arXiv subjects

Guanwen Fang

Publications and source records attributed to Guanwen Fang.

At least 37 records · Page 2Linked to original sources

arXiv2023

Multiwavelength Analysis of a Nearby Heavily Obscured AGN in NGC 449

We presented the multiwavelength analysis of a heavily obscured active galactic nucleus (AGN) in NGC 449. We first constructed a broadband X-ray spectrum using the latest NuSTAR and XMM-Newton data. Its column density ($\simeq 10^{24} \rm{cm}^{-2}$) and photon index ($Γ\simeq 2.4$) were reliably obtained by analyzing the broadband X-ray spectrum. However, the scattering fraction and the intrinsic X-ray luminosity could not be well constrained. Combined with the information obtained from the mid-infrared (mid-IR) spectrum and spectral energy distribution (SED) fitting, we derived its intrinsic X-ray luminosity ($\simeq 8.54\times 10^{42} \ \rm{erg\ s}^{-1}$) and scattering fraction ($f_{\rm{scat}}\simeq 0.26\%$). In addition, we also derived the following results: (1). The mass accretion rate of central AGN is about $2.54 \times 10^{-2} \rm{M}_\odot\ \rm{yr}^{-1}$, and the Eddington ratio is $8.39\times 10^{-2}$; (2). The torus of this AGN has a high gas-to-dust ratio ($N_{\rm H}/A_{\rm V}=8.40\times 10^{22}\ \rm{cm}^{-2}\ \rm{mag}^{-1}$); (3). The host galaxy and the central AGN are both in the early stage of co-evolution.

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arXiv2022

The physical properties of massive green valley galaxies as a function of environments at $0.5<z<2.5$ in 3D-\textit{HST}/CANDELS fields

To investigate the effects of environment in the quenching phase, we study the empirical relations for green valley (GV) galaxies between overdensity and other physical properties (i.e., effective radius $r_{\rm e}$, Sérsic indices $n$, and specific star formation rate sSFR). Based on five 3D-{\it HST}/CANDELS fields, we construct a large sample of 2126 massive ($M_{\star} > 10^{10} M_{\sun}$) GV galaxies at $0.5<z<2.5$ and split it into the higher overdensity quarter and the lower overdensity quarter. The results shows that GV galaxies in denser environment have higher $n$ values and lower sSFR at $0.5< z <1$, while there is no discernible distinction at $1 < z < 2.5$. No significant enlarging or shrinking is found for GV galaxies in different environments within the same redshift bin. It suggests that a dense environment would promote the growth of bulge and suppress star formation activity of GV galaxies at $0.5< z <1.5$, but would not affect the galaxy size. We also study the dependence of the fraction of three populations (Blue Cloud, Green Valley, and Red Sequence) on both environments and $M_{\star}$. At a given $M_{\star}$, blue cloud fraction goes down with increasing environment density, while red sequence fraction is opposite. For the most massive GV galaxies, a sharp drop appears in the denser environment. Coupled with the mass dependence of three fractions in different redshift bins, our result implies that stellar mass and environments jointly promote the quenching process. Such dual effect is also confirmed by re-calculating the new effective GV fraction as the number of GV galaxies over the number of non-quiescent galaxies.

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arXiv2021

AGN lifetimes in UV-selected galaxies: a clue to supermassive black hole-host galaxy coevolution

The coevolution between supermassive black holes (SMBHs) and their host galaxies has been proposed for more than a decade, albeit with little direct evidence about black hole accretion activities regulating galaxy star formation at $z>1$. In this paper, we study the lifetimes of X-ray active galactic nuclei (AGNs) in $UV$-selected red sequence (RS), blue cloud (BC) and green valley (GV) galaxies, finding that AGN accretion activities are most prominent in GV galaxies at $z\sim1.5-2$, compared with RS and BC galaxies. We also compare AGN accretion timescales with typical color transition timescales of $UV$-selected galaxies. We find that the lifetime of GV galaxies at $z\sim1.5-2$ is very close to the typical timescale when the AGNs residing in them stay in the high-accretion-rate mode at these redshifts; for BC galaxies, the consistency between the color transition timescale and the black hole strong accretion lifetime is more likely to happen at lower redshifts ($z<1$). Our results support the scenario where AGN accretion activities govern $UV$ color transitions of host galaxies, making galaxies and their central SMBHs coevolve with each other.

