arXiv subjects

H. L. Gomez

Publications and source records attributed to H. L. Gomez.

At least 37 records · Page 2Linked to original sources

arXiv2017

Using dust, gas and stellar mass selected samples to probe dust sources and sinks in low metallicity galaxies

We combine samples of nearby galaxies with Herschel photometry selected on their dust, metal, HI, and stellar mass content, and compare these to chemical evolution models in order to discriminate between different dust sources. In a companion paper, we used a HI-selected sample of nearby galaxies to reveal a sub-sample of very gas rich (gas fraction > 80 per cent) sources with dust masses significantly below predictions from simple chemical evolution models, and well below $M_d/M_*$ and $M_d/M_{gas}$ scaling relations seen in dust and stellar-selected samples of local galaxies. We use a chemical evolution model to explain these dust-poor, but gas-rich, sources as well as the observed star formation rates (SFRs) and dust-to-gas ratios. We find that (i) a delayed star formation history is required to model the observed SFRs; (ii) inflows and outflows are required to model the observed metallicities at low gas fractions; (iii) a reduced contribution of dust from supernovae (SNe) is needed to explain the dust-poor sources with high gas fractions. These dust-poor, low stellar mass galaxies require a typical core-collapse SN to produce 0.01 - 0.16 $M_{\odot}$ of dust. To match the observed dust masses at lower gas fractions, significant grain growth is required to counteract the reduced contribution from dust in SNe and dust destruction from SN shocks. These findings are statistically robust, though due to intrinsic scatter it is not always possible to find one single model that successfully describes all the data. We also show that the dust-to-metals ratio decreases towards lower metallicity.

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arXiv2017

ALMA spectral survey of Supernova 1987A --- molecular inventory, chemistry, dynamics and explosive nucleosynthesis

We report the first molecular line survey of Supernova 1987A in the millimetre wavelength range. In the ALMA 210--300 and 340--360 GHz spectra, we detected cold (20--170 K) CO, 28SiO, HCO+ and SO, with weaker lines of 29SiO from ejecta. This is the first identification of HCO+ and SO in a young supernova remnant. We find a dip in the J=6--5 and 5--4 SiO line profiles, suggesting that the ejecta morphology is likely elongated. The difference of the CO and SiO line profiles is consistent with hydrodynamic simulations, which show that Rayleigh-Taylor instabilities cause mixing of gas, with heavier elements much more disturbed, making more elongated structure. We obtained isotopologue ratios of 28SiO/29SiO>13, 28SiO/30SiO>14, and 12CO/13CO>21, with the most likely limits of 28SiO/29SiO>128, 28SiO/30SiO>189. Low 29Si and 30Si abundances in SN 1987A are consistent with nucleosynthesis models that show inefficient formation of neutron-rich isotopes in a low metallicity environment, such as the Large Magellanic Cloud. The deduced large mass of HCO+ (~5x10^-6 Msun) and small SiS mass (<6x10^-5 Msun) might be explained by some mixing of elements immediately after the explosion. The mixing might have caused some hydrogen from the envelope to sink into carbon and oxygen-rich zones after the explosion, enabling the formation of a substantial mass of HCO+. Oxygen atoms may have penetrated into silicon and sulphur zones, suppressing formation of SiS. Our ALMA observations open up a new window to investigate chemistry, dynamics and explosive-nucleosynthesis in supernovae.

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arXiv2017

The dust mass in Cassiopeia A from a spatially resolved Herschel analysis

Theoretical models predict that core-collapse supernovae (CCSNe) can be efficient dust producers (0.1-1.0 Msun), potentially accounting for most of the dust production in the early Universe. Observational evidence for this dust production efficiency is however currently limited to only a few CCSN remnants (e.g., SN1987A, Crab Nebula). In this paper, we revisit the dust mass produced in Cassiopeia A (Cas A), a ~330-year old O-rich Galactic supernova remnant (SNR) embedded in a dense interstellar foreground and background. We present the first spatially resolved analysis of Cas A based on Spitzer and Herschel infrared and submillimetre data at a common resolution of ~0.6 arcmin for this 5 arcmin diameter remnant following a careful removal of contaminating line emission and synchrotron radiation. We fit the dust continuum from 17 to 500 micron with a four-component interstellar medium (ISM) and supernova (SN) dust model. We find a concentration of cold dust in the unshocked ejecta of Cas A and derive a mass of 0.3-0.5 Msun of silicate grains freshly produced in the SNR, with a lower limit of >=0.1-0.2 Msun. For a mixture of 50% of silicate-type grains and 50% of carbonaceous grains, we derive a total SN dust mass between 0.4 Msun and 0.6 Msun. These dust mass estimates are higher than from most previous studies of Cas A and support the scenario of supernova dominated dust production at high redshifts. We furthermore derive an interstellar extinction map for the field around Cas A which towards Cas A gives average values of A_V=6-8 mag, up to a maximum of A_V=15 mag.

