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From the Cluster Temperature Function to the Mass Function at Low Z

The project consists of three X-ray Multimirror Mission (XMM) observations of the hot, nearby galaxy cluster Triangulum Australis (one pointing to the cluster center and two offsets). The major goal of the project is to obtain an accurate radial gas temperature profile out to large radii and derive the total gravitating mass within the radius of over density 500. A secondary goal is to obtain a detailed two-dimensional gas temperature map of the central region. We have recently received the central pointing (of the two CDROMs received, one contains the central pointing and the other, a zero-exposure dummy observation). The analysis of the data at hand is underway with the aim of deriving the two-dimensional temperature map. At present, we have to limit quantitative analysis to the EPIC-MOS data because of the absence of the necessary calibration products for the Extended Full Frame mode used for EPIC-pn in these observations. We anticipate the arrival of the data from the remaining two offset pointings in the near future, after which we will be able to undertake the total mass derivation. Money from this year's installment of the grant was partially spent on an upgrade of the computer used for the above data analysis, and on travel to a conference that extensively discussed the XMM calibration issues.

Markevitch, Maxim

From the Cluster Temperature Function to the Mass Function at Low Z

This XMM project consisted of three observations of the nearby, hot galaxy cluster Triangulum Australis, one of the cluster center and two offsets. The goal was to measure the radial gas temperature profile out to large radii and derive the total gravitating mass within the radius of average mass overdensity 500. The central pointing also provides data for a detailed two-dimensional gas temperature map of this interesting cluster. We have analyzed all three observations. The derivation of the temperature map using the central pointing is complete, and the paper is soon to be submitted. During the course of this study and of the analysis of archival XMM cluster observations, it became apparent that the commonly used XMM background flare screening techniques are often not accurate enough for studies of the cluster outer regions. The information on the cluster's total masses is contained at large off-center distances, and it is precisely the temperatures for those low-brightness regions that are most affected by the detector background anomalies. In particular, our two offset observations of the Triangulum have been contaminated by the background flares ("bad cosmic weather") to a degree where they could not be used for accurate spectral analysis. This forced us to expand the scope of our project. We needed to devise a more accurate method of screening and modeling the background flares, and to evaluate the uncertainty of the XMM background modeling. To do this, we have analyzed a large number of archival EPIC blank-field and closed-cover observations. As a result, we have derived stricter background screening criteria. It also turned out that mild flares affecting EPIC-pn can be modeled with an adequate accuracy. Such modeling has been used to derive our Triangulum temperature map. The results of our XMM background analysis, including the modeling recipes, are presented in a paper which is in final preparation and will be submitted soon. It will be useful not only for our future analysis but for other XMM cluster observations as well.

Mushotzky, Richard

Development of the Functional Flow Block Diagram for the J-2X Rocket Engine System

The J-2X program calls for the upgrade of the Apollo-era Rocketdyne J-2 engine to higher power levels, using new materials and manufacturing techniques, and with more restrictive safety and reliability requirements than prior human-rated engines in NASA history. Such requirements demand a comprehensive systems engineering effort to ensure success. Pratt & Whitney Rocketdyne system engineers performed a functional analysis of the engine to establish the functional architecture. J-2X functions were captured in six major operational blocks. Each block was divided into sub-blocks or states. In each sub-block, functions necessary to perform each state were determined. A functional engine schematic consistent with the fidelity of the system model was defined for this analysis. The blocks, sub-blocks, and functions were sequentially numbered to differentiate the states in which the function were performed and to indicate the sequence of events. The Engine System was functionally partitioned, to provide separate and unique functional operators. Establishing unique functional operators as work output of the System Architecture process is novel in Liquid Propulsion Engine design. Each functional operator was described such that its unique functionality was identified. The decomposed functions were then allocated to the functional operators both of which were the inputs to the subsystem or component performance specifications. PWR also used a novel approach to identify and map the engine functional requirements to customer-specified functions. The final result was a comprehensive Functional Flow Block Diagram (FFBD) for the J-2X Engine System, decomposed to the component level and mapped to all functional requirements. This FFBD greatly facilitates component specification development, providing a well-defined trade space for functional trades at the subsystem and component level. It also provides a framework for function-based failure modes and effects analysis (FMEA), and a rigorous baseline for the functional architecture.

White, Thomas

A Generalized, Compactly-Supported Correlation Function for Data Assimilation Applications

Correlation functions play an essential role in modern data assimilation, where they are used to model covariances given a set of tunable parameters or applied as tapering functions to localize covariances in ensemble-based schemes. One of the most widely-used correlation functions in data assimilation is the Gaspari and Cohn (1999) piecewise-rational, compactly-supported parametric correlation function (hereafter referred to as GC99). The GC99 correlation function is useful due to its tunable cut-off parameter c and Gaussian-like shape achieved when the parameter a is set to one-half. These properties are attractive for tapering functions in data assimilation applications. However, the GC99 correlation function is homogeneous over Euclidean 3-space and isotropic when restricted to the sphere, properties that may be less than ideal for some geophysical applications. GC99 is also compactly-supported on a sphere of fixed radius, which requires tuning of the cut-off parameter c that can depend on the specific application. This work presents a generalization of the GC99 correlation function that allows the cut-off parameter c and shape parameter a to vary over space to gain more flexibility in shape while maintaining its compact support property. The function, which we call the Generalized Gaspari Cohn (GenGC) correlation function, introduces inhomogeneity in Euclidean 3-space and anisotropy when restricted to the sphere by allowing both parameters c and a to vary, as functions, over the spatial domain. The GC99 correlation function is a special case of GenGC where the functions c and a are held constant, as fixed parameters rather than functions. The GenGC correlation function also generalizes the follow-on to the work of Gaspari and Cohn (1999) presented in Gaspari et al. (2006), which allowed a to vary while keeping c fixed. We illustrate through simple one- and two-dimensional examples the variety of inhomogeneous and anisotropic correlation functions GenGC can produce by varying c and a over space, and suggest applications where they may be useful in data assimilation, such as covariance modeling or localization. In particular, we describe how the GenGC correlation function can be used to construct covariances using correlation length and variance fields derived from dynamics. For example, the correlation length field for advective dynamics is governed by a partial differential equation (PDE) in N spatial dimensions, where N is the number of space dimensions of the state. Correlation length fields can be determined from this PDE and used with GenGC to construct the corresponding correlations. We can then approximate the full covariance by rescaling by the variance, which also satisfies a PDE in N spatial dimensions for advective dynamics. Thus we can approximate the full covariance without solving the covariance PDE, which is in 2N spatial dimensions, by solving just two PDEs each in only N spatial dimensions. This approach to evolving the correlation length and variance fields, then reconstructing the correlations using GenGC, is suggested as an alternative to current methods of covariance modeling in data assimilation algorithms.

GC99