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Barry Rothberg

Publications and source records attributed to Barry Rothberg.

At least 37 records · Page 2Linked to original sources

Lunar-like silicate material forms the Earth quasisatellite (469219) 2016 HO3 Kamo`oalewa

Little is known about Earth quasi-satellites, a class of near-Earth small solar system bodies that orbit the sun but remain close to the Earth, because they are faint and difficult to observe. Here we use the Large Binocular Telescope (LBT) and the Lowell Discovery Telescope (LDT) to conduct a comprehensive physical characterization of quasi-satellite (469219) Kamo`oalewa and assess its affinity with other groups of near-Earth objects. We find that (469219) Kamo`oalewa rotates with a period of 28.3 (+1.8/-1.3) minutes and displays a reddened reflectance spectrum from 0.4-2.2 microns. This spectrum is indicative of a silicate-based composition, but with reddening beyond what is typically seen amongst asteroids in the inner solar system. We compare the spectrum to those of several material analogs and conclude that the best match is with lunar-like silicates. This interpretation implies extensive space weathering and raises the prospect that Kamo`oalewa could comprise lunar material.

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Relics of Supermassive Black Hole Seeds: The Discovery of an Accreting Black Hole in an Optically Normal, Low Metallicity Dwarf Galaxy

The detection and characterization of supermassive black holes (SMBHs) in local low mass galaxies is crucial to our understanding of the origins of SMBHs. This statement assumes that low mass galaxies have had a relatively quiet cosmic history, so that their black holes have not undergone significant growth and therefore can be treated as relics of the original SMBH seeds. While recent studies have found optical signatures of active galactic nuclei (AGNs) in a growing population of dwarf galaxies, these studies are biased against low metallicity and relatively merger-free galaxies, thus missing precisely the demographic in which to search for the relics of SMBH seeds. Here, we report the detection of the [\ion{Si}{6}]1.963~$μ$m coronal line (CL), a robust indicator of an AGN in the galaxy SDSS~J160135.95+311353.7, a nearby ($z=0.031$) low metallicity galaxy with a stellar mass approximately an order of magnitude lower than the LMC ($M_*\approx10^{8.56}$~M$_\odot$) and no optical evidence for an AGN. The AGN bolometric luminosity implied by the CL detection is $\approx10^{42}$~erg~s$^{-1}$, precisely what is predicted from its near-infrared continuum emission based on well-studied AGNs. Our results are consistent with a black hole of mass $\approx~10^5$~M$_\odot$, in line with expectations based on its stellar mass. This is the first time a near-infrared CL has been detected in a low mass, low metallicity galaxy with no optical evidence for AGN activity, providing confirmation of the utility of infrared CLs in finding AGNs in low mass galaxies when optical diagnostics fail. These observations highlight a powerful avenue of investigation to hunt for low mass black holes in the JWST era.

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Multi-wavelength observations of SDSS J105621.45+313822.1, a broad-line, low-metallicity AGN

In contrast to massive galaxies with Solar or super-Solar gas phase metallicities, very few Active Galactic Nuclei (AGN) are found in low-metallicity dwarf galaxies. Such a population could provide insight into the origins of supermassive black holes. Here we report near-infrared spectroscopic and X-ray observations of SDSS J105621.45+313822.1, a low-mass, low-metallicity galaxy with optical narrow line ratios consistent with star forming galaxies but a broad H$α$ line and mid-infrared colors consistent with an AGN. We detect the [Si VI] 1.96$μ$m coronal line and a broad Pa$α$ line with a FWHM of $850 \pm 25$~km~s$^{-1}$. Together with the optical broad lines and coronal lines seen in the SDSS spectrum, we confirm the presence of a highly accreting black hole with mass $(2.2 \pm 1.3) \times 10^{6}$~M$_{\odot}$, with a bolometric luminosity of $\approx10^{44}$~erg~s$^{-1}$ based on the coronal line luminosity, implying a highly accreting AGN. Chandra observations reveal a weak nuclear point source with $L_{\textrm{X,2-10 keV}} = (2.3 \pm 1.2) \times 10^{41}$~erg~s$^{-1}$, $\sim 2$ orders of magnitude lower than that predicted by the mid-infrared luminosity, suggesting that the AGN is highly obscured despite showing broad lines in the optical spectrum. The low X-ray luminosity and optical narrow line ratios of J1056+3138 highlight the limitations of commonly employed diagnostics in the hunt for AGNs in the low metallicity low mass regime.

