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Y. Sophia Dai

Publications and source records attributed to Y. Sophia Dai.

At least 37 records · Page 2Linked to original sources

Dust and Cold Gas Properties of Starburst HyLIRG-Quasars at $z \sim 2.5$

Some high-z active galactic nuclei (AGNs) are found to reside in extreme star-forming galaxies, such as hyper-luminous infrared galaxies (HyLIRGs), with AGN-removed $L_{\rm{IR}}$ of $>10^{13} L_{\rm{\odot}}$. In this paper, we report NOEMA observations of six apparent starburst HyLIRGs associated with optical quasars at $z\sim2-3$ in the Stripe 82 field, to study their dust and molecular CO properties. Five out of the six candidates are detected with CO(4-3) or CO(5-4) emission, and four in 2mm dust continuum. Based on the linewidth-$L'_{\rm{CO(1-0)}}$ diagnostics, we find that four galaxies are likely unlensed or weakly lensed sources. The molecular gas mass is in the range of $μM_{\rm{H_2}} \sim0.8-9.7\times10^{10} M_{\odot}$ (with $α= 0.8 M_{\odot} (\rm{K km s^{-1} pc^2})^{-1}$ and $μ$ is the unknown possible gravitational magnification factor). We fit their SEDs, after including the observed 2mm fluxes and upper limits, and estimate their apparent (uncorrected for possible lensing effect) star formation rates ($μ$SFRs) to be $\sim400-2500$ $M_{\rm{\odot}} \rm{yr^{-1}}$ with depletion time of $\sim20-110$ Myr. We notice interesting offsets, of $\sim10-40$ kpc spatially or $\sim1000-2000$ km s$^{-1}$ spectroscopically, between the optical quasar and the mm continuum or CO emissions. The observed velocity shift is likely related to the blueshifted broad-emission-line region of quasars, though mergers or recoiling black holes are also possible causes, which can explain the spatial offset and the high intrinsic SFRs in the HyLIRG-quasar systems.

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Active Galactic Nuclei in a Mid-Infrared Selected Galaxy Sample at z>0.13: [Ne V]3426 Line Emission as a Benchmark

We present a 24 um-selected spectroscopic sample z > 0.13 (median z = 0.41) in the Lockman Hole field, consisting of 4035 spectra. Our aim is to identify AGNs and determine their fraction in this mid-infrared selected sample. In this work, we use the [Ne V]3426 emission line to spectroscopically identify AGNs. Combined with broad-line Type I AGNs selected in our previous study, our sample consists of 887 (22%) spectroscopically confirmed AGNs. We perform a stacking analysis on the remaining spectra, and find that in various MIR-wedge-selected AGN candidates, the stacked spectra still show significant [Ne V]3426 emission, In contrast, no clear [Ne V]3426 signal is detected in non-AGN candidates falling outside the wedges. Assuming a range of AGN mid-IR SED slope of 0.3< alpha <0.7, and an average star-forming relation derived from 65 star-forming templates, we develop a robust method to separate the AGN and star-forming contributions to the mid-IR SEDs using the rest-frame L12 /L1.6 vs L4.5 /L1.6 diagram. We separate the objects into bins of L12 , and find that AGN fraction increases with increasing L12. We also find that the stacked [Ne V]3426 strength scales with L12 . The pure AGN luminosity at 12 um exhibits a positive correlation with the star formation rates, indicating possible co-evolution and common gas supply between the AGN and their host galaxies. Varying population properties across the redshift range explored contribute to the observed correlation.

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UV-Bright Star-Forming Clumps and Their Host Galaxies in UVCANDELS at 0.5 $\leq$ z $\leq$ 1

