On understanding the figures of merit for detection and measurement of x-ray polarization

The prospects for accomplishing X-ray polarization measurements appear to have grown in recent years after a more than 35-year hiatus. Unfortunately, this long hiatus has brought with it some confusion over the statistical uncertainties associated with polarization measurements of astronomical sources. The heart of this confusion stems from a misunderstanding (or potential misunderstanding) of a standard figure of merit-the minimum detectable polarization (MDP)-that one of us introduced many years ago. We review the relevant statistics, and quantify the differences between the MDP and the uncertainty of an actual polarization measurement. We discuss the implications for future missions.Comment: 5 pages, 2 figures, to be presented at SPIE conference 7732 (paper 13), corrected typo

Measuring x-ray polarization in the presence of systematic effects: Known background

The prospects for accomplishing x-ray polarization measurements of astronomical sources have grown in recent years, after a hiatus of more than 37 years. Unfortunately, accompanying this long hiatus has been some confusion over the statistical uncertainties associated with x-ray polarization measurements of these sources. We have initiated a program to perform the detailed calculations that will offer insights into the uncertainties associated with x-ray polarization measurements. Here we describe a mathematical formalism for determining the 1- and 2-parameter errors in the magnitude and position angle of x-ray (linear) polarization in the presence of a (polarized or unpolarized) background. We further review relevant statistics-including clearly distinguishing between the Minimum Detectable Polarization (MDP) and the accuracy of a polarization measurement.Comment: 12 pages, 4 figures, for SPIE conference proceeding

The Anderson model of localization: a challenge for modern eigenvalue methods

We present a comparative study of the application of modern eigenvalue algorithms to an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for the large, sparse, real, symmetric, and indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation of Cullum and Willoughby with the implicitly restarted Arnoldi method coupled with polynomial and several shift-and-invert convergence accelerators as well as with a sparse hybrid tridiagonalization method. We demonstrate that for our problem the Lanczos implementation is faster and more memory efficient than the other approaches. This seemingly innocuous problem presents a major challenge for all modern eigenvalue algorithms.Comment: 16 LaTeX pages with 3 figures include

Methods of optimizing X-ray optical prescriptions for wide-field applications

We are working on the development of a method for optimizing wide-field X-ray telescope mirror prescriptions, including polynomial coefficients, mirror shell relative displacements, and (assuming 4 focal plane detectors) detector placement along the optical axis and detector tilt. With our methods, we hope to reduce number of Monte-Carlo ray traces required to search the multi-dimensional design parameter space, and to lessen the complexity of finding the optimum design parameters in that space. Regarding higher order polynomial terms as small perturbations of an underlying Wolter I optic design, we begin by using the results of Monte-Carlo ray traces to devise trial analytic functions, for an individual Wolter I mirror shell, that can be used to represent the spatial resolution on an arbitrary focal surface. We then introduce a notation and tools for Monte-Carlo ray tracing of a polynomial mirror shell prescription which permits the polynomial coefficients to remain symbolic. In principle, given a set of parameters defining the underlying Wolter I optics, a single set of Monte-Carlo ray traces are then sufficient to determine the polymonial coefficients through the solution of a large set of linear equations in the symbolic coefficients. We describe the present status of this development effort.Comment: 14 pages, to be presented at SPIE conference 7732 (paper 93

Measurements with the Chandra X-Ray Observatory's flight contamination monitor

NASA's Chandra X-ray Observatory includes a Flight Contamination Monitor (FCM), a system of 16 radioactive calibration sources mounted to the inside of the Observatory's forward contamination cover. The purpose of the FCM is to verify the ground-to-orbit transfer of the Chandra flux scale, through comparison of data acquired during the ground calibration with those obtained in orbit, immediately prior to opening the Observatory's sun-shade door. Here we report results of these measurements, which place limits on the change in mirror--detector system response and, hence, on any accumulation of molecular contamination on the mirrors' iridium-coated surfaces.Comment: 7pages,8figures,for SPIE 4012, paper 7

Unbiased methods for removing systematics from galaxy clustering measurements

Measuring the angular clustering of galaxies as a function of redshift is a powerful method for extracting information from the three-dimensional galaxy distribution. The precision of such measurements will dramatically increase with ongoing and future wide-field galaxy surveys. However, these are also increasingly sensitive to observational and astrophysical contaminants. Here, we study the statistical properties of three methods proposed for controlling such systematics – template subtraction, basic mode projection, and extended mode projection – all of which make use of externally supplied template maps, designed to characterize and capture the spatial variations of potential systematic effects. Based on a detailed mathematical analysis, and in agreement with simulations, we find that the template subtraction method in its original formulation returns biased estimates of the galaxy angular clustering. We derive closed-form expressions that should be used to correct results for this shortcoming. Turning to the basic mode projection algorithm, we prove it to be free of any bias, whereas we conclude that results computed with extended mode projection are biased. Within a simplified setup, we derive analytical expressions for the bias and discuss the options for correcting it in more realistic configurations. Common to all three methods is an increased estimator variance induced by the cleaning process, albeit at different levels. These results enable unbiased high-precision clustering measurements in the presence of spatially varying systematics, an essential step towards realizing the full potential of current and planned galaxy surveys

Algebraic results for the values ϑ3(mτ)\vartheta_3(m\tau) and ϑ3(nτ)\vartheta_3(n\tau) of the Jacobi theta-constant

Let $\vartheta_3(\tau) = 1+2\sum_{\nu=1}^{\infty} e^{\pi i\nu^2\tau}$ denote the classical Jacobi theta-constant. We prove that the two values $\vartheta_3(m\tau)$ and $\vartheta_3(n\tau)$ are algebraically independent over $\mathbb{Q}$ for any $\tau$ in the upper half-plane such that $q=e^{\pi i\tau}$ is an algebraic number, where $m,n\geq2$ are distinct integers

Study of HST counterparts to Chandra X-ray sources in the Globular Cluster M71

We report on archival Hubble Space Telescope (HST) observations of the globular cluster M71 (NGC 6838). These observations, covering the core of the globular cluster, were performed by the Advanced Camera for Surveys (ACS) and the Wide Field Planetary Camera 2 (WFPC2). Inside the half-mass radius (r_h = 1.65') of M71, we find 33 candidate optical counterparts to 25 out of 29 Chandra X-ray sources while outside the half-mass radius, 6 possible optical counterparts to 4 X-ray sources are found. Based on the X-ray and optical properties of the identifications, we find 1 certain and 7 candidate cataclysmic variables (CVs). We also classify 2 and 12 X-ray sources as certain and potential chromospherically active binaries (ABs), respectively. The only star in the error circle of the known millisecond pulsar (MSP) is inconsistent with being the optical counterpart. The number of X-ray faint sources with L_x>4x10^{30} ergs/s (0.5-6.0 keV) found in M71 is higher than extrapolations from other clusters on the basis of either collision frequency or mass. Since the core density of M71 is relatively low, we suggest that those CVs and ABs are primordial in origin.Comment: 12 pages, 6 figures. Accepted for publication in Astronomy and Astrophysic