Physical Properties of Dense Cores in the Rho Ophiuchi Main Cloud and A Significant Role of External Pressures in Clustered Star Formation
Hajime Maruta, Fumitaka Nakamura, Ryoichi Nishi, Norio Ikeda, Yoshimi Kitamura
aa r X i v : . [ a s t r o - ph . S R ] J a n Physical Properties of Dense Cores in the ρ Ophiuchi Main Cloud and ASignificant Role of External Pressures in Clustered Star Formation
Hajime Maruta , Fumitaka Nakamura , , Ryoichi Nishi , Norio Ikeda , Yoshimi Kitamura ABSTRACT
Using the archive data of the H CO + ( J = 1 −
0) line emission taken with theNobeyama 45 m radio telescope with a spatial resolution of ∼ . ρ Ophiuchi main cloud. The H CO + data also indicates that the fractional abundance of H CO + relative to H is roughlyinversely proportional to the square root of the H column density with a mean of1 . × − . The mean radius, FWHM line width, and LTE mass of the identified coresare estimated to be 0.045 ± ± − , and 3.4 ± M ⊙ , respectively.The majority of the identified cores have subsonic internal motions. The virial ratio,the ratio of the virial mass to the LTE mass, tends to decrease with increasing the LTEmass and about 60 percent of the cores have virial ratios smaller than 2, indicating thatthese cores are not transient structures but self-gravitating. The detailed virial analysissuggests that the surface pressure often dominates over the self-gravity and thus plays acrucial role in regulating core formation and evolution. By comparing the ρ Oph coreswith those in the Orion A molecular cloud observed with the same telescope, we foundthat the statistical properties of the core physical quantities are similar between the twoclouds if the effect of the different spatial resolutions is corrected. The line widths ofthe ρ Oph cores appear to be nearly independent of the core radii over the range of 0.01 − Subject headings:
ISM: clouds — ISM: individual ( ρ Ophiuchi) — ISM: structure —stars: formation — turbulence
1. Introduction
Observations of young stellar populations in the Galaxy have revealed that the majority of starsform in clusters (Lada & Lada 2003; Allen et al. 2006). For example, Lada et al. (1991) performed Department of Physics, Niigata University, 8050 Ikarashi-2, Niigata 950-2181, Japan Astrophysics Lab., Faculty of Education, Niigata University, 8050 Ikarashi-2, Niigata 950-2181, Japan Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Sagami-hara, Kanagawa 229-8510, Japan −
90 % of all the young stellar populations associated with the molecular cloud formonly in three rich embedded clusters. Using the 2MASS point source catalog, Carpenter (2000)estimated the fraction of young stellar populations contained in clusters to be 50 −
100 % fornearby cluster forming molecular clouds such as Perseus, Orion A, Orion B, and MonR2. Recentnear-infrared surveys of young stellar populations using the Spitzer Space Telescope have confirmedthat clustered star formation is the dominant mode of star formation in the Galaxy (Allen et al.2006; Poulton et al. 2008; Rom´an-Z´uniga et al. 2008). However, previous studies of star formationhave focused on star formation in relative isolation, for which the environmental effects that playa crucial role in clustered star formation are thought to be minor or at least secondary. Thus, howstars form in the cluster environment remains only poorly understood.An important clue to understanding star formation in clusters has come from recent millimeterand submillimeter observations of the nearby cluster-forming region, the ρ Ophiuchi molecularcloud (L1688), which uncovered dense cores with a mass spectrum that resembles the Salpeter IMF(Motte et al. 1998; Johnstone et al. 2000a; Stanke et al. 2006). Similar studies in other star formingregions have confirmed that such a resemblance between the core mass spectrum and the stellarIMF seems to be common in nearby star forming regions (Testi & Sargent 1998; Motte et al. 2001;Reid & Wilson 2006; Ikeda et al. 2007, 2009). These observations suggest that the stellar IMF maybe determined to a large extent by the core mass distribution (e.g., Andr´e et al. 2007), althoughmass accretion from ambient gas onto forming stars may sometimes plays an important role indetermining the final masses of stars (Bonnell et al. 2001; Bate et al. 2003; Wang et al. 2009).Therefore, it is important to investigate the formation and evolution of dense cores in the clusterenvironment.Recent theoretical studies on core formation in the cluster environment have emphasized therole of supersonic turbulence (Klessen et al. 1998; Tilley & Pudritz 2004; V´azquez-Semadeni et al.2005; Dib et al. 2007). These studies have demonstrated that dense cores are formed in regionscompressed by converging turbulent flows. However, the origin of this supersonic turbulence andits role in core formation are still a matter of debate. Two main scenarios have been proposed. Inthe first scenario, formation of dense cores (and thus star formation) is considered to be completedonly in one turbulent crossing time, i.e., one dynamical time, so that the cluster forming regionsare destroyed within one dynamical time (Elmegreen 2000; Hartmann 2001; Heitsch et al. 2008).In this case, the mass spectrum of dense cores is shaped by the cascade of primordial interstellarturbulence that has a power law form (Padoan & Nordlund 2002). On the other hand, in the secondscenario, star formation is considered to continue at least over several dynamical times. In this case,additional turbulent motions must be replenished to keep supersonic turbulence in molecular cloudsbecause supersonic turbulence dissipates quickly in one dynamical time (Stone et al. 1998; Mac Low1999). In such circumstances, dense cores are continuously created (and destroyed) in molecular gasdisturbed strongly by the additional turbulent motions. In the cluster environment, stellar feedbacksuch as stellar winds, HII regions, and protostellar outflows has been discussed as the dominant 3 –source of additional supersonic turbulence (Norman & Silk 1980; Matzner 2007; Li & Nakamura2006). Other sources of turbulence such as supernova blastwaves, external cloud shearing, orconverging flows of cloud formation may also play a role (e.g., McKee & Ostriker 2007). As for theprotostellar outflow-driven turbulence, Swift & Welch (2008) identified the characteristic energyinjection scale at ≈ ρ Ophiuchi molecular cloud.The ρ Ophiuchi molecular cloud is the nearest cluster forming regions at a distance of ∼ ρ Ophiuchi molecular cloud (hereafter the ρ Ophiuchi main cloud) is known to harbor a rich clusterof young stellar objects (YSOs) in different evolutionary stages (Wilking et al. 2008; Enoch et al.2008; Jørgensen et al. 2008). Recent millimeter and submillimeter dust continuum observations ofthe ρ Ophiuchi main cloud have revealed that a large number of dense cores are concentrated inthe central dense part of the cloud and their mass spectra are similar in shape to the stellar IMF(Motte et al. 1998; Johnstone et al. 2000a). By comparing with isothermal, pressure-confined, self-gravitating Bonnor-Ebert spheres, Johnstone et al. (2000a) argued that the dense cores detectedby the dust continuum emission are likely to be self-gravitating and expected to form stars (seealso Johnstone et al. 2004). However, to accurately assess the dynamical states of the dense cores,it is necessary to perform molecular line observations using high density tracers such as H CO + ( J = 1 −
0) and N H + ( J = 1 − cm − . Recently, the Nobeyama observatory has released the archive data of the H CO + ( J = 1 −
0) line emission toward the ρ Ophiuchi main cloud taken with the 45 m radio telescope.At a distance of 125 pc, the spatial resolution of the data of 21”, corresponding to ∼ .
01 pc, iscomparable to those of the dust continuum observations, e.g., the SCUBA at the JCMT with abeam size 14”at 850 µ m. Thus, the Nobeyama archive data is suitable for analyzing the dynamicsof small scale structures comparable to the ones detected by the dust continuum observations. Ouranalysis is complementary to that of Andr´e et al. (2007) who observed several subclumps (OphA, B1, B2, C, E, and F) with the IRAM 30 m radio telescope using N H + ( J = 1 −
0) line tomeasure the line widths of the dust cores detected by Motte et al. (1998). The H CO + data ofthe Nobeyama 45 m telescope was taken in a larger area covering the whole dense region of the ρ Ophiuchi main cloud, including all the subclumps observed by Andr´e et al. (2007). Since the densegas detected by molecular line emissions such as H CO + ( J = 1 −
0) and N H + ( J = 1 −
0) maynot be always well associated with the dust cores and therefore the line widths of the dust coresestimated by Andr´e et al. (2007) may not be accurate, we identified the dense cores directly fromthe molecular line data cube and estimated their physical quantities such as the core radius, mass, 4 –and line width.The rest of the paper is organized as follows. First, the description of the archive data ispresented in §
2. In § CO + emission, which indicates very clumpy structures. Then, in § CO + relative to H are derived directly by comparing the H CO + column densities with the H column densities obtained from the 850 µ m dust continuum emission.In § §
2. Data
We used the archive data of the ρ Ophiuchi main cloud taken in H CO + ( J = 1 −
0) molecularline (86.75433 GHz) with the Nobeyama 45 m radio telescope. The fits data is available from theweb page of the Nobeyama Radio Observatory at the National Astronomical Observatory of Japan( ). The observations were carried out in the period from 2002 Marchto 2003 May. All spectra were obtained in the position-switching mode. The data has 118 × × α - δ - v LSR space, covering the whole dense region of the ρ Ophiuchi main cloudwith size of about 1 . × . − and the average Root MeanSquare (rms) noise determined from signal-free channels is σ = 0.11 K in T ∗ A .
3. Distribution of Dense Molecular Gas in the ρ Ophiuchi molecular cloud3.1. Overall Distribution of the H CO + emission Here, we present the overall distribution of dense gas observed by the H CO + ( J = 1 − CO + emission toward the ρ Ophiuchi main cloud. For comparison, the 850 µ m image obtained with the SCUBA at the JCMTis shown in Figure 1b. While the distribution of the H CO + emission appears to cover a largerarea than that of the 850 µ m emission, the overall spatial distribution of the H CO + emissionis similar to that of the 850 µ m emission. We note that it is very difficult to observe extended 5 –structures with bolometers mounted on ground-based telescope due to the atmospheric emission.The extended structures larger than ∼ ≈ .
