Investigating the Spitzer
Instability Using AMUSE
Michael Brewer and Stephen McMillan
Department of Physics, Drexel University
We find that, using two-component systems of total size N = 5,000 and a number
fraction for the heavy population of 0.01, the timescale for segregations, in which
the heavy population separates from the background and come into temporary
equilibrium near the center of the cluster, scales as 1/(μ-1), where μ is the ratio of
the mass of a heavy star to the mass of a background star. This scaling is in
agreement with theoretical predictions and published results from other numerical
simulations2. The blue line in Figure 1 is a function of that form fitted to our data.
Figure 2 shows two sample plots (µ=10) of mass segregation, one performed with
multiples enabled (left) and the same run without multiples (right). We observe the
same behavior for the first half of each simulation, confirming that close encounters
are not important during mass segregation, although binary activity becomes
important afterwards as the core evolves during the second half of these simulations
where the results diverge. The mass segregation times discussed above were
determined by eye from plots similar to Figure 2 (see the arrow in that figure).
The AMUSE codebase is a modular package of codes for astrophysical research
simulations. Intended to replace traditional kitchen sink types of codes, which
require that the end-user accept all of the code creators choices for implementations
of multiple different types of physics, the modular structure of AMUSE allows users
to easily select which codes are desired for each type of physics required, and
because the modules share a common interface, this allows users to change their
choices of modules without significant changes to their code. Major physics codes
are implemented as high-performance modules which are called from and controlled
by user built Python scripts.
and we refer to the quantity on the left hand side as the Spitzer parameter. In our
numerical investigation of this threshold, we use both the core masses and total
masses of each population to calculate the Spitzer parameter; it is not yet clear
which is more physically meaningful. We started our survey of the instability with 27
parameter combinations, varying µ, W0 (the King parameter), and Q = N2/N1. Figure
3 below is a pair of plots of the parameter space against the Spitzer criterion; the
current results are in rough agreement with Spitzer (all runs used 32,000 particles).
Figure 4a shows a typical unstable run, and figure 4b shows a stable run.
Figure 1: Mass Ratio Vs Segregation Time. Data points shown as blue circles; calculated fit shown as blue line.
N = 5,000 for these runs.
Mass segregation time.
Spitzer Parameter (red), Spitzer Criterion (green), and
number fraction of heavy stars inside system half-mass radius
Mass Segregation time (code units)
Mass ratio µ
Figure 3: Spitzer Criterion plotted as a green line with data results. Unstable runs are
marked with a red X, stable runs marked with a blue O. Left plot uses total population
masses, while the right plot computes the total mass only from stars within the core.
Mass ratio µ
Number fraction of heavy stars
outside system half-mass radius
For two-component star clusters, where the more numerous light stars have
individual mass m1 and total mass M1 and the heavy stars have individual mass m2
and total mass M2 with mass ratio µ = m2/m1, dynamical evolution toward
equipartition of kinetic energy causes the heavier stars to sink towards the center. If
the total mass of heavy stars is sufficiently large relative to the light population, the
core will become unstable and undergo rapid dynamical evolution. The Spitzer3
criterion for stability is:
The purpose of this project was to run simulations with a new module, “multiples”
designed to manage close encounters in detail, allowing for more careful study of
small-N systems (e.g. binaries and triples) and freeing the main “large-N” codes
from having to follow such interactions carefully. Figure 5 (left) shows timing and
encounter data for a 64,000 particle run with a long duration in system time. Long
runs like these allow us to profile the code. Spikes from encounter activity track the
spikes in simulation run time. The new comparison run with 16,000 particles (Figure
5 right) shows significantly fewer spikes, indicating significant improvements in
performance, although work on improving the speed of the multiples module
continues.
Radius (code units), Log10 of resolved encounters,
and clock time since last step (Log10 of minutes)
We examine the behavior of a new module in the AMUSE framework designed for
detailed handling of binary stars and other few-body systems within large N-body
codes, applying the module to the study of mass segregation in two-component star
clusters and a numerical investigation of the Spitzer mass segregation instability.
Working with AMUSE
M2/M1
Introduction
Spitzer Instability
System time (code units)
Figure 5: Long profile of system behavior on left with new comparison run on right. Median
radius of heavy particles in black, clock time since last system time in mustard-brown, and
encounters since last step in cyan. Left: N = 64,000. Right: N = 16,000
Both: µ = 7.5, W0 = 3, Q = 5×10-4
System time (code units)
Figure 4a: Time evolution of the Spitzer parameter, the unstable
case. µ = 15, W0 = 5, Q = 1×10-3 . There is a sharp increase in
parameter as segregation occurs and sharp fall as objects are
ejected from core.
References
1. Pelupessy, F. I. et al.,”The Astrophysical Multipurpose Software Environment”, Astronomy and Astrophysics 557,
84, 2013. <http://adsabs.harvard.edu/abs/2013A%26A...557A..84P>
2. J. M. Fregeau, K. J. Joshi, S. F. Portegies Zwart, and F. A. Rasio. “Mass Segregation In Globular Clusters.” The
Astrophysical Journal, 570, pg 171-183, May 2002.
3. Spitzer, Lyman. “Dynamical Evolution of Globular Clusters”. Princeton University Press. 1987.
4. Eisenstein, DJ, Hut, P, “HOP: A new group-finding algorithm for N-body simulations”, ApJ 498 (1998)
System time (code units)
Figure 2: comparison of a run with multiples managing encounters (left) to a softened run without multiples
(right). As expected, runs behave similarly during segregation, but diverge afterwards as binaries form in the
core and the Spitzer instability sets in. N = 16,000, µ = 10, W0 = 3, Q = 1×10-3 Arrow in left figure points to
estimated mass segregation time
System time (code units)
Figure 4b: Time evolution of the Spitzer parameter, the stable
case. µ = 5, W0 = 3, Q = 5×10-4. There is a sharp increase in
parameter as segregation occurs, then slower increase as core
evolves.