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arXiv2021

Morphological Transformation and Star Formation Quenching of Massive Galaxies at 0.5 < z < 2.5 in 3D-HST/CANDELS

To figure out the effect of stellar mass and local environment on morphological transformation and star formation quenching in galaxies, we use the massive ($M_* \geq 10^{10} M_{\odot}$) galaxies at $0.5 \leq z \leq 2.5$ in five fields of 3D-HST/CANDELS. Based on the {\it UVJ} diagnosis and the possibility of possessing spheroid, our sample of massive galaxies are classified into four populations: quiescent early-type galaxies (qEs), quiescent late-type galaxies (qLs), star-forming early-type galaxies (sEs), and star-forming late-type galaxies (sLs). It is found that the quiescent fraction is significantly elevated at the high ends of mass and local environmental overdensity, which suggests a clear dependence of quenching on both mass and local environment. Over cosmic time, the mass dependence of galaxy quiescence decreases while the local environment dependence increases. The early-type fraction is found to be larger only at high-mass end, indicating a evident mass dependence of morphological transformation. This mass dependence becomes more significant at lower redshifts. Among the four populations, the fraction of active galactic nucleus (AGN) in the qLs peaks at $2<z \leq 2.5$, and rapidly declines with cosmic time. The sEs are found to have higher AGN fractions of $20-30\%$ at $0.5\leq z<2$ . The redshift evolution of AGN fractions in the qLs and sEs suggests that the AGN feedback could have played important roles in the formation of the qLs and sEs.

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arXiv2021

The effect of environment on star formation activity and morphology at 0.5 < z < 2.5 in CANDELS

To explore the effect of environment on star-formation and morphological transformation of high-redshift galaxies, we present a robust estimation of localized galaxy overdensity using a density estimator within the Bayesian probability framework.The maps of environmental overdensity at $0.5< z< 2.5$ are constructed for the five CANDELS fields. In general, the quiescent fraction increases with overdensity and stellar mass. Stellar mass dominates the star formation quenching for massive galaxies, while environmental quenching tends to be more effective for the low-mass galaxies at $0.5< z < 1$. For the most massive galaxies ($M_* > 10^{10.8} M_{\odot}$), the effect of environmental quenching is still significant up to $z \sim 2.5$. No significant environmental dependence is found in the distributions of Sérsic index and effective radius for SFGs and QGs separately. The primary role of environment might be to control the quiescent fraction. And the morphological parameters are primarily connected with star formation status. The similarity in the trends of quiescent fraction and Sérsic index along with stellar mass indicates that morphological transformation is accompanied with star formation quenching.

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arXiv2021

The effect of the morphological quenching mechanism on star formation activity at 0.5 < z < 1.5 in 3D-HST/CANDELS

Several mechanisms for the transformation of blue star-forming to red quiescent galaxies have been proposed, and the green valley (GV) galaxies amid them are widely accepted in a transitional phase. Thus, comparing the morphological and environmental differences of the GV galaxies with early-type disks (ETDs; bulge dominated and having a disk) and late-type disks (LTDs; disk dominated) is suitable for distinguishing the corresponding quenching mechanisms. A large population of massive ($M_* \geqslant 10^{10}M_\odot$) GV galaxies at $0.5 \leqslant z \leqslant 1.5$ in 3D-HST/CANDELS is selected using extinction-corrected $(U-V)_{\rm rest}$ color. After eliminating any possible active galactic nucleus candidates and considering the "mass-matching", we finally construct two comparable samples of GV galaxies with either 319 ETD or 319 LTD galaxies. Compared to the LTD galaxies, it is found that the ETD galaxies possess higher concentration index and lower specific star formation rate, whereas the environments surrounding them are not different. This may suggest that the morphological quenching may dominate the star formation activity of massive GV galaxies rather than the environmental quenching. To quantify the correlation between the galaxy morphology and the star formation activity, we define a dimensionless morphology quenching efficiency $Q_{\rm mor}$ and find that $Q_{\rm mor}$ is not sensitive to the stellar mass and redshift. When the difference between the average star formation rate of ETD and LTD galaxies is about 0.7 $M_\odot \rm \;yr^{-1}$, the probability of $Q_{\rm mor}\gtrsim 0.2$ is higher than 90\%, which implies that the degree of morphological quenching in GV galaxies might be described by $Q_{\rm mor}\gtrsim 0.2$.

