Compact Groups of Galaxies in Sloan Digital Sky Survey and LAMOST Spectral Survey. II. Dynamical Properties of Isolated and Embedded Groups
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Compact Groups of Galaxies in Sloan Digital Sky Survey and LAMOST Spectral Survey. II.Dynamical properties of Isolated and Embedded Groups
Yun-Liang Zheng ( 郑 云亮 )
1, 2 and Shi-Yin Shen ( 沈 世 银 )
1, 3 Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 NandanRoad, Shanghai, China, 200030 University of the Chinese Academy of Sciences, No.19A Yuquan Road, Beijing, China, 100049 Key Lab for Astrophysics, Shanghai, China, 200034 (Received 2020.11.5; Revised 2021.2.23; Accepted -)
Submitted to ApJABSTRACTCompact groups (CGs) of galaxies appear to be the densest galaxy systems containing a few lumi-nous galaxies in close proximity to each other, which have a typical size of a few tens kilopacsec inobservation. On the other hand, in the modern hierarchical structure formation paradigm, galaxies areassembled and grouped in dark matter haloes, which have a typical size of a few hundreds of kiloparsec.Few studies have explored the physical connection between the observation based CGs and halo modelbased galaxy groups to date. In this study, by matching the largest local CG catalog of Zheng &Shen (2020) to the halo based group catalog of Yang et al. (2007), we find that the CGs are physicallyheterogenous systems and can be mainly separated into two categories, the isolated systems and thoseembedded in rich groups or clusters. By examining the dynamical features of CGs, we find that theisolated CGs have systematically lower dynamical masses than that of non-compact ones at the samegroup luminosity, indicating a more evolved stage of isolated CGs. On the other hand, the embeddedCGs are mixtures of chance alignments in poor clusters and recent infalling groups (sub-structures) ofrich clusters.
Keywords: keyword1 – keyword2 – keyword3 INTRODUCTIONAccording to the hierarchical diagram, over half ofthe galaxies are clustered into the group systems viagravitational instability. Tidal interaction (Toomre &Toomre 1972; Barnes & Hernquist 1992), ram pressures(Gunn & Gott 1972), and galaxy harassment (Farouki &Shapiro 1981; Moore et al. 1996) are expected to be morefrequently happened in these bound systems. Groupsystems of galaxies are therefore widely used to studythe environmental dependence of galaxy evolution (e.g.,Dressler 1980, Butcher & Oemler 1984, Goto et al. 2003,Coenda et al. 2012, Alpaslan et al. 2015, Cluver et al.2020).
Corresponding author: Shi-Yin [email protected]
Among various of the group systems, compact groupsof galaxies (hereafter CGs) are special cases which typ-ically contain a few luminous galaxies in close proxim-ity to each other. Unlike the massive systems wheretheir higher velocity dispersion favoring more rapid fly-byes, the relatively low velocity dispersion ( ∼
250 kms − , Hickson et al. 1992) of CGs make the occurrenceof tidal interactions and mergers to be more frequent(e.g., Mendes de Oliveira & Hickson 1994, Ostriker etal. 1995).Historically, Hickson (1982) identified 100 CGs fromPalomar Observatory Sky Survey, introducing a set ofphotometric based criteria: 1) richness, 2) isolation, 3)compactness. With the follow-up spectroscopic surveys,redshift information has also been absorbed as a crite-rion (e.g., Hickson et al. 1992; D´ıaz-Gim´enez et al. 2012;Sohn et al. 2015) and optimized for CG selection (D´ıaz-Gim´enez et al. 2018; Zheng & Shen 2020). In Zheng& Shen (2020, hereafter Paper I), we have identified a a r X i v : . [ a s t r o - ph . GA ] F e b Zheng & Shen large spectroscopically confirmed CG sample via modi-fied Hickson criteria:1. Richness: 3 ≤ N (14 . ≤ r ≤ . ≤ θ n ≥ θ G
3. Compactness: µ ≤ . −
4. Velocity Difference: | V − V med | ≤ − where N is the number of members with galactic ex-tinction corrected r -band Petrosian magnitude 14 . ≤ r ≤ . µ is the r -band effective surface brightness(compactness) averaged over the smallest enclosing cir-cle with angular radius θ G , θ n is the angular radius ofthe largest concentric circle that contains no externalgalaxies, V is the recessional velocity of each memberand V med is the median of them.Among the above Hickson Criteria, the compactness( µ ≤ .
0) criterion ensures the typical separation ofCG members within a few tens of kpc and the size ofCGs ( θ G ) typically being less than 100 kpc. However, inΛCDM cosmology, the virial radii of the host dark mat-ter halos of galaxy groups are much more extended. Forexample, at redshift z ∼
0, a group of galaxies with halomass M h ∼ M (cid:12) has a virial radius out to R v ∼ R v than θ G implies that CGsmight be identified or embedded in larger groups as sub-systems. Indeed, previous studies (e.g., Rood & Struble1994; Barton et al. 1998; Palumbo et al. 1995; Ander-nach & Coziol 2005; de Carvalho et al. 2005; Mendel etal. 2011; D´ıaz-Gim´enez & Zandivarez 2015) found a fairproportion of CGs ( ∼ − H = 70 km s − Mpc − and Ω m = 0 .
