Large-scale periodic velocity oscillation in the filamentary cloud G350.54+0.69
MMNRAS , 000–000 (0000) Preprint 22 May 2019 Compiled using MNRAS L A TEX style file v3.0
Large-scale periodic velocity oscillation in the filamentary cloudG350.54 + Hong-Li Liu (cid:63) , , , Amelia Stutz † , , Jing-Hua Yuan Chinese Academy of Sciences South America Center for Astronomy, China-Chile Joint Center for Astronomy, Camino El Observatorio Department of Physics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong SAR Departamento de Astronom´ıa, Universidad de Concepci´on, Av. Esteban Iturra s/n, Distrito Universitario, 160-C, Chile Max-Planck-Institute for Astronomy, K¨onigstuhl 17, 69117 Heidelberg, Germany National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, China
22 May 2019
ABSTRACT
We use APEX mapping observations of CO, and C O (2-1) to investigate the internal gaskinematics of the filamentary cloud G350.54 + + ∼ . pc and an amplitude of ∼ . km s − . Comparing with gravitational-instability induced core formation models, we conjecture that this periodic velocity oscillationcould be driven by a combination of longitudinal gravitational instability and a large-scaleperiodic physical oscillation along the filament. The latter may be an example of an MHDtransverse wave. This hypothesis can be tested with Zeeman and dust polarization measure-ments. Key words:
ISM: individual objects: G350.5 – ISM: clouds – ISM: structure – ISM:molecules – stars: formation – infrared: ISM
Filamentary structures in the interstellar medium (ISM) have longbeen recognized (e.g., Schneider & Elmegreen 1979) but their rolein the process of star formation has received focused attention onlyrecently thanks to long-wavelength Herschel data (e.g., Molinari etal. 2010; Andr´e et al. 2010). With these data, recent studies havedemonstrated the ubiquity of the filaments in the cold ISM of ourMilky Way (e.g., Molinari et al. 2010; Andr´e et al. 2010; Stutz &Kainulainen 2015; Stutz 2018). Meanwhile, analysis of Herscheldata has revealed a close connection between filamentary cloudsand star formation (e.g., Andr´e et al. 2010, 2014; K¨onyves & Andr´e2015; Stutz & Kainulainen 2015; Stutz & Gould 2016; Liu, Stutz,& Yuan 2018a). For instance, most cores are detected in filamentaryclouds (Andr´e et al. 2014; K¨onyves & Andr´e 2015; Stutz & Kainu-lainen 2015). Moreover, with gas velocity information, protostellarcores are found kinematically coupled to the dense filamentary en-vironments in Orion (Stutz & Gould 2016); that is, the protostellarcores have similar radial velocities as the gas filament.The overarching goal of obtaining empirical observational in-formation of the physical state of observed filaments has radicallyimproved with the gradual availability and subsequent exploitationof gas radial velocities, which reveal internal kinematics of the fil- (cid:63)
E-mail: [email protected] † E-mail: [email protected], [email protected] aments in great detail (e.g., Hacar & Tafalla 2011; Tackenberg etal. 2014; Henshaw et al. 2014; Beuther et al. 2015; Tafalla & Hacar2015; Hacar et al. 2017, 2018; Lu et al. 2018). Indeed, the kinemat-ics is the only way to probe not only where the mass is presently, butalso most importantly how it is moving under the influence of vari-ous possible forces. In short, the gas kinematics provide a powerfulwindow into the physical processes at play in the conversion of gasmass into stellar mass in filaments. Velocity gradients have beenfor example interpreted as being driven by gravity (e.g., Hacar &Tafalla 2011; Tackenberg et al. 2014; Henshaw, Longmore, & Krui-jssen 2016; Williams et al. 2018; Lu et al. 2018; Yuan, et al. 2018).For instance, velocity gradients along the filament may be relatedto longitudinal collapse leading to core formation (e.g., Hacar &Tafalla 2011; Kirk et al. 2013; Peretto et al. 2013, 2014; Tacken-berg et al. 2014; Hacar et al. 2017; Williams et al. 2018; Lu et al.2018), or those perpendicular to the filament may result from thefilament rotation or radial collapse (e.g., Ragan et al. 2012; Kirk etal. 2013; Beuther et al. 2015; Wu et al. 2018; Liu, et al. 2018b).With the goal of understanding the link between dense coresand star formation in filamentary clouds, we have investigated thefilamentary cloud G350.54 + Her-schel continuum data (see Fig. 1 of Liu, Stutz, & Yuan 2018a, here-after Paper I). G350.5 is a straight and isolated in morphology andcomposed of two distinct filaments, G350.5-N in the north andG350.5-S in the south. G350.5-N(S) is ∼ . pc ( ∼ . pc) longwith a mass of ∼ M (cid:12) ( ∼ M (cid:12) ). The nine gravitationally c (cid:13) a r X i v : . [ a s t r o - ph . GA ] M a y H.L. Liu et al. − − − V lsr (km s − )012 T m b ( K ) COC O Figure 1.