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arXiv2021

Estimating lifetimes of uv-selected massive galaxies at $0.5 \le z \le 2.5$ in the cosmos/ultravista field through clustering analyses

We present an estimation of lifetimes of massive galaxies with distinct UV colors at $0.5 \le z \le 2.5$ in the COSMOS/UltraVISTA field. After dividing the galaxy sample into subsamples of red sequence (RS), blue cloud (BC), and green valley (GV) galaxies in different redshift bins, according to their rest-frame extinction-corrected UV colors, we derive their lifetimes using clustering analyses. Several essentials that may influence the lifetime estimation have been explored, including the dark matter (DM) halo mass function (HMF), the width of redshift bin, the growth of DM halos within each redshift bin, and the stellar mass. We find that the HMF difference results in scatters of $\sim0.2$ dex on lifetime estimation; adopting a redshift bin width of $Δz = 0.5$ is good enough to estimate the lifetime; and no significant effect on lifetime estimation is found due to the growth of DM halos within each redshift bin. The galaxy subsamples with higher stellar masses generally have shorter lifetimes; however, the lifetimes among different subsamples at z > 1:5 tend to be independent of stellar mass. Consistently, the clustering-based lifetime for each galaxy subsample agrees well with that inferred using the spectral energy distribution modeling. Moreover, the lifetimes of the RS and BC galaxies also coincide well with their typical gas depletion timescales attributed to the consumption of star formation. Interestingly, the distinct lifetime behaviors of the GV galaxies at $z \le 1.5$ and $z>1.5$ can not be fully accounted for by their gas depletion timescales. Instead, this discrepancy between the lifetimes and gas depletion timescales of the GV galaxies suggests that there are additional physical processes, such as feedback of active galactic nuclei, accelerating the quenching of GV galaxies at high redshifts.

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arXiv2020

Dust Temperature of Compact Star-forming Galaxies at $ z \sim 1-3$ in 3D-{\it HST}/CANDELS

Recent simulation studies suggest that the compaction of star-forming galaxies (SFGs) at high redshift might be a critical process, during which the central bulge is being rapidly built, followed by quenching of the star formation. To explore dust properties of SFGs with compact morphology, we investigate the dependence of dust temperature, $T_{\rm{dust}}$, on their size and star formation activity, using a sample of massive SFGs with $\log (M_{\ast}/M_{\odot}) > 10$ at $1 < z < 3$, drawn from the 3D-{\it HST}/CANDELS database in combination with deep {\it Herschel} observations. $T_{\rm{dust}}$ is derived via fitting the mid-to-far-infrared photometry with a mid-infrared power law and a far-infrared modified blackbody. We find that both extended and compact SFGs generally follow a similar $T_{\rm{dust}}-z$ evolutionary track as that of the main-sequence galaxies. The compact SFGs seem to share similar dust temperature with extended SFGs. Despite the frequent occurrence of AGNs in compact SFGs, we do not observe any effect on dust caused by the presence of AGN in these galaxies during the compaction. Our results disfavor different ISM properties between compact and extended SFGs, suggesting that a rapid and violet compaction process might be not necessary for the formation of compact SFGs.

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arXiv2020

Piercing through Highly Obscured and Compton-thick AGNs in the Chandra Deep Fields. II. Are Highly Obscured AGNs the Missing Link in the Merger-Triggered AGN-Galaxy Coevolution Models?

