construct a new model of Type Ia Supernovae (SNe Ia), based on the single degenerate scenario, taking accountof the metallicity dependences of white dwarf (WD) wind and the mass-stripping effect on the binary companion
star. Our model naturally predicts that SN Ia lifetime distribution spans a range of 0.1–20 Gyr with the double peaks
at ∼0.1 and 1 Gyr. While the present SN Ia rate in elliptical galaxies can be reproduced with the old population of
the red giants+WD systems, the large SN Ia rate in radio galaxies could be explained with the young populationof the main-sequence+WD systems. Because of the metallicity effect, i.e., because of the lack of winds from WDs
in the binary systems, the SN Ia rate in the systems with [Fe/H]  −1, e.g., high-z spiral galaxies, is supposed tobe very small. Our SN Ia model can give better reproduction of the [(α, Mn, Zn)/Fe]–[Fe/H] relations in the solar
neighborhood than other models such as the double-degenerate scenario. The metallicity effect is more strongly
required in the presence of the young population of SNe Ia. We also succeed in reproducing the galactic supernova
rates with their dependence on the morphological type of galaxies, and the cosmic SN Ia rate history with a peak atz ∼ 1. At z  1, the predicted SN Ia rate decreases toward higher redshifts and SNe Ia will be observed only in thesystems that have evolved with a short timescale of chemical enrichment. This suggests that the evolution effect inthe supernova cosmology can be small.
Key words: galaxies: abundances – galaxies: evolution – supernovae: general
Online-only material: color figures

1994; Hillebrandt & Niemeyer 2000). The progenitors of the
majority of SNe Ia are most likely the Chandrasekhar (Ch) mass
WDs (e.g., Nomoto et al. 1997 for a review; see also Nomoto
et al. 2007), although the sub-Ch mass models might correspondto some peculiar subluminous SNe Ia. The early time spectraof the majority of SNe Ia are in excellent agreement with thesynthetic spectra of the Ch mass models, while the spectra of
the sub-Ch mass models are too blue to be compatible withobservations (Höflich & Khokhlov 1996; Nugent et al. 1997).
For the evolution of accreting C+O WDs toward the Ch mass,
two scenarios have been proposed; One is a double-degenerate(DD) scenario, i.e., merging of double C+O WDs with a
combined mass surpassing the Ch mass limit (Iben & Tutukov
1984; Webbink 1984). From the long-term observational search(SPY project) of the WD binaries for the DD systems, more thanthousand WDs are found. However, their combined masses are
smaller than the Ch mass, except for a few candidates near
the Ch mass (Napiwotzki 2007; Geier et al. 2007; Tovmassian
et al. 2008). Theoretically, it has been suggested that themerging of WDs leads to accretion-induced collapse rather
than SNe Ia (Saio & Nomoto 1985, 1998). The other is asingle-degenerate (SD) scenario, i.e., the WD mass grows by
accretion of hydrogen-rich matter via mass transfer from abinary companion (e.g., Nomoto et al. 2000; Livio 2001, forreviews). Two progenitor systems have been found for red giant(RG) and near main-sequence (MS) companions (e.g., Hachisu
et al. 1999b). The SD scenario has predicted the detection ofhydrogen-rich circumstellar matter from the companion star fora certain class of SN Ia binary systems, and such circumstellar
matter has been actually detected for some SNe Ia (e.g., Hamuy
et al. 2003; Patat et al. 2007).

1. INTRODUCTION
The star formation histories of galaxies are imprinted on stellar populations of the galaxies. Because of the age–metallicity
degeneracy for stellar populations, it is difficult to determine“age” from their colors and spectra alone. However, the elemental abundance ratios can provide independent information
on “age,” because different heavy elements are produced fromdifferent supernovae with different timescales (e.g., Pagel 1997;
Matteucci 2001). To derive the information of “ages” from elemental abundance patterns as a cosmic clock, the most important uncertainty is the lifetime of Type Ia Supernovae (SNe Ia),i.e., the evolutionary time (delay-time) of the progenitor fromthe main sequence through the explosion, which is mainly determined by the lifetime of the relatively low-mass (8 M )companion star of the exploding white dwarf (WD).
There exist two distinct types of supernova explosions (e.g.,
Arnett 1996; Filippenko 1997); One is Type II supernovae(SNe II), which are the core collapse-induced explosions ofmassive stars (8 M ) with short lifetimes of 106 –107 yr, andproduce more αelements (O, Mg, Si, S, Ca, and Ti) relative to
Fe with respect to the solar ratios (i.e., [α/Fe] > 0). Type Ib
and Ic Supernovae (SNe Ib and Ic) are also the core-collapse
supernovae, but have lost the envelope material before theexplosions. Recently, it was found that bright core-collapse
supernovae, hypernovae (HNe) produce an important amountof Fe (e.g., Nomoto et al. 2006). In chemical evolution models,the contributions of SNe Ib, Ic, and HNe can be included in thatof SNe II (Kobayashi et al. 2006). The other is SNe Ia, which are
the thermonuclear explosions of accreting WDs in close binaries
and produce mostly Fe and little αelements (e.g., Nomoto et al.
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ROLE OF SNe Ia IN CHEMICAL EVOLUTION. I.

The lifetime of SNe Ia has been estimated from different
aspects as follows, and these estimates conflict each other.
1. Population synthesis. The lifetime distribution of SNe Ia
has been predicted with the population synthesis models
for the evolution of binary systems (e.g., Yungelson 2005;
Belczynski et al. 2005; Tutukov & Fedorova 2007). The
predicted rate of SD systems in those models is muchsmaller than DD systems, and the majority of SNe Ia havethe lifetime as short as ∼0.1 Gyr (e.g., Ruiz-Lapuente et al.
1995; Yungelson & Livio 1998). However, the opticallythick winds from the WD, which were not taken into
account in the above models, have been shown to play
an essential role in the evolution of the accreting WDs(Hachisu et al. 1996, 1999a, 1999b). Furthermore, thestripping effect may increase the parameter space for the
SD systems (Hachisu et al. 2008). If these effects are taken
into account in the population synthesis models, the DD
rate could be smaller, and the SD rate could be larger.
2. Chemical evolution. The typical lifetime of SNe Ia could
have been constrained from the chemical evolution of the
Milky Way Galaxy, most notably in the [α/Fe]–[Fe/H]
relations. Metal-poor stars with [Fe/H]  −1 have [α/Fe]∼ 0.5 for Mg, Si, and Ca on an average (e.g., McWilliam
et al. 1995; Cayrel et al. 2004), while disk stars with [Fe/H]
 −1 show a decrease in [α/Fe] with increasing metallicity(e.g., Edvardsson et al. 1993; Bensby et al. 2004). Such an
evolutionary change in [α/Fe] against [Fe/H] has been
explained with the early heavy-element production by SNe
II and the delayed enrichment of Fe by SNe Ia (Matteucci& Greggio 1986). Conversely, chemical evolution models
can constrain the nature of progenitor systems of SNe Ia;
for example, Yoshii et al. (1996) estimated the lifetime of
SN Ia progenitors to be as long as 0.5–3 Gyr.
3. Supernova rates. The redshift evolution of the SN Ia rate
can give a rough estimate of the typical lifetime of SNe
Ia (i.e., delay-time) for the assumed star formation history.
The decrease of the SN Ia rate seen at z ∼ 1 corresponds tothe delay-time of ∼3 Gyr (Strolger et al. 2004). However,
the observed cosmic star formation rates (SFRs) involve
uncertainties from dust extinction and completeness. The
environmental dependence on galaxy evolution must be
important if the SNe Ia rate depends on metallicity. SN
Ia rates depending on the galaxy type can give stronger
constraint on the SN Ia progenitor models (Calura &Matteucci 2006). Della Valle et al. (2005) suggested that the
short lifetime of ∼0.1 Gyr is required to explain the highSN Ia rate in radio-loud elliptical galaxies, and Sullivan
et al. (2006) suggested the similar lifetime distribution fromthe relation between the SN Ia rate and the specific SFR.
Therefore, the bimodal distribution of the lifetime of SNeIa, so-called A+B model, has been proposed (Scannapieco& Bildsten 2005; Mannucci et al. 2006).
In addition, Kobayashi et al. (1998, hereafter K98) have foundthe metallicity effect on the WD winds, consequently, on the occurrence of SNe Ia, based on the simulations of the progenitor
evolution by Hachisu et al. (1999a, 1999b). If the iron abundance of the progenitors is as low as [Fe/H]  −1.1, then thewind is too weak for SNe Ia to occur. Thus, the SN Ia rate in thesystem with [Fe/H]  −1.1 is supposed to be “very small.” Thisprediction has not been observationally confirmed yet, and several SNe Ia in metal-poor environment have been reported (e.g.,
Prieto et al. 2008). Statistical studies of a large sample are required, since there must be a spread in the metallicity distribution

