The Globular Cluster System in the Inner Region of the Giant Elliptical Galaxy NGC 4472

We present a study of globular clusters in the inner region of the giant elliptical galaxy NGC 4472, based on the HST WFPC2 archive data. We have found about 1560 globular cluster candidates at the galactocentric radius r < 4 arcmin. V-(V-I) diagram of these objects shows a dominant vertical structure which consists obviously of two components: blue globular clusters (BGCs) and red globular clusters (RGCs). The luminosity function of the globular clusters is derived to have a peak at V(max)=23.50+/-0.16 from Gaussian fitting. The distance to NGC 4472 is estimated to be d=14.7+/-1.3 Mpc.The peak luminosity for the RGCs is similar to that for the BGCs, which indicates that the RGCs may be several Gyrs younger than the BGCs. The mean luminosity of the bright BGCs decreases by 0.2 mag with increasing galactocentric radius over the range of 9 arcmin, while that of the RGCs does not. The observed color distribution of these globular clusters is distinctively bimodal with peaks at (V-I) = 0.98 and 1.23. The mean observed color of all the globular clusters with V < 23.9 mag is derived to be (V-I)=1.11. These colors are exactly the same as those for the globular clusters in M87. It is found that the relative number of the BGCs to the RGCs is increasing with the increasing galactocentric radius. Surface number density profiles of both the BGCs and RGCs get flat in the central region, and the core radii of the globular cluster systems are measured to be r_c = 1.9 arcmin for the BGCs, r_c = 1.2 arcmin for the RGCs, and r_c = 1.3 arcmin for the total sample, which are much larger than the stellar core of the galaxy. In general the properties of the globular clusters in the inner region of NGC 4472 are consistent with those of the globular clusters in the outer region of NGC 4472.Comment: 27 pages (AASLaTex), 22 Postscript Figures, Accepted for Publication in the Astronomical Journal, Jul. 31st, 200

On the Self-Consistent Response of Stellar Systems to Gravitational Shocks

We study the reaction of a globular star cluster to a time-varying tidal perturbation (gravitational shock) using self-consistent N-body simulations and address two questions. First, to what extent is the cluster interior protected by adiabatic invariants. Second, how much further energy change does the postshock evolution of the cluster potential produce and how much does it affect the dispersion of stellar energies. We introduce the adiabatic correction} as ratio of the energy change, , to its value in the impulse approximation. When the potential is kept fixed, the numerical results for the adiabatic correction for stars with orbital frequency \omega can be approximated as (1 + \omega^2 \tau^2)^{-\gamma}. For shocks with the characteristic duration of the order the half-mass dynamical time of the cluster, \tau < t_{dyn,h}, the exponent \gamma = 5/2. For more prolonged shocks, \tau > 4 t_{dyn,h}, the adiabatic correction is shallower, \gamma = 3/2. When we allow for self-gravity and potential oscillations which follow the shock, the energy of stars in the core changes significantly, while the total energy of the system is conserved. Paradoxically, the postshock potential fluctuations reduce the total amount of energy dispersion, . The effect is small but real and is due to the postshock energy change being statistically anti-correlated with the shock induced heating. These results are to be applied to Fokker-Planck models of the evolution of globular clusters.Comment: 20 pages; ApJ 513 (in press

The Goldman-Rota identity and the Grassmann scheme

We inductively construct an explicit (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. The main step is a constructive, linear algebraic interpretation of the Goldman-Rota recurrence for the number of subspaces of a finite vector space. This interpretation shows that the up operator on subspaces has an explicitly given recursive structure. Using this we inductively construct an explicit orthogonal symmetric Jordan basis with respect to the up operator and write down the singular values, i.e., the ratio of the lengths of the successive vectors in the Jordan chains. The collection of all vectors in this basis of a fixed rank forms a (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. We also pose a bijective proof problem on the spanning trees of the Grassmann graphs.Comment: 19 Page

A Computationally Efficient Projection-Based Approach for Spatial Generalized Linear Mixed Models

Inference for spatial generalized linear mixed models (SGLMMs) for high-dimensional non-Gaussian spatial data is computationally intensive. The computational challenge is due to the high-dimensional random effects and because Markov chain Monte Carlo (MCMC) algorithms for these models tend to be slow mixing. Moreover, spatial confounding inflates the variance of fixed effect (regression coefficient) estimates. Our approach addresses both the computational and confounding issues by replacing the high-dimensional spatial random effects with a reduced-dimensional representation based on random projections. Standard MCMC algorithms mix well and the reduced-dimensional setting speeds up computations per iteration. We show, via simulated examples, that Bayesian inference for this reduced-dimensional approach works well both in terms of inference as well as prediction, our methods also compare favorably to existing "reduced-rank" approaches. We also apply our methods to two real world data examples, one on bird count data and the other classifying rock types