arXiv subjects

K. E. Johnson

Publications and source records attributed to K. E. Johnson.

At least 37 records · Page 2Linked to original sources

arXiv2017

Legacy ExtraGalactic UV Survey with The Hubble Space Telescope. Stellar cluster catalogues and first insights into cluster formation and evolution in NGC 628

We report the large effort which is producing comprehensive high-level young star cluster (YSC) catalogues for a significant fraction of galaxies observed with the Legacy ExtraGalactic UV Survey (LEGUS) Hubble treasury program. We present the methodology developed to extract cluster positions, verify their genuine nature, produce multiband photometry (from NUV to NIR), and derive their physical properties via spectral energy distribution fitting analyses. We use the nearby spiral galaxy NGC628 as a test case for demonstrating the impact that LEGUS will have on our understanding of the formation and evolution of YSCs and compact stellar associations within their host galaxy. Our analysis of the cluster luminosity function from the UV to the NIR finds a steepening at the bright end and at all wavelengths suggesting a dearth of luminous clusters. The cluster mass function of NGC628 is consistent with a power-law distribution of slopes $\sim -2$ and a truncation of a few times $10^5$ M$_\odot$. After their formation YSCs and compact associations follow different evolutionary paths. YSCs survive for a longer timeframe, confirming their being potentially bound systems. Associations disappear on time scales comparable to hierarchically organized star-forming regions, suggesting that they are expanding systems. We find mass-independent cluster disruption in the inner region of NGC628, while in the outer part of the galaxy there is little or no disruption. We observe faster disruption rates for low mass ($\leq$ $10^4$ M$_\odot$) clusters suggesting that a mass-dependent component is necessary to fully describe the YSC disruption process in NGC628.

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arXiv2017

The Hierarchical Distribution of the Young Stellar Clusters in Six Local Star Forming Galaxies

We present a study of the hierarchical clustering of the young stellar clusters in six local (3--15 Mpc) star-forming galaxies using Hubble Space Telescope broad band WFC3/UVIS UV and optical images from the Treasury Program LEGUS (Legacy ExtraGalactic UV Survey). We have identified 3685 likely clusters and associations, each visually classified by their morphology, and we use the angular two-point correlation function to study the clustering of these stellar systems. We find that the spatial distribution of the young clusters and associations are clustered with respect to each other, forming large, unbound hierarchical star-forming complexes that are in general very young. The strength of the clustering decreases with increasing age of the star clusters and stellar associations, becoming more homogeneously distributed after ~40--60 Myr and on scales larger than a few hundred parsecs. In all galaxies, the associations exhibit a global behavior that is distinct and more strongly correlated from compact clusters. Thus, populations of clusters are more evolved than associations in terms of their spatial distribution, traveling significantly from their birth site within a few tens of Myr whereas associations show evidence of disruption occurring very quickly after their formation. The clustering of the stellar systems resembles that of a turbulent interstellar medium that drives the star formation process, correlating the components in unbound star-forming complexes in a hierarchical manner, dispersing shortly after formation, suggestive of a single, continuous mode of star formation across all galaxies.

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arXiv2016

The properties, origin and evolution of stellar clusters in galaxy simulations and observations

We investigate the properties and evolution of star particles in two simulations of isolated spiral galaxies, and two galaxies from cosmological simulations. Unlike previous numerical work, where typically each star particle represents one `cluster', for the isolated galaxies we are able to model features we term `clusters' with groups of particles. We compute the spatial distribution of stars with different ages, and cluster mass distributions, comparing our findings with observations including the recent LEGUS survey. We find that spiral structure tends to be present in older (100s Myrs) stars and clusters in the simulations compared to the observations. This likely reflects differences in the numbers of stars or clusters, the strength of spiral arms, and whether the clusters are allowed to evolve. Where we model clusters with multiple particles, we are able to study their evolution. The evolution of simulated clusters tends to follow that of their natal gas clouds. Massive, dense, long-lived clouds host massive clusters, whilst short-lived clouds host smaller clusters which readily disperse. Most clusters appear to disperse fairly quickly, in basic agreement with observational findings. We note that embedded clusters may be less inclined to disperse in simulations in a galactic environment with continuous accretion of gas onto the clouds than isolated clouds and correspondingly, massive young clusters which are no longer associated with gas tend not to occur in the simulations. Caveats of our models include that the cluster densities are lower than realistic clusters, and the simplistic implementation of stellar feedback.

