CHEMICAL YIELDS IN SPH SIMULATIONS
A data-driven statistical approach
Francisco J. Martínez-Serrano1, Arturo Serna1, Mercedes Molla2 , Rosa DomínguezTenreiro3
1 Dpt. Física y A.C., Universidad Miguel Hernández, E-03206 Elche, Alicante, Spain
2 Dpt. Fusión y Física de Partículas, CIEMAT, Avda. Complutense 22, E-28040 Madrid, Spain
3 Dpt. Física Teórica C-XI, Universidad Autónoma de Madrid, E-28049 Cantoblanco, Madrid,
Spain
1.
Introduction
It is important to trace the evolution of the chemical abundances in the universe. The main ingredients in order to solve it are the results from the theory
of stellar evolution, and the equations of chemical evolution giving the diffusion and increase of metals of the interstellar medium (ISM).
Several approaches have been devised to deal with the problem of element
production and its mixing into the ISM. Chemo–dynamical models for individual galaxies, considering a multi–phase ISM have been developed by Ferrini et al. [1], Samland et al. [2] or Molla et al. [3] among others. Gravitohydrodynamic codes also allow us to follow the evolution of the abundances
through the motion of the gas and diffusion. It has also been more recently
addressed at cosmic scales by Mosconi et al. [4], Lia et al. [5], Cora & White
[6], Portinari et al. [7], Kobayashi [8] and Aguirre et al. [9].
2.
The simulation algorithm
In this work we follow the approach by Lia et al. [5], who adapted the
Smoothed Particle Hydrodynamics (SPH) algorithm to handle the production
and diffusion of the different metals. Their approach is probabilistic, because
the creation and destruction of star-like particles happens at random, according
to probabilities computed to match the rate of gas ejection of an stellar population given by the theory of stellar evolution. This approach differs from the one
followed by Mosconi et al. [4] because it does not consider that each baryonic
particle is composed of a gas and a star part: they are either gas or stars. This
is physically motivated since star particles are fully collisionless and do not
2
follow gas particles, allowing the introduction of metal diffusion. It is based
on the SPH translation of the diffusion equation, however with this approach
more particles are needed to map the same configuration.
We go a step further and develop an adaptation of a multistep AP3M-likeSPH code, DEVA (Serna et al. [10]), to follow the evolution of abundances of
17 isotopes as they are produced in stars and diffuse later. DEVA pays special
attention towards conservation laws by including the so-called ∇h terms in the
SPH equations, which improve energy and entropy conservation. DEVA also
conserves angular momentum for a low number of particles, something which
is crucial in disk formation and evolution. The ∇h terms proved to be important in high density regions, associated either to central cores of collapsed
objects or shock fronts, that are the regions where star formation and metal
enrichment first happen and are more efficient. Since metal enrichment determines the evolution of subsequent stellar generations, the ∇h terms can help
to increase the accuracy of the metallicity distributions obtained.
Our approach starts from that of Lia et al. [5], and takes into account the
metallicity dependence of the yields and stellar lifetimes. This biparametric
(on time and metallicity) approach forces us to abandon the simple analytical
relations for the representation of the metal yields used by those authors and
to develop instead a method based on tables of computed values as we will
describe.
In the SPH algorithm each particle represents a fluid element whose position, velocity, energy, density, chemical composition, etc. are followed in time.
The properties of the fluid at an arbitrary position are locally estimated by interpolation of the properties of its neighbors, the chemical abundances are one
of these properties.
As explained in Serna et al. [10], a phenomenological parameterization
based on the empirical Kennicutt-Schmidt law is used to trigger star formation.
A probability to become a star-like particle is assigned to each gas particle, according to a Schmidt-like transformation rule based on the local gas density.
Particles are transformed into stars according to this probability.
Once a particle becomes a star-like one, it no longer interacts hydrodynamically with its neighbors. A star particle can be considered to represent a starburst, effectively hosting a Single Stellar Population (SSP). The metallicity of
this SSP influences its later release of metals and its probability of becoming
a gas-again particle. As time passes since its birth, stars begin to die, according to τM (z, M ), which is taken from Portinari et al. [11], and the (hidden)
gas fraction of the particle increases, thus effectively increasing its probability
gt (t) of becoming a gas-again particle:
3
Chemical yields in SPH simulations
gt (∆t) =
Z
t+∆t
e(t)dt
t
(1)
1 − E(t)
where e(t) = dE(t)
dt represents the ejection rate of gas at time t, and ∆t stands
for the chemical simulation step (see details in Lia et al. [5]).
Once it is decided, according to gt (t), that a star-like particle becomes a
gas-again particle, it has to be assigned new abundances, which are computed
by using the method described in Section 3.
Diffusion of metals in the gas is achieved by the SPH translation of the
diffusion equation
dZ
= −κ∇2 Z
(2)
dt
where the diffusion coefficient κ is computed from the size of a supernova
remnant 106 yr after the explosion, given a typical velocity of 50 km s−1 .
3.
The stellar yields
Following the basic equations of the chemical evolution field (Ferrini et al.
