Download ONE- AND MULTI- ZONE MODELS: Chemical properties

Chapter 5
ONE- AND MULTI- ZONE
MODELS:
Chemical properties
5.1 Why the Infall Scheme ?
The closed-box approximation: review
Bressan et al. (1994) chemo-spectro-photometric models for elliptical galaxies which were
particularly designed to match the UV excess observed in these systems (Burstein et al., 1988)
together with its dependence on the galaxy luminosity, mass, metallicity, Mg2 index, and age. In
brief, Bressan et al. (1994) models stands on the closed-box approximation and the enrichment
law Y=Z=2.5, allow for the occurrence of galactic winds powered by the energy input from
supernov and stellar winds from massive stars, and use the color-magnitude relation of elliptical
galaxies in Virgo and Coma clusters (Bower et al., 1992a Bower et al., 1992b) as one of the main
constraints.
In the Bressan et al. (1994) models the history of star formation consists an initial period
of activity followed by quiescence after the onset of the galactic winds, whose duration depends
on the galactic mass, being longer in the high mass galaxies and shorter in the low mass ones.
With the adoption of the closed-box description of chemical evolution, the maximum eciency of
star formation occurred at very initial stage leading to a metallicity distribution in the models
(relative number of star per metallicity bin, the so-called partition function N(Z)) skewed towards
low metallicities.
The comparison of the theoretical integrated energy distribution (ISED) of the models with
those of prototype galaxies with dierent intensity of the UV excess, for instance NGC 4649 with
strong UV emission and (1550{V) 2.24 and NGC 1404 with intermediate UV excess and (1550{
V) 3.30, indicated that a good agreement was possible but for the region 2,000
A to about 3,500
A,
where the theoretical ISED exceeded the observed one.
53
54
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Figure 5.1: The observed spectrum of the elliptical galaxy NGC 4649 (lled squares) kindly provided by L.M. Buson (1995, private communication). The vertical bars show the global error on
the (1550{V) colors from Burstein et al. (1988). The two lines show theoretical spectra from the
galactic model of Bressan et al. (1994) with 3 1012M characterized by the parameters k = 1,
= 20 and = 0:40 and age 15 Gyr, obtained with dierent assumptions concerning the contribution from stars in dierent metallicity bins. The solid line is the regular spectrum in which
_
all the components are present. Note the discrepancy in the wavelength interval 2,000-4,000
AThe
dashed line is the same but without the contribution from all stars with metallicity Z 0:008.
Note the agreement reached in this case. All the spectra are normalized to coincide in the ux at
5,000
A.
The analysis of the reason of disagreement led Bressan et al. (1994) to conclude that this is a
consequence of the closed-box approximation. Indeed this type of model is known to predict an
excess of low-metal stars. The feature is intrinsic to the model and not to the particular numerical
algorithm used to follow the chemical history of a galaxy.
In order to check their suggestion, Bressan et al. (1994) performed several numerical experiments in which they articially removed from the mix of stellar populations predicted by the
closed-box models the contribution to the ISED from stars in dierent metallicity bins. Looking
at the paradigmatic case of the ISED of NGC 4649, Bressan et al. (1994) found that this can be
matched by a mix of stellar populations in which no stars with metallicity lower than Z = 0:008
are present. The results of these experiments are shown here in Fig. 5.1 for the sake of clarity.
It goes without saying that one may attribute the discrepancy of predicted and observed spectra
in the region 2,000 to 3,500
A to uncertainties in the theoretical spectra. However, the following
considerations can be made. Firstly, those experiments claried that the main contributors to
the excess of ux are the turn-o stars of low metallicity (0:0004 Z 0:008) whose eective
5.1. WHY THE INFALL SCHEME ?
55
temperature at the canonical age of 15 Gyr ranges from 6,450 K to 5,780 K. Secondly, the Kurucz
(1992) spectra (at the base of the library adopted by Bressan et al. (1994)) t the Sun and Vega,
two stars of dierent metallicity but whose eective temperatures encompass the above values.
Finally, the disagreement in question implies that the theoretical spectra overestimate the ux in
this region by a factor of about 4, which is hard to accept. On the basis of the above considerations
Bressan et al. (1994) concluded that the excess is real and that the discrepancy in question is the
\Analog of the G-Dwarf Problem" in the solar vicinity.
The G-Dwarf Problem
The metallicity distribution of long-lived disk dwarf leads to a problem originally discovered
by van den Bergh (1962) and Schmidt (1963). The most straightforward assumptions, to study
the solar vicinity, are the following:
1. the solar neighborhood can be modeled as a closed system
2. it started as 100% metal-free gas
3. the IMF is constant
4. the gas is chemically homogeneous at all times.
The model based on these assumptions is named \simple closed-box model". When this model
is applied to study the chemical history of the solar vicinity it fails to explain the metallicity
distribution observed among the old eld stars. While the observations indicate an extreme
paucity of the low-metal stars, the opposite is predicted by the closed-box model giving rise to the
so-called \G-Dwarf Problem" (Tinsley, 1980a Tinsley, 1980b).
The rst accurate data on G-dwarfs (stars with lifetimes of the order of or larger than the age
of the Galaxy) were obtained by Pagel & Patchett (1975) revised in the following by Pagel (1989).
Sommer-Larsen (1991) applied a correction to the distribution of Pagel & Patchett (1975) taking
into account the fact that metal poor stars are older and have larger scale heights than metalrich stars. Very recently, Wyse & Gilmore (1995) and Rocha-Pinto & Maciel (1996) have derived
new metallicity distribution using uvby photometry and up-to-date parallaxes. These data again
conrm the existence of a G-dwarf problem but dier considerably from the classic distribution of
Pagel & Patchett (1975). In particular, the new data show a prominent peak around Fe/H]={0.2
dex which was not present in the previous distributions.
The G-dwarf problem suggests the progressive building-up of the Galaxy through accretion of
primordial gas, in contrast to the simple scheme closed-box approximation. Alternative solutions
are: prompt initial enrichment from the halo or bulge providing a nite initial metallicity in the
ISM, higher yields due to an IMF skewed toward massive stars in the early galactic phase.
The Infall
56
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
The most popular solution of the G-dwarf dilemma are models with infall (Larson, 1972
Lynden-Bell, 1975 Tinsley, 1980a Chiosi, 1981), i.e. models in which the total mass of the disk
is let increase with time at a suitable rate starting from a much lower value. With the current
law of star formation (proportional to the gas mass) the competition between gas accretion by
infall and gas depletion by star formation gives rise to a non monotonic time dependence of the
star formation rate which instead of steadily decreasing from the initial stage as in the closed-box
model starts small, increase to a peak value, and than declines over a time scale which is a sizable
fraction of the infall time scale. The advantage of the infall model with respect to the closed-box
one, is that the metallicity increase faster, and very few stars are formed at very low metallicity.
Since the excess of very low-metal stars is avoided, the G-dwarf problem is naturally solved.
The existence of infall of gas on the Galactic disk is inferred from high velocity and very high
velocity clouds (Mirabel, 1989) and is claimed as a natural consequence of galaxy formation from
extended halos. Infall is also desiderable to prevent gas consumption in spirals in times shorter
than their ages (Tinsley, 1980a). The total rate of infall of gas in our Galaxy as estimate by Mirabel
(1989) is between 0.2 and 0.5M per year. An estimate of the infall time scale, accordingly to the
previous value, give an infall time scale () about 0.11 Gyr.
Applied to an elliptical galaxy, the infall model of chemical evolution can closely mimic the collapse of the parental proto-galaxy from a very extended size to the one observed today. Plausibly,
as gas falls into the gravitational potential well at a suitable rate and the galaxy shrinks, the gas
density increases so that star formation begins. As more gas ows in, the more ecient star formation gets. Eventually, the whole gas content is exhausted and turned into stars, thus quenching
further star formation. Like in the disk, the star formation rate starts small, rise a maximum, and
then declines. Because of the more ecient chemical enrichment of the infall model, the initial
metallicity for the bulk of star forming activity is signicantly dierent from zero. This type of
models are consistent with the chemo-dynamical models by Theis et al. (1992) and are advocated
by Phillips (1993) to interpret the galaxy counts at faint magnitudes, which are known to exceed
those expected from standard evolutionary models with an initial spike of star formation followed
by quiescence.
5.2 The One-Zone Model
In this section we describe in detail how the infall scheme is adapted to follow elliptical galaxies
when they are conceived as single entities, i.e. one-zone models. In short, the galaxy is made
of two components: dark-matter and baryonic (luminous) material. Dark-matter is supposed to
remain constant in time and to aect only the gravitational potential. The luminous material,
originally in form of primordial gas is supposed to increase at a suitable rate (Eq. 2.1) from a
small value of the present-day total luminous mass.
5.2. THE ONE-ZONE MODEL
57
5.2.1 Binding energies: One-Zone model
In the supernova driven galactic winds models, described in the previous chapter x 2, the onset
of galactic winds occur when the condition Eq. 2.25 is veried. To this aim we need to know the
thermal energy and gravitational potential of the gaseous component. Since the thermal energy has
already been amply described in section x 2.3.1, we limit ourselves to present here the derivation
of the gravitational potential.
The dynamical structure of the luminous and dark matter follows the picture Bertin et al.
(1992) and Saglia et al. (1992), in which the mass and radius of the dark component, MD T and
RD T , respectively, are related to those of the luminous material, ML T and RL T , by the relation
ML T (t) 1 RL T (t)
MD T
2 RD T
R
(t)
L
T
1 + 1:37 R
DT
(5.1)
The mass of the dark component is assumed be constant in time and equal to MD T = ML T (TG ) with = 5.
Furthermore, the binding gravitational energy of the gas is given by
#g (t) = ;LG Mg R(t)ML(t)T (t) ; G MRg (t)M(t)D T #0LD
LT
LT
(5.2)
where Mg (t) is the current value of the gas mass, L is numerical factor ' 0:5, and
1 RL T (t)
LD = 2 R
DT
#0
1 + 1:37 RRL T (t)
DT
(5.3)
is the contribution to the gravitational energy given by the presence of dark matter. Following
Bertin et al. (1992) and Saglia et al. (1992) in the above relations the ratio ML T =MD T and
RL T =RD T are xed both equal to 0.2.
In order to apply the above equations for the binding gravitational energy of the gas to a model
galaxy is necessary an estimate of the radius RL T . Including the assumed ratios ML T =MD T
and RL T =RD T in the Eq. 5.3 the contribution to gravitational energy by the dark matter is
#0LD = 0:04 and the total correction to the gravitational energy of the gas (Eq. 5.2) does not
exceed 0.3 of the term for the luminous mass.
Given these premises, and assuming that at each stage of the infall process the amount of
luminous mass that has already accumulated gets soon virialized and turned into stars, the total
gravitational energy and radius of the material already settled onto the equilibrium conguration
may be approximate with the relations for the total gravitational energy and radius as function
of the mass (in this case ML T (t)) obtained by Saito (1979a),(1979b) for elliptical galaxies whose
spatial distribution of stars is such that the global luminosity prole obeys the R1=4 law. In such
a case the relation between RL T (t) and ML T (t) is
58
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
L T (t)
RL T (t) = 26:1 M
1012M
(2;)
kpc
(5.4)
with = 1:45 (cf. also Arimoto & Yoshii (1987)).