27 (Komatsu et al. 2011). SAMPLES & DATA2.1.
Compact Group Sample
In Paper I, CGs were derived from the latest ver-sion of New York University Value-Added Galaxy Cat-alog (VAGC, Blanton et al. 2005) which based on SDSSlegacy survey with a set of improved reduction. Theredshift incompleteness due to fiber collision has beenreduced by SDSS-DR14 (Abolfathi et al. 2018), GAMA-DR2 (Liske et al. 2015), and LAMOST-DR7 (Luo et al.2015). As presented in Paper I, we selected CGs in red-shift slices to reduce the bias against nearby groups andderived 6,144 conservative CGs (hereafter cCGs) con-taining 19,465 galaxies with complete redshifts.Also, asmentioned in Appendix A of Paper I, 74 cCGs have as-sociation with bright galaxies ( r < .
00 mag). Whenwe join these bright galaxies into the 74 cCGs, 26 of ompact Groups of Galaxies in SDSS & LAMOST. II. L . (see Section 2.4) would be updated, while the other48 violate the CG criteria and thus would be removedfrom the CG sample used in this paper. In addition, wefurther remove 16 CGs after a careful inspection of thespectroscopic data of all group members, where the spec-troscopic redshifts of few member galaxies have been in-correctly reused during the CG construction and there-fore could potentially bias the velocity dispersion finallymeasured. This results in a sample of 6,080 cCGs with19,273 member galaxies.2.2. Halo-based Group Sample: Y07 group catalog
In this paper, we adopt the galaxy group catalog con-structed by Yang et al. (2007, 2012, hereafter Y07),which is also based on VAGC of SDSS-DR7. Y07 has ap-plied a halo-based group finder to assign each galaxy inthe SDSS-DR7 Main Galaxy Sample (MGS) within red-shift range 0 . < z < .
20 to a unique group. In Y07,three versions of group catalog have been constructedbased on different redshift sources. In this work, we usesample III of Y07 group catalog, where a small fractionof SDSS MGS ( ∼ , ∼ , ∼ ∼ N = 2) and update theirgroup luminosity L . accordingly (see Section 2.4). Forthe other groups that still have members with assignedredshifts, we keep them unchanged. As we will showin next section, these assigned redshifts have negligibleeffects on our study.2.3. CG categories: match with Y07 groups
We cross match the members in cCGs with the up-dated Sample III group catalog shown above. Most ofthe galaxies ( ∼ ∼
6% galaxies have no cor-responding Y07 groups due to one of the two followingreasons:
Table 1.
The classification of CG subsamplesCG Subsamples Sample SizeIsolated CGs 1667Predominant CGs 1570Embedded CGs: 1370
Single Embedded Group 901Multiple Embedded Groups 469
Split CGs 1282Unmatched CGs 191Overall 6080
1. The sky coverage of the galaxy catalog we used isslightly larger than that of Y07, because Y07 hasdiscarded the galaxies located near the survey edgeor very low completeness regions. This results 118cCGs have no counterparts in Y07.2. In Y07, the faint-end magnitude cuts are var-ied with positions, ranged from 17.62 to 17.72 inextinction-corrected Petrosian magnitude, whichare the results of different versions of target se-lection of the MGS for spectroscopic observationthrough the period covered by Early Data Release(Stoughton et al. 2002), while we adopt the latestversion of a fixed value r f = 17 .
77 mag for CGselection. This operation results in the updatingof ∼
470 Y07 groups. We have tested that al-most all of the extra members indeed belong tothe same Y07 groups according to the Y07 groupfinder. However, there are also 73 cCGs having atleast 2 members without matching in Y07 groups.Fore safety, we discard them from further investi-gation.As a result, 5889 out of 6080 cCGs with all of theirmembers could be matched with the update Y07 groups.Among them, 1667 have their members being the sameas Y07 groups, while 2940 cCGs are the subsets of Y07groups. Apart from these one-to-one matches, there arealso 1282 cCGs with their members matched to differentY07 groups. These split CGs are mainly attributed tothe large velocity difference cut ∆
V < − used in our CG selection, while the velocity differencecut used in Y07 groups is dynamically linked to theirvirial mass.In this study, we ignore the split CGs (see Appendix Afor a detailed discussion). We define these 1667 CGswith the same memberships as the Y07 groups as iso-lated CGs. For the cCGs being subsets of Y07 groups, Zheng & Shen
Y07
Figure 1.