06 pc) are therefore suppressed in removing theatmospheric emission during the data reduction process, making the 850 µ m map mostly devoid ofextended emission (Johnstone et al. 2000b). Our H CO + map shows that the well-known densesubclumps (Oph A, B1, B2, C, E, and F. Note that the Oph D region is out of our area.), previouslyidentified by DCO + and other observations, are very clumpy as recent submillimeter observationshave revealed (Motte et al. 1998; Johnstone et al. 2000a). In the next section, we identify theclumpy structures as dense cores applying a clump-finding algorithm, clumpfind (Williams et al.1994), and derive physical properties of the identified cores. Blow-up total integrated intensity maps toward the Oph A and Oph B1, B2 and C regionsare shown in Figures 2a and 2b, respectively. Iso antenna temperature surfaces of the same areasin the 3D data space ( α - δ - v LSR ) are also shown in Figure 3. Our intensity maps can be comparedwith Figure 2 of Andr´e et al. (2007), the N H + ( J = 1 −
0) intensity maps obtained with theIRAM 30 m telescope. The beam sizes for both the telescopes are comparable: the HPBW ofthe IRAM 30 m, 26.4”, is about 1.5 times as large as that of the Nobeyama 45 m telescope, 18”.Both the molecules are also expected to trace dense molecular gas in nearly the same density rangebecause the critical densities for excitation of H CO + ( J = 1 −
0) and N H + ( J = 1 −
0) lines arecomparable ( n cr , H CO + = 8 × cm − and n cr , N H + = 2 × cm − ). In fact, the distribution ofthe H CO + emission is remarkably similar to that of the N H + emission.The most intense line emission of H CO + ( J = 1 −
0) comes from the Oph A region. Themaximum antenna temperature reaches T ∗ A ≃ µ m map. The starless objects SM1, SM1N, and SM2 identified by Andr´e et al. (1993) are locatedjust at the densest part of the ridge. Although relatively strong H CO + emission is detectedtoward VLA1623, the prototypical Class 0 object, it is difficult to distinguish the H CO + emissionassociated with VLA 1623 from the ambient component because of our resolution as large as 21”.In the 850 µ m map, Wilson et al. (1999) found two filaments in the Oph A region [see Figure1 of Wilson et al. (1999)]. Similar structures can be recognized in our H CO + map. In the 850 µ m map, the two filaments (the dashed lines in Figure 2a) intersect at the position (RA, Dec) =(16 h m s , − ◦ ′ ′′ ; J2000). In the H CO + map, the north-east filament is not seen, whereasthe north-west filament, a part of the Oph A ridge, has strong H CO + emission. Such a feature isconsistent with that of the N H + map by Di Francesco et al. (2004), indicating that the north-eastfilament presumably consist of diffuse warm gas. The filament is reminiscent of the Orion bar wherethe dust continuum emission enhanced by heating from PDRs and young stars is prominent butthe H CO + line emission is not seen (Ikeda et al. 2007). In fact, the location of the north-east 6 –filament agrees well with the edge of the PDR seen in ISO images (Abergel et al. 1996).In the western side of the Oph A ridge, Wilson et al. (1999) found another interesting structure:two arcs that are labeled with No. 1 and 2 in Figures 2a and 3a. In addition to these two, weidentified 3 new arcs in the Oph A region (labeled with No. 3 through 5). Although the No. 1 arcis not clearly recognized as an arc in the total intensity map of Figure 2a owing to the presenceof the extended diffuse component, the No. 1 arc as well as the No. 2 and 3 can be clearly seenin the iso-temperature surfaces in Figure 3a. These arcs may be created by the stellar wind fromthe young B3 star S1. Another possibility is the effect of three outflows detected in this regionwhose outflow axes are almost parallel in the α - δ space and inclination angles are around 60 − CO + emission is detected, appear to be parts of a cavity created by the VLA 1623 outflow.On the other hand, in the eastern side of the Oph A ridge, we can recognize two new arcs. Thesearcs (labeled with No. 4 and 5) appear to be related to a giant protostellar outflow that has beenrecently discovered by Nakamura et al. (2009, in preparation) on the basis of the CO ( J = 3 − CO + map shows a very clumpystructure with ∼
10 local peaks in the region. Many dust cores identified by Motte et al. (1998)are distributed mainly in the bright area of the H CO + map, where some cores identified bythe clumpfind in the data cube (see §
4) are overlapped on the plane of the sky. It is difficult todistinguish such an overlapped core from the other in the H CO + total integrated intensity mapor the dust continuum map that does not have velocity information. In the southern part of theOph B2 region (labeled with No. 1), a hole is seen in both the H CO + and 850 µ m maps. Thehole is particularly clear in the data cube in Figure 3b. Two other holes are also seen near thenorthern (No. 2) and southern (No. 3) edges of the B2 region. These holes are likely to be createdby the giant protostellar outflow recently found by Nakamura et al. (2009). The blue lobe of thisoutflow is extended beyond the B2 region, reaching the No. 4 and 5 arcs. The apparent luminosityand the total mass of this outflow are largest in ρ Oph. The driving source is likely to be a classI YSO Elias 32 or 33. No. 1 and 2 holes are likely to be created by the red and blue lobes,respectively. The interaction between the protostellar outflows and the dense gas will be discussedelsewhere (Nakamura et al. 2009, in preparation). Cavities that could be created by protostellaroutflows are also found in NGC1333, a nearby cluster forming clump in the Perseus molecularcloud (Knee & Sandell 2000; Sandell & Knee 2000; Quillen et al. 2005), suggesting that densitystructures in cluster forming regions are strongly affected by protostellar outflows. Nakamura etal. (2009) also found other examples of the dynamical interaction between the dense gas and theoutflows in ρ Oph, which will be shown elsewhere. 7 –
4. Physical Properties of Dense Cores
We identified 68 H CO + cores in the ρ Ophiuchi main cloud, using the clumpfind methoddescribed in § § CO + relative to H , which is important to the determination of core mass, is estimated in § § µ m cores listed in Table 7 ofJørgensen et al. (2008). Finally, we show in § To identify dense cores from the H CO + data cube, we applied the clumpfind method de-scribed in § T A, th ) and stepsize (∆ T A )of the 2 σ noise level for core identification, following Williams et al. (1994) who recommendedusing this set of the parameters based on the results of their test simulations. To identify the coreswith reasonable accuracy, we imposed the following 3 criteria: (1) a core must contain two or morecontinuous velocity channels, each of which has at least 3 pixels whose intensities are above the3 σ noise level, and (2) the pixels must be connected to one another in both the space and velocitydomains. (3) Furthermore, we rejected core candidates located at the edge of the observed region.The total intensity of the minimum mass core identified by this method has about 60 σ levels. Usingour definition of core mass (eq.[1]), a minimum core mass is estimated to be 0.28 M ⊙ and 0.38 M ⊙ for T ex = 12 K and 18 K, respectively. The total mass of the identified 68 cores is estimated to be228 M ⊙ , 60 % of the total mass detected above the 2 σ level.In general, the physical properties of the identified cores depend on the core identificationscheme and its parameters. In our case, we expect that our clumpfind can detect the real 3Ddensity structures in the cloud from the H CO + ( J = 1 −
0) spectral line position-position-velocity data cube because of the following reasons. According to Williams et al. (1994), thephysical properties of the cores identified by clumpfind are affected siginificantly by the adoptedthreshold and stepsize unless they are carefully chosen. They recommend using the 2 σ noise levelfor both the threshold and stepsize, for which the clumpfind could detect and accurately measurethe individual core properties in their test simulations. We adopt this set of the parameters in ourclumpfind. Recently, several authors revisted this problem (Pineda et al. 2009; Ikeda & Kitamura2009). Pineda et al. (2009) reexamined the effect of varying the values of the threshold ( T A, th =3, 5,and 7 σ ) and stepsize (3 σ ≤ ∆ T A ≤ σ ) for the core identification, and confirmed that the physicalquantities of the identified cores depend on these parameters especially when the distribution ofthe emission is not localized in the data cube and the larger values of the parameters are used 8 –[see also Schneider & Brooks (2004) for the case of the low-spatial resolution ( ∼ CO ( J = 1 −
0) data with T A, th = ∆ T A = 3 σ ]. Ikeda & Kitamura (2009) explored morereasonable parameter ranges of T A, th = 2 − σ and 2 σ ≤ ∆ T A ≤ σ suitable for their core studyin the OMC-1 region, and demonstrated that the power-law index of a core mass spectrum doesnot depend on the threshold and stepsize. Williams et al. (1994) also discussed the “overlap” effectfor which multiple cores that almost overlap along the line-of-sight are not separately identifiedby the clumpfind when they have nearly the same line-of-sight velocities. This overlap effect islikely to be significant when the gas is highly turbulent and the emission tends to be distributedin the large volume of the 3D data cube (Ballesteros-Paredes & Mac Low 2002; Smith et al. 2008).For example, Ostriker et al. (2001) performed 3D MHD turblent simulations and claimed that theemission lines are often singly peaked even in the presence of multiple condensations along theline of sight. We note that their model clouds are highly turbulent, gravitationally unbound andthe condensations have extremely large virial ratios, ranging from 10 to 10 . This suggests thatthe overlap effect may be crucial for less dense gravitationally unbound molecular gas componentsobserved by CO and CO (see Ossenkopf & Mac Low 2002, for the case of the Polaris molecularcloud, a nearby diffuse, gravitationally unbound molecular cloud.). In contrast, for our H CO + data, the overlap effect is expected to be minor because the H CO + emission can trace only densegas with densities greater than ∼ cm − and appears to be reasonably localized in the 3D datacube, indicating that the dense gas is likely to have a small volume filling factor in the cloud andmoderately quiescent (the Mach number ∼ − § CO + data.A caveat of our analysis is that the cores identified by the above procedure (and any othercore identification schemes) are resolution dependent in the sense that the substructures smallerthan the telescope beam size cannot be resolved. It remains unclear whether an identified core isa unit that is separated dynamically from the background media. However, we expect that thestructures identified from high density tracers such as H CO + ( J = 1 − H + ( J = 1 − ∼ cm − (Li & Nakamura 2006;Nakamura & Li 2007). Therefore, in the following, we discuss the statistical properties of the coresidentified by the above procedure, assuming that the identified cores can be regarded as units thatare reasonably separated dynamically from the background media. Further investigation of thisproblem should be done when higher spatial resolution observations are available. 9 – We derive physical properties of the cores using the definitions described in § T ∗ A, peak , within the core. The core radius, R core , isdetermined by taking the projected area enclosed by the 2 σ level contour and computing the radiusof the circle required to reproduce the area, taking into account the correction for the telescopebeam (18”). The aspect ratio is computed as the ratio of the major to minor axis lengths that aredetermined by the two-dimensional Gaussian fitting to the total integrated intensity distribution ofthe core. The FWHM line width, dv core , is corrected for the velocity resolution of the spectrometers(= 0 .
13 km s − ) and for line broadening due to hyperfine splitting of H CO + . The LTE mass isestimated as M LTE = 1 . × − (cid:18) X H CO + . × − (cid:19) − T ex exp (4 . /T ex ) × (cid:18) D (cid:19) (cid:18) ∆ θ ′′ (cid:19) (cid:16) η . (cid:17) − (cid:18) P i T ∗ A,i ∆ v i K km s − (cid:19) M ⊙ , (1)by assuming that the H CO + emission is optically thin. Here, X H CO + is the fractional abundanceof H CO + relative to H , T ex is the excitation temperature, D is the distance to the ρ Ophiuchimain cloud, P i T ∗ A,i ∆ v i is the total integrated intensity of the core and ∆ v i = 0 .