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arXiv2015

Exploring X-ray Binary Populations in Compact Group Galaxies with $Chandra$

We obtain total galaxy X-ray luminosities, $L_X$, originating from individually detected point sources in a sample of 47 galaxies in 15 compact groups of galaxies (CGs). For the great majority of our galaxies, we find that the detected point sources most likely are local to their associated galaxy, and are thus extragalactic X-ray binaries (XRBs) or nuclear active galactic nuclei (AGNs). For spiral and irregular galaxies, we find that, after accounting for AGNs and nuclear sources, most CG galaxies are either within the $\pm1σ$ scatter of the Mineo et al. (2012) $L_X$ - star formation rate (SFR) correlation or have higher $L_X$ than predicted by this correlation for their SFR. We discuss how these "excesses" may be due to low metallicities and high interaction levels. For elliptical and S0 galaxies, after accounting for AGNs and nuclear sources, most CG galaxies are consistent with the Boroson et al. (2011) $L_X$ - stellar mass correlation for low-mass XRBs, with larger scatter, likely due to residual effects such as AGN activity or hot gas. Assuming non-nuclear sources are low- or high-mass XRBs, we use appropriate XRB luminosity functions to estimate the probability that stochastic effects can lead to such extreme $L_X$ values. We find that, although stochastic effects do not in general appear to be important, for some galaxies there is a significant probability that high $L_X$ values can be observed due to strong XRB variability.

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arXiv2015

The Spatial Distribution of the Young Stellar Clusters in the Star Forming Galaxy NGC 628

We present a study of the spatial distribution of the stellar cluster populations in the star forming galaxy NGC 628. Using Hubble Space Telescope broad band WFC3/UVIS UV and optical images from the Treasury Program LEGUS (Legacy ExtraGalactic UV Survey), we have identified 1392 potential young (<100 Myr) stellar clusters within the galaxy, identified from a combination of visual inspection and automatic selection. We investigate the clustering of these young stellar clusters and quantify the strength and change of clustering strength with scale using the two-point correlation function. We also investigate how image boundary conditions and dust lanes affect the observed clustering. The distribution of the clusters is well fit by a broken power law with negative exponent $α$. We recover a weighted mean index of $α$ ~ -0.8 for all spatial scales below the break at 3".3 (158 pc at a distance of 9.9 Mpc) and an index of $α$ ~ -0.18 above 158 pc for the accumulation of all cluster types. The strength of the clustering increases with decreasing age and clusters older than 40 Myr lose their clustered structure very rapidly and tend to be randomly distributed in this galaxy whereas the mass of the star cluster has little effect on the clustering strength. This is consistent with results from other studies that the morphological hierarchy in stellar clustering resembles the same hierarchy as the turbulent interstellar medium.

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arXiv2015

The Brightest Young Star Clusters in NGC 5253

The nearby dwarf starburst galaxy NGC5253 hosts a number of young, massive star clusters, the two youngest of which are centrally concentrated and surrounded by thermal radio emission (the `radio nebula'). To investigate the role of these clusters in the starburst energetics, we combine new and archival Hubble Space Telescope images of NGC5253 with wavelength coverage from 1500 Ang to 1.9 micron in 13 filters. These include H-alpha, P-beta, and P-alpha, and the imaging from the Hubble Treasury Program LEGUS (Legacy Extragalactic UV Survey). The extraordinarily well-sampled spectral energy distributions enable modeling with unprecedented accuracy the ages, masses, and extinctions of the 9 optically brightest clusters (M_V < -8.8) and the two young radio nebula clusters. The clusters have ages ~1-15 Myr and masses ~1x10^4 - 2.5x10^5 M_sun. The clusters' spatial location and ages indicate that star formation has become more concentrated towards the radio nebula over the last ~15 Myr. The most massive cluster is in the radio nebula; with a mass 2.5x10^5 M_sun and an age ~1 Myr, it is 2-4 times less massive and younger than previously estimated. It is within a dust cloud with A_V~50 mag, and shows a clear nearIR excess, likely from hot dust. The second radio nebula cluster is also ~1 Myr old, confirming the extreme youth of the starburst region. These two clusters account for about half of the ionizing photon rate in the radio nebula, and will eventually supply about 2/3 of the mechanical energy in present-day shocks. Additional sources are required to supply the remaining ionizing radiation, and may include very massive stars.