Average spectra of CO, and C O (2-1) over the G350.5 fila-mentary cloud. The black color stands for the observed spectrum. The redline is the Gaussian fitting to the spectrum of CO, which is a sum of thetwo Gaussian components (blue lines), the weak-emission one from − . to . km s − , and the major one from − . to − km s − . The green lineis the single-component Gaussian fitting to the spectrum of C O, whichmatches the major component of the CO spectrum. The vertical dashedblack line represents a systematic velocity of − . km s − . bound dense cores associated with low-mass protostars suggest asite of ongoing low-mass star formation. In this work, we shift thefocus to the gas kinematic properties. The nature of its simple mor-phology would be helpful in reducing the velocity ambiguities re-sulting from projection effects of potentially complex morpholo-gies. This paper is organized as follows: observations are describedin Section 2, analysis results are presented in Section 3, the discus-sion is given in Section 4 to a large-scale periodic velocity fluctu-ation along the main structure of the filament G350.5, and a sum-mary is put in Section 5. Observations of CO, and C O (2-1) toward G350.5 weremade simultaneously using the Atacama Pathfinder Experiment(APEX ) 12–m telescope (G¨usten et al. 2006) at Llano de Cha-jnantor (Chilean Andes) in the on-the-fly mode on Septem-ber 24, 2017. The observations were centered at α =17 h m . s , δ = − ◦ (cid:48) . (cid:48)(cid:48) with a mapping size of (cid:48) × (cid:48) , rotated by ◦ relative to the RA decreasing direction. Aneffective spectral resolution of 114 KHz or 0.15 km s − is reachedat a tuned central frequency of 220 GHz between both for CO (2-1) and for C O (2-1). The angular resolution at this frequency is ∼ (cid:48)(cid:48) , which corresponds to ∼ . pc at the distance of G350.5, . ± . pc (see Paper I). The reduced spectra finally present atypical rms value of 0.42 K. More details on the data reduction canbe found in Paper I. Before addressing the kinematic structure of G350.5, we firstdescribe the spatial distribution of molecular emission. Figure 1presents the average spectra of CO, and C O (2-1) over thewhole G350.5 system. The CO profile has two velocity compo-nents, one from − . to . km s − (hereafter weak-emission com-ponent) and the other from − . to − km s − (hereafter majorcomponent); C O has a single-peak profile corresponding to themajor component of CO. The systemic velocity is determined tobe − . km s − from the average spectrum of C O (2-1).Figure 2 shows the velocity-integrated intensity maps of CO, and C O (2-1). The spatial distribution of the major com-ponent of CO (see Fig. 2a) matches the column density ( N H )distribution (black contours, derived from Herschel data in Paper Ithrough the pixel-wise spectral energy distribution fitting, e.g., Liu,et al. 2016, 2017), showing two discontinuous filaments as indi-cated by the two cyan curves. In addition, some small-scale struc-tures (i.e., filament ‘branches’) stretch perpendicular to the mainstructure of the cloud G350.5, which coincide with the perpendicu-lar dust striations surrounding the filament as observed in Herschelcontinuum images (see Paper I). Emission of the major componentof CO (2-1) has a C O counterpart (in Fig. 2c), but the latter ismore narrowly distributed than the former. In contrast, the weak-emission component of CO (2-1) in Fig. 2b has no detectableC O counterpart. However, it can be seen that the two compo-nents of CO are indeed associated with each other. In the southand center of the filament, the two most prominent clumps in theintensity map of the CO weak-emission component can also befound in that of the major component. This is supportive of the as-sociation between the weak-emission and major components.