1467

function of stars in a galaxy and also the [O/Fe] is uncertain (seeSection 2.3 for the details).
If the lifetime distribution function of SNe Ia depends on
metallicity, the detailed calculation of chemical evolution is evenmore important to understand the observations of supernova
rates and abundance ratios. The redshift evolution of the cosmic
SN Ia rate is affected by chemical enrichment histories in various
types of galaxies. The estimates of the star formation histories
of galaxies from their abundance ratios are affected by themetallicity effect on the SN Ia rate. In addition, Fe can beproduced by the four sources (Kobayashi et al. 2006): (1) SNeIa, (2) relatively large Fe production from low-mass SNe II with
M = 13–15 M , (3) large Fe production from HNe, or (4) the
SN I.5 explosion of some asymptotic giant branch (AGB) stars.
Therefore, not only [α/Fe] but also other elements should beconsidered to discuss the contribution of SNe Ia.
This may be the case for dwarf spheroidal (dSph) galaxies; theobserved low [α/Fe] at low [Fe/H] in dSphs is often suggested
to be due to the significant contribution of SNe Ia at such low
metallicity (e.g., Venn et al. 2004). However, there is a large
variety among galaxies, and a decreasing trend in [Mg/Fe] at[Fe/H] < −1 may be seen in Draco (Shetrone et al. 2001) and
Sculper (Shetrone et al. 2003), but not for other α elements
such as Si and Ca. For more metal-poor stars, some stars showhigh [α/Fe] (Koch et al. 2008; Frebel et al. 2009a), but others
show low [α/Fe] (Aoki et al. 2009; Frebel et al. 2009b). Since[Mn/Fe] in dSph galaxies is as low as in Milky Way halo stars,
the abundance pattern can be better explained with low-mass
SNe II than SNe Ia (Tolstoy et al. 2003; Kobayashi 2003;
Kobayashi et al. 2006, hereafter K06). This is also supportedby the low [Ba/Fe] in Aoki et al. (2009). Chemodynamical
evolution of dSph galaxies are complicated because of the low
SFR and large dark matter content, and can be largely affected
only by a few supernovae. At least, the low [α/Fe] in low
metallicity stars does not necessarily mean the contribution of
SNe Ia, and certainly more detailed studies are required under
the reasonable model of SN Ia progenitors.
Similarly, it is argued that damped Lyα (DLA) systems showlow [α/Fe] at low [Fe/H] (e.g., Pettini et al. 1999). Prochaska &Wolfe (2002), however, concluded that [Si/Fe] ∼ 0.3 at [Fe/H]< −1.5, and also similar values were presented in DessaugesZavadsky et al. (2007). The elemental abundance ratios in DLA
systems are affected by the dust depletion. The dust depletionhas been often estimated using [Zn/Fe] ratio, but there is no
assurance that [Zn/Fe] ∼ 0 for low metallicity. In fact, an
increasing trend of [Zn/Fe] has been observed in the Milky Way
Galaxy (Primas et al. 2000; Nissen et al. 2007). Nissen et al.
(2007) showed that [S/Zn], which is free from the dust effect,of halo stars is similar to that in DLA systems. Such [S/Zn] and[Si/Fe] could be explained by hypernovae. Since the observation
is for neutral gas in galaxies, it can be easily affected by dilutionof hydrogen in chemodynamical simulations (Kobayashi et al.
2007). Again, more detailed studies are required.
The modeling of SNe Ia in the chemical evolution was firstmade by Greggio & Renzini (1983), where the SN Ia rate is
calculated from the distribution of binaries based on Whelan &Iben (1973), and then extended by Greggio (1996) including theefficiency of accretion of the envelope of the secondary to the
WD. This model has recently been updated in Greggio (2005)
and Matteucci et al. (2006). An alternative model was proposed
by K98, fully based on the SD scenario, where the metallicity
effect is essential to explain the [O/Fe]–[Fe/H] relation in thesolar neighborhood. Kobayashi et al. (2000, hereafter K00)

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applied this model for different types of galaxies, and succeededin reproducing the present SN II and Ia rates in different types ofgalaxies. Matteucci & Recchi (2001, hereafter MR01), however,argued that they could not reproduce K98 results and that thelack of SNe Ia at low metallicities produces results at variancewith the observed [O/Fe]–[Fe/H] relation. However, this is dueto their misunderstanding of the parameterization of the K98
models, as described in detail in Section 5.
In this paper, we first show the lifetime distribution functions
derived from different SN Ia models. We then introduce our
new SN Ia model by taking into account the stripping effect(Hachisu et al. 2008) and show the lifetime distribution function
as a function of metallicity (Section 2). We then evaluate
the SN Ia models from the comparison with observations.
The evolution of the elemental abundance ratios in the solar
neighborhood (Section 3) and the galactic and cosmic supernova
rates (Section 4) can be better reproduced with our SN Ia model
including the metallicity effect. In Section 5, we discuss thetwo formulations in K98 and MR01 in detail, and reproduce
both results using the alternative formulation. In Section 6,
we summarize our conclusions. The detailed discussion on
the elemental abundance ratios in other galaxies will be in thesecond paper.
2. LIFETIME DISTRIBUTION
2.1. Previous Models of Type Ia Supernovae
Lifetime (i.e., delay-time) of SNe Ia is not a single parameter.
The SN Ia lifetime, tIa , can be determined from the lifetimeof companion stars in C+O WD binary systems, because thetime interval from the start of mass accretion to the explosion
is as short as 107 yr. Since the companion’s mass ranges aredifferent among various SN Ia scenarios, the typical lifetime isdifferent for the different SN Ia models. In chemical evolution
models, two formulations have been proposed in K00 and
MR01 (see Section 5 for details, Equations (2) and (1)).
For the DD scenario, the timescale of gravitational radiation
until the two WDs start to merge is also important, and thelifetime distribution functions are provided from the population
synthesis of the binary evolution.
Figure 1 shows the distribution functions of the companion’s
mass md (upper panel) and lifetime tIa (lower panel) for the three
SN Ia models. These correspond to the SN Ia rate of the simple
stellar population, which is defined as a single generation of stars
with the same age and metallicity. We adopt the Salpeter initial
mass function (IMF) φ(m) ∝ m−1.35 for m = 0.07–50 M ,and the metallicity dependent main-sequence lifetimes, whichis calculated by Kodama & Arimoto (1997) with the stellar
evolution code described in Iwamoto & Saio (1999).

Figure 1. Distribution functions of the progenitor mass (upper panel) andlifetime (lower panel), i.e., the SN Ia rates of stars with the same ages, for
K98 model with Z = 0.002 (red solid line) and Z = 0.02 (red short-dashedline), MR01 model (blue long-dashed line; Matteucci & Recchi 2001), and DD
model (green dot-dashed line; Tutukov & Yungelson 1994).
(A color version of this figure is available in the online journal.)that the spikes5 around ∼1 Gyr are due to the change inthe stellar lifetime across the upper mass limit (∼2 M depending on metallicity) for the occurrence of the He core
flash.
2. The green dot-dashed lines are for the DD scenario, where
the Tutukov & Yungelson’s (1994) function is adopted inour chemical evolution model. The corresponding mass
distribution (converted with the stellar lifetime function ford
Z = 0.02) spans a range of 1–5 M .6 Because dm
is larger
dtIa
for larger md , a typical lifetime is tIa ∼ 0.1 Gyr, the fraction
with 1 Gyr is very small, and the fraction with 10 Gyris almost zero.
3. The blue long-dashed lines are for the MR01-like model,
calculated with Equation (1), but with the same IMF as
in our model. The mass distribution (converted with thestellar lifetime function for Z = 0.02) extends over 0–8 Mhaving a peak at ∼1.5 M . The decrease of the rate toward
larger m2 is due to the IMF, and the decrease toward smaller
m2 is caused by m1 + m2  3 M . A typical lifetime istIa ∼ 0.3 Gyr, which is longer than the DD model. The
fraction with 1 Gyr is 2 times larger than the DD model,
and the fraction with 10 Gyr is small but not zero.

1. The red solid and short-dashed lines are the distributions
adopted in K98, which are calculated with Equation (2) andthe mass ranges at Z = 0.004.4 Because of the metallicitydependent stellar lifetimes, the resultant lifetime distribution RIa,t slightly depends on metallicity (solid lines,
Z = 0.002; short-dashed lines, Z = 0.02). The two components appear at (md , tIa ) = (∼2 M , 0.5–1.5 Gyr) and(∼1 M , 2–20 Gyr), which correspond to the two typesof binary systems, the MS+WD and RG+WD systems, respectively. A kind of bi-modality, which is observationallyproposed, has been naturally realized in our model. Note
54The same as in Figure 1 of K00, but multiplied by the number fraction ofWDs, m11 φ(m1 )dm1 = 0.019.

In Figure 1 of K00, those are not seen due to a coarse mass grid.
The drop of RIa,m in the DD model is caused by the same reason as the
spikes.

6

No. 2, 2009

ROLE OF SNe Ia IN CHEMICAL EVOLUTION. I.

1469

Table 1
−1
SN Ia Rate of Simple Stellar Population (in 10−4 M) as a Function of Companion’s Mass md (M )This WorkZ = 0.002

Z = 0.004

K98
Z = 0.02

Z = 0.05

MR01

DD[Fe/H]  −1.1

md

RIa,m
RG

RIa,m
MS

RIa,m
RG

RIa,m
MS

RIa,m
RG

RIa,m
MS

RIa,m
RG

RIa,m
MS

RIa,m
RG

RIa,m
MS

RIa,m

RIa,m

0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.3
3.6
4.0
4.3
4.6
5.0
5.5
6.0
6.5
7.0
7.5
8.0

0
0
0
2.193
5.966
5.246
4.664
4.187
3.788
2.802
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.218
3.247
3.030
2.837
2.664
2.509
2.369
2.242
2.036
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
2.193
5.966
5.246
4.664
4.187
3.788
3.451
3.163
2.915
2.698
2.508
2.156
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.218
3.247
3.030
2.837
2.664
2.509
2.369
2.242
2.126
2.020
1.924
1.835
1.753
1.541
1.370
0.216
0
0
0
0
0
0
0
0
0

0
0
0
2.193
5.966
5.246
4.664
4.187
3.788
3.451
3.163
2.915
2.698
2.508
2.340
2.191
2.058
1.938
1.830
1.732
1.642
1.561
1.486
1.417
0.095
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.218
3.247
3.030
2.837
2.664
2.509
2.369
2.242
2.126
2.020
1.924
1.835
1.753
1.541
1.370
1.189
1.078
0.984
0.879
0.734
0
0
0
0
0

0
0
2.874
6.878
5.966
5.246
4.664
4.187
3.788
3.451
3.163
2.915
2.698
2.508
2.340
2.191
2.058
1.938
1.830
1.732
1.642
1.561
1.486
1.417
1.354
1.190
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.218
3.247
3.030
2.837
2.664
2.509
2.369
2.242
2.126
2.020
1.924
1.835
1.753
1.541
1.370
1.189
1.078
0.984
0.879
0.773
0.228
0
0
0
0

0
0
0
2.841
7.729
6.796
6.043
5.424
4.907
3.630
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.756
11.27
10.52
9.845
9.246
8.708
8.221
7.780
7.067
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

1.439
1.976
2.550
3.283
4.020
4.922
5.787
6.862
7.847
8.977
7.822
6.751
5.870
5.140
4.527
4.009
3.567
3.187
2.859
2.574
2.324
2.105
1.912
1.740
1.587
1.219
0.948
0.688
0.545
0.433
0.319
0.216
0.142
0.088
0.049
0.020
0

0
0
0
0.479
0.864
1.244
1.770
2.148
2.598
2.755
2.927
2.886
2.347
1.460
1.512
1.586
1.947
2.458
2.529
2.616
2.695
2.789
2.833
2.875
2.587
3.090
3.049
3.407
2.615
2.162
0.733
0.041
0.003
0
0
0
0

