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Simultaneous Chandra X-ray, HST Ultraviolet, and Ulysses Radio Observations of Jupiter's Aurora

Observations of Jupiter carried out by the Chandra ACIS-S instrument over 24-26 February, 2003, show that the auroral X-ray spectrum consists of line emission consistent with high-charge states of precipitating ions, and not a continuum as might be expected from bremsstrahlung. The part of the spectrum due to oxygen peaks around 650 eV, which indicates a high fraction of fully-stripped oxygen in the precipitating ion flux. A combination of the OVIII emission lines at 653 eV and 774 eV, as well as the OVII emission lines at 561 eV and 666 eV, are evident in the measure auroral spectrum. There is also line emission at lower energies in the spectral region extending from 250 to 350 eV, which could be from sulfur and/or carbon. The Jovian auroral X- ray spectra are significantly different from the X-ray spectra of comets. The charge state distribution of the oxygen ions implied by the measured auroral X-ray spectra strongly suggests that, independent of the source of the energetic ions - magnetospheric or solar wind - the ions have undergone additional acceleration. This spectral evidence for ion acceleration is also consistent with the relatively high intensities of the X-rays compared to the available phase space density of the (unaccelerated) source populations of solar wind or magnetospheric ions at Jupiter, which are orders of magnitude too small to explain the observed emissions. The Chandra X-ray observations were executed simultaneously with observations at ultraviolet wavelengths by the Hubble Space Telescope and at radio wavelengths by the Ulysses spacecraft. These additional data sets suggest that the source of the X-rays is magnetospheric in origin, and that the precipitating particles are accelerated by strong field-aligned electric fields, which simultaneously create both the several-MeV energetic ion population and the relativistic electrons observed in situ by Ulysses that are correlated with approximately 40 minute quasi-periodic radio outbursts.

Elsner, R. F.↗

XMM-Newton Observations of the 2003 X-Ray Minimum of Eta Carinae

The XMM-Newton X-ray observatory took part in the multi-wavelength observing campaign of the massive, evolved star Eta Carinae in 2003 during its recent X-ray minimum in June 2003. This paper reports on the first results of these observations, which were performed (1) before the minimum (five times in January, 2003), (2) near the X-ray maximum just before the minimum (two times in June) and (3) during the minimum (four times in July-August). Hard X-ray emission from the point source of Eta Carinae was detected even during the minimum. The observed flux above 3 keV was approx. 3x10(exp -12) ergs cm(exp -2)/s, which is about one percent of the flux before the minimum. Light curves from the individual observations show no time variability on the scale of a few kilo-seconds. Changes in the spectral shape occurred, but these changes were smaller than expected if the minimum is produced solely by an increase of hydrogen column density. Fits of the hard X-Ray source by an absorbed 1T model show a constant plasma temperature at around 5 keV and an increase of column density from 5x10(exp 22) cm(exp -2) to 2x10(exp 23) cm(exp -2). The spectra below 6 keV significantly deviate from the models that fit the higher energy emission. The X-ray minimum seems to be dominated by an apparent decrease of the emission measure, suggesting that the brightest part of the X-ray emitting region is completely obscured during the minimum in the form of an eclipse. Partial covering plasma emission models might be considered for the spectral variation. The spectra also showed strong iron K line emission from both hot and cold gases, and weak line emission from Ni, Ca, Ar, S and Si.

Hamaguchi, K.↗

Calculating the X-Ray Fluorescence from the Planet Mercury Due to High-Energy Electrons

The least-studied terrestrial planet is Mercury due to its proximity to the Sun, which makes telescopic observations and spacecraft encounters difficult. Our lack of knowledge about Mercury should change in the near future due to the recent launching of MESSENGER, a Mercury orbiter. Another mission (BepiColombo) is currently being planned. The x-ray spectrometer on MESSENGER (and planned for BepiColombo) can characterize the elemental composition of a planetary surface by measuring emitted fluorescent x-rays. If electrons are ejected from an atom s inner shell by interaction with energetic particles such as photons, electrons, or ions, electrons from an outer shell can transfer to the inner shell. Characteristic x-rays are then emitted with energies that are the difference between the binding energy of the ion in its excited state and that of the ion in its ground state. Because each element has a unique set of energy levels, each element emits x-rays at a unique set of energies. Electrons and ions usually do not have the needed flux at high energies to cause significant x-ray fluorescence on most planetary bodies. This is not the case for Mercury where high-energy particles were detected during the Mariner 10 flybys. Mercury has an intrinsic magnetic field that deflects the solar wind, resulting in a bow shock in the solar wind and a magnetospheric cavity. Electrons and ions accelerated in the magnetosphere tend to follow its magnetic field lines and can impact the surface on Mercury s dark side Modeling has been done to determine if x-ray fluorescence resulting from the impact of high-energy electrons accelerated in Mercury's magnetosphere can be detected by MESSENGER. Our goal is to understand how much bulk chemical information can be obtained from x-ray fluorescence measurements on the dark side of Mercury.