Example SDSS images of cCGs where their members are one-to-one correspondent with the members of Y07 groups:a). Isolated CGs with no external galaxies host in the same halo. b). Predominant CGs with other fainter galaxies sharing thesame halo. c). Embedded CGs with brighter galaxies occupying the same halo as non-dominant subsystems. d). Split CGswhose members belong to at least two different halos of Y07. The inner white dashed circles represent the smallest enclosedcircles θ G , the outer white dashed circles represent the concentric circles 3 θ G . Green dashed circles represent the smallestenclosed circles for Y07 groups, which is manually enlarged for clarity. Solid circles mark the member galaxies of cCGs (white)or their corresponding Y07 groups (green). The IDs of cCG and its corresponding Y07 group are labelled at the top-right cornerof each image. we define them as ‘embedded systems’, where their hostY07 groups are referred as ‘parent groups’. To bet-ter distinguish the dynamical effects induced by parentgroups, we define the embedded CGs as the subgroupsdo not dominate the luminosity of their parent groupsaccording to: N par (cid:88) i =1 L i ≥ N emb (cid:88) j =1 L j (1)where L j is the luminosity of the j th member of embed-ded CGs and L i is the luminosity of the i th member of their parent groups. For each galaxy we compute their . r -band luminosity using: LL (cid:12) = 10 − . [ r − DM( z ) − K . r ( z ) − . ] (2)where DM( z ) is the bolometric distance modulus, K . r ( z ) is K-Correction value at z = 0 . KCORRECT package of Blanton & Roweis (2007),4 .
64 is the r -band magnitude of the sun in AB system.This leaves 1370 embedded CGs being hosted by 1084parent groups, while the 1570 remainders are referred toas predominant CGs. We show example images for each ompact Groups of Galaxies in SDSS & LAMOST. II. Redshift . . . . . F r a c t i o n Richness . . . . F r a c t i o n Embedded CGsIsolated CGsnon-compact groups log[ L /( h L )] . . . . . . . . F r a c t i o n log[ LOS (km/s)] . . . . . . . . F r a c t i o n Figure 2.
The distributions of the isolated (red hatched) CGs, embedded (blue filled) CGs, and control sample of non-compactgroups (grey opened). Upper Left: Redshift. Upper Right: Richness. Lower Left: Group Luminosity, L . Lower Right: LOSVelocity Dispersion, σ LOS . type of CGs in figure 1. Table 1 summarizes the resultsof the classification. In the following, we do not considerthe predominant CGs since it is difficult to distinguishthe dynamical effects between the embedded system andhost groups. A basic comparison of the parent groupsof predominant CGs and embedded CGs is presented inAppendix B.For embedded CGs, there are cases where multipleCGs exist in a single host galaxy group. In our sam-ple, there are 469 CGs in this situation, hosted by atotal of 183 parent groups. We have tested that thesemultiple embedded CGs and single embedded CGs didnot differ statistically in their dynamical properties (seeAppendix B for detail). Therefore, for statistical signif-icance, we do not distinguish between these two cases in the later sections and uniformly refer to them as em-bedded CGs.It is worthy of mentioning that ∼
45% of the par-ent groups have few member galaxies with redshifts as-signed from their nearest neighbors (Section 2.2). Theseredshift assigned members would be neglected when wecalculate the velocity dispersion of the parent groups.On the other hand, these members are included in cal-culating their total luminosity (Section 2.4). Since mostof the parent groups of embedded CGs contain N (cid:38) µ > . − from the Y07 group Zheng & Shen M r h R e l a t i v e f r a c t i o n M = - . ± . = -0.57±0.03 M = - . ± . = -0.85±0.03 M = - . ± . = -0.91±0.02 Yang+2009: ( , M ) = (-1.12, -20.61)Isolated CGsEmbbedded CGsnon-compact groups Figure 3.
The . r -band luminosity functions of iso-lated (red), embbeded (blue) CGs, and non-compact groups(black) derived via both non-parametric (stepwise) and para-metric maximum likelihood estimator with ( α , M (cid:63) ) quotedinside the figure. The dashed line represents the LFs of allthe galaxies in Y07 catalog given by Yang et al. (2009). Allthe LFs are normalized to 1.0 at . M r − h = − . catalog. To do that, we first exclude the groups withincomplete spectroscopic redshifts and containing anygalaxies in cCGs. Then we match their richness andredshift distribution to the isolated CG sample. Foreach isolated CG, we match 3 unique Y07 groups whichhave the same richness and redshift within a tolerance of z ∼ .
01 by means of a Monte Carlo sampling. Finally,we get a control sample of 3 × Total Group Luminosity: L . For an unbiased comparison of galaxy groups at differ-ent redshifts, a characteristic total luminosity of galaxygroups need to be defined. In paper I, we simplysummed up the luminosities of the members for all CGsand argued that this ‘apparent total group luminos-ity’ is a good proxy of their real total luminosity. Inthis study, following Y07, we use L . to character-ize the total luminosity of each galaxy group, which is defined as the sum of the luminosities of all mem-bers brighter than . M r − h < − . z ≤ .
09, the faint-end flux limit r f ∼ . . M r − h < − . L . of these groups are obtained by sum-ming up the luminosities of the group members brighterthan . M r − h < − . z > .
