Brandon C. Kelly

Some Aspects of Measurement Error in Linear Regression of Astronomical Data

I describe a Bayesian method to account for measurement errors in linear regression of astronomical data. The method allows for heteroscedastic and possibly correlated measurement errors and intrinsic scatter in the regression relationship. The method is based on deriving a likelihood function for the measured data, and I focus on the case when the intrinsic distribution of the independent variables can be approximated using a mixture of Gaussian functions. I generalize the method to incorporate multiple independent variables, nondetections, and selection effects (e.g., Malmquist bias). A Gibbs sampler is described for simulating random draws from the probability distribution of the parameters, given the observed data. I use simulation to compare the method with other common estimators. The simulations illustrate that the Gaussian mixture model outperforms other common estimators and can effectively give constraints on the regression parameters, even when the measurement errors dominate the observed scatter, source detection fraction is low, or the intrinsic distribution of the independent variables is not a mixture of Gaussian functions. I conclude by using this method to fit the X-ray spectral slope as a function of Eddington ratio using a sample of 39 z 0.8 radio-quiet quasars. I confirm the correlation seen by other authors between the radio-quiet quasar X-ray spectral slope and the Eddington ratio, where the X-ray spectral slope softens as the Eddington ratio increases. IDL routines are made available for performing the regression.

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.

MODELING THE TIME VARIABILITY OF SDSS STRIPE 82 QUASARS AS A DAMPED RANDOM WALK

We model the time variability of ~9000 spectroscopically confirmed quasars in SDSS Stripe 82 as a damped random walk (DRW). Using 2.7 million photometric measurements collected over 10 yr, we confirm the results of Kelly et al. and Kozlowski et al. that this model can explain quasar light curves at an impressive fidelity level (0.01-0.02 mag). The DRW model provides a simple, fast (O(N) for N data points), and powerful statistical description of quasar light curves by a characteristic timescale (τ) and an asymptotic rms variability on long timescales (SF∞). We searched for correlations between these two variability parameters and physical parameters such as luminosity and black hole mass, and rest-frame wavelength. Our analysis shows SF∞ to increase with decreasing luminosity and rest-frame wavelength as observed previously, and without a correlation with redshift. We find a correlation between SF∞ and black hole mass with a power-law index of 0.18 ± 0.03, independent of the anti-correlation with luminosity. We find that τ increases with increasing wavelength with a power-law index of 0.17, remains nearly constant with redshift and luminosity, and increases with increasing black hole mass with a power-law index of 0.21 ± 0.07. The amplitude of variability is anti-correlated with the Eddington ratio, which suggests a scenario where optical fluctuations are tied to variations in the accretion rate. However, we find an additional dependence on luminosity and/or black hole mass that cannot be explained by the trend with Eddington ratio. The radio-loudest quasars have systematically larger variability amplitudes by about 30%, when corrected for the other observed trends, while the distribution of their characteristic timescale is indistinguishable from that of the full sample. We do not detect any statistically robust differences in the characteristic timescale and variability amplitude between the full sample and the small subsample of quasars detected by ROSAT. Our results provide a simple quantitative framework for generating mock quasar light curves, such as currently used in LSST image simulations.

Atmospheric Parameters of Field L and T Dwarfs

We present an analysis of the 0.95-14.5 ?m spectral energy distributions of nine field ultracool dwarfs with spectral types ranging from L1 to T4.5. Effective temperatures, gravities, and condensate cloud sedimentation efficiencies are derived by comparing the data to synthetic spectra computed from atmospheric models that self-consistently include the formation of condensate clouds. Overall, the model spectra fit the data well, although the agreement at some wavelengths remains poor due to remaining inadequacies in the models. Derived effective temperatures decrease steadily through the L1-T4.5 spectral types, and we confirm that the effective temperatures of ultracool dwarfs at the L/T transition are nearly constant, decreasing by only ~200 K from spectral types L7.5 to T4.5. The condensate cloud properties vary significantly among the L dwarfs in our sample, ranging from very thick clouds to relatively thin clouds with no particular trend with spectral type. The two objects in our sample with very red -->J ? Ks colors are, however, best fitted with synthetic spectra that have thick clouds, which hints at a possible correlation between the near-infrared colors of L dwarfs and the condensate cloud properties. The fits to the two T dwarfs in our sample (T2 and T4.5) also suggest that the clouds become thinner in this spectral class, in agreement with previous studies. Restricting the fits to narrower wavelength ranges (i.e., individual photometric bands) almost always yields excellent agreement between the data and models. Limitations in our knowledge of the opacities of key absorbers such as FeH, VO, and CH4 at certain wavelengths remain obvious, however. The effective temperatures obtained by fitting the narrower wavelength ranges can show a large scatter compared to the values derived by fitting the full spectral energy distributions; deviations are typically ~200 K and, in the worst cases, up to 700 K.