For each SN Ia progenitor model, the distribution function of
the companion’s mass and lifetime, i.e., the SN Ia rate in 1 Mof simple stellar population is given in Tables 1–3. In our SD
scenario, the SN Ia rate is set to be 0 at [Fe/H] < −1.1.
2.2. A New SN Ia Model
Hachisu et al. (2008, hereafter HKN08) introduced thestripping effect, where a part of the envelope mass of thecompanion stars is stripped by the interaction with the WD
winds. The efficiency is described by one parameter c, which
depends mainly on the wind velocity. The effect of geometry
can be given as a function of the mass ratio, and is included inc. The value of c can be determined from the observed binary
systems such as supersoft X-ray sources, and is ∼1.5–10 for
one source (Hachisu & Kato 2003a, 2003b) and will be ∼3 on
the average.
Although the binary evolution is calculated only for Z = 0.02
in HKN08, the wind velocity depends on the metallicity, andthe case with higher metallicity should be similar to the casewith larger c (I. Hachisu 2007, private communication). For
Z = 0.002, the mass stripping is supposed to be inefficient, andthus the mass ranges of companion stars are the same as in K98
and K00. For higher metallicity, the mass ranges are wider. Here,
we take the results with c = 0, 1, and 3 for Z = 0.002, 0.004,
and 0.02, respectively. In the chemical evolution model, theupper and lower mass of the binary systems, mu and m , aregiven as a function of [Fe/H] (Table 4), since iron is the mosteffective element for the opacity of the WD winds. While themetallicity effect is introduced as the simple cutoff the SN Ia
rate in K98 and K00, the metallicity effect is included in theparameter space of the binary systems that leads to SNe Ia in
the present work.
Figure 2 shows the SN Ia rate in the simple stellar population
as functions of companion’s mass md (upper panel) and lifetimetIa (lower panel) for the metallicity Z = 0.002 (blue), 0.004(green), 0.02 (orange), and 0.05 (red). The solid and dashed lines
indicate the MS+WD and RG+WD systems that correspondto the young and old population of SNe Ia, respectively. The
total number of SNe Ia is determined to reproduce the chemical
evolution in the solar neighborhood ([bRG , bMS ] = [0.023, 0.023]
at Z = 0.004, see Section 3 for the details), and also adopted
for the other types of galaxies (Section 4). We should note

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Table 2
−1
SN Ia Rate of Simple Stellar Population (in 10−4 M) as a Function of Time tIa (Gyr)This WorkZ = 0.002

Z = 0.004

Z = 0.02

Z = 0.05

tIa

RIa,t
RG

RIa,t
MS

tIa

RIa,t
RG

RIa,t
MS

tIa

RIa,t
RG

RIa,t
MS

tIa

RIa,t
RG

RIa,t
MS

45.15
27.16
17.08
11.33
7.974
5.891
4.533
3.563
2.812
2.271
1.807
1.502
1.262
1.063
0.903
0.836
0.777
0.720
0.669
0.596
0.533
0.475
0.425
0.386
0.351
0.268
0.212
0.161
0.135
0.114
0.094
0.075
0.062
0.052
0.045
0.039
0.034

0
0
0
0.048
0.219
0.305
0.401
0.487
0.587
0.558
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.099
1.812
2.678
4.513
4.581
4.637
3.804
3.279
3.374
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

57.50
32.39
19.56
12.53
8.413
5.929
4.265
3.165
2.519
1.998
1.629
1.478
1.349
1.225
1.117
0.979
0.864
0.766
0.683
0.612
0.550
0.497
0.450
0.409
0.374
0.296
0.239
0.184
0.157
0.135
0.112
0.092
0.077
0.065
0.056
0.050
0.044

0
0
0
0.039
0.181
0.253
0.338
0.480
0.649
0.776
1.217
2.083
2.127
2.164
1.758
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.172
2.801
2.470
2.238
2.498
2.774
3.066
3.375
3.701
4.045
4.408
4.789
6.163
6.864
8.623
1.847
0
0
0
0
0
0
0
0
0

84.76
48.76
29.17
18.59
12.36
8.575
6.026
4.501
3.462
2.783
2.234
1.845
1.540
1.415
1.305
1.198
1.105
0.981
0.876
0.783
0.702
0.630
0.568
0.514
0.467
0.356
0.278
0.206
0.171
0.144
0.116
0.093
0.075
0.063
0.053
0.046
0.041

0
0
0
0.026
0.119
0.166
0.229
0.327
0.441
0.562
0.675
0.840
1.253
2.129
2.166
2.196
1.896
1.691
1.840
1.993
2.160
2.330
2.554
2.789
0.239
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.101
2.756
2.803
2.842
2.455
2.189
2.382
2.580
2.796
3.017
3.306
3.610
4.431
4.899
6.398
7.788
10.41
12.54
15.43
17.97
0
0
0
0
0

78.29
44.78
27.04
17.24
11.00
7.596
5.563
4.151
3.322
2.607
2.082
1.793
1.557
1.412
1.287
1.119
0.979
0.862
0.763
0.676
0.603
0.538
0.482
0.434
0.392
0.295
0.228
0.167
0.137
0.114
0.090
0.071
0.057
0.048
0.040
0.035
0.030

0
0
0.021
0.086
0.124
0.193
0.271
0.374
0.491
0.557
0.777
1.109
1.418
1.864
1.597
1.421
1.599
1.790
1.973
2.164
2.372
2.587
2.852
3.132
3.912
4.371
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.115
2.413
2.068
1.840
2.070
2.317
2.554
2.802
3.070
3.349
3.692
4.055
5.063
5.658
7.479
9.124
12.17
14.86
18.65
23.34
9.631
0
0
0
0

that this function is different from HKN08 because the adopted
IMF, stellar lifetimes, and binary parameters are different. Thisfunction in Figure 2 is better for chemical evolution models
since it is calibrated to meet the observations in the solar
neighborhood.
Figure 3 shows the summation of the young and old components of the SN Ia rate as functions of companion’s mass(upper panel) and lifetime (lower panel). The bimodal distribution is realized at (md , tIa ) = (∼2 M , 0.1–0.5 Gyr) and(∼1 M , 1–3 Gyr) corresponding to the MS+WD and RG+WD
systems, respectively. The lifetimes at the two peaks dependon the metallicity: tIa ∼ 0.5 and 3 Gyr for Z = 0.002 (blueshort-dashed line), tIa ∼ 0.2 and 1 Gyr for Z = 0.004 (green
solid lines), tIa ∼ 0.1 and 1 Gyr for Z = 0.02 (orange dottedline) and 0.05 (red long-dashed line). Note that the spikes appears around ∼1 Gyr because the stellar lifetime is substantiallylonger for smaller mass stars that undergo the He core flash.
This enhanced rate might cause the SN Ia driven galactic winds
in elliptical galaxies.
At higher metallicity, the lifetime distribution extends toboth shorter and longer lifetimes. Such a short lifetime as

tIa ∼ 0.1 Gyr is caused by the stripping effect that enables
the accreting WDs with massive (∼5–6 M ) companions to
explode. The lifetime as long as tIa ∼ 20 Gyr is caused by theelongation of the lifetime of small-mass companion stars, which
is dominant in present-day elliptical galaxies.
HKN08 model predicts the existence of various types ofcircumstellar matter (CSM) around the WDs just before making
SN Ia explosions depending on the mass accretion rates (andthus on the binary parameters) at the last stages. (1) case WIND:
If the accretion rate is high enough for the optically thick WD
wind continue to blow, the stripped-off material forms CSM very
near the WD because the velocity of CSM is rather low; this casecorresponds to SNe 2002ic (e.g., Deng et al. 2004) and 2005gj(Aldering et al. 2006), where SN Ia undergoes circumstellarinteraction. (2) If the accretion rate has decreased below thecritical rate for the wind to occur at the explosion, the strippedoff material forms CSM with various speed, but it has beendispersed far from the WDs. Depending on the distance fromthe WDs and geometry of the materials, CSM features may or
may not be observed as in SN 2006X (Patat et al. 2007) or SN
2007af (Simon et al. 2007). We also note that the speed of theNo. 2, 2009

ROLE OF SNe Ia IN CHEMICAL EVOLUTION. I.

1471

Table 3
−1
SN Ia Rate of Simple Stellar Population (in 10−4 M) as a Function of TimetIa (Gyr)K98
Z = 0.002

Figure 2. Distribution functions of the progenitor mass (upper panel) andlifetime (lower panel) for the metallicity Z = 0.002 (blue), 0.004 (green), 0.02
(orange), and 0.05 (red). The solid and dashed lines indicate the MS+WD and
RG+WD systems, respectively, that correspond to the young and old populationof SNe Ia.
(A color version of this figure is available in the online journal.)

stripped-off materials is as low as 10–100 km s−1 , so that thesweeping effect of very fast wind discussed by Badenes et al.
(2007) is not expected.
2.3. Metallicity Effect
After our theoretical prediction of the metallicity effect, theSNe Ia in low-luminous galaxies have been paid attention. K00
commented on SN1895B and SN1972E in NGC 5253, andSN1937C in IC 4182 that the metallicities of the host galaxies
are higher than [Fe/H] = −1.1. Hamuy et al. (2000) showedthat no host galaxies are found at [Fe/H] < −1 (Figure 1 in
Hamuy et al. 2001) for 44 supernovae. However, SN1999aw
is in the galaxy with MB = −12.2 (Strolger et al. 2002),
which roughly corresponds to [O/H] ∼ −1.2 (Tremonti et al.
2004). SN2005cg is in the host galaxy with Mr = −16.75 and[O/H] = −0.4 to −0.1 (Quimby et al. 2006). SN2007bk isin the galaxy with [O/H] = −0.59, but located at a distance of
8.7 kpc (Prieto et al. 2008), where [O/H] is estimated as ∼−0.7(see below for the details). Badenes et al. (2009) showed thestellar population near the supernova remnants (SNRs) of SNe
Ia in the Small Magellanic Cloud from the color–magnitude
diagram. The mean stellar metallicities are [Fe/H] = −0.96,−0.94, −1.15, and −0.80, but there is also a young metal-

MR01

DD

Z = 0.02

tIa

RIa,t
RG

RIa,t
MS

tIa

RIa,t
RG

RIa,t
MS

RIa,t

RIa,t

45.15
27.16
17.08
11.33
7.974
5.891
4.533
3.563
2.812
2.271
1.807
1.502
1.262
1.063
0.903
0.836
0.777
0.720
0.669
0.596
0.533
0.475
0.425
0.386
0.351
0.268
0.212
0.161
0.135
0.114
0.094
0.075
0.062
0.052
0.045
0.039
0.034

0
0
0
0.062
0.284
0.395
0.519
0.630
0.760
0.722
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.345
6.289
9.295
15.66
15.90
16.09
13.20
11.38
11.71
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

84.76
48.76
29.17
18.59
12.36
8.575
6.026
4.501
3.462
2.783
2.234
1.845
1.540
1.415
1.305
1.198
1.105
0.981
0.876
0.783
0.702
0.630
0.568
0.514
0.467
0.356
0.278
0.206
0.171
0.144
0.116
0.093
0.075
0.063
0.053
0.046
0.041