By using a large highly obscured ($N_{\rm H} > 10^{23}\ \rm cm^{-2}$) AGN sample (294 sources at $z \sim 0-5$) selected from detailed X-ray spectral analyses in the deepest Chandra surveys, we explore distributions of these X-ray sources in various optical/IR/X-ray color-color diagrams and their host-galaxy properties, aiming at characterizing the nuclear obscuration environment and the triggering mechanism of highly obscured AGNs. We find that the refined IRAC color-color diagram fails to identify the majority of X-ray selected highly obscured AGNs, even for the most luminous sources with ${\rm log}\,L_{\rm X}\, \rm (erg\ s^{-1}) > 44$. Over 80% of our sources will not be selected as heavily obscured candidates using the flux ratio of $f_{\rm 24 μm}\, /\,f_R > 1000$ and $R - K > 4.5$ criteria, implying complex origins and conditions for the obscuring materials that are responsible for the heavy X-ray obscuration. The average star formation rate of highly obscured AGNs is similar to that of stellar mass- ($M_*$-) and $z$-controlled normal galaxies, while the lack of quiescent hosts is observed for the former. Partial correlation analyses imply that highly obscured AGN activity (traced by $L_{\rm X}$) appears to be more fundamentally related to $M_*$, and no dependence of $N_{\rm H}$ on either $M_*$ or SFR is detected. Morphology analyses reveal that 61% of our sources have a significant disk component, while only 27% of them exhibit irregular morphological signatures. These findings together point toward a scenario where secular processes (e.g., galactic-disk instabilities), instead of mergers, are most probable to be the leading mechanism that triggers accretion activities of X-ray-selected highly obscured AGNs.

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arXiv2020

The Compact Star-Forming Galaxies at $2<z<3$ in 3D-HST/CANDELS: AGN and Non-AGN Physical Properties

We investigate the differences in the stellar population properties, the structure, and the environment between massive compact star-forming galaxies (cSFGs) with or without active galactic nucleus (AGN) at $2<z<3$ in the five 3D-HST/CANDELS fields. In a sample of 221 massive cSFGs, we constitute the most complete AGN census so far, identifying 66 AGNs by the X-ray detection, the mid-infrared color criterion, and/or the SED fitting, while the rest (155) are non-AGNs. Further dividing these cSFGs into two redshift bins, i.e., $2<z<2.5$ and $2.5 \leq z<3$, we find that in each redshift bin the cSFGs with AGNs have similar distributions of the stellar mass, the specific star formation rate, and the ratio of $L_{\rm IR}$ to $L_{\rm UV}$ to those without AGNs. After having performed a two-dimensional surface brightness modeling for those cSFGs with X-ray-detected AGNs (37) to correct for the influence of the central point-like X-ray AGN on measuring the structural parameters of its host galaxy, we find that in each redshift bin the cSFGs with AGNs have comparable distributions of all concerned structural parameters, i.e., the Sersic index, the 20\%-light radius, the Gini coefficient, and the concentration index, to those without AGNs. With a gradual consumption of available gas and dust, the structure of cSFGs, indicated by the above structural parameters, seem to be slightly more concentrated with decreasing redshift. At $2<z<3$, the similar environment between cSFGs with and without AGNs suggests that their AGN activities are potentially triggered by internal secular processes, such as gravitational instabilities or/and dynamical friction.

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arXiv2020

Structures, stellar population properties, AGN fractions, and environments of massive compact galaxies at $1 < z < 2$ in 3D--{\it HST}/CANDELS

We present a study on structures and physical properties of massive ($M_* >10^{10} M_{\sun} $) compact galaxies at $1.0<z<2.0$ in five 3D--{\it HST}/CANDELS fields. Compared with the extended star-forming galaxies (eSFGs), compact star-forming galaxies (cSFGs) are found to have the lower level of star formation, and mainly distribute in the quiescent region of the {\it UVJ} diagram. The distributions of dust attenuation and S{é}rsic index support that the progenitors of cQGs are cSFGs, and cSFGs are at a transitional phase between eSFGs and cQGs. The prevalence of X-ray selected AGNs ($\sim 28\%$) is confirmed in the cSFGs at $1<z<2$ which indicates that the violent gas-rich processes such as merger and disk instability could drive the structure to be more compact, and trigger both star formation and black hole growth in the central regions. Our results support the "two-step" scenario that the cSFGs at $1<z<2$ are the intermediate population after compaction but before a quick quenching. Our analysis of parametric and nonparametric morphologies shows that cQGs (eQGs) are more concentrated and have less substructures than cSFGs (eSFGs), and quenching and compactness should be associated with each other. The cSFGs at $1.5<z<2$ ($1<z<1.5$) prefer to be in higher (lower) density environment, similar as cQGs (eSFGs). It suggests that merger or strong interaction might be the main driving mechanism of compaction at higher redshifts, whereas the disk instability of individual galaxies might play a more important role on the formation of cSFGs at lower redshifts.