Burbine, T. H.↗

Scattering of Solar X-Rays by Jupiter and Saturn

Soft X-ray emission has been observed from the disk of both Jupiter and Saturn as well as from the auroral regions of these planets. The low-latitude disk emission as observed by ROSAT, the Chandra X-Ray Observatory, and XMM-Newton appears to be uniformly distributed across the disk and to be correlated with solar activity. These characteristics suggest that the source of the disk x-rays are: (1) elastic scattering of solar X-rays by atmospheric neutrals (2) absorption of solar X-rays in the carbon K-shell followed by fluorescent emission. The carbon atoms are found in methane molecules located below the homopause. In this paper we present the results of calculations of the scattering albedo and of the emitted x-ray intensity for a range of atmospheric abundances and for a number of solar irradiance spectra. The model calculations are compared with recent x-ray observations of Jupiter and Saturn.

Cravens, T. E.↗

Discovery of Oxygen Kalpha X-ray Emission from the Rings of Saturn

Using the Advanced CCD Imaging Spectrometer (ACIS), the Chandra X-ray Observatory (CXO) observed the Saturnian system for one rotation of the planet (approx.37 ks) on 20 January, 2004, and again on 26-27 January, 2004. In this letter we report the detection of X-ray emission from the rings of Saturn. The X-ray spectrum from the rings is dominated by emission in a narrow (approx.130 eV wide) energy band centered on the atomic oxygen Ka fluorescence line at 0.53 keV. The X-ray power emitted from the rings in the 0.49-0.62 keV band is about one-third of that emitted from Saturn disk in the photon energy range 0.24-2.0 keV. Our analysis also finds a clear detection of X-ray emission from the rings in the 0.49-0.62 keV band in an earlier (14-15 April, 2003) Chandra ACIS observation of Saturn. Fluorescent scattering of solar X-rays from oxygen atoms in the H20 icy ring material is the likely source mechanism for ring X-rays, consistent with the scenario of solar photo-production of a tenuous ring oxygen atmosphere and ionosphere recently discovered by Cassini.

Bhardwaj, Anil↗

Numerical Model for Cosmic Rays Species Production and Propagation in the Galaxy

In recent years, considerable progress has been made in studying the propagation and origin of cosmic rays, as new and more accurate data have become available. Many models have been developed to study cosmic ray interactions and propagation showed flexibility in resembling various astrophysical conditions and good agreement with observational data. However, some astrophysical problems cannot be addressed using these models, such as the stochastic nature of the cosmic rays source, small-scale structures and inhomogeneities in the interstellar gas that can affect radioactive secondary abundance in cosmic rays. We have developed a new model and a corresponding computer code that can address some of these limitations. The model depends on the expansion of the backward stochastic solution of the general diffusion transport equation (Zhang 1999) starting from an observer position to solve a group of diffusion transport equations each of which represents a particular element or isotope of cosmic ray nuclei. In this paper we are focusing on key abundance ratios such as B/C, sub-Fe/Fe, (10)Be/(9)Be, (26)Al/(27)Al, (36)Cl/(37)Cl and (54)Mn/(55)Mn, which all have well established cross sections, to evaluate our model. The effect of inhomogeneity in the interstellar medium is investigated. The contribution of certain cosmic ray nuclei to the production of other nuclei is addressed. The contribution of various galactic locations to the production of cosmic ray nuclei observed at solar system is also investigated.

