Aneta Siemiginowska

STATISTICS, HANDLE WITH CARE: DETECTING MULTIPLE MODEL COMPONENTS WITH THE LIKELIHOOD RATIO TEST

The likelihood ratio test (LRT) and the related F-test, popularized in astrophysics by Eadie and coworkers in 1971, Bevington in 1969, Lampton, Margon, & Bowyer, in 1976, Cash in 1979, and Avni in 1978, do not (even asymptotically) adhere to their nominal χ2 and F-distributions in many statistical tests common in astrophysics, thereby casting many marginal line or source detections and nondetections into doubt. Although the above authors illustrate the many legitimate uses of these statistics, in some important cases it can be impossible to compute the correct false positive rate. For example, it has become common practice to use the LRT or the F-test to detect a line in a spectral model or a source above background despite the lack of certain required regularity conditions. (These applications were not originally suggested by Cash or by Bevington.) In these and other settings that involve testing a hypothesis that is on the boundary of the parameter space, contrary to common practice, the nominal χ2 distribution for the LRT or the F-distribution for the F-test should not be used. In this paper, we characterize an important class of problems in which the LRT and the F-test fail and illustrate this nonstandard behavior. We briefly sketch several possible acceptable alternatives, focusing on Bayesian posterior predictive probability values. We present this method in some detail since it is a simple, robust, and intuitive approach. This alternative method is illustrated using the gamma-ray burst of 1997 May 8 (GRB 970508) to investigate the presence of an Fe K emission line during the initial phase of the observation. There are many legitimate uses of the LRT and the F-test in astrophysics, and even when these tests are inappropriate, there remain several statistical alternatives (e.g., judicious use of error bars and Bayes factors). Nevertheless, there are numerous cases of the inappropriate use of the LRT and similar tests in the literature, bringing substantive scientific results into question.

ARE THE VARIATIONS IN QUASAR OPTICAL FLUX DRIVEN BY THERMAL FLUCTUATIONS

We analyze a sample of optical light curves for 100 quasars, 70 of which have black hole mass estimates. Our sample is the largest and broadest used yet for modeling quasar variability. The sources in our sample have z < 2.8, 1042 λL λ(5100 A) 1046, and 106 M BH/M ☉ 1010. We model the light curves as a continuous time stochastic process, providing a natural means of estimating the characteristic timescale and amplitude of quasar variations. We employ a Bayesian approach to estimate the characteristic timescale and amplitude of flux variations; our approach is not affected by biases introduced from discrete sampling effects. We find that the characteristic timescales strongly correlate with black hole mass and luminosity, and are consistent with disk orbital or thermal timescales. In addition, the amplitude of short-timescale variations is significantly anticorrelated with black hole mass and luminosity. We interpret the optical flux fluctuations as resulting from thermal fluctuations that are driven by an underlying stochastic process, such as a turbulent magnetic field. In addition, the intranight variations in optical flux implied by our empirical model are 0.02 mag, consistent with current microvariability observations of radio-quiet quasars. Our stochastic model is therefore able to unify both long- and short-timescale optical variations in radio-quiet quasars as resulting from the same underlying process, while radio-loud quasars have an additional variability component that operates on timescales 1 day.

Sherpa : a mission-independent data analysis application

The ever-increasing quality and complexity of astronomical data underscores the need for new and powerful data analysis applications. This need has led to the development of Sherpa, a modeling and fitting program in the CIAO software package that enables the analysis of multi-dimensional, multi-wavelength data. In this paper, we present an overview of Sherpas features, which include: support for a wide variety of input and output data formats, including the new Model Descriptor List (MDL) format; a model language which permits the construction of arbitrarily complex model expressions, including ones representing instrument characteristics; a wide variety of fit statistics and methods of optimization, model comparison, and parameter estimation; multi-dimensional visualization, provided by ChIPS; and new interactive analysis capabilities provided by embedding the S-Lang interpreted scripting language. We conclude by showing example Sherpa analysis sessions.