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arXiv2015

The Physical Conditions in a Pre Super Star Cluster Molecular Cloud in the Antennae Galaxies

We present an analysis of the physical conditions in an extreme molecular cloud in the Antennae merging galaxies. This cloud has properties consistant with those required to form a globular cluster. We have obtained ALMA CO and 870$μ$m observations of the Antennae galaxy system with $\sim 0".5$ resolution. This cloud stands out in the data with a radius of $\lesssim 24$~pc and mass of $>5\times 10^6$~M$_\odot$. The cloud appears capable of forming a globular cluster, but the lack of associated thermal radio emission indicates that star formation has not yet altered the environment. The lack of thermal radio emission places the cloud in an early stage of evolution, which we expect to be short-lived ($\lesssim 1$~Myr) and thus rare. Given its mass and kinetic energy, for the cloud to be confined (as its appearance strongly suggests) it must be subject to an external pressure of P/$k_B \gtrsim 10^8$~K~cm$^{-3}$ -- 10,000 times higher than typical interstellar pressure. This would support theories that high pressures are required to form globular clusters and may explain why extreme environments like the Antennae are preferred environments for generating such objects. Given the cloud temperature of $\sim 25$~K, the internal pressure must be dominated by non-thermal processes, most likely turbulence. We expect the molecular cloud to collapse and begin star formation in $\lesssim 1$~Myr.

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arXiv2015

A Comprehensive HST BVI Catalogue Of Star Clusters In Five Hickson Compact Groups Of Galaxies

We present a photometric catalogue of star cluster candidates in Hickson compact groups (HCGs) 7, 31, 42, 59, and 92, based on observations with the Advanced Camera for Surveys and the Wide Field Camera 3 on the Hubble Space Telescope. The catalogue contains precise cluster positions (right ascension and declination), magnitudes, and colours in the BVI filters. The number of detected sources ranges from 2200 to 5600 per group, from which we construct the high-confidence sample by applying a number of criteria designed to reduce foreground and background contaminants. Furthermore, the high-confidence cluster candidates for each of the 16 galaxies in our sample are split into two sub-populations: one that may contain young star clusters and one that is dominated by globular older clusters. The ratio of young star cluster to globular cluster candidates varies from group to group, from equal numbers to the extreme of HCG 31 which has a ratio of 8 to 1, due to a recent starburst induced by interactions in the group. We find that the number of blue clusters with $M_V < -9$ correlates well with the current star formation rate in an individual galaxy, while the number of globular cluster candidates with $M_V < -7.8$ correlates well (though with large scatter) with the stellar mass. Analyses of the high-confidence sample presented in this paper show that star clusters can be successfully used to infer the gross star formation history of the host groups and therefore determine their placement in a proposed evolutionary sequence for compact galaxy groups.

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arXiv2014

Legacy ExtraGalactic UV Survey (LEGUS) with The Hubble Space Telescope. I. Survey Description

The Legacy ExtraGalactic UV Survey (LEGUS) is a Cycle 21 Treasury program on the Hubble Space Telescope, aimed at the investigation of star formation and its relation with galactic environment in nearby galaxies, from the scales of individual stars to those of ~kpc-size clustered structures. Five-band imaging, from the near-ultraviolet to the I-band, with the Wide Field Camera 3, plus parallel optical imaging with the Advanced Camera for Surveys, is being collected for selected pointings of 50 galaxies within the local 12 Mpc. The filters used for the observations with the Wide Field Camera 3 are: F275W(2,704 A), F336W(3,355 A), F438W(4,325 A), F555W(5,308 A), and F814W(8,024 A); the parallel observations with the Advanced Camera for Surveys use the filters: F435W(4,328 A), F606W(5,921 A), and F814W(8,057 A). The multi-band images are yielding accurate recent (<~50 Myr) star formation histories from resolved massive stars and the extinction-corrected ages and masses of star clusters and associations. The extensive inventories of massive stars and clustered systems will be used to investigate the spatial and temporal evolution of star formation within galaxies. This will, in turn, inform theories of galaxy evolution and improve the understanding of the physical underpinning of the gas-star formation relation and the nature of star formation at high redshift. This paper describes the survey, its goals and observational strategy, and the initial science results. Because LEGUS will provide a reference survey and a foundation for future observations with JWST and with ALMA, a large number of data products are planned for delivery to the community.