09) groups, we make a correction to the observedtotal luminosity using L . = (cid:82) ∞ L cut L Φ ( L ) dL (cid:82) ∞ L f ( z ) L Φ ( L ) dL N (cid:88) i =1 L i (3)where L cut is the luminosity that corresponds to . M r − h = − . L f ( z ) is the faint luminosity limitof a galaxy that can be observed at the redshift of thatgroup, Φ ( L ) is the . r-band luminosity function (here-after LF) of the group members being considered. Weuse two canonical methods to derive the LFs of thesesamples: the non-parametric stepwise maximum likeli-hood (Efstathiou et al. 1988) for binned LF and themaximum likelihood estimator (Tammann et al. 1979)to calculate the best-fit Schechter (1976) function:Φ ( L ) dL = Φ (cid:63) (cid:18) LL (cid:63) (cid:19) α exp (cid:18) − LL (cid:63) (cid:19) d (cid:18) LL (cid:63) (cid:19) (4)where Φ (cid:63) is the overall amplitude, L (cid:63) is the character-istic luminosity and α is the faint-end slope.We calculate the LFs of the members for the isolated,embedded CGs, and non-compact groups respectively.We plot them in figure 3 where the LF of overall Y07galaxies by Yang et al. (2009) is also shown for compari-son. Here we focus only on the shape of these LFs and allthe LFs are normalized to 1.0 at . M r − h = − . ompact Groups of Galaxies in SDSS & LAMOST. II. , we show thedistributions of the final L . of the isolated CGs, eme-beed CGs, and control non-compact groups in the lowerleft panel of figure 2. As a result of lower redshift dis-tribution (upper left panel of figure 2), the embeddedCGs show systematically lower L . distribution thanisolated ones. 2.5. Velocity Dispersion
The LOS veocity dispersions of the groups are com-puted using a variant of the gapper estimator describedby Beers et al. (1990), which is less biased for smallgroups (D´ıaz-Gim´enez & Zandivarez 2015). The methodinvolves ordering the set of recessional velocities { V i } ofthe N member galaxies and defining gaps as g i = V i +1 − V i , i = 1 , , · · · , N − σ gap = √ π (1 + z g ) N ( N − N − (cid:88) i =1 w i g i (6)where z g is the group redshift and w i is the Gaussianweight defined as: w i = i ( N − i ).In practice, we assume one of the members is static atthe center-of-mass velocity of that group, the estimated σ gap therefore should be multiplied by an extra factor (cid:112) N/ ( N −
1) following Eke et al. (2004). Also, the red-shift measurement errors increase the estimate of σ LOS in quadrature, thus the final σ LOS of each group is givenby: σ LOS = (cid:115) max (cid:18) , N σ N − − V (cid:19) V = 1 N N (cid:88) i =1 V ,i (7) Note that there are 77 CGs without L . calculated because thatnone of their members is brighter than . M r − h < − . log[ L /( h L )] . . . . . . . . . l o g [ L O S ( k m / s )] Embedded CGsEmbedded CGs( L par as x-axis)Isolated CGsnon-compact Groups Figure 4.
Group velocity dispersion ( σ LOS ) as a function ofgroup luminosity ( L . ) for isolated CGs (red), embeddedCGs (blue), and control non-compact groups (black) on alogarithmic scale with bin size of 0 . σ LOS in each L . bin, whereas the 16 th and 84 th percentiles are coveredby shaded areas respectively. Only the data bins with atleast 10 groups are plotted. The green line represents thesame scale relation for embedded CGs but their L . arereplaced by L par19 . . The vertical error bars show the errors ofthe median log ( σ LOS ) and the horizontal error bars indicatethe median absolute deviation of log ( L . ) in each bin. where V err ,i is the recessional velocity error of the i th member of the group. In most of the cases, the contri-bution from V err ,i is negliable. The typical value of V err of the galaxies in SDSS and LAMOST Spectral Surveyis at the level of ∼
10 km s − . For the redshifts takenfrom alternative surveys (e.g., 2dFGRS & GAMA) andwithout errors for individual galaxies, we use the typicaluncertainty: V err ,i ∼
33 km s − for GAMA (Baldry etal. 2014); ∼ −
120 km s − for 2dFGRS (depends onspectroscopic quality)(Colless et al. 2001).There is significant randomness in calculating σ LOS ofsmall groups, which results and dominates the error of σ LOS of each CG. We denote the error of σ LOS as σ err .We estimate the σ err of each CG by performing a simpleMonte Carlo simulation. More specifically, for a groupwith N members and estimated σ LOS , we randomly gen-erate ∼ N (cid:0) , σ LOS (cid:1) . Wethen calculate the velocity dispersion for each mockgroup using Equation 7 and take the scatter of 100,000mock groups as the expected value of σ err . Zheng & Shen
We show the distributions of σ LOS of the isolated CGs,embedded CGs, and the control non-compact groups inthe lower right panel of figure 2. Although the embeddedCGs have similar richness distribution (upper left panelof figure 2) and even lower redshift distribution (upperright panel of figure 2) than isolated CGs, they have sys-tematically higher σ LOS distribution than isolated ones.This systematical difference implies a different physicalorigins of these two types of CGs which we will discussnext. RESULTS3.1. σ LOS − L . Relation
