637 research outputs found
New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)
We consider hypo-elliptic diffusion and convection-diffusion on , the quotient of the Lie group of rigid body motions SE(3) in
which group elements are equivalent if they are equal up to a rotation around
the reference axis. We show that we can derive expressions for the convolution
kernels in terms of eigenfunctions of the PDE, by extending the approach for
the SE(2) case. This goes via application of the Fourier transform of the PDE
in the spatial variables, yielding a second order differential operator. We
show that the eigenfunctions of this operator can be expressed as (generalized)
spheroidal wave functions. The same exact formulas are derived via the Fourier
transform on SE(3). We solve both the evolution itself, as well as the
time-integrated process that corresponds to the resolvent operator.
Furthermore, we have extended a standard numerical procedure from SE(2) to
SE(3) for the computation of the solution kernels that is directly related to
the exact solutions. Finally, we provide a novel analytic approximation of the
kernels that we briefly compare to the exact kernels.Comment: Revised and restructure
Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D
We propose an efficient approach for the grouping of local orientations
(points on vessels) via nilpotent approximations of sub-Riemannian distances in
the 2D and 3D roto-translation groups and . In our distance
approximations we consider homogeneous norms on nilpotent groups that locally
approximate , and which are obtained via the exponential and logarithmic
map on . In a qualitative validation we show that the norms provide
accurate approximations of the true sub-Riemannian distances, and we discuss
their relations to the fundamental solution of the sub-Laplacian on .
The quantitative experiments further confirm the accuracy of the
approximations. Quantitative results are obtained by evaluating perceptual
grouping performance of retinal blood vessels in 2D images and curves in
challenging 3D synthetic volumes. The results show that 1) sub-Riemannian
geometry is essential in achieving top performance and 2) that grouping via the
fast analytic approximations performs almost equally, or better, than
data-adaptive fast marching approaches on and .Comment: 18 pages, 9 figures, 3 tables, in review at JMI
The Relation Between the Globular Cluster Mass and Luminosity Functions
The relation between the globular cluster luminosity function (GCLF,
dN/dlogL) and globular cluster mass function (GCMF, dN/dlogM) is considered.
Due to low-mass star depletion, dissolving GCs have mass-to-light (M/L) ratios
that are lower than expected from their metallicities. This has been shown to
lead to an M/L ratio that increases with GC mass and luminosity. We model the
GCLF and GCMF and show that the power law slopes inherently differ (1.0 versus
0.7, respectively) when accounting for the variability of M/L. The observed
GCLF is found to be consistent with a Schechter-type initial cluster mass
function and a mass-dependent mass-loss rate.Comment: 4 pages, 2 figures. To appear in the proceedings of "Galaxy Wars:
Stellar Populations and Star Formation in Interacting Galaxies" (Tennessee,
July 2009
Black hole mergers in the universe
Mergers of black-hole binaries are expected to release large amounts of
energy in the form of gravitational radiation. However, binary evolution models
predict merger rates too low to be of observational interest. In this paper we
explore the possibility that black holes become members of close binaries via
dynamical interactions with other stars in dense stellar systems. In star
clusters, black holes become the most massive objects within a few tens of
millions of years; dynamical relaxation then causes them to sink to the cluster
core, where they form binaries. These black-hole binaries become more tightly
bound by superelastic encounters with other cluster members, and are ultimately
ejected from the cluster. The majority of escaping black-hole binaries have
orbital periods short enough and eccentricities high enough that the emission
of gravitational radiation causes them to coalesce within a few billion years.
We predict a black-hole merger rate of about per year per
cubic megaparsec, implying gravity wave detection rates substantially greater
than the corresponding rates from neutron star mergers. For the first
generation Laser Interferometer Gravitational-Wave Observatory (LIGO-I), we
expect about one detection during the first two years of operation. For its
successor LIGO-II, the rate rises to roughly one detection per day. The
uncertainties in these numbers are large. Event rates may drop by about an
order of magnitude if the most massive clusters eject their black hole binaries
early in their evolution.Comment: 12 pages, ApJL in pres
On the Interpretation of the Age Distribution of Star Clusters in the Small Magellanic Cloud
We re-analyze the age distribution (dN/dt) of star clusters in the Small
Magellanic Cloud (SMC) using age determinations based on the Magellanic Cloud
Photometric Survey. For ages younger than 3x10^9 yr the dN/dt distribution can
be approximated by a power-law distribution, dN/dt propto t^-beta, with
-beta=-0.70+/-0.05 or -beta=-0.84+/-0.04, depending on the model used to derive
the ages. Predictions for a cluster population without dissolution limited by a
V-band detection result in a power-law dN/dt distribution with an index of
~-0.7. This is because the limiting cluster mass increases with age, due to
evolutionary fading of clusters, reducing the number of observed clusters at
old ages. When a mass cut well above the limiting cluster mass is applied, the
dN/dt distribution is flat up to 1 Gyr. We conclude that cluster dissolution is
of small importance in shaping the dN/dt distribution and incompleteness causes
dN/dt to decline. The reason that no (mass independent) infant mortality of
star clusters in the first ~10-20 Myr is found is explained by a detection bias
towards clusters without nebular emission, i.e. cluster that have survived the
infant mortality phase. The reason we find no evidence for tidal (mass
dependent) cluster dissolution in the first Gyr is explained by the weak tidal
field of the SMC. Our results are in sharp contrast to the interpretation of
Chandar et al. (2006), who interpret the declining dN/dt distribution as rapid
cluster dissolution. This is due to their erroneous assumption that the sample
is limited by cluster mass, rather than luminosity.Comment: 8 pages, 4 figures, accepted for publication in Ap
Monte-Carlo Simulations of Globular Cluster Evolution - I. Method and Test Calculations
We present a new parallel supercomputer implementation of the Monte-Carlo
method for simulating the dynamical evolution of globular star clusters. Our
method is based on a modified version of Henon's Monte-Carlo algorithm for
solving the Fokker-Planck equation. Our code allows us to follow the evolution
of a cluster containing up to 5x10^5 stars to core collapse in < 40 hours of
computing time. In this paper we present the results of test calculations for
clusters with equal-mass stars, starting from both Plummer and King model
initial conditions. We consider isolated as well as tidally truncated clusters.