[1], Pagel [12], Portinari et al. [11], Lia et al. 2002 [5]), the ejected mass of a
SSP is:
E(t) =
Z
t
e(t0 )dt0
(3)
τmin
where:
e(t) = (M − Mr )Φ(M ) −
dM
dt
,
(4)
M =M (t)
τmin stands for the life time of the most massive star, Mr the remnant of a
star of mass M , shown in Fig.1, as the function M(t). Φ(M ) is the Initial
Mass Function (IMF). As this work constitutes a first approximation, only a
Salpeter IMF, with a power law x = −2.35, was considered, but it is trivial to
implement another one, such as the one given by Elmegreen [13].
The ejection of each element z, equivalently is:
Ez (t) =
Z
t
ez (t0 )dt0
(5)
τmin
where
ez (t) = ((M − Mr )Xz,0 + yz )Φ(M ) −
which we may decompose into two terms:
dM
dt
(6)
M =M (t)
4
ez (t) = Xz,0 e(t) + pz (t)
(7)
The first term takes into account the elements already present in the star
particle at the moment of its creation, while the second one pz (t) is the stellar
production of new metals:
dM
pz (t) = yz Φ(M ) −
dt
(8)
M =M (t)
The stellar yields yz are the new elements which each star creates in its interior by the nucleosynthesis process, and they are given by different authors
from the stellar evolution field. In this case we take those from the models of
Portinari et al. [11] for the high-mass stars and Marigo [14], for the intermediate and low-mas stars. The calculation will be repeated in the near future for
other set of yields, mainly the ones by Molla, Gavilán & Buell [15], which take
into account the effects of the convective dredge up and the hot bottom burning , incorporating the most recent improvements in the TP-AGB processes.
As both sets of yields are metallicity dependent, our computation of the new
abundances is also metallicity dependent.
Therefore, the calculations for computing abundances of a SSP reduce to
compute e(t) and pz (t). This method involves manipulation, including derivatives, of the input data (remnants and stellar yields), which are usually given
as tables. Since they are metallicity dependent, our functions are actually
e(z, t) and pz (1, t). To calculate a continuous function of the two parameters
(metallicity and time), an interpolation between known points must be performed. Furthermore, although the given models represent a significant part
of the whole function, particles may happen to form in the simulation with a
metallicity outside the known range, thus extrapolation may be needed.
Once the given functions, e(z, t) and pz (z, t), have analytical approximations, we can compute the total ejection E(t), following Eq. 3 and the integrated yields Pz (z, t), integrating them along time:
Pz (z, t) =
Z
t
τmin
pz (z, t)dt
(9)
and give tabulated values for the application range of the models. The integration is performed numerically using proper integrating algorithms. The
integrated yields Pz (z, t) for some metals are plotted in Fig. 2 along with the
integrated total ejection E(t).
4.
DEVA calculation results for a SSP
To test the metal production implementation in DEVA, the same test as in
Lia et al. [5] has been implemented. The test consists of studying the production of metals in a single burst of star formation. 5000 particles are turned into
5
Chemical yields in SPH simulations
MΤ Hz,log10 HtLL
Mr Hz,ML
100
75
50 M HM L
Τ
Ÿ
25
0
10
0.05
0.04
0.03
Mr HMŸ L
5
0
100
0.01
0.02
0.03
0.04
z
z 0.02
80
9
60
40
M0 HMŸ L
20
7
8
log10 HtL
0.01
10
0.05
SnIa
0.003
rSnIa
0.002
0.001
0
0
10
0.01
9
log10 HtL
0.02
0.03
z
0.04
8
0.05
Figure 1.
Functions Mr (z, M ), Mτ (z, t) and SnIa rate
stars and its gas fraction (fraction of gas-again particles) and gas metallicity are
studied. The tabulated values for Pz (z, t) and E(z, t) are used by the DEVA
code by means of bilinear interpolation.
When, according to the probability gt (∆t), a star particle transforms into a
gas-again particle, it is assigned new abundances, namely
Pz (t + ∆t) − Pz (t) + MzIa (RSnIa (t + ∆t) − RSnIa(t))
+ Xz0
E(t + ∆t) − E(t)
(10)
here t is the time since the gas became a star and ∆t is the chemical timestep,
RSnIa is the integrated rate of SnIa as given in Fig. 1, MzIa represents the
chemical ejecta for the given metal in a SnIa, and Xz0 = z represents the intial
metallicity of the stellar particle. The W7 model from Iwamoto et al. [16] is
used for this parameter.
Since 5000 SSPs together are a SSP, we can make the same computation
analytically as follows:
Xz (∆t) =
[Z] =
Pz (z, t) + MzIa RSnIa (z, t) + E(z, t)Z0
E(z, t)
where Z stands for any metal.