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arXiv2014

Herschel imaging of the dust in the Helix Nebula (NGC 7293)

In our series of papers presenting the Herschel imaging of evolved planetary nebulae, we present images of the dust distribution in the Helix nebula (NGC 7293). Images at 70, 160, 250, 350, and 500 micron were obtained with the PACS and SPIRE instruments on board the Herschel satellite. The broadband maps show the dust distribution over the main Helix nebula to be clumpy and predominantly present in the barrel wall. We determined the spectral energy distribution of the main nebula in a consistent way using Herschel, IRAS, and Planck flux values. The emissivity index of 0.99 +/- 0.09, in combination with the carbon rich molecular chemistry of the nebula, indicates that the dust consists mainly of amorphous carbon. The dust excess emission from the central star disk is detected at 70 micron and the flux measurement agree with previous measurement. We present the temperature and dust column density maps. The total dust mass across the Helix nebula (without its halo) is determined to be 0.0035 solar mass at a distance of 216 pc. The temperature map shows dust temperatures between 22 and 42 K, which is similar to the kinetic temperature of the molecular gas, strengthening the fact that the dust and gas co-exist in high density clumps. Archived images are used to compare the location of the dust emission in the far infrared (Herschel) with the ionized (GALEX, Hbeta) and molecular hydrogen component. The different emission components are consistent with the Helix consisting of a thick walled barrel-like structure inclined to the line of sight. The radiation field decreases rapidly through the barrel wall.

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arXiv2014

A stubbornly large mass of cold dust in the ejecta of Supernova 1987A

We present new Herschel photometric and spectroscopic observations of Supernova 1987A, carried out in 2012. Our dedicated photometric measurements provide new 70 micron data and improved imaging quality at 100 and 160 micron compared to previous observations in 2010. Our Herschel spectra show only weak CO line emission, and provide an upper limit for the 63 micron [O I] line flux, eliminating the possibility that line contaminations distort the previously estimated dust mass. The far-infrared spectral energy distribution (SED) is well fitted by thermal emission from cold dust. The newly measured 70 micron flux constrains the dust temperature, limiting it to nearly a single temperature. The far-infrared emission can be fitted by 0.5+-0.1 Msun of amorphous carbon, about a factor of two larger than the current nucleosynthetic mass prediction for carbon. The observation of SiO molecules at early and late phases suggests that silicates may also have formed and we could fit the SED with a combination of 0.3 Msun of amorphous carbon and 0.5 Msun of silicates, totalling 0.8 Msun of dust. Our analysis thus supports the presence of a large dust reservoir in the ejecta of SN 1987A. The inferred dust mass suggests that supernovae can be an important source of dust in the interstellar medium, from local to high-redshift galaxies.

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arXiv2014

The Herschel exploitation of local galaxy Andromeda (HELGA) V: Strengthening the case for substantial interstellar grain growth

In this paper we consider the implications of the distributions of dust and metals in the disc of M31. We derive mean radial dust distributions using a dust map created from Herschel images of M31 sampling the entire far-infrared (FIR) peak. Modified blackbodies are fit to approximately 4000 pixels with a varying, as well as a fixed, dust emissivity index (beta). An overall metal distribution is also derived using data collected from the literature. We use a simple analytical model of the evolution of the dust in a galaxy with dust contributed by stellar sources and interstellar grain growth, and fit this model to the radial dust-to-metals distribution across the galaxy. Our analysis shows that the dust-to-gas gradient in M31 is steeper than the metallicity gradient, suggesting interstellar dust growth is (or has been) important in M31. We argue that M31 helps build a case for cosmic dust in galaxies being the result of substantial interstellar grain growth, while the net dust production from stars may be limited. We note, however, that the efficiency of dust production in stars, e.g., in supernovae (SNe) ejecta and/or stellar atmospheres, and grain destruction in the interstellar medium (ISM) may be degenerate in our simple model. We can conclude that interstellar grain growth by accretion is likely at least as important as stellar dust production channels in building the cosmic dust component in M31.

