Dynamical Evolution of Dense Star Clusters
The dynamical evolution of dense star clusters is a problem of fundamental
importance in astrophysics, but many aspects of the problem have remained
unresolved in spite of several decades of theoretical work and many recent
improvements in the quality of observational data (especially with the
Hubble Space Telescope and the
Chandra Xray Observatory).
Starting around 1998, my students and I embarked on a new theoretical study of dense star
cluster dynamics using a parallel supercomputer code based on Monte Carlo
techniques for solving the FokkerPlanck equation in spherical symmetry.
Although the initial focus of our work was on globular clusters, our code is
very general and has been used more recently for the study of other systems,
such as young, dense star clusters in galactic nuclei and starbursts. Our
code was developed and tested thoroughly during the first few years of this
project
[61]
[74],
and was used in several initial studies of systems
containing single stars only. In particular, we studied cluster lifetimes in
the Galactic tidal field and their dependence on the stellar initial mass
function
[74],
and we investigated the development of the “Spitzer
masssegregation instability” in twocomponent systems
[60].
A key feature of our new study, and the main present goal of this work, is
the inclusion of primordial binaries in the dynamical calculations. The
realization in the early 1990's that primordial binaries are present in
globular clusters in dynamically significant numbers has completely
revolutionized our theoretical perspective on these systems. Most
importantly, dynamical interactions between hard primordial binaries and
other single stars or binaries are now thought to be the primary mechanism
for supporting a globular cluster against core collapse
[88].
In addition, exchange interactions between primordial binaries and compact objects can
explain very naturally the formation of large numbers of Xray binaries and
recycled pulsars in globular cluster cores
[57].
Resonant interactions
involving primordial binaries can also result in dramatically increased
stellar collision rates
[94].
Direct observational evidence for stellar collisions and mergers in globular
clusters comes from the detection (mainly with HST) of large numbers of blue
stragglers concentrated in the dense cluster cores (see below).
The main goal of our current work is to design Monte Carlo simulations of
cluster dynamics treating both the cluster itself and all relevant
interactions selfconsistently, including direct physical collisions and
dynamical interactions involving primordial binaries. This idea is
particularly timely because the latest generation of parallel supercomputers
now makes it possible to perform such simulations for a number of objects
equal to the actual number of stars in a large cluster (N ~ 10^{5} – 10^{6} for
a globular cluster; up to ~ 10^{8} for galactic nuclei). Most previous studies
of globular cluster dynamics incorporating primordial binaries have used
direct N–body simulations. These simulations have been limited to very small
N ~ 10^{3} – 10^{4}. Specialpurpose computing hardware, such as the Japanese
GRAPE, has recently made possible direct N–body simulations with up to N ~
10^{5} stars, although the computing time for the largest simulations with
binaries can be many months. In contrast, our parallel Monte Carlo code
allows us to perform selfconsistent dynamical simulations of clusters
containing N ~ 10^{5} – 10^{6} stars with ~ 10% binaries in typically less than
a day of computing time. Even higher binary fractions, which may be more
realistic for globular clusters
[98]
can now be handled very efficiently through direct
integrations of all 3–body and 4–body encounters
[94].
A closely related new project, done in collaboration with Dr. Marc Freitag
(ARI, Heidelberg, Germany) concerns the formation of massive black holes in
dense clusters through runaway collisions and mergers of young, massive
mainsequence stars during the first ~ 10^{7} yr of dynamical evolution
[90],
[92].
Runaway collisions may provide the most natural path for the
formation of “intermediatemass” black holes in globular clusters and super
star clusters. Another related new project, led by Northwestern postdoc
Natasha Ivanova, studies the effects of dynamical interactions in dense
clusters on the stellar evolution of binary star populations
[98].
Postdoctoral fellows who worked with me on this include
Simon Portegies Zwart (while a Hubble Fellow at MIT), and Northwestern postdocs
Natasha Ivanova
and John Fregeau.
Three graduate students have been associated with this work:
Kriten Joshi (MIT PhD '00),
John Fregeau (MIT PhD '04),
and Atakan Gurkan (Northwestern PhD '05).
Several undergraduate students (MIT UROP, NSF REU, and senior
thesis students) have also been involved, and in many cases coauthored some
of our papers:
Eric Ford
[56],
Boris Zbarsky
[56],
Wes Watters
[60],
Cody Nave
[74],
Pokman Cheung
[94].
This work has been
supported by several grants from
the NASA Astrophysics Theory Program and the NSF Stellar Astronomy and
Astrophysics program.
Coalescing compact binaries
Coalescing compact binaries
with neutron star or black hole components
(NSNS or NSBH) are the most promising sources of gravitational waves for
detection by current laser interferometers, such as LIGO
and VIRGO.
