Galactic Chemical Evolution in a One Zone Model

Joachim Köppen Kiel/Strasbourg/Illkirch April 2002


About the Applet and what it does

WithIn a galaxy, stars are born from the gas by gravitational instabilities. These stars live for some time, being powered by thermonuclear reactions in their centre which synthesize from the primordial elements hydrogen and helium all the other elements, such as carbon, nitrogen, oxygen, and iron. When the stars finish their lives either by exploding as a supernova of by expelling an planetary nebula, these thermonuclear 'ashes' are ejected into the interstellar gas, and are going to be used in the next generation of stars. In this fashion, the abundances of the heavy elements increase continually over the time, until there is sufficient of them available to make planets and create life.

The Applet computes this cycle of chemical evolution in a representative volume of a galaxy. This volume may be considerd as a closed system, where star formation consumes eventually all the gas. But also we can allow for an accretion (infall or inflow) of gas from another volume or from outside the galaxy, or for a mass loss by galactic winds. For both proceses, the user can input graphically how the rates should depend on the age of the system. The initial conditions (the amount of gas and its metallicity) can be changed. The recipe for the star formation rate - how it depends on the available gas mass - can be graphically given by the user. And how the production rates (yields) of two elements (called A and B) depend on metallicity (called Z) can be specified arbitrarily. The `element' Z is assumed to be produced with a constant yield by each stellar generation (`primary production'), and is used as a tracer of the overall evolution.

Any of the variables of the system may be plotted against another one, so their evolution in time, age, or redshift may be inspected, or the relation between two of them. The redshifts are computed with a Friedman type cosmological model, with the Hubble constant and the deceleration parameter. Furthermore, the distribution of the metallicity of the long-lived stars (`G-dwarfs') can be shown.

Some technical details: we neglect the finite stellar lifetimes (i.e. we use the Instantaneous Recycling Approximation) which is good for the elements produced only in the short-lived massive stars, such as oxygen. Elements produced in intermediate mass stars, such as carbon and nitrogen, or in type Ia supernovae (iron), cannot reliably computed with the applet! The equations are solved numerically with an automatically adjusted time step, which is controlled by the value given for the accuracy on the page with initial conditions.


When you start up the applet, I recommend that you hit the Start button, and you see something like the following screen.

On the left hand side at the top, there is a Choice to navigate between the various input pages. It now says Initial Cond. which is the page just below. It contains mostly the parameters for the initial situation:

controls the numerical accuracy of the results. A value of 0.01 is a compromise between accuracy and computation time for normal models. It is the maximum relative change permitted during one time step for the masses and the metallicities. When in doubt about your results, rerun them with a smaller value. When in haste, you may use a larger value, but keep below 1.0.
initial lg(gas mass)
is the mass of gas - in the mass units you have decided - at the beginning
initial lg Z
the initial metallicity
initial lg Z(A)
is the initial abundance of the element A
initial lg Z(B)
is the initial abundance of the element B
inflow: lg Z
the metallicity of in the inflowing gas
inflow: lg Z(A)
the abundance of element A in the inflowing gas
inflow: lg Z(B)
the abundance of element B in the inflowing gas
Hubble const.
in km/s per Mpc
Deceler. parm q0
the deceleration parameter. We use the latter two parameters for a Friedman model of the Universe to compute also redshifts. model. For simplicity, all models will start at the beginning of the Universe, and not at a redshift of say 5. Note that the models will always end at the present epoch.

On the right hand side, at the top, there are three Buttons:

brings up the x-y plots you see just below
tells that each new simulation will be plotted by a fresh single curve with previous plots wiped out. Clicking the button will change over to allow that the curves of all models are superposed on the same plot
brings up the histogram of the metallicities of long-lived stars (the "G dwarfs")

The page with the plots has several controls:

lg(gas fraction)
was chosen for the X-axis
gas Z abund.
was chosen for the Y-axis
wipes the plot, e.g. after making a new choice of the axes
starts the model simulation whose results are displayed in the plot
stops the current simulation
continues with a stopped simulation
drag & zoom
allows to zoom in any portion of the plot: click the button, then drag the mouse across the desired rectangle to be zoomed (from top left to bottom right). Release of the mouse button will bring up the zoomed view. Note that any curves present will be wiped, and need to be recalculated.
brings back the full view

