Massimiliano Bonamente

The Extreme-Ultraviolet Excess Emission of the Virgo and A1795 Clusters: Reobservation with in Situ Background Measurements

The Virgo and A1795 clusters of galaxies were reobserved by the Extreme Ultraviolet Explorer with in situ background measurements by pointing at small offsets. Earlier, a similar reobservational strategy applied to the cluster A2199 revealed that the background radial profile was consistent with a flat distribution, and therefore the original method of extracting cluster EUV signals by the subtraction of an asymptotically determined background was valid. It is shown here that the same conclusions hold for the current sample. A model of the background was obtained from its known properties and the in situ measurements, and the subtracted cluster fluxes remain in agreement with those reported in our discovery papers. They are also consistent with results from the most conservative procedure of direct point-to-point subtraction of the in situ background and proper error propagation, which still preserves the existence of the EUV excess and its rising radial trend. We present evidence that argues against the soft excess as due to peculiarities in the line-of-sight Galactic absorption. The data appear to favor a thermal origin of the emission.

ROSAT and BeppoSAX Evidence of Soft X-Ray Excess Emission in the Shapley Supercluster: A3571, A3558, A3560, and A3562

Excess soft X-ray emission in clusters of galaxies has so far been detected for sources that lie along lines-of-sight to very low Galactic HI column density (such as Coma, A1795, A2199 and Virgo, N_H 0.9-2.0 10^{20} cm-2). We show that the cluster soft excess emission can be investigated even at higher N_H, which provides an opportunity for investigating soft X-ray emission characteristics among a large number of clusters. The ROSAT PSPC analysis of some members of the Shapley concentration (A3571, A3558, A3560 and A3562, at N_H 4-4.5 10^{20} cm-2) bears evidence for excess emission in the 1/4 keV band. We were able to confirm the finding for the case of A3571 by a pointed SAX observation. Within the current sample the soft X-ray flux is again found to be consistently above the level expected from a hot virialized plasma. The data quality is however insufficient to enable a discrimination between alternative models of the excess low energy flux.

Statistics and analysis of scientific data

Theory of Probability.- Random Variables and Their Distribution.- Sum and Functions of Random Variables.- Estimate of Mean and Variance and Confidence Intervals.- Distribution Function of Statistics and Hypothesis Testing.- Maximum Likelihood Fit to a Two-Variable Dataset.- Goodness of Fit and Parameter Uncertainty.- Comparison Between Models.- Monte Carlo Methods.- Markov Chains and Monte Carlo Markov Chains.- A: Numerical Tables.- B: Solutions.

Estimate of Mean and Variance and Confidence Intervals

In this chapter we study the problem of estimating parameters of the distribution function of a random variable when N observations of the variable are available. We discuss methods that establish what sample quantities must be calculated to estimate the corresponding parent quantities. This establish a firm theoretical framework that justifies the definition of the sample variance as an unbiased estimator of the parent variance, and the sample mean as an estimator of the parent mean. One of these methods, the maximum likelihood method, will later be used in more complex applications that involve the fit of two–dimensional data and the estimation of fit parameters. The concepts introduced in this chapter constitute the core of the statistical techniques for the analysis of scientific data.

Three Fundamental Distributions: Binomial, Gaussian, and Poisson

There are three distributions that play a fundamental role in statistics. The binomial distribution describes the number of positive outcomes in binary experiments, and it is the “mother” distribution from which the other two distributions can be obtained. The Gaussian distribution can be considered as a special case of the binomial, when the number of tries is sufficiently large. For this reason, the Gaussian distribution applies to a large number of variables, and it is referred to as the normal distribution. The Poisson distribution applies to counting experiments, and it can be obtained as the limit of the binomial distribution when the probability of success is small.

Introduction to Markov Chains

The theory of Markov chains is rooted in the work of Russian mathematician Andrey Markov and has an extensive body of literature to establish its mathematical foundations. The availability of computing resources has recently made it possible to use Markov chains to analyze a variety of scientific data. Monte Carlo Markov chains are now one of the most popular methods of data analysis. This chapter presents the key mathematical properties of Markov chains, necessary to understand its implementation as Monte Carlo Markov chains.

Fitting Two-Variable Datasets with Bivariate Errors

The maximum likelihood method for the fit of a two-variable dataset described in Chap. 8 assumes that one of the variables (the independent variable X) has negligible errors. There are many applications where this assumption is not applicable and uncertainties in both variables must be taken into account. This chapter expands the treatment of Chap. 8 to the fit of a two-variable dataset with errors in both variables.

Mean, Median, and Average Values of Variables

The data analyst often faces the question of what is the “best” value to report from N measurements of a random variable. In this chapter we investigate the use of the linear average, the weighted average, the median and a logarithmic average that may be applicable when the variable has a log-normal distribution. The latter may be useful when a variable has errors that are proportional to their measurements, avoiding the inherent bias arising in the weighted average from measurements with small values and small errors. We also introduce a relative-error weighted average that can be used as an approximation for the logarithmic mean for log-normal distributions.

Functions of Random Variables and Error Propagation

Sometimes experiments do not directly measure the quantity of interest, but rather associated variables that can be related to the one of interest by an analytic function. It is therefore necessary to establish how we can infer properties of the interesting variable based on properties of the variables that have been measured directly. This chapter explains how to determine the probability distribution function of a variable that is function of other variables of known distribution, and how to measure its mean and variance, the latter usually referred to as error propagation formulas. We also establish two fundamental results of the theory of probability, the central limit theorem and the law of large numbers.

Maximum Likelihood Methods for Two-Variable Datasets

One of the most common tasks in the analysis of scientific data is to establish a relationship between two quantities. Many experiments feature the measurement of a quantity of interest as function of another control quantity that is varied as the experiment is performed. In this chapter we use the maximum likelihood method to determine whether a certain relationship between the two quantities is consistent with the available measurements and the best-fit parameters of the relationship. The method has a simple analytic solution for a linear function but can also be applied to more complex analytic functions.