Figure 4 displays the scale relations between the me-dian σ LOS and L . for isolated CGs, embedded CGsand non-compact groups, where each small dot repre-sents a group with their categories color-coded. Weshow the median of σ LOS at each L . bin with a binwidth equal to ∆ L . = 0 . th and 84 th percentile respectively. For eachdata bin, the error of the median values are estimate bymultiplying the standard error of the mean by a constantof 1.25.As can be seen, the σ LOS − L . relations show simi-lar monotonic trends for isolated CGs and non-compactgroups. However, there is a systematical offset that,at given L . , the compact groups show systematicallylarger σ LOS than the non-compact ones. This result in-dicates that the compactness (size) of groups might playan important role in describing the dynamics of groupsof galaxies. Indeed, the galaxy groups are known to bedistributed on a fundamental plane (FP) in the loga-rithm space of L − σ − R parameters (e.g., Adami etal. 1998; Fritsch & Buchert 1999; D´ıaz & Muriel 2005;D’Onofrio et al. 2020). We will make a more detaileddiscussion on FP of isolated groups in Section 3.2.Moreover, the median σ LOS of embedded CGs are sig-nificantly larger than that of isolated CGs at given L . .The very large offset of the embedded CGs at given L . implies that the embedded CGs might not be dynami-cally bound system. Based on the scatter of σ LOS variedwith L . , we suppose that the dynamical status of suchembedded systems are more likely to be dependent ontheir parent groups because the parent groups of em-bedded CGs span a very large range in their dynamicalmass. To verify this hypothesis, for each embedded CG,we take the total luminosity of its parent group L par19 . and plot the median σ LOS − L par19 . relation for em-bedded CGs in figure 4. In this case, we see that the σ LOS − L par19 . relation becomes to be in consistence withthe σ LOS − L . relation of isolated CGs. This resultchallenges our view of the dynamical nature of theseembedded CGs. Are they distinct sub-systems of largerhost groups? If so, what determines their dynamicalproperties? Or even, is it possible that such systems arenot dynamically unique in any way and are just formedor selected as a result of chance alignment? We willmake a more detailed discussion in Section 3.3.3.2. Dynamical status of isolated CGs
For a gravitationally bound system, σ LOS is related toits total dynamical mass in a following way: M dyn = σ R dyn G (8)where σ = 3 σ is 3D velocity dispersion based onisotropic assumption and R dyn is its dynamical radius.Following Diaferio et al. (1994), we take the mean har-monic radius R H to characterize the dynamical radiusgiven by: 1 R H = 2 N ( N − (cid:88) i 05 and1 . ± . 03 for isolated CGs and non-compact groupsrespectively, which are consistent with each other inside1 − σ errors. The slopes of both type groups are largerthan 1, which are qualitatively in agreement with earlyfindings for normal galaxy groups (e.g., Girardi et al.2000; Popesso et al. 2005). We argue that a larger slopethan 1 in M dyn − L . relation is a result of system-atical larger mass-to-light ratio M dyn /L . for highermass groups (see the lines of constant M dyn /L . ratios We note that the L . of the parent groups are directly takenfrom the sample III catalog of Y07. As we have discussed in Sec-tion 2.3, the richness of the parent galaxies is significantly largerthan CG themselves, the memberships of few galaxies with as-signed redshifts from the nearest neighbour have negligible effectson final L . . ompact Groups of Galaxies in SDSS & LAMOST. II. log[ L /( h L )] . . . . . . . . l o g [ M d y n / ( h M )] l o g ( M d y n h L . ) = . l o g ( M d y n h L . ) = . l o g ( M d y n h L . ) = . l o g ( M d y n h L . ) = . l o g ( M d y n h L . ) = . l o g ( M d y n h L . ) = . l o g ( M d y n h L . ) = . l o g ( M d y n h L . ) = . Isolated CGsnon-compact Groupsnon-compact ( 26 < < 27.2 )non-compact ( > 27.2 ) Figure 5. Symbols and shaded areas are median and 16 th to 84 th percentile of M dyn as a function of group luminosity( L . ) for isolated CGs (red), and non-compact group sam-ples (black) on a logarithmic scale with bin size of 0 . µ (cid:46) . 2) and loose ( µ (cid:38) . 2) subsam-ples of non-compact groups are also shown in green and bluerespectively. The solid lines are linear fit for these samples.Bold dashed lines are M dyn − L . relation based on virialequilibrium ( M dyn ∝ L ) with various M dyn L values. Thevertical error bars show the errors of the median log ( M dyn )and the horizontal error bars indicate the median absolutedeviation of log ( L . ) in each bin. Only the data bins withat least 10 groups are plotted. in figure 5 for reference), which also has been suggestedby early studies (e.g., Girardi et al. 2000, 2002; Eke etal. 2004; Popesso et al. 2007). In this study, we are fo-cusing on the comparison between the CGs and normalgroups and therefore do not further explore the physi-cal implications of the exact slopes of the M dyn − L . relation.For isolated CGs, comparing with the L − σ relationshown in figure 4, when R H is taken into considera-tion, their median M dyn becomes systematically smallerthan that of the non-compact groups. That is to say,the R H of CGs are significantly smaller than that ofnon-compact groups with the same L . , even smallerthan the prediction of Equation 8. To further iden-tify the effect of R H in calculation of M dyn , we furtherdivide the non-compact groups into two sub-sampleswith equal numbers at their median surface brightness µ ∼ . − . Here, the surface brightnessof non-compact groups is calculated following the sameway as that for CGs, i.e., the mean surface brightness of galaxies