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Enhanced Seeing Mode at the LBT: A Method to Significantly Improve Angular Resolution over a 4' x 4' Field of View

Since 2014, the LBT's First Light Adaptive Optics (FLAO) system has also included a seldom used capability, known as Enhanced Seeing Mode (ESM), that can improve the angular resolution over a 4' x 4' field of view (FOV). In full AO operation, FLAO provides diffraction limited (DL) capabilities over a small (30" x 30") FOV. By comparison, ESM can achieve significantly enhanced resolution, over natural seeing, across a far larger FOV. This improves operational efficiency over standard seeing limited (SL) observations and is applicable across a broader range of scientific targets. ESM uses 11 modes of correction (including tip and tilt) to remove residual aberrations and jitter which significantly improves angular resolution over the full FOV. While this mode does not reach the DL, it can achieve uniform angular resolutions as good as 0".22 over the FOV. Furthermore, it allows for the use of multi-object spectroscopy with R~10,000 or imaging with angular resolution similar to that achieved by the Wide-Field Camera 3 infrared channel on the Hubble Space Telescope, but powered by 11.6 meters of effective aperture in binocular mode. As part of the on-going characterization of ESM, we have demonstrated that even in poor seeing conditions (1".5-2") the image quality delivered to the focal station is improved by factors of 2-3. Here, we present the first results of the characterization of ESM, including systematic tests of the delivered PSF across the FOV as a function of the brightness of, and distance from, the AO Reference Star. We present a range of galactic and extra-galactic targets showing the improvements obtained over a broad range of seeing conditions and propose ESM as a standard observational mode for near-Infrared observations.

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A Triple AGN in a Mid-Infrared Selected Late Stage Galaxy Merger

The co-evolution of galaxies and the supermassive black holes (SMBHs) at their centers via hierarchical galaxy mergers is a key prediction of $Λ$CDM cosmology. As gas and dust are funneled to the SMBHs during the merger, the SMBHs light up as active galactic nuclei (AGNs). In some cases, a merger of two galaxies can encounter a third galaxy, leading to a triple merger, which would manifest as a triple AGN if all three SMBHs are simultaneously accreting. Using high-spatial resolution X-ray, near-IR, and optical spectroscopic diagnostics, we report here a compelling case of an AGN triplet with mutual separations < 10 kpc in the advanced merger SDSS J084905.51+111447.2 at z = 0.077. The system exhibits three nuclear X-ray sources, optical spectroscopic line ratios consistent with AGN in each nucleus, a high excitation near-IR coronal line in one nucleus, and broad Pa$α$ detections in two nuclei. Hard X-ray spectral fitting reveals a high column density along the line of sight, consistent with the picture of late-stage mergers hosting heavily absorbed AGNs. Our multiwavelength diagnostics support a triple AGN scenario, and we rule out alternative explanations such as star formation activity, shock-driven emission, and emission from fewer than three AGN. The dynamics of gravitationally bound triple SMBH systems can dramatically reduce binary SMBH inspiral timescales, providing a possible means to surmount the "Final Parsec Problem." AGN triplets in advanced mergers are the only observational forerunner to bound triple SMBH systems and thus offer a glimpse of the accretion activity and environments of the AGNs prior to the gravitationally-bound triple phase.

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Buried Black Hole Growth in IR-selected Mergers: New Results from Chandra

Observations and theoretical simulations suggest that a significant fraction of merger-triggered accretion onto supermassive black holes is highly obscured, particularly in late-stage galaxy mergers, when the black hole is expected to grow most rapidly. Starting with the Wide-Field Infrared Survey Explorer all-sky survey, we identified a population of galaxies whose morphologies suggest ongoing interaction and which exhibit red mid-infrared colors often associated with powerful active galactic nuclei (AGNs). In a follow-up to our pilot study, we now present Chandra/ACIS and XMM-Newton X-ray observations for the full sample of the brightest 15 IR-preselected mergers. All mergers reveal at least one nuclear X-ray source, with 8 out of 15 systems exhibiting dual nuclear X-ray sources, highly suggestive of single and dual AGNs. Combining these X-ray results with optical line ratios and with near-IR coronal emission line diagnostics, obtained with the near-IR spectrographs on the Large Binocular Telescope, we confirm that 13 out of the 15 mergers host AGNs, two of which host dual AGNs. Several of these AGNs are not detected in the optical. All X-ray sources appear X-ray weak relative to their mid-infrared continuum, and of the nine X-ray sources with sufficient counts for spectral analysis, eight reveal strong evidence of high absorption with column densities of $N_\mathrm{H} \gtrsim 10^{23}$~cm$^{-2}$. These observations demonstrate that a significant population of single and dual AGNs are missed by optical studies, due to high absorption, adding to the growing body of evidence that the epoch of peak black hole growth in mergers occurs in a highly obscured phase.