0
0
0
0.034
0.154
0.215
0.297
0.423
0.571
0.592
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0.351
9.565
9.730
9.865
8.520
7.596
8.267
8.956
9.293
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0.003
0.007
0.017
0.039
0.080
0.156
0.284
0.535
0.913
1.463
1.668
1.945
2.727
4.363
4.189
4.017
3.287
2.780
2.875
2.963
3.057
3.143
3.285
3.425
4.014
3.873
4.426
4.510
5.267
5.522
5.598
5.274
4.807
4.066
2.949
1.532
0

0
0
0
0.006
0.017
0.039
0.087
0.168
0.302
0.449
0.624
0.832
1.090
1.239
1.399
1.589
1.794
2.144
2.543
3.012
3.543
4.165
4.868
5.657
6.542
9.824
14.24
22.33
25.27
27.56
12.86
1.004
0.086
0
0
0
0

Table 4Upper and Lower Mass of the Binary Companion of SN Ia Progenitor Systems
Depending on the Iron Abundance, in the Mass Unit of M
Fe/H

−1.1

−1.0

−0.7

0

0.4

mRG,
mRG,u
mMS,
mMS,u

0.9
0.9
1.8
1.8

0.9
1.5
1.8
2.6

0.9
2.0
1.8
4.0

0.9
3.0
1.8
5.5

0.8
3.5
1.8
6.0

rich population as they commented. From the statistical studies
of a large sample such as the Sloan Digital Sky Survey andthe Supernova Legacy Survey, no such metallicity effect has
been seen in observations yet (e.g., Gallagher et al. 2005, 2008;
Howell et al. 2009), and rather opposite effect (i.e., decreasingof SN Ia rates toward higher metallicity) has been suggested by
Cooper et al. (2009).
Note, however, that the following uncertainties are involvedin these arguments; (1) The observation is the emission weighted
gas phase [O/H], and can be different from the [Fe/H] of the1472

KOBAYASHI & NOMOTO

Vol. 707[O/H] ∼ [Fe/H] ∼ −0.7 with Δ[Fe/H]/Δ log r/re ∼ −0.2 andre ∼ 1.5–2 kpc.
3. [X/FE]–[FE/H] RELATIONS

Figure 3. Distribution functions of the progenitor mass (upper panel) andlifetime (lower panel), i.e., the SN Ia rates of stars with the same ages, for our
new models with the metallicity Z = 0.002 (blue short-dashed line), Z = 0.004(green solid line), Z = 0.02 (orange dotted line), and Z = 0.05 (red long-dashedline).
(A color version of this figure is available in the online journal.)

progenitor system; gas metallicity can be larger than stellar
metallicity in chemically evolved and closed systems, or can besmaller due to the dilution by gas accretion and/or inhomogeneous effects (i.e., stars tend to be formed from metal-enrichedgas). (2) The metallicity correlates with the galaxy mass (themass–metallicity relation, e.g., Tremonti et al. 2004), but the
scatter is larger for smaller galaxies. (3) The mean stellar [α/Fe]
correlates with the galaxy mass in the sense that [α/Fe] is larger
for more massive galaxies ([α/Fe] ∼ 0–0.3, e.g., Thomas &Maraston 2003), although the errorbar is quite large. (4) Thereexits a radial gradient of metallicity, where the metallicity islower in outside (e.g., Kobayashi & Arimoto 1999). (5) Thereexits a metallicity distribution of stars even at a certain locationin galaxies (Figure 4).
Föerster & Schawinski (2008) reported that there is noradial dependence of SN Ia rate found in early-type galaxies.
However, this work also did not reach the metallicity threshold.
The metallicity gradient of early-type galaxies is typically
Δ[Fe/H]/Δ log r ∼ −0.3, and the metallicity at the effective
radius is [Fe/H]e ∼ −0.3 to 0 (Kobayashi & Arimoto 1999).
The metallicity effect would be seen at r  10re in intermediatemass galaxies. In dwarf galaxies, SNe Ia can be found atr ∼ 50 kpc, because the effective radius does not becomeso small (Binggeli et al. 1984), and because the metallicity
gradient is shallower for smaller galaxies (Spolaor et al. 2009).
For SN2007bk, the local metallicity is roughly estimated asUsing the one-zone chemical evolution model, we show theevolution of elemental abundance ratios in the solar neighborhood for different SN Ia models. The star formation history isdetermined from the following observations in the solar neighborhood. The formulation is described in K00; we use the modelthat allows the infall of primordial gas from outside the disk region. The SFR is assumed to be proportional to the gas fraction.
For the infall rate, we adopt the same formula as in K98 andK06 that is proportional to t exp[− τti ]. The input parameters are
the Galactic age of 13 Gyr and infall timescale τi = 5 Gyr. The
star formation timescale τs = 2.2 Gyr is determined from thepresent gas fraction fg = 0.15 (see K00 for the definition). The
same parameters are set for the other SN Ia models, because
the contribution of SNe Ia on the gas fraction is negligible. The
nucleosynthesis yields as functions of mass, metallicity, and explosion energy are given in K06 for HNe, SNe II, and SNe Ia.
Photometric evolution is also calculated as in K00.
Figure 4 shows (a) the SFR, (b) the age–metallicity relations,and (c) the metallicity distribution function (MDF) for different
SN Ia models. The resultant SFR peaks at t ∼ 8 Gyr, butdoes not very much change for t ∼ 5–13 Gyr, and the present
SFR is consistent with the observational estimate. The red solid,
cyan dotted, green short-dashed, blue long-dashed, and magenta
dot-dashed lines are for our new model with and without
the metallicity effect, the DD, MR01-like, and K98 models,
respectively. All models can give excellent agreement withthese observations. Compared with the K98 model (magenta
dot-dashed line), our new SN Ia model (red solid line) givesslightly slower increase in [Fe/H] after [Fe/H] ∼ −1 (panelb), and slightly larger number of metal-poor stars around[Fe/H] ∼ −1 (panel c). This is because the metallicity effectis included not with a sharp cutoff but with a gradual decreasein our new model. Recently, Nordström et al. (2004) showed anarrower MDF than the plotted observations (Edvardsson et al.
1993; Wyse & Gilmore 1995), which may suggest that a sharp
cutoff exist in the SN Ia rate at [Fe/H] ∼ −1.
The total number of SNe Ia is chosen to meet the MDF; For
the DD model, the total fraction of SNe Ia is set to be 0.001. For
the MR01-like model, the binary fraction (see Section 5 for thedefinition) of A = 0.035 is adopted. In our models, the binaryparameter b denotes the fraction of primary stars that eventually
explode as SNe Ia for the RG+WD and MS+WD systems.
The combination of bRG and bMS can be determined from themetal-rich end of the MDF and the [O/Fe]–[Fe/H] relation at[Fe/H]  −1. For our new SN Ia model, b is normalized at
Z = 0.004, and [bRG , bMS ] = [0.023, 0.023] are chosen for thebest model. (Note that the models with [0.015, 0.028], [0.010,
0.032] and [0.005, 0.035] are also possible.) Since the mass
range expands for higher metallicity, the fraction of SNe Ia
is 0.040 and 0.032 at Z = 0.05 for the RG+WD and MS+WD
systems, respectively. For the case without the metallicity effect,
the companion’s mass ranges for Z = 0.02 and [bRG , bMS ] =[0.027, 0.027] are adopted. For the K98 model, [bRG , bMS ] =[0.02, 0.04] are chosen in K06 for the updated nucleosynthesis
yields.
In the solar neighborhood, elemental abundance ratios areobserved for a number of stars over a wide range of metallicity,which gives one of the most stringent constraints on theNo. 2, 2009

ROLE OF SNe Ia IN CHEMICAL EVOLUTION. I.

1473

Figure 4. Chemical evolution of the solar neighborhood: (a) the star formation rate, (b) the age–metallicity relation, and (c) the metallicity distribution function fordifferent SN Ia models: our SD model (red solid line), SD model without metallicity effect (cyan dotted line), DD model (green short-dashed line), MR01 model (bluelong-dashed line), and K98 model (magenta dot-dashed line). Observational data sources are: Matteucci (1997), errorbar in panel (a); Edvardsson et al. (1993), filled
circles in panels (b) and (c); Wyse & Gilmore (1995), open circles in panel (c).
(A color version of this figure is available in the online journal.)

progenitors of SNe Ia. Figures 5–7 show the evolutions ofelemental abundance ratios [X/Fe] for O, Mg, Mn, and Zn,
against the iron abundance [Fe/H].
In the early stage of the galaxy formation, only SNe II
explode, and [α/Fe] stays constant. Because of the delayed
Fe production by SNe Ia, [α/Fe] decreases toward 0. The
observations with UV OH lines show a monotonic increase in[O/H] toward lower metallicity (Israelian et al. 1998; Boesgaard
et al. 1999), but these are largely affected by the threedimensional effect of the atmosphere model and are not plotted
here. The O observations with the forbidden line and infraredlines are more reliable, and show the plateau rather than
the increase as well as the other α elements (Mg, Si, Ca,
and Ti).
Manganese shows a characteristic behavior. Observational
data show that [Mn/Fe] increases toward higher metallicity.
Feltzing et al. (2007) concluded that the Mn trend is likely tobe due to the metallicity dependent SNe II, but the trend canbe naturally reproduced by the delayed enrichment by SNe Ia
since SNe Ia produce [Mn/Fe] > 0 (K06). This is consistent
with that this increasing trend of [Mn/Fe] appears from the same
[Fe/H] as the decreasing trends of [α/Fe]. In the observation, alarger increase of [Mn/Fe] may be seen at [Fe/H] ∼ 0 than ourmodels, which may be due to the metallicity dependence of SN
Ia yields, which is not included in our models.
The species of zinc depend on metallicity. At low metallicity,64
Zn is produced by complete Si-burning in HNe (Umeda &Nomoto 2002), which are assumed to be a half of massive(20 M ) SNe II with the lifetime of 107 yr (K06). At highmetallicity, neutron-rich isotopes of zinc 66–70 Zn are producedby neutron-capture in He and C burning, which is larger for
higher metallicity (Figure 5 in K06) and ejected by SNe II.
Because of the combination of the lifetime and metallicity
effects, [Zn/Fe] shows an interesting track.
1. In the DD model (green short-dashed line), [α/Fe] decreases and [Mn/Fe] increases too early and too quickly
from [Fe/H] ∼ −2 compared with the observations. Thisis because the lifetime of the majority of SNe Ia is shorter
than 0.3 Gyr (Figure 1). The SN Ia lifetime is so shortthat [Zn/Fe] decreases by 0.25 dex from [Fe/H] ∼ −2to ∼−0.6. Because the SN Ia rate sharply decreases forlonger lifetime, and because the SFR does not change very
much, the contribution of HNe becomes larger than SNe
Ia at later time. This results in the [Zn/Fe] increase from
[Fe/H] ∼ −0.6 to ∼0, which is not seen in the observations.
2. In the MR01-like model reproduced with our code (bluelong-dashed line), the decrease in [α/Fe] and the increase
in [Mn/Fe] are seen at [Fe/H] ∼ −2 as early as in the DD
model. The slope of [α/Fe] and [Mn/Fe] against [Fe/H] isa little shallower than the DD model because the fractionof lifetime with 1 Gyr is larger. The [Zn/Fe] decreases
only by 0.15 dex from [Fe/H] ∼ −2 to ∼ − 0.6.
3. Our SN Ia model with the metallicity effect (red solid line)
gives the best agreement with the observations. The decrease in [α/Fe] and the increase in [Mn/Fe] are determined not from the lifetime effect but from the metallicity effect on the SN Ia rate. Since the lowest metallicity