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arXiv2019

Color dependence of clustering of massive galaxies at 0.5$\le z \le$2.5: similar spatial distributions between green valley galaxies and AGNs

We present a measurement of the spatial clustering of rest-frame UV-selected massive galaxies at $0.5\le z \le 2.5$ in the COSMOS/UltraVISTA field. Considering four separate redshift bins with $Δz=0.5$, we construct three galaxy populations, i.e., red sequence (RS), blue cloud (BC), and green valley (GV) galaxies, according to their rest-frame extinction-corrected UV colors. The correlation lengths of these populations are confirmed to be dependent on their rest-frame UV color and redshift: UV redder galaxies are found to be more clustered. In all redshift bins, the GV galaxies generally have medium clustering amplitudes and are hosted within dark matter halos whose masses are more or less between those of RS and BC galaxies; and the clustering amplitude of GV galaxies is close to that of AGNs in the same redshift bin, suggesting that AGN activity may be responsible for transforming galaxy colors. After carefully examining their stellar masses, we find that the clustering amplitudes of galaxy samples with different colors are all similar once they have a similar median stellar mass and that the median stellar mass alone may be a good predictor of galaxy clustering.

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arXiv2019

The mass dependence of structure, star formation rate, and mass assembly mode at 0.5<z< 2.5

To investigate the mass dependence of structural transformation and star formation quenching, we construct three galaxy samples using massive ($M_* > 10^{10} M_{\odot}$) red, green, and blue galaxy populations at $0.5<z<2.5$ in five 3D--{\it HST}/CANDELS fields. The structural parameters, including effective radius ($r_{\rm e}$), galaxy compactness ($Σ_{1.5}$), and second order moment of 20\% brightest pixels ($M_{20}$) are found to be correlated with stellar mass. Sérsic index ($n$), concentration ($C$), and Gini coefficient ($G$) seem to be insensitive to stellar mass. The morphological distinction between blue and red galaxies is found at a fixed mass bin, suggesting that quenching processes should be accompanied with transformations of galaxy structure and morphology. Except for $r_e$ and $Σ_{1.5}$ at high mass end, structural parameters of green galaxies are intermediate between red and blue galaxies in each stellar mass bin at $z < 2$, indicating green galaxies are at a transitional phase when blue galaxies are being quenched into quiescent statuses. The similar sizes and compactness for the blue and green galaxies at high-mass end implies that these galaxies will not appear to be significantly shrunk until they are completely quenched into red QGs. For the green galaxies at $0.5<z<1.5$, a morphological transformation sequence of bulge buildup can be seen as they are gradually shut down their star formation activities, while a faster morphological transformation is verified for the green galaxies at $1.5<z<2.5$.

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arXiv2018

The morphological evolution, AGN fractions, dust content, environments, and downsizing of massive green valley galaxies at 0.5<z<2.5 in 3D-HST/CANDELS

To explore the evolutionary connection among red, green, and blue galaxy populations, based on a sample of massive ($M_* > 10^{10} M_{\odot} $) galaxies at 0.5 2. A significant environmental difference is found between green and red galaxies at z<1.5. Green and blue galaxies at z>0.5 seem to have similar local density distributions, suggesting that environment quenching is not the major mechanism to cease star formation at z>0.5. The fractions of three populations as functions of mass support a "downsizing" quenching picture that the bulk of star formation in more massive galaxies is completed earlier than that of lower mass galaxies.

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arXiv2018

M101: Spectral Observations of HII Regions and Their Physical Properties

By using the Hectospec 6.5 m Multiple Mirror Telescope (MMT) and the 2.16 m telescope of National Astronomical Observatories, Chinese Academy of Sciences (NAOC), we obtained 188 high signal-to-noise ratio (S/N) spectra of HII regions in the nearby galaxy M101, which are the largest spectroscopic sample of HII regions for this galaxy so far. These spectra cover a wide range of regions on M101, which enables us to analyze two dimensional distributions of its physical properties. The physical parameters are derived from emission lines or stellar continuum, including stellar population age, electron temperature, oxygen abundance and etc. The oxygen abundances are derived using two empirical methods based on O3N2 and R$_{23}$ indicators, as well as the direct Te method when OIII$\lambda4363$ is available. By applying the harmonic decomposition analysis to the velocity field, we obtained line-of-sight rotation velocity of 71 km s$^{-1}$ and a position angle of 36 degree. The stellar age profile shows an old stellar population in galaxy center and a relative young stellar population in outer regions, suggesting an old bulge and a young disk. Oxygen abundance profile exhibits a clear break at $\sim$18 kpc, with a gradient of $-$0.0364 dex kpc$^{-1}$ in the inner region and $-$0.00686 dex kpc$^{-1}$ in the outer region. Our results agree with the "inside-out" disk growth scenario of M101.