Finally, the volume density of gas g (t) is given by
3Mg (t)
g (t) = 4R
(t)3
LT
(5.5)
It is worth recalling that if dark and luminous matter are supposed to have the same spatial
distribution, Eqs. 5.2 and 5.3 are no longer required. Indeed, the binding energy of the gas can
be simply obtained from the total gravitational energy of Saito (1979b) provided that the mass
ML (t) is replaced by the sum ML T (t) + MD T . In such a case, the binding energy of the gas is
#g (t) = #L+D (t)Mg (t)2 ; Mg (t)]:
(5.6)
5.2.2 Galactic winds: One-Zone model
As described in the Eqs. 2.15, 2.16 and 2.17 the function SN (t) is cooling law governing the
energy content of supernova remnants. In order to calculate SN (t) for the one-zone model, the
time variation of the supernov thermal content is taken from Cox (1972)
SN (tSN ) = 7:2 1050
0 erg
(5.7)
for 0 tSN tc and
SN (tSN ) = 2:2 10 0
50
tSN ;0:62 erg
tc
(5.8)
for tSN tc , where 0 is the initial blast wave energy of the supernov in units of 1051 ergs,
which is assumed equal to unity for all supernova types, tSN = t ; t0 is the time elapsed since the
supernova explosion, and tc is the cooling time of a supernova remnant, i.e.
tc = 5:7 104 40=17n;0 9=17 yr
(5.9)
where n0 (t) = g (t)=mH is the number density of the interstellar medium as a function of time,
and g (t) is the corresponding gas density.
Stellar winds from massive stars can also contribute to the heating of the ISM, although their
eective contribution with respect that of supernov is subject for recent discussion (Bressan
et al., 1994 Gibson, 1994 Tantalo et al., 1996 Tantalo et al., 1998c) and section x 5.4. On the
other hand, stellar winds can be very important for the energetics of the ISM in very early stages
of a starburst (Leitherer et al., 1992 Leitherer, 1997).
5.3. ONE-ZONE MODELS: COMPARING CLOSED- AND OPEN-BOX MODELSS
59
Figure 5.2: The (1550{V) versus (V{K) relationship for closed-box models. The shaded area shows
the observational data. The models are calculated for dierent values of as indicated, and for all
the values of listed in Table 5.1. The solid line shows models with =0.40 and the same values
of as before but in which the low-metallicity component is taken away. All the models refer to
the 3ML T 12 galaxy.
5.3 One-Zone models: Comparing Closed- and Open-box
modelss
In order to test the eects of the new SSPs, we have performed a preliminary analysis using both
closed-box and infall models.
The guide line is to search the combination of the parameters and for which the most massive
galactic model (3ML T 12) at the canonical age of 15 Gyr matches the maximum observational
values for (V{K) and (1550{V) colors, 3.5 and 2 respectively (cf. Figs.6.1 and 6.5 chapter x 6).
To this aim, assuming k = 1, dierent values of and are explored.
In the closed-box approximation, simply xes the time scale of star formation. The values
under consideration are listed in Table 5.1. Furthermore, for each value of , six values of are
taken into account, i.e. =0.25, 0.30, 0.35, 0.40, 0.50, and 0.60. The analysis is limited to the
case of the 3ML T 12 galaxy. Table with the details of these models will be available on request.
The results are shown in Fig. 5.2 which displays the correlation between (V{K) and (1550{V) and
the corresponding observational data (shaded area). The (1550{V) colors are from Burstein et al.
(1988), whereas the (V{K) colors are from Bower et al. (1992a), (1992b).
Clearly no combination of the parameters is found yielding models satisfying the above constraint (upper left corner of the shaded area in Fig. 5.2). Varying the age within the range allowed
60
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Table 5.1: Adopted values for the parameter and associated time scale of star formation tSF
;1 )
(Gyr
tSF (Gyr)
20
15
10
5
1
0.050
0.067
0.100
0.200
1.000
by the CMR (say 15 2 Gyr) does not help to match both colors of the reference galactic model.
Similar results are obtained for the other galactic masses. There is a tight relationship between
the theoretical (1550{V) and (V{K) which can be easily understood in terms of intrinsic behaviour
of the closed-box model and the dependence of the two colors on the metallicity. As shown by
Bressan et al. (1994), the maximum metallicity determines the UV intensity and hence (1550{V)
color, whereas the mean metallicity governs the (V{K) color.
In a closed-box system, the current (maximum) metallicity obeys the relation:
Z(t) = Yz ln ;1
(5.10)
where = Mg (t)=ML T (TG ) is the fractionary mass gas and yz is the chemical yield per stellar
generation. This relation holds for Z 1. The mean metallicity is given by:
R
R
hZ(t)i = (t)Z(t)dt
(5.11)
(t)dt
With the aid of Eqs. 5.10, 5.11 and 2.12 can be pointed out relationship between hZ(t)i and
Z(t) shown in Fig. 5.3. Owing to the low-metal skewed distribution typical of the closed-box
models, if the maximum metallicity rises to values at which the H-HB and AGB manqu$e stars are
formed and in turn the dominant source of the UV radiation is switched on, the mean metallicity
is to low to generate the right (V{K) colors. In order to prove this statement, in Fig. 5.2 are shown
models in which all the stars with Z 0:008 are articially removed (solid line). As expected,
solutions are now possible. On the one hand this result conrms the conclusions by Bressan et al.
(1994) about the intrinsic diculty of the closed-box models, on the other hand it makes evident
that the success or failure of a particular spectro-photometric model in matching the observational
data depends very much on details of the adopted spectral library.
In order to check whether infall models can cope with the above diculty, a preliminary analysis
of the problem is performed calculating infall models with the same parameters and as for
the closed-box case. In these models k = 1, = 0:10 Gyr, and the galactic mass is 3ML T 12.
The results are shown in Fig. 5.4. With the infall scheme many models exist whose (V{K) and
(1550{V) match the observational data.
5.4. WHY GRADIENTS ?
61
Figure 5.3: The Z(t) versus hZ(t)i for the same models as in Fig. 5.2. Note the linear relationship
between the two metallicities for models with the same . See the text for more details
The reason of the success is due to the law of chemical enrichment of infall models (cf. Tinsley
(1980a))
1
Z(t) = yz log ; 1
(5.12)
where all the symbols have the same meaning as in Eq. 5.10, which leads to the relationship
between the maximum and mean metallicity shown in Fig. 5.5. Since the metallicity distribution
of infall models is skewed towards the high metallicity side, at given maximum metallicity higher
mean metallicities are found that allow to match of the observational (V{K) and (1550{V) colors
at the same time.
5.4 Why Gradients ?
Looking at the results to be presented in chapter x 6 below for the one-zone models, which are
forced to simultaneously t the CMR, (V{K) versus V, and the (1550{V) versus Mg2 relation, one
may argue that the one-zone approximation is not fully adequate to interpret observational data
that refers to dierent regions of a galaxy. Infact, while the data on the UV-excess, color (1550{V)
and Mg2 index essentially refer to the central part of a galaxy (see Burstein et al. (1988)), the
integrated magnitudes and colors, such as V, (B{V), (V{K) etc., refer to the whole galaxy. This is
particularly true with the data for galaxies in the Virgo and Coma cluster we are going to discuss.
62
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Figure 5.4: The (1550{V) versus (V{K) for all infall models. The shaded area indicates the range
spanned by the observational data. The models are calculated for dierent values of as indicated,
and for all the values of listed in Table 5.1. All the models refer to the 3ML T 12 galaxy
In addition to this, the presence of spatial gradients in broad-band colors and line strength
indices observed in elliptical galaxies (Worthey, 1992 Gonz$ales, 1993 Schombert et al., 1993
Davies et al., 1993 Carollo et al., 1993 Carollo & Danziger, 1994a Carollo & Danziger, 1994b
Balcells & Peletier, 1994 Fisher et al., 1995 Fisher et al., 1996) indicates that the one-zone
models (either closed or with infall) ought to abandoned in favor of models in which the spatial
distribution of mass density and star formation rate is taken into account.
Finally, since radial variations in colors and line strength indices can be eventually reduced to
variations in age and chemical composition (metallicity), or both, of the underlying stellar populations, the interpretation of the gradients bears very much on the general mechanism of galaxy
formation and evolution. Unfortunately, separating age from metallicity eects is a cumbersome
aair, otherwise known as the age-metallicity degeneracy (cf. Worthey et al. (1994) and references therein) which makes it dicult to trace back the history of star formation and chemical
enrichment both in time and space.
The need of a simple tool to follow the chemical history of a galaxy with gradients in mass
density and star formation, thus abandoning the widely adopted one-zone approximation without
embarking in a fully dynamical description, has spurred the kind of model presented in this chapter
and then utilized in this thesis (chapter 8).
These models stand on the infall scheme but include the radial dependence of gas density and
star formation, and adopt a simple scheme to simulate the collapse of luminous material into the
5.5. THE MULTI-ZONE MODEL
63
Figure 5.5: The Z(t) against hZ(t)i for the open models as in Fig. 5.4. Note the linear relationship
between the two metallicities for models with the same . See the text for more details
potential well of dark matter.
5.5 The Multi-Zone Model
Elliptical galaxies are assumed to be made of baryonic and dark material both with spherical
distributions but dierent density proles. Let ML T (TG ) and MD T (TG ) be the total luminous and
dark mass, respectively, existing in the galaxy at the present time (TG is the galaxy age). The two
components have dierent eective (half mass) radii, named RL e(TG ) and RD e(TG ) (thereinafter
shortly indicated as RL e and RD e), and their masses are in the ratio MD T (TG )=ML T (TG ) = .
Although may vary from galaxy to galaxy, for the purposes of this study it is thought to be
constant.
An essential feature of the models is that while dark matter is assumed to have remained
constant in time, the luminous material is assumed to have accreted at a suitable rate (to be
dened below) into the potential well of the former. Owing to this hypothesis, no use will be made
of dark matter but for the calculation of the gravitational potential and the whole formulation of
the problem will stand on the mass and density of luminous material which are let grow with time
from zero to their present day value.
The model whose radial density prole upon integration over radius and time yields the mass
ML T (TG ) is referred to as the asymptotic model. If L (r t) is the radial density prole of luminous
matter at any age t and _L (r t) is the rate of variation by gas accretion, the following relation
64
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
holds
ML T (TG ) =
ZT
0
G
dt
Z
RL TG
0
4 r2 _L (r t) dr:
(5.13)
The asymptotic model is divided in a number of spherical shells with equal value of the asymptotic luminous mass, typically 5% of ML T (TG ). Since the density L (r TG) is changing with
radius (decreasing outward), the thickness and volume of the shells are not the same. They are
indicated by
0 = r0 ; r0
rj=
j +1 j
2
0 ) = 4 (r03 ; r03 )
V (rj=
2
3 j +1 j
where rj0 +1 and rj0 are the outer and inner radii of the shells, and j = 0 ::J-1 with r00 = 0 (the
center) and rJ0 = RL T (the total radius). The radii rj0 are not yet dened.