(11)
6
1
EHz,log10 HtLL
log10 HtL 10
10
H
log10 HtL
9
9
8
8
7
7
0.3
0
-0.01
P
z
-0.02
-0.03
-0.04
E0.2
0.1
0
0.01
0.01
0.02
0.02
0.03
0.04
z
0.03
0.04
z
0.05
3
10
0.05
12
He
10
log10 HtL
9
C
log10 HtL
9
8
8
7
7
0.008
0.006
Pz
0.004
0.002
0
0.00004
Pz
0.00002
0
0.01
0.01
0.02
0.02
0.03
0.04
z
0.03
0.04
z
0.05
16
10
0.05
56
O
10
log10 HtL
9
Fe
log10 HtL
9
8
8
7
7
0.01
0.0008
0.0006
Pz
0.0004
0.0002
0
0.0075
Pz
0.005
0.0025
0
0.01
0.01
0.02
0.02
0.03
0.04
z
0.03
0.04
z
0.05
Figure 2.
0.05
Functions E(z, t) and Pz (z, t) for 1 H, 2 He, 12 C, 16 O and 56 Fe
The results of this test are presented in Fig. 3, where the evolution of the
total abundance of the returned gas (computed as 1 − zH − zHe ), and abundances of 12 C, 16 O and 56 Fe for a metal-free SSP can be appreciated. As it
can be seen, metallicity is high at the beginning, when the highly enriched gas
from massive stars is expelled. Later on, this metallicity is diluted by the less
metallic gas ejected by long lived stars. The abundance of iron is an exception,
as it grows later due to the contribution of type Ia supernovae. In spite of the
statistical fluctuations of the probabilistic process, it follows very closely the
analytical prediction.
Our results agree, in shape and magnitude to those in Lia et al. [5], and in
fact they match closely when the stellar evolution data for solar metallicity are
taken, except for 56 Fe which increases strongly for later times in our case. We
note, however, that the same initial metallicity must be taken for the selection
7
Chemical yields in SPH simulations
of the stellar yield table and for calculations of Eq.(10) and (11). Otherwise,
the calculations result to be inconsistent. In fact, the lowest metallicity available in the yield tables must be used when a metal free gas is assumed in the
simulation, as both Lia et al. and we assume. Since those authors do not use
metallicity dependent yields, they must perfom their calculations as if an effective solar abundance was actually used for yields, although a low Z is included
in Eq.(10). We explain the existing differences between both result sets, as an
effect of this inconsistent selection of metallicities, as it can be seen by looking
at Fig. 2, where the integrated yields of 16 O and 56 Fe are represented. These
yields are greater for metal-free stars.
Z
Z
12
@ 12 CD
C
0.25
0.02
0.2
0.015
0.15
0.01
0.1
0.005
0.05
7.5
6.5
8
8.5
16
@ 16 OD
9
9.5
10
log10 HtL
7.5
6.5
56
@ 56 FeD
O
8
8.5
9
9.5
10
log10 HtL
9
9.5
10
log10 HtL
Fe
0.2
0.01
0.15
0.008
0.006
0.1
0.004
0.05
0.002
6.5
7.5
8
8.5
9
9.5
10
log10 HtL
6.5
7.5
8
8.5
Figure 3. Abundances in the returned gas for the SSP test. Staired line represents the DEVA
approach, continuous line represents the analytical result.
5.
Conclusions
We have extended the method proposed in Lia et al. [5]to take into account
the inicial metallicity of stellar particles. This algorithm has been implemented
in a computational efficient way in DEVA, a SPH code that focuses on conservation laws.
Our test produces systematically greater yields that can be explained by the
evolution of metal free stars, which tend to be more efficient in metal production (see Fig. 2). This means that the dependence on metallicity needs to be
taken into account for a correct modeling of chemical evolution. In simulations of galaxy formation where dynamical evolution is coupled to chemical
evolution, such differences could be important.
8
References
[1] Ferrini, Matteucci, Pardi & Penco ApJ, 387, 138 (1992)
[2] Samland M., Hensler G., Theis C., 1997, ApJ 476, 544
[3] Molla M., Ferrini F., Díaz A.I., 1997, ApJ 475, 519
[4] Mosconi et al. , MNRAS, 325, 34M (2001)
[5] Lia C., Portinari L., Carraro G., 2002, MNRAS, 330, 821
[6] Cora & White, Ap&SS, 284, 425 (2003)
[7] Portinari et al. astro-ph/0402427 preprint
[8] Kobayashi, C., MNRAS, 347, 740 (2004)
[9] Aguirre et al. astro-ph/0411076 preprint
[10] Serna A., Domínguez-Tenreiro R., Sáiz A., 2003, ApJ, 597, 878
[11] Portinari L., Chiosi C., Bressan A., 1998, A&A 334, 505 (PCB98)
[12] Pagel, B. E. J. 1997, Nucleosynthesis and chemical evolution of galaxies, Cambridge :
Cambridge University Press, 1997. ISBN 0521550610,
[13] Elmegreen B.G., 2004, 2004MNRAS.tmp..342E
[14] Marigo, P. 2001, A&A, 370, 194
[15] Gavilán, Buell & Mollá, 2004, A&A, in press
[16] Iwamoto, K., Brachwitz, F., Nomoto, K., Kishimoto, N., Umeda, H., Hix, W. R., & Thielemann, F. 1999, ApJS, 125, 439