1474

KOBAYASHI & NOMOTO

Vol. 707

Figure 5. Evolution of the elemental abundance ratios, [X/Fe]–[Fe/H] relations, in the solar neighborhood for different SN Ia models: our SD model (red solid line),
DD model (green short-dashed line), and MR01 model (blue long-dashed line). Observational data sources are: For disk stars, Edvardsson et al. (1993), small open
circles; thin disk stars in Bensby et al. (2003, 2004), small filled circles; accretion component in Gratton et al. (2003), triangles. For halo stars, McWilliam et al.
(1995), large open circles; Ryan et al. (1996), filled squares; Cayrel et al. (2004), large filled circles; Honda et al. (2004), filled pentagons. For Mn, Gratton (1989),
open squares; Feltzing et al. (2007), small filled squares. For Zn, Sneden et al. (1991), open squares; Primas et al. (2000), eight-pointed asterisks; Nissen et al. (2004,
2007), open circles.
(A color version of this figure is available in the online journal.)

to produce SNe Ia is [Fe/H] = −1.1, the evolutionarychange is caused at [Fe/H] ∼ −1 by the companion stars
with MMS,u = 2.6 M after the main-sequence lifetime of∼0.5 Gyr. In the K98 model, the metallicity effect is included more sharply, and thus the evolutionary change appears a bit more sharply.
Our model predicts an interesting evolution of [Zn/Fe]
around [Fe/H] ∼ −1, as a contrast to other models;[Zn/Fe] increases from [Fe/H] ∼ −1.5 by the metallicity effect on Zn yield, and decreases from [Fe/H]∼ −1 by the contribution of SNe Ia. This trend seems
to be consistent with the observations, although the scatter is not small (Nissen et al. 2004 at [Fe/H] < −1
and Nissen et al. 2007 with the NLTE correction at[Fe/H] > −1). Further observations of these elements at[Fe/H]  −1.5 is critically important in order to identify
the SN Ia progenitors and to clarify the metallicity dependence on the nucleosynthesis yields.
4. If we do not include the metallicity effect (dotted line in
Figure 6), [α/Fe] starts to decrease at [Fe/H] ∼ −2. This
corresponds to the shortest SN Ia lifetime of ∼0.1 Gyr for
MMS,u = 5.5 M . This is too early to be compatible withthe observations. In the K98 model, the [α/Fe] decrease
starts at [Fe/H] ∼ −1.5 (see Figure 3 in K98), because
the shortest lifetime (∼0.5 Gyr for MMS,u = 2.6 M ) is
longer than our new model. In other words, without themetallicity effect, the existence of the young population of
SNe Ia cause the [α/Fe] decrease at much smaller [Fe/H],
and the discrepancy with the observations becomes larger.

5. If there is the metallicity effect, these results do not dependvery much on the c parameter (Figure 7). Even with c = 10
that leads to the shortest lifetime of 0.03 Gyr, the difference
is not visible. This is because the majority of stars have
low metallicity as Z < Z , and the metal-rich component
can contribute only after ∼10 Gyr. Even if the stripping iseffective even for Z = 0.002 (long dashed line), the results
does not change very much. Therefore, the evolutionarychange in the [X/Fe]–[Fe/H] relations is simply caused by
the limit of the WD winds, as introduced in K98.
Figure 8 shows the SN Ia rate history in the solar neighborhood. Even with the same present SN Ia rate, different SN Ia
models produce different SN Ia rate histories. For the DD model,
the lifetimes of majority of SNe Ia lifetime are short, and the SN
Ia rate shows a rapid increase at t < 1 Gyr and then a constantuntil the present time. For the SD model, the lifetimes are a bit
longer, and the SN Ia rate shows a slower increase at t < 5 Gyr.
With the metallicity effect, SNe Ia start to occur at 3 Gyr when
[Fe/H] reaches −1.1 (Figure 4(b)). Compared with our previous
work, our new model shows a more gradual increase due to thesmoother SN Ia lifetime distribution as a function of metallicity.
In Figure 9, the supernova rate is plotted against the metallicity indicators: (a) the iron abundance, (b) the gas-phase metallicity, and (c) the mean stellar metallicity (see Equation (15)in K00 for the definition). Since the SN Ia rate is assumed to
be 0 at [Fe/H] < −1.1, the rapid decrease is seen at [Fe/H]∼ −1 after the short time-delay of the chemical enrichment(panel (a)). Because of the oxygen enhancement at lower metallicity, the decrease is seen at higher [O/H] as ∼ − 0.6. In dwarf

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Figure 6. Same as Figure 5 but for our SD model with (solid line) and without (dotted line) metallicity effect.
(A color version of this figure is available in the online journal.)

Figure 7. Same as Figure 5 but with different c; c = (0, 1, 3, 5) (solid line, main model), c = (0, 2, 5, 10) (short dashed line), and c = (2, 3, 5, 10) (long-dashed line),
respectively for Z = 0.002, 0.004, 0.02, 0.05.
(A color version of this figure is available in the online journal.)

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Figure 8. Same as Figure 4 but for the SN Ia rate history in the solar
neighborhood with different SN Ia models: our SD model (red solid line),
SD model without metallicity effect (cyan dotted line), DD model (green shortdashed line), MR01 model (blue long-dashed line), and K98 model (magenta
dot-dashed line). The dot-dashed line shows the SN II rate. The dot shows theobserved SN Ia rate in S0a-Sb type galaxies (Cappellaro et al. 1999).
(A color version of this figure is available in the online journal.)Vol. 707

Figure 10. Cumulative function of SN Ia lifetimes in the solar neighborhoodwith our SD model at the galactic ages of t = 3 (clue dotted line), 5 (cyan
short-dashed line), 10 (green long-dashed line), and 13 Gyr (red solid line).
(A color version of this figure is available in the online journal.)
Table 5
Input Parameters of Galaxy Models: Timescales of Star Formation and Inflow
in GyrGalaxy TypeE
S0a/SaSab/SbSbc/ScScd/Sdτs (Gyr)τi (Gyr)

0.1
1.8
1.75
2.0
2.1

0.1
2.0
4.5
8.3
23.0

0.1 Gyr is possible only for such higher metallicity as Z > Z .
Figure 10 shows the time evolution of the cumulative functionof SNe Ia lifetime in the solar neighborhood, where the starformation and chemical enrichment histories are taken into
account. At the galactic age of t = 3 Gyr, only the MS+WD
system can produce SNe Ia, and thus the lifetime of all SNe Ia isless than 1 Gyr. The RG+WD systems contribute at later time.
At present (t = 13 Gyr), the fraction of SNe Ia that have shorter
lifetime than 1 Gyr is ∼50%. The total fraction with 0.1 Gyris only 10% in our model.
4. GALACTIC SUPERNOVAE RATE
4.1. SN Ia Rate in Ellipticals
Figure 9. Supernova rates against the metallicity indicators, (a) the ironabundance, (b) the gas-phase metallicity, and (c) the mean stellar metallicity,with the star formation history in the solar neighborhood.
(A color version of this figure is available in the online journal.)galaxies, however, the decrease will be seen at [O/H] ∼ −1since the observed dwarf galaxies have low [O/Fe]. For themean metallicity (panel c), which can be observed with the absorption lines of integrated stellar populations, the decrease isseen at lower [Fe/H] as ∼ −1.2. With the mass–metallicity
relation of galaxies (Tremonti et al. 2004), the metallicity threshold corresponds to MB = −15 mag and 3 × 108 M .
Mannucci et al. (2006) suggested that ∼50% of SNe Ia should
be composed of the systems with lifetimes as short as 0.1 Gyr.
On the other hand, Matteucci et al. (2006) claimed that thefraction should be less than ∼35% in order to explain the[O/Fe]–[Fe/H] relation and iron abundance in the intracluster
medium. In our model, although 50% of SNe Ia eventually
come from the MS+WD systems, the lifetime shorter thanThe present SN Ia rate in elliptical galaxies gives also
a stringent constraint on the SN Ia progenitor models.
Figure 11 shows the SN Ia rate history in ellipticals, which
is similar to Figure 5 of K00, but for the cosmological parameters of H0 = 70 km s−1 Mpc−1 , Ω0 = 0.3, λ0 = 0.7, the galactic
wind epoch of 1.5 Gyr, and the galactic age of 13 Gyr. (Note thatthe observed SN Ia rate is proportional to H02 .) The star formation history is assumed as in K00; the bulk of stars in ellipticals
are formed at z  3, and thus having ages older than 10 Gyr.
The adopted infall and star formation timescales are summarizedin Table 5. The present B–V color of 0.9 (Roberts & Haynes
1994), the mean stellar metallicity of ∼−0.2 (Kobayashi &Arimoto 1999), and the mean stellar [α/Fe] of 0.2 (Spolaor
et al. 2009) are reproduced. The present stellar mass-to-light ratio is M/LK = 1.8 and M/LB = 9.3 in our model. The resultant
SFR is similar to the SN II rate (black dot-dashed line).
1. In the DD model (green short-dashed line), the SN Ia
lifetimes are too short to explain the observed SN Ia rate at

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Figure 11. SN Ia rate history in ellipticals with different SN Ia models: our SD
model (red solid line), DD model (green short-dashed line), and MR01 model(blue long-dashed line). The dot-dashed line shows the SN II rate. The dot shows
the observed SN Ia rate in E-S0 type galaxies (Cappellaro et al. 1999).
(A color version of this figure is available in the online journal.)