inside the innermost circle θ G . Therefore, com-paring with the isolated CGs ( µ < . µ > . . < µ < . M dyn − L . relations for these two sub-samplesas green and blue triangles in Fig. 5 respectively. As canbe seen, there are negligible differences of M dyn − L . relation between the two sub-samples of non-compactgroups, which implies that they might be in a quasi-virial equilibrium state so that the dynamical mass keepswhen R H varies.On the other hand, the systematical lower M dyn of iso-lated CGs implies that they have deviated from quasi-dynamic equilibrium and entered into a phase of galaxymerging. Specifically, when galaxy groups evolve andreach compact status of µ ∼ 26 mag arcsec − , the inter-nal frequent close encounters of galaxies cause dynami-cal friction and shrink the mean separation between thegroup members significantly, which therefore results ina smaller M dyn being measured. We present a more de-tailed discussion on the dynamical evolution of galaxygroups in Appendix C.3.3. Dynamical nature of embedded CGs To have a better understanding of the dynamical na-ture of the embedded CGs, we directly compare the re-lation between σ LOS and σ parLOS for embedded CGs andshow the results in figure 6.There is a good one-to-one correlation for these groupswith σ parLOS (cid:46) 500 km s − . As we have mentioned in Sec-tion 2.3, all parent groups have total luminosities at leasttwice of their embedded CGs (Equation 1) and theirtypical richness are N (cid:38) σ LOS and σ parLOS is caused by the dominance ofthe embedded CG members in parent groups. The excel-lent consistence between σ LOS and σ parLOS implies that thedynamics of these apparent sub-systems are only deter-mined by their host groups. That is to say, these embed-ded CGs are not dynamically distinct sub-systems butmore likely to be the consequence of chance alignmentswithin larger systems. We will verify this hypothesisusing a detailed Monte-Carlo simulation below.For massive clusters of galaxies ( σ parLOS (cid:38) 500 km s − ),the median σ LOS are on average 20% − 40% below theone-to-one prediction. This might be caused by the biasof the velocity difference criterion ( | V − V med | < − ) used in the CG selection where the galaxieswith large relative velocities chosen randomly from large σ parLOS groups are more likely to violate. By excludingsuch systems from CG selection, the resulted embeddedCGs would certainly have σ LOS < σ parLOS on average.0 Zheng & Shen parLOS ( km / s ) L O S ( k m / s ) Embedded cCGsSimulation parLOS ( km / s ) L O S ( k m / s ) Embedded cCGs ( R / R med < R / R med < R / R med > R / R med > . . . . . R / R m e d Figure 6. The comparison of the embedded CGs and mock CGs from Monte-Carlo simulation. Left: Velocity dispersion ofthe embedded CGs ( σ LOS ) versus that of their parent groups ( σ parLOS ). The open hexagons represent the median σ LOS of theembedded CGs in each σ par LOS bin, whose vertical error bars represent the errors of the median σ LOS and horizontal error barsrepresent the median absolute deviation of σ parLOS in each bin. The shaded area shows the 16 th and 84 th percentiles of σ LOS distribution at given σ parLOS , whereas the three dashed lines show the 16 th , 50 th and 84 th percentiles for the mock CGs. Inthe bottom shows the histograms of the radial distance of observed (filled) and mock (open) CGs within the parent groups ineach σ parLOS bin. Right: The CGs located at R/R med > R/R med < σ LOS of mockCGs respectively. The bold lines in both panels are one-to-one correspondence between σ LOS and σ parLOS . On the other hand, it is also possible that these embed-ded CGs are gravitationally bound sub-systems recentlyfalling into large clusters. In this case, the dynamicsof their members could be partially heated but stillbe kinematically colder than the host clusters (e.g.,Choque-Challapa et al. 2019; Benavides et al. 2020)).To distinguish these two different scenarios, we alsoneed a Monte-Carlo simulation. Monte-Carlo Simulation: We run Monte-Carlo simula-tion to test the hypothesis that the embedded CGs arepurely selected from chance aliment of the galaxy mem-bers in parent groups. To make the simulation as real-istic as possible, we build the mock CGs from the par-ent groups of embedded CGs. In specific, for each par-ent group, we keep the projected radial distances of allmembers with respect to the luminosity weighted cen-ter defined by Y07 and keep their radial velocities un-changed, only randomized their projected azimuthal po-sitions. We then apply the CG selection criteria used inPaper I and embedded criterion (Equation 1) to search the mock CGs. For each embedded CG, we performmultiple runs by randomizing its parent group until 100mock CGs derived (100 × th and 84 th percentiles of σ LOS of the mock CGs as black dashed lines in the leftpanel of figure 6. In addition to σ LOS , the projected ra-dial position distribution of the mock CGs can also beused to test the chance alignment hypothesis since theradial positions of all group members have been keptduring simulation. Here we use R/R med to character-ize the projected radial position of the embedded CGsin parent groups, where R is the projected distance ofthe CG luminosity weighted center to the parent groupcenter and R med is the median projected distance of allmembers of the parent groups. The R/R med distribu-tions for both of the