276 research outputs found
Illustrating chaos: A schematic discretization of the general three-body problem in Newtonian gravity
We present a formalism for constructing schematic diagrams to depict chaotic
three-body interactions in Newtonian gravity. This is done by decomposing each
interaction in to a series of discrete transformations in energy- and angular
momentum-space. Each time a transformation is applied, the system changes state
as the particles re-distribute their energy and angular momenta. These diagrams
have the virtue of containing all of the quantitative information needed to
fully characterize most bound or unbound interactions through time and space,
including the total duration of the interaction, the initial and final stable
states in addition to every intervening temporary meta-stable state. As shown
via an illustrative example for the bound case, prolonged excursions of one of
the particles, which by far dominates the computational cost of the
simulations, are reduced to a single discrete transformation in energy- and
angular momentum-space, thereby potentially mitigating any computational
expense. We further generalize our formalism to sequences of (unbound)
three-body interactions, as occur in dense stellar environments during binary
hardening. Finally, we provide a method for dynamically evolving entire
populations of binaries via three-body scattering interactions, using a purely
analytic formalism. In principle, the techniques presented here are adaptable
to other three-body problems that conserve energy and angular momentum.Comment: 9 pages, 4 figures; accepted for publication in MNRA
A novel mechanism for the distance-redshift relation
We consider a novel mechanism to account for the observed distance-redshift
relation. This is done by presenting a toy model for the large-scale matter
distribution in a static Universe. Our model mainly concerns particles with
masses far below those in the Standard Model of Particle Physics. The model is
founded on three main assumptions: (1) a mass spectrum dN/dm m (where and are both
positive constants) for low-mass particles with m 10 eV
M, where M is the Planck mass; (2) a particle
mass-wavelength relation of the form /mc, where m and
and are both constants; and (3) For such low-mass particles, locality
can only be defined on large spatial scales, comparable to or exceeding the
particle wavelengths.
We use our model to derive the cosmological redshift characteristic of the
Standard Model of Cosmology, which becomes a gravitational redshift in our
model. We compare the results of our model to empirical data and show that, in
order to reproduce the sub-linear form of the observed distance-redshift
relation, our model requires + 0. We further place our
toy model in the context of the Friedmann Universe via a superposition of
Einstein Universes, each with its own scale factor a. Given the
overwhelming evidence supporting an expanding Universe, we then address
possible modifications to our base model that would be required to account for
the available empirical constraints, including the addition of some initial
expansion. Finally, we consider potentially observable distinctions between the
cosmological redshift and our proposed mechanism to account for the observed
distance-redshift relation.Comment: 23 pages, 7 figures; accepted for publication in Classical and
Quantum Gravit
Back to the Future: Estimating Initial Globular Cluster Masses from their Present Day Stellar Mass Functions
We use N-body simulations to model the 12 Gyr evolution of a suite of star
clusters with identical initial stellar mass functions over a range of initial
cluster masses, sizes, and orbits. Our models reproduce the distribution of
present-day global stellar mass functions that is observed in the Milky Way
globular cluster population. We find that the slope of a star cluster's stellar
mass function is strongly correlated with the fraction of mass that the cluster
has lost, independent of the cluster's initial mass, and nearly independent of
its orbit and initial size. Thus, the mass function - initial mass relation can
be used to determine a Galactic cluster's initial total stellar mass, if the
initial stellar mass function is known. We apply the mass function - initial
mass relation presented here to determine the initial stellar masses of 33
Galactic globular clusters, assuming an universal Kroupa initial mass function.
Our study suggests that globular clusters had initial masses that were on
average a factor of 4.5 times larger than their present day mass, with three
clusters showing evidence for being 10 times more massive at birth.Comment: Revised Version: 11 pages, 4 figures, Accepted for publication in
MNRA
Interrupted Stellar Encounters in Star Clusters
Strong encounters between single stars and binaries play a pivotal role in
the evolution of star clusters. Such encounters can also dramatically modify
the orbital parameters of binaries, exchange partners in and out of binaries,
and are a primary contributor to the rate of physical stellar collisions in
star clusters. Often, these encounters are studied under the approximation that
they happen quickly enough and within a small enough volume to be considered
isolated from the rest of the cluster. In this paper, we study the validity of
this assumption through the analysis of a large grid of single - binary and
binary - binary scattering experiments. For each encounter we evaluate the
encounter duration, and compare this with the expected time until another
single or binary star will join the encounter. We find that for lower-mass
clusters, similar to typical open clusters in our Galaxy, the percent of
encounters that will be "interrupted" by an interloping star or binary may be
20-40% (or higher) in the core, though for typical globular clusters we expect
<1% of encounters to be interrupted. Thus, the assumption that strong
encounters occur in relative isolation breaks down for certain clusters.