To investigate the large-scale kinematics, we show the velocity cen-troid map of C O (2-1) of the cloud G350.5 in Fig. 3. Velocityinformation is shown only in the main structure of the filament,beyond which C O (2-1) emission is too noisy ( < σ ), overlaidwith the N H contours (Paper I) for comparison. Strikingly, a large-scale periodic velocity fluctuation appears along the main structure,which is ∼ pc long (Paper I). The nature of the large-scale andperiodic signal suggests that the observed velocity fluctuation isreal, otherwise, this periodicity will be hardly maintained on a largescale if it happens by chance. Specifically, the red and blue colorsin Fig. 3 represent the red and blue-shifted velocities relative to thesystemic velocity of − . km s − . Comparing the N H contourswith the velocity centroid map, we find a spatial correspondencebetween the velocity extremes and the density enhancements forsome dense cores (i.e., C2, C5, C6, C7a,b, and C8). The possibleorigins of this correspondence will be discussed further in Sect. 4.We do not present here the velocity centroid map of CO (2-1),which does not show the periodic fluctuation as in C O (2-1) emis-sion. Since CO tends to trace more extended emission than C O,the velocity characteristics revealed by both species are not neces-sarily the same. This publication is based on data acquired with the Atacama PathfinderExperiment (APEX). APEX is a collaboration between the Max-Planck-Institut fur Radioastronomie, the European Southern Observatory, and theOnsala Space Observatory. MNRAS000
To investigate the large-scale kinematics, we show the velocity cen-troid map of C O (2-1) of the cloud G350.5 in Fig. 3. Velocityinformation is shown only in the main structure of the filament,beyond which C O (2-1) emission is too noisy ( < σ ), overlaidwith the N H contours (Paper I) for comparison. Strikingly, a large-scale periodic velocity fluctuation appears along the main structure,which is ∼ pc long (Paper I). The nature of the large-scale andperiodic signal suggests that the observed velocity fluctuation isreal, otherwise, this periodicity will be hardly maintained on a largescale if it happens by chance. Specifically, the red and blue colorsin Fig. 3 represent the red and blue-shifted velocities relative to thesystemic velocity of − . km s − . Comparing the N H contourswith the velocity centroid map, we find a spatial correspondencebetween the velocity extremes and the density enhancements forsome dense cores (i.e., C2, C5, C6, C7a,b, and C8). The possibleorigins of this correspondence will be discussed further in Sect. 4.We do not present here the velocity centroid map of CO (2-1),which does not show the periodic fluctuation as in C O (2-1) emis-sion. Since CO tends to trace more extended emission than C O,the velocity characteristics revealed by both species are not neces-sarily the same. This publication is based on data acquired with the Atacama PathfinderExperiment (APEX). APEX is a collaboration between the Max-Planck-Institut fur Radioastronomie, the European Southern Observatory, and theOnsala Space Observatory. MNRAS000 , 000–000 (0000) arge-scale periodic velocity oscillation in G350.54 + h m s m s s s RA (J2000) − ◦ D ec ( J ) (a) I (K km s − ) 17 h m s m s s s RA (J2000)(b) I (K km s − ) 17 h m s m s s s RA (J2000)(c) . . . . . . . I (K km s − ) Figure 2. (a–b:) Velocity-integrated intensity maps of the two velocity components of CO (2-1) toward G350.5, the major one from − . to − km s − ,and the weak-emission one from − to − . km s − . (c:) Velocity-integrated intensity of C O (2-1), integrated over − to − km s − . The pixel size of thethree maps is reduced to 7 (cid:48)(cid:48) , a quarter of the beam size, for a better visualization. The cyan curves in all panels represent the spines of the two discontinuousfilaments as identified in Paper I and the black contours stand for the N H column density (see Paper I), starting from . × cm − with a step of . × cm − ( σ = 0 . × cm − ). h m s m s s s s RA (J2000) − ◦ D ec ( J ) C8C7BC7A C6 C5 C4 C3 C2 C1
G350.5-N G350.5-S − . − . − . − . − . − . V (km s − ) Figure 3.
