Galactic Chemical Evolution with Gas Flows
Joachim Köppen Kiel/Strasbourg/Illkirch Oct.97
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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 metalpoor 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
massweighted stellar metallicity (<Z>):
$g\text{'}\; =\; A\; \; SFR$
$s\text{'}\; =\; SFR$
$(gZ)\text{'}\; =\; (y\; \; Z)\; SFR$
$(gZ\_s)\text{'}\; =\; ((Z\; y\_s)\; \; Z\_s)\; SFR$
$d<Z>/ds\; =\; (Z\; \; <Z>)s$
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 metalfree 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 timedependent
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,
rerun 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 longlived
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 massaveraged 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 xaxis
for this plot must be metallicity, one must first select lg(Z/y)
this for the xaxis, only then one can chose ds/dZ
among the items for the yaxis.
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 yaxis.
The controls:
 Plot on Xaxis
 select the variable to be plotted
on the xaxis; the choice will be used for the next click
of Clear or Start
 Plot on Yaxis
 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
pointbypoint 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
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