Thereinafter the following notation are used and change of the radial coordinate
G
Each zone of a model is identied by its mid radius rj0 +1=2 = (rj0 +1 + rj0 )=2 shortly indicated
0 .
by rj=
2
Radial distances are expressed in units of the eective radius of the luminous material in the
0 =RL e.
asymptotic model, i.e rj=2 = rj=
2
All masses are expressed in units of 1012 M . Finally galactic models are labelled by their
asymptotic total luminous mass ML T (TG ) in the same units, shortly indicated by ML T 12.
For each shell, the mean density is briey indicated as L (rj=2 t), s (rj=2 t), and g (rj=2 t),
for total luminous material, for stars and gas, respectively, so that the corresponding masses are
ML (rj=2 t) = L (rj=2 t) V (rj=2) R3L e
Mg (rj=2 t) = g (rj=2 t) V (rj=2) R3L e
Ms(rj=2 t) = s (rj=2 t) V (rj=2) R3L e:
By denition
;1ML (rj=2 TG ) = ML T (TG )
%Jj =0
(5.14)
Mg (rj=2 t) + Ms (rj=2 t) = ML (rj=2 t):
(5.15)
and
Identical relationships can be dened for the dark matter by substituting its constant density
prole. Since there would be no direct use of these relations, the space zoning of the dark matter
5.5. THE MULTI-ZONE MODEL
65
distribution is taken to be the same as for the luminous component, so that the contribution of
dark matter to the total gravitational potential in each zone is properly calculated (see below).
The infall scheme for this set of model follows the law given by Eq. 2.1 however adapted to
the density formalism. This means that the luminous component in each shell is let increase with
time according to
d (r t) t
L j=2
=
(r
)exp
;
L0 j=2
dt
(rj=2)
(5.16)
where (rj=2) is the local time scale of gas accretion for which a suitable prescription is required,
the function L0(rj=2) is xed by imposing that at the present galactic age TG the density of
luminous material in each shell has grown to the value given by the adopted prole L (r TG ).
It follows that the time dependence for L (rj=2 t) is given again by the Eq. 2.3 with the density
formalism. For the asymptotic mass density in each shell the geometric mean of the values at
inner and outer radii are adopted, i.e.
q
L (rj=2 TG ) = L (rj +1 TG ) L (rj TG ):
(5.17)
To summarize, each shell is characterized by:
the radius rj=2
the asymptotic mass ML(rj=2 TG ), which is a suitable fraction of the total asymptotic
luminous mass ML T 12
the mass of dark matter MD (rj=2 TG ). Since this mass is constant with time no other
specication is required
the asymptotic mean density L (rj=2 TG ) of baryonic mass (gas and stars)
the gravitational potential for the luminous component 'L (rj=2 t) varying with time, and
the corresponding gravitational potential of dark-matter 'D (rj=2 TG ), constant with time.
Both will be dened below.
5.5.1 Binding energies: Multi-Zone model
Density pro le of the luminous matter. The asymptotic spatial distribution of luminous
matter is supposed to follow the Young (1976) density prole. This is derived from assuming that
the r1=4-law holds and the mass to luminosity ratio is constant throughout the galaxy (Poveda
et al., 1960 Young, 1976 Ciotti, 1991). It is worth to remind the reader that the density L (r TG)
and the gravitational potential 'L (r TG) are expressed by Young (1976) as a function of the
66
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
eective radius for which a suitable relationship with the total luminous mass is required (see
below).
The adoption of the Young (1976) density prole imposes that the resulting model at the age
TG has (i) a radially constant mass to luminosity ratio (ii) a luminosity prole obeying the r1=4
law.
Density pro le of the dark matter. The mass distribution and gravitational potential of the
dark-matter are derived from the density proles by Bertin et al. (1992) and Saglia et al. (1992)
however adapted to the Young formalism for the sake of internal consistency. In brief starting
from the density law
r4
D (r) = (rD2 0 + rD2 )02
(5.18)
D0
where rD 0 and D 0 are two scale factors of the distribution. The density scale factor D 0 is
derived from imposing the relation MD T = ML T and the denition of MD T by means of its
density law
Z1
MD T = 4
r2 (r)dr = 4D 0 rD3 0m(1)
(5.19)
0
with
m(1) =
Z1
0
rD 0
3
r2
r 22 dr
1+ r 0
(5.20)
D
This integral is solved numerically. Finally, the density prole of dark-matter is
MD T 1
D (r) = m(
1) 4r3
D0
1
r
2 2 :
1+ r 0
The radial dependence of the gravitational potential of dark matter is
Zr
'D (r) = ;G MDr2(r) dr
0
which upon integration becomes
r
2
'D (r) = ;4GD 0 rD 0'fD r
D0
r
where 'fD r0 is given by
Z r=r 0 m(r=r )
Dr 02 dr
0
rD 0 r 0
(5.21)
D
D
D
(5.22)
(5.23)
(5.24)
5.5. THE MULTI-ZONE MODEL
67
Figure 5.6: The density proles of baryonic and dark material in the prototype galaxy of 1ML T 12.
For the baryonic component the asymptotic density is displayed (see the text for more details)
This integral is solved numerically and stored as a look-up table function of r=rD 0. It is
assumed rD 0 = 21 RD e, where RD e indicate the eective radius of dark matter. This can be
derived from relation (5.19) looking for the radial distance within which half of the dark-matter
mass is contained. Finally, all the models below are calculated adopting = 5 in the ratio
MD T (TG )=ML T (TG ) = .
For sake of comparison, in Fig. 5.6 are show the density proles of baryonic and dark material in
the prototype galaxy of 1ML T 12.
The gravitational binding energies. The binding gravitational energy for the gas in each shell
is given by
#g (rj=2 t) = g (rj=2 t)V (rj=2)'L (rj=2 t) + g (rj=2 t)V (rj=2)'D (rj=2 TG )
(5.25)
5.5.2 Galactic winds: Multi-Zone model
Gibson (1994) has pointed out that the standard formalism to deal with the energy deposited by
supernova explosions and stellar winds (the same we have adopted in the one-zone models) may
actually underestimate the contribution by supernova explosions and overestimated that by stellar
winds from massive stars.
68
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
The same formalism described in 2.3 for the one-zone models is also applied to the models with
gradients in mass density, with several major changes: (i) density is used instead of mass (ii) the
cooling law for supernova remnants strictly follows the prescription by Gibson (1994) and Gibson
(1996b) (iii) and a minor revision of the energy deposit by stellar winds from massive stars.
Following Gibson (1994) and Gibson (1996b) and references therein, the evolution of a SNR
can be characterized by three dynamical phases: (i) free expansion (until the mass of the swept
up interstellar material reaches that of the SN ejecta) (ii) adiabatic expansion until the radiative
cooling time of newly shocked gas equals the expansion time of the remnant (iii) formation of
a cold dense shell (behind the front) which begins when some sections of the shocked gas have
radiate most of their thermal energy, begin further compressed by the pressure of the remainder
of the shocked material.
In the earliest phase the evolution of the supernova remnant is governed by the Sedov-Taylor
solution for a self-similar adiabatic shock (Ostriker & McKee, 1988)
1=5
E
0
t2=5
Rs (t) = 1:15 (t)
g
(5.26)
where Rs(t) is the radius of the outer edge of the SNR shock front, E0 is the initial blast energy
in units of 1050 ergs (or equivalently E0 = 10 0 , where 0 is the same energy in units of 1051
erg), and g (t) is the gas mass density of the environment.
Radiative cooling of the shocked material leads to the formation of a thin, dense shell at time
tsf
4 3=14
0
tsf = 3:61 10 n;4=7
0
Z ;5=14
Z
yr
(5.27)
where n0 is the hydrogen number density, Z is the metallicity of the interstellar medium, and Z =
0.016. The blast wave decelerates until the radiative energy lost in the shell's material starts to
dominate. At this point, the shell enters the so-called pressure-driven snow-plow (PDS) phase at
the time tpds 0:37tsf .
The evolution of the thermal energy in the hot, dilute interior of the supernova remnants can
be taken equal to
SN (tSN ) = 0:717 E0
erg
(5.28)
when tSN tpds i.e. during the adiabatic phase. Note that tSN = t ; t0 is the time elapsed since
the supernova explosion. During the early PDS-phase, when tpds tSN 1:17 tsf , the thermal
energy evolution is given by
5.5. THE MULTI-ZONE MODEL
69
"
"
14=5#
10 #;1=5 " t 4 #;1=9
0:86
t
R
s
SN
SN
+ 0:43 E0 R
+1
erg
SN (tSN ) = 0:29 E0 1 ;
tsf
tsf + 1
sf
(5.29)
and the radius changes according to
3=10
Rs(tSN ) = Rpds 43 ttSN ; 13
pds
pc
(5.30)
where Rpds is the radius at the beginning of the PDS-stage
Rpds = 14:0 20=7 n30=7 ( ZZ );1=7 pc
(5.31)
The interior continues to lose energy by pushing the shell through the interstellar medium and
by radiative cooling. At time tmerge
=49 ( Z )15=49
tmerge = 21:1 tsf 50=49 n10
0
Z
yr
(5.32)
the remnants merge with the interstellar medium and lose their identity as separate entities.
The thermal energy during the time interval 1:17tsf tSN tmerge is given by the second
term of Eq. 5.29. The evolution after the tmerge time is described again by the second term of
Eq. 5.29, but the radius Rs is given by
Rs = Rmerge = 3:7 Rpds 30=98 n30=49 ( ZZ )9=98 pc
(5.33)
tcool = 203 tsf ( ZZ );9=14 yr
(5.34)
Finally, when tSN tcool in which
the thermal energy is given by the second term of Eq. 5.29 but with Rs = Rmerge .
The time dependence of the cooling law for interstellar media with dierent metallicities is shown
in Fig. 5.7 and it is compared with the classical one of Cox (1972). It is soon evident that this more
elaborated scheme for the cooling of supernova remnants supplies more energy to the interstellar
medium than the old one. The adoption of the Cox (1972) cooling law by Bressan et al. (1994),
Tantalo et al. (1996) and in this thesis for the one-zone models (see section x 5.2.2 above) may
also explain why other sources of energy to power galactic winds were invoked (see the remark in
x 2.3.2).
As far as stellar winds are concerned the formalismwe have adopted has already been described
in section 2.3.2.
70
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Figure 5.7: The cooling law of supernova remnants as a function of the gas metallicity as indicated
Finally, when the total thermal energy of the gas in each shell exceeds the total gravitational
energy, star formation over there is supposed to halt and the local gas content to be ejected by
the galaxy.
5.5.3 The mass-radius relationships
To proceed further it is necessary to adopt suitable relationships between the RL e and ML T , so
that once the total baryonic mass is assigned, the eective radius is derived, and all the other
quantities are properly re-scaled.
For the purposes of this study and limited to the case of H0 = 50 km sec;1 Mpc;1 , the following
relation are derived from the data of Carollo et al. (1993), and Goudfrooij et al. (1994)
RL e = 17:13 ML0:557
T 12
(5.35)
where RL e is in Kpc.