13 km s − . Weadopt the fractional abundance X H CO + = 1 . × − (see § D = 125 pc onthe basis of the values recently updated by Lombardi et al. (2008) and Loinard et al. (2008) (seealso Wilking et al. 2008). The excitation temperature is assumed to be T ex = 12 K (Motte et al.1998) except for the area shown in Figure 2a, i.e., the Oph A region, where T ex = 18 K followingthe N H + observations by Di Francesco et al. (2004).The virial mass M vir is calculated as M vir = 5 a − R core dv G = 209 a − (cid:18) R core pc (cid:19) (cid:18) dv tot km s − (cid:19) M ⊙ , (2)and dv tot = (cid:20) dv + 8 ln 2 k B T (cid:18) µm H − m obs (cid:19)(cid:21) / (3)where k B is the Boltzmann constant, T is the kinetic temperature of the molecular gas, µ isthe mean molecular weight of 2.33, m H is the mass of a hydrogen atom, m obs is the mass of aH CO + molecule, and a is a dimensionless parameter of order unity which measures the effectsof a nonuniform or nonspherical mass distribution (Bertoldi & McKee 1992). For a uniform sphereand a centrally-condensed sphere with ρ ∝ r − , a = 1 and 5/3, respectively. For our cores, theeffects of the nonspherical mass distribution appear to be small because the aspect ratios are not 10 –so far from unity (see also Figure 2 of Bertoldi & McKee 1992). The virial ratio, the ratio betweenthe virial mass to the LTE mass, is defined as α vir ≡ M vir /M LTE , which is equal to twice the ratioof the internal kinetic energy to the gravitational energy.The mean number density of the core, ¯ n , is calculated as the LTE mass divided by the volumeof a sphere with radius, R core , ¯ n = 3 M LTE πµm H R . (4) CO + To evaluate the fractional abundance of H CO + , we smoothed the 850 µ m map with theangular resolution of 21” to match the H CO + total integrated intensity map and computed thecolumn densities of H CO + ( N H CO + ) and H ( N H ) at each pixel in the maps. Here, we computed N H from the relation N H = S beam850 / [Ω beam µm H κ B ( T )] , (5)where S beam850 is the 850 µ m flux density per beam, Ω beam is the main beam solid angle, B ( T )is the Plank function for a dust temperature T , and κ (= 0.01 cm g − ) is the dust opacityat 850 µ m (Henning et al. 1995; Ossenkopf & Henning 1994; Johnstone et al. 2000a). The dusttemperature is assumed to be equal to the gas temperature. In Figure 4 the fractional abundances X H CO + computed at all the pixels above the 3 σ levels for both the H CO + and 850 µ m mapsare plotted against N H . We note that the H column density derived from the 850 µ m map is verysensitive to the temperature and the dust opacity assumed, and therefore have uncertainty of atleast a factor of a few. The fractional abundance of H CO + tends to decrease with increasing H column density. The best-fit power-law is given bylog X H CO + = ( − . ± .
01) + ( − . ± . N H / cm − ) (6)with a correlation coefficient R = 0 .
68, indicating that X H CO + is nearly proportional to N − / .This tendency is in good agreement with a theoretical consideration of the ionization fraction inmolecular clouds if the column density is proportional to the local volume density for each molecule.When the ionization rate by cosmic rays ( ∝ n n , where n n is the density of neutral gas) balanceswith the recombination rate ( ∝ n i n e , where n i and n e are the densities of positive ion and electron,respectively, and n i ≈ n e ), the ionization fraction is inversely proportional to the square root of theneutral gas density (Elmegreen 1979). In typical regions in molecular clouds, the most abundantpositive molecular ion is expected to be HCO + . Therefore, the fractional abundance of H CO + aswell as that of HCO + is expected to be inversely proportional to the square root of the neutral gasdensity if the fractional abundance of H CO + relative to HCO + is almost constant. More detailedstudies indicate that the relation X H CO + ∝ n − / n is a reasonable approximation of the fractionalabundance in the density range of n . cm − , although metal ions such as Mg + may be themost abundant positive ions (Nishi et al. 1991; Nakano et al. 2002). We note that as mentioned 11 –in §
3, in the 850 µ m data the structures larger than 1.5’ ( ≈ .
06 pc) are removed during the datareduction process. To evaluate how the effect of this artificial filtering affects the estimate of theH CO + abundance, we repeated the same analysis by filtering out all the structures larger than1.5’ in the H CO + map and confirmed that the effect of this artificial filtering is negligible. Thisis because the dense gas detected by the 850 µ m and H CO + is spatially well-localized.In nearby low-mass star forming regions, Butner et al. (1995) found that X H CO + ranges from3 × − to 4 × − for dense cores with 2 × cm − . N H . × cm − , where theyestimated the H column densities from C O ( J = 1 −
0) observations. For comparison, their valuesare indicated by crosses in Figure 4. Although their X H CO + tends to decrease with increasing N H in the same way as for our data, their X H CO + tend to be somewhat larger than the valuespredicted from eq. [6]. If X H CO + derived by Butner et al. (1995) is fitted by a power-law with thesame power index as in eq. [6], the coefficient becomes about twice that of eq. [6]. It is difficult,however, to judge whether this difference originates from the different environments because boththe data have uncertainty at least by a factor of a few. If both the data are fitted by a power-lawwith a single index, then the best-fit function is given bylog X H CO + = ( − . ± .
01) + ( − . ± . N H / cm − ) (7)with R = 0 .
87. The mean fractional abundance h X H CO + i is estimated to 1 . × − witha standard deviation of 0 . × − . In the present paper, we assume the constant fractionalabundance of X H CO + = 1 . × − for the entire area. In future analysis, the spatial variationin fractional abundance will be taken into account.van Dishoeck et al. (1995) measured a smaller fractional abundance X H CO + ≈ . × − toward IRAS 16293 − ρ Ophiuchi molecular cloud complex (see Andr´e et al. 2000). They obtained the H columndensity of 2 × cm − toward this YSO from the C O observations, somewhat larger than themean H column density of ρ Oph evaluated from the 850 µ m data. Their value, plotted in Fig. 4by a filled square, is almost on our best-fit power-law of eq. [7]. Thus, we believe that our adoptedfractional abundance of 1 . × − is a plausible representative value for the ρ Ophiuchi maincloud.We note that several high-resolution interferometric observations of starless cores in relativelyquiescent, low temperature star forming regions like Taurus have revealed that several gas-phasemolecules including CO, HCO + , and their isotopes tend to deplete at the densities higher than10 cm . One might interpret that the decrease in the H CO + abundance with increasing columndensity is due to the effect of the depletion. However, the effect of the depletion should be smallbecause of the following reasons. In cluster forming regions like ρ Oph, relatively high turbulentmotions, protostellar outflows and stellar radiation tend to increase the temperature steadily and/ortemporarily, preventing or slowing down the molecular depletion in starless cores. In addition, thehigh density part where the molecular depletion is significant is spatially localized and thereforethe interferometric observations with high spatial resolution are needed to resolve the effect of 12 –depletion in a core. However, our beam size ( ∼ ′′ ) is not enough to well resolve the structuresinside each core (Aikawa et al. 1995). Furthermore, even if the depletion of gas-phase CO andHCO + is significant in ρ Oph, the fractional abundance of HCO + relative to CO does not changewith increasing column density because these carbon-bearing molecules tend to equally deplete(Aikawa et al. 1995). In contrast, the H CO + fractional abundances of Butner et al. (1995) andvan Dishoeck et al. (1995), for both of which the H column densities are measured from the COisotopes, do not show such a constant abundance, but decrease significantly with increasing columndensity, indicating that the dependence of the H CO + abundance on the column density comesfrom the different physics from the molecular depletion. In Table 1, we present the physical properties of the 68 identified cores. In Table 2, wesummarize the minimum, maximum, mean, and median values of each quantity. The histogramsof the radius ( R core ), LTE mass ( M LTE ), and mean density (¯ n ) of the H CO + cores are shown inFigure 5. For comparison, the radius, mass, and mean density of the 850 µ m cores listed in Table7 of Jørgensen et al. (2008) are indicated by the grey histograms in Figure 5. We selected only the850 µ m cores located inside the same observed area as our H CO + map. We also removed coreswhose radii are smaller than the SCUBA beam size of 14” and corrected the radii of the other coresfor the telescope beam size by using eq. [2] of Ikeda et al. (2007). In this procedure, 43 dust coreswere selected using the 2 σ noise level of 0.04 Jy beam − . The minimum, maximum, mean, andmedian values of each quantity of the 850 µ m cores are summarized in Table 3. We note that asmentioned in §
3, in the 850 µ m data the structures larger than 1.5’ ( ≈ .
06 pc) are removed duringthe data reduction process and therefore the number of dust cores with radii larger than about 0.03pc is likely to be underestimated.The radius of the H CO + cores ranges from 0.022 to 0.069pc and its distribution has a singlepeak at around the mean of 0.045pc (see Figure 5a). The maximum radius of the H CO + coresis only about three times as large as the minimum radius. In other words, most of the cores havesimilar size. On the other hand, the distribution of the LTE mass is somewhat broad comparedwith that of the core radius. The LTE mass of the H CO + cores ranges from 0.4 to 22 M ⊙ and itsdistribution has a single peak at around the mean of 3.3 M ⊙ (see Figure 5b). The maximum massis about 56 times as large as the minimum. The mean density of the H CO + cores also shows asimilar broad distribution, ranging from 3 . × to 5 . × cm − , with a single peak at aroundthe mean density of 1 . × cm − (see Figure 5c).These distributions of the physical quantities of the H CO + cores are qualitatively similarto those of the 850 µ m cores, although the mean radius and mass of the H CO + cores are abouttwice those of the 850 µ m cores. On the other hand, the mean density of the H CO + cores issmaller than that of the 850 µ m cores by a factor of 5. This large difference in the mean density 13 –probably comes from the fact that the 850 µ m emission tends to trace the higher density parts ofthe H CO + cores (see Ikeda et al. 2009). Another reason is due to the overlap effect. It is likelythat dense cores often overlap one another along the line of sight. Although such overlapped corescan be separated in some degree for the H CO + cores because of their velocity information, suchseparation is impossible for the two-dimensional clumpfind method that was used for the 850 µ mmap. In fact, the distribution of the core radius for the 850 µ m cores has a tail toward the largerradius. A more prominent tail can be seen in the distribution of the core mass shown in Figure5b, where the mean core mass is about 2.6 times as large as the median. The tail in the coremass, however, can be caused by the temperature difference. For all the dust cores, the uniformtemperature of T = 15 K is assumed in Jørgensen et al. (2008). For several massive dust cores,however, the temperatures may be underestimated because they are located in the Oph A regionwhere the temperatures are likely to be higher because of active star formation.The distribution of the FWHM line width of the H CO + cores are shown in Figure 6. Theline width ranges from 0.19 to 0.77 km s − with a mean of 0.49 km s − . Following Myers et al.(1991), we classified our cores into two groups: “thermal core” and “turbulent core”, on the basisof the critical line width, dv cr , expressed by dv cr = (cid:20) k B T (cid:18) µm H + 1 m obs (cid:19)(cid:21) / . (8)The critical line width of H CO + is estimated to be 0.50 and 0.62 km s − for T = 12 K and 18 K,respectively. Figure 6 indicates that about 40 % of the cores are classified as the turbulent core.However, almost all the turbulent cores have transonic turbulent motions, and highly turbulentcores ( dv core & − ) found in massive star forming regions such as Orion A (Ikeda et al. 2007)are not seen in the ρ Ophiuchi main cloud.