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arXiv2013

Stellar Populations in Compact Galaxy Groups: a Multi-Wavelength Study of HCGs 16, 22, and 42, their Star Clusters and Dwarf Galaxies

We present a multi-wavelength analysis of three compact galaxy groups, HCGs 16, 22, and 42, which describe a sequence in terms of gas richness, from space- (Swift, HST, Spitzer) and ground-based (LCO, CTIO) imaging and spectroscopy. We study various signs of past interactions including a faint, dusty tidal feature about HCG 16A, which we tentatively age-date at <1 Gyr. This represents the possible detection of a tidal feature at the end of its phase of optical observability. Our HST images also resolve what were thought to be double nuclei in HCG 16C and D into multiple, distinct sources, likely to be star clusters. Beyond our phenomenological treatment, we focus primarily on contrasting the stellar populations across these three groups. The star clusters show a remarkable intermediate-age population in HCG 22, and identify the time at which star formation was quenched in HCG 42. We also search for dwarf galaxies at accordant redshifts. The inclusion of 33 members and 27 'associates' (possible members) radically changes group dynamical masses, which in turn may affect previous evolutionary classifications. The extended membership paints a picture of relative isolation in HCGs 16 and 22, but shows HCG 42 to be part of a larger structure, following a dichotomy expected from recent studies. We conclude that (a) star cluster populations provide an excellent metric of evolutionary state, as they can age-date the past epochs of star formation; and (b) the extended dwarf galaxy population must be considered in assessing the dynamical state of a compact group.

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arXiv2011

The merger history, AGN and dwarf galaxies of Hickson Compact Group 59

Compact group galaxies often appear unaffected by their unusually dense environment. Closer examination can, however, reveal the subtle, cumulative effects of multiple galaxy interactions. Hickson Compact Group (HCG) 59 is an excellent example of this situation. We present a photometric study of this group in the optical (HST), infrared (Spitzer) and X-ray (Chandra) regimes aimed at characterizing the star formation and nuclear activity in its constituent galaxies and intra-group medium. We associate five dwarf galaxies with the group and update the velocity dispersion, leading to an increase in the dynamical mass of the group of up to a factor of 10 (to 2.8e13 Msun), and a subsequent revision of its evolutionary stage. Star formation is proceeding at a level consistent with the morphological types of the four main galaxies, of which two are star-forming and the other two quiescent. Unlike in some other compact groups, star-forming complexes across HCG 59 closely follow mass-radius scaling relations typical of nearby galaxies. In contrast, the ancient globular cluster populations in galaxies HCG 59A and B show intriguing irregularities, and two extragalactic HII regions are found just west of B. We age-date a faint stellar stream in the intra-group medium at ~1 Gyr to examine recent interactions. We detect a likely low-luminosity AGN in HCG 59A by its ~10e40 erg/s X-ray emission; the active nucleus rather than star formation can account for the UV+IR SED. We discuss the implications of our findings in the context of galaxy evolution in dense environments.

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arXiv2011

Star Clusters in the Tidal Tails of Interacting Galaxies: Cluster Populations Across a Variety of Tail Environments

We have searched for compact stellar structures within 17 tidal tails in 13 different interacting galaxies using F606W- and F814W- band images from the Wide Field Planetary Camera 2 (WFPC2) on the Hubble Space Telescope (HST). The sample of tidal tails includes a diverse population of optical properties, merging galaxy mass ratios, HI content, and ages. Combining our tail sample with Knierman et al. (2003), we find evidence of star clusters formed in situ with Mv < -8.5 and V-I < 2.0 in 10 of 23 tidal tails; we are able to identify cluster candidates to Mv = -6.5 in the closest tails. Three tails offer clear examples of "beads on a string" star formation morphology in V-I color maps. Two tails present both tidal dwarf galaxy (TDG) candidates and cluster candidates. Statistical diagnostics indicate that clusters in tidal tails may be drawn from the same power-law luminosity functions (with logarithmic slopes ~ -2 - -2.5) found in quiescent spiral galaxies and the interiors of interacting systems. We find that the tail regions with the largest number of observable clusters are relatively young (< 250 Myr old) and bright (V < 24 mag arcsec^(-2)), probably attributed to the strong bursts of star formation in interacting systems soon after periapse. Otherwise, we find no statistical difference between cluster-rich and cluster-poor tails in terms of many observable characteristics, though this analysis suffers from complex, unresolved gas dynamics and projection effects.