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FIGS: Spectral fitting constraints on the star formation history of massive galaxies since Cosmic Noon

We constrain the stellar population properties of a sample of 52 massive galaxies, with stellar mass log Ms>10.5, over the redshift range 0.5<z<2 by use of observer-frame optical and near-infrared slitless spectra from HST's ACS and WFC3 grisms. The deep exposures (~100 ks) allow us to target individual spectra of massive galaxies to F160W=22.5AB. Our spectral fitting approach uses a set of six base models adapted to the redshift and spectral resolution of each observation, and fits the weights of the base models, including potential dust attenuation, via an MCMC method. Our sample comprises a mixed distribution of quiescent (19) and star-forming galaxies (33). We quantify the width of the age distribution (Dt) that is found to dominate the variance of the retrieved parameters according to Principal Component Analysis. The population parameters follow the expected trend towards older ages with increasing mass, and Dt appears to weakly anti-correlate with stellar mass, suggesting a more efficient star formation at the massive end. As expected, the redshift dependence of the relative stellar age (measured in units of the age of the Universe at the source) in the quiescent sample rejects the hypothesis of a single burst (aka monolithic collapse). Radial colour gradients within each galaxy are also explored, finding a wider scatter in the star-forming subsample, but no conclusive trend with respect to the population parameters.

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A Two-Dimensional Spectroscopic Study of Emission Line Galaxies in the Faint Infrared Grism Survey (FIGS) I: Detection Method and Catalog

We present the results from the application of a two-dimensional emission line detection method, EMission-line two-Dimensional (EM2D), to the near-infrared G102 grism observations obtained with the Wide-Field Camera 3 (WFC3) as part of the Cycle 22 {\em Hubble Space Telescope} Treasury Program: the Faint Infrared Grism Survey (FIGS). Using the EM2D method, we have assembled a catalog of emission line galaxies (ELGs) with resolved star formation from each of the four FIGS fields. Not only can one better assess the global properties of ELGs, but the EM2D method allows for the analysis and an improved study of the individual emission-line region {\it within} each galaxy. This paper includes a description of the methodology, advantages, and the first results of the EM2D method applied to ELGs in FIGS. The advantage of 2D emission line measurements includes significant improvement of galaxy redshift measurements, approaching the level of accuracy seen in high-spectral-resolution data, but with greater efficiency; and the ability to identify and measure the properties of multiple sites of star-formation and over scales of $\sim$ 1 kpc within individual galaxies out to z $\sim$ 4. The EM2D method also significantly improves the reliability of high-redshift ($z\sim7$) Lyman-$α$ detections. Coupled with the wide field of view and high efficiency of space-based grism observations, EM2D provides a noteworthy improvement on the physical parameters that can be extracted from grism observations.

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Discovery of a z = 7.452 High Equivalent Width Lyman-α Emitter from the Hubble Space Telescope Faint Infrared Grism Survey

