The chemical evolution, providing a star formation history, is a preliminary step
for our spectrophotometric model.
The code describes one-zone (no dependence on space, only on time) open
models including the infall of primordial gas, according to the
standard equations of galactic chemical evolution, which we outline
below.
In order to simulate
the chemical evolution of galaxies, we have to express the time derivative of
total gas mass Mg(t) and of the mass of gas in the form of element i,
Mg,i(t) = Xi(t) Mg(t), in terms of computable quantities at time t,
i.e. as a function of the previous history of the galaxy, keeping into
account all the processes intervening in the evolution:
|
|
.
M
|
g,i
|
= |
.
M
|
g,i
|
|
ê
ê
|
SF
|
+ |
.
M
|
g,i
|
|
ê
ê
|
FB
|
+ |
.
M
|
g,i
|
|
ê
ê
|
Inf
|
|
| (1) |
where the 3 terms on the right hand side represent, respectively, the
depletion of gas due to the formation of new stars, the
feedback to the ISM due to the final stages and death of
stars, and the infall of primordial gas forming the galaxy.
The rate of gas depletion due to star formation can
be written as:
|
|
.
M
|
g,i
|
|
ê
ê
|
SF
|
= - Xi(t) Y(t) = - Mg,i(t) |
Y(t)
Mg(t)
|
|
| (2) |
where Y(t) is the SFR, which we write as the sum of two
terms, a Schmidt-type one and, when needed, an analytical one,
explicit function of time (e.g. constant or exponential, used for
instance to add a burst of star formation over a quiet evolution):
|
Y(t) = n Mg(t)k + f(t) M\odot yr-1 |
| (3) |
The stellar mass return to the ISM is computed under the assumption that
it takes place entirely at the stars death. With this (good) approximation,
the mass of gas in the form of element i given back to the ISM by all stars
which have died before time t is:
|
Mg,i(t) |FB = |
ó
õ
|
t
0
|
dt¢ |
ó
õ
|
¥
M(t-t¢,Z(t¢))
|
dm Y(t¢) F(m)Ri(m, Z(t¢)) |
| (4) |
where M(t,Z) is the mass of the star of metallicity Z and
lifetime t, F(m) is the IMF (see below) and Ri(m, Z) the mass
fraction in the form of element i ejected by the star of mass
m and metallicity Z (we adopt stellar ejecta from Portinari et al. 1998).
The time derivative of this integral yields the rate of stellar
feedback to the ISM:
|
|
.
M
|
g,i
|
(t) |
ê
ê
|
FB
|
= - |
ó
õ
|
t
0
|
|
¶M(t-t¢,Z(t¢))
¶t
|
Y(t¢) [ F(m)Ri(m, Z(t¢))]m = M(t-t¢,Z(t¢)) dt¢ |
| (5) |
The IMF is a power-law, F(m) µ m-x in different mass intervals,
normalized to unitary total mass.
Of course, since the purpose is the spectral synthesis, we adopt those IMF for
which our SSP library has been computed (see Sec. ).
An equation analogous to the latter one holds for the type II supernova rate,
i.e. the death rate of stars more massive than Mup = 5-8 M\odot
(depending on stellar models having or not overshooting):
|
SNR(t) = - |
ó
õ
|
t
0
|
|
¶M(t-t¢,Z(t¢))
¶t
|
Y(t¢) [ F(m)/m ]m = max(Mup,M(t-t¢,Z(t¢))) dt¢ |
| (6) |
In practice, since only stars born during the last few 107 yr
contribute to the above integral (the lifetime of the star with initial
mass Mup), it is convenient to integrate from t-t(Mup,Z(t)).
This is a good approximation since stellar lifetimes are weak functions
of Z and the timescales involved are short.
We do not include type Ia supernovae.
This is justified for our purposes: we are not interested at this stage in
the detailed chemical evolution of galaxies, just on the global metal content
and its evolutionary rate, since they provide a hint to the amount of dust in
the ISM and its variation in time (but we are working to improve this point in
order to predict the detailed evolution of heavy elements
making up dust grains). Moreover the SNR is computed only to estimate the
radio emission from galaxies (Sec. ), not the energetics of the
ISM ruling for instance the star formation efficiency and the onset of
galactic winds (see below).
We assume, as has been usually done, that the gas accretes
into galaxies at an exponential rate:
|
|
.
M
|
g,i
|
|
ê
ê
|
Inf
|
= Xi,Inf AInf exp(-t/tinf) |
| (7) |
normalized in order to accrete a mass of gas MInf at time
tInf. Xi,Inf is the mass fraction of element i in the
infalling gas.
We allow for the possibility of interrupting the star formation activity and
the infall at a time twind. In models for the chemical evolution of
galaxies (see Matteucci 1996), this break is introduced to reproduce the
chemical and spectrophotometric properties of elliptical galaxies. The
computation is usually performed igniting galactic winds when the amount of
thermal energy from supernova explosions and stellar winds into the ISM
overhelms the gravitational binding energy of the gas. Actually this requires
a complicated treatment for a very simple and unrealistic scheme, i.e. the
sudden break, all over the galaxy, of every activity. The many uncertain
factors required for this computation (i.e. amount and distribution of dark
matter, interactions of supernovae and stellar winds with ISM) are introduced
as parameters to set in order to find the value of twind reproducing the
observations.
Therefore we set twind directly as a parameter (Mihara & Takahara 1994,
Matteucci 1996), with the typical values that fit the observed properties
of elliptical galaxies. At that time the galaxy is supposed to be totally
depleted of gas, and possibly replenished due to stellar feedback: a fraction
fwind of the stellar mass from dying stars remains within the galaxy while
1-fwind continues to be lost. We introduced this further parameter in order
to reproduce the amount of dust and IR emission observed in local giant
ellipticals (see Sec. ), that (with the ejecta we have adopted) require
fwind ~ 10-3.
File translated from TEX by TTH, version 2.25.
On 25 Jan 2000, 16:32.