Recent observations of the Ophiuchi molecular cloud complex with the Spitzer Space Telescope(Padgett et al. 2008) allow us to compare the physical properties of H CO + cores associated withand without YSOs (Jørgensen et al. 2008). Previous studies have suggested that the physicalproperties of dense cores associated with YSOs are different from those of cores without YSOs(e.g., Benson & Myers 1989; Jijina et al. 1999; Walsh et al. 2007). Based on the spectral indicesat the near- and mid-infrared wavelengths, Jørgensen et al. (2008) classified YSOs in L1688 into 4classes that represent the following evolutionary stages of YSOs: Class I, Flat Spectrum, Class II,and Class III. We use their YSO catalog to identify the H CO + cores with YSOs. We note that intheir classification, the Class 0 YSOs, the youngest objects, are included in Class I. For example,the prototypical Class 0 object, VLA 1623, is classified as Class I in their list.The spatial distribution of the YSOs is shown in Figure 7, which indicates that the distributionof the Class I and Flat Spectrum sources follows that of the H CO + ( J = 1 −
0) emission. On 14 –the other hand, more evolved YSOs, the Class II and III sources, tend to be distributed in theentire observed area with no strong correlation with the distribution of the H CO + ( J = 1 − CO cores into the following twogroups: protostellar and starless cores. If either a Class I or Flat Spectrum object is located withinthe extent of a H CO + core, we regard the core as a protostellar core. The others are classifiedas starless cores, even if Class II or III sources are located within the cores. In this classification, aClass I or Flat Spectrum source is sometimes located within the extents of two or more cores whichpartly overlap one another along the line of sight. We classified all such cores as protostellar coresbecause we have no information on the line of sight velocities of the YSOs. Therefore, the numberof protostellar cores is likely to be overestimated. According to the above procedure, we found 34protostellar and 34 starless cores.Figure 8 compares the physical properties of the starless and protostellar cores. The minimum,maximum, mean, and median values are summarized in Table 4. The radii (Figures 8a) and masses(Figures 8b) of the protostellar cores tend to be larger than those of the starless cores. In particular,the mean LTE mass of the protostellar cores is about twice as large as that of the starless cores.These results are consistent with those of the N H + cores in NGC1333 (Walsh et al. 2007) and theH CO + cores in Orion A (Ikeda et al. 2007).The line widths of the protostellar cores tend to be larger than those of the starless cores(Figures 8c). A similar tendency is seen in the NH cores observed by Benson & Myers (1989)and Jijina et al. (1999), who interpreted that stellar winds and protostellar outflows create localturbulence. If stellar winds and protostellar outflows enhance internal turbulent motions for theprotostellar cores, such cores must have had virial ratios smaller than the current values in theprestellar phase. Although the distribution of the virial ratios for the protostellar cores appears tobe broader than that of the starless cores, there seems to be no significant difference in the virialratio between the protostellar and starless cores (Figures 8d): for both the starless and protostellarcores, the virial ratio has a mean of about 2 −
3, ranging from 0.4 to 8. Here, the dimensionlessparameter a in eq.[2] is adopted to be unity, the value of a uniform sphere, for both the starless andprotostellar cores. If for protostellar cores, the value of a centrally condensed density distribution, a = 5 /
3, is adopted, the virial ratios of the protostellar cores tend to be somewhat smaller thanthose of the starless cores that are expected to have flattened density distributions. This appearsto contradict the idea that outflows enhance the virial ratios of the parent cores. About 60 % of thecores have virial ratios smaller than 2, for which the net internal energies are negative. Therefore,we conclude that the majority of the cores are likely to be self-gravitating.Figure 8e shows that the distributions of the mean densities of the protostellar and starlesscores are similar to each other, although some of the protostellar cores have very large densities.Figure 8f indicates that the starless cores tend to have larger aspect ratios than those of theprotostellar cores. For example, for the starless cores, 5 (15%) out of 34 have aspect ratios largerthan 2, whereas for the protostellar cores, no cores have such large aspect ratios. The starless cores 15 –with aspect ratios larger than 2 have large virial ratios ( α vir &
3) and are likely to be gravitationallyunbound.
The line width measured from molecular line emission provides us with information on kineticproperties such as thermal motion and turbulence. Based on data in the literature, Larson (1981)found that the line width of molecular clouds and cores is well correlated with the radius andmass, and the correlations are approximately of power-law form. He suggested that the power-law relations stem from processes of interstellar turbulence cascade. The strong correlations of thepower-law relations also suggest that the molecular clouds and cores are nearly in virial equilibrium.Since his pioneering work, many studies have been carried out to investigate the dependence of theline width on its radius for various molecular clouds and cores using different molecular emissionlines (e.g., Sanders et al. 1985; Dame et al. 1986; Heyer & Brunt 2004). Observations based mainlyon low-density tracers such as CO ( J = 1 −
0) and CO ( J = 1 −
0) have generally supportedLarson’s idea (e.g., Myers 1983; Fuller & Myers 1992). In contrast, observations based on higher-density tracers such as C O, CS, and H CO + have often shown that the dependence of the linewidth on its radius tends to be very weak (e.g., Tachihara et al. 2002; Ikeda et al. 2007, 2009;Saito et al. 2008). For massive star forming regions, the shallower power-law relations have alsobeen found (Caselli & Myers 1995; Plume et al. 1997).The line width-radius relation for the identified cores is presented in Figure 9a. The filledsquares and open circles indicate the starless and protostellar cores, respectively. As mentioned in § dv core / km s − ) = (0 . ± . . ± . R core / pc) , (9)with R = 0 . CO observations. The line widths of theH CO + cores tend to be significantly larger than those of the Heyer & Brunt relation and somewhatsmaller than those of the Larson relation. We note that the line widths of the Larson relation tendto be larger over the scale 0.01 − CO + cores in Orion A (Ikeda et al. 2007). In § CO + cores in more detail.To separate the turbulent motion from the thermal motion within each core, we plot in Figure9b the nonthermal line width dv NT [ ≡ ( dv − k B T /µm H ) / ] against core radius. The best-fitpower-law for all the cores, shown by the solid line in Figure 9b, is given bylog( dv NT / km s − ) = (0 . ± . . ± . R/ pc) (10)with R = 0 . dv NT / km s − ) = 1 . R/ pc) . , is indicated by the dashedline, where we replaced the radius of Caselli & Myers (1995) ( R CM ) by the radius of our defini-tion. See the Appendix in detail. Although our relation has a power-law index similar to that ofCaselli & Myers (1995), the coefficient is somewhat larger. This suggests that turbulence withinour cores is not as much dissipated as the Caselli & Myers relation predicts. This tendency isdifferent from that of the H CO + cores in Orion A where the dv NT - R core relation agrees well withthe Caselli & Myers relation (Ikeda et al. 2007). In § § M LTE /M ⊙ ) = (3 . ± .
34) + (2 . ± .
25) log( R core / pc) (11)where R = 0 . M LTE = 4 πµm H ¯ nR / n = 1 . × cm − . The power-law index of ∼ CO + ( J = 1 − n cr ≃ × cm − . The coefficient of our M LTE - R core relation is an order of magnitude larger thanthat of Larson (1981), which is mainly based on CO ( J = 1 −
0) whose critical density is about10 cm − . This tendency is in good agreement with that of the Orion A cores (Ikeda et al. 2007,2009). 17 – The boundedness of a core is often estimated using the virial ratio. In Figure 10, we plot againstLTE mass the virial ratio calculated using eq.[2], where for both the protostellar and starless coresa dimensionless parameter a is set to unity, the value of a uniform sphere. The best-fit power-lawof our cores is given bylog α vir = (0 . ± . − . ± . M LTE /M ⊙ ) (12)with R = 0 . α vir = 2 . M LTE /M J ) − / ,where M J is the Jeans mass defined by eq. [2.13] of Bertoldi & McKee (1992). For comparison,the virial ratio of a self-gravitating core confined by ambient pressure is plotted by the dashed linein Figure 10. Here, using the velocity dispersion of 0 .
35 km s − and the ambient gas density of10 cm − (see the derivation of these values in § . ⊙ , almost equal to the median mass of the identified cores. The best-fit power-law is very closeto the virial ratio of a self-gravitating core confined by ambient pressure. The virial ratios of theidentified cores tend to be, however, smaller than that of a non self-gravitating, pressure-confinedcore with α vir = 2 . M LTE /M J ) − / , for a given core mass (Bertoldi & McKee 1992). Therefore,for our cores, both the self-gravity and ambient pressure play an important role in dynamics of thecores. In §
5. Discussion5.1. Core Mass Spectrum in the ρ Ophiuchi Main Cloud
The mass spectrum of our cores is plotted in Figure 11. The core mass spectrum appears tobe fitted by a two-component power-law. There seems to be a break at around 7 M ⊙ , about twicethe mean LTE mass of 3.4 M ⊙ . Above the break, the mass spectrum is steeper, while below thebreak, it is flattened. The mass spectrum can be fitted by the following two-component power-law: dN/dM ∝ M − . ± . for M LTE . M ⊙ (13) ∝ M − . ± . for M LTE & M ⊙ . (14)This mass spectrum is broadly consistent with those of the dust cores in ρ Oph, although for the dustcores the low-mass parts are somewhat steeper and the break masses are smaller (Motte et al. 1998;Johnstone et al. 2000a; Stanke et al. 2006; Stamatellos et al. 2007; Enoch et al. 2008; Simpson et al.2008). For example, Stamatellos et al. (2007) revised the mass spectrum of the dust cores identifiedby Motte et al. (1998) by reestimating the dust temperatures. The revised mass spectrum is fitted 18 –by a two-component power-law with a low-mass index of − .