Giant star-forming clumps are a prominent feature of star-forming galaxies (SFGs) and contain important clues on galaxy formation and evolution. However, basic demographics of clumps and their host galaxies remain uncertain. Using the HST/WFC3 F275W images from the Ultraviolet Imaging of the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (UVCANDELS), we detect and analyze giant star-forming clumps in galaxies at 0.5 $\leq$ z $\leq$ 1, connecting two epochs when clumps are common (at cosmic high-noon, z $\sim$ 2) and rare (in the local universe). We construct a clump sample whose rest-frame 1600 Å luminosity is 3 times higher than the most luminous local HII regions (M$_{UV} \leq -$16 AB). In our sample, 35 $\pm$ 3$\%$ of low-mass galaxies (log[M$_{*}$/M$_{\odot}$] $<$ 10) are clumpy (i.e., containing at least one off-center clump). This fraction changes to 22 $\pm$ 3$\%$ and 22 $\pm$ 4$\%$ for intermediate (10 $\leq$ log[M$_{*}$/M$_{\odot}$] $\leq$ 10.5) and high-mass (log[M$_{*}$/M$_{\odot}$] $>$ 10.5) galaxies in agreement with previous studies. When compared to similar-mass non-clumpy SFGs, low- and intermediate-mass clumpy SFGs tend to have higher SFRs and bluer rest-frame U-V colors, while high-mass clumpy SFGs tend to be larger than non-clumpy SFGs. However, clumpy and non-clumpy SFGs have similar Sérsic index, indicating a similar underlying density profile. Furthermore, we investigate how UV luminosity of star-forming regions correlates with the physical properties of host galaxies. On average, more luminous star-forming regions reside in more luminous, smaller, and/or higher-specific SFR galaxies and are found closer to their hosts' galactic center.

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Investigating the Dominant Environmental Quenching Process in UVCANDELS/COSMOS Groups

We explore how the fraction of quenched galaxies changes in groups of galaxies with respect to the distance to the center of the group, redshift, and stellar mass to determine the dominant process of environmental quenching in $0.2 < z < 0.8$ groups. We use new UV data from the UVCANDELS project in addition to existing multiband photometry to derive new galaxy physical properties of the group galaxies from the zCOSMOS 20k Group Catalog. Limiting our analysis to a complete sample of log$(M_*/M_{\odot})>10.56$ group galaxies we find that the probability of being quenched increases slowly with decreasing redshift, diverging from the stagnant field galaxy population. A corresponding analysis on how the probability of being quenched increases with time within groups suggests that the dominant environmental quenching process is characterized by slow ($\sim$Gyr) timescales. We find a quenching time of approximately $4.91^{+0.91}_{-1.47} $Gyrs, consistent with the slow processes of strangulation (Larson et al. 1980) and delayed-then-rapid quenching (Wetzel et al. 2013 arXiv:1206.3571v2 [astro-ph.CO]), although more data are needed to confirm this result.

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A spatially resolved analysis of star-formation burstiness by comparing UV and H$α$ in galaxies at z$\sim$1 with UVCANDELS

The UltraViolet imaging of the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey Fields (UVCANDELS) program provides HST/UVIS F275W imaging for four CANDELS fields. We combine this UV imaging with existing HST/near-IR grism spectroscopy from 3D-HST$+$AGHAST to directly compare the resolved rest-frame UV and H$α$ emission for a sample of 979 galaxies at $0.7 1.5$), suggesting that bursty star-formation is likely prevalent in the outskirts of even the most massive galaxies but is likely over-shadowed by their brighter cores. Furthermore, we present the UV-to-H$α$ ratio as a function of galaxy surface brightness, a proxy for stellar mass surface density, and find that regions below $\sim$10$^{7.5}$ M$_\odot$ kpc$^{-2}$ are consistent with bursty star-formation, regardless of their galaxy stellar mass, potentially suggesting that local star-formation is independent of global galaxy properties at the smallest scales. Lastly, we find galaxies at $z>1.1$ to have bursty star-formation regardless of radius or surface brightness.

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Fraction of Clumpy Star-Forming Galaxies at $0.5\leq z\leq 3$ in UVCANDELS: Dependence on Stellar Mass and Environment