BAYESIAN ESTIMATION OF HARDNESS RATIOS: MODELING AND COMPUTATIONS

A commonly used measure to summarize the nature of a photon spectrum is the so-called hardness ratio, which compares the numbers of counts observed in different passbands. The hardness ratio is especially useful to distinguish between and categorize weak sources as a proxy for detailed spectral fitting. However, in this regime classical methods of error propagation fail, and the estimates of spectral hardness become unreliable. Here we develop a rigorous statistical treatment of hardness ratios that properly deals with detected photons as independent Poisson random variables and correctly deals with the non-Gaussian nature of the error propagation. The method is Bayesian in nature and thus can be generalized to carry out a multitude of source-population-based analyses. We verify our method with simulation studies and compare it with the classical method. We apply this method to real-world examples, such as the identification of candidate quiescent low-mass X-ray binaries in globular clusters and tracking the time evolution of a flare on a low-mass star.

Constraints on black hole growth, quasar lifetimes, and Eddington ratio distributions from the SDSS broad-line quasar black hole mass function

We present an estimate of the black hole mass function of broad-line quasars (BLQSOs) that self-consistently corrects for incompleteness and the statistical uncertainty in the mass estimates, based on a sample of 9886 quasars at 1 1 it is highly incomplete at M BH 109 M ☉ and L/L Edd 0.5. We estimate a lower limit on the lifetime of a single BLQSO phase to be t BL > 150 ± 15 Myr for black holes at z = 1 with a mass of M BH = 109 M ☉, and we constrain the maximum mass of a black hole in a BLQSO to be ~3 × 1010 M ☉. Our estimated distribution of BLQSO Eddington ratios peaks at L/L Edd ~ 0.05 and has a dispersion of ~0.4 dex, implying that most BLQSOs are not radiating at or near the Eddington limit; however, the location of the peak is subject to considerable uncertainty. The steep increase in number density of BLQSOs toward lower Eddington ratios is expected if the BLQSO accretion rate monotonically decays with time. Furthermore, our estimated lifetime and Eddington ratio distributions imply that the majority of the most massive black holes spend a significant amount of time growing in an earlier obscured phase, a conclusion which is independent of the unknown obscured fraction. These results are consistent with models for self-regulated black hole growth, at least for massive systems at z > 1, where the BLQSO phase occurs at the end of a fueling event when black hole feedback unbinds the accreting gas, halting the accretion flow.

Accretion Disk Model of Short-Timescale Intermittent Activity in Young Radio Sources

We associate the existence of short-lived compact radio sources with the intermittent activity of the central engine caused by a radiation pressure instability within an accretion disk. Such objects may constitute a numerous sub-class of Giga-Hertz Peaked Spectrum sources, in accordance with the population studies of radio-loud active galaxies, as well as detailed investigations of their radio morphologies. We perform the model computations assuming the viscosity parametrization as proportional to a geometrical mean of the total and gas pressure. The implied timescales are consistent with the observed ages of the sources. The duration of an active phase for a moderate accretion rate is short enough (< 10 3 − 10 4 years) that the ejecta are confined within the host galaxy and thus these sources cannot evolve into large size radio galaxies unless they are close to the Eddington limit.

ANALYSIS OF ENERGY SPECTRA WITH LOW PHOTON COUNTS VIA BAYESIAN POSTERIOR SIMULATION

Over the past 10 years Bayesian methods have rapidly grown more popular in many scientific disciplines as several computationally intensive statistical algorithms have become feasible with increased computer power. In this paper we begin with a general description of the Bayesian paradigm for statistical inference and the various state-of-the-art model-fitting techniques that we employ (e.g., the Gibbs sampler and the Metropolis-Hastings algorithm). These algorithms are very flexible and can be used to fit models that account for the highly hierarchical structure inherent in the collection of high-quality spectra and thus can keep pace with the accelerating progress of new space telescope designs. The methods we develop, which will soon be available in the Chandra Interactive Analysis of Observations (CIAO) software, explicitly model photon arrivals as a Poisson process and thus have no difficulty with high-resolution low-count X-ray and γ-ray data. We expect these methods to be useful not only for the recently launched Chandra X-Ray Observatory and XMM but also for new generation telescopes such as Constellation X, GLAST, etc. In the context of two examples (quasar S5 0014+813 and hybrid-chromosphere supergiant star α TrA), we illustrate a new highly structured model and how Bayesian posterior sampling can be used to compute estimates, error bars, and credible intervals for the various model parameters. Application of our method to the high-energy tail of the ASCA spectrum of α TrA confirms that even at a quiescent state, the coronal plasma on this hybrid-chromosphere star is indeed at high temperatures (>10 MK) that normally characterize flaring plasma on the Sun. We are also able to constrain the coronal metallicity and find that although it is subject to large uncertainties, it is consistent with the photospheric measurements.