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arXiv2010

Galaxy evolution in a complex environment: a multi-wavelength study of HCG 7

[Abridged] The environment where galaxies are found heavily influences their evolution. Close groupings, like the cores of galaxy clusters or compact groups, evolve in ways far more dramatic than their isolated counterparts. We have conducted a multiwavelength study of HCG7, consisting of four giant galaxies: 3 spirals and 1 lenticular. We use Hubble Space Telescope (HST) imaging to identify and characterize the young and old star cluster populations. We find young massive clusters (YMC) mostly in the three spirals, while the lenticular features a large, unimodal population of globular clusters (GC) but no detectable clusters with ages less than ~Gyr. The spatial and approximate age distributions of the ~300 YMCs and ~150 GCs thus hint at a regular star formation history in the group over a Hubble time. While at first glance the HST data show the galaxies as undisturbed, our deep ground-based, wide-field imaging that extends the HST coverage reveals faint signatures of stellar material in the intra-group medium. We do not detect the intra-group medium in HI or Chandra X-ray observations, signatures that would be expected to arise from major mergers. We find that the HI gas content of the individual galaxies and the group as a whole are a third of the expected abundance. The appearance of quiescence is challenged by spectroscopy that reveals an intense ionization continuum in one galaxy nucleus, and post-burst characteristics in another. Our spectroscopic survey of dwarf galaxy members yields one dwarf elliptical in an apparent tidal feature. We therefore suggest an evolutionary scenario for HCG7, whereby the galaxies convert most of their available gas into stars without major mergers and result in a dry merger. As the conditions governing compact groups are reminiscent of galaxies at intermediate redshift, we propose that HCGs are appropriate for studying galaxy evolution at z~1-2.

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arXiv2010

UV+IR Star Formation Rates: Hickson Compact Groups with Swift and Spitzer

We present Swift UVOT (1600-3000A) 3-band photometry for 41 galaxies in 11 nearby (<4500km/s) Hickson Compact Groups (HCGs) of galaxies. We use the uvw2-band (2000A) to estimate the dust-unobscured component, SFR_UV, of the total star-formation rate, SFR_T. We use Spitzer MIPS 24-micron photometry to estimate SFR_IR, the dust-obscured component of SFR_T. We obtain SFR_T=SFR_UV+SFR_IR. Using 2MASS K_s band based stellar mass, M*, estimates, we calculate specific SFRs, SSFR=SFR_T/M*. SSFR values show a clear and significant bimodality, with a gap between low (<~3.2x10^-11 / yr) and high SSFR (>~1.2x10^-10 / yr) systems. All galaxies with MIR activity index a_IRAC <= 0 (>0) are in the high- (low-) SSFR locus, as expected if high levels of star-formation power MIR emission from polycyclic aromatic hydrocarbon molecules and a hot dust continuum. All elliptical/S0 galaxies are in the low-SSFR locus, while 22 out of 24 spirals/irregulars are in the high-SSFR locus, with two borderline cases. We divide our sample into three subsamples (I, II and III) according to decreasing HI-richness of the parent galaxy group to which a galaxy belongs. Consistent with the SSFR and a_IRAC bimodality, 12 out of 15 type-I (11 out of 12 type-III) galaxies are in the high- (low-) SSFR locus, while type II galaxies span almost the full range of SSFR values. Unlike HCG galaxies, galaxies in a comparison quiescent SINGS sub-sample are continuously distributed both in SSFR and a_IRAC. Any uncertainties can only further enhance the SSFR bimodality. These results suggest that an environment characterized by high galaxy number-densities and low galaxy velocity-dispersions, such as the one found in compact groups, plays a key role in accelerating galaxy evolution by enhancing star-formation processes in galaxies and favoring a fast transition to quiescence.(abridged)

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arXiv2010

Hierarchical Structure Formation and Modes of Star Formation in Hickson Compact Group 31