For the same objects and using the diameters from the RC3 catalogue the relation between total
radius and mass of the luminous material is also derived
RL T = 39:10 ML0:402
T 12
(5.36)
in the same units as above.
The relations above are displayed in Fig. 5.8 and compared with the mass radius relation by Saito
(1979a) and (1979b). Finally, Table 5.2 lists RL e and RL T as assigned to each model galaxy.
5.6. FREE PARAMETERS OF THE MODELS
71
Table 5.2: Eective and total radii (in Kpc) assigned to model galaxies of dierent ML T 12
ML T
12
3
1
0.5
0.1
0.05
0.005
RL T
RL e
60.78
39.10
29.60
15.51
11.74
4.66
31.60
17.13
11.64
4.75
3.23
0.90
5.5.4 Mass zoning of the models
The mass zoning of the models is chosen in such a way that within each shell about 5% of the
luminous mass ML T 12 is contained. From the tabulations of Young (1976) the corresponding
fractionary radius r0 =RL e is derived, and from the mass-radius relationships above can be x the
eective radius RL e, and the real inner and outer radii of each shell. The results are given in
Table 5.3, whereby the meaning of the various symbols is self-explanatory.
Since the observational data for the gradients do not extend beyond 2RL e (see Carollo &
Danziger (1994a),(1994b)), the models are limited to the rst eleven regions of the galaxy, i.e.
to fractionary radii r0 =RL e = 1:6 or equivalently the inner sphere containing 55% of the total
luminous mass ML T 12. Care must be paid when comparing integrated observational quantities,
such as magnitudes and colors (see chapter x 6 for more details), with model results.
5.6 Free Parameters of the Models
Primary parameters of the models are:
1) The galactic mass ML T (TG ). This is a mere label identifying the models. In the closedbox approximation it is the initial luminous mass of the galaxy, whereas in the infall scheme it
represents the asymptotic value. In any case galactic winds lower the galactic mass to the value
already stored in stars at the age of wind ejection. The models (one-zone and multi-zone) presented
in this thesis are calculated for the following values of ML T (TG ): 3 1012M , 1 1012M ,
5 1011M , 1 1011M , 5 1010M , and 1 1010 M . Thereinafter, all the models will be
labelled by the mass ML T (TG ) in units of 1012M in turn shortly indicated by ML T 12.
2) The galaxy age TG . This could be constrained to the model of the universe, i.e. to H0, q0,
and red-shift of galaxy formation zfor . However, because of the uncertainty on the cosmological
parameters, galaxy ages are let vary in the range of current estimates of the globular cluster ages,
i.e. 13 15 2 Gyr (cf. Chiosi et al. (1992) for a recent review of the subject).
3) The ratio ML T =MD T which x the gravitational potential and the eect of dark matter, is a
72
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Figure 5.8: The mass-radius relationships derived from the observational data by Carollo et al.
(1993): the open circles are the total radius, whereas lled squares are the eective radius. The
dotted, and dashed lines show the relationships ML (RT ) and ML (Re). Finally, the long-dashed
line displays the relation by Saito (1979a),(1979b) for purposes of comparison
free-parameter for both the set of models as described in sections x 5.2.1 and 5.5.1, which is been
taken equal to 5. The ratio RL T =RD T is a free-parameter only for the one-zone model and is
taken equal to 5.
4) The exponent k of the star formation rate has been choice equal to the unity for both the set
of model presented in this thesis, only a set with k = 2 has been calculated for the case of the
one-zone models (see Tables 5.8 and 5.9).
5) The initial mass function (slope and ). The slope is kept constant. , which is the fraction of
the IMF stars more massive than 1M , determining the chemical yields per stellar generation and
the nal mean metallicity of the galaxy, is let vary. The higher , the easier the galaxy builds up
high metallicity stars. According to the current nucleosynthesis prescriptions, IMF (0:3 < < 0:50
cf. Matteucci (1991), Rana (1991), and Trimble (1991)), and in order to get models with M=LB
ratios in agreement with the observational data (see below), the value of = 0:50 has been adopted
for both the kind of models.
6) The time scale of gas accretion (r).
(a) In the one-zone model, the infall time scale for the does not dependent on the radial coordinate.
Even if is a free parameter, we have adopted the typical collapse time scale of a galaxy
with mass ML T (TG ). Assuming a homogeneous distribution of matter in the volumes lled
by the dark and luminous components, the total mass density T can be expressed as
5.6. FREE PARAMETERS OF THE MODELS
73
Table 5.3: Percentage of luminous mass contained in the sphere of fractionary radius r0 =RL e, and
actual radius r0 (in Kpc) for model galaxies with dierent ML T 12 as indicated
ML T
%ML T 12
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
3
12
j
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
r
0
Re
0.1106
0.2005
0.2954
0.3983
0.5127
0.6405
0.7833
0.9464
1.1330
1.3490
1.6020
1.9030
2.2690
2.7240
3.3060
4.0870
4.9580
r
0
3.49
6.33
9.33
12.58
16.20
20.24
24.75
29.90
35.80
42.62
50.62
60.13
71.69
86.07
104.45
129.13
156.65
1
0.5
0
r
r
1.89
3.43
5.06
6.82
8.78
11.05
13.42
16.21
19.41
23.11
27.45
32.60
38.87
46.67
56.64
70.02
84.94
0
1.29
2.33
0.69
4.64
5.97
7.46
9.12
11.02
13.19
15.71
18.65
22.16
26.42
31.72
38.49
47.59
57.73
0.01
r
0
0.53
0.95
1.40
1.89
2.44
3.04
3.72
4.50
5.38
6.41
7.61
9.04
10.78
12.94
15.70
19.41
23.55
0.05
r
0
0.36
0.65
0.95
1.29
1.66
2.07
2.53
3.06
3.66
4.36
5.17
6.14
7.33
8.79
10.67
13.19
16.01
0.005
r
0
0.10
0.18
0.26
0.36
0.46
0.57
0.70
0.85
1.01
1.21
1.43
1.70
2.03
2.44
2.96
3.66
4.44
T = (1 + 2 )L
(5.37)
where L can be determined with the aid of equation 5.5. Accordingly, the collapse time
scale is (see also Arimoto & Yoshii (1987))
;0:5
0:325
1
8 ML T (TG )
(ML T (TG )) = 1 + 2
1:26 10 1012M
yr
(5.38)
This relation yields the collapse time scales reported in Table 5.4 as function of the galactic
mass. The time scale of mass accretion drives the infall process. For tending to 0, the model
reduces to an initial burst of star formation, while for tending to 1, star formation does
never stop. However, all the models below are calculated with = 0:10 Gyr independently
of the galactic mass. This choice reduces the number of parameters and appears to be fully
appropriate to the present purposes.
(b) In the multi-zone model a dierent reasoning has been followed. A successful description of
the gas accretion phase as a function of the galacto-centric distance, is possible adapting
to galaxies the radial velocity law describing the nal collapse of the core in a massive star
(Bethe & Davidsen, 1990), which obeys the following scheme
74
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Table 5.4: Free-Fall time scale for the various masses
ML T 12
(Gyr)
3
1
0.5
0.1
0.05
0.01
0.18
0.12
0.10
0.06
0.05
0.03
homologous collapse in all regions internal to a certain value of the radius (r ): v(r) / r
free-fall outside: v(r) / r; 21 where r is the radius at which the maximum velocity occurs. This picture is also conrmed
by Tree-SPH technique (cf. Carraro et al. (1998) and references therein). In brief, looking at
the paradigmatic case of the adiabatic collapse of a galaxy (dark plus baryonic material) of
1012M , initial mean density of 1:6 10;25 g cm;3, free-fall time scale of 0.25 Gyr, and age
of 0.22 Gyr, can be notice that the radial velocity v(r) as a function of the radial distance
r starts from zero at the center, increases to a maximum at a certain distance, and then
decreases again moving further out. The situation is shown in Fig. 5.9, where the velocity is
p
in units of v = GM=R2]), the distance is units of the initial radius (100 Kpc), and the
maximum occurs at r=R ' 0:4 (for this particular model).
How does the above simple scheme compare with the results of numerical calculations? To
this aim, in Fig. 5.9 are shown the best t of the data from the numerical model for the
two branches of the velocity curve and compare them with the above relationships. In this
particular example, the slope along the ascending branch (r=R < 0:4) is 1:5 2 instead of
1, whereas that along the descending branch (r=R > 0:4) is {0.87 instead of {0.5.
A close inspection of the numerical Tree-SPH models reveals that neither the slopes of the
velocity branches nor the radius of the peak velocity are constant in time. Therefore all
these will consider as free parameters.
The velocity v(r) is expressed as:
v(r) = c1 r
for r r
v(r) = c2 r;
for r > r
(where c1 , c2, and are suitable constants), and the time scale of accretion as:
r
(rj=2) / v(r)
5.6. FREE PARAMETERS OF THE MODELS
75
Figure 5.9: The radial velocity v(r) versus radius r relationship for a Tree-SPH model of the
adiabatic collapse of a galaxy with total mass (baryonic and dark matter) of 1012M from Carraro
et al. (1998). The velocity and radius are normalized to v and R as described in the text. The
full dots are the results of the numerical calculations. The solid and dotted lines are the best ts
of the data: v(r) / r1:5 for the inner core and v(r) / r;0:87 for the external regions. The dashed
lines are the same but for the strict homologous collapse and free-fall description
Many preliminary models calculated with above receipt, of which no detail is given here for
the sake of brevity, indicate that = 2 and = 0:5 are good choices. The value = 2
is indeed taken from the Tree-SPH models whereas = 0:5 follows from the core collapse
analogy. The determination of the constants c1 and c2 is not required as long as for scaling
relationships. Therefore the time scale of gas accretion can be written as proportional to
some arbitrary time scale, modulated by a correction term arising from the scaling law for
the radial velocity. The free-fall time scale tff referred to the whole system is taken for the
time scale base-line.
(rj=2) = tff rr
if rj=2 r
(5.39)
3=2
(rj=2) = tff rrj=2
if rj=2 > r
(5.40)
j=2
For the free-fall time scale tff has been adopted the relation by Arimoto & Yoshii (1987)
tff = 0:0697 ML0:325
T 12
Gyr:
(5.41)
76
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Figure 5.10: The accretion time scale (r) as a function of the galacto-centric distance for the
models with dierent asymptotic mass ML T 12 as indicated
Finally, for the sake of simplicity, r is taken equal to 12 RL e. Other choices are obviously
possible. They would not change the main qualitative results of this study.
Fig. 5.10 shows the values of (rj=2) (Table 5.5) as a function of the galacto-centric distance.
7) The eciency (r) of the star formation rate.
(a) As far as the one-zone model is concerned, this parameter ( = 1=tSF ) is constant and is
temptatively varied in order to t the major constraints for elliptical galaxies. The combined
eect of all the observational constraints (see section x 5.3 above) connes in the range
1 < < 12. More precisely, from = 12 for the 3ML T 12 galaxy to = 1 for the 0.01ML T 12
object (see Tables 5.8 and 5.9).