the present epoch (Cappellaro et al. 1999). Toward z ∼ 4,
the SN Ia rate rapidly increases, and the SN Ia rate at z ∼ 4
is 10 times larger than the present rate.
2. In the MR01-like model (blue long-dashed line), the fraction of SNe Ia with tIa  10 Gyr is larger than the DD
model, so that the present rate is not too small comparedwith the observation. Toward z ∼ 3, the SN Ia rate gradually increases, and the SN Ia rate at z ∼ 1 and ∼ 3 is larger
by a factor of 2 and 4, respectively, than the present rate.
3. In ellipticals, the chemical enrichment takes place so early
that the metallicity effect on SNe Ia is not effective. Thus
the SN Ia rate depends almost only on the lifetime. Ourmodel (red solid line) includes the RG+WD systems withtIa  10 Gyr as the SN Ia progenitors, and thus the observed
SN Ia rate in ellipticals can be well reproduced. Towardhigher redshift, the predicted SN Ia rate does not changevery much (in the supernova units, SNu).
4. In the K98 model (see Figure 5 in K00), the lifetime
distribution shows a gap between the two components of
the MS+WD and RG+WD systems (Figure 1), so thattwo peaks are clearly seen in the SN Ia rate in ellipticals.
Such peaks are not clearly seen in our new SN Ia model
because the lifetime ranges get gradually wider for higher
metallicity (Figure 3).
The sharp peak at t ∼ 1.5 Gyr is caused both by the initialstar burst assumed in the SFR and by the spikes seen in Figure 3,
which are caused by the enhancement of the stellar lifetime fromthe occurrence of the He core flash. The galactic winds might
be driven by this enhancement of the SN Ia rate in ellipticalgalaxies. Therefore, we set the galactic wind epoch of 1.5 Gyrin our model of elliptical galaxies.
4.2. SN Ia Rate Depending on Galaxy Type
Since the star formation history is different for different typesof galaxies, the SN Ia and II rates depend on the morphologicaltype of galaxies. In the observations, the present rates areavailable for various types of galaxies, which can give a
constraint on the SN Ia models. In Figure 4 of K00, the observed
ratios between SN Ia and II rates have been reproduced with
our SN Ia model. We show the similar figure to compare withthe updated observation (Mannucci et al. 2005). We should

Figure 12. Time evolution of (a) the SFR, (b) the SN II and Ia rates, (c) thegas fraction per luminosity, and (d) the B−V color for four types of spirals (reddotted line, S0a/Sa; yellow solid line, Sab/Sb; green long-dashed line, Sbc/Sc;
blue short-dashed line, Scd/Sd). In the panel (b), the upper and lower fourlines are for the SN II and Ia rates per B-band luminosity (the supernova units,SNu), respectively. Observational data sources are: Cappellaro et al. (1999),
open circles (SN II) and filled circles (SN Ia) of S0a-Sb and Sbc-Sd in panel(b); Roberts & Haynes (1994), crosses in panels (c) and (d).
(A color version of this figure is available in the online journal.)

note, however, that the absolute rates per B-band luminosity
involves an uncertainty from dust extinction, and the rates permass involve an uncertainty in the mass-to-light ratio.
We update the evolution models of four types of spiralgalaxies, in order to meet the observational constraints on thepresent colors and the gas fractions, assuming the galactic ageof 13 Gyr. The adopted infall and star formation timescales are
summarized in Table 5. Figure 12 shows the time evolution of(a) the SFR, (c) the gas fraction per luminosity, and (d) the B −V
color for four types of spirals (red dotted line for S0a/Sa; yellow
solid line for Sab/Sb; green long-dashed line for Sbc/Sc; blue
short-dashed line for Scd/Sd). In earlier type of spirals, starformation takes place earlier, and thus the earlier-type of spirals
have older ages of stellar populations, smaller gas fractions,and redder colors at present. In the panel (b), the upper andlower four lines respectively show the SN II and Ia rates per Bband luminosity (in SNu). The observational data is taken from
Cappellaro et al. (1999) for S0a-Sb and Sbc-Sd. The present SN
II+Ibc rate is larger for later type of spirals, while the SN Ia
rate in SNu is not different so much among the various types ofgalaxies, which are roughly consistent with the observation.
Figure 13 shows the present supernova rates per K-band
luminosity (upper panel) and per mass (lower panel) against
the morphological type of galaxies. The supernova rates arelarger for later type of spirals because the SFR is higher. Almost
parallel relations between the observed rate and the galaxy typeare seen for SNe II and Ibc, which may suggest the binary
fraction is universal being independent of the galaxy type. In
our models, among core-collapse supernovae, we assume that85% and 15% are SNe II (blue dashed line) and SNe Ibc (green
dotted line), respectively. Thus, it seems to be reasonable toadopt the same b parameters as SNe Ia for all types of galaxies.
As a result, our SN Ia model (red solid line) gives excellent

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Figure 13. Present supernova rates per K-band luminosity (upper panel) and permass (lower panel) against the morphological type of galaxies for SNe II (bluedashed line), Ia (red solid line), and Ibc (green dotted line). The observational
data is taken from Mannucci et al. (2005) for SNe II (open circles), Ia (filled
circles), and Ibc (crosses).
(A color version of this figure is available in the online journal.)Vol. 707

Figure 14. Same as Figure 13, but for different SN Ia models: our SD modelwith (red solid line) and without (cyan dotted line) metallicity effect, DD model(green short-dashed line), and MR01 model (blue long-dashed line).
(A color version of this figure is available in the online journal.)

agreement with the observed SN Ia rates. In our SN Ia models,there are two types of progenitor systems; The SN Ia rate is larger
for later-type spirals, which is due to the young population of
the MS+WD systems. The slope of the SN Ia rate against thegalaxy type is flatter than that of the SN II and Ibc rates, whichis due to the old population of the RG+WD systems.
The observational data are taken from Mannucci et al.
(2005), where the adopted mass-to-light ratios are M/LK ∼0.8, 0.7, 0.5, and 0.4 for E/S0, S0a/b, Sbc/d, and Irr in their
Table 2. In our models, the “stellar” mass-to-light ratios are
M/LK ∼ 1.29, 1.14, 1.07, and 1.03 for S0a/Sa, Sab/Sb, Sbc/
Sc, and Scd/Sd, which are ∼2 times larger than their ratios.
This difference might come from the difference in the IMF. We
thus plot the observational data multiplied by a factor of 2 forthe rates per mass.
Figure 14 compares the galactic supernova rates for different
SN Ia models.
1. In the DD model (green short-dashed line), the lifetime of
the majority of SNe Ia is short, no large difference between
the SN II and Ia rates is seen in the trend against the galaxy
type, which is not consistent with the observations.
2. The MR01-like model (blue long-dashed line) and our SD
models (red solid line and cyan dotted line) give almost thesame trends, regardless of the metallicity effect, as far as
the lifetime distributions in this work and K98 are applied.
4.3. Cosmic Supernovae Rate
Galaxies have various timescales for star formation andchemical enrichment, and the occurrence of SNe Ia depends
on the metallicity therein. Therefore, we should calculate thecosmic supernova rate by summing up the supernova rates
in spirals and ellipticals with the ratios of the relative masscontribution in the Universe (see K00 for the detail). As in
Figure 6 of K00, the resultant cosmic SFRs shows a peak atz ∼ 2. Figure 15 shows the cosmic supernova rates with our

Figure 15. Cosmic SN Ia rate histories (green lines) as a composite of thosein spirals (blue lines) and ellipticals (red lines) in the supernova units (upper
panel) and in the units of the comoving density (lower panel) for our SD modelwith (solid lines) and without (dotted lines) metallicity effect. Observational
data sources are: Cappellaro et al. (1999), open triangles (for total, Ss, andEs); Hamuy & Pinto (1999), asterisk; Hardin et al. (2000), four-pointed star;
Pain et al. (1996), open pentagon; Pain et al. (2002), filled pentagon; Riess(1996), open diamonds; Madgwick et al. (2003), cross; Blanc et al. (2004),
filled square; Cappellaro et al. (2005), inverted open triangle; Botticella et al.
(2008), three-pointed stars; Neil et al. (2006), filled triangle; Tonry et al. (2003),
filled diamond; Dahlen et al. (2004), filled circles; Barris & Tonry (2006), open
squares; Kuznetsova et al. (2008), open circles; Poznanski et al. (2007), stars.
The SN II rate history is shown with the green dot-dashed lines on the top, andthe observed SN II rates are shown with gray color.
(A color version of this figure is available in the online journal.)

new SN Ia model (solid lines), as a composite of those in spirals(blue lines) and ellipticals (red lines) in SNu (upper panel) andin the unit of the comoving density (lower panel), respectively.
The SN II rate (green dot-dashed lines) traces the star formation

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seen in our new SN Ia model, because the lifetime ranges
get gradually wider for higher metallicity (Figure 3).
The observed cosmic SFR shows an increase by a factor of∼10 from z = 0 to ∼1 (e.g., Schiminovich et al. 2005), andsuch a large increase is not seen in the composite SFRs of thesemodels for ellipticals and spirals. Such a large increase in the
SFR results in the rapid increase in SN II and Ia rates from z = 0
to ∼1 that may be seen in the observed SN Ia rates. By modifying
the SFRs in spirals as well as including the contribution ofirregular galaxies, there can be a better fit (Calura & Matteucci2006). However, we do not apply such parameter-tuning in thispaper because other effects are more important in the cosmic
chemical enrichment: (1) the number evolution of each type ofgalaxies, (2) merging of galaxies, (3) internal structure, namely,
metallicity gradients within galaxies, (4) the contribution of
dwarf galaxies. We will rather show the cosmic supernova rates
with fully self-consistent cosmological simulations (Kobayashi
et al. 2007; Kobayashi 2008). Nonetheless, our result with thesimple assumptions is robust; the SN Ia rate is high in ellipticals
and drops in spirals at high redshift.
Figure 16. Same as Figure 15, but for DD model (short-dashed lines) and MR01
model (long-dashed lines).
(A color version of this figure is available in the online journal.)