Most calculations of gravitational wave
emission from coalescing binaries have focused on the waveforms emitted
during the last few thousand orbits, as the frequency sweeps upwards from
about ~10Hz to ~1000Hz. The waveforms in this regime can be calculated
fairly accurately by performing highorder postNewtonian (PN) expansions of
the equations of motion for two point masses. However, at the end of the
inspiral, when the binary separation becomes comparable to the stellar radii
(and, ultimately, when the two stars merge), hydrodynamics becomes important
and the character of the waveforms must change. In
advanced LIGO,
special purpose narrowband detectors that can sweep up
frequency in real time may be used to try to catch the corresponding final
burst of gravitational waves. In this terminal phase of the coalescence, the
waveforms contain information not just about the effects of relativity, but
also about the interior structure of a NS and the nuclear equation of state
(EOS) at high density. Extracting this information from observed waveforms,
however, requires detailed theoretical knowledge about all relevant
hydrodynamic processes. Since 1992 I have been developing both numerical and
analytical tools that have allowed us to develop this knowledge, in
preparation for the arrival of the first gravitationalwave data. Additional
motivation for this work is provided by the many theoretical models of
gammaray bursts (GRBs) that rely on coalescing compact binaries to provide
the ~ 10 ^{53} erg of energy required to power bursts observed over
cosmological distances.
The final merger of the two stars is driven by a combination of
relativistic effects and global hydrodynamic
instabilities, which can drive
the binary system to rapid coalescence once the tidal interaction between the
two components becomes sufficiently strong.
Using numerical hydrodynamic calculations in Newtonian gravity, Rasio
& Shapiro [10,
21,
27] *
demonstrated for the first time the existence of these global instabilities
for double NS systems. Our results were later confirmed by many other groups,
sometimes using very different numerical approaches.
A detailed analytic study of dynamical and secular
instabilities in compact binaries
was also undertaken (extending the classic work of Chandrasekhar
for an incompressible fluid), using a new
energy variational method
[11,
14,
17].
This analytic work confirmed
the importance of hydrodynamic instabilities for the terminal
evolution of coalescing compact binaries.
Until recently, most hydrodynamic calculations of compact binary
coalescence had been performed, for simplicity, in Newtonian gravity.
However, it is clear that the coalescence of NSNS and, even more NSBH
binaries, will be strongly influenced by PN corrections to gravity. In
particular, we have shown analytically that PN effects tend to make compact
binaries even more unstable than their Newtonian counterparts
[46].
Northwestern postdoc Josh Faber (formerly my MIT graduate student) and I
have developed a new version of our 3D hydrodynamic SPH (Smoothed Particle
Hydrodynamics) code StarCrash
incorporating a full treatment of PN gravity. We have used this code for a comprehensive new study of NSNS
binary coalescence in relativistic gravity
[62]
[73]
[79].
We performed detailed comparisons with our previous Newtonian results
[21,
27,
54]
and we studied systematically the dependence of the gravitationalwave
signals on the NS EOS, NS spins, and on the binary mass ratio.
Traditional PN expansions capture qualitatively many new relativistic
effects (such as compactness effects and gravitational radiation reaction),
but, quantitatively, they provide a rather poor approximation to the
structure of a NSNS binary.
Over the last few years, with Northwestern postdocs Josh Faber and Philippe
Grandclément, we have developed the first fully relativistic version
of our SPH code. Gravity is now treated using the equations of general
relativity (rather than a PN expansion) in the socalled conformal
flatness (CF) approximation, which suppresses some the dynamic degrees
of freedom of the field. Our new CFSPH code
[93]
provides, for the first time, a completely consistent
treatment of the initial conditions for close NSNS binaries on
quasicircular orbits (always constructed in the CF approximation of GR) and
the hydrodynamic merger calculation. A key result of our work is that NS EOS
signatures may be present in the gravitationalwave signals at much lower
frequencies than previously believed
[81]
[93].
Our new CFSPH code will also allow us for the first time to study
realistically the hydrodynamic coalescence of NSBH systems. These systems
may be even more important than NSNS binaries as sources of gravitational
waves. Stable mass transfer from a NS to a BH, and the tidal disruption of a
NS by a BH in a compact binary, have also been discussed recently by many
authors in the context of GRBs. However, very little is known theoretically
about these systems, especially when compared to the large amount of work
done on NSNS binaries. In the next few years of this project we plan to
perform a systematic survey of the parameter space for NSBH binaries, and,
in particular, we will try to identify the boundary between stable mass
transfer and rapid unstable coalescence. These hydrodynamic studies will
also complement the many “Grand Challenge” efforts currently underway to try
to simulate numerically the collision of two BH or the mergers of BHBH
binaries (which are purely gravitational problems in numerical relativity,
without any fluid sources).
Northwestern postdocs Josh Faber and Philippe Grandclément, as well as several
undergraduate (NSF REU) students have been working with me on this project, which has been
funded by several grants from the NSF Gravitational Physics Program. REU
students who have coauthored some of our papers include Justin Manor
[73].
Movies of many of our 3D hydrodynamic calculations of binary mergers have
been developed by REU student Rocky Jones [3d Astronomical Visualization].
* Numbers in square brackets refer to papers in the list of publications.