(b) In order to derive the specic eciency of star formation (r) for the multi-zone model, the
simple scale relations developed by Arimoto & Yoshii (1987) is utilized however adapted
to the density formalism. At the typical galactic densities (10;22 10;24g cm;3) and
considering hydrogen as the dominant coolant (Silk, 1977) the critical Jeans length is much
smaller than the galactic radius, therefore the galaxy gas can be considered as made of
many c loud lets whose radius is as large as the Jeans scale. If these clouds collapse nearly
isothermal without suering from mutual collisions, they will proceed through subsequent
fragmentation processes till opaque small subunits (stars) will eventually be formed. In such
a case the stars are formed on the free-fall time scale. In contrast, if mutual collisions occur,
5.6. FREE PARAMETERS OF THE MODELS
77
they will be supersonic giving origin to layers of highly cooled and compressed material
the Jeans scale will then fall below the thickness of the compressed layer and fragmentation
occurs on the free-fall time scale of the high density layers and nally the whole star forming
process is driven by the collision time scale. On the basis of these considerations, the ratio
s
1
(5.42)
tff tcol
is taken as a measure of the net eciency of star formation.
Let (r) to be express as the product of a suitable yet arbitrary specic eciency referred
to the whole galaxy times a dimensionless quantity F(r) describing as the above ratio varies
with the galacto-centric distance. An obvious expression for F (r) is the ratio (5.42) itself
normalized to its central value.
According to Arimoto & Yoshii (1987) the mean collision time scale referred to the whole
galaxy can be written as
tcol = 0:0072 ML0:1T 12
Gyr
(5.43)
With the aid of this and the relation for the free-fall time scale above, can be calculated
as
=
"s
1
#
tff tcol gal :
(5.44)
Extending by analogy the above denition of free-fall and collision time scales to each individual region, the following expression for F (r) can be obtained
r 3 (r T ) F (r) = r1=2 g (r1=2 TG)
g j=2 G
j=2
(5.45)
with obvious meaning of the symbols. In principle, the exponent could be derived from
the mass dependence of tff and tcol , i.e. ' 0:2. However, a preliminary analysis of the
problem has indicated that F (r) must vary with the radial distance more strongly than this
simple expectation. The following relation for has been found to give acceptable results
as far as gradients in star formation, metallicity, colors, etc. are concerned
= 0:98 (ML 12)0:02
Finally, the total expression for (r) is
(5.46)
78
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Figure 5.11: The specic eciency of star formation (r) as a function of the galacto-centric
distance for the models with dierent asymptotic mass ML T 12 as indicated. See the text for
more details
r1=2 3 g (r1=2 TG ) (r) = t t
r
(r T ) Gyr;1
ff col gal
g j=2 G
j=2
1
0:5
(5.47)
Fig. 5.11 shows the values of (rj=2) (Table 5.5), corresponding to the mean point of each
shell, as a function of the galacto-centric distance, for all the models under consideration.
As expected, in a galaxy the specic eciency of star formation increase going outward.
5.7 Chemical Properties
The main properties of each model at the stage of galactic wind are summarized in the rst part
of Tables 5.8, 5.9 and in Table 5.10. Both tables give: the asymptotic mass ML T 12 (column
1), the eciency of star formation (r) (column 2) column (3) the IMF parameter column
(4) is the time scale of mass accretion (r) in Gyr. Column (5) through (7) are the age in Gyr
at which the galactic wind occurs, and the corresponding dimensionless mass of gas G(r t) and
living stars S(r t), respectively. According to their denition, in order to obtain the real mass in
gas and stars (in solar units) one has to multiply them by the normalization mass, i.e. ML T 12
and ML(r TG ) for each shell (column 13, Tab. 5.10) for the one-zone and multi-zone model,
respectively. Likewise, to get from G(r t) and S(r t) the corresponding densities for the multi-zone
model, the multiplicative factor is L (rj=2 TG). Columns (8) and (9) are the maximum and mean
5.7. CHEMICAL PROPERTIES
79
Table 5.5: The radial dependence of (rj=2) and (rj=2) in galactic models of dierent asymptotic
luminous mass as indicated. The collapse time scales (rj=2) are in Gyr. The galactic baryonic
masses are in units of 1012M
Region
j
0
1
2
3
4
5
6
7
8
9
10
rj +1=2
r1=2
r3=2
r5=2
r7=2
r9=2
r11=2
r13=2
r15=2
r17=2
r19=2
r21=2
3ML T 12
0.74
0.29
0.18
0.13
0.10
0.14
0.20
0.27
0.35
0.46
0.59
1ML T 12
7.1
50.0
111.6
198.6
325.5
501.4
753.8
1116.0
1632.9
2383.7
3493.2
0.52
0.20
0.13
0.09
0.07
0.10
0.14
0.19
0.24
0.32
0.41
0:5ML T 12
9.0
60.6
132.8
233.3
378.3
577.3
860.3
1262.8
1832.5
2653.1
3855.9
0.42
0.16
0.10
0.07
0.06
0.08
0.11
0.15
0.20
0.25
0.33
0:1ML T 12
10.4
68.3
148.3
258.5
416.3
631.6
936.1
1366.9
1973.4
2842.6
4110.1
0.25
0.10
0.06
0.04
0.03
0.05
0.07
0.09
0.12
0.15
0.20
0:05ML T 12
14.7
90.6
191.9
328.7
521.4
780.8
1142.8
1648.8
2352.9
3350.2
4787.8
0.20
0.08
0.05
0.03
0.03
0.04
0.05
0.07
0.09
0.12
0.16
17.0
102.4
214.6
364.9
575.2
856.5
1247.1
1790.2
2542.1
3602.2
5122.6
0:005ML T 12
0.09
0.04
0.03
0.03
0.03
0.03
0.03
0.03
0.04
0.06
0.07
27.7
154.1
312.3
518.5
800.7
1171.1
1676.6
2367.9
3309.8
4616.9
6462.5
metallicity, Z(r t) and hZ(r t)i reached at the onset of the galactic wind. Column (10) contains
the rate of star formation (r t) in units of M /yr. Columns (11) and (12) are the gravitational
binding energy of the gas #g (r t), the total thermal energy of this Eg (r t), respectively. All energy
0 =Re.
are in units of 1050 ergs. Column (14) in Tab. 5.10 is the mid shell fractionary radius rj=
2
The second part of Tables 5.8 and 5.9 present the integrated magnitudes and colors of the
model galaxies at ve dierent ages. Columns (1), (2) and (3) identify the model by means of
(r), , and ML T 12, respectively column (4) is the age in Gyr, columns (5) and (6) are the
integrated absolute bolometric (Mbol ) and visual magnitude (MV ) nally, columns (7) through
(11) are the integrated colors (U{B), (B{V), (V{R), (V{K) and (1550{V), respectively. It is worth
to remind that all the integrated magnitudes refer to the current amount of the galaxy mass in
form of living stars (see below for more details).
5.7.1 Gas content, metallicity, SFR, and N (Z )
The fractionary gas content G(r t) and metallicity Z(r t) for both the one-zone and central region
of multi-zone models, as function of time are shown in Figs. 5.12 and 5.13, respectively. In all
the models, the fractionary gas density Gg (r t) starts small, increases up to a maximum and then
decreases exponentially to zero as a result of the combined eect of gas accretion by infall and gas
consumption by star formation, but owing to the dierent value of (r) from model to model, the
peak occurs later at increasing galaxy mass ML T (TG ).
The top panel of Fig. 5.14 shows the rate of star formation (in units of M /yr) as a function
80
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Figure 5.12: The gas fraction G(RL T t) (Panel a) and metallicity Z(RL T t) (Panel b) as a
function of the age in Gyr for the one-zone model galaxies with = 0:50, (RL T ) increasing with
the mass, and galactic mass from 3ML T 12 to 0.01ML T 12. In Panel (a), the intersection of the
curves with the vertical lines indicates the stage of the onset of galactic winds and consequent
drop-o of the gas content. In Panel (b) the horizontal lines show the constant metallicity after
the interruption of star formation by galactic winds
of time (in Gyr) for the central core of the multi-zone models, up to the onset of galactic winds,
very similar results are obtained for the one-zone model (see Tantalo et al. (1996) for all details].
As expected, the rate of star formation starts very small, grows to a maximum, and then declines
exponentially with time, closely mimicking the gas content. The initial period of very low activity
is the reason why infall models avoid the so called G-Dwarf problem. The gas liberated by evolving
stars (supernova explosions, stellar winds, and PN) in subsequent epochs is not shown here as all
this gas is supposed to be rapidly heated up to the escape velocity.
The bottom panels of the same gure display the comparison between the thermal and the
binding energy of the gas, Eth(r t) and #g (r t), respectively, as a function of time for the nuclear
regions. All the energies are in units of 1050 erg. The intersection between #g (r t) and Eth(r t)
corresponds to the onset of the galactic wind for the innermost region. Similar diagrams can be
drawn for all the remaining shells and for the one-zone models. They are not displayed for the
sake of brevity.
The chemical structure of the models is best understood looking at the fractionary cumulative
mass distribution of living stars, %Z0 SZ =S, where S is the mass fraction in stars, and SZ is the
mass fraction of stars with metallicity up to Z, and at the so-called partition function N(Z), i.e.
the relative number of living stars per metallicity bin. Within a galaxy (or region of it) both
5.7. CHEMICAL PROPERTIES
81
Figure 5.13: The gas fraction G(rj=2 t) (top panel) and metallicity Z(rj=2 t) (bottom panel) as a
function of the age in Gyr for the central core of the multi-zone models with dierent asymptotic
mass ML T 12 as indicated
distributions vary as a function of the age. The fractionary, cumulative mass distribution as a
function of Z is shown in Fig. 5.15 limited to the central core (r1=2, left panel) and the rst shell
(r3=2, right panel) for all the multi-zone models calculated at the present age (TG = 15 Gyr). The
vertical line corresponds to the solar metallicity. In the core and the rst shell of the most massive
galaxy, about 10% of the stars have metallicity lower than solar. In contrast, the central region
of the lowest mass galaxy has about 25% of its stellar content with metallicity lower than solar.
This percentage increases to about 36% in the rst shell. In all galaxies the percentage of stars
with metallicity lower than solar increases moving further out. For sake of brevity I do not show
here the results for the one-zone models which are very similar (see Tantalo et al. (1996)).
The partition function N(Z) of the simplest one-zone models compared with the that of one
Closed-box model is shown in Fig. 5.16. As expected the Closed-box contains more metal-poor
object than the it infall models, that conrms the natural solution of the G-Dwarf problem.
The partition function N(Z) for multi-zone galaxy models at the age of 15 Gyr is shown in
Fig. 5.17 limited to the central core (left panel) and rst shell (right panel). From this diagram
can be pointed out that the mean metallicity of the stars in the core goes from Z ' 0:03 to
Z ' 0:04 the peak value tends to slightly shift toward higher metallicities at increasing galaxy
mass and there are wings toward both low and high metallicities. The distribution tends to be
more concentrated in the rst shell, where we see a more abundant population of low metallicity
stars and a sharper cut-o at the high metallicity edge caused by the action of galactic winds.