Instead, many strong encounters develop into more complex "mini-clusters",
which must be accounted for in studying, for example, the internal dynamics of
star clusters, and the physical stellar collision rate.Comment: 5 pages, 4 figures, accepted for publication in The Astrophysical
Journal Letter
Small-N collisional dynamics II: Roaming the realm of not-so-small-N
We develop a formalism for calculating probabilities for the outcomes of
stellar dynamical interactions, based on results from -body scattering
experiments. We focus here on encounters involving up to six particles and
calculate probabilities for direct stellar collisions; however our method is in
principle valid for larger particle numbers. Our method relies on the binomial
theorem, and is applicable to encounters involving any combination of particle
radii. We further demonstrate that our base model is valid to within a few
percent for any combination of particle masses, provided the minimum mass ratio
is within a factor of a few from unity. This method is particularly suitable
for models of collisional systems involving large numbers of stars, such as
globular clusters, old open clusters and galactic nuclei, where small subsets
of stars may regularly have very close encounters, and the direct integration
of all such encounters is computationally expensive. Variations of our method
may also be used to treat other encounter outcomes, such as ejections and
exchanges.Comment: 10 pages, 4 figures, 3 tables; accepted for publication in MNRA
Formation of runaway stars in a star-cluster potential
We study the formation of runaway stars due to binary-binary (2+2)
interactions in young star-forming clusters and/or associations. This is done
using a combination of analytic methods and numerical simulations of 2+2
scattering interactions, both in isolation and in a homogeneous background
potential. We focus on interactions that produce two single stars and a binary,
and study the outcomes as a function of the depth of the background potential,
within a range typical of cluster cores. As reference parameters for the
observational properties, we use those observed for the system of runaway stars
AE Aur and Col and binary Ori. We find that the outcome fractions
have no appreciable dependence on the depth of the potential, and neither do
the velocities of the ejected single stars. However, as the potential gets
deeper and a larger fraction of binaries remain trapped, two binary populations
emerge, with the escaped component having higher speeds and shorter semi-major
axes than the trapped one. Additionally, we find that the relative angles
between the ejected products are generally large. In particular, the angle
between the ejected fastest star and the escaped binary is typically , with a peak at around . However, as the
potential gets deeper, the angle distribution becomes broader. Finally, we
discuss the implications of our results for the interpretation of the
properties of the runaway stars AE Aur and Col.Comment: 20 pages, 16 figures, 4 tables, accepted to MNRA
Modifying two-body relaxation in N-body systems by gas accretion
We consider the effects that accretion from the interstellar medium onto the
particles of an N-body system has on the rate of two-body relaxation. To this
end, we derive an accretion-modified relaxation time by adapting Spitzer's
two-component model to include the damping effects of accretion. We consider
several different mass-dependencies and efficiency factors for the accretion
rate, as well as different mass ratios for the two components of the model.