High-resolution imaging of galaxies in rest-frame UV has revealed the existence of giant star-forming clumps prevalent in high redshift galaxies. Studying these sub-structures provides important information about their formation and evolution and informs theoretical galaxy evolution models. We present a new method to identify clumps in galaxies' high-resolution rest-frame UV images. Using imaging data from CANDELS and UVCANDELS, we identify star-forming clumps in an HST/F160W$\leq 25$ AB mag sample of 6767 galaxies at $0.5\leq z\leq 3$ in four fields, GOODS-N, GOODS-S, EGS, and COSMOS. We use a low-pass band filter in Fourier space to reconstruct the background image of a galaxy and detect small-scale features (clumps) on the background-subtracted image. Clumpy galaxies are defined as those having at least one off-center clump that contributes a minimum of 10$\%$ of the galaxy's total rest-frame UV flux. We measure the fraction of clumpy galaxies ($\rm f_{clumpy}$) as a function of stellar mass, redshift, and galaxy environment. Our results indicate that $\rm f_{clumpy}$ increases with redshift, reaching $\sim 65\%$ at $z\sim 1.5$. We also find that $\rm f_{clumpy}$ in low-mass galaxies ($\rm 9.5\leq log(M_*/M_\odot)\leq 10$) is 10$\%$ higher compared to that of their high-mass counterparts ($\rm log(M_*/M_\odot)>10.5$). Moreover, we find no evidence of significant environmental dependence of $\rm f_{clumpy}$ for galaxies at the redshift range of this study. Our results suggest that the fragmentation of gas clouds under violent disk instability remains the primary driving mechanism for clump formation, and incidents common in dense environments, such as mergers, are not the dominant processes.

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The JCMT BISTRO Survey: A Spiral Magnetic Field in a Hub-filament Structure, Monoceros R2

We present and analyze observations of polarized dust emission at 850 $μ$m towards the central 1 pc $\times$ 1 pc hub-filament structure of Monoceros R2 (Mon R2). The data are obtained with SCUBA-2/POL-2 on the James Clerk Maxwell Telescope (JCMT) as part of the BISTRO (B-fields in Star-forming Region Observations) survey. The orientations of the magnetic field follow the spiral structure of Mon R2, which are well-described by an axisymmetric magnetic field model. We estimate the turbulent component of the magnetic field using the angle difference between our observations and the best-fit model of the underlying large-scale mean magnetic field. This estimate is used to calculate the magnetic field strength using the Davis-Chandrasekhar-Fermi method, for which we also obtain the distribution of volume density and velocity dispersion using a column density map derived from $Herschel$ data and the C$^{18}$O ($J$ = 3-2) data taken with HARP on the JCMT, respectively. We make maps of magnetic field strengths and mass-to-flux ratios, finding that magnetic field strengths vary from 0.02 to 3.64 mG with a mean value of 1.0 $\pm$ 0.06 mG, and the mean critical mass-to-flux ratio is 0.47 $\pm$ 0.02. Additionally, the mean Alfvén Mach number is 0.35 $\pm$ 0.01. This suggests that in Mon R2, magnetic fields provide resistance against large-scale gravitational collapse, and magnetic pressure exceeds turbulent pressure. We also investigate the properties of each filament in Mon R2. Most of the filaments are aligned along the magnetic field direction and are magnetically sub-critical.

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SDSS-IV MaNGA: How Galaxy Interactions Influence Active Galactic Nuclei

We present a comparative study of active galactic nuclei (AGN) between galaxy pairs and isolated galaxies with the final data release of the MaNGA integral field spectroscopic survey. We build a sample of 391 kinematic galaxy pairs within the footprint of the survey and select AGN using the survey's spectra. We use the comoving volume densities of the AGN samples to quantify the effects that tidal interactions have on the triggering of nuclear accretion. Our hypothesis is that the pair sample contains AGN that are triggered by not only stochastic accretion but also tidally induced accretion and correlated accretion. With the level of stochastically triggered AGN fixed by the control sample, we model the strength of tidally induced accretion and correlated accretion as a function of projected separation (rp) and compare the model expectations with the observed volume densities of dual AGN and offset AGN (single AGN in a pair). At rp ~ 10 kpc, we find that tidal interactions induce ~30% more AGN than stochastic fueling and cause ~12% of the offset AGN to become dual AGN because of correlations. The strength of both these effects decreases with increasing rp. We also find that the OIII luminosities of the AGN in galaxy pairs are consistent with those found in isolated galaxies, likely because stochastically fed AGN dominate even among close pairs. Our results illustrates that while we can detect tidally induced effects statistically, it is challenging to separate tidally induced AGN and stochastically triggered AGN in interacting galaxies.