We present the results of an unbiased search for Lyα emission from continuum-selected 6 < z < 8 galaxies. Our dataset consists of 160 orbits of G102 slitless grism spectroscopy obtained with the Hubble Space Telescope (HST) Wide Field Camera 3 as part of the Faint Infrared Grism Survey (FIGS; PI: Malhotra), which obtains deep slitless spectra of all sources in four fields, and was designed to minimize contamination in observations of previously-identified high-redshift galaxy candidates. The FIGS data can potentially spectroscopically confirm the redshifts of galaxies, and as Lyα emission is resonantly scattered by neutral gas, FIGS can also constrain the ionization state of the intergalactic medium (IGM) during the epoch of reionization. These data have sufficient depth to detect Lyα emission in this epoch, as Tilvi et al. (2016) have published the FIGS detection of previously known (Finkelstein et al. 2013) Lyα emission at z = 7.51. The FIGS data use five separate roll-angles of HST to mitigate the contamination by nearby galaxies. We created a method that accounts for and removes the contamination from surrounding galaxies, and also removes any dispersed continuum light from each individual spectrum (Pirzkal et al. 2017). We searched for significant (> 4σ) emission lines using two different automated detection methods, free of any visual inspection biases. Applying these methods on photometrically-selected high-redshift candidates between 6 < z < 8 we find two emission lines, one previously published by Tilvi et al. (2016), and a new line at 1.028μm. We identify this lines as Lyα at z = 7.452 +/- 0.003. This newly spectroscopically confirmed galaxy has the highest Lyα rest-frame equivalent width (EW Lyα) yet published at z > 7 (140.3 +/- 19.0Å).

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Buried AGNs in Advanced Mergers:Mid-infrared color selection as a dual AGN finder

A direct consequence of hierarchical galaxy formation is the existence of dual supermassive black holes (SMBHs), which may be preferentially triggered as active galactic nuclei (AGN) during galaxy mergers. Despite decades of searching, however, dual AGNs are extremely rare, and most have been discovered serendipitously. Using the all-sky WISE survey, we identified a population of over 100 morphologically identified interacting galaxies or mergers that display red mid-infrared colors often associated in extragalactic sources with powerful AGNs. The vast majority of these advanced mergers are optically classified as star-forming galaxies suggesting that they may represent an obscured population of AGNs that cannot be found through optical studies. In this work, we present Chandra/ACIS observations and near-infrared spectra with the Large Binocular Telescope of six advanced mergers with projected pair separations less than ~ 10 kpc. The combined X-ray, near-infrared, and mid-infrared properties of these mergers provide confirmation that four out of the six mergers host at least one AGN, with four of the mergers possibly hosting dual AGNs with projected separations less than ~10 kpc, despite showing no firm evidence for AGNs based on optical spectroscopic studies. Our results demonstrate that 1) optical studies miss a significant fraction of single and dual AGNs in advanced mergers, and 2) mid-infrared pre-selection is extremely effective in identifying dual AGN candidates in late-stage mergers. Our multi-wavelength observations suggest that the buried AGNs in these mergers are highly absorbed, with intrinsic column densities in excess of N_H >10^24cm^-2, consistent with hydrodynamic simulations.

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FIGS -- Faint Infrared Grism Survey: Description and Data Reduction

The Faint Infrared Grism Survey (FIGS) is a deep Hubble Space Telescope (HST) WFC3/IR (Wide Field Camera 3 Infrared) slitless spectroscopic survey of four deep fields. Two fields are located in the Great Observatories Origins Deep Survey-North (GOODS-N) area and two fields are located in the Great Observatories Origins Deep Survey-South (GOODS-S) area. One of the southern fields selected is the Hubble Ultra Deep Field. Each of these four fields were observed using the WFC3/G102 grism (0.8$μm$-1.15$μm$ continuous coverage) with a total exposure time of 40 orbits (~ 100 kilo-seconds) per field. This reaches a 3 sigma continuum depth of ~26 AB magnitudes and probes emission lines to $\approx 10^{-17}\ erg\ s^{-1} \ cm^{-2}$. This paper details the four FIGS fields and the overall observational strategy of the project. A detailed description of the Simulation Based Extraction (SBE) method used to extract and combine over 10000 spectra of over 2000 distinct sources brighter than m_F105W=26.5 mag is provided. High fidelity simulations of the observations is shown to significantly improve the background subtraction process, the spectral contamination estimates, and the final flux calibration. This allows for the combination of multiple spectra to produce a final high quality, deep, 1D-spectra for each object in the survey.

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Enabling New ALMA Science with Improved Support for Time-Domain Observations

While the Atacama Large Millimeter/submillimeter Array (ALMA) is a uniquely powerful telescope, its impact in certain fields of astrophysics has been limited by observatory policies rather than the telescope's innate technical capabilities. In particular, several observatory policies present challenges for observations of variable, mobile, and/or transient sources --- collectively referred to here as "time-domain" observations. In this whitepaper we identify some of these policies, describe the scientific applications they impair, and suggest changes that would increase ALMA's science impact in Cycle 6 and beyond. Parties interested in time-domain science with ALMA are encouraged to join the ALMA Time-domain Special Interest Group (ATSIG) by signing up for the ATSIG mailing list at https://groups.google.com/group/alma-td-sig .