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arXiv2014

Herschel-ATLAS: Properties of dusty massive galaxies at low and high redshifts

We present a comparison of the physical properties of a rest-frame $250μ$m selected sample of massive, dusty galaxies from $0 1$ SMGs have an average SFR of $390^{+80}_{-70}\,$M$_\odot$yr$^{-1}$ which is 120 times that of the low-redshift sample matched in stellar mass to the SMGs (SFR$=3.3\pm{0.2}$ M$_\odot$yr$^{-1}$). The SMGs harbour a substantial mass of dust ($1.2^{+0.3}_{-0.2}\times{10}^9\,$M$_\odot$), compared to $(1.6\pm0.1)\times{10}^8\,$M$_\odot$ for low-redshift dusty galaxies. At low redshifts the dust luminosity is dominated by the diffuse ISM, whereas a large fraction of the dust luminosity in SMGs originates from star-forming regions. At the same dust mass SMGs are offset towards a higher SFR compared to the low-redshift H-ATLAS galaxies. This is not only due to the higher gas fraction in SMGs but also because they are undergoing a more efficient mode of star formation, which is consistent with their bursty star-formation histories. The offset in SFR between SMGs and low-redshift galaxies is similar to that found in CO studies, suggesting that dust mass is as good a tracer of molecular gas as CO.

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arXiv2014

The dust budget crisis in high-redshift submillimetre galaxies

We apply a chemical evolution model to investigate the sources and evolution of dust in a sample of 26 high-redshift ($z>1$) submillimetre galaxies (SMGs) from the literature, with complete photometry from ultraviolet to the submillimetre. We show that dust produced only by low-intermediate mass stars falls a factor 240 short of the observed dust masses of SMGs, the well-known `dust-budget crisis'. Adding an extra source of dust from supernovae can account for the dust mass in 19 per cent of the SMG sample. Even after accounting for dust produced by supernovae the remaining deficit in the dust mass budget provides support for higher supernova yields, substantial grain growth in the interstellar medium or a top-heavy IMF. Including efficient destruction of dust by supernova shocks increases the tension between our model and observed SMG dust masses. The models which best reproduce the physical properties of SMGs have a rapid build-up of dust from both stellar and interstellar sources and minimal dust destruction. Alternatively, invoking a top-heavy IMF or significant changes in the dust grain properties can solve the dust budget crisis only if dust is produced by both low mass stars and supernovae and is not efficiently destroyed by supernova shocks.

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arXiv2013

Detection of a Noble Gas Molecular Ion, {36}ArH^+, in the Crab Nebula

Noble gas molecules have not hitherto been detected in space. From spectra obtained with the Herschel Space Observatory, we report the detection of emission in the 617.5 GHz and 1234.6 GHz J = 1-0 and 2-1 rotational lines of {36}ArH^+ at several positions in the Crab Nebula, a supernova remnant known to contain both H2 molecules and regions of enhanced ionized argon emission. {36}Ar is believed to have originated from explosive nucleosynthesis in massive stars during core-collapse supernova events. Its detection in the Crab Nebula, the product of such a supernova event, confirms this expectation. The likely excitation mechanism for the observed {36}ArH^+ emission lines is electron collisions in partially ionized regions with electron densities of a few hundred per centimeter cubed.

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arXiv2013

H-ATLAS: estimating redshifts of Herschel sources from sub-mm fluxes

Upon its completion the Herschel ATLAS (H-ATLAS) will be the largest submillimetre survey to date, detecting close to half-a-million sources. It will only be possible to measure spectroscopic redshifts for a small fraction of these sources. However, if the rest-frame spectral energy distribution (SED) of a typical H-ATLAS source is known, this SED and the observed Herschel fluxes can be used to estimate the redshifts of the H-ATLAS sources without spectroscopic redshifts. In this paper, we use a subset of 40 H-ATLAS sources with previously measured redshifts in the range 0.5 1 we estimate that the rms of Delta z/(1+z) = 0.12. We have used this template to estimate the redshift distribution for the sources detected in the H-ATLAS equatorial fields, finding a bimodal distribution with a mean redshift of 1.2, 1.9 and 2.5 for 250, 350 and 500 um selected sources respectively. \end{abstract}