history, and shows a peak at z ∼ 2 in the comoving density,
being consistent with the observations (Cappellaro et al. 2005;
Botticella et al. 2008).
As in spirals, the total SN Ia rate (green solid lines) gradually
decreases toward z ∼ 2.6 in the unit per luminosity, while thetotal SN Ia rate slightly increases to z ∼ 1 and then decreases
in the unit per volume. In ellipticals, the chemical enrichment
takes place so quickly that the metallicity is large enough to
produce SNe Ia at z  2. These are broadly consistent withthe observations (e.g., Pain et al. 2002, Supernova Cosmology
Project, 38 SNe Ia; Tonry et al. 2003, High-z Supernova Search
Team, 8 SNe Ia; Dahlen et al. 2004, Great Observatories
Origins Deep Survey, 25 SNe Ia; Barris & Tonry 2006, Institutefor Astronomy Deep Survey, 98 SNe Ia?; Neil et al. 2006,
Supernova Legacy Survey, 58 SNe Ia). Around z ∼ 0.5, thetotal SN Ia rate may be smaller than the observations, although
Poznanski et al. (2007) shows a significant lower rate with the
Subaru Deep Field.
The cosmic SN Ia rate history for different SN Ia models arealso shown in Figures 15 and 16.
1. If we do not include the metallicity effect (dotted lines in
Figure 15), the SN Ia rate in spirals does not decrease somuch, and the total SN Ia rate per luminosity is almost
constant. The rate per volume slightly increases towardhigher redshifts, and is larger than the present rate by afactor of ∼2 at z ∼ 2. In the observations, the decrease in
the total rate is seen, which supports the existence of themetallicity effect.
2. In the DD (short-dashed lines in Figure 16) and MR01-like(long-dashed lines in Figure 16) models, the lifetime of the
majority of SNe Ia is so short that the rate in ellipticals
keeps on increasing at z  4, and that the rate in spirals
shows a peak at z ∼ 1.5. As a result, the total SN Ia rate
per luminosity is almost constant, and the rate per volume
keeps on increasing toward higher redshifts. The total SN
Ia rate is larger than the present rate by a factor of ∼2 and∼4 at z ∼ 1 and 2, respectively.
3. In the K98 model (see Figure 6 in K00), two peaks are seenin the SN Ia rate history in ellipticals, which are not clearly

5. DISCUSSION
5.1. SN Ia Formulation
In galactic chemical evolution models, the following two formulations have been proposed. Their parameterizations are very
different, and it would be worth summering here. Observationsof binary systems could give a clue to improve these formulations.
1. The formulation of the SN Ia rate has been first proposedby Greggio & Renzini (1983). The SN Ia rate is describedas MB,sup μmax
RIa = A
φ(MB )
f (μ) ψ(t − τM2 ) dμ dMB .
MB,inf

μmin(1)
The first integral is for the total mass of the binary
MB = M1 + M2 , and the second integral is for the massfraction of the secondary star, μ = M2 /MB . In MR01, theScalo IMF φ is adopted, and the distribution function of
the mass fraction is described as f (μ) = 21+γ (1 + γ )μγ .
γ = 2 is adopted in their successful models, which gives
weight for massive secondaries in the distribution functionof mass ratios (see Figure 19). The upper and lower
limits of integrals are MB,inf = max[2M2 (t), MBm ] andMB,sup = MBM /2 + M2 (t), where the limits to the total
mass of the binary are MBm = 3 M and MBM = 16 M .
Therefore, the initial mass range of the primary stars is
1.5 M  M1  8 M and the secondary stars are AGB
stars with the initial mass of M2  8 M . The lower limit of
the primary, however, should not 1.5 M but ∼3 M , since
the primary should be a C+O WD. Although A is called
as the binary fraction, A actually indicates the fraction ofbinary systems with suitable separation to become SNe Ia(Greggio & Renzini 1983).
2. Based on the SD scenario, an alternative formulation has
been proposed by K98 and K00. The SN Ia rate is given as mp,u1
φ(m)dm
RIa = bmax[mp, , mt ] m
 md,u1×ψ(t − τm ) φd (m)dm. (2)max[md, , mt ] m

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These integrals are calculated separately for the primary andsecondary stars (mp and md ), respectively. In our works, theSalpeter IMF φ is adopted, and the distribution functionof the companion’s mass is assumed to be the powerlaw as φd (m) ∝ m−0.35 . The slope −0.35 is chosen tomeet the observations (e.g., Duquennoy & Mayor 1991),
and such negative slope gives the distribution function
weighted for less-massive secondaries (see Figure 19).
φd is normalized to unity in the integrated mass rangesm
as md,d,u m1 φd (m) dm = 1. Since the primary stars are
C+O WDs, the initial mass range of the primary stars is
mp = 3–8 M . The initial mass ranges of the companion
stars, md, and md,u , are given by the simulation of binary
evolution (Hachisu et al. 1999a, 1999b) for the RG+WD
and MS+WD systems. The binary parameter b denotes thetotal fraction of primary stars that eventually explode as
SNe Ia, and is determined from the chemical evolution
in the solar neighborhood. b includes the binary fraction
and the efficiency for each binary system. In addition,
the metallicity effect is taken into account (Section 2).
The lowest metallicity to produce SNe Ia was set to be[Fe/H] = −1.1 both in K98 and K00.7 In our new model(Section 2.2), the mass stripping effect is taken into account,and md, and md,u are given as a function of metallicity as
in Table 4.
Strictly speaking, md, and md,u depend on m1 , but it canbe neglected. For MWD = 0.7 M , the mass range is a little
narrower, and thus the shortest lifetime is 0.2 Gyr that islonger than 0.1 Gyr for MWD = 1.0 M (Z = 0.02). The
range of the orbital period is much narrower, so that thecontribution of MWD < 1.0 M is smaller. On the other hand,for MWD = 1.1 M , the shortest lifetime is a little shorter,
but the number of such WD supposed to be very small. As aresult, the [X/Fe]–[Fe/H] relation is mainly determined by thecontribution for MWD = 1.0 M .
The results slightly depend to some extent on the stellar lifetime function τm of low and intermediate mass stars.
Figure 17 shows the adopted stellar lifetime function as a function of initial mass and metallicity, which is provided by Kodama& Arimoto (1997). The stellar evolution tracks are calculatedwith the code described in Iwamoto & Saio (1999) with the coreovershooting. Because of the coarse mass grid, there is a tiny
irregularity at 1.8 M (Z = 0.004, 0.008), 2.0 M (Z = 0.02),
2.2 M (Z = 0.0001–0.002), and 3.3 M (Z = 0.0002) in thetable, which we have smoothed with interpolation. The stellar
lifetime function overall agrees with more recent models suchas Karakas & Lattanzio (2007). The long and short dashed-lines
are for the analytical functions adopted in MR01 and HKN08,
respectively, where the metallicity effect is neglected. With thoseanalytical functions, the stellar lifetime is systematically shorter.
We adopt the Salpeter IMF with a slope of −1.35. Kroupa(2008) showed a similar slope (−1.3 ± 0.5) for m  0.5 M ,which is flatter than in the Scalo IMF. For m < 0.5 M , the slope
is shallower and the number of brown dwarfs are smaller thanin the Salpeter IMF. In chemical evolution models, however,metal enrichment is proceeded only by stars with m  0.5 M .
If a suitable value is chosen for the lower mass-cut of the
IMF, the Salpeter IMF can give equal results with the Kroupa
IMF. Therefore, we adopt the single slope for a mass range ofm = 0.07–50 M . This does not necessarily mean that there
7 The note in MR01 is incorrect. See K98, .16 in the right column, p.156,
and K00, .15 in the right column, p.27.

Vol. 707

Figure 17. Stellar lifetime as a function of initial mass and metallicity. Thelong and short dashed-lines are for the analytic functions adopted in MR01 andHachisu et al. (2008), respectively.
(A color version of this figure is available in the online journal.)

is no star at m < 0.07 M . The requirement is that the massfraction of m = 0.5–50 M is 45%.
5.2. Binary Parameters
In the two formulations, Equations (1) and (2), the treatmentof binary systems is different for the following three points:(1) binary fraction, (2) mass ratio, and (3) binary IMF. These
differences may reflect the formation processes of binary systems. Theoretically, the prompt fragmentation scenario seem to
be more feasible than captures or delayed breakup of accretion
disks (Tohline 2002), and these scenarios will be tested withthe observed distribution functions of the binary parameters. In
chemical evolution models, however, these uncertainties can becancelled out by the choice of b or A.
1. Binary fraction. The total number of SNe Ia, namely, thebinary fraction A or binary parameters b, could be determined from the binary population synthesis. However, theevolution of binary systems is complicated and the effect of
the WD winds is not included. In Equation (2), we treat b as
parameters being determined for the RG+WD and MS+WD
systems, independently, from the requirement to meet theobserved MDF of G-dwarf stars and the [O/Fe]–[Fe/H]
relation in the solar neighborhood. In K98, we adoptedbRG = bMS = 0.04 from the Kolmogorov–Smirnov (KS)test of the [O/Fe]–[Fe/H] relation, but in K00, we adopted[bRG , bMS ] = [0.02, 0.05] from the χ 2 test.8 In K06, we
adopted smaller values of [0.02, 0.04] because Fe are also
produced by HNe in the updated nucleosynthesis yields.
For our new SN Ia model, [bRG , bMS ] = [0.023, 0.023] arechosen at Z = 0.004, which becomes [0.040, 0.032] at
Z = 0.05, as discussed in Section 3. In total, the fraction of
8

There are two misprints in the appendix of K00, not[bMS = 0.02, bRG = 0.05] but [bRG = 0.02, bMS = 0.05].

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Figure 19. Distribution function of mass ratios: (q) ∝ m−(1+x) with x = 0.35
(solid line, this work), constant f (μ) (dotted line), constant f (q) (short-dashedline), and f (μ) ∝ μγ with γ = 2 (long-dashed line, MR01). The histogramshows the observational data (Duquennoy & Mayor 1991).
(A color version of this figure is available in the online journal.)

Figure 18. (a) The [O/Fe]–[Fe/H] relation in the solar neighborhood for
different SN Ia models. The solid line is for our original K98 model. The dottedline is calculated with MR01 formula in Equation (1) with the appropriate
parameters for K98 model (γ = −1.35, A = 0.45). The dashed and longdashed lines correspond to Model 1 and 3 in MR01, respectively. Observational
data sources are: for disk stars, Edvardsson et al. (1993), small open circles;
thin disk stars in Bensby et al. (2004), small filled circles; accretion component
in Gratton et al. (2003), triangles. For halo stars, Cayrel et al. (2004), large
filled circles. (b) The time evolution of SN Ia rate in SNu. The dot is for theobservational data for S0a-Sb (Cappellaro et al. 1999).
(A color version of this figure is available in the online journal.)