The net effect of accretion is to accelerate mass segregation by increasing
the average mass , since the relaxation time is inversely proportional
to . Under the assumption that the accretion rate increases with the
accretor mass, there are two additional effects that accelerate mass
segregation. First, accretion acts to increase the range of any initial mass
spectrum, quickly driving the heaviest members to even higher masses. Second,
accretion acts to reduce the velocities of the accretors due to conservation of
momentum, and it is the heaviest members that are affected the most. Using our
two-component model, we quantify these effects as a function of the accretion
rate, the total cluster mass, and the component masses. We conclude by
discussing the implications of our results for the dynamical evolution of
primordial globular clusters, primarily in the context of black holes formed
from the most massive stellar progenitors.Comment: 9 pages, 3 figures, accepted for publication in MNRAS; edited to
match proof correction
When does a star cluster become a multiple star system? I. Lifetimes of equal-mass small-N systems
What is the difference between a long-lived unstable (or quasi-stable)
multiple star system and a bona fide star cluster? In this paper, we present a
possible framework to address this question, by studying the distributions of
disruption times for chaotic gravitational encounters as a function of the
number of interacting particles. To this end, we perform a series of numerical
scattering experiments with the \texttt{FEWBODY} code, to calculate the
distributions of disruption times as a function of both the particle number N
and the virial coefficient k. The subsequent distributions are fit with a
physically-motivated function, consisting of an initial exponential decay
followed by a very slowly decreasing tail at long encounter times due to
long-lived quasi-stable encounters. We find three primary features
characteristic of the calculated distributions of disruption times. These are:
(1) the system half-life increases with increasing particle number, (2) the
fraction of long-lived quasi-stable encounters increases with increasing
particle number and (3) both the system half-life and the fraction of
quasi-stable encounters increase with decreasing virial coefficient. We discuss
the significance of our results for collisional dynamics, and consider the
extrapolation of our results to larger-N systems. We suggest that this could
potentially offer a clear and unambiguous distinction between star clusters and
(unstable or quasi-stable) multiple star systems. Although we are limited by
very small-number statistics, our results tentatively suggest that (for our
assumptions) this transition occurs at a critical particle number of order 100.Comment: 7 pages, 4 figures, 2 tables; accepted for publication in MNRA
A Statistical Solution to the Chaotic, Non-Hierarchical Three-Body Problem
The three-body problem is arguably the oldest open question in astrophysics,
and has resisted a general analytic solution for centuries. Various
implementations of perturbation theory provide solutions in portions of
parameter space, but only where hierarchies of masses or separations exist.
Numerical integrations show that bound, non-hierarchical triples of Newtonian
point particles will almost always disintegrate into a single escaping star and
a stable, bound binary, but the chaotic nature of the three-body problem
prevents the derivation of tractable analytic formulae deterministically
mapping initial conditions to final outcomes. However, chaos also motivates the
assumption of ergodicity, suggesting that the distribution of outcomes is
uniform across the accessible phase volume. Here, we use the ergodic hypothesis
to derive a complete statistical solution to the non-hierarchical three-body
problem, one which provides closed-form distributions of outcomes (e.g. binary
orbital elements) given the conserved integrals of motion. We compare our
outcome distributions to large ensembles of numerical three-body integrations,
and find good agreement, so long as we restrict ourselves to "resonant"
encounters (the ~50% of scatterings that undergo chaotic evolution). In
analyzing our scattering experiments, we identify "scrambles" (periods in time
where no pairwise binaries exist) as the key dynamical state that ergodicizes a
non-hierarchical triple. The generally super-thermal distributions of survivor
binary eccentricity that we predict have notable applications to many
astrophysical scenarios. For example, non-hierarchical triples produced
dynamically in globular clusters are a primary formation channel for black hole
mergers, but the rates and properties of the resulting gravitational waves
depend on the distribution of post-disintegration eccentricities.Comment: 54 pages, 8 figure
Stellar dynamics in gas: The role of gas damping
In this paper, we consider how gas damping affects the dynamical evolution of
gas-embedded star clusters. Using a simple three-component (i.e. one gas and
two stellar components) model, we compare the rates of mass segregation due to
two-body relaxation, accretion from the interstellar medium, and gas dynamical
friction in both the supersonic and subsonic regimes. Using observational data
in the literature, we apply our analytic predictions to two different
astrophysical environments, namely galactic nuclei and young open star
clusters. Our analytic results are then tested using numerical simulations
performed with the NBSymple code, modified by an additional deceleration term
to model the damping effects of the gas.
The results of our simulations are in reasonable agreement with our analytic
predictions, and demonstrate that gas damping can significantly accelerate the
rate of mass segregation. A stable state of approximate energy equilibrium
cannot be achieved in our model if gas damping is present, even if Spitzer's
Criterion is satisfied. This instability drives the continued dynamical
decoupling and subsequent ejection (and/or collisions) of the more massive
population. Unlike two-body relaxation, gas damping causes overall cluster
contraction, reducing both the core and half-mass radii. If the cluster is mass
segregated (and/or the gas density is highest at the cluster centre), the
latter contracts faster than the former, accelerating the rate of core
collapse.Comment: 17 pages, 8 figures, 1 table; accepted for publication in MNRA
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