82
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Figure 5.14: Panel a: the star formation rate as a function of time for the central core of the
galaxy models with dierent asymptotic mass ML T 12 as indicated. Panel b: the gravitational
binding energy #g (r t) and thermal content of the gas Eth (r t), for the same models as above.
Energies are in units of 1050 ergs. Analog results for the one-zone models (Tantalo et al., 1996)
Likewise for the remaining shells not displayed here.
5.7.2 The onset of galactic winds
Dierent results have been obtained for the one-zone and multi-zone models as far as the onset of
galactic winds are concerned.
In the one-zone models the onset of galactic winds occurs within the rst 0.5Gyr for all the
masses under consideration (cf. entries of Table 5.8). In contrast to the old models of elliptical
galaxies by Bressan et al. (1994), in which the onset of galactic winds occured later in massive
galaxies, in the present models with infall the winds tend to occur earlier in more massive galaxies
than in the low mass ones. This can be understood as the result of the eciency of star formation
per unit mass of gas () increasing from = 1 to = 12 as the galactic mass goes from 0.01 to
3ML T 12. This trend is imposed by the simultaneous t of the slope and mean colors of the CMR
and the dependence of the UV excess on the galaxy luminosity and hence mass (see chapter 6).
The result agrees with the suggestion by Matteucci (1994) to explain the high value of the
Mg/Fe] observed in the brightest elliptical galaxies and its decrease toward the solar value at
decreasing luminosity of the galaxy. We will come back to this topic in more detail in chapter x 7.
In the multi-zone models the galactic winds occur earlier in the external regions than in the
center, or at given relative distance from the center it occurs later at increasing galaxy mass. This
5.7. CHEMICAL PROPERTIES
83
Figure 5.15: The cumulative fractionary mass of living stars as a function of the metallicity for
the galaxy models with mass 3, 0.5, 0.05, and 0.005 ML T 12. Panels a and b corresponds to the
central core and rst shell, respectively. Same results have been obtained for the one-zone models
(Tantalo et al., 1996)
is shown in Fig. 5.18 which displays the age of the galactic wind tgw as a function of the galactocentric distance. The stratication in metallicity, and relative percentage of stars in dierent
metallicity bins resulting from the above trend in tgw bears very much on inferring chemical
abundances from local or integrated photometric properties of elliptical galaxies.
As far as the metallicity is concerned, this increases more slowly at increasing galaxy mass up
to the maximum value reached in coincidence of the galactic wind. As expected the maximum
metallicity increases with the galaxy mass, because in this type of model galactic winds occur
later at increasing galaxy mass (cf. the entries of Tables 5.10 and Fig. 5.18). Fig. 5.19 shows
the maximum (Zmax , top panel) and mean (Zmean , bottom panel) metallicity as a function of
the radial distance from the center (normalized to the eective radius of each galaxy) for all the
multi-zone models as in Figs. 5.10 and 5.11. The mean gradient in the maximum metallicity,
dZmax =dlog(r), within 1:5RL e ranges from {0.064 to {0.042 going from massive to dwarf galaxies,
whereas the mean gradient in mean metallicity, dZmean=dlog(r), over the same radial distance and
galaxy mass interval goes from {0.021 to {0.019.
Finally, we note that owing to the onset of galactic winds and subsequent interruption of the
star formation activity, the remnant galaxy made of stars has a mass which is only a fraction of
its asymptotic value (ML T 12).
84
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Figure 5.16: The relative number of alive stars per metallicity bin normalized to ML (TG ). The
solid and dotted lines refer to one-zone models with infall of 3ML T 12 and 0.5ML T 12 with = 12
and = 5:2, respectively. Both are calculated with = 0:50. The thick line displays the closed-box
model with 3ML T 12, = 20, and = 0:40
5.7.3 Internal consistency of the Multi-zone models
The scheme elaborated to obtain a multi-zone structure is self-contained in absence of galactic
winds, because in such case at the galaxy age TG all shells have reached their asymptotic mass
and the eective radius RL e (the basic scale factor associating the asymptotic density of the
Young prole to each radius) is consistent with ML T (TG ).
At the stage of galactic wind it is supposed that all the gas contained in the shell, the one still
in the infall process and the one expelled by supernova explosions and stellar winds are ejected
into the intergalactic medium and never re-used to form stars. This implies that at the stage of
galactic wind the real mass of each shell (the fraction of gas turned into long-lived stars up to this
stage), is smaller than the corresponding asymptotic mass ML(rj=2 TG ). Indeed in each shell
the luminous mass has grown up to the value ML (rj=2 tgw ), where tgw is the local value of the
age at the onset of the galactic wind. Therefore:
;1 ML (rj=2 tgw ) < ML T (TG )
%Jj =0
(5.48)
Recalling that all the calculations refer to the innermost part of the galaxy (the one containing
55% of the mass ML T (TG )), the Eq. 5.48 should be replaced by:
5.7. CHEMICAL PROPERTIES
85
Figure 5.17: Relative number of living stars per metallicity bin in the central core panel a) and
rst shell panel b) for the multi-zone models with 3, 0.5, 0.05 and 0.005 ML T 12 (the same models
as in Fig. 5.15)
%10
j =0 ML(rj=2 tgw ) < 0:55 ML T (TG )
(5.49)
Looking at the case of the 3ML T 12 galaxy, the sphere which has been followed in detail has
total asymptotic mass of 1:65 1012M , each shell containing about 0:15 1012M (cf. Column
(5) of Table 5.10). In contrast, the total mass reached in the same sphere at the onset of the
galactic wind amounts only to 0:66 1012M , i.e. some 40% of the expected mass. Even more
important, while the innermost shells were able to convert in stars about 0.8 of their asymptotic
mass, this is not the case of the outermost shells in which only about 2% of the potential mass has
been turned into stars, all the rest being dispersed by a very early wind. Considering that owing
to the very low densities in regions above the last shell (approximately 1:5RL e), the galactic winds
would occur even earlier than in the last computed shell, this means that starting with 3ML T 12 of
gas eligible to star formation only 22% of it has been actually turned into long-lived stars visible
today. The situation gets slightly better at decreasing ML T (TG ) because of the much shorter
mean infall time scale (cf. Tables 5.6 and 5.5).
Furthermore, looking at the radial prole of L (rj=2 tgw ) and comparing it with L (rj=2 TG ),
the former is steeper than the latter, over the shells external RL e in particular. However, when the
comparison is limited to the shells inside RL e (up to j = 8 in the actual notation), the dierence
is remarkably smaller. This implies that the region inside RL e does not depart too much from the
86
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Figure 5.18: The age at which galactic winds occur in regions of increasing distance from the
galactic center. The models are the same as in Figs. 5.10 and 5.11
basic hypothesis. Finally, the eective radius RL e used to interpolate in the Young (1976) density
prole and to assign L (rj=2 TG ) referred to the asymptotic mass ML T (TG ). Since the actual
present-day mass of the galaxy is smaller than this, the actual eective radius should be smaller
than the originally adopted value. With the aid of relation (5.35) above, the 3ML T 12 galaxy has
RL e ' 31:9 Kpc, whereas the 0.66ML T 12 daughter should have RL e ' 13:7 Kpc (a factor 2.3
smaller). This means that the ratio of the mean density (inside RL e) of the parent to daughter
galaxy is about 0.5. It is as if the models are calculated under-estimating their real density by a
factor of about two. Considering that even within the eective radius passing from the center to
the periphery the density of luminous mass drops by orders of magnitude, cf. Young (1976), and
all other uncertainties aecting the models, the above discrepancy can be perhaps tolerate. The
results presented in this thesis perhaps constitute the best justication of these models, which do
not dare to replace more sophisticated, physically grounded formulations in literature, but simply
aim at providing simple tool to investigate the chemo-spectro-photometric properties and their
spatial gradients of spherical systems roughly simulating elliptical galaxies.
Table 5.6 summarizes the data relative to the above discussion for all the model galaxies under
examination. It lists the asymptotic total mass ML T (TG ) (column 1), the corresponding eective
radius RL e(TG ) (column 2), the asymptotic mass ML (1:5RL e TG ) within 1:5RL e (the studied
model, column 3), the actual mass ML T (tgw ) of the galaxy within 1:5RL e at the age tgw (column
4), the actual mass ML T (RL e tgw ) of the galaxy within RL e at the age tgw (column 5), and the
5.7. CHEMICAL PROPERTIES
87
Figure 5.19: The gradients in maximum (top panel) and mean metallicity (bottom panel) for the
model galaxies with dierent ML T 12 as indicated
real eective radius RL e(tgw ) (column 6).
Having assumed the Young (1976) density prole imposes that the resulting model at the age
TG must possess (i) a radially constant mass to luminosity ratio (ii) a luminosity prole obeying
the r1=4 law.
In brief to check the radial dependence of the (M=LB ) , the cumulative (M=LB ) (rj=2) have
been calculated moving from the center up to the last computed zone. The results are presented
in Table 5.7 limited to a few selected radii and the 3 and 0.1 ML T 12 galaxies. The selected
radii rj=2 correspond to the central core, 0:6RL e, RL e and 1:5RL e. It is soon evident that the
(M=LB )(rj=2) ratio is nearly constant (within about 10%) passing from the center to the external
regions. This implies that the rst condition imposed by the choice of the Young (1976) density
prole for the luminous material, i.e. radially constant mass to luminosity ratio, is almost fully
veried.
Reference : Spectro-Photometric evolution of elliptical galaxies. II. Models with Infall
R. Tantalo, C. Chiosi, A. Bressan and F. Fagotto
1996, Astron. Astrophys. 311, 361, astro-ph/9602003
Reference : Spectro-Photometric evolution of elliptical galaxies. III. Infall models with gradients
in mass density and star formation
R. Tantalo, C. Chiosi, A. Bressan, P. Marigo and L. Portinari
1997, Astron. Astrophys. Accepted, astro-ph/9710079
88
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
Table 5.6: Fractionary masses of gas and stars components in units of 1012M for the models
presented in this work.
ML T
(TG)
3
1
0.5
0.1
0.05
0.005
(
RL e TG
)
ML
(1:5RL e TG )
31.6
17.1
11.7
4.7
3.2
0.9
ML T
1.65000
0.55000
0.27500
0.05500
0.02750
0.00275
(1:5RL e tgw )
(
ML RL e tgw
0.65700
0.21800
0.10900
0.02200
0.01090
0.00102
0.65600
0.21500
0.10700
0.02100
0.01070
0.00097
)
(
RL e tgw
)
13.55
7.33
4.98
2.04
1.38
0.37
Table 5.7: The cumulative mass to blue-luminosity ratio log(M=LB ) at the age of 15 Gyr and
as a function of the galacto-centric distance. ML (rj=2) in the mass in units of 1012M contained
in the sphere of radius rj=2. The magnitudes and colors are the integrated values within the same
sphere. The radii rj=2 correspond to the central core, 0:6RL e, RL e and 1:5RL e.