Farahat, Ashraf↗

Nondestructive Evaluation of Advanced Materials with X-ray Phase Mapping

X-ray radiation has been widely used for imaging applications since Rontgen first discovered X-rays over a century ago. Its large penetration depth makes it ideal for the nondestructive visualization of the internal structure and/or defects of materials unobtainable otherwise. Currently used nondestructive evaluation (NDE) tools, X-ray radiography and tomography, are absorption-based, and work well in heavy-element materials where density or composition variations due to internal structure or defects are high enough to produce appreciable absorption contrast. However, in many cases where materials are light-weight and/or composites that have similar mass absorption coefficients, the conventional absorption-based X-ray methods for NDE become less useful. Indeed, the light-weight and ultra-high-strength requirements for the most advanced materials used or developed for current flight mission and future space exploration pose a great challenge to the standard NDE tools in that the absorption contrast arising from the internal structure of these materials is often too weak to be resolved. In this presentation, a solution to the problem, the use of phase information of X-rays for phase contrast X-ray imaging, will be discussed, along with a comparison between the absorption-based and phase-contrast imaging methods. Latest results on phase contrast X-ray imaging of lightweight Space Shuttle foam in 2D and 3D will be presented, demonstrating new opportunities to solve the challenging issues encountered in advanced materials development and processing.

Hu, Zhengwei↗

"Short, Hard Gamma-Ray Bursts - Mystery Solved?????"

After over a decade of speculation about the nature of short-duration hard-spectrum gamma-ray bursts (GRBs), the recent detection of afterglow emission from a small number of short bursts has provided the first physical constraints on possible progenitor models. While the discovery of afterglow emission from long GRBs was a real breakthrough linking their origin to star forming galaxies, and hence the death of massive stars, the progenitors, energetics, and environments for short gamma-ray burst events remain elusive despite a few recent localizations. Thus far, the nature of the host galaxies measured indicates that short GRBs arise from an old (> 1 Gyr) stellar population, strengthening earlier suggestions and providing support for coalescing compact object binaries as the progenitors. On the other hand, some of the short burst afterglow observations cannot be easily explained in the coalescence scenario. These observations raise the possibility that short GRBs may have different or multiple progenitors systems. The study of the short-hard GRB afterglows has been made possible by the Swift Gamma-ray Burst Explorer, launched in November of 2004. Swift is equipped with a coded aperture gamma-ray telescope that can observe up to 2 steradians of the sky and can compute the position of a gamma-ray burst to within 2-3 arcmin in less than 10 seconds. The Swift spacecraft can slew on to this burst position without human intervention, allowing its on-board x ray and optical telescopes to study the afterglow within 2 minutes of the original GRB trigger. More Swift short burst detections and afterglow measurements are needed before we can declare that the mystery of short gamma-ray burst is solved.

Parsons, A.↗

X-ray Mapping of Terrestrial and Extraterrestrial Materials Using the Electron Microprobe

Lunar samples returned from the Apollo program motivated development of the Bence-Albee algorithm for the rapid and accurate analysis of lunar materials, and established interlaboratory comparability through its common use. In the analysis of mineral and rock fragments it became necessary to combine micro- and macroscopic analysis by coupling electron-probe microanalysis (EPMA) with automated stage point counting. A coarse grid that included several thousand points was used, and initially wavelength-dispersive (WDS) and later energydispersive (EDS) data were acquired at discrete stage points using approx. 5 sec count times. A approx 50 micrometer beam diameter was used for WDS and up to 500 micrometer beam diameter for EDS analysis. Average analyses of discretely sampled phases were coupled with the point count data to calculate the bulk composition using matrix algebra. Use of a defocused beam resulted in a contribution from multiple phases to each analytical point, and the analytical data were deconvolved relative to end-member phase chemistry on the fly. Impressive agreement was obtained between WDS and EDS measurements as well as comparison with bulk chemistry obtained by other methods. In the 30 years since these methods were developed, significant improvements in EPMA automation and computer processing have taken place. Digital beam control allows routine collection of x-ray maps by EDS, and stage mapping for WDS is conducted continuously at slew speed and incrementally by sampling at discrete points. Digital pulse processing in EDS systems has significantly increased the throughput for EDS mapping, and the ongoing development of Si-drift detector systems promises mapping capabilities rivaling WDS systems. Spectrum imaging allows a data cube of EDS spectra to be acquired and sophisticated processing of the original data is possible using matrix algebra techniques. The study of lunar and meteoritic materials includes the need to conveniently: (1) Characterize the sample at microscopic and macroscopic scales with relatively high sensitivity, (2) Determine the modal abundance of minerals, and (3) Identify and relocate discrete features of interest in terms of size and chemistry. The coupled substitution of cations in minerals can result in significant variation in mineral chemistry, but at similar average Z, leading to poor backscattered-electron (BSE) contrast discrimination of mineralogy. It is necessary to discriminate phase chemistry at both the trace element level and the major element level. To date, the WDS of microprobe systems is preferred for mapping due to high throughput and the ability to obtain the necessary intensity to discriminate phases at both trace and major element concentrations. It is desirable to produce fully quantitative compositional maps of geological materials, which requires the acquisition of k-ratio maps that are background and dead-time corrected, and which have been corrected by phi(delta z> or an equivalent algorithm at each pixel. To date, turnkey systems do not allow the acquisition of k-ratio maps and the rigorous correction in this manner. X-ray maps of a chondrule from the Ourique meteorite, and a comb-layered xenolith from the San Francisco volcanic field, have been analyzed and processed to extract phase information. The Ourique meteorite presents a challenge due to relatively low BSE contrast, and has been studied using spectrum imaging. X-ray maps for Si, Mg, and FeK(alpha) were used to produce RGB images. The xenolith sample contains sector-zoned augite, olivine, plagioclase, and basaltic glass. X-ray maps were processed using Lispix and ImageJ software to produce mineral phase maps. The x-ray maps for Mg, Ca, and Ti were used with traceback to generate binary images that were converted to RGB images. These approaches are successful in discriminating phases, but it is desirable to achieve the methods that were used on lunar samples 30 years ago on current microprobe systems. Curnt research includes x-ray mapping analysis of the Dalgety Downs chondrite by micro x-ray fluorescence and spectrum imaging, in collaboration with Kenny Witherspoon of IXRF Systems and Dale Newbury of NIST.

