Generation of potential/surface density pairs in flat disks Power law distributions

We report a simple method to generate potential/surface density pairs in flat axially symmetric finite size disks. Potential/surface density pairs consist of a ``homogeneous'' pair (a closed form expression) corresponding to a uniform disk, and a ``residual'' pair. This residual component is converted into an infinite series of integrals over the radial extent of the disk. For a certain class of surface density distributions (like power laws of the radius), this series is fully analytical. The extraction of the homogeneous pair is equivalent to a convergence acceleration technique, in a matematical sense. In the case of power law distributions, the convergence rate of the residual series is shown to be cubic inside the source. As a consequence, very accurate potential values are obtained by low order truncation of the series. At zero order, relative errors on potential values do not exceed a few percent typically, and scale with the order N of truncation as 1/N**3. This method is superior to the classical multipole expansion whose very slow convergence is often critical for most practical applications.Comment: Accepted for publication in Astronomy & Astrophysics 7 pages, 8 figures, F90-code available at http://www.obs.u-bordeaux1.fr/radio/JMHure/intro2applawd.htm

Scalable N-body code for the modelling of early-type galaxies

Early-type galaxies exhibit a wealth of photometric and dynamical structures. These signatures are fossil records of their formation and evolution processes. In order to examine these structures in detail, we build models aimed at reproducing the observed photometry and kinematics. The developed method is a generalization of the one introduced by Syer and Tremaine (1996), consisting in an N-body representation, in which the weights of the particles are changing with time. Our code is adapted for integral-field spectroscopic data, and is able to reproduce the photometric as well as stellar kinematic data of observed galaxies. We apply this technique on SAURON data of early-type galaxies, and present preliminary results on NGC 3377.Comment: 6 pages, 2 figures. Original version printed in the Proceedings of "Science perspective for 3D spectroscopy", 2005, Eds Kissler-Patig, Walsh, Roth, ES0, Springe

Resonant Orbits in Triaxial Galaxies

Box orbits in triaxial potentials are generically thin, that is, they lie close in phase space to a resonant orbit satisfying a relation of the form l\omega_1 +m\omega_2+n\omega_3=0 between the three fundamental frequencies. Resonant orbits are confined to a membrane; they play roughly the same role, in three dimensions, that closed orbits play in two. Stable resonant orbits avoid the center of the potential; orbits that are thick enough to pass near the center are typically stochastic. Very near the center, where the gravitational potential is dominated by the black hole, resonant orbits continue to exist, including at least one family whose elongation is parallel to the long axes of the triaxial figure.Comment: 20 Latex pages, 11 Postscript figures. Submitted to The Astronomical Journa

Relativistic and Newtonian core-shell models: analytical and numerical results

We make a detailed analysis of Newtonian as well as relativistic core-shell models recently proposed to describe a black hole or neutron star surrounded by shells of matter, and in a seminal sense also galaxies, supernovae and star remnants since there are massive shell-like structures surrounding many of them and also evidences for many galactic nuclei hiding black holes. We discuss the unicity of the models in relation to their analyticity at the black hole horizon and also to the full elimination of conical singularities. Secondly, we study the role played by the presence/lack of discrete reflection symmetries about equatorial planes in the chaotic behavior of the orbits, which is to be contrasted with the almost universal acceptance of reflection symmetries as default assumptions in galactic modeling. We also compare the related effects if we change a true central black hole by a Newtonian central mass. The numerical findings are: 1- The breakdown of the reflection symmetry about the equatorial plane in both Newtonian and relativistic core-shell models does i) enhance in a significant way the chaoticity of orbits in reflection symmetric oblate shell models and ii) inhibit significantly also the occurrence of chaos in reflection symmetric prolate shell models. In particular, in the prolate case the lack of the reflection symmetry provides the phase space with a robust family of regular orbits that is otherwise not found at higher energies. 2- The relative extents of the chaotic regions in the relativistic cases (i. e. with a true central black hole) are significantly larger than in the corresponding Newtonian ones (which have just a $-1/r$ central potential).Comment: AASTEX, 22 pages plus 28 postscript figures, to appear in Ap.

