Tanuka Chattopadhyay

Statistical Evidence for Three Classes of Gamma-Ray Bursts

Two different multivariate clustering techniques, the K-means partitioning method and the Dirichlet process of mixture modeling, have been applied to the BATSE gamma-ray burst (GRB) catalog, to obtain the optimum number of coherent groups. In the standard paradigm, GRBs are classified into only two groups, the long and short bursts. However, for both of the clustering techniques, the optimal number of classes was found to be three, a result that is consistent with previous statistical analysis. In this classification, the long bursts are further divided into two groups that are primarily differentiated by their total fluence and duration and hence are called low- and high-fluence GRBs. Analysis of GRBs with known redshifts and spectral parameters suggests that low-fluence GRBs have nearly constant isotropic energy output of 1052 ergs, while for the high-fluence ones the energy output ranges from 1052 to 1054 ergs. It is speculated that the three kinds of GRBs reflect three different origins: mergers of neutron star systems, mergers between white dwarfs and neutron stars, and collapse of massive stars.

STUDY OF NGC 5128 GLOBULAR CLUSTERS UNDER MULTIVARIATE STATISTICAL PARADIGM

An objective classification of the globular clusters (GCs) of NGC 5128 has been carried out by using a model-based approach of cluster analysis. The set of observable parameters includes structural parameters, spectroscopically determined Lick indices and radial velocities from the literature. The optimum set of parameters for this type of analysis is selected through a modified technique of principal component analysis, which differs from the classical one in the sense that it takes into consideration the effects of outliers present in the data. Then a mixture model based approach has been used to classify the GCs into groups. The efficiency of the techniques used is tested through the comparison of the misclassification probabilities with those obtained using the K-means clustering technique. On the basis of the above classification scheme three coherent groups of GCs have been found. We propose that the clusters of one group originated in the original cluster formation event that coincided with the formation of the elliptical galaxy, and that the clusters of the two other groups are of external origin, from tidally stripped dwarf galaxies on random orbits around NGC 5128 for one group, and from an accreted spiral galaxy for the other.

Structures in the fundamental plane of early‐type galaxies

The fundamental plane of early-type galaxies is a rather tight three-parameter correlation discovered more than twenty years ago. It has resisted a both global and precise physical interpretation despite a consequent number of works, observational, theoretical or using numerical simulations. It appears that its precise properties depend on the population of galaxies in study. Instead of selecting a priori these populations, we propose to objectively construct homologous populations from multivariate analyses. We have undertaken multivariate cluster and cladistic analyses of a sample of 56 low-redshift galaxy clusters containing 699 early-type galaxies, using four parameters: effective radius, velocity dispersion, surface brightness averaged over effective radius, and Mg2 index. All our analyses are consistent with seven groups that define separate regions on the global fundamental plane, not across its thickness. In fact, each group shows its own fundamental plane, which is more loosely defined for less diversified groups. We conclude that the global fundamental plane is not a bent surface, but made of a collection of several groups characterizing several fundamental planes with different thicknesses and orientations in the parameter space. Our diversification scenario probably indicates that the level of diversity is linked to the number and the nature of transforming events and that the fundamental plane is the result of several transforming events. We also show that our classification, not the fundamental planes, is universal within our redshift range (0.007 -- 0.053). We find that the three groups with the thinnest fundamental planes presumably formed through dissipative (wet) mergers. In one of them, this(ese) merger(s) must have been quite ancient because of the relatively low metallicity of its galaxies, Two of these groups have subsequently undergone dry mergers to increase their masses. In the k-space, the third one clearly occupies the region where bulges (of lenticular or spiral galaxies) lie and might also have formed through minor mergers and accretions. The two least diversified groups probably did not form by major mergers and must have been strongly affected by interactions, some of the gas in the objects of one of these groups having possibly been swept out. The interpretation, based on specific assembly histories of galaxies of our seven groups, shows that they are truly homologous. They were obtained directly from several observables, thus independently of any a priori classification. The diversification scenario relating these groups does not depend on models or numerical simulations, but is objectively provided by the cladistic analysis. Consequently, our classification is more easily compared to models and numerical simulations, and our work can be readily repeated with additional observables.