The Radio-Loud Fraction of Quasars is a Strong Function of Redshift and Optical Luminosity

Using a sample of optically selected quasars from the Sloan Digital Sky Survey, we have determined the radio-loud fraction (RLF) of quasars as a function of redshift and optical luminosity. The sample contains more than 30,000 objects and spans a redshift range of 0 10 to define radio-loud quasars, we find that b0 = -0.132 ± 0.116, bz = -2.052 ± 0.261, and bM = -0.183 ± 0.025. The RLF at z = 0.5 declines from 24.3% to 5.6% as luminosity decreases from M2500 = -26 to -22, and the RLF at M2500 = -26 declines from 24.3% to 4.1% as redshift increases from 0.5 to 3, suggesting that the RLF is a strong function of both redshift and luminosity. We also examine the impact of flux-related selection effects on the RLF determination using a series of tests and find that the dependence of the RLF on redshift and luminosity is highly likely to be physical, and that the selection effects we considered are not responsible for the dependence.

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.

Gemini Near-Infrared Spectroscopy of Luminous z?~?6 Quasars: Chemical Abundances, Black Hole Masses, and Mg II Absorption

We present Gemini near-infrared spectroscopic observations of six luminous quasars at z = 5.8 to ~6.3. Five of them were observed using Gemini South GNIRS, which provides a simultaneous wavelength coverage of 0.9-2.5 ?m in cross-dispersion mode. The other source was observed in the K band with Gemini North NIRI. We calculate line strengths for all detected emission lines and use their ratios to estimate gas metallicity in the broad-line regions of the quasars. The metallicity is found to be supersolar, with a typical value of ~4 Z?, and a comparison with low-redshift observations shows no strong evolution in metallicity up to z ~ 6. The Fe II/Mg II ratio of the quasars is 4.9 ? 1.4, consistent with low-redshift measurements. We estimate central black hole masses of 109-1010 M? and Eddington luminosity ratios of order unity. We identify two Mg II ??2796, 2803 absorbers with rest equivalent width W > 1 ? at 2.2 1.5 ? at z > 3 in the spectra, with the two most distant absorbers at z = 4.8668 and 4.8823, respectively. The redshift number densities (dN/dz) of Mg II absorbers with W > 1.5 ? are consistent with no cosmic evolution up to z > 4.

A DESCRIPTION OF QUASAR VARIABILITY MEASURED USING REPEATED SDSS AND POSS IMAGING

We provide a quantitative description and statistical interpretation of the optical continuum variability of quasars. The Sloan Digital Sky Survey (SDSS) has obtained repeated imaging in five UV-to-IR photometric bands for 33,881 spectroscopically confirmed quasars. About 10,000 quasars have an average of 60 observations in each band obtained over a decade along Stripe 82 (S82), whereas the remaining ~25,000 have 2-3 observations due to scan overlaps. The observed time lags span the range from a day to almost 10 years, and constrain quasar variability at rest-frame time lags of up to 4 years, and at rest-frame wavelengths from 1000 A to 6000 A. We publicly release a user-friendly catalog of quasars from the SDSS Data Release 7 that have been observed at least twice in SDSS or once in both SDSS and the Palomar Observatory Sky Survey, and we use it to analyze the ensemble properties of quasar variability. Based on a damped random walk (DRW) model defined by a characteristic timescale and an asymptotic variability amplitude that scale with the luminosity, black hole mass, and rest wavelength for individual quasars calibrated in S82, we can fully explain the ensemble variability statistics of the non-S82 quasars such as the exponential distribution of large magnitude changes. All available data are consistent with the DRW model as a viable description of the optical continuum variability of quasars on timescales of ~5-2000 days in the rest frame. We use these models to predict the incidence of quasar contamination in transient surveys such as those from the Palomar Transient Factory and Large Synoptic Survey Telescope.