Stellar Collisions and Blue Stragglers
Blue stragglers are objects that appear as mainsequence (MS)
stars above the turnoff point in the colormagnitude
diagram of a globular cluster, i.e., they appear too massive for
their environment. They are generally thought to be
formed through collisions and mergers of lowermass MS
stars in the dense cluster core. Although this merger hypothesis
has been with us for more than two decades, and is by now
well accepted, detailed calculations of the merger process were
lacking until a few years ago.
In 1995 I initiated a large collaborative effort
to study theoretically the formation and evolution of blue
stragglers resulting from stellar collisions, using a combination
of 3D numerical hydrodynamics calculations with SPH
[31,
34]
and stellar evolution calculations.
For the first time in a study of stellar collisions,
the results of hydrodynamic calculations were combined with
a stateoftheart stellar evolution code (the
YREC code developed at Yale by Demarque and collaborators). This allowed
us, in particular,
to study chemical mixing and stellar evolution of collision products in a
completely selfconsistent way
[47].
This is crucial for obtaining accurate
theoretical models of blue stragglers and quantitative predictions for
their observable properties. The YREC code is used to construct
realistic initial conditions
(spherical stellar models) for our SPH collision calculations.
In turn, the SPH code
produces final merged configurations to be used as initial conditions for
stellar evolution calculations of blue stragglers using YREC.
The initial phase of our work has focused
on hydrodynamic mixing during the merger process as well as the
subsequent evolution of
merger remnants, first on a thermal timescale as they contract to the main
sequence, and then on a stellar evolution timescale. We include a
treatment of rotation and rotationallyinduced mixing in the stellar
evolution calculations (as provided by YREC).
Our goal is to calculate accurately the entire evolution of blue stragglers
from the moment of their formation in a dynamical merger, through their
premainsequence and mainsequence phases. This must be done for many
different combinations of stellar masses and orbital parameters for the
collisions.
This initial phase of the project is now nearing completion
[67,
68],
and our results lead to detailed theoretical predictions for the
observable characteristics of blue stragglers (colors, luminosities, numbers).
Comparisons with observations (in particular, many large samples of blue
stragglers detected with HST in several different clusters)
can now be made to test dynamical models of
star cluster evolution including stellar collisions, and to
test the merger hypothesis for the origin of blue stragglers.
This ongoing collaboration includes J. Lombardi (Vassar) and A.
Sills (McMaster), as well as Northwestern postdocs J. Fregeau and N.
Ivanova.
Extrasolar Planetary Systems
A new era in astronomy began a few years ago with the first clear
detections of several Jupitertype planets around nearby solarlike
stars. These discoveries of extrasolar planets will no doubt lead to
significant improvements in our understanding of many processes
related to planet formation, structure and evolution, as well as
deeper questions such as the existence of extraterrestrial life in
the Universe.
NASA has identified the search for and characterization of planetary
systems as one of its highest future priorities.
Beginning in 1996 I have been interested in exploring some of the
theoretical implications of these recent discoveries. With Eric
Ford (at the time a sophomore in Physics at MIT!) I wrote the first
paper
[44]
proposing that the development of dynamical instabilities
during the early stages of planet formation could
provide a natural explanation for the very surprising orbital
characteristics of the observed systems (which include giant planets
in highly eccentric orbits, or in very tight circular orbits with
periods as short as a few days). This idea is by now well accepted
(and has received further observational support with the recent
detection of three giant planets of comparable
masses in a marginally stable configuration in the Upsilon Andromedae
system). The longterm stability of the Solar
System, in spite of its chaotic nature,
may have been necessary for the development of intelligent life.
However, it may also be very atypical, and may in fact require
very special conditions during the early stages of planet formation.
One of these special conditions may be the formation of a single, dominant
giant planet (Jupiter in the Solar System). Our work, based on numerical
integrations using simplectic methods, examines the
dynamical consequences of the presence of several giant planets of
comparable masses in the same system. As a first case, we have recently
completed a systematic study of the dynamical evolution of unstable
planetary systems containing two identical giant planets
[69].
As a corollary, we have also studied the tidal interaction (including
circularization of a highly eccentric orbit) between
a giant planet and a solarlike star
[40,
52].
Extrasolar planets have also been detected around radio pulsars.
To this day, the only example we know of a system of earthmass
planets orbiting any star other than our own Sun is the system
of planets around the millisecond radio pulsar PSR 1257+12,
discovered by Wolszczan in 1992. Clear confirmation for at least
two planets in this system came in 1994 when
my theoretical prediction
[8]
for a nearresonant gravitational
perturbation effect in the system was verified. More recent
theoretical modeling of the pulsar timing data has revealed the
presence of a more distant, giant planet in the system
[45].
A giant planet has also been detected in orbit around PSR 162026,
a binary millisecond pulsar located near the center of the globular
cluster M4
[19,
26].
Among the many theoretical implications of this extraordinary system
[56]
is that planets must also be
common in older stellar populations such as those found in globular
clusters and the Galactic halo.