ML T
12
rj=2
3
0.06
0.58
1.04
1.48
0.1
0.06
0.58
1.04
1.48
(B{V)
MB
-21.07
-22.60
-22-69
-22.70
1.00
1.00
0.99
0.99
-20.07
-21.64
-21.70
-21.70
0.123
0.632
0.656
0.660
7.448
8.976
8.856
8.803
-17.31
-18.96
-19.08
-19.10
1.01
0.93
0.91
0.91
-16.30
-18.03
-18.17
-18.19
0.004
0.021
0.022
0.022
8.466
8.141
7.511
7.462
MV
(
ML rj=2
)
M=LB
12
(1)
3.000
1.000
0.500
0.100
0.050
0.010
ML T
Table 5.8:
(
12.00
7.20
5.20
3.00
2.50
1.00
12.00
7.20
5.20
3.00
2.50
1.00
12.00
7.20
5.20
3.00
2.50
1.00
12.00
7.20
5.20
3.00
2.50
1.00
12.00
7.20
5.20
3.00
2.50
1.00
12.00
7.20
5.20
3.00
2.50
1.00
RL T
(2)
12.0
7.2
5.2
3.0
2.5
1.0
RL T
(
)
)
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
(3)
0.50
0.50
0.50
0.50
0.50
0.50
(4)
0.10
0.10
0.10
0.10
0.10
0.10
)
12
3.000
1.000
0.500
0.100
0.050
0.010
3.000
1.000
0.500
0.100
0.050
0.010
3.000
1.000
0.500
0.100
0.050
0.010
3.000
1.000
0.500
0.100
0.050
0.010
3.000
1.000
0.500
0.100
0.050
0.010
3.000
1.000
0.500
0.100
0.050
0.010
ML T
(
RL T
17.0000
17.0000
17.0000
17.0000
17.0000
17.0000
15.0000
15.0000
15.0000
15.0000
15.0000
15.0000
12.0000
12.0000
12.0000
12.0000
12.0000
12.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
8.0000
8.0000
8.0000
8.0000
8.0000
8.0000
5.0000
5.0000
5.0000
5.0000
5.0000
5.0000
Age
(5)
0.26
0.31
0.35
0.34
0.29
0.43
tg!
)
tg!
-24.616
-23.366
-22.517
-20.402
-19.363
-17.292
-24.694
-23.448
-22.603
-20.492
-19.458
-17.402
-24.866
-23.627
-22.786
-20.682
-19.648
-17.590
-25.088
-23.839
-22.988
-20.852
-19.810
-17.752
-25.269
-24.026
-23.178
-21.048
-20.012
-17.968
-25.699
-24.456
-23.608
-21.463
-20.401
-18.322
Mbol
-23.696
-22.501
-21.697
-19.673
-18.690
-16.709
-23.778
-22.587
-21.787
-19.773
-18.794
-16.824
-23.950
-22.764
-21.968
-19.966
-18.989
-17.019
-24.144
-22.950
-22.148
-20.132
-19.152
-17.181
-24.349
-23.153
-22.349
-20.335
-19.359
-17.397
-24.776
-23.584
-22.782
-20.753
-19.757
-17.774
MV
(7)
0.696
0.648
0.589
0.408
0.310
0.218
S(RL T
)
tg!
0.605
0.596
0.576
0.475
0.418
0.330
0.598
0.588
0.568
0.473
0.420
0.336
0.554
0.548
0.533
0.448
0.399
0.318
0.525
0.523
0.509
0.422
0.375
0.299
0.483
0.480
0.467
0.386
0.344
0.276
0.445
0.421
0.398
0.323
0.280
0.215
(U{B)
(8)
0.0710
0.0629
0.0543
0.0328
0.0235
0.0158
Z(RL T
)
(
0.998
1.000
0.995
0.948
0.916
0.861
1.001
1.002
0.998
0.955
0.928
0.880
0.981
0.985
0.983
0.946
0.922
0.880
0.967
0.973
0.971
0.932
0.908
0.868
0.936
0.946
0.946
0.909
0.887
0.850
0.910
0.908
0.902
0.862
0.837
0.792
(B{V)
(9)
0.0360
0.0307
0.0265
0.0166
0.0122
0.0080
)
hZ RL T tg! i
Evolution of the Integrated Colors
tg!
(6)
0.232
0.312
0.381
0.560
0.640
0.768
G(RL T
(
tg!
0.742
0.735
0.728
0.704
0.690
0.664
0.742
0.735
0.727
0.704
0.690
0.667
0.732
0.725
0.719
0.697
0.685
0.663
0.727
0.720
0.713
0.688
0.676
0.656
0.711
0.705
0.699
0.676
0.665
0.646
0.692
0.683
0.674
0.651
0.638
0.615
(V{R)
(10)
8.25E+03
2.22E+03
9.82E+02
1.67E+02
7.97E+01
7.67E+00
RL T
One-zone model: Chemical properties at the Stage of Wind Ejection. k=1
)
tg!
3.406
3.329
3.261
3.093
2.983
2.792
3.404
3.327
3.258
3.082
2.974
2.796
3.414
3.340
3.273
3.090
2.979
2.798
3.464
3.390
3.318
3.105
2.985
2.804
3.439
3.373
3.306
3.098
2.983
2.811
3.467
3.386
3.310
3.100
2.970
2.770
(V{K)
(11)
2.60E+10
7.13E+09
3.20E+09
4.55E+08
1.90E+08
2.22E+07
g (RL T
)
tg!
2.064
2.713
3.401
3.365
3.248
3.039
2.211
2.933
3.794
3.891
3.820
3.752
2.279
3.150
4.589
5.406
5.376
5.382
2.146
3.060
4.704
5.940
5.917
5.806
2.624
3.526
5.090
6.067
5.997
5.794
6.106
6.080
6.032
5.816
5.638
5.229
(1550{V)
(12)
2.65E+10
7.40E+09
3.23E+09
4.67E+08
1.93E+08
2.25E+07
(
Eg RL T
)
5.7. CHEMICAL PROPERTIES
89
12
(1)
3.000
1.000
0.500
0.100
0.050
0.010
ML T
Table 5.9:
(
12.00
7.20
5.20
3.00
2.50
1.00
12.00
7.20
5.20
3.00
2.50
1.00
12.00
7.20
5.20
3.00
2.50
1.00
12.00
7.20
5.20
3.00
2.50
1.00
12.00
7.20
5.20
3.00
2.50
1.00
12.00
7.20
5.20
3.00
2.50
1.00
RL T
(2)
12.0
7.2
5.2
3.0
2.5
1.0
RL T
(
)
)
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
(3)
0.50
0.50
0.50
0.50
0.50
0.50
(4)
0.10
0.10
0.10
0.10
0.10
0.10
)
12
3.000
1.000
0.500
0.100
0.050
0.010
3.000
1.000
0.500
0.100
0.050
0.010
3.000
1.000
0.500
0.100
0.050
0.010
3.000
1.000
0.500
0.100
0.050
0.010
3.000
1.000
0.500
0.100
0.050
0.010
3.000
1.000
0.500
0.100
0.050
0.010
ML T
(
RL T
17.0000
17.0000
17.0000
17.0000
17.0000
17.0000
15.0000
15.0000
15.0000
15.0000
15.0000
15.0000
12.0000
12.0000
12.0000
12.0000
12.0000
12.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
8.0000
8.0000
8.0000
8.0000
8.0000
8.0000
5.0000
5.0000
5.0000
5.0000
5.0000
5.0000
Age
(5)
7.71
4.58
3.12
0.82
0.52
0.61
tg!
tg!
)
tg!
(7)
0.846
0.856
0.831
0.578
0.420
0.265
S(RL T
)
tg!
(8)
0.2002
0.1686
0.1409
0.0578
0.0351
0.0203
Z(RL T
)
(
(9)
0.0697
0.0620
0.0544
0.0263
0.0171
0.0100
-25.357
-24.090
-23.248
-20.868
-19.750
-17.564
-25.421
-24.151
-23.306
-20.955
-19.841
-17.670
-25.620
-24.354
-23.511
-21.144
-20.034
-17.863
-25.844
-24.564
-23.715
-21.345
-20.205
-18.022
-26.123
-24.777
-23.928
-21.550
-20.404
-18.233
-26.639
-25.329
-24.406
-22.000
-20.830
-18.625
Mbol
-24.154
-22.930
-22.139
-20.058
-19.023
-16.942
-24.240
-23.010
-22.213
-20.149
-19.123
-17.055
-24.482
-23.254
-22.448
-20.336
-19.317
-17.253
-24.722
-23.496
-22.681
-20.518
-19.483
-17.414
-25.020
-23.728
-22.923
-20.729
-19.689
-17.628
-25.434
-24.201
-23.317
-21.173
-20.115
-18.028
MV
0.406
0.424
0.444
0.550
0.473
0.365
0.393
0.414
0.440
0.543
0.471
0.369
0.341
0.338
0.362
0.509
0.446
0.349
0.330
0.333
0.340
0.484
0.420
0.327
0.330
0.377
0.392
0.446
0.385
0.301
0.217
0.332
0.373
0.384
0.322
0.242
(U{B)
0.845
0.864
0.883
0.981
0.946
0.883
0.839
0.863
0.888
0.985
0.954
0.899
0.785
0.802
0.833
0.971
0.945
0.896
0.758
0.770
0.796
0.958
0.930
0.882
0.707
0.762
0.785
0.934
0.907
0.862
0.665
0.734
0.800
0.893
0.861
0.810
(B{V)
)
hZ RL T tg! i
Evolution of the Integrated Colors
(6)
0.031
0.068
0.116
0.415
0.574
0.731
G(RL T
(
tg!
0.749
0.746
0.742
0.722
0.702
0.671
0.746
0.743
0.741
0.722
0.702
0.674
0.727
0.722
0.721
0.713
0.694
0.669
0.720
0.713
0.710
0.706
0.686
0.660
0.701
0.711
0.706
0.693
0.673
0.649
0.675
0.697
0.707
0.671
0.649
0.622
(V{R)
(10)
3.42E+01
3.28E+01
3.42E+01
5.13E+01
4.12E+01
5.35E+00
RL T
One-zone model: Chemical properties at the Stage of Wind Ejection k=2
)
tg!
3.719
3.670
3.609
3.240
3.091
2.878
3.692
3.646
3.592
3.239
3.083
2.877
3.635
3.577
3.535
3.252
3.092
2.882
3.645
3.567
3.523
3.293
3.107
2.883
3.674
3.598
3.540
3.289
3.100
2.881
3.805
3.756
3.701
3.310
3.107
2.871
(V{K)
(11)
3.55E+09
1.57E+09
9.77E+08
3.39E+08
1.72E+08
2.12E+07
g (RL T
)
tg!