5, a high-mass index of −
2, and a breakat around 1 M ⊙ . Simpson et al. (2008) reanalyzed the SCUBA 850 µ m archive data and obtainedthe core mass spectrum that can be fitted by a three-component power-law with a low-mass index of − .
7, intermediate-mass index of − .
3, high-mass index of − .
35, and two breaks at around 0.7 M ⊙ and 2 M ⊙ . We note that the slopes and the break masses of the core mass spectra are affected bythe number of bins, if it is not appropriately set (Rosolowsky 2005). Thus, the cumulative form ofthe core mass spectra is sometimes used when the number of cores is small, although it requiresa more complicated uncertainty analysis (see e.g., Reid & Wilson 2006). As for the parameters ofthe clumpfind, the slopes are almost independent of the threshold and stepsize of the clumpfind aslong as they are appropriately set (Ikeda & Kitamura 2009). For the core mass spectra mentionedabove, the difference in the break mass is likely to come from the fact that the H CO + emissiontraces more extended, less dense structures than the dust continuum emissions. The break massmay also be affected by the “confusion” which means the situation that multiple cores with similar v LSR are overlapped along the same line of sight and therefore cannot be separated into individualcores using the clumpfind (see Ikeda et al. 2007, 2009).Based on the ISOCAM observations, Bontemps et al. (2001) identified over 200 YSOs in ρ Oph and derived the mass function of 123 Class II YSOs that is well fitted by a two-componentpower-law with a low-mass index of − .
15, a high-mass index of − .
7, and a break at around0.55 M ⊙ . The mass spectrum of the H CO + cores is roughly similar in shape to the stellar IMF in ρ Oph, although the slopes in the low mass and high mass parts of the H CO + core mass spectrumare somewhat shallower and the break mass is one order of magnitude larger. CO + Cores in the ρ Ophiuchi Main Cloud
Virial theorem is useful for analyzing the dynamical states of dense cores. The virial equationfor a uniform spherical core is given by12 ∂ I∂t = U + W + S (15)where the terms, I , U , W , and S , denote the moment of inertia, internal kinetic energy, gravitationalenergy including the effect of magnetic field, and surface pressure, respectively, and are given asfollows (Nakano 1998): U = 3 M dv W = − GM R core " − (cid:18) ΦΦ cr (cid:19) (17) S = − πR P ex . (18)The values Φ and Φ cr are, respectively, the magnetic flux penetrating the core and the criticalmagnetic flux above which the magnetic field can support the core against the self-gravity. Here, 19 –the core is assumed to have condensed from much lower density medium and the radius of themagnetic flux tube penetrating the core is much smaller than that before contraction [see § P ex is the surface pressure including both thermal and turbulent components. M is the core mass and is chosen to be the LTE mass of the core.All the above terms in the virial equation except P ex and Φ can be estimated from the physicalquantities listed in Table 1. It is difficult to estimate the surface pressures exerted on individualcores directly from our data. Instead of deriving the surface pressures of individual cores, we adopta representative surface pressure from the average densities and velocity dispersions that weremeasured in subclupms Oph A, B, C, E, and F: h P ex i ≈ h ρ i h σ i . Here, h ρ i is the average densityand h σ i is the average velocity dispersion including both the thermal and turbulent components as h σ i = h σ NT i + c s , where h σ NT i and c s are the velocity dispersion of the nonthermal componentand the isothermal sound speed, respectively. The average density and velocity dispersion wereevaluated as h ρ i / (2 . m H ) ≈ (0 . − × cm − and h σ i ≈ . − . − , respectively. Theaverage surface pressure is then given by h P ex i /k B ≈ × K cm − . This value is about a fewtimes as large as the thermal pressure [ ≈ cm − × (12 −
18) K ≈ (1 − × K cm − ]. In thefollowing, we adopt the above h P ex i as a representative value for our cores. This surface pressureis about a half the average internal pressure inside the cores ( ≈ × K cm − ) and close to thelower limit of the critical ambient pressure for the dust cores obtained by Johnstone et al. (2000a).On the other hand, there is only one reliable measurement of the magnetic field strength towardthe ρ Ophiuchi main cloud. Crutcher et al. (1993) performed OH Zeeman effect measurementstoward two positions in ρ Oph with a beam size 18’. For one of the two positions that well coversthe entire observed area of our data, they derived the line-of-sight magnetic field strength of about10 µ G at the low density cloud envelope of N H ≈ × cm − (see also Troland et al. 1996).Based on this measurement, Crutcher (1999) estimated the magnetic flux normalized to the criticalvalue to be Φ / Φ cr ≃ .
4. If we adopt this value as a representative value of Φ / Φ cr for our cores,the magnetic effect is likely to be minor: it reduces the gravitational energy term only by 16%.However, recent turbulent simulations have demonstrated that cores formed out of turbulent cloudscan have much larger values of Φ / Φ cr than the cloud initial values (e.g., Dib et al. 2007) and thus itis very difficult to assess the values of Φ / Φ cr for the individual cores without direct measurementsof the magnetic fields associated with the cores. In the following, we simply assume Φ / Φ cr = 0.The equilibrium line, U + W + S = 0, is shown by the solid line in Fig. 12, where the surfaceterm ( S ) is plotted against the gravitational energy term ( W ), both normalized to the internalenergy term ( U ). For the cores that lie below the solid line, the value of U + W + S is negativeand thus expected to be bound. All the others are unbound and expected to disperse away, ifthey do not gain more mass through accretion and/or merging with other cores, or reduce internalsupport through turbulence dissipation. More than a half the cores lie below the equilibrium line,and are thus bound: they are expected to collapse and form stars. This is also true even in thepresence of magnetic field as long as the magnetic flux does not exceed about a half the criticalvalue. Furthermore, the majority of the cores lie below the line of S = W (dashed line), indicating 20 –that the surface term is more important than the gravitational energy term. Even for the coreslying above the line S = W , the surface pressure appears to be dynamically important for almostall such cores because the deviation from the line is small. We note that this plot should be validstatistically, but for any individual points the true surface pressures could be higher or lower thanthe representative surface pressure.Our virial analysis indicates that the formation and evolution of dense cores in the clusterenvironment are likely to be strongly influenced by the external compression due to local turbulentmotions (see also Dobashi et al. 2001 for importance of external pressures at cloud scales). Suchlocal compression may be responsible for the formation of binary and multiple stars or formationof substellar objects (brown dwarfs or planetary mass objects) in each core under the clusterenvironment (Hennebelle et al. 2003; Gomez et al. 2007; Whitworth et al. 2007). The Orion A molecular cloud is the nearest giant molecular cloud located at a distance of480 pc (Genzel et al. 1981), about 4 times as distant as the ρ Ophiuchi main cloud. Recently,Ikeda et al. (2007) carried out the H CO + ( J = 1 −
0) core survey in the whole region of OrionA using the Nobeyama 45 m telescope and identified 236 cores by the same method as adoptedin the present paper. Their core sample is ideal for comparison with our cores because the samemolecular emission line, the same telescope, and the same core identification procedure are used.In this subsection, we compare the physical properties of our cores in ρ Oph with those in OrionA. Figures 13 and 14 compare the physical properties of the ρ Oph cores with those of the OrionA cores. The open squares and crosses indicate the ρ Oph cores and the Orion A cores, respectively.The mean radius of the ρ Oph cores, 0.045 pc, is about three times smaller than that of the OrionA cores, 0.14 pc. This probably reflects the different spatial resolutions (or different distances) forboth the observations. In fact, the area where the ρ Oph cores are distributed is well separatedfrom that of the Orion A cores on the plots of the line width-radius and mass-radius relations (seeFigure 13). On the other hand, the mean mass of the ρ Oph cores, 3.35 M ⊙ , is only about 3.6times smaller than that of the Orion A cores, 12 M ⊙ . This is because the mean density of theOrion A cores, 1 . × cm − , is much smaller than that of the ρ Oph cores, 1 . × cm − . Inparticular, the mean density of the Orion A cores is smaller than the critical density of the H CO + ( J = 1 − × cm − , and therefore the Orion A cores are likely to contain substructures thatcould not be spatially resolved.The mean line width of the ρ Oph cores, 0.488 km s − , is almost the same as that of the OrionA cores, 0.52 km s − , in spite of the different core radii. The ranges of the line widths are alsosimilar to each other (see Figure 13). On the other hand, the virial ratio of the Orion A cores seemsthree times larger than that of the ρ Oph cores for a given LTE mass, as shown in Figure 14. This 21 –difference may be explained also by the different spatial resolutions as follows. The virial ratio isproportional to both the radius and the square of the line width for a given LTE mass. Since theline widths of both the cores are comparable to each other and the radii of the Orion A cores areroughly three times larger than those of the ρ Oph cores, the virial ratio of Orion A becomes threetimes larger than that of ρ Oph for a given LTE mass.To elucidate whether or not the different spatial resolutions dominate the different core prop-erties in the two clouds, we try to eliminate the effect of the different spatial resolutions as follows.We smoothed the original ρ Oph data on a coarser grid with a grid spacing of 80” by convolvingwith a 2D Gaussian kernel with a FWHM of 80” in the α - δ space. The smoothed data correspondto the data that would be observed by the Nobeyama 45 m telescope if the ρ Ophiuchi main cloudwere located at the same distance as the Orion A molecular cloud. Then, applying the clumpfindto the smoothed data, we reidentified 16 dense cores, which are plotted by the filled circles inFigures 13 and 14. In Figures 13a and 13b, the ρ Oph cores identified from the smoothed data arelocated just in the areas where the Orion A cores are distributed. In contrast, the virial ratios ofthe reidentified ρ Oph cores are somewhat smaller than those of Orion A for a given LTE mass (seeFigure 14a). The larger virial ratios for the Orion A cores may reflect the larger X H CO + adoptedby Ikeda et al. (2007), 4 . × − , which is derived using the CO abundance measured in thecloud envelope of Taurus, which has much lower column densities than Orion A. If we use the COabundance obtained by Frerking et al. (1982) for ρ Oph, which has column densities comparableto the region observed by Ikeda et al. (2007), the H CO + abundance is reduced by a factor of2, close to our value. Note that our value is in agreement with that of Ikeda et al. (2007) withinuncertainty mentioned by them. Using our value of X H CO + , the distribution of the Orion A coresin the virial ratio-LTE mass plot becomes consistent with that of the ρ Oph cores identified fromthe smoothed data, as presented in Figure 14b. Although in that case, the LTE masses of the OrionA cores shift upward in Figure 13b, the distributions of the ρ Oph and Orion A cores in the LTEmass-radius relation plot are still well overlapped with each other. Therefore, we conclude thatthere are no clear differences in the core properties between the ρ Oph and Orion A clouds. Theapparent differences in the core properties between the two clouds are likely to be caused mainlyby the different spatial resolutions. CO + cores and Implication for Turbulent Generation Figure 13 shows that there are no highly turbulent cores having dv core & − in ρ Oph.According to Ikeda et al. (2007), all the highly turbulent cores in Orion A are located within 1pc from the M42 HII region and are probably influenced greatly by the nearby OB stars. Theturbulent cores are expected to be responsible for massive star formation (see the discussion of § ρ Oph may imply that massiveO stars will not form in ρ Oph under the current environment.It is also interesting that the range of the line widths of the ρ Oph cores identified from the 22 –smoothed data almost coincide with that of the original ρ Oph cores. If we fit a single power-lawto both the ρ Oph cores identified from the original and smoothed data, the best-fit result becomes dv core = (0 . ± . R . ± . with a very small correlation coefficient of 0.137, which is shownby the solid line in Figure 13. In other words, the line width appears to be almost independent ofthe core radius. Since the cores identified from the smoothed data contain adjacent several coresidentified from the original data, the nearly independent relationship between the line width andcore radius, or the nearly flat line width-radius relation, suggests that the inter-core motions amongthe neighboring cores are almost comparable to the internal motions in the individual cores. Sucha feature is pointed out by Andr´e et al. (2007) who measured the velocity difference among theneighboring cores from the centroid velocities of the cores observed by N H + ( J = 1 − dv core ≈ const.) and the core mass is proportional to R , then thevirial ratio is scaled as α vir ∝ R core dv / ( GM LTE ) ∝ M − / , a similar power-law to that derivedby Bertoldi & McKee (1992). This again suggests the importance of ambient turbulent pressure indynamics of the cores as discussed in § ρ Oph cores appears to be inconsistent with one of the mostreliable measurements of the line width-size relation, recently obtained by Heyer & Brunt (2004),who found a strong correlation between the line width and size based on the CO observationstoward 27 nearby giant molecular clouds, despite the large differences in cloud environments andlocal star formation activity. Their relation, plotted by the dashed line in Figure 13 [( dv/ kms − ) =1 . R/ pc) . ], leads to the interpretation that most of the turbulent energy comes from the largestscale that is comparable to the cloud size (see the Appendix). Heyer & Brunt (2007) claimed thatthe driving mechanisms of turbulence at small scales such as protostellar outflows and stellar windsmay not play an important role in dynamics of molecular clouds. The FWHM line width (∆ V ∼ . − ) and radius ( R ∼ . CO ( J = 1 −
0) line toward the whole ρ Ophiuchi main cloud are in good agreement with the Heyer & Brunt relation. In contrast, the ρ Ophcores identified from the original data tend to deviate upwards from the Heyer & Brunt relation.The nearly flat line width-radius relation of the ρ Oph cores may also suggest that turbulent energyhas been injected at the scales smaller than R ∼ .