Carpenter, P.↗

Simultaneous Chandra X ray, Hubble Space Telescope Ultraviolet, and Ulysses Radio Observations of Jupiter's Aurora

Observations of Jupiter carried out by the Chandra Advanced CCD Imaging Spectrometer (ACIS-S) instrument over 24-26 February 2003 show that the auroral X-ray spectrum consists of line emission consistent with high-charge states of precipitating ions, and not a continuum as might be expected from bremsstrahlung. The part of the spectrum due to oxygen peaks around 650 eV, which indicates a high fraction of fully stripped oxygen in the precipitating ion flux. A combination of the OVIII emission lines at 653 eV and 774 eV, as well as the OVII emission lines at 561 eV and 666 eV, are evident in the measure auroral spectrum. There is also line emission at lower energies in the spectral region extending from 250 to 350 eV, which could be from sulfur and/or carbon. The Jovian auroral X-ray spectra are significantly different from the X-ray spectra of comets. The charge state distribution of the oxygen ions implied by the measured auroral X-ray spectra strongly suggests that independent of the source of the energetic ions, magnetospheric or solar wind, the ions have undergone additional acceleration. This spectral evidence for ion acceleration is also consistent with the relatively high intensities of the X rays compared with the available phase space density of the (unaccelerated) source populations of solar wind or magnetospheric ions at Jupiter, which are orders of magnitude too small to explain the observed emissions. The Chandra X-ray observations were executed simultaneously with observations at ultraviolet wavelengths by the Hubble Space Telescope and at radio wavelengths by the Ulysses spacecraft. These additional data sets suggest that the source of the X rays is magnetospheric in origin and that the precipitating particles are accelerated by strong field-aligned electric fields, which simultaneously create both the several-MeV energetic ion population and the relativistic electrons observed in situ by Ulysses that are correlated with approx.40 min quasi-periodic radio outbursts.

Elsner, R. F.↗

Canadian Led X-ray Polarimeter Mission CXP

We propose a Canadian-led X-ray Polarimetry Mission (CXP), to include a scattering X-ray Polarimeter and sensitive All-Sky X-ray Monitor (ASXM). Polarimetry would provide a new observational window on black holes, neutron stars, accretion disks and jets, and the ASXM would offer sensitive monitoring of the volatile X-ray sky. The envisioned polarimeter consists of a hollow scattering beryllium cone surrounded by an annular proportional counter, in a simple and elegant design that is reliable and low-risk. It would be sensitive in the 6-30 keV band to approx. 3% polarization in approx. 30 Galactic sources and 2 AGN in a baseline 1-yr mission, and have sensitivity greater than 10 times that of the previous X-ray polarimeter flown (NASA's OSO-8, 1975-78) for most sources. This X-ray polarimeter would tackle questions like, Do black holes spin?, How do pulsars pulse?, What is the geometry of the magnetic field in accreting neutron stars? Where and how are jets produced in microquasars and AGN?, What are the geometries of many of the most famous accretion-disk systems in the sky? This will be done using a novel and until-now unexploited technique that will greatly broaden the available observational phase space of compact objects by adding to timing and spectroscopy observations of polarization fraction and position angle as a function of energy. The All-Sky X-ray Monitor would scan for transients, both as potential targets for the polarimeter but also as a service to the worldwide astronomical community. The entire CXP mission could be flown for $40- 60M CDN, according to estimates by ComDev International, and could be built entirely in Canada. It would fall well within the CSA's SmallSat envelope and would empower the growing and dynamic Canadian High-Energy Astrophysics community with world leadership in a potentially high impact niche area.