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arXiv2013

A Herschel study of NGC 650

As part of the Herschel Guaranteed Time Key Project MESS (Mass loss of Evolved StarS) we have imaged a sample of planetary nebulae. In this paper we present the PACS and SPIRE images of the classical bipolar planetary nebula NGC 650. We used these images to derive a temperature map of the dust. We also constructed a photoionization and dust radiative transfer model using the spectral synthesis code Cloudy. To constrain this model, we used the PACS and SPIRE fluxes and combined these with hitherto unpublished IUE and Spitzer IRS spectra as well as various other data from the literature. The temperature map combined with the photoionization model were used to study various aspects of the central star, the nebula, and in particular the dust grains in the nebula. The central star parameters are determined to be T_eff = 208 kK and L = 261 L_sol assuming a distance of 1200 pc. The stellar temperature is much higher than previously published values. We confirm that the nebula is carbon-rich with a C/O ratio of 2.1. The nebular abundances are typical for a type IIa planetary nebula. With the photoionization model we determined that the grains in the ionized nebula are large (assuming single-sized grains, they would have a radius of 0.15 micron. Most likely these large grains were inherited from the asymptotic giant branch phase. The PACS 70/160 micron temperature map shows evidence for two radiation components heating the grains. The first component is direct emission from the central star, while the second component is diffuse emission from the ionized gas (mainly Ly alpha). We show that previous suggestions that there is a photo-dissociation region surrounding the ionized region are incorrect. The neutral material resides in dense clumps inside the ionized region. These may also harbor stochastically heated very small grains in addition to the large grains.

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arXiv2013

Carbon Monoxide in the Cold Debris of Supernova 1987A

We report spectroscopic and imaging observations of rotational transitions of cold CO and SiO in the ejecta of SN1987A, the first such emission detected in a supernova remnant. In addition to line luminosities for the CO J=1-0, 2-1, 6-5, and 7-6 transitions, we present upper limits for all other transitions up to J=13-12, collectively measured from the Atacama Large Millimeter Array (ALMA), the Atacama Pathfinder EXperiment (APEX), and the Herschel Spectral and Photometric Imaging REceiver (SPIRE). Simple models show the lines are emitted from at least 0.01 solar masses of CO at a temperature > 14 K, confined within at most 35% of a spherical volume expanding at ~ 2000 km/s. Moreover, we locate the emission within 1'' of the central debris. These observations, along with a partial observation of SiO, confirm the presence of cold molecular gas within supernova remnants and provide insight into the physical conditions and chemical processes in the ejecta. Furthermore, we demonstrate the powerful new window into supernova ejecta offered by submillimeter observations.

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arXiv2013

GAMA/H-ATLAS: Linking the Properties of Sub-mm Detected and Undetected Early-Type Galaxies: I. z$\le$0.06 Sample

We present two large, nearby (0.013$\le$z$\le$0.06) samples of Early-Type Galaxies (ETGs): a visually classified sample of 220 ETGs, created using source-matched data from the Galaxy and Mass Assembly (GAMA) database with FIR/sub-mm detections from $Herschel$-ATLAS; and a visually classified sample of 551 ETGs which are undetected with $Herschel$-ATLAS. Active galactic nuclei (AGN) are removed from our samples using optical emission line diagnostics. These samples are scrutinised to determine characteristics of sub-mm detected versus undetected ETGs. We find similarities in the stellar mass distributions of the two ETG samples but testing other properties uncovers significant differences. The sub-mm detected sample is shown to have lower concentration and Sérsic indices than those in the undetected sample - a result which may be linked to the presence of dust in the former. Optical and UV-optical colours are also shown to be much bluer, indicating that the dust is linked with recent star formation. The intrinsic effective radii are on average 1.5 times larger for the sub-mm detected ETGs. Surface densities and groups data from the GAMA database are examined for the two samples, leading to the conclusion that dusty ETGs inhabit sparser environments than non-dusty ETGs in the nearby universe, although environments of the brightest ETGs are shown to differ the least. Modified Planck functions are fit to the H-ATLAS detected PACS and SPIRE fluxes for ETGs with sub-mm flux densities of at least 3$σ$ in the 350$μ$m SPIRE band, giving a resultant mean cold dust temperature of T$_{d}$=22.1K, with a range of 9-30K. The corresponding mean dust mass is 1.8$\times10^{7}$M$_{\odot}$, with a range of (0.08-35.0)$\times10^{7}$M$_{\odot}$. The dust masses calculated from these fits, normalised by stellar mass, are shown to increase with decreasing stellar mass and bluer colours.