35 pc, at which our best-fit power-law intersectswith the Heyer & Brunt relation. This scale is in reasonable agreement with the characteristic lengthscale of the outflow-driven turbulence estimated from the theoretical consideration (Matzner 2007;Nakamura & Li 2007). Based on CO and C O observations, Swift & Welch (2008) derived asimilar characteristic scale of the outflow-driven turbulence of ≈ .
05 pc toward L1551. However,the universality of the Heyer & Brunt relation implies that the outflow-driven turbulence may belimited to the localized dense regions where active star formation occurs. A caveat is that it isunclear whether the Heyer & Brunt relation, based on the CO observations, is still applicableto dense gas where stars are forming. Therefore, the deviation of the line widths for ρ Oph fromthe Heyer-Brunt relation may be apparent. However, it is worth noting that the nearly flat linewidth-radius relation is consistent with the virial ratio-mass relation of the pressure-confined, self-gravitating cores ( α vir ∝ M − / ). This may suggest that the characteristic length scale of 0.35 pcis related to the formation of dense self-gravitating structures. In any case, if such a small scale 23 –driving of turbulence is common in the cluster environment, star formation in clusters is likely tocontinue over several dynamical times.On the other hand, the line widths of our cores deviate downward from the Larson relationwhich may correspond to the upper bound of the observed line width-radius relations. Until now,many authors have derived the line width-radius relations for various molecular clouds and cores.Those studies show a scatter in the index and coefficient of the power-law line width-radius relations.The line widths of the Larson relation, (∆ v/ kms − ) = 1 . R/ pc) . , tend to be somewhat largerover the range of 0.01 − v/ kms − ) = 2 . R/ pc) . , notfar from the Heyer & Brunt relation. In nearby low-mass star forming regions Fuller & Myers (1992)compiled the data of dense cores and derived the power-law relation of (∆ v/ kms − ) = 1 . R/ pc) . for starless cores. If the line width-radius relation of the low-density gas in ρ Oph follows the Larsonrelation, instead of the Heyer & Brunt relation, the smaller line widths of the H CO + cores maysuggest that the cores have formed preferentially in regions where supersonic turbulence dissipatedand thus the cores so formed have smaller line widths compared to the Larson relation. If this isthe case, the nearly independent relationship between the line width and core radius implies thatthe turbulent field at the small scales has not been relaxed yet, and therefore the timescale of coreformation may be of the order of a dynamical time. In that case, it is unclear how protostellaroutflows contribute to maintain supersonic turbulence in the cluster environment.
6. Summary
Using the archive data of the H CO + ( J = 1 −
0) molecular line emission taken with theNobeyama 45 m radio telescope, we analyzed the molecular gas distribution in the central denseregion of the ρ Ophiuchi main cloud. We summarize the primary results of the paper as follows:1. We compared the global distribution of the H CO + emission with that of the 850 µ mdust continuum emission and found that the overall spatial distributions are similar to each other,while the H CO + emission appears to cover a larger area than the dust continuum emission. Bycomparing between the H CO + and 850 µ m maps, we revealed that the fractional abundance ofH CO + relative to H decreases with increasing the H column density as X H CO + ∝ N − / .This tendency is consistent with the theoretical prediction (e.g., Nakano et al. 2002). The meanfractional abundance in the region is estimated to be 1 . × − .2. From the 3D data cube of the H CO + emission we identified 68 dense cores using theclumpfind method. From comparison with the positions of YSOs recently identified by the Spitzerspace telescope, the cores are classified into the following two groups: 34 protostellar and 34 starlesscores. The radii, masses, and line widths of the protostellar cores tend to be larger than those of thestarless cores. The virial ratio tends to increase with decreasing the LTE mass, although we found 24 –no significant difference in the virial ratio between the protostellar and starless cores. Furthermore,the virial ratios of the cores can be well described by the model of a self-gravitating core confinedby ambient pressure derived by Bertoldi & McKee (1992), suggesting that both self-gravity andambient pressure play an important role in dynamics of the cores.3. The mass spectrum of the H CO + cores can be fitted by a two-component power-law thatresembles the stellar IMF in ρ Oph. The core mass spectrum has a break at around 7 M ⊙ , close tothe mean LTE mass of the cores. It implies that the stellar IMF may be at least partly determinedby the core mass distribution.4. Applying the virial analysis, we conclude that most of the cores are bound and are expectedto collapse. Furthermore, for the majority of the cores, the surface pressure term is more impor-tant than the gravitational energy term. This result suggests that in the cluster environment theformation and evolution of the cores may be regulated largely by the surface pressures.5. We compared the physical properties of the H CO + cores in ρ Oph with those in OrionA. To eliminate the effect of the different distances, we smoothed the original ρ Oph data into acoarser grid so that the spatial resolutions coincide with each other. We identified 16 cores fromthe smoothed data using the clumpfind. These cores would be observed by the Nobeyama 45 mtelescope if the ρ Ophiuchi main cloud were located at the same distance as the Orion A molecularcloud. The physical properties of the ρ Oph cores identified from the smoothed data appear toresemble those of Orion A. The fact that the mean density of the Orion A cores (1 . × cm − )is somewhat smaller than the critical density of H CO + ( J = 1 −
0) transition of 8 × cm − suggests that the Orion A cores contain clumpy substructures that could not be spatially resolved.6. The range of the line widths of the ρ Oph cores identified from the smoothed data almostcoincides with that of the original ρ Oph cores. This suggests that the line width-radius relation ismore or less flat over the range of 0.01 − CO + . Such a flatrelation is inconsistent with that derived from the CO observations by Heyer & Brunt (2004) whofound that the line width correlates strongly with the size for the low-density molecular gas. At thescales below 0.3 pc, the line widths of the ρ Oph cores deviate upwards from the Heyer & Bruntrelation. This may be due to turbulence driven by protostellar outflows. However, if the actual linewidths of the low-density gas in ρ Oph agree with the values of the Larson relation, then the nearlyindependent relationship between the line width and core radius may be interpreted as evidencethat the cores have formed preferentially in regions where supersonic turbulence dissipated.