SNe Ia is 5%–7%, which is consistent with the constraints
in Maoz (2008) except for the X-ray observations.
In Equation (1), the definition of A is totally different
from that of b, and A can be simply determined from
the MDF. This is misunderstood in Matteucci & Recchi(2001), where they claimed that they failed to reproduce theK98 results (Mod.1 in MR01). We would demonstrate in
Figure 18 that we can reproduce their and our results. For thecomparison, we adopt the same IMF, stellar lifetimes, andnucleosynthesis yields as in our models. Adopting MR01’s
formula Equation (1) and parameters, we almost reproduce
Mod.1 and Mod.3 in MR01 (short- and long-dashed lines,
respectively). Here we adopt A = 0.035 to meet the MDF,
which is better than A = 0.05 originally adopted in MR01.
In Mod.3 in MR01, A = 0.02 and 0.05 are adopted for the
RG+WD and MS+WD systems, respectively, which are thesame values of the binary parameter b in K00. However,since φd is normalized to unity not for m = 0.07–50 M but
for md ∼ 1–3 M , the value of A should be 10 times larger
than the value of b. Adopting A = 0.45, we reproduce ouroriginal K98 model (solid line) even with the MR01 formula(dotted line). With the MR01 formula, the slope of [O/Fe]
is slightly different, and [O/Fe] around [Fe/H] ∼ −0.5 isslightly higher than the K98 model. The best choice of A
depends on the two other assumptions (i and ii), and varies
in the range of A = 0.10–0.45 as shown below. This valueof A is larger than in the MR01-like model because theadopted mass ranges both for primary and secondary stars
are narrower.
2. Mass ratio. In Figure 19, we show several functions of
the mass ratio f (q) or the mass fraction f (μ) (Note thatμ12q ≡ m2 /m1 = 1−μand dμ= (1+q)2 = (1 − μ) ). The
dqfunction that is introduced in MR01 is weighted for massive
secondaries (i.e., the positive value of γ , long-dashedline). Among the binary population synthesis, several
functions are adopted: f (q) ∝ (1 + q)−2 , i.e., constantf (μ) (dotted line, Portegies Zwart 1995), and constantf (q), i.e., f (μ) ∝ 1/(1 − μ)2 (short-dashed line; Han et al.
1995). Observations of binary systems, however, suggestthat the function is weighted for less-massive secondaries,
as in our function (solid line). The histogram shows theobservation of G-dwarfs by Duquennoy & Mayor (1991).
For B-type stars (Shatsky & Tokovinin 2002; Kouwenhoven
et al. 2005), the power-law distributions with the slopes of−0.33 to −0.5 are reported, and the random paring of stars
is not supported. Note that the shape of f (q) is affectedby selection bias (Hogeveen 1992), and may depend on thedensity and composition of the stellar aggregate where thebinaries have been formed (Kroupa 1995). Our function
is between these observations; the secondary IMF φd isweighted for less-massive secondaries, and the slope isshallower than that of the primary IMF φ.
In Figure 20, we show that the mass ratio function does notchange our results. In other words, it is possible to obtain
almost identical results with different sets of A and f (μ).
For our SN Ia model, although MR01 adopted γ = −0.35,
γ = −1.35 is correct because the number function shouldbe the mass function φd multiplied by 1/m. Since f (μ)
with γ = −1.35 is weighted for low-mass secondaries,
the relative contribution of the MS+WD systems is small,

1482

KOBAYASHI & NOMOTO

Vol. 707
Table 6
Test Results and Predictions of the SN Ia Models
SD

Metallicity Effect

Yes

MR01

DD

No

Lifetime (Gyr)0.1–20 peak at 0.1, 1
∼0.3
∼0.1
[α/Fe]–[Fe/H] relation
×××Present SN Ia rate in Es×Cosmic SN Ia rate history
Peak at z ∼ 1 Constant Increase Increase
SNe Ia observed at z  2 in Ss?
few
Yes
Yes
Yes

As a summary, once a suitable b or A is adopted, almost
identical results can be produced with either of these two
formulations. The most important factor is the mass rangesof companion stars, which changes the distribution functionof SN Ia lifetimes. The total number of SNe Ia, b or A, aredetermined from the observational constraints in this paper, but
may be larger than what can be obtained with binary population
synthesis models (e.g., Maoz 2008). Other assumptions, themass ratio function and IMF, are not very important since these
two effects can be cancelled out by the choice of the total numberof SNe Ia. Observations of binary systems and the theory ofbinary formation could give a clue to solve this degeneracy, andmay put more constraints on SN Ia models.
6. CONCLUSIONS

Figure 20. Same as Figure 18 but for different SN Ia models. The solid line iscalculated with MR01 formula in Equation (1) with the appropriate parametersfor K98 model. The dotted and short-dashed lines are for the different functionsof mass ratios, constant f (μ) and constant f (q), respectively. The long-dashedline is for the same f (μ) as in MR01, but with our parameters of binary
companions.
(A color version of this figure is available in the online journal.)and A should be as large as 0.45. With the constant f (μ)
or constant f (q), the contribution of the MS+WD systems
is as large as that of the RG+WD systems, and thus A
becomes 0.30 and 0.25, respectively. With γ = 2 as in
MR01, A = 0.20 gives the best fit.
3. Binary IMF. The IMF is applied to the total mass of thebinary in Equation (1), while it is applied to the mass ofthe primary star in Equation (2). This may correspondto different approaches in the formation of binaries: acommon or individual formation. This treatment also does
not change our results. If we adopt not φ(mB ) but φ(m1 ),the normalization is changed by a factor. For γ = −1.35,
constant f (μ), constant f (q), and γ = 2, A becomes 0.25,
0.15, 0.12, and 0.10, respectively.
Needless to say, for the models that give the same [O/Fe][Fe/H] relation, the SN Ia rate history is almost the same. In
the lower panels of Figures 18 and 20, the time evolution of
SN Ia rate is shown. Only 0.05 dex difference is seen in thepresent SN Ia rate among the models with different binary massfractions. The present SN Ia rate with γ = −1.35 is 0.1 dex
larger than with γ = 2. It seems not possible to put constraint
on the binary mass fraction and binary IMF from the observed
SN Ia rate because of the large errorbar.

We construct a new SN Ia model, based on the SD scenario
for the SN Ia progenitors, taking account of the metallicity
dependences of the WD wind (K98) and the mass-stripping
effect on the binary companion star (HKN08). The lifetime
distribution of SNe Ia is determined from the main-sequencemass range of the companion stars in the WD binary systems,
and is given as a function of metallicity (Figure 3). The
SN Ia rate in the systems with [Fe/H]  −1, e.g., r 
10re in intermediate-mass galaxies, r  50 kpc in dwarf
galaxies, and high-z spiral galaxies, is supposed to be very small(Section 2.3).
Our model naturally predicts that the SN Ia lifetime distribution spans a range of tIa ∼ 0.1–20 Gyr with the double peaks
at ∼0.1 and 1 Gyr reflecting the two systems of the companion
stars: the MS+WD and RG+WD systems, respectively. With thechemical enrichment history in the Galactic disk, the fraction of
the young population of SNe Ia is 10% and 50% for tIa < 0.1
and 1 Gyr, respectively (Figure 10). In the galaxies with a initialstar burst, the SN Ia rate shows a strong peak at t ∼ 1.5 Gyr,which may generate the SN Ia driven galactic winds.
We make a comparison of the lifetime distribution functions
derived from different SN Ia models, and evaluate the resultsfrom the observational constraints as summarized in Table 6.
1. Contrary to the argument in Matteucci & Recchi (2001,
MR01), our SN Ia models give better reproduction of the[α/Fe]-[Fe/H] relation in the solar neighborhood. In the
DD and MR01-like models, the typical lifetimes of SNe Ia
are ∼0.1 and 0.3 Gyr, respectively, which results in the
too early decrease in [α/Fe] at [Fe/H] ∼ −2. If we do notinclude the metallicity effect in our model, [α/Fe] decreases
too early because of the shortest lifetime, ∼0.1 Gyr, of the
MS+WD systems. In other words, the metallicity effectis more strongly required in the presence of the young
population of SNe Ia, to be consistent with the chemical
evolution of the solar neighborhood.
2. At the same [Fe/H], [Mn/Fe] starts to increase towardhigher metallicity, because SNe Ia produce [Mn/Fe] > 0.

No. 2, 2009

ROLE OF SNe Ia IN CHEMICAL EVOLUTION. I.

This evolutionary change is started from the same [Fe/H] as
the decreasing trends of [α/Fe]. [Zn/Fe] evolves differently
for the different SNe Ia models, because of the combinationof the different lifetime of SNe Ia and the metallicity effecton Zn yields of SNe II. Further observations of theseelements at [Fe/H]  −1.5 are critically important inorder to identify the SN Ia progenitors and to clarify themetallicity dependence on the nucleosynthesis yields. The
lack of the similar [Mn/Fe] increase in dSphs suggests thatthose are not enriched by SNe Ia.
3. To explain the observed SN Ia in ellipticals at z = 0, thelifetimes of a sufficient fraction of SNe Ia should be longer
than ∼10 Gyr, which is satisfied in our SD model andthe MR01-like model, but not in the DD model. Our SN
Ia models are also successful in reproducing the SN II,
Ibc, and Ia rates with their dependence on the galaxy type
(Mannucci et al. 2006); This cannot be reproduced with the
DD model. The large SN Ia rate in radio galaxies could
be explained with the young population of the MS+WD
systems in our model.
4. We also provide the cosmic supernova rate history as acomposite of those in spirals and ellipticals. The cosmic
SN Ia rate in comoving density shows a peak at z ∼ 1 anddecreases toward higher redshift in our model. In contrast,
this rate increases toward higher redshifts in the DD and
MR01-like models. Because of the metallicity effect, i.e.,
because of the lack of winds from WDs in the binary
systems at [Fe/H] < −1.1, the SN Ia rate in the lowmetallicity systems, e.g., high-z spiral galaxies, is supposed
to be very small in our model. This metallicity effect willappear only around [Fe/H] ∼ −1 (Figure 9). In contrast,
the SN Ia rate in such systems is as high as present in the
DD and MR01-like models. In our models, at z  1, SNe Ia
will be observed only in the systems that have evolved witha short timescale of chemical enrichment. This suggeststhat the evolution effect in the supernova cosmology can besmall.
This work has been supported in part by World PremierInternational Research Center Initiative (WPI Initiative), theMinistry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, and by the Grant-in-Aid for Scientific Research of the Japan Society for Promotion of Science (JSPS) (18104003, 18540231, 20244035, 20540226) andMEXT (19047004, 20040004), and also by the National Science
Foundation under Grant No. PHY05-51164 at Kavli Institute for
Theoretical Physics of USA. C.K. thanks JSPS for a financial
support. We thank I. Hachisu, M. Kato, M. Tanaka, F. Matteucci,
S. Recchi, M. Della Valle, P. Podsiadlowski, B. Schmidt, R.
Ellis, L. Greggio, and A. Renzini for fruitful discussion. We areparticularly grateful to the late Prof. B. E. J. Pagel for generous
suggestion and encouragement.
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