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Was 49b: An Overmassive AGN in a Merging Dwarf Galaxy?

We present a combined morphological and X-ray analysis of Was 49, an isolated, dual AGN system notable for the presence of a dominant AGN Was 49b in the disk of the primary galaxy Was 49a, at a projected radial distance of 8 kpc from the nucleus. Using X-ray data from Chandra, NuSTAR, and Swift, we find that this AGN has a bolometric luminosity of L_bol ~ 2 x 10^45 erg/s, with a black hole mass of M_BH=1.3^{+2.9}_{-0.9} x 10^8 M_Sol. Despite its large mass, our analysis of optical data from the Discovery Channel Telescope shows that the supermassive black hole is hosted by a stellar counterpart with a mass of only 5.6^{+4.9}_{-2.6} x 10^9 M_Sol, making the SMBH potentially larger than expected from SMBH-galaxy scaling relations, and the stellar counterpart exhibits a morphology that is consistent with dwarf elliptical galaxies. Our analysis of the system in the r and K bands indicates that Was 49 is a minor merger, with a mass ratio of Was 49a to Was 49b between 1:7 and 1:15. This is in contrast with findings that the most luminous merger-triggered AGNs are found in major mergers, and that minor mergers predominantly enhance AGN activity in the primary galaxy.

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Current Status of the Facility Instrumentation Suite at The Large Binocular Telescope Observatory

We review the current status of the facility instrumentation for the Large Binocular Telescope (LBT). The LBT has 2x 8.4m primary mirrors on a single mount with an effective collecting area of 11.8m or 23m when interferometrically combined. The facility instruments are: 1) the Large Binocular Cameras (LBCs), each with a 23'x25' field of view (FOV). The blue and red optimized optical LBCs are mounted at the prime focus of the left and right primary mirrors, respectively. The filter suite of the two LBCs covers 0.3-1.1μm, including the new TiO (0.78μm) and CN (0.82μm) filters; 2) the Multi-Object Double Spectrograph (MODS), two identical optical spectrographs each mounted at a straight through f/15 Gregorian mount. MODS-1 & -2 can do imaging with Sloan filters and medium resolution (R~2000) spectroscopy, each with 24 interchangeable masks (multi-object or longslit) over a 6'x6' FOV. Each MODS is capable of blue (0.32-0.6μm) and red (0.5-1.05μm) wavelength only coverage or, using a dichroic, 0.32-1.05μm coverage; and 3) the two LBT Utility Camera in the Infrared instruments (LUCIs), each mounted at a bent-front Gregorian f/15 port. LUCI-1 & 2 are designed for seeing-limited (4'x4'FOV) and AO (0.5'x0.5' FOV) imaging & spectroscopy over 0.95-2.5μm with spectroscopic resolutions of R~400-11000, including 32 interchangeable cryogenically cooled masks. All facility instruments are on the LBT and, for the first time, have been on-sky for science. We also report on the first science use of "mixed-mode" (differently paired instruments). While both primary mirrors reside on a single fixed mount, they are capable of operating independently within a defined "co-pointing" limit. This provides users with the additional capability to independently dither each mirror or center observations on two different sets of spatial coordinates within this limit. (ABRIDGED)

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Globular Cluster Systems in Brightest Cluster Galaxies. II: NGC 6166

We present new deep photometry of the globular cluster system (GCS) around NGC 6166, the central supergiant galaxy in Abell 2199. HST data from the ACS and WFC3 cameras in F475W, F814W are used to determine the spatial distribution of the GCS, its metallicity distribution function (MDF), and the dependence of the MDF on galactocentric radius and on GC luminosity. The MDF is extremely broad, with the classic red and blue subpopulations heavily overlapped, but a double-Gaussian model can still formally match the MDF closely. The spatial distribution follows a Sersic-like profile detectably to a projected radius of at least $R_{gc} = 250$ kpc. To that radius, the total number of clusters in the system is N_{GC} = 39000 +- 2000, the global specific frequency is S_N = 11.2 +- 0.6, and 57\% of the total are blue, metal-poor clusters. The GCS may fade smoothly into the Intra-Cluster Medium of A2199; we see no clear transition from the core of the galaxy to the cD halo or the ICM. The radial distribution, projected ellipticity, and mean metallicity of the red (metal-richer) clusters match the halo light extremely well for R > 15 kpc, both of them varying as σ_{MRGC} ~ σ_{light} ~ R^-1.8. By comparison, the blue (metal-poor) GC component has a much shallower falloff σ_{MPGC} ~ R^-1.0 and a more nearly spherical distribution. This strong difference in their density distributions produces a net metallicity gradient in the GCS as a whole that is primarily generated by the population gradient. With NGC 6166 we appear to be penetrating into a regime of high enough galaxy mass and rich enough environment that the bimodal two-phase description of GC formation is no longer as clear or effective as it has been in smaller galaxies.