Using the Fundamental Plane of black hole activity to distinguish X-ray processes from weakly accreting black holes

The Fundamental Plane of black hole activity is a relation between X-ray luminosity, radio luminosity and black hole mass for hard-state Galactic black holes and their supermassive analogues. The Fundamental Plane suggests that, at low-accretion rates, the physical processes regulating the conversion of an accretion flow into radiative energy could be universal across the entire black hole mass scale. However, there is still a need to further refine the Fundamental Plane in order to better discern the radiative processes and their geometry very close to the black hole, in particular the source of hard X-rays. Further refinement is necessary because error bars on the best-fitting slopes of the Fundamental Plane are generally large, and also the inferred coefficients can be sensitive to the adopted sample of black holes. In this work, we regress the Fundamental Plane with a Bayesian technique. Our approach shows that sub-Eddington black holes emit X-ray emission that is predominantly optically thin synchrotron radiation from the jet, provided that their radio spectra are flat or inverted. X-ray emission dominated by very radiatively inefficient accretion flows is excluded at the >3σ level. We also show that it is difficult to place Fanaroff-Riley type I (FR I) galaxies on to the Fundamental Plane because their X-ray jet emission is highly affected by synchrotron cooling. On the other hand, BL Lac objects (i.e. relativistically beamed sub-Eddington AGN) fit on to the Fundamental Plane. Including a uniform subset of high-energy peaked BL Lac objects from the Sloan Digital Sky Survey, we find sub-Eddington black holes with flat/inverted radio spectra follow log LX= (1.45 ± 0.04)log LR− (0.88 ± 0.06)log MBH− 6.07 ± 1.10, with σint= 0.07 ± 0.05 dex. Finally, we discuss how the effects of synchrotron cooling of jet emission from the highest black hole masses can bias Fundamental Plane regressions, perhaps leading to incorrect inferences on X-ray radiation mechanisms.

THE COSMOS ACTIVE GALACTIC NUCLEUS SPECTROSCOPIC SURVEY. I. XMM-NEWTON COUNTERPARTS ∗

We present optical spectroscopy for an X-ray and optical flux-limited sample of 677 XMM-Newton selected targets covering the 2 deg^2 Cosmic Evolution Survey field, with a yield of 485 high-confidence redshifts. The majority of the spectra were obtained over three seasons (2005-2007) with the Inamori Magellan Areal Camera and Spectrograph instrument on the Magellan (Baade) telescope. We also include in the sample previously published Sloan Digital Sky Survey spectra and supplemental observations with MMT/Hectospec. We detail the observations and classification analyses. The survey is 90% complete to flux limits of f_(0.5-10 keV) > 8 × 10^(–16) erg cm^(-2) s^(–1) and i^+_(AB) 3 × 10^(42) erg s^(–1)) to z < 1, of both optically obscured and unobscured types. We find statistically significant evidence that the obscured-to-unobscured AGN ratio at z < 1 increases with redshift and decreases with luminosity.We present optical spectroscopy for an X-ray and optical flux-limited sample of 677 XMM-Newton selected targets covering the 2 deg^2 COSMOS field, with a yield of 485 high-confidence redshifts. The majority of the spectra were obtained over three seasons (2005-2007) with the IMACS instrument on the Magellan (Baade) telescope. We also include in the sample previously published Sloan Digital Sky Survey spectra and supplemental observations with MMT/Hectospec. We detail the observations and classification analyses. The survey is 90% complete to flux limits of f_{0.5-10 keV}>8 x 10^-16 erg cm^-2 s^-1 and i_AB+ 3 x 10^42 erg s^-1) to z<1, of both optically obscured and unobscured types. We find statistically significant evidence that the obscured to unobscured AGN ratio at z<1 increases with redshift and decreases with luminosity.