A. Comparison with Other Line Width-Radius Relations
In the present paper, the FWHM line width and core radius defined by our clumpfind methodare used for showing the line width-radius relation of the identified cores. Since the different authorsuse the different definitions of the line width and core radius (Heyer & Brunt 2004; Caselli & Myers1995; Larson 1981), we here convert the other line width-radius relations using our definitions. 25 –Heyer & Brunt (2004) applied the PCA analysis to derive the relationship between the linewdith and the size inside the molecular clouds. Their line width ∆ v HB and size L HB correspond tothe autocorrelation widths on both the spatial and the velocity axes. In the following, we transferthem into our FWHM line width and radius to do a direct comparison with our result.If both a line and intensity profiles of a core can be approximated by Gaussian profiles,then the line width and size defined by Heyer & Brunt (2004) are given by ∆ v HB = 2 σ v and L HB = 2 σ L , where σ v and σ L are the dispersions of the line and intensity profiles, respectively (seeHeyer & Brunt 2004). On the other hand, the FWHM line width is defined as ∆ v FWMH = √ σ v .Thus, the FWMH line width can be rewritten as ∆ v FWHM = 1 . v HB .In the present paper, the core radius, R core , is determined by taking the projected area enclosedby the 2 σ level contour and by computing the radius of the circle required to reproduce the area.For a core with a Gaussian profile, our core radius is given by R core = L HB p ln( T peak /T th ), where T peak and T th are the peak antenna temperature and the threshold temperature of the 2 σ level,respectively. From Table 1, the mean peak antenna temperature of the 68 identified cores isestimated to be h T peak i = 1 . h T peak i as a representative value, R core = 0 . L HB .According to Heyer & Brunt (2004), the linewidth-size relation for GMCs can be fitted bythe power-law of (∆ v HB / km s − ) = 0 . L HB / pc) . . Using ∆ v FWHM = 1 . v HB and R core =0 . L HB , the Heyer & Brunt relation can be rewritten as (∆ v FWHM / km s − ) = 1 . R core / pc) . ,which is shown by the dashed line in Figure 13a.Caselli & Myers (1995) also used the different definition for the core radius. They used thehalf-maximum contour to define the core size. Assuming that a core has a Gaussian shape, thehalf-maximum radius is equal to R CM = p ln 2 / ln( T peak /T th ) R core , and R CM = 0 . R core for themean peak antenna temperature of h T peak i = 1 . dv NT / km s − ) = 1 . R CM / pc) . , with ( dv NT / km s − ) = 1 . R core / pc) . .Larson (1981) used the 3D velocity dispersion ( σ ) and the maximum core size ( L max ). As-suming that L max = 2 R core , his original relation, ( σ / km s − ) = 1 . L max / pc) . ), is rewrittenas (∆ v FWHM / km s − ) = 1 . R core / pc) . .This work is supported in part by a Grant-in-Aid for Scientific Research of Japan (19204020,20540228) and a Grant for Promotion of Niigata University Research Projects. We thank JesJørgensen for kindly giving us the data of YSOs in L1688. We also thank Zhi-Yun Li, DougJohnstone, and Philippe Andr´e for valuable comments, and Chris Brunt and Mark Heyer for helpfulcomments that improved the presentation of § REFERENCES
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31 –Fig. 1.— (a): H CO + ( J = 1 −
0) total integrated intensity map in the velocity range of v LSR = 1 − − toward the ρ Ophiuchi main cloud. The grey scale indicates the integrated intensity in unitsof K km s − . The contours start from 0.3 K km s − , corresponding to the 3 σ noise level, at intervalsof 0.2 K km s − . (b): 850 µ m image of the same area as in panel (a), obtained with the SCUBAat the JCMT. The grey scale indicates the intensity in linear scale from -0.4 to 4 Jy beam − .The contours start from 0.2 Jy beam − at intervals of 0.2 Jy beam − . The dense subclumps aredesignated by A, B1, B2, C, E, and F. 32 – SM2SM1SM1N VLA1623 (a) Oph A 1 3 245beam (b) Oph B1, B2, C
Oph B2 Oph COph B1
Fig. 2.— H CO + ( J = 1 −
0) total integrated intensity maps taken with the Nobeyama 45 mtelescope toward (a) the Oph A region and (b) the Oph B1, B2, and C regions. The grey scale andcontours are the same as those of Figure 1a. In panel (a), the two dahed lines indicate the positionof the filaments seen in the 850 µ m map (Wilson et al. 1999). The positions of the submillimetersources, SM1, SM1N, and SM2, and the prototypical Class 0 object, VLA 1623, are indicated bycrosses. In each panel the arcs and holes discussed in the text are indicated by numbers. The beamsize of the Nobeyama 45 m telescope is shown at the lower left corner of each panel. 33 –Fig. 3.— Three-dimensional representation of the antenna temperature ( T ∗ A ) in the α - δ - v LSR spacetoward (a) the Oph A region, (b) the Oph B1, B2, and C regions, and (c) the whole observed area.For panels (a) and (b), the areas projected on the plane of the sky are the same as those of Figures2a and 2b, respectively. The color shows the iso-temperature surfaces. The color bar indicates theantenna temperature in units of K. In each panel the numbers have the same meaning as in Figure2. (Mpeg animations of Figs. 2a, 2b, and 2c are available in the online journal.) 34 –Fig. 4.— Fractional abundance of H CO + relative to H against the H column density in the ρ Ophiuchi main cloud. The solid line shows the best-fit power-law for the ρ Oph data. For compari-son, the fractional abundances of H CO + derived by Butner et al. (1995) and van Dishoeck et al.(1995) are plotted by crosses and a filled square, respectively. The dashed line shows the best-fitpower-law for both their data and ours. 35 – (a) (c)(b) dust H CO
13 +
Fig. 5.— Histograms of (a) the radius, (b) core mass, (c) mean density of the H CO + cores ( openhistograms ) and the 850 µ m dust cores ( grey histograms ) in the ρ Ophiuchi main cloud. 36 – thermalturbulent
Fig. 6.— Histogram of the FWHM line width of the H CO + cores in the ρ Ophiuchi main cloud.The open and grey histogams indicate the thermal and turbulent cores, respectively. Note thatthe gas temperature is assumed to be 12 K for all the cores except for those located in the Oph Aregion, for which T = 18 K. 37 –Fig. 7.— Distribution of YSOs identified with the Spitzer Space telescople. Contours indicate thetotal integrated intensity of the H CO + ( J = 1 −
0) emission, as shown in Figure 1a. The red andblue circles are for Class I sources including Class 0 and Flat Spectrum sources, respectively. Thegreen and cyan squares are for Class II and Class III sources, respectively. The cores associatedwith the circles are defined as protostellar cores in this study. 38 – α (a) (b) (c) LTE without YSOswith YSOs
N N N (e) (f) NN (d) N Fig. 8.— Histograms of (a) the radius, (b) LTE mass, (c) FWHM line width, (d) virial ratio,(e) mean density, and (f) aspect ratio of the H CO + cores. For each panel, the open and greyhistograms are for the cores without and with YSOs, respectively. (a) (c)(b) Fig. 9.— (a) Line width-radius relation, (b) Nonthermal line width-radius relation, and (c) Mass-radius relation of the H CO + cores. The filled squares and open circles are for the cores withoutand with YSOs, respectively. In each panel, the best-fit power-law is indicated by the solid line.In panel (a) the Larson and Heyer & Brunt relations are plotted by the dotted and dashed lines,respectively. Note that we extrapolated these relations down to 0.01 pc, for comparison. In panels(b) and (c), the dashed lines indicate the Caselli & Myers relation and the relation of M LTE =4 πµm H ¯ nR / ∝ R ), respectively. 39 – α Fig. 10.— Virial ratio-mass relation of the H CO + cores. The filled squares and open circlesare the same as those in Figure 9. The virial ratio of a self-gravitating core confined by ambientpressure is indicated by the dashed line (Bertoldi & McKee 1992). 40 – ∆∆ Fig. 11.— Mass spectrum of the H CO + cores in the ρ Ophiuchi main cloud. The error barscorrespond to √ N counting statistics, where N is the number of cores in each mass bin. The solidlines are the least square fits in the interval of 1 M ⊙ . M LTE . M ⊙ (∆ N/ ∆ M ∝ M − . ) and M LTE & M ⊙ (∆ N/ ∆ M ∝ M − . ). 41 –Fig. 12.— Relationship between the surface term, S , and the gravitational term, W , in the virialequation. They are normalized to the internal kinetic energy term, U [Note that W/U = − ( α vir ) − ].The solid line indicates the virial equilibrium, U + W + S = 0. The dashed line indicates the lineat which W = S . For the cores that lie below the solid line, the value of U + W + S is negativeand thus expected to be bound. All others are unbound and expected to disperse away, if they donot gain more mass through accretion and/or merging with other cores, or reduce internal supportthrough turbulence dissipation. The filled squares and open circles are the same as those in Figure9. 42 –Fig. 13.— Line width-radius relation ( upper panel ) and mass-radius relation ( lower panel ) of theH CO + cores toward the ρ Ophiuchi main cloud ( open squares ) and toward the Orion A molecularcloud ( crosses ). The ρ Oph cores identified from the smoothed data are indicated by filled circles,for comparison. The solid line in the upper panel indicates the best-fit power-law for the ρ Ophcores identified from both the original and smoothed data. The dotted and dashed lines indicatethe Larson and the Heyer & Brunt relations, respectively. 43 – α α
Fig. 14.— Virial ratio-mass relation of the ρ Oph cores and the Orion A cores. For Orion A,the fractional abundances of H CO + are set to 4 . × − (adopted by Ikeda et al. 2007) and1 . × − (adopted for ρ Oph in the present paper) in panels (a) and (b), respectively. Theopen squares, crosses, and filled circles are the same as those of Figure 13. The dashed lines arethe same as that of Fig. 10. Note that the Jeans mass given by eq. [2.13] of Bertoldi & McKee(1992) is almost the same for both the original and smoothed ρ Oph cores.