Velocity centroid map of C O (2-1) for the filamentary cloudG350.5. Only the main filament is shown because of sufficient detectionof C O (2-1) therein. The pixel size of the map is 14 (cid:48)(cid:48) , half of the beamsize, as recommended by the APEX observatory. All pixels correspond todetection greater than σ (i.e., σ = 0 . K km s − ). The red and blue colorscales can be regarded as the red and blue-shifted velocities with respect tothe systematic velocity − . km s − . A large-scale periodic velocity fluctu-ation appears almost along the main structure. The black contours representthe N H column density, starting from . × cm − with a step of . × cm − . The dense cores identified in Paper I are also labelled asC1–C8. CO and C O The observed velocity dispersion ( σ obs ) is measured from both CO and C O (2-1) in the main structure of the filament whereboth species are well-detected. In principle, σ obs can be estimatedfrom the second-order moment of the PPV data cube. However,this method will cause additional uncertainties due to the simpletreatment of multiple-velocity components as one. To better esti-mate σ obs especially from CO 2-1, we define 47 positions alongthe ridgeline of the filament, 41 in G350.5-N and 6 in G350.5-S.Neighbour positions are separated by one pixel, half of the beamsize of both CO and C O maps. σ obs is then obtained from theGaussian fitting to the spectra of both CO, and C O (2-1) at theselected 47 positions. The fitting plots are shown in Fig. C1.The velocity dispersion combines the contributions from boththermal and non-thermal gas motions in turbulent clouds. However,their contributions can be overestimated due to the line broadeningby optical depth. Such overestimate can be expressed analyticallyas a function of the optical depth (Phillips et al. 1979): σ obs = 1 √ ln (cid:34) ln (cid:32) τln (cid:0) e − τ +1 (cid:1) (cid:33)(cid:35) / σ int , (1)where σ int is the intrinsic velocity dispersion unaffected by opticaldepth τ (see Appendix A). For comparison, we calculate the gasdispersions (i.e., non-thermal dispersion σ NT , and the total disper-sion σ tot ) based on the observed, and intrinsic velocity dispersionsmeasured from both CO and C O, respectively, as follows: σ NT = (cid:113) σ / int − kT k m ,σ th = (cid:113) kT k µ p m H ,σ tot = (cid:112) σ + σ , (2)where σ obs / int can be replaced either with σ obs or with σ int (seeEq. 1), k is Boltzmanns constant, T k is the gas kinetic tempera-ture, m is the mass of the two species, m H is the mass of atomic MNRAS , 000–000 (0000)
H.L. Liu et al. σ tot , ob /σ th N u m b e r C O CO (a) . . . . . σ tot , int /σ th N u m b e r C O CO M (b) Figure 4.
Histogram of the total velocity dispersion measured from both CO (red) and C O (2-1) (blue) for G350.5. In panel (a) is the observed velocitydispersion. The mean values in the observed velocity dispersions measured from CO and C O are . ± . σ th (red dashed line), and . ± . σ th (blue dashed line), respectively. In panel (b) is the opacity-corrected (intrinsic) velocity dispersion. The mean values are . ± . σ th from CO (red dashedline), and . ± . σ th from C O (blue dashed line). The different distributions of the velocity dispersions of CO before and after the opacity correctiondemonstrate that the line opacity can cause the broadening of the intrinsic velocity dispersion. The intrinsic velocity dispersion distributions indicate that theoverall turbulent motions of the filament G350.5 are supersonic (i.e., the 3D Mach number follows M D > ). hydrogen, and µ p = 2 . is the mean molecular weight per freeparticle for an abundance ratio N ( H ) /N ( He ) = 10 and a negligi-ble admixture of metals (e.g., Kauffmann et al. 2008). The gas ki-netic temperature here was assumed to be equal to the average dusttemperature ( T d ∼ K ) of the filament instead of the excitationtemperature of each species since the latter is found to be ∼ K onaverage lower than the former, indicating that the two species mightnot be fully thermalized in the filament (see Appendix B for the cal-culation of the excitation temperature). The assumed T k gives riseto the thermal sound speed of molecular gas σ th ∼ . km s − .Figure 4 presents the histograms of both the observed ( σ tot , obs in panel a) and intrinsic ( σ tot , int in panel b) total dispersions ofgas for the entire filament represented by 41 positions in G350.5-Nand 6 in G350.5-S. The intrinsic gas dispersion is overall smallerthan the observed one without the optical-depth correction, indi-cating that the optical depth correction is important, especially for CO. In view of this, we analyse only the intrinsic gas disper-sion (in Fig. 4b) in what follows. The average total dispersionsare . ± . σ th , and . ± . σ th , measured from CO, andC O, respectively. These values indicate that the filament G350.5as a whole is supersonic with a mach number of M D > (seeFig. 