On the trapping of stars by a newborn stellar supercluster

Numerical experiments conducted by Fellhauer et al. (MNRAS, 372, 338, 2006) suggest that a supercluster may capture up to about 40 per cent of its mass from the galaxy where it belongs. Nevertheless, in those experiments the cluster was created making appear its mass out of nothing, rather than from mass already present in the galaxy. Here we use a thought experiment, plus a few simple computations, to show that the difference between the dynamical effects of these two scenarios (i.e., mass creation vs. mass concentration) is actually very important. We also present the results of new numerical experiments, simulating the formation of the cluster through mass concentration, that show that trapping depends critically on the process of cluster formation and that the amounts of gained mass are substantially smaller than those obtained from mass creation.Comment: 6 pages, 3 figures. Submitted to MNRA

Models of cuspy triaxial stellar systems. III: The effect of velocity anisotropy on chaoticity

In several previous investigations we presented models of triaxial stellar systems, both cuspy and non cuspy, that were highly stable and harboured large fractions of chaotic orbits. All our models had been obtained through cold collapses of initially spherical $N$--body systems, a method that necessarily results in models with strongly radial velocity distributions. Here we investigate a different method that was reported to yield cuspy triaxial models with virtually no chaos. We show that such result was probably due to the use of an inadequate chaos detection technique and that, in fact, models with significant fractions of chaotic orbits result also from that method. Besides, starting with one of the models from the first paper in this series, we obtained three different models by rendering its velocity distribution much less radially biased (i.e., more isotropic) and by modifying its axial ratios through adiabatic compression. All three models yielded much higher fractions of regular orbits than most of those from our previous work. We conclude that it is possible to obtain stable cuspy triaxial models of stellar systems whose velocity distribution is more isotropic than that of the models obtained from cold collapses. Those models still harbour large fractions of chaotic orbits and, although it is difficult to compare the results from different models, we can tentatively conclude that chaoticity is reduced by velocity isotropy.Comment: 11 pages, 14 figures. Accepted for publication in MNRA

Models of cuspy triaxial stellar systems. II. Regular orbits

In the first paper of this series we used the N--body method to build a dozen cuspy (gamma ~ 1) triaxial models of stellar systems, and we showed that they were highly stable over time intervals of the order of a Hubble time, even though they had very large fractions of chaotic orbits (more than 85 per cent in some cases). The models were grouped in four sets, each one comprising models morphologically resembling E2, E3, E4 and E5 galaxies, respectively. The three models within each set, although different, had the same global properties and were statistically equivalent. In the present paper we use frequency analysis to classify the regular orbits of those models. The bulk of those orbits are short axis tubes (SATs), with a significant fraction of long axis tubes (LATs) in the E2 models that decreases in the E3 and E4 models to become negligibly small in the E5 models. Most of the LATs in the E2 and E3 models are outer LATs, but the situation reverses in the E4 and E5 models where the few LATs are mainly inner LATs. As could be expected for cuspy models, most of the boxes are resonant orbits, i.e., boxlets. Nevertheless, only the (x, y) fishes of models E3 and E4 amount to about 10 per cent of the regular orbits, with most of the fractions of the other boxlets being of the order of 1 per cent or less.Comment: Accepted for publication in the Monthly Notices of the Royal Astronomical Societ

On the correct computation of all Lyapunov exponents in Hamiltonian dynamical systems

The Lyapunov Characteristic Exponents are a useful indicator of chaos in astronomical dynamical systems. They are usually computed through a standard, very efficient and neat algorithm published in 1980. However, for Hamiltonian systems the expected result of pairs of opposite exponents is not always obtained with enough precision. We find here why in these cases the initial order of the deviation vectors matters, and how to sort them in order to obtain a correct result.Comment: 8 pages, 3 figure

A three dimensional investigation of two dimensional orbits

Orbits in the principal planes of triaxial potentials are known to be prone to unstable motion normal to those planes, so that three dimensional investigations of those orbits are needed even though they are two dimensional. We present here an investigation of such orbits in the well known logarithmic potential which shows that the third dimension must be taken into account when studying them and that the instability worsens for lower values of the forces normal to the plane. Partially chaotic orbits are present around resonances, but also in other regions. The action normal to the plane seems to be related to the isolating integral that distinguishes regular form partially chaotic orbits, but not to the integral that distinguishes partially from fully chaotic orbits.Comment: Accepted for publication in CMD

Models of cuspy triaxial stellar systems. I. Stability and chaoticity

We used the N-body code of Hernquist and Ostriker (1992) to build a dozen cuspy ({\gamma}\approx 1) triaxial models of stellar systems through dissipationless collapses of initially spherical distributions of 10^6 particles. We chose four sets of initial conditions that resulted in models morphologically resembling E2, E3, E4 and E5 galaxies, respectively. Within each set, three different seed numbers were selected for the random number generator used to create the initial conditions, so that the three models of each set are statistically equivalent. We checked the stability of our models using the values of their central densities and of their moments of inertia, which turned out to be very constant indeed. The changes of those values were all less than 3 per cent over one Hubble time and, moreover, we show that the most likely cause of those changes are relaxation effects in the numerical code. We computed the six Lyapunov exponents of nearly 5,000 orbits in each model in order to recognize regular, partially and fully chaotic orbits. All the models turned out to be highly chaotic, with less than 25 per cent of their orbits being regular. We conclude that it is quite possible to obtain cuspy triaxial stellar models that contain large fractions of chaotic orbits and are highly stable. The difficulty to build such models with the method of Schwarzschild (1979) should be attributed to the method itself and not to physical causes.Comment: 10 pages, 5 figures, published in MNRAS (The definitive version is available at www.blackwell-synergy.com

Large-scale study of the NGC 1399 globular cluster system in Fornax

We present a Washington C and Kron-Cousins R photometric study of the globular cluster system of NGC 1399, the central galaxy of the Fornax cluster. A large areal coverage of 1 square degree around NGC 1399 is achieved with three adjoining fields of the MOSAIC II Imager at the CTIO 4-m telescope. Working on such a large field, we can perform the first indicative determination of the total size of the NGC 1399 globular cluster system. The estimated angular extent, measured from the NGC 1399 centre and up to a limiting radius where the areal density of blue globular clusters falls to 30 per cent of the background level, is 45 +/- 5 arcmin, which corresponds to 220 - 275 kpc at the Fornax distance. The bimodal colour distribution of this globular cluster system, as well as the different radial distribution of blue and red clusters, up to these large distances from the parent galaxy, are confirmed. The azimuthal globular cluster distribution exhibits asymmetries that might be understood in terms of tidal stripping of globulars from NGC 1387, a nearby galaxy. The good agreement between the areal density profile of blue clusters and a projected dark-matter NFW density profile is emphasized.Comment: 9 pages, 9 figures. Accepted for publication in A&

Granular discharge and clogging for tilted hoppers

We measure the flux of spherical glass beads through a hole as a systematic function of both tilt angle and hole diameter, for two different size beads. The discharge increases with hole diameter in accord with the Beverloo relation for both horizontal and vertical holes, but in the latter case with a larger small-hole cutoff. For large holes the flux decreases linearly in cosine of the tilt angle, vanishing smoothly somewhat below the angle of repose. For small holes it vanishes abruptly at a smaller angle. The conditions for zero flux are discussed in the context of a {\it clogging phase diagram} of flow state vs tilt angle and ratio of hole to grain size