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The JCMT BISTRO-2 Survey: Magnetic Fields of the Massive DR21 Filament

We present 850 $μ$m dust polarization observations of the massive DR21 filament from the B-fields In STar-forming Region Observations (BISTRO) survey, using the POL-2 polarimeter and the SCUBA-2 camera on the James Clerk Maxwell Telescope. We detect ordered magnetic fields perpendicular to the parsec-scale ridge of the DR21 main filament. In the sub-filaments, the magnetic fields are mainly parallel to the filamentary structures and smoothly connect to the magnetic fields of the main filament. We compare the POL-2 and Planck dust polarization observations to study the magnetic field structures of the DR21 filament on 0.1--10 pc scales. The magnetic fields revealed in the Planck data are well aligned with those of the POL-2 data, indicating a smooth variation of magnetic fields from large to small scales. The plane-of-sky magnetic field strengths derived from angular dispersion functions of dust polarization are 0.6--1.0 mG in the DR21 filament and $\sim$ 0.1 mG in the surrounding ambient gas. The mass-to-flux ratios are found to be magnetically supercritical in the filament and slightly subcritical to nearly critical in the ambient gas. The alignment between column density structures and magnetic fields changes from random alignment in the low-density ambient gas probed by Planck to mostly perpendicular in the high-density main filament probed by JCMT. The magnetic field structures of the DR21 filament are in agreement with MHD simulations of a strongly magnetized medium, suggesting that magnetic fields play an important role in shaping the DR21 main filament and sub-filaments.

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The Velocity Map Asymmetry of Ionized Gas in MaNGA. I. The Catalog and General Properties

The SDSS-IV MaNGA survey has measured two-dimensional maps of emission line velocities for a statistically powerful sample of nearby galaxies. The asymmetric features of these kinematics maps reflect the non-rotational component of a galaxy's internal motion of ionized gas. In this study, we present a catalog of kinematic asymmetry measurement of $Hα$ velocity map of a sample of 5353 MaNGA galaxies. Based on this catalog, we find that `special' galaxies (e.g. merging galaxies, barred galaxies, and AGN host galaxies) contain more galaxies with highly asymmetric velocity maps. However, we notice that more than half of galaxies with high kinematic asymmetry in our sample are quite `regular'. For those `regular' galaxies, kinematic asymmetry shows a significant anti-correlation with stellar mass at $\log M_\star < 9.7$, while such a trend becomes very weak at $\log M_\star>9.7$. Moreover, at a given stellar mass, the kinematic asymmetry shows weak correlations with photometric morphology, star formation rate, and environment, while it is independent of HI gas content. We also have quantified the observational effects in the kinematic asymmetry measurement. We find that both the signal-to-noise ratio of $Hα$ flux and disk inclination angle contribute to the measures of kinematic asymmetry, while the physical spatial resolution is an irrelevant factor inside the MaNGA redshift coverage.

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Bright z~9 Galaxies in Parallel: The Bright End of the Rest-UV Luminosity Function from HST Parallel Programs

The abundance of bright galaxies at z>8 can provide key constraints on models of galaxy formation and evolution, as the predicted abundance varies greatly when different physical prescriptions for gas cooling and star formation are implemented. We present the results of a search for bright z=9-10 galaxies selected from pure-parallel Hubble Space Telescope imaging programs. We include 132 fields observed as part of the Brightest of Reionizing Galaxies survey, the Hubble Infrared Pure Parallel Imaging Extragalactic Survey, and the WFC3 Infrared Spectroscopic Parallel survey. These observations cover a total of 620 sq. arcmin, about 70% of which is also covered with Spitzer Space Telescope infrared imaging. We identify thirteen candidate galaxies in the range 8.3<z<11 with 24.5 < m_H < 26.5 (-22.9 < M_UV < -21.2). This sample capitalizes on the uncorrelated nature of pure parallel observations to overcome cosmic variance and leverages a full multi-wavelength selection process to minimize contamination without sacrificing completeness. We perform detailed completeness and contamination analyses, and present measurements of the bright end of the UV luminosity function using a pseudo-binning technique. We find a number density consistent with results from Finkelstein et al. (2022) and other searches in HST parallel fields. These bright candidates likely reside in overdensities, potentially representing some of the earliest sites of cosmic reionization. These new candidates are excellent targets for follow-up with JWST, and four of them will be observed with the NIRSpec prism in Cycle 1.