Chandra Discovery of a 300 Kiloparsec X-Ray Jet in the Gigahertz-peaked Spectrum Quasar PKS 1127–145

We have discovered an X-ray jet with Chandra imaging of the z=1.187 radioloud quasar PKS 1127-145. In this paper we present the Chandra X-ray data, follow-up VLA observations, and optical imaging using the HST WFPC2. The X-ray jet contains 273±5 net counts in 27 ksec and extends ∼ 30 from the quasar core, corresponding to a minimum projected linear size of ∼ 330h 50 kpc. The evaluation of the X-ray emission processes is complicated by the observed offsets between X-ray and radio brightness peaks. We discuss the problems posed by these observations to jet models. In addition, PKS 1127-145 is a Giga-Hertz Peaked Spectrum radio source, a member of the class of radio sources suspected to be young or “frustrated” versions of FRI radio galaxies. However the discovery of an X-ray and radio jet extending well outside the host galaxy of PKS 1127145 suggests that activity in this and other GPS sources may be long-lived and complex. Subject headings: Quasars: individual (PKS 1127-145) – galaxies: jets – X-Rays:

A Stochastic Model for the Luminosity Fluctuations of Accreting Black Holes

In this work we have developed a new stochastic model for the fluctuations in lightcurves of accreting black holes. The model is based on a linear combination of stochastic processes and is also the solution to the linear diffusion equation perturbed by a spatially correlated noise field. This allows flexible modeling of the power spectral density (PSD), and we derive the likelihood function for the process, enabling one to estimate the parameters of the process, including break frequencies in the PSD. Our statistical technique is computationally efficient, unbiased by aliasing and red noise leak, and fully accounts for irregular sampling and measurement errors. We show that our stochastic model provides a good approximation to the X-ray lightcurves of galactic black holes, and the optical and X-ray lightcurves of AGN. We use the estimated time scales of our stochastic model to recover the correlation between characteristic time scale of the high frequency X-ray fluctuations and black hole mass for AGN, including two new ‘detections’ of the time scale for Fairall 9 and NGC 5548. We find a tight anti-correlation between the black hole mass and the amplitudeof the driving noise field, which is proportional to the amplitude of the high frequency X-ray PSD, and we estimate that this parameter gives black hole mass estimates to within ∼ 0.2 dex precision, potentially the most accurate method for AGN yet. We also find evidence that ≈ 13% of AGN optical PSDs fall off flatter than 1/f 2 , and, similar to previous work, find that the optical fluctuations are more suppressed on short time scales compared to the X-rays, but are larger on long time scales, suggesting the optical fluctuations are not solely due to reprocessing of X-rays. Subject headings: accretion, accretion disks — galaxies: active — methods: data analysis — quasars: general

Chandra Survey of Radio-quiet, High-Redshift Quasars

We observed 17 optically selected, radio-quiet, high-redshift quasars with the Chandra ACIS and detected 16 of them. The quasars have redshifts between 3.70 and 6.28 and include the highest-redshift quasars known. When compared with low-redshift quasars observed with ROSAT, these high-redshift quasars are significantly more X-ray-quiet. We also find that the X-ray spectral index of the high-redshift objects is flatter than the average at lower redshift. These trends confirm the predictions of models in which the accretion flow is described by a cold, optically thick accretion disk surrounded by a hot, optically thin corona, provided the viscosity parameter α ≥ 0.02. The high-redshift quasars have supermassive black holes, with masses of ~1010 M☉, and are accreting material at ~0.1 times the Eddington limit. We detect 10 X-ray photons from the z = 6.28 quasar SDSS 1030+0524, which might have a Gunn-Peterson trough and be near the redshift of reionization of the intergalactic medium. The X-ray data place an upper limit on the optical depth of the intergalactic medium, τ(IGM) 20.