A six-parameter space to describe galaxy diversification

Galaxy diversification proceeds by transforming events like accretion, interaction or mergers. These explain the formation and evolution of galaxies that can now be described with many observables. Multivariate analyses are the obvious tools to tackle the datasets and understand the differences between different kinds of objects. However, depending on the method used, redundancies, incompatibilities or subjective choices of the parameters can void the usefulness of such analyses. The behaviour of the available parameters should be analysed before an objective reduction of dimensionality and subsequent clustering analyses can be undertaken, especially in an evolutionary context. We study a sample of 424 early-type galaxies described by 25 parameters, ten of which are Lick indices, to identify the most structuring parameters and determine an evolutionary classification of these objects. Four independent statistical methods are used to investigate the discriminant properties of the observables and the partitioning of the 424 galaxies: Principal Component Analysis, K-means cluster analysis, Minimum Contradiction Analysis and Cladistics. (abridged)

Independent Component Analysis for the objective classification of globular clusters of the galaxy NGC 5128

Independent Component Analysis (ICA) is closely related to Principal Component Analysis (PCA) and factor analysis. Whereas ICA finds a set of source data that are mutually independent, PCA finds a set of data that are mutually uncorrelated. The assumption that data from different physical processes are uncorrelated does not always imply the reverse case that uncorrelated data are coming from different physical processes. This is because lack of correlation is a weaker property than independence.In the present case an objective classification of the globular clusters (GCs) of NGC 5128 has been carried out. Components responsible for significant variation have been obtained through both Principal Component Analysis (PCA) and Independent Component Analysis (ICA) and the classification has been done by K -means clustering. The set of observable parameters includes structural parameters, spectroscopically determined Lick indices and radial velocities from the literature.We propose that GCs of NGC 5128 consist of two groups. One group originated in the original cluster formation event that coincided with the formation of the elliptical galaxy and the other group emerged from an accreted spiral galaxy. This is unlike the previous result (Chattopadhyay et?al., 2009) which accounts for a third group originating from the accretion of tidally stripped dwarf galaxies.

UNCOVERING THE FORMATION OF ULTRACOMPACT DWARF GALAXIES BY MULTIVARIATE STATISTICAL ANALYSIS

We present a statistical analysis of the properties of a large sample of dynamically hot old stellar systems, from globular clusters (GCs) to giant ellipticals, which was performed in order to investigate the origin of ultracompact dwarf galaxies (UCDs). The data were mostly drawn from Forbes et al. We recalculated some of the effective radii, computed mean surface brightnesses and mass-to-light ratios, and estimated ages and metallicities. We completed the sample with GCs of M31. We used a multivariate statistical technique (K-Means clustering), together with a new algorithm (Gap Statistics) for finding the optimum number of homogeneous sub-groups in the sample, using a total of six parameters (absolute magnitude, effective radius, virial mass-to-light ratio, stellar mass-to-light ratio, and metallicity). We found six groups. FK1 and FK5 are composed of high- and low-mass elliptical galaxies, respectively. FK3 and FK6 are composed of high-metallicity and low-metallicity objects, respectively, and both include GCs and UCDs. Two very small groups, FK2 and FK4, are composed of Local Group dwarf spheroidals. Our groups differ in their mean masses and virial mass-to-light ratios. The relations between these two parameters are also different for the various groups. The probability density distributions of metallicity for the four groups of galaxies are similar to those of the GCs and UCDs. The brightest low-metallicity GCs and UCDs tend to follow the mass-metallicity relation like elliptical galaxies. The objects of FK3 are more metal-rich per unit effective luminosity density than high-mass ellipticals.