Kaspi, V.↗

Gamma-ray Observations with Swift and their Impact on the TeV Community

The Swift gamma-ray burst explorer was launched on Nov. 20, 2004 from Cape Canaveral, Florida. The first instrument onboard became fully operational less than a month later. Since that time the Burst Alert Telescope (BAT) on Swift has detected more than 150 gamma-ray bursts (GRBs), most of which have also been observed within two minutes by the Swift narrow-field instruments: the X-Ray Telescope (XRT) and the Ultra-Violet and Optical Telescope (UVOT). Swift trigger notices are distributed worldwide within seconds of the trigger through the Gamma-ray burst Coordinates Network (GCN) and a substantial fraction of GRBs have been followed up by ground and space-based telescopes, ranging in wavelength from radio to TeV. Results have included the first rapid localization of a short GRB and further validation of the theory that short and long bursts have different origins; detailed observations of the power-law decay of burst afterglows leading to an improved understanding of the fireball and afterglow models; and detection of the most distant GRB ever found. Swift is also a sensitive X-ray observatory with capabilities to monitor galactic and extragalactic transients on a daily basis, carry out the first all-sky hard X-ray survey since HEAO-1, and study in detail the spectra of X-ray transients. The talk will emphasize the connection between Swift/BAT GRB observations and source monitoring and TeV observations.

Krimm, Hans↗

INTEGRAL and XMM-Newton Observations of the X-Ray Pulsar IGR J16320-4751/AX J1691.9-4752

We report on observations of the X-ray pulsar IGR J16320-4751 (also known as AX J1631.9-4752) performed simultaneously with International Gamma-Ray Astrophysics Laboratory (INTEGRAL) and XMM-Newton. We refine the source position and identify the most likely infrared counterpart. Our simultaneous coverage allows us to confirm the presence of X-ray pulsations at approximately 1300 s, that we detect above 20 keV with INTEGRAL for the first time. The pulse fraction is consistent with being constant with energy, which is compatible with a model of polar accretion by a pulsar. We study the spectral properties of IGR J16320-4751 during two major periods occurring during the simultaneous coverage with both satellites, namely a flare and a non-flare period. We detect the presence of a narrow 6.4 keV iron line in both periods. The presence of such a feature is typical of supergiant wind accretors such as Vela X-1 or GX 301-2. We inspect the spectral variations with respect to the pulse phase during the non-flare period, and show that the pulse is solely due to variations of the X-ray flux emitted by the source and not due to variations of the spectral parameters. Our results are therefore compatible with the source being a pulsar in a High Mass X-ray Binary. We detect a soft excess appearing in the spectra as a blackbody with a temperature of approximately 0.07 keV. We discuss the origin of the X-ray emission in IGR J16320-4751: while the hard X-rays are likely the result of Compton emission produced in the close vicinity of the pulsar, based on energy argument we suggest that the soft excess is likely the emission by a collisionally energized cloud in which the compact object is embedded.

Rodriquez, J.↗

GLAST and Ground-Based Gamma-Ray Astronomy

The launch of the Gamma-ray Large Area Space Telescope together with the advent of a new generation of ground-based gamma-ray detectors such as VERITAS, HESS, MAGIC and CANGAROO, will usher in a new era of high-energy gamma-ray astrophysics. GLAST and the ground based gamma-ray observatories will provide highly complementary capabilities for spectral, temporal and spatial studies of high energy gamma-ray sources. Joint observations will cover a huge energy range, from 20 MeV to over 20 TeV. The LAT will survey the entire sky every three hours, allowing it both to perform uniform, long-term monitoring of variable sources and to detect flaring sources promptly. Both functions complement the high-sensitivity pointed observations provided by ground-based detectors. Finally, the large field of view of GLAST will allow a study of gamma-ray emission on large angular scales and identify interesting regions of the sky for deeper studies at higher energies. In this poster, we will discuss the science returns that might result from joint GLAST/ground-based gamma-ray observations and illustrate them with detailed source simulations.