Table 1. Properties of the H CO + cores in the ρ Ophiuchi main cloud
ID R.A. Decl. v LSR T ∗ A, peak R core R core Aspect dv core M LTE M vir α vir ¯ n YSOs a (J2000.0) (J2000.0) (km s − ) (K) (arcsec) (pc) Ratio (km s − ) ( M ⊙ ) ( M ⊙ ) (10 cm − )1 16 26 7.3 -24 20 27.6 3.15 1.50 72.9 0.0442 2.33 0.476 6.89 5.10 0.74 3.31 N2 16 26 10.3 -24 23 12.0 3.41 0.87 99.7 0.0604 1.19 0.764 7.07 11.6 1.64 1.33 Y3 16 26 13.3 -24 20 6.7 3.02 0.70 72.7 0.0441 1.67 0.502 2.78 5.32 1.91 1.34 N4 16 26 14.8 -24 21 8.3 3.41 0.78 78.8 0.0478 1.05 0.551 3.85 6.28 1.63 1.47 N5 16 26 19.4 -24 23 32.0 2.76 0.69 90.0 0.0545 1.52 0.528 4.57 6.89 1.51 1.17 Y6 16 26 22.4 -24 24 54.0 3.28 1.05 76.8 0.0466 1.34 0.730 4.68 8.35 1.79 1.92 Y7 16 26 23.8 -24 20 26.5 3.41 0.99 73.3 0.0444 1.07 0.639 4.06 6.82 1.68 1.92 N8 16 26 26.7 -24 16 40.1 3.80 0.53 36.5 0.0221 1.42 0.453 0.49 2.46 5.03 1.86 N9 16 26 26.9 -24 22 29.8 3.28 3.27 83.2 0.0504 1.34 0.547 17.88 6.59 0.37 5.77 Y10 16 26 26.9 -24 23 52.0 3.80 3.99 90.0 0.0545 1.20 0.627 21.78 8.19 0.38 5.56 Y11 16 26 28.4 -24 23 31.4 2.76 0.79 65.9 0.0399 1.62 0.585 1.97 5.58 2.83 1.28 Y12 16 26 33.0 -24 24 12.2 3.02 0.99 67.6 0.0410 1.28 0.415 2.43 4.27 1.76 1.46 N13 16 26 33.0 -24 26 15.6 3.15 1.12 61.3 0.0371 1.26 0.609 2.73 5.41 1.98 2.21 N14 16 26 34.5 -24 25 55.0 3.54 1.39 86.3 0.0523 1.11 0.442 6.25 5.70 0.91 1.81 N15 16 26 36.0 -24 25 34.3 3.15 0.69 51.2 0.0311 1.23 0.267 1.03 2.58 2.52 1.42 N16 16 26 43.5 -24 25 54.3 3.80 0.78 56.1 0.0340 1.19 0.279 1.38 2.87 2.08 1.46 N17 16 26 45.1 -24 26 56.0 3.93 0.83 37.4 0.0227 1.96 0.186 0.72 1.71 2.39 2.55 N18 16 26 45.3 -24 33 26.8 3.80 1.17 113.8 0.0690 1.24 0.528 7.02 7.15 1.02 0.89 Y19 16 26 46.5 -24 23 50.7 3.02 0.63 50.1 0.0304 2.28 0.388 0.93 3.02 3.24 1.38 N20 16 26 46.6 -24 27 57.6 3.80 0.53 50.3 0.0305 1.27 0.316 0.39 2.02 5.13 0.58 Y21 16 26 49.6 -24 26 55.7 3.93 0.85 43.9 0.0266 1.11 0.454 0.79 2.36 2.98 1.73 N22 16 26 49.7 -24 30 0.8 3.80 1.23 99.4 0.0602 1.03 0.385 4.14 4.60 1.11 0.79 Y23 16 26 52.9 -24 36 52.0 4.45 0.92 58.2 0.0353 1.37 0.550 0.49 3.83 7.76 0.47 N24 16 26 52.9 -24 37 12.5 4.71 0.98 50.7 0.0307 2.10 0.337 0.77 2.13 2.76 1.10 N25 16 26 55.9 -24 36 31.2 4.58 0.95 63.6 0.0386 1.37 0.438 1.03 3.30 3.20 0.75 N26 16 26 57.0 -24 23 50.0 3.80 1.37 79.1 0.0480 1.02 0.393 2.41 3.72 1.54 0.91 Y27 16 26 57.3 -24 31 22.6 3.93 1.52 101.9 0.0618 1.21 0.543 7.63 6.61 0.87 1.34 Y28 16 26 57.4 -24 37 53.4 4.58 0.66 48.6 0.0295 1.06 0.666 0.74 4.07 5.54 1.19 Y29 16 27 1.9 -24 34 47.9 3.67 1.56 100.5 0.0609 1.24 0.669 6.09 8.46 1.39 1.11 Y30 16 27 3.4 -24 36 10.1 4.97 0.95 58.1 0.0352 1.28 0.379 1.19 2.66 2.23 1.13 Y Table 1—Continued
ID R.A. Decl. v LSR T ∗ A, peak R core R core Aspect dv core M LTE M vir α vir ¯ n YSOs a (J2000.0) (J2000.0) (km s − ) (K) (arcsec) (pc) Ratio (km s − ) ( M ⊙ ) ( M ⊙ ) (10 cm − )31 16 27 3.5 -24 38 54.7 4.58 1.29 56.5 0.0343 1.65 0.427 1.89 2.87 1.51 1.95 N32 16 27 4.7 -24 26 54.7 3.67 0.47 72.2 0.0438 1.07 0.547 0.94 4.73 5.05 0.46 Y33 16 27 6.3 -24 32 44.2 4.06 1.52 87.0 0.0528 1.14 0.523 4.53 5.41 1.20 1.28 N34 16 27 9.6 -24 39 35.4 4.32 1.03 78.5 0.0476 1.11 0.587 1.69 5.59 3.31 0.65 Y35 16 27 10.7 -24 27 35.4 4.32 0.71 81.4 0.0493 1.29 0.516 1.85 4.99 2.70 0.64 Y36 16 27 10.8 -24 29 18.2 3.80 1.38 73.9 0.0448 1.19 0.765 3.92 7.51 1.92 1.81 Y37 16 27 12.2 -24 25 52.4 3.41 0.77 78.8 0.0478 1.15 0.472 1.18 4.39 3.72 0.45 Y38 16 27 12.3 -24 29 59.3 3.80 1.37 69.2 0.0420 1.49 0.722 3.17 6.48 2.04 1.78 N39 16 27 14.0 -24 38 12.8 4.71 0.65 53.9 0.0327 2.27 0.616 0.69 4.07 5.94 0.81 N40 16 27 15.3 -24 30 40.2 3.67 1.37 100.0 0.0606 1.21 0.678 5.61 8.58 1.53 1.05 Y41 16 27 18.2 -24 27 55.5 3.41 0.78 92.1 0.0558 1.13 0.723 2.35 8.63 3.67 0.56 Y42 16 27 21.1 -24 23 48.4 3.15 1.15 79.3 0.0481 1.08 0.471 2.87 4.41 1.54 1.07 Y43 16 27 21.2 -24 26 53.6 4.45 1.26 98.0 0.0594 1.15 0.596 5.02 7.11 1.42 0.99 Y44 16 27 21.6 -24 39 55.2 4.84 0.96 70.6 0.0428 1.11 0.735 1.66 6.78 4.07 0.88 Y45 16 27 24.3 -24 27 55.1 3.80 1.76 82.2 0.0498 1.04 0.527 4.33 5.15 1.19 1.45 Y46 16 27 24.7 -24 40 56.7 4.84 0.83 58.3 0.0353 1.26 0.631 0.95 4.55 4.81 0.89 Y47 16 27 25.7 -24 24 29.3 3.02 1.70 95.7 0.0580 1.15 0.438 5.37 4.96 0.92 1.14 Y48 16 27 25.8 -24 27 13.8 3.80 1.93 71.7 0.0435 1.12 0.595 5.47 5.19 0.95 2.76 Y49 16 27 26.0 -24 33 24.1 3.54 0.97 100.9 0.0612 1.56 0.449 3.02 5.35 1.77 0.55 N50 16 27 28.8 -24 27 34.2 4.45 1.87 86.9 0.0527 1.21 0.405 4.58 4.20 0.92 1.30 Y51 16 27 29.2 -24 41 37.5 3.67 0.79 86.1 0.0522 1.13 0.525 2.27 5.37 2.37 0.66 Y52 16 27 30.7 -24 40 35.7 4.32 1.38 84.0 0.0509 1.62 0.316 2.53 3.37 1.33 0.79 N53 16 27 33.3 -24 26 11.6 3.93 1.63 114.2 0.0692 1.46 0.563 8.99 7.72 0.86 1.12 Y54 16 27 33.3 -24 26 52.7 4.58 1.97 95.6 0.0579 1.01 0.361 4.96 4.21 0.85 1.06 N55 16 27 33.3 -24 28 35.6 3.80 0.76 92.1 0.0558 1.12 0.675 2.01 7.85 3.90 0.48 N56 16 27 36.5 -24 34 4.5 3.41 0.51 76.6 0.0465 2.33 0.603 0.90 5.64 6.29 0.37 N57 16 27 38.3 -24 41 16.4 3.54 0.53 57.7 0.0350 1.61 0.331 0.47 2.39 5.10 0.45 N58 16 27 42.9 -24 42 38.3 3.80 1.06 76.8 0.0466 1.51 0.444 3.19 4.03 1.26 1.31 Y59 16 27 46.0 -24 44 41.6 3.67 1.31 80.8 0.0490 1.53 0.302 2.97 3.16 1.06 1.05 Y60 16 27 51.9 -24 42 58.3 3.93 1.02 75.1 0.0455 1.02 0.341 1.70 3.17 1.87 0.75 N Table 1—Continued
ID R.A. Decl. v LSR T ∗ A, peak R core R core Aspect dv core M LTE M vir α vir ¯ n YSOs a (J2000.0) (J2000.0) (km s − ) (K) (arcsec) (pc) Ratio (km s − ) ( M ⊙ ) ( M ⊙ ) (10 cm − )61 16 27 57.7 -24 35 4.8 4.19 0.97 56.9 0.0345 1.44 0.398 1.31 2.71 2.06 1.32 N62 16 27 59.2 -24 34 23.6 4.19 0.75 41.0 0.0248 1.01 0.353 0.64 1.78 2.80 1.72 N63 16 28 0.6 -24 33 1.2 4.32 1.34 73.0 0.0442 1.05 0.402 2.43 3.51 1.44 1.16 N64 16 28 3.7 -24 34 43.8 4.19 0.75 48.5 0.0294 1.68 0.238 0.60 1.68 2.79 0.99 N65 16 28 6.7 -24 34 23.1 4.32 0.71 62.9 0.0381 1.14 0.350 0.91 2.71 2.97 0.68 N66 16 28 15.8 -24 35 44.7 4.19 1.06 69.5 0.0421 1.04 0.306 1.38 2.74 1.98 0.77 N67 16 28 20.4 -24 36 25.6 4.19 1.60 52.6 0.0319 1.11 0.324 1.94 2.15 1.11 2.48 Y68 16 28 26.4 -24 36 4.6 4.19 1.20 97.4 0.0590 1.34 0.271 3.42 3.58 1.05 0.69 NNote. — Units of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes, and arcseconds. CoresNo. 1 through 17 and 19 are located in the Oph A region and the temperature is set to be 18K. For the other cores, T = 12K. The typicaluncertainties of R core and dv core are, respectively, about 0.01 pc, derived from the uncertainty in the estimation of the core projected area, andthe velocity resolution of 0.13 km s − . a Y and N mean, respectively, the cores associated with and without YSOs that are classified as either Class I or Flat Spectrum objects basedon the Spitzer observations.
47 –Table 2. Summary of the Physical Properties of the H CO + coresProperty Minimum Maximum Mean a Median R core (pc) 0.0221 0.0692 0.0450 ± dv core (km s − ) 0.186 0.765 0.488 ± M LTE (M ⊙ ) 0.39 21.79 3.35 ± M vir /M LTE ± n (cm − ) 3 . × . × (1 . ± . × . × Aspect Ratio 1.01 2.33 1.34 ± a With standard deviationTable 3. Summary of the Physical Properties of the 850 µm coresProperty Minimum Maximum Mean a Median R core (pc) 0.00494 0.0601 0.0204 ± M LTE (M ⊙ ) 0.0868 8.25 1.42 ± n (cm − ) 1 . × . × (6 . ± . × . × With standard deviationNote. — The uniform temperature of T = 15 K is assumed for all the cores in Jørgensen et al.(2008). 48 –Table 4. Summary of the Physical Properties of the H CO + cores with and without YSOsProperty Minimum Maximum Mean a MedianCores without YSOs R core (pc) 0.0221 0.0612 0.0403 ± dv core (km s − ) 0.186 0.722 0.431 ± M LTE (M ⊙ ) 0.469 6.89 1.54 ± M vir /M LTE ± n (cm − ) 3 . × . × (1 . ± . × . × Aspect Ratio 1.01 2.33 1.45 ± R core (pc) 0.0295 0.0692 0.0497 ± dv core (km s − ) 0.302 0.765 0.544 ± M LTE (M ⊙ ) 0.394 21.8 4.55 ± M vir /M LTE ± n (cm − ) 0 . × . × (1 . ± . × . × Aspect Ratio 1.02 1.62 1.22 ± aa