The handful of low-mass, late-type galaxies that comprise Hickson Compact Group 31 is in the midst of complex, ongoing gravitational interactions, evocative of the process of hierarchical structure formation at higher redshifts. With sensitive, multicolor Hubble Space Telescope imaging, we characterize the large population of <10 Myr old star clusters that suffuse the system. From the colors and luminosities of the young star clusters, we find that the galaxies in HCG 31 follow the same universal scaling relations as actively star-forming galaxies in the local Universe despite the unusual compact group environment. Furthermore, the specific frequency of the globular cluster system is consistent with the low end of galaxies of comparable masses locally. This, combined with the large mass of neutral hydrogen and tight constraints on the amount of intragroup light, indicate that the group is undergoing its first epoch of interaction-induced star formation. In both the main galaxies and the tidal-dwarf candidate, F, stellar complexes, which are sensitive to the magnitude of disk turbulence, have both sizes and masses more characteristic of z=1-2 galaxies. After subtracting the light from compact sources, we find no evidence for an underlying old stellar population in F -- it appears to be a truly new structure. The low velocity dispersion of the system components, available reservoir of HI, and current star formation rate of ~10 solar masses per year, indicate that HCG31 is likely to both exhaust its cold gas supply and merge within ~1 Gyr. We conclude that the end product will be an isolated, X-ray-faint, low-mass elliptical.

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arXiv2009

Tidal Tails in Interacting Galaxies: Formation of Compact Stellar Structures

We have used V- and I- band images from the Hubble Space Telescope (HST) to identify compact stellar clusters within the tidal tails of twelve different interacting galaxies. The seventeen tails within our sample span a physical parameter space of HI/stellar masses, tail pressure and density through their diversity of tail lengths, optical brightnesses, mass ratios, HI column densities, stage on the Toomre sequence, and tail kinematics. Our preliminary findings in this study indicate that star cluster demographics of the tidal tail environment are compatible with the current understanding of star cluster formation in quiescent systems, possibly only needing changes in certain parameters or normalization of the Schechter cluster initial mass function (CIMF) to replicate what we observe in color-magnitude diagrams and a brightest absolute magnitude -- log N plot.

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arXiv2009

Probing Star Formation at Low Metallicity: The Radio Emission of Super Star Clusters in SBS0335-052

We present high-resolution radio continuum observations of the nascent starburst in the metal-poor galaxy SBS 0335-052. These radio data were taken with the Very Large Array and include observations at 0.7cm, 1.3cm, 2cm, 3.6cm, and 6cm. These observations enable us to probe the thermal radio nebulae associated with the extremely young star-forming regions in this galaxy. Two discrete and luminous star-forming regions are detected in the south of the galaxy that appear to be associated with massive star clusters previously identified at optical wavelengths. However, the remaining optically-identified massive star clusters are not clearly associated with radio emission (either thermal or non-thermal) down to the sensitivity limits of these radio data. The spectral energy distributions of the two radio-detected clusters are consistent with being purely thermal, and the entire region has an inferred ionizing flux of ~1.2 x 10^ 53 s^-1, which is equivalent to ~12,000 "typical" O-type stars (type O7.5 V). The observations presented here have resolved out a significant contribution from diffuse non-thermal emission detected previously, implying a previous episode of significant star formation. The current star formation rate (SFR) for this southern region alone is ~1.3 M_sun yr^-1, or ~ 23M_sun yr^-1 kpc^-2. This SFR derived from thermal radio emission also suggests that previous optical recombination line studies are not detecting a significant fraction of the current star formation in SBS 0335-052. From model fits to the radio spectral energy distribution, we infer a global mean density in the two youngest clusters of n_e > 10^3-10^4 cm^-3. In addition, a comparison between the compact and diffuse radio emission indicates that up to ~50% of the ionizing flux could be leaking out of the compact HII regions.

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arXiv2008

Structure of the interstellar medium around Cas A

We present a three-year series of observations at 24 microns with the Spitzer Space Telescope of the interstellar material in a 200 x 200 arcmin square area centered on Cassiopeia A. Interstellar dust heated by the outward light pulse from the supernova explosion emits in the form of compact, moving features. Their sequential outward movements allow us to study the complicated three-dimensional structure of the interstellar medium (ISM) behind and near Cassiopeia A. The ISM consists of sheets and filaments, with many structures on a scale of a parsec or less. The spatial power spectrum of the ISM appears to be similar to that of fractals with a spectral index of 3.5. The filling factor for the small structures above the spatial wavenumber k ~ 0.5 cycles/pc is only ~ 0.4%.

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