McEnery, Julie↗

Gamma-Ray Emission from the Broad-Line Radio Galaxy 3C 111

The broad-line radio galaxy 3C 111 has been suggested as the counterpart of the y-ray source 3EG J0416+3650. While 3C 111 meets most of the criteria for a high-probability identification, like a bright flat-spectrum radio core and a blazar-like broadband SED, in the Third EGRET Catalog, the large positional offset of about 1.5' put 3C 111 outside the 99% probability region for 3EG J0416+3650, making this association questionable. We present a re-analysis of all available archival data for 3C 111 from the EGRET archives, resulting in detection of variable hard-spectrum high-energy gamma-ray emission above 1000 MeV from a position close to the nominal position of 3C 111, in three separate viewing periods (VPs), at a 3sigma level in each. A second variable hard-spectrum source is present nearby. At >100 MeV, one variable soft-spectrum source seems to account for most of the EGRET-detected emission of 3EG J0416+3650. A follow-up Swift UVOT/XRT observation reveals one moderately bright X-ray source in the error box of 3EG J0416+3650, but because of the large EGRET position uncertainty, it is not certain that the X-ray and gamma-ray sources are associated. Another Swift observation near the second (unidentified) hard gamma-ray source detected no X-ray source nearby.

Hartman, Robert C.↗

The X-ray Variability of Eta Car, 1996-2010

X-ray photometry in the 2-10 keY band of the the supermassive binary star Eta Car has been measured with the Rossi X-ray Timing Explorer from 1996-2010. The ingress to X-ray minimum is consistent with a period of 2024 days. The 2009 X-ray minimum began on January 162009 and showed an unexpectedly abrupt recovery starting after 12 Feb 2009. The X-ray colors become harder about half-way through all three minima and continue until flux recovery. The behavior of the fluxes and X-ray colors for the most recent X-ray minimum, along with Chandra high resolution grating spectra at key phases suggests a significant change in the inner wind of Eta Car, a possible indicator that the star is entering a new unstable phase of mass loss.

Corcoran, Michael F.↗

X-Ray Optics: Past, Present, and Future

X-ray astronomy started with a small collimated proportional counter atop a rocket in the early 1960s. It was immediately recognized that focusing X-ray optics would drastically improve both source location accuracy and source detection sensitivity. In the past 5 decades, X-ray astronomy has made significant strides in achieving better angular resolution, large photon collection area, and better spectral and timing resolutions, culminating in the three currently operating X-ray observatories: Chandra, XMM/Newton, and Suzaku. In this talk I will give a brief history of X-ray optics, concentrating on the characteristics of the optics of these three observatories. Then I will discuss current X-ray mirror technologies being developed in several institutions. I will end with a discussion of the optics for the International X-ray Observatory that I have been developing at Goddard Space Flight Center.

Zhang, William W.↗

Optimizing X-Ray Optical Prescriptions for Wide-Field Applications

X-ray telescopes with spatial resolution optimized over the field of view (FOV) are of special interest for missions, such as WFXT, focused on moderately deep and deep surveys of the x-ray sky, and for solar x-ray observations. Here we report on the present status of an on-going study of the properties of Wolter I and polynominal grazing incidence designs with a view to gain a deeper insight into their properties and simply the design process. With these goals in mind, we present some results in the complementary topics of (1) properties of Wolter I x-ray optics and polynominal x-ray optic ray tracing. Of crucial importance for the design of wide-field x-ray optics is the optimization criteria. Here we have adopted the minimization of a merit function, M, which measures the spatial resolution averaged over the FOV: M= ((integral of d phi) between the limits of 0 and 2 pi) (integral of d theta theta w(theta) sigma square (theta,phi) between the limits of 0 and theta(sub FOV)) (integral of d phi between the limits of 0 and phi/4) (Integral of d theta theta w(theta) between the limits of 0 and theta(sub FOV) where w(theta(sub 1) is a weighting function and Merit function: sigma-square (theta, phi) = summation of (x,y,z) [ -<(x,y,z)> (exp 2)] is the spatial variance for a point source on the sky at polar and azimuthal off-axis angles (theta,phi).

Elsner, R. F.↗