embedded CGs and mock CGs ineach σ parLOS bin are also shown in the left panel of figure 6.As can be seen, both σ LOS and R/R med distributionsof the embedded CGs with σ parLOS (cid:46) 300 km s − can bewell reproduced by the mock CGs, which validates thechance alignment hypothesis. However, it is worth to ompact Groups of Galaxies in SDSS & LAMOST. II. σ parLOS would bedominated by the σ LOS of embedded CGs, and we couldnot distinguish a dynamical bound CG from the chancealignment hypotheses.There is a slight deviation between the observed andmock CGs within 300 km s − < σ parLOS < 500 km s − groups and tend to be very significant for σ parLOS > − groups. For the groups with σ parLOS > 500 kms − , although mock CGs have already passed the veloc-ity filter ( V − V med > − ), there still exist thesystematic deviation from observation. Moreover, thepredicted R/R med distribution also show significant dif-ferences from the observations. The embedded CGs weidentified in σ parLOS > 300 km s − groups are evidentlyand gradually biased to the outer regions of the par-ent groups. Combing these two effects, we infer that,for the large-scale environment like group systems with σ parLOS > 300 km s − , the CGs we selected using tradi-tional Hickson-like criteria might not be fully explainedby the chance alignment effect. Some of these compactsub-systems might be bound objects that could have en-tered into the larger systems in the last 1 - 2 Gyr as ar-gued by Lisker et al. (2018). For these infalling systems,they might have yet to complete their first pericentricpassage due to the short time-scale, which makes themmainly be located at the outer regions of the parent sys-tem and with their morphology being kept in a compactstate.To further verify this conclusion, we separate em-bedded CGs into two categories according to their ra-dial position and compare their σ LOS . We show themedian σ LOS of the inner ( R/R med < 1) and outer( R/R med > 1) CGs as the green and red triangles inthe right panel of figure 6, the standard deviation of themedian σ LOS of corresponding mock CGs obtained from1000 bootstrap resamplings with the same sample sizeas observed CGs in each bin are shown in the green filledand red dashed areas respectively. Clearly, the σ LOS ofinner CGs can be well reproduced by Monte-Carlo sim-ulation. This result implies that the inner CGs might bedominated by chance alignments along the line of sightwithin their parent groups (Mamon 1986, 2008), wherethe high number density of galaxy members in centralregions of galaxy groups could easily trigger such a se-lection bias. Conversely, the σ LOS of the outer CGsare systematically below the mock samples. This resultindicates that these outer CGs, at least, might consist of (or include) newly-accreted groups. Indeed, numer-ical simulations (e.g., Cohn 2012; Choque-Challapa etal. 2019) have predicted that these newly accreted sys-tems have not passed through the cluster center and aremainly located at the outskirts of host clusters. Despiteof experiencing dynamical heating, these sub-systems re-main dynamically colder and more compact than thehost cluster (Benavides et al. 2020). After the first peri-centric passage, they would soon be disassembled andvirialized within the host cluster. Such a scenario is alsoconsistent with the early finding that the substructuresof rich clusters appear to decrease towards their centralregions (Biviano et al. 2002). CONCLUSIONSIn this paper, we use a large sample of CGs taken fromPaper I to explore their spatial relation with Y07 groupsdefined by halo-based model. We show that ∼ 27% CGshave one-to-one correspondence with the groups in a sin-gle dark matter halo, which we refer to as ‘isolated CGs’.The remaining CGs have complex associations with darkmatter haloes. After removing the CGs which domi-nate the luminosity of the haloes ( ∼ ∼ ∼ 23% of themembedded within large clusters as non-dominant com-ponents, we refer to as ‘embedded CGs’. The relativelylow percentage of isolated CGs we found is a result ofour careful inspection of the relations between CGs andhalo-based group sample. In our result, the fraction ra-tio of isolated CGs to embedded CGs is about 1 : 1,which is consistent with early finding of Mendel et al.(2011). If we consider these CGs dominate the luminos-ity of host halos also to be isolated CGs, we will get aratio of 2 . σ LOS )and group luminosity ( L . ) for isolated CGs ismonotonic and similar as that for non-compactgroups. However, the σ LOS of isolated CGs aresystematically higher than that of non-compactgroups at given L . . By considering group ra-dius, we find that the dynamical mass of isolatedCGs are systematically smaller than that of non-compact groups. But for non-compact groups,their dynamical mass show negligible dependenceof group compactness. This result implies thatthe isolated CGs are more likely to be dynam-ically more evolved systems which have entered2 Zheng & Shen into the orbital dissipation phase induced by dy-namical friction.2. For embedded CGs, the correlation between σ LOS and L . of themselves is much weaker than with L . of their parent groups ( L par19 . A. THE σ LOS − L . RELATION FOR SPLIT,SINGLE EMBEDDED AND MULTIPLEEMBEDDED CGSIn this appendix, we show the σ LOS − L . relationsfor two sub-types of CGs that we have not discussed indetail in the main text of the manuscript, the