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Not In Our Backyard: Spectroscopic Support for the CLASH z=11 Candidate MACS0647-JD

We report on our first set of spectroscopic Hubble Space Telescope observations of the z~11 candidate galaxy strongly lensed by the MACSJ0647.7+7015 galaxy cluster. The three lensed images are faint and we show that these early slitless grism observations are of sufficient depth to investigate whether this high-redshift candidate, identified by its strong photometric break at ~1.5 micron, could possibly be an emission line galaxy at a much lower redshift. While such an interloper would imply the existence of a rather peculiar object, we show here that such strong emission lines would clearly have been detected. Comparing realistic, two-dimensional simulations to these new observations we would expect the necessary emission lines to be detected at >5 sigma while we see no evidence for such lines in the dispersed data of any of the three lensed images. We therefore exclude that this object could be a low redshift emission line interloper, which significantly increases the likelihood of this candidate being a bona fide z~11 galaxy.

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An Optically Obscured AGN in a Low Mass, Irregular Dwarf Galaxy: A Multi-Wavelength Analysis of J1329+3234

Supermassive black holes (SMBHs) are found ubiquitously in large, bulge-dominated galaxies throughout the local universe, yet little is known about their presence and properties in bulgeless and low-mass galaxies. This is a significant deficiency, since the mass distribution and occupation fraction of nonstellar black holes provide important observational constraints on SMBH seed formation theories and many dwarf galaxies have not undergone major mergers that would erase information on their original black hole population. Using data from the Wide-field Infrared Survey Explorer, we discovered hundreds of bulgeless and dwarf galaxies that display mid-infrared signatures of extremely hot dust highly suggestive of powerful accreting massive black holes, despite having no signatures of black hole activity at optical wavelengths. Here we report, in our first follow-up X-ray investigation of this population, that the irregular dwarf galaxy J132932.41+323417.0 (z = 0.0156) contains a hard, unresolved X-ray source detected by XMM-Newton with luminosity L_2-10 keV = 2.4 x 10^40 erg/s, over two orders of magnitude greater than that expected from star formation, strongly suggestive of the presence of an accreting massive black hole. While enhanced X-ray emission and hot dust can be produced in extremely low metallicity environments, J132932.41+323417.0 is not extremely metal poor (~40% solar). With a stellar mass of 2.0 x 10^8 M_Sol, this galaxy is similar in mass to the Small Magellanic Cloud, and is one of the lowest mass galaxies with evidence for a massive nuclear black hole currently known.

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Globular Cluster Systems in Brightest Cluster Galaxies: A Near-Universal Luminosity Function?

We present the first results from our HST Brightest Cluster Galaxy (BCG) survey of seven central supergiant cluster galaxies and their globular cluster (GC) systems. We measure a total of 48000 GCs in all seven galaxies, representing the largest single GC database. We find that a log-normal shape accurately matches the observed luminosity function (LF) of the GCs down to the GCLF turnover point, which is near our photometric limit. In addition, the LF has a virtually identical shape in all seven galaxies. Our data underscore the similarity in the formation mechanism of massive star clusters in diverse galactic environments. At the highest luminosities (log L > 10^7 L_Sun) we find small numbers of "superluminous" objects in five of the galaxies; their luminosity and color ranges are at least partly consistent with those of UCDs (Ultra-Compact Dwarfs). Lastly, we find preliminary evidence that in the outer halo (R > 20 kpc), the LF turnover point shows a weak dependence on projected distance, scaling as L_0 ~ R^-0.2, while the LF dispersion remains nearly constant.

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