Our results are compared to those obtained from approximate, self-similar
analytic solutions, from direct numerical integrations of the Fokker-Planck
equation, and from direct N-body integrations performed on a GRAPE-4
special-purpose computer with N=16384. In all cases we find excellent agreement
with other methods, establishing our new code as a robust tool for the
numerical study of globular cluster dynamics using a realistic number of stars.Comment: 35 pages, including 8 figures, submitted to ApJ. Revised versio
How many young star clusters exist in the Galactic center?
We study the evolution and observability of young compact star clusters
within about 200pc of the Galactic center. Calculations are performed using
direct N-body integration on the GRAPE-4, including the effects of both stellar
and binary evolution and the external influence of the Galaxy. The results of
these detailed calculations are used to calibrate a simplified model applicable
over a wider range of cluster initial conditions. We find that clusters within
200 pc from the Galactic center dissolve within about 70 Myr. However, their
projected densities drop below the background density in the direction of the
Galactic center within 20 Myr, effectively making these clusters undetectable
after that time. Clusters farther from the Galactic center but at the same
projected distance are more strongly affected by this selection effect, and may
go undetected for their entire lifetimes. Based on these findings, we conclude
that the region within 200 pc of the Galactic center could easily harbor some
50 clusters with properties similar to those of the Arches or the Quintuplet
systems.Comment: ApJ Letters in pres
A runaway collision in a young star cluster as the origin of the brightest supernova
Supernova 2006gy in the galaxy NGC 1260 is the most luminous one recorded
\cite{2006CBET..644....1Q, 2006CBET..647....1H, 2006CBET..648....1P,
2006CBET..695....1F}. Its progenitor might have been a very massive (
\msun) star \cite{2006astro.ph.12617S}, but that is incompatible with hydrogen
in the spectrum of the supernova, because stars \msun are believed to
have shed their hydrogen envelopes several hundred thousand years before the
explosion \cite{2005A&A...429..581M}. Alternatively, the progenitor might have
arisen from the merger of two massive stars \cite{2007ApJ...659L..13O}. Here we
show that the collision frequency of massive stars in a dense and young cluster
(of the kind to be expected near the center of a galaxy) is sufficient to
provide a reasonable chance that SN 2006gy resulted from such a bombardment. If
this is the correct explanation, then we predict that when the supernova fades
(in a year or so) a dense cluster of massive stars becomes visible at the site
of the explosion
Progenitors of Supernovae Type Ia
Despite the significance of Type Ia supernovae (SNeIa) in many fields in
astrophysics, SNeIa lack a theoretical explanation. The standard scenarios
involve thermonuclear explosions of carbon/oxygen white dwarfs approaching the
Chandrasekhar mass; either by accretion from a companion or by a merger of two
white dwarfs. We investigate the contribution from both channels to the SNIa
rate with the binary population synthesis (BPS) code SeBa in order to constrain
binary processes such as the mass retention efficiency of WD accretion and
common envelope evolution. We determine the theoretical rates and delay time
distribution of SNIa progenitors and in particular study how assumptions affect
the predicted rates.Comment: 6 pages, 6 figures, appeared in proceedings for "The 18th European
White Dwarf Workshop
The Evolution of Globular Clusters in the Galaxy
We investigate the evolution of globular clusters using N-body calculations
and anisotropic Fokker-Planck (FP) calculations. The models include a mass
spectrum, mass loss due to stellar evolution, and the tidal field of the parent
galaxy. Recent N-body calculations have revealed a serious discrepancy between
the results of N-body calculations and isotropic FP calculations. The main
reason for the discrepancy is an oversimplified treatment of the tidal field
employed in the isotropic FP models. In this paper we perform a series of
calculations with anisotropic FP models with a better treatment of the tidal
boundary and compare these with N-body calculations. The new tidal boundary
condition in our FP model includes one free parameter. We find that a single
value of this parameter gives satisfactory agreement between the N-body and FP
models over a wide range of initial conditions.
Using the improved FP model, we carry out an extensive survey of the
evolution of globular clusters over a wide range of initial conditions varying
the slope of the mass function, the central concentration, and the relaxation
time. The evolution of clusters is followed up to the moment of core collapse
or the disruption of the clusters in the tidal field of the parent galaxy. In
general, our model clusters, calculated with the anisotropic FP model with the
improved treatment for the tidal boundary, live longer than isotropic models.
The difference in the lifetime between the isotropic and anisotropic models is
particularly large when the effect of mass loss via stellar evolution is rather
significant. On the other hand the difference is small for relaxation-
dominated clusters which initially have steep mass functions and high central
concentrations.Comment: 36 pages, 11 figures, LaTeX; added figures and tables; accepted by
Ap
- …