0.112
0.267
0.454
3.131
3.369
3.115
0.152
0.295
0.517
3.472
3.909
3.794
0.188
0.199
0.343
3.940
5.434
5.441
0.447
0.529
0.605
3.994
5.940
5.853
1.385
1.633
1.902
4.451
6.064
5.869
0.622
5.452
6.170
5.962
5.798
5.365
(1550{V)
(12)
3.58E+09
1.58E+09
9.78E+08
3.48E+08
1.76E+08
2.14E+07
(
Eg RL T
)
90
CHAPTER 5. MODELS: CHEMICAL PROPERTIES
(rj=2 )
(2)
7.1
50.0
111.6
198.6
325.5
501.4
753.8
1116.0
1632.9
2383.7
3493.2
9.0
60.6
132.8
233.3
378.3
577.3
860.3
1262.8
1832.5
2653.1
3855.9
10.4
68.3
148.3
258.5
416.3
631.6
936.1
1366.9
1973.4
2842.6
4110.1
ML T 12
(1)
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
(3)
0.42
0.16
0.10
0.07
0.06
0.08
0.11
0.15
0.20
0.25
0.33
0.52
0.20
0.13
0.09
0.07
0.10
0.14
0.19
0.24
0.32
0.41
0.74
0.29
0.18
0.13
0.10
0.14
0.20
0.27
0.35
0.46
0.59
(4)
(rj=2 )
2.68
0.40
0.19
0.10
0.06
0.03
0.02
0.01
0.01
0.01
0.01
3.60
0.54
0.25
0.13
0.07
0.04
0.02
0.01
0.01
0.01
0.01
5.12
0.79
0.38
0.21
0.11
0.05
0.02
0.01
0.01
0.01
0.01
(5)
tg!
0.006
0.015
0.018
0.020
0.022
0.017
0.010
0.005
0.003
0.001
0.001
0.005
0.012
0.014
0.018
0.020
0.015
0.009
0.004
0.002
0.001
0.001
0.004
0.010
0.011
0.013
0.015
0.012
0.007
0.004
0.002
0.001
0.001
(6)
G(rj=2 tg! )
0.879
0.860
0.803
0.723
0.593
0.305
0.118
0.049
0.032
0.022
0.016
0.864
0.875
0.818
0.722
0.592
0.309
0.112
0.044
0.027
0.019
0.014
0.845
0.874
0.834
0.764
0.637
0.284
0.102
0.038
0.022
0.016
0.011
(7)
S(rj=2 tg! )
0.0876
0.0386
0.0335
0.0311
0.0293
0.0269
0.0227
0.0095
0.0071
0.0056
0.0049
0.0947
0.0412
0.0344
0.0311
0.0290
0.0268
0.0228
0.0105
0.0082
0.0068
0.0065
0.0964
0.0439
0.0362
0.0323
0.0293
0.0261
0.0236
0.0148
0.0102
0.0092
0.0079
(8)
Z(rj=2 tg! )
0.0338
0.0266
0.0248
0.0231
0.0208
0.0167
0.0087
0.0095
0.0071
0.0056
0.0049
0.0351
0.0275
0.0255
0.0237
0.0216
0.0184
0.0108
0.0105
0.0082
0.0068
0.0065
0.0365
0.0286
0.0266
0.0251
0.0232
0.0197
0.0134
0.0050
0.0102
0.0092
0.0079
(9)
hZ (rj=2 tg! )i
Table 5.10: Multi-zone model: Chemical properties at the Stage of Wind Ejection
1.41E+00
2.52E+01
6.25E+01
1.24E+02
2.24E+02
2.55E+02
2.20E+02
1.68E+02
1.28E+02
9.87E+01
7.65E+01
1.92E+00
3.41E+01
9.21E+01
2.01E+02
3.64E+02
4.11E+02
3.60E+02
2.71E+02
2.07E+02
1.59E+02
1.23E+02
3.67E+00
6.97E+01
1.74E+02
3.63E+02
6.97E+02
9.04E+02
7.79E+02
5.83E+0
4.41E+02
3.38E+02
2.61E+02
(10)
(rj=2 tg! )
3.54E+07
2.78E+07
2.48E+07
2.46E+07
2.49E+07
1.74E+07
9.55E+06
4.81E+06
2.51E+06
1.34E+06
7.21E+05
7.59E+07
5.80E+07
5.55E+07
6.01E+07
6.05E+07
4.19E+07
2.31E+07
1.14E+07
5.91E+06
3.13E+06
1.68E+06
2.99E+08
2.34E+08
2.04E+08
2.08E+08
2.20E+08
1.73E+08
9.28E+07
4.54E+07
2.30E+07
1.21E+07
6.41E+06
(11)
g (rj=2 tg! )
3.60E+07
2.81E+07
2.71E+07
2.71E+07
2.60E+07
1.77E+07
9.56E+06
4.98E+06
3.18E+06
1.52E+06
8.70E+05
8.37E+07
6.56E+07
6.20E+07
6.03E+07
6.22E+07
4.50E+07
2.35E+07
1.15E+07
6.70E+06
3.47E+06
2.13E+06
7.99E+07
2.34E+08
2.28E+08
2.30E+08
2.41E+08
1.81E+08
9.53E+07
4.56E+07
2.30E+07
1.26E+07
6.56E+06
(12)
Eg (rj=2 tg! )
0.02425
0.02500
0.02501
0.02498
0.02500
0.02500
0.02499
0.02500
0.02503
0.02498
0.02503
0.04850
0.05001
0.05002
0.04995
0.05001
0.05001
0.04999
0.05000
0.05006
0.04997
0.05007
0.14550
0.15000
0.15010
0.14990
0.15000
0.15000
0.15000
0.15000
0.15020
0.14990
0.15020
(13)
ML (rj=2 TG )
0.060
0.156
0.248
0.347
0.456
0.577
0.712
0.865
1.040
1.241
1.476
0.060
0.156
0.248
0.347
0.456
0.577
0.712
0.865
1.040
1.241
1.476
0.060
0.156
0.248
0.347
0.456
0.577
0.712
0.865
1.040
1.241
1.476
(14)
rj=2
5.7. CHEMICAL PROPERTIES
91
(rj=2 )
(2)
14.7
90.6
191.9
328.7
521.4
780.8
1142.8
1648.8
2352.9
3350.2
4787.8
17.0
102.4
214.6
364.9
575.2
856.5
1247.1
1790.2
2542.1
3602.2
5122.6
27.7
154.1
312.3
518.5
800.7
1171.1
1676.6
2367.9
3309.8
4616.9
6462.5
ML T 12
(1)
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
(3)
0.09
0.04
0.03
0.03
0.03
0.03
0.03
0.03
0.04
0.06
0.07
0.20
0.08
0.05
0.03
0.03
0.04
0.05
0.07
0.09
0.12
0.16
0.25
0.10
0.06
0.04
0.03
0.05
0.07
0.09
0.12
0.15
0.20
(4)
(rj=2 )
0.44
0.07
0.03
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.00
1.08
0.16
0.08
0.05
0.03
0.02
0.01
0.01
0.01
0.00
0.00
1.45
0.22
0.10
0.05
0.04
0.02
0.01
0.01
0.01
0.00
0.00
(5)
tg!
0.018
0.048
0.049
0.044
0.035
0.027
0.020
0.012
0.007
0.004
0.002
0.010
0.028
0.031
0.032
0.033
0.024
0.014
0.008
0.004
0.002
0.001
0.008
0.022
0.027
0.029
0.028
0.021
0.013
0.007
0.004
0.002
0.001
(6)
G(rj=2 tg! )
0.920
0.785
0.686
0.521
0.376
0.265
0.199
0.120
0.085
0.068
0.047
0.912
0.825
0.759
0.689
0.591
0.325
0.140
0.077
0.051
0.037
0.026
0.902
0.850
0.766
0.681
0.617
0.326
0.129
0.067
0.041
0.029
0.024
(7)
S(rj=2 tg! )
0.0583
0.0358
0.0360
0.0301
0.0241
0.0115
0.0059
0.0020
0.0012
0.0015
0.0004
0.0720
0.0354
0.0327
0.0331
0.0322
0.0255
0.0108
0.0055
0.0035
0.0027
0.0019
0.0765
0.0367
0.0327
0.0315
0.0323
0.0260
0.0118
0.0066
0.0034
0.0027
0.0033
(8)
Z(rj=2 tg! )
0.0269
0.0210
0.0172
0.0126
0.0070
0.0115
0.0059
0.0020
0.0012
0.0015
0.0004
0.0299
0.0241
0.0221
0.0199
0.0159
0.0102
0.0108
0.0055
0.0035
0.0027
0.0019
0.0312
0.0251
0.0229
0.0208
0.0184
0.0127
0.0040
0.0066
0.0034
0.0027
0.0033
(9)
hZ (rj=2 tg! )i
1.18E-01
1.81E+00
3.76E+00
5.52E+00
6.85E+00
7.72E+00
8.23E+00
6.69E+00
5.27E+00
4.12E+00
3.23E+00
3.90E-01
7.04E+00
1.59E+01
2.82E+01
4.54E+01
4.98E+01
4.31E+01
3.38E+01
2.61E+01
2.02E+01
1.58E+01
5.74E-01
9.43E+00
2.48E+01
4.64E+01
6.98E+01
8.11E+01
7.04E+01
5.48E+01
4.22E+01
3.26E+01
2.54E+01
(10)
(rj=2 tg! )
Table 5.10: (Continue) Multi-zone model: Chemical properties at the Stage of Wind Ejection
1.43E+05
1.14E+05
9.10E+04
7.02E+04
5.11E+04
3.69E+04
2.59E+04
1.44E+04
7.98E+03
4.48E+03
4.48E+03
2.15E+06
1.86E+06
1.56E+06
1.42E+06
1.31E+06
9.04E+05
5.04E+05
2.67E+05
1.43E+05
7.81E+04
4.30E+04
4.99E+06
3.84E+06
3.71E+06
3.53E+06
3.03E+06
2.20E+06
1.22E+06
6.39E+05
3.38E+05
1.83E+05
1.01E+05
(11)
g (rj=2 tg! )
1.57E+05
1.21E+05
9.52E+04
7.26E+04
5.12E+04
3.72E+04
2.72E+04
1.57E+04
1.07E+04
1.07E+04
1.07E+04
2.34E+06
1.86E+06
1.72E+06
1.52E+06
1.36E+06
9.14E+05
5.07E+05
3.15E+05
2.06E+05
1.16E+05
4.73E+04
5.33E+06
4.36E+06
3.93E+06
3.67E+06
3.54E+06
2.29E+06
1.22E+06
7.26E+05
3.58E+05
1.89E+05
1.63E+05
(12)
Eg (rj=2 tg! )
0.00024
0.00025
0.00025
0.00025
0.00025
0.00025
0.00025
0.00025
0.00025
0.00025
0.00025
0.00243
0.00250
0.00250
0.00250
0.00250
0.00250
0.00250
0.00250
0.00250
0.00250
0.00250
0.00485
0.00500
0.00500
0.00500
0.00500
0.00500
0.00500
0.00500
0.00501
0.00500
0.00501
(13)
ML (rj=2 TG )
0.060
0.156
0.248
0.347
0.456
0.577
0.712
0.865
1.040
1.241
1.476
0.060
0.156
0.248
0.347
0.456
0.577
0.712
0.865
1.040
1.241
1.476
0.060
0.156
0.248
0.347
0.456
0.577
0.712
0.865
1.040
1.241
1.476
(14)
rj=2
92
CHAPTER 5. MODELS: CHEMICAL PROPERTIES