The plot shows the relation between the lg(-ln f) which is a weird but useful form of the gas fraction with the metallicity Z. The blue diagonal line is the so called Simple Model, the red curve traces our model. What is behind our model, anyway? Choosing the SFR page brings up the relation between the star formation rate and the gas mass:

The red dots connected with red lines are the chosen recipe for the star formation rate: the linear relation was chosen. You may move a dot to a different position by grabbing it with the mouse and dragging it over. Also, clicking at an empty location will place a new dot there. On the top there are two buttons which have become enabled:

rem.last pt.
this will remove the right-most point
rem all. pts
will wipe all the points

Chosing the Accretion Ratio brings this page:

which is another interactive plot showing how the ratio of the rates of infall (= accretion = inflow) and star formation depends on the age. Here we had chosen a constant ratio of 3.16. Linear Accretion Model thus the evoltuion stops at Z = 0.316 y

There are three more input plots: for galactic winds, and for the nucleosynthesis recipe of elements A and B. see below

But let us explore a bit further the Linear Accretion Models: This is a plot with the superposed curves from all models with accretion ratios 0.01, 0.03, 0.1, 0.3,1.0 etc. They show that the metallicity towards which such a model tends, is the inverse of the accretion ratio:

The next screen shows the age-metallicity relationships from these models, from which the saturating metallicities are even more obvious:

The next screen shows how some arbitrary history of the accretion ratio results in a relation between star formation rate (red curve) and accretion rate (blue) with redshift: Obviously the earlier accretion causes an accumulation of gas which then is comsumed to make stars. The SFR peaks at lower redshifts than the accretion rate.

For the same model, we show the stellar metallicity histogram (red curve) in comparison with the histogram of the Simple Model (cyan curve).

When one wants to display this histogram, one may select

selects which element (Z, A, or B) is used for the x-axis
clears the plot
starts the simulation. Then you will notice that at the bottom of the plot a blue line moves from the left to the right. This indicates the current metallicity. When it does not move anymore, or when ever you want, you can ...
stop the simulation and display the histogram. BUT: if you stop too early, and therefore the histogram does not contain enough data points, you may get a weird plot. Be more patient, or increase the numerical accuracy.

In the model, we follow the abundances of three "elements": Z is assumed to be produced with a constant yield, and we use it as a reference metallicity. Elements A and B will be synthesized with yields that depend in a user-defined way on the metallicity Z. This is done with the pages Yield A and Yield B . Initially, the yield of A is assumed to be constant ("primary" production), while the yield of B depends linearily on Z ("secondary" production).

For the following screen, we made the yield of B constant (=1.0), but we played with the yield of A: the left point is always kept at the position shown (1.0), but the right point was moved from the top (100) to the bottom (0.01) in half-decade steps. The plot on the right hand side shows the effects of the various metallicity-dependent yields for A in the diagram of abundance ratio B/A plotted against the abundance of A:

We see that if A's yield declines with metallicity (as shown) one gets the top curve, the B/A ratio increases with A. This kind of diagram is used to judge on the nucleosynthesis origins of the elements, usually one takes for element A one which is produced with constant yield, such as the N/O vs. O/H plots. But if the oxygen yield were not constant ...

For the screen below, we changed the slope of the yield, but keeping the point in the centre (1.0, 1.0) fixed:

The blue circles are data used for an exercise. If one plays simply with the yield curve - without bothering about its physical reality - one finds that this curve is quite well mapped from the yield plot into the abundance ratio plot:

The Units

Apart from the units as shown in the plot axes, one word about the unit of mass for the gas, stars, and the rates of star formation, infall and wind loss: There is no preferred unit, so one may think of one unit as the entire mass of a galaxy, or its mass in solar mases, or the mass contained in some given volume (a cubic kpc ...). In models without inflow of gas, the natural mass unit would be the mass initially present in the zone. In models with inflow, it may be more practical to set the initial gas mass to a tiny portion of the mass unit, and one may end up with a final mass much greater than that unit. In the applet, the ranges of the masses are taken rather large, in order to permit the user to treat any problem one might like to investigate.

While the type of a model and its solution depends on certain ratios of the rates (e.g. the accretion ratio) rather than the rates themselves, the real values of the present star formation and accretion rates are of interest. They depend on the assumed mass unit: if one had chosen the mass of the galaxy, then the proper rates are in "galaxy masses per Gyr", of course.

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