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Groups and protocluster candidates in the CLAUDS and HSC-SSP joint deep surveys

Using the extended halo-based group finder developed by Yang et al. (2021), which is able to deal with galaxies via spectroscopic and photometric redshifts simultaneously, we construct galaxy group and candidate protocluster catalogs in a wide redshift range ($0 < z < 6$) from the joint CFHT Large Area $U$-band Deep Survey (CLAUDS) and Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) deep data set. Based on a selection of 5,607,052 galaxies with $i$-band magnitude $m_{i} < 26$ and a sky coverage of $34.41\ {\rm deg}^2$, we identify a total of 2,232,134 groups, within which 402,947 groups have at least three member galaxies. We have visually checked and discussed the general properties of those richest groups at redshift $z>2.0$. By checking the galaxy number distributions within a $5-7\ h^{-1}\mathrm{Mpc}$ projected separation and a redshift difference $Δz \le 0.1$ around those richest groups at redshift $z>2$, we identified a list of 761, 343 and 43 protocluster candidates in the redshift bins $2\leq z<3$, $3\leq z<4$ and $z \geq 4$, respectively. In general, these catalogs of galaxy groups and protocluster candidates will provide useful environmental information in probing galaxy evolution along the cosmic time.

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The Seventeenth Data Release of the Sloan Digital Sky Surveys: Complete Release of MaNGA, MaStar and APOGEE-2 Data

This paper documents the seventeenth data release (DR17) from the Sloan Digital Sky Surveys; the fifth and final release from the fourth phase (SDSS-IV). DR17 contains the complete release of the Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey, which reached its goal of surveying over 10,000 nearby galaxies. The complete release of the MaNGA Stellar Library (MaStar) accompanies this data, providing observations of almost 30,000 stars through the MaNGA instrument during bright time. DR17 also contains the complete release of the Apache Point Observatory Galactic Evolution Experiment 2 (APOGEE-2) survey which publicly releases infra-red spectra of over 650,000 stars. The main sample from the Extended Baryon Oscillation Spectroscopic Survey (eBOSS), as well as the sub-survey Time Domain Spectroscopic Survey (TDSS) data were fully released in DR16. New single-fiber optical spectroscopy released in DR17 is from the SPectroscipic IDentification of ERosita Survey (SPIDERS) sub-survey and the eBOSS-RM program. Along with the primary data sets, DR17 includes 25 new or updated Value Added Catalogs (VACs). This paper concludes the release of SDSS-IV survey data. SDSS continues into its fifth phase with observations already underway for the Milky Way Mapper (MWM), Local Volume Mapper (LVM) and Black Hole Mapper (BHM) surveys.

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Independent Core Rotation in Massive Filaments in Orion

We present high-angular-resolution ALMA (Atacama Large Millimeter Array) images of N$_{2}$H$^{+}$ (1--0) that has been combined with those from the Nobeyama telescope toward OMC-2 and OMC-3 filamentary regions. The filaments (with typical widths of $\sim$ 0.1 pc) and dense cores are resolved. The measured 2D velocity gradients of cores are between 1.3 and 16.7 km\,s$^{-1}$\,pc$^{-1}$, corresponding to a specific angular momentum ($J/M$) between 0.0012 and 0.016 pc\,km\,s$^{-1}$. With respect to the core size $R$, the specific angular momentum follows a power law $J/M \propto R^{1.52~\pm~0.14}$. The ratio ($β$) between the rotational energy and gravitational energy ranges from 0.00041 to 0.094, indicating insignificant support from rotation against gravitational collapse. We further focus on the alignment between the cores' rotational axes, which is defined to be perpendicular to the direction of the velocity gradient ($θ_{G}$), and the direction of elongation of filaments ($θ_{f}$) in this massive star-forming region. The distribution of the angle between $θ_{f}$ and $θ_{G}$ was f ound to be random, i.e. the cores' rotational axes have no discernible correlation with the elongation of their hosting filament. This implies that, in terms of angular momentum, the cores have evolved to be dynamically independent from their natal filaments.