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arXiv2017

Spectroscopic Observation and Analysis of HII regions in M33 with MMT: Temperatures and Oxygen Abundances

The spectra of 413 star-forming (or HII) regions in M33 (NGC 598) were observed by using the multifiber spectrograph of Hectospec at the 6.5-m Multiple Mirror Telescope (MMT). By using this homogeneous spectra sample, we measured the intensities of emission lines and some physical parameters, such as electron temperatures, electron densities, and metallicities. Oxygen abundances were derived via the direct method (when available) and two empirical strong-line methods, namely, O3N2 and N2. In the high-metallicity end, oxygen abundances derived from O3N2 calibration were higher than those derived from N2 index, indicating an inconsistency between O3N2 and N2 calibrations. We presented a detailed analysis of the spatial distribution of gas-phase oxygen abundances in M33 and confirmed the existence of the axisymmetric global metallicity distribution widely assumed in literature. Local variations were also observed and subsequently associated with spiral structures to provide evidence of radial migration driven by arms. Our O/H gradient fitted out to 1.1 $R_{25}$ resulted in slopes of $-0.17\pm0.03$, $-0.19\pm0.01$, and $-0.16\pm0.17$ dex $R_{25}^{-1}$ utilizing abundances from O3N2, N2 diagnostics, and direct method, respectively.

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arXiv2016

How accurate are infrared luminosities from monochromatic photometric extrapolation?

Template-based extrapolations from only one photometric band can be a cost-effective method to estimate the total infrared (IR) luminosities ($L_{\mathrm{IR}}$) of galaxies. By utilizing multi-wavelength data that covers across 0.35--500\,$\mathrm{μm}$ in GOODS-North and GOODS-South fields, we investigate the accuracy of this monochromatic extrapolated $L_{\mathrm{IR}}$ based on three IR spectral energy distribution (SED) templates (\citealt[CE01]{Chary2001}; \citealt[DH02]{Dale2002}; \citealt[W08]{Wuyts2008a}) out to $z\sim 3.5$. We find that the CE01 template provides the best estimate of $L_{\mathrm{IR}}$ in {\it Herschel}/PACS bands, while the DH02 template performs best in {\it Herschel}/SPIRE bands. To estimate $L_{\mathrm{IR}}$, we suggest that extrapolations from the available longest wavelength PACS band based on the CE01 template can be a good estimator. Moreover, if PACS measurement is unavailable, extrapolations from SPIRE observations but based on the \cite{Dale2002} template can also provide a statistically unbiased estimate for galaxies at $z\lesssim 2$. The emission of rest-frame 10--100\,$\mathrm{μm}$ range of IR SED can be well described by all the three templates, but only the DH02 template shows nearly unbiased estimate of the emission of the rest-frame submillimeter part.

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arXiv2016

Compact star forming galaxies as the progenitors of compact quiescent galaxies: clustering result

We present a measurement of the spatial clustering of massive compact galaxies at $1.2\le z \le 3$ in CANDELS/3D-HST fields. We obtain the correlation length for compact quiescent galaxies (cQGs) at $z\sim1.6$ of $r_{0}=7.1_{-2.6}^{+2.3}\ h^{-1}Mpc$ and compact star forming galaxies (cSFGs) at $z\sim2.5$ of $r_{0}=7.7_{-2.9}^{+2.7}\ h^{-1}Mpc$ assuming a power-law slope $γ=1.8$. The characteristic dark matter halo masses $M_H$ of cQGs at $z\sim1.6$ and cSFGs at $z\sim2.5$ are $\sim7.1\times 10^{12}\ h^{-1} M_\odot$ and $\sim4.4\times10^{12}\ h^{-1} M_\odot$, respectively. Our clustering result suggests that cQGs at $z\sim1.6$ are possibly the progenitors of local luminous ETGs and the descendants of cSFGs and SMGs at $z>2$. Thus an evolutionary connection involving SMGs, cSFGs, QSOs, cQGs and local luminous ETGs has been indicated by our clustering result.

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