A cerebellar mechanism for learning prior distributions of time intervals

Knowledge about the statistical regularities of the world is essential for cognitive and sensorimotor function. In the domain of timing, prior statistics are crucial for optimal prediction, adaptation and planning. Where and how the nervous system encodes temporal statistics is, however, not known. Based on physiological and anatomical evidence for cerebellar learning, we develop a computational model that demonstrates how the cerebellum could learn prior distributions of time intervals and support Bayesian temporal estimation. The model shows that salient features observed in human Bayesian time interval estimates can be readily captured by learning in the cerebellar cortex and circuit level computations in the cerebellar deep nuclei. We test human behavior in two cerebellar timing tasks and find prior-dependent biases in timing that are consistent with the predictions of the cerebellar model

The Kinematics of the Outer Halo of M87

Radial velocities are presented for a new sample of globular clusters in the outer halo of M87 at a distance of 300 to 540 arcsec (24 to 43 kpc) from the center of this galaxy. These are used to augment our previously published data and an analysis of the rotation and velocity dispersion of the M87 globular cluster system is carried out. The rotation is $\sim300$ \kms at R = 32 kpc, at which point the velocity dispersion is also still quite high, $\sim450$ \kms. The high rotation is interesting. The outer halo of M87 is, as was found in our previous kinematic analysis, very massive.Comment: Accepted for publication in the AJ. 13 pages with 3 figure

The internal structure and formation of early-type galaxies: the gravitational--lens system MG2016+112 at z=1.004

[Abridged] We combine our measurements of the velocity dispersion and the surface brightness profile of the lens galaxy D in the system MG2016+112 (z=1.004) with constraints from gravitational lensing to study its internal mass distribution. We find that: (i) dark matter accounts for >50% of the total mass within the Einstein radius (99% CL), excluding at the 8-sigma level that mass follows light inside the Einstein radius with a constant mass-to-light ratio (M/L). (ii) the total mass distribution inside the Einstein radius is well-described by a density profile ~r^-gamma' with an effective slope gamma'=2.0+-0.1+-0.1, including random and systematic uncertainties. (iii) The offset of galaxy D from the local Fundamental Plane independently constrains the stellar M/L, and matches the range derived from our models, leading to a more stringent lower limit of >60% on the fraction of dark matter within the Einstein radius (99%CL). Under the assumption of adiabatic contraction, the inner slope of the dark matter halo before the baryons collapsed is gamma_i<1.4 (68 CL), marginally consistent with the highest-resolution cold dark matter simulations that indicate gamma_i~1.5. This might indicate that either adiabatic contraction is a poor description of E/S0 formation or that additional processes play a role as well. Indeed, the apparently isothermal density distribution inside the Einstein radius, is not a natural outcome of adiabatic contraction models, where it appears to be a mere coincidence. By contrast, we argue that isothermality might be the result of a stronger coupling between luminous and dark-matter, possibly the result of (incomplete) violent relaxation processes. Hence, we conclude that galaxy D appears already relaxed 8 Gyr ago.Comment: 8 pages, 4 figures, ApJ, in press, minor change

Chaos and Elliptical Galaxies

Recent results on chaos in triaxial galaxy models are reviewed. Central mass concentrations like those observed in early-type galaxies -- either stellar cusps, or massive black holes -- render most of the box orbits in a triaxial potential stochastic. Typical Liapunov times are 3-5 crossing times, and ensembles of stochastic orbits undergo mixing on time scales that are roughly an order of magnitude longer. The replacement of the regular orbits by stochastic orbits reduces the freedom to construct self-consistent equilibria, and strong triaxiality can be ruled out for galaxies with sufficiently high central mass concentrations.Comment: uuencoded gziped PostScript, 12 pages including figure

A SAURON look at galaxy bulges

Kinematic and population studies show that bulges are generally rotationally flattened systems similar to low-luminosity ellipticals. However, observations with state-of-the-art integral field spectrographs, such as SAURON, indicate that the situation is much more complex, and allow us to investigate phenomena such as triaxiality, kinematic decoupling and population substructure, and to study their connection to current formation and evolution scenarios for bulges of early-type galaxies. We present the examples of two S0 bulges from galaxies in our sample of nearby galaxies: one that shows all the properties expected from classical bulges (NGC5866), and another case that presents kinematic features appropriate for barred disk galaxies (NGC7332).Comment: 4 pages, 3 figures, accepted for publishing in AN (refereed conf. proc. of the Euro3D Science workshop, IoA Cambridge, May 2003