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arXiv2012

The enigmatic nature of the circumstellar envelope and bow shock surrounding Betelgeuse as revealed by Herschel. I. Evidence of clumps, multiple arcs, and a linear bar-like structure

Context. The interaction between stellar winds and the interstellar medium (ISM) can create complex bow shocks. The photometers on board the Herschel Space Observatory are ideally suited to studying the morphologies of these bow shocks. Aims. We aim to study the circumstellar environment and wind-ISM interaction of the nearest red supergiant, Betelgeuse. Methods. Herschel PACS images at 70, 100, and 160 micron and SPIRE images at 250, 350, and 500 micron were obtained by scanning the region around Betelgeuse. These data were complemented with ultraviolet GALEX data, near-infrared WISE data, and radio 21 cm GALFA-HI data. The observational properties of the bow shock structure were deduced from the data and compared with hydrodynamical simulations. Results. The infrared Herschel images of the environment around Betelgeuse are spectacular, showing the occurrence of multiple arcs at 6-7 arcmin from the central target and the presence of a linear bar at 9 arcmin. Remarkably, no large-scale instabilities are seen in the outer arcs and linear bar. The dust temperature in the outer arcs varies between 40 and 140 K, with the linear bar having the same colour temperature as the arcs. The inner envelope shows clear evidence of a non-homogeneous clumpy structure (beyond 15 arcsec), probably related to the giant convection cells of the outer atmosphere. The non-homogeneous distribution of the material even persists until the collision with the ISM. A strong variation in brightness of the inner clumps at a radius of 2 arcmin suggests a drastic change in mean gas and dust density some 32 000 yr ago. Using hydrodynamical simulations, we try to explain the observed morphology of the bow shock around Betelgeuse. Conclusions: [abbreviated]

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arXiv2012

A Cool Dust Factory in the Crab Nebula: A Herschel Study of the Filaments

Whether supernovae are major sources of dust in galaxies is a long-standing debate. We present infrared and submillimeter photometry and spectroscopy from the Herschel Space Observatory of the Crab Nebula between 51 and 670 micron as part of the Mass Loss from Evolved StarS program. We compare the emission detected with Herschel with multiwavelength data including millimeter, radio, mid-infrared and archive optical images. We carefully remove the synchrotron component using the Herschel and Planck fluxes measured in the same epoch. The contribution from line emission is removed using Herschel spectroscopy combined with Infrared Space Observatory archive data. Several forbidden lines of carbon, oxygen and nitrogen are detected where multiple velocity components are resolved, deduced to be from the nitrogen-depleted, carbon-rich ejecta. No spectral lines are detected in the SPIRE wavebands; in the PACS bands, the line contribution is 5% and 10% at 70 and 100 micron and negligible at 160 micron. After subtracting the synchrotron and line emission, the remaining far-infrared continuum can be fit with two dust components. Assuming standard interstellar silicates, the mass of the cooler component is 0.24(+0.32)(-0.08) Msolar for T = 28.1(+5.5)(-3.2)K. Amorphous carbon grains require 0.11 +/- 0.01 Msolar of dust with T = 33.8(+2.3)(-1.8) K. A single temperature modified-blackbody with 0.14Msolar and 0.08Msolar for silicate and carbon dust respectively, provides an adequate fit to the far- infrared region of the spectral energy distribution but is a poor fit at 24-500 micron. The Crab Nebula has condensed most of the relevant refractory elements into dust, suggesting the formation of dust in core-collapse supernova ejecta is efficient.