split CGsand multiple embedded CGs. The split CGs are thoseinhabit multiple Y07 groups (an example is shown inthe panel d) of figure 1), whereas the multiple embeddedCGs are the cases that at least two CGs embed the sameY07 group. For split CGs, their L . are corrected bythe LF of isolated CGs for simplicity. The results areshown in the left and right panel of figure 7 respectively.At given group luminosity L . , the split cCGs showsignificantly higher σ LOS than isolated CGs, and evensingle embedded CGs. As we have already mentionedin the main text, the high velocity dispersion of split CGs are mainly attributed to the large velocity differ-nce cut ∆ V < − used in CG selection (PaperI). Therefore, we conclude that these split CGs are ap-parent systems and are not gravitational bound.In the right panel of figure 7, we compare σ LOS − L . and σ LOS − L par19 . for single embedded and multiple em-bedded CGs. Apparently, at given L . of CGs, themultiple embedded CGs show higher σ LOS than singleembedded one. When consider the luminosity of par-ent groups L par19 . , the multiple and single embedded CGsfollow the same L − σ relation. Therefore, the σ LOS ofboth single and multiple embedded CGs are dominatedby their parent groups, and the higher σ LOS of multipleembedded CGs is simply a result of their richer parentgroups. ompact Groups of Galaxies in SDSS & LAMOST. II. log[ L /( h L )] . . . . . . . . . l o g [ L O S ( k m / s )] Embedded CGsIsolated CGsnon-compact GroupsSplit CGs log[ L /( h L )] . . . . . . . . . l o g [ L O S ( k m / s )] Single Embedded CGsSingle Embedded CGs( L par as x-axis)Multiple Embedded CGsMultiple Embedded CGs( L par as x-axis) Figure 7. The L . − σ LOS relation for the sub-types of CGs that we have not discussed in detail in the main text. LeftPanel: Split CGs (gold), comparing with non-compact groups (black), isolated (red), and embedded CGs (blue) . Right Panel:Single (teal) and Multiple (purple) Embedded CGs, where the symbols are connected with dashed lines represent the same scalerelation but use L par19 . as x-axis. The vertical error bars show the errors of the median log ( M dyn ) and the horizontal error barsindicate the median absolute deviation of log ( L . ) in each bin. Only the data bins with at least 10 groups are plotted.B. THE BASIC STATISTICAL PROPERTIES OFPARENT GROUPSIn figure 2, we have presented basic statistical prop-erties of different categories of CGs when matched withY07 groups. Here we show the statistical properties ofthe Y07 groups host our CG samples.As mentioned in section 2.3, apart from 1667 isolatedCGs and 1282 split CGs, there are 1570 and 1370 pre-dominant and embedded CGs respectively, the latter ofwhich include 469 multiple embedded CGsand 901 sin-gle embedded CGs. Figure 8 shows the histograms ofredshift, richness, L . , and σ LOS for the parent groupsof predominant and both types of embedded CGs.These three categories of parent groups shown sig-nificantly different redshift and richness distributions,which, however, are obviously results of selection ef-fects. From predominate CGs, single embedded CGsto multiple embedded CGs, because of their decreas-ing dominance and similar richness distribution, theirparent groups certainly have increasing richness (upperright panel) and L . (lower left panel) distributions.Their different σ LOS distributions (lower right panel) arethen simply the result of the σ LOS − L . relation. Forthe redshift distribution, because of the selection effectsin flux limited sample, the higher richness parent groupsare certainly biased to lower redshifts (upper left panel). C. THE EVOLUTIONARY SCHEME FOR GROUPSMamon (1993, 2007) has provided an analytical modelof the dimensionless mass bias, M dyn /M , where M refers to the true mass of the galaxy systems, versus di-mensionless crossing time t cr /t , where t refers to theage of universe, for an isolated system at different evolu-tionary stage. Figure 9 shows the solid track that galaxysystems should follow, the arrows indicate the evolu-tionary direction of a galaxy system: expands alongsidewith the Hubble flow at first, then decouples from thisflow, turns around when reaches maximum expansion,collapses subsequently and finally virializes with contin-uous orbital energy dissipation induced by dynamicalfriction.In this appendix, we plot the M dyn /M versus t cr /t relation for isolated CGs and non-compact groups so asto further compare their dynamical status. The isolatedCGs and non-compact groups are plotted as red andgray dots respectively in figure 9 where their crossingtime are given by: t cr = R dyn /σ . Unfortunately, truemasses of galaxy systems, M , are unknown. Here, wehave made a simple assumption of M / ( hL . ) ∼ M ), figure 94 Zheng & Shen Redshift . . . . . . F r a c t i o n Richness . . . . F r a c t i o n Multiple Embedded CGSingle Embedded CGsPredominant CGs log[ L /( h L )] . . . . . . . . F r a c t i o n log[ LOS (km/s)] . . . . . F r a c t i o n Figure 8. The distributions of the parent groups of predominant (open), single embedded (hatched), and multiple (filled)embedded CGs. Upper Left: Redshift. Upper Right: Richness. Lower Left: L . . Lower Right: LOS velocity dispersion. shows that many of the isolated CGs are more likely tobe in the phase of dynamic friction.REFERENCES Abolfathi, B., Aguado, D. 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