| Background | The variables | How to start | The controls | Applet | Applet Index |

The gaseous and stellar matter in galaxies undergoes an evolution of its chemical composition, as stars are born from the interstellar gas. Within the stars thermonuclear reactions generate not only the energy that makes the stars shine, but also produce synthesize heavy elements from hydrogen and helium. When the stars finish their life by exploding as a supernova or by ejecting a planetary nebula, much of this fusion products are ejected into and mixed with the interstellar gas.

In the Applet we simulate this chemical evolution of galaxy represented as a homogeneous volume of gas and stars, but permitting the inflow of metal-poor gas whose rate follows a completely arbitrary prescription which is specified by the user. The evolution of such a system is described by these equations for mass in gas (g), stars (s), gas metallicities of a primary element (Z) and a secondary element (Z_s), and the mass-weighted stellar metallicity (<Z>):

$g\text{'}\; =\; A\; -\; SFR$

$s\text{'}\; =\; SFR$

$(gZ)\text{'}\; =\; (y\; -\; Z)\; SFR$

$(gZ\_s)\text{'}\; =\; ((Z\; y\_s)\; -\; Z\_s)\; SFR$

with the star formation rate SFR, the rate A of the infalling gas, the yield y for the primary element, and the yield factor y_s for the secondary element. The finite lifetimes of the stars are neglected, which is a good approximation for elements produced only in massive stars (e.g. oxygen).

Since in the first four equations the time dependence can be eliminated by changing over to s: ds = SFR dt, the mass in stars becomes the independent variable. Furthermore, the (constant) yields can be eliminated by dividing the metallicities by the yields. Thus, only one free parameter remains: the ratio a = A/SFR of infall and star formation rate. This parameter and how it changes with time determines the character of the possible solutions. In this applet, the user can specify freely the dependence of a(s), and investigate its influence on the solution.

How to start: After the applet is loaded, click Start, the only button accessible. This brings you to the display of lg(a) vs. lg(s) which is where you specify the accretion ratio a: click enter data then by clicking the mouse on the plot area enter as many points as you wish to describe the accretion ratio's dependence on s, i.e. essentially time. erase removes the last entry, and erase all removes all points. A constant ratio can be entered by clicking just one point. The interpolating curve that will be used in the calculation can be displayed by clicking show the AR.

Then select with the Plot on ... buttons the variables one wants to plot against each other. For example metallicity lg(Z/y) as a function of the gas fraction lg(-ln f): the plot shows the solution of the Simple Model (closed box a=0)

$Z\; =\; y\; ln(f)$

as a blue straight line. The locus of the steady states for models with constant accretion ratio (cf. our paper) is shown as the green curve. The model starts with 100 percent metal-free gas. In this plot, one can also start the model from any arbitrary point in the plot, by clicking first Go from here then on the plot area. One will find:

• all models with accretion decreasing monotonically in time stay in the region between the blue and green curves
• models with constant accretion either get stuck on the green curve (if a is larger than 1) or reach a limiting metallicity
• starting with metallicities higher than the Simple Model, all models evolve out of that region
• models which have an increase of the accretion rate when most of the gas is already used up, perform small loops in the diagram.
• if that late accretion event is strong (a larger than 1), the loop may enter the region of metallicities smaller than predicted by the green curve

In the other plots, one will find the corresponding behaviour. In a few important diagrams, the solutions of the Simple Model and constant accretion models are also shown as blue and green curves. In the plot of lg(Z_s/Z) vs. lg(Z/y) the grey curve is the approximate asympotic solution for constant a (cf. our paper).

Numerical Method: rather than solving the equations as a function of s, the applet solves the time-dependent system, assuming a star formation rate depending linearly on the current gas density (linear SFR). Choosing one of the other possible laws yields the same results, except for the dependence on Time. A simple Euler method with constant time step is employed to solve the equations. This is fast and usually sufficiently accurate. When in doubt about the results, re-run with a smaller time step or use the automatic adaptive step.

The variables:

Time, lg(Time)

is measured in arbitrary units; for a linear SFR, it is the star formation timescale

lg(a)

a or AR is the instantaneous ratio of the accretion rate of the gas and the star formation rate

lg(M/M0)

is the current ratio of the total mass (gas plus stars) and the initial gas mass

lg(gas), lg(stars)

are the masses in the form of gas and long-lived stars - including stellar remnants

lg(-ln f)

f = gas/(gas+stars) is the current gas fraction. In this slightly complicated form, this quantity is simply proportional to lg(Z) from the Simple Model

lg(Z/y)

is the metallicity of the gas for a primarily produced element, measured in terms of the true yield y

lg(y(eff)/y)

this factor between the effective yield and the true yield measures how strongly the system deviates from a Simple Model: y(eff) = -Z/ln(f)

lg(<Z>/y)

the mass-averaged stellar metallicity - for the primary element Z - in units of the true yield

lg(Z_s/y_s)

is the metallicity in the gas of a secondarily produced element, measured in terms of the true secondary yield y_s

lg(Z_s/Z)

the abundance ratio in the gas of a secondarily and a primarily produced element, normalized to their true yield ratio

ds/dZ, 0.1 * ds/dZ

the stellar metallicity distribution, i.e. the mass contained in stars per unit logarithmic interval in Z. Since the x-axis for this plot must be metallicity, one must first select lg(Z/y) this for the x-axis, only then one can chose ds/dZ among the items for the y-axis. The distribution function is displayed as a continuous curve. It may fold back, e.g. when an event of strong accretion takes place; then the total distribution would be given by the sum of all its branches. The second item presents the computed distribution scaled down by a factor of ten in the y-axis.

The controls:

Plot on X-axis

select the variable to be plotted on the x-axis; the choice will be used for the next click of Clear or Start

Plot on Y-axis

ditto

Keep this

the present plot is marked so that it can be called back simply by the next button:

Show kept

this shows the plot that had been marked, as before

Start

starts the model calculation

Pause

halts the calculation

Carry on

continues with the present model

Clear

wipes the plot area, and draws the axes for the new plot

Go from here

click here, then click at that position on the plot where you want to start the calculation with a different initial condition (possible only for certain plots)

linear SFR

switches between different dependences of the star formation rate on gas mass: linear, quadratic, cubic, or constant

constant dt=

toggles between a constant time step - as to be entered in the field - or an automatic variation of the time step to assure a certain accuracy - as shown in the field

enter data

when showing the lg(a) vs. lg(s) plot, one may enter point-by-point by (any number of) mouse clicks how the accretion ratio a depends on the stellar mass, i.e. essentially time. The points are shown as small open circles. Outside the range in s covered by the points, the ratio is taken to be constant. For all other plots, one may enter (for each plot separately) any number of data points (black dots) to compare the models with observed data or to mark certain points

grab and drop a circle

by grabbing a circle near its centre, one can move it about in the plot

erase

removes the last entered data point or mark

erase all

removes all the points of the accretion ratio a(stars)

set Accretion Ratio AR

this button permits direct jump to the plot where one may change or enter the accretion ratio. When in that mode, a click will show the interpolated curve through the data points

log(AR) shift

for the calculations, the accretion ratios is increased or decreased by the specified amount, while preserving the time dependence