When are projections also embeddings?

We study an autonomous four-dimensional dynamical system used to model certain geophysical processes.This system generates a chaotic attractor that is strongly contracting, with four Lyapunov exponents $\lambda_i$ that satisfy $\lambda_1+ \lambda_2+\lambda_3<0$, so the Lyapunov dimension is $D_L=2+|\lambda_3|/\lambda_1 < 3$ in the range of coupling parameter values studied. As a result, it should be possible to find three-dimensional spaces in which the attractors can be embedded so that topological analyses can be carried out to determine which stretching and squeezing mechanisms generate chaotic behavior. We study mappings into $R^3$ to determine which can be used as embeddings to reconstruct the dynamics. We find dramatically different behavior in the two simplest mappings: projections from $R^4$ to $R^3$. In one case the one-parameter family of attractors studied remains topologically unchanged for all coupling parameter values. In the other case, during an intermediate range of parameter values the projection undergoes self-intersections, while the embedded attractors at the two ends of this range are topologically mirror images of each other

A Comparison of Tests for Embeddings

It is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test. The classical tests, which depend on real numbers (fractal dimensions, Lyapunov exponents) averaged over an attractor, are compared with a topological test that depends on integers. The comparison can only be done for mappings into three dimensions. We find that the classical tests fail to predict when a mapping is an embedding and when it is not. We point out the reasons for this failure, which are not restricted to three dimensions

Faint dwarf galaxies in nearby clusters

Besides giant elliptical galaxies, a number of low-mass stellar systems inhabit the cores of galaxy clusters, such as dwarf elliptical galaxies (dEs/dSphs), ultra-compact dwarf galaxies (UCDs), and globular clusters. The detailed morphological examination of faint dwarf galaxies has, until recently, been limited to the Local Group (LG) and the two very nearby galaxy clusters Virgo and Fornax. Here, we compare the structural parameters of a large number of dEs/dSphs in the more distant clusters Hydra I and Centaurus to other dynamically hot stellar systems.Comment: 2 pages, 1 figure; to appear in "A Universe of Dwarf Galaxies: Observations, Theories, Simulations", held in Lyon, France (June 14-18, 2010), eds. M. Koleva, P. Prugniel & I. Vauglin, EAS Series (Paris: EDP

The tidal stripping of satellites

We present an improved analytic calculation for the tidal radius of satellites and test our results against N-body simulations. The tidal radius in general depends upon four factors: the potential of the host galaxy, the potential of the satellite, the orbit of the satellite and {\it the orbit of the star within the satellite}. We demonstrate that this last point is critical and suggest using {\it three tidal radii} to cover the range of orbits of stars within the satellite. In this way we show explicitly that prograde star orbits will be more easily stripped than radial orbits; while radial orbits are more easily stripped than retrograde ones. This result has previously been established by several authors numerically, but can now be understood analytically. For point mass, power-law (which includes the isothermal sphere), and a restricted class of split power law potentials our solution is fully analytic. For more general potentials, we provide an equation which may be rapidly solved numerically. Over short times (\simlt 1-2 Gyrs $\sim 1$ satellite orbit), we find excellent agreement between our analytic and numerical models. Over longer times, star orbits within the satellite are transformed by the tidal field of the host galaxy. In a Hubble time, this causes a convergence of the three limiting tidal radii towards the prograde stripping radius. Beyond the prograde stripping radius, the velocity dispersion will be tangentially anisotropic.Comment: 10 pages, 5 figures. Final version accepted for publication in MNRAS. Some new fully analytic tidal radii have been added for power law density profiles (including the isothermal sphere) and some split power law

The mass of dwarf spheroidal galaxies and the missing satellite problem

We present the results from a suite of N-body simulations of the tidal stripping of two-component dwarf galaxies comprising some stars and dark matter. We show that recent kinematic data from the local group dwarf spheroidal (dSph) galaxies suggests that dSph galaxies must be sufficiently massive ($10^9 - 10^{10}$M$_\odot$) that tidal stripping is of little importance for the stars. We discuss the implications of these massive dSph galaxies for cosmology and galaxy formation.Comment: 4 pages, 1 figure, to appear in the proceedings of the IAUC198 "Near-Field Cosmology with Dwarf Elliptical Galaxies", H. Jerjen & B. Binggeli (eds.). Comments welcom

The effect of stellar-mass black holes on the structural evolution of massive star clusters

We present the results of realistic N-body modelling of massive star clusters in the Magellanic Clouds, aimed at investigating a dynamical origin for the radius-age trend observed in these systems. We find that stellar-mass black holes, formed in the supernova explosions of the most massive cluster stars, can constitute a dynamically important population. If a significant number of black holes are retained (here we assume complete retention), these objects rapidly form a dense core where interactions are common, resulting in the scattering of black holes into the cluster halo, and the ejection of black holes from the cluster. These two processes heat the stellar component, resulting in prolonged core expansion of a magnitude matching the observations. Significant core evolution is also observed in Magellanic Cloud clusters at early times. We find that this does not result from the action of black holes, but can be reproduced by the effects of mass-loss due to rapid stellar evolution in a primordially mass segregated cluster.Comment: Accepted for publication in MNRAS Letters; 2 figures, 1 tabl

A new low mass for the Hercules dSph: the end of a common mass scale for the dwarfs?

We present a new mass estimate for the Hercules dwarf spheroidal galaxy (dSph), based on the revised velocity dispersion obtained by Aden et al. (2009, arXiv:0908.3489). The removal of a significant foreground contamination using newly acquired Stromgren photometry has resulted in a reduced velocity dispersion. Using this new velocity dispersion of 3.72 +/- 0.91 km/s, we find a mass of M_300=1.9^{+1.1}_{-0.8} 10^6 M_sun within the central 300 pc, which is also the half-light radius, and a mass of M_433=3.7_{-1.6}^{+2.2} 10^6 M_sun within the reach of our data to 433 pc, significantly lower than previous estimates. We derive an overall mass-to-light ratio of M_433/L=103^{+83}_{-48} M_sun/L_sun. Our mass estimate calls into question recent claims of a common mass scale for dSph galaxies. Additionally, we find tentative evidence for a velocity gradient in our kinematic data of 16 +/- 3 km/s/kpc, and evidence of an asymmetric extension in the light distribution at about 0.5 kpc. We explore the possibility that these features are due to tidal interactions with the Milky Way. We show that there is a self-consistent model in which Hercules has an assumed tidal radius of r_t = 485 pc, an orbital pericentre of r_p = 18.5 +/- 5 kpc, and a mass within r_t of M_{tid,r_t}=5.2 +/- 2.7 10^6 M_sun. Proper motions are required to test this model. Although we cannot exclude models in which Hercules contains no dark matter, we argue that Hercules is more likely to be a dark matter dominated system which is currently experiencing some tidal disturbance of its outer parts.Comment: 10 pages, 3 figures, Accepted for publication by ApJ