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An IFU View of the Active Galactic Nuclei in MaNGA Galaxy Pairs

The role of active galactic nuclei (AGNs) during galaxy interactions and how they influence the star formation in the system are still under debate. We use a sample of 1156 galaxies in galaxy pairs or mergers (hereafter `pairs') from the MaNGA survey. This pair sample is selected by the velocity offset, projected separation, and morphology, and is further classified into four cases along the merger sequence based on morphological signatures. We then identify a total of 61 (5.5%) AGNs in pairs based on the emission-line diagnostics. No evolution of the AGN fraction is found, either along the merger sequence or compared to isolated galaxies (5.0%). We observe a higher fraction of passive galaxies in galaxy pairs, especially in the pre-merging cases, and associate the higher fraction to their environmental dependence. The isolated AGN and AGN in pairs show similar distributions in their global stellar mass, star formation rate (SFR), and central [OIII] surface brightness. AGNs in pairs show radial profiles of increasing specific SFR and declining Dn4000 from center to outskirts, and no significant difference from the isolated AGNs. This is clearly different from star-forming galaxies (SFGs) in our pair sample, which show enhanced central star formation, as reported before. AGNs in pairs have lower Balmer decrements at outer regions, possibly indicating less dust attenuation. Our findings suggest that AGNs likely follow an inside-out quenching and the merger impact on the star formation in AGNs is less prominent than in SFGs.

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Spectroscopically Identified Emission Line Galaxy Pairs in the WISP survey

We identify a sample of spectroscopically measured emission line galaxy (ELG) pairs up to z=1.6 from the WFC3 Infrared Spectroscopic Parallels (WISP) survey. WISP obtained slitless, near-infrared grism spectroscopy along with direct imaging in the J and H bands by observing in the pure-parallel mode with the Wide Field Camera Three (WFC3) on the Hubble Space Telescope (HST). From our search of 419 WISP fields covering an area of ~0.5 deg$^{2}$, we find 413 ELG pair systems, mostly Halpha emitters. We then derive reliable star formation rates (SFRs) based on the attenuation-corrected Halpha fluxes. Compared to isolated galaxies, we find an average SFR enhancement of 40%-65%, which is stronger for major pairs and pairs with smaller velocity separations (Delta_v < 300 km/s). Based on the stacked spectra from various subsamples, we study the trends of emission line ratios in pairs, and find a general consistency with enhanced lower-ionization lines. We study the pair fraction among ELGs, and find a marginally significant increase with redshift $f \propto (1+z)^α$, where the power-law index α=0.58$\pm$0.17 from $z\sim$0.2 to $z\sim$1.6. The fraction of Active galactic Nuclei (AGNs), is found to be the same in the ELG pairs as compared to isolated ELGs.

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Searching for low-redshift faint galaxies with MMT/Hectospec

We present redshifts for 2753 low-redshift galaxies between $0.03 \lesssim z_{\rm spec}\lesssim0.5$ with 18 $\leq$ $r$ $\leq$ 22 obtained with Hectospec at the Multi-Mirror Telescope (MMT). The observations targeted the XMM-LSS, ELAIS-N1 and DEEP2-3 fields, each of which covers $\sim$ 1 deg$^2$. These fields are also part of the recently completed CFHT Large Area U-band Deep Survey (CLAUDS) and on-going Hyper Suprime-Cam deep fields surveys. The efficiency of our technique for selecting low-redshift galaxies is confirmed by the redshift distribution of our sources. In addition to redshifts, these high S/N spectra are used to measure ages, metallicities, and nuclear activity levels. In combination with the photometric catalogue in $u$, $g$, $r$, $i$, $z$, $y$ down to 27 AB mag, we are able to study the galaxy population down to stellar masses of $\sim$ 10$^8 M_\odot$ . This paper presents the observational strategy, the reduction procedure and properties of the galaxy sample.

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Spectral energy distribution similarity of the local galaxies and the 3.6um selected galaxies from the Spitzer Extended Deep Survey

The Spitzer Extended Deep Survey (SEDS) as a deep and wide mid-infrared (MIR) survey project provides a sample of 500000+ sources spreading 1.46 square degree and a depth of 26 AB mag (3$σ$). Combining with the previous available data, we build a PSF-matched multi-wavelength photometry catalog from u band to 8$μ$m. We fit the SEDS galaxies spectral energy distributions by the local galaxy templates. The results show that the SEDS galaxy can be fitted well, indicating the high redshift galaxy ($z \sim 1$) shares the same templates with the local galaxies. This study would facilitate the further study of the galaxy luminosity and high redshift mass function.

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