While I observed the bright, rich
globular clusters M15 and M2 in autumn 2011, I wondered how the
trajectories of stellar members might look in such a
million body system. I assumed a somehow chaotic appearance. Not
quite sure, I decided to make computer models for such systems to gain
M15 image from 2010/10/10
The ~150 globular clusters in our Galaxy are among the oldest objects
known. How could they survive as stable objects over times of more than
12 billion years? I previously had read about a core collapse and
a black hole in the center of M15, but I was suprised to read what a
complex system such a cluster is. Read:
A Thousand Blazing Suns: The Inner Life of Globular
Other galaxies have globular clusters as
well. With a midrange Dobsonian it is possible to observe the brightest
in M31, the nearby Andromeda Galaxy. One of the most impressive images
have seen yet, is from William E. Harris' website. About 16000
globular clusters in the halo of giant elliptical galaxy NGC 3311
, the centrally dominant cD
galaxy in the Hydra I cluster.
Clusters from Starlab
A star cluster is composed of N
interacting with each other due to the gravitational force of every
body in the system.
It is not easy to find a starting configuration for a
cluster close to equilibrium. Two solutions are commonly used,
Plummer's model (1911) or King's model
(1966). All models below have the same number of stars, N = 4096.
Plummer's model above is favored under theoretical aspects,
more than a dozen characteristic functions can be used to describe it.
King's model comes closer to observed globulars. The 3 images below
show increasing stellar
concentration towards the center, which is controlled
here by a single parameter w, which measures the radius of the clusters
halo in units of the core radius.
w = 11
Following the evolution of such a system means to integrate
directly the N
individual equations of motion, this is the N
It is not advisable to start this numerical work from scratch, using
Runge-Kutta integrators or something comparable. It took decades to
develop state of the art methods, used in special software like NBODY4,
NBODY6 or the starlab-code which is used here.
Although a real globular cluster has about ten to hundred times more
the modells below, you will get an impression about the dynamics. It
might be interesting to note, that clusters of a million members,
consisting of a realistic fraction of binaries are still beyond reach
of current computer power. Special treatment is used for close
encounters and binary or multiple subsystems that form either
dynamically or exist in the initial configuration.
Six examples of three body encounters. In all cases a binary (m1=0.55,
red and m2=0.45, green) and a field star (M=0.77, blue) interact
in a more or less complicated way. In the first and second case the
configuration is preserved. In all other cases the original binary is
disrupted and the light m2 component (green) escapes. With
a gain in kinetic energy=velocity it often leaves the
cluster. Frequency of close interactions is highest in dense regions.
The rearrangement of energies among the participants in many cases
enables the binary to leave the core also. This is a mechanism to
delay, prevent or reverse a core-collaps (see below).
If such an exchange process happens in the cluster's core, chances are
good, that a fresh formed couple consists of a main-sequence star
and a heavy, compact object like a white dwarf or a neutron star, which
dominate the stellar population in this region. The
dynamical interaction between the partners, mass transfer in close
binary systems causing enforced stellar evolution ..., enriches the
core region with all sorts of relativistic binaries.
and binary evolution, concentration of massive stars in the core,
stellar collisions in crowded regions, possible presence of a central
black hole and the
tidal field of the Galaxy must also be considered, when the model
should give a realistic match to an existing globular.
The following videos are best viewed
in full screen
mode. Works with Firefox and Chrome, Internet-Explorer will offer a download ...
A larger globular cluster: King
W12, 4096 bodies, t=0..15
While it took a few minutes to compute the first animation, the
computer ran for nearly two hours to finish the 4096 film.
The following film
shows a small open cluster of 36 stars evolving for a while. At the
end, two very
different trajectories of members 3 and 10 are shown in 3D.
These simulations were made with the
KIRA-Integrator available for
download from the Starlab site.
It is quite easy to use the Starlab tools to make simple things:
Create a 5000-particle W0 = 7 King
model, with numbered stars, unscaled. The next example creates a plot
of the cluster, while the third uses kira to evolve it for 10 timesteps
makeking -n 5000 -w 7 -i -u
makeking -n 5000 -w 7 -i -u | xstarplot
makeking -n 5000 -w 7 -i -u | kira -t 10 | hxstarplot
and really complicatet things:
Create a King model with a power-law mass spectrum and a binary
population, then evolve it with stellar and binary evolution. Use all
4 CPUs for integration
makeking -n 5000 -w 7 -i -u \
| makemass -f 1 -x -2.0 -l 0.1 -u 20 \
| makesecondary -f 0.1 -l 0.1 \
| add_star -Q 0.5 -R 5 \
| scale -M 1 -E -0.25 -Q 0.5 \
| makebinary -f 1 -l 1 -u 1000 -o 2 \
| kira -T 4 -t 100 -d 1 -D 10 -f 0.3 \
-n 10 -q 0.5 -Q -G 2 -B
Almost all commands display detailed help for their usage. Try:
Unfortunately starlab does not make use of the spectacular power of
todays gpgpu-computing. To benefit from recent developments, I have to
switch to another package: NBODY6.
processing units (GPUs) can speed up computations by more than 100
Clusters and results from NBODY6
The above shown examples represent
the state with beginning of 2012. Meanwhile it is February and I
a Nvidia GeForce GTX460 GPU to speed things up.
It now takes 31 seconds instead of 2 hours to compute the above results.
Below is a first core collapse of a cluster shown, computed with
nbody6.gpu, the code provided by Sverre
Aarseth's Institute of Astronomy N-Body and Downloads Page
The achieved GFLPOS depend on the number of particles in the system. In NBODY6 it
goes asymptotically to ca. 500 GFLOPS when N > 50000.
Later I found,
that best perfomance of the GTX460 results, when the number of stars is
choosen such that N=7 x 2**k.
Measurements were made with help of the nbody-code from the NVIDIA-GPU-Computing-SDK.
It took about a hour to compute the result below. Starting model was a
Plummer sphere of equal mass stars, computed with:
mcluster -P0 -N 4096 -f 0 -t 0 -O 5.0 -T 1500.0 -C 0 -G 1 -u 0
The mcluster code is developed by Andreas Küpper : http://www.astro.uni-bonn.de/~akuepper/mcluster/mcluster.html
Without this code you will hardly have a chance to generate desired
NBODY6 input files.
Documentation of NBODY6 is out of date and you need the remembering
brain of an elephant to manage the combinations of ~100 parameters
(numbers) that drive the computations.
The curve in red shows the evolution of the clusters half-mass radius
the green one gives the core radius.
While the half-mass radius increases, the core shrinks
The first collapse is observed
between t=1000 and 1100, a second one at t=1200.
Meanwhile I use a comfortable shell script to obtain results ...
pushing Fermi to the limits ...
The new version runs 2 concurrent kernels on the GTX480. It achieves
nearly 800 GFLOPs/s. Now my CPU (Phenom II) is the limiting factor.
Click into the following image to get a merging double cluster in
Click into the following image to get a new window with a "living"
double cluster in three dimensions !!!
This new state-of-the-art 3D-technology is known as X3DOM and requires
latest browser versions. Best results are obtained with
google chrome vers. 17 or higher.
If you plan to fly through the cluster, Firefox (Version < 11) is currently not a
responsive space ship.