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arXiv2012

The Herschel Exploitation of Local Galaxy Andromeda (HELGA). I: Global far-infrared and sub-mm morphology

We have obtained Herschel images at five wavelengths from 100 to 500 micron of a ~5.5x2.5 degree area centred on the local galaxy M31 (Andromeda), our nearest neighbour spiral galaxy, as part of the Herschel guaranteed time project "HELGA". The main goals of HELGA are to study the characteristics of the extended dust emission, focusing on larger scales than studied in previous observations of Andromeda at an increased spatial resolution, and the obscured star formation. In this paper we present data reduction and Herschel maps, and provide a description of the far-infrared morphology, comparing it with features seen at other wavelengths. We use high--resolution maps of the atomic hydrogen, fully covering our fields, to identify dust emission features that can be associated to M31 with confidence, distinguishing them from emission coming from the foreground Galactic cirrus. Thanks to the very large extension of our maps we detect, for the first time at far-infrared wavelengths, three arc-like structures extending out to ~21, ~26 and ~31 kpc respectively, in the south-western part of M31. The presence of these features, hosting ~2.2e6 Msol of dust, is safely confirmed by their detection in HI maps. Overall, we estimate a total dust mass of ~5.8e7 Msol, about 78% of which is contained in the two main ring-like structures at 10 and 15 kpc, at an average temperature of 16.5 K. We find that the gas-to-dust ratio declines exponentially as a function of the galacto-centric distance, in agreement with the known metallicity gradient, with values ranging from 66 in the nucleus to ~275 in the outermost region. [Abridged]

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arXiv2012

Herschel-ATLAS: Multi-wavelength SEDs and physical properties of 250 micron-selected galaxies at z < 0.5

We present a pan-chromatic analysis of an unprecedented sample of 1402 250 micron-selected galaxies at z < 0.5 (mean z = 0.24) from the Herschel-ATLAS survey. We complement our Herschel 100-500 micron data with UV-K-band photometry from the Galaxy And Mass Assembly (GAMA) survey and apply the MAGPHYS energy-balance technique to produce pan-chromatic SEDs for a representative sample of 250 micron selected galaxies spanning the most recent 5 Gyr of cosmic history. We derive estimates of physical parameters, including star formation rates, stellar masses, dust masses and infrared luminosities. The typical H-ATLAS galaxy at z < 0.5 has a far-infrared luminosity in the range 10^10 - 10^12 Lsolar (SFR: 1-50 Msolar/yr) thus is broadly representative of normal star forming galaxies over this redshift range. We show that 250 micron-selected galaxies contain a larger mass of dust at a given infra-red luminosity or star formation rate than previous samples selected at 60 micron from IRAS. We derive typical SEDs for H-ATLAS galaxies, and show that the emergent SED shape is most sensitive to specific star formation rate. The optical-UV SEDs also become more reddened due to dust at higher redshifts. Our template SEDs are significantly cooler than existing infra-red templates. They may therefore be most appropriate for inferring total IR luminosities from moderate redshift submillimetre selected samples and for inclusion in models of the lower redshift submillimetre galaxy populations.

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arXiv2012

The Herschel Exploitation of Local Galaxy Andromeda (HELGA) II: Dust and Gas in Andromeda

We present an analysis of the dust and gas in Andromeda, using Herschel images sampling the entire far-infrared peak. We fit a modified-blackbody model to ~4000 quasi-independent pixels with spatial resolution of ~140pc and find that a variable dust-emissivity index (beta) is required to fit the data. We find no significant long-wavelength excess above this model suggesting there is no cold dust component. We show that the gas-to-dust ratio varies radially, increasing from ~20 in the center to ~70 in the star-forming ring at 10kpc, consistent with the metallicity gradient. In the 10kpc ring the average beta is ~1.9, in good agreement with values determined for the Milky Way (MW). However, in contrast to the MW, we find significant radial variations in beta, which increases from 1.9 at 10kpc to ~2.5 at a radius of 3.1kpc and then decreases to 1.7 in the center. The dust temperature is fairly constant in the 10kpc ring (ranging from 17-20K), but increases strongly in the bulge to ~30K. Within 3.1kpc we find the dust temperature is highly correlated with the 3.6 micron flux, suggesting the general stellar population in the bulge is the dominant source of dust heating there. At larger radii, there is a weak correlation between the star formation rate and dust temperature. We find no evidence for 'dark gas' in M31 in contrast to recent results for the MW. Finally, we obtained an estimate of the CO X-factor by minimising the dispersion in the gas-to-dust ratio, obtaining a value of (1.9+/-0.4)x10^20 cm^-2 [K kms^-1]^-1.

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