Horizontal Branch Morphology of Globular Clusters: A Multivariate Statistical Analysis

The proper interpretation of horizontal branch (HB) morphology is crucial to the understanding of the formation history of stellar populations. In the present study a multivariate analysis is used (principal component analysis) for the selection of appropriate HB morphology parameter, which, in our case, is the logarithm of effective temperature extent of the HB (log Teff HB). Then this parameter is expressed in terms of the most significant observed independent parameters of Galactic globular clusters (GGCs) separately for coherent groups, obtained in a previous work, through a stepwise multiple regression technique. It is found that, metallicity ([Fe/H]), central surface brightness (μv), and core radius (rc) are the significant parameters to explain most of the variations in HB morphology (multiple R 2 ∼ 0.86) for GGC elonging to the bulge/disk while metallicity ([Fe/H]) and absolute magnitude (Mv) are responsible for GGC belonging to the inner halo (multiple R 2 ∼ 0.52). The robustness is tested by taking 1000 bootstrap samples. A cluster analysis is performed for the red giant branch (RGB) stars of the GGC belonging to Galactic inner halo (Cluster 2). A multi-episodic star formation is preferred for RGB stars of GGC belonging to this group. It supports the asymptotic giant branch (AGB) model in three episodes instead of two as suggested by Carretta et al. for halo GGC while AGB model is suggested to be revisited for bulge/disk GGC.

Statistical Methods for Astronomical Data Analysis

Introduction to Astrophysics.- Introduction to Statistics.- Sources of Astronomical Data.- Statistical Inference.- Advanced Regression and its Application with Measurement Error.- Missing Observations and Imputation.- Dimension Reduction and Clustering.- Clustering, Classification and Data Mining.- Time Series Analysis.- Monte Carlo Simulation.- Uses of Softwares.- Appendix.

Globular Clusters in the Milky Way and Dwarf Galaxies: A Distribution-Free Statistical Comparison

It has been found that globular clusters (GCs) in dwarf galaxies and those in the Milky Way (MW) outer halo mostly have the same parent distributions, while GCs in the MW disk and inner halo have a different origin from those in dwarf galaxies. Thus, these dwarf galaxies did not play a crucial role in the formation of the Galactic disk or inner halo. In order to investigate this phenomenon in a more objective manner, a statistical comparison of the GCs of our Galaxy and those of neighboring dwarf galaxies has been carried out by a multivariate nonparametric method. For the various parameters of GCs in the MW and in dwarf galaxies, the multivariate Gaussian assumption fails, so a nonparametric method of comparison (instead of multivariate analysis of variance [MANOVA]) has been chosen. The test is performed on GCs of the MW disk, inner halo, and outer halo separately, with GCs from neighboring dwarf galaxies Canis Major, Fornax, and Sculptor, and the LMC dwarf irregular galaxy. The test is also performed for GCs from dwarf spheroidal galaxies in the neighborhood of M31: M33, NGC 147, NGC 185, and NGC 205.

COSMIC HISTORY OF THE INTEGRATED GALACTIC STELLAR INITIAL MASS FUNCTION: A SIMULATION STUDY

ABSTRACT Theoretical as well as observational studies suggest that the stellar initial mass func-tion (IMF) might become top heavy with increasing redshift. Embedded cluster massfunction is a power law having index β, whose value still remains controversial. Inthe present work, we investigate the effect of evolving IMF and varying indices ofβ for the integrated galactic initial mass function, in relation to several measures ofstar formation rates of galaxies at various redshifts by random simulation. The result-ing IGIMF is segmented power law at various redshifts having slopes α 1,IGIMF andα 2,IGIMF with a turnover at a characteristic mass m c ′ . These differ from the stel-lar initial mass functions with slopes α 1,IMF , α 2,IMF , and characteristic masses m c for different values of redshift z, β, minimum and maximum masses of the embeddedclusters.Key words: galaxies:star clusters: general - galaxies:evolution 1 INTRODUCTIONThe form of stellar initial mass function is of considerabledebate in the present era as it describes the nature ofstellar population, the ratio of high mass to low mass starsand influences the dynamical evolution of star clusters aswell as star formation history of the whole galaxy. Usuallyit is derived using observed luminosity function togetherwith an assumed mass-to-light ratio for the stars underconsideration. Generally, IMFs, as suggested by variousauthors, are either of Salpeter type (Salpeter 1955) orconsists of segmented power laws (Scalo 1986; Kroupa 2001;Chabrier 2003) with a turnover at some characteristic massm