4b), where the 3D Mach number is M D = √ σ NT , int /σ th given the 1D measurement σ NT , int , and assuming isotropic turbu-lence in three dimensions (e.g., Kainulainen & Federrath 2017). The complex interactions between turbulence, gravity and mag-netic fields happen everywhere in molecular clouds and regulatestar formation therein. We can describe these interactions throughthe virial theorem. According to Fiege & Pudritz (2000a), thevirial equation of self-gravitating, magnetized turbulent filamentaryclouds can be written in the form: T net + W + M = 0 . (3)Therefore, the virial equilibrium of the clouds depends on the com-petition between the net kinetic energy ( T net ), magnetic energy ( M ), and gravitational potential ( W ). T net accounts for the dif-ference between the internal ( T int ) and external ( T ext ) turbulentenergy, assuming that molecular clouds are confined by the exter-nal pressure rather than completely isolated. Note that all energieshere are measured per unit length.The internal kinetic energy T int is calculated as: T int = 12 M line σ , int , (4)where M line is the line mass along the filament, which can beobtained in Fig. 3 of Paper I. While σ tot , int measured from COand C O (2-1) are similar, the one from CO (2-1) was finallyadopted in the calculation since CO emission tends to tracelarger-scale gas. The external turbulent energy T ext can be ex-pressed as a function of the external pressure P ext : T ext = k P ext πR , (5)where k is the Boltzmann constant and R fil is the filament ra-dius, ∼ . pc as measured in Paper I. A conservative value of P ext = 5 × K cm − is assumed, which is in the range be-tween K cm − for the general ISM (Chromey, Elmegreen &Elmegreen 1989) and K cm − for several molecular clouds as-sociated with HI complexes (Boulares & Cox 1990).The gravitational potential energy W is derived from the linemass of the filament: W = − M G, (6)where G is the gravitational constant. The magnetic energy M iswritten as: M = B µ πR , (7)where B is the magnetic field strength and µ is the permeability offree space. B is estimated following the empirical linear relation-ship between the field strength and gas column density. This rela-tionship was summarized by Crutcher (2012) from Zeeman mea-surements in the form: B = C × N H − µG, (8) MNRAS000
Histogram of the total velocity dispersion measured from both CO (red) and C O (2-1) (blue) for G350.5. In panel (a) is the observed velocitydispersion. The mean values in the observed velocity dispersions measured from CO and C O are . ± . σ th (red dashed line), and . ± . σ th (blue dashed line), respectively. In panel (b) is the opacity-corrected (intrinsic) velocity dispersion. The mean values are . ± . σ th from CO (red dashedline), and . ± . σ th from C O (blue dashed line). The different distributions of the velocity dispersions of CO before and after the opacity correctiondemonstrate that the line opacity can cause the broadening of the intrinsic velocity dispersion. The intrinsic velocity dispersion distributions indicate that theoverall turbulent motions of the filament G350.5 are supersonic (i.e., the 3D Mach number follows M D > ). hydrogen, and µ p = 2 . is the mean molecular weight per freeparticle for an abundance ratio N ( H ) /N ( He ) = 10 and a negligi-ble admixture of metals (e.g., Kauffmann et al. 2008). The gas ki-netic temperature here was assumed to be equal to the average dusttemperature ( T d ∼ K ) of the filament instead of the excitationtemperature of each species since the latter is found to be ∼ K onaverage lower than the former, indicating that the two species mightnot be fully thermalized in the filament (see Appendix B for the cal-culation of the excitation temperature). The assumed T k gives riseto the thermal sound speed of molecular gas σ th ∼ . km s − .Figure 4 presents the histograms of both the observed ( σ tot , obs in panel a) and intrinsic ( σ tot , int in panel b) total dispersions ofgas for the entire filament represented by 41 positions in G350.5-Nand 6 in G350.5-S. The intrinsic gas dispersion is overall smallerthan the observed one without the optical-depth correction, indi-cating that the optical depth correction is important, especially for CO. In view of this, we analyse only the intrinsic gas disper-sion (in Fig. 4b) in what follows. The average total dispersionsare . ± . σ th , and . ± . σ th , measured from CO, andC O, respectively. These values indicate that the filament G350.5as a whole is supersonic with a mach number of M D > (seeFig. 4b), where the 3D Mach number is M D = √ σ NT , int /σ th given the 1D measurement σ NT , int , and assuming isotropic turbu-lence in three dimensions (e.g., Kainulainen & Federrath 2017). The complex interactions between turbulence, gravity and mag-netic fields happen everywhere in molecular clouds and regulatestar formation therein. We can describe these interactions throughthe virial theorem. According to Fiege & Pudritz (2000a), thevirial equation of self-gravitating, magnetized turbulent filamentaryclouds can be written in the form: T net + W + M = 0 . (3)Therefore, the virial equilibrium of the clouds depends on the com-petition between the net kinetic energy ( T net ), magnetic energy ( M ), and gravitational potential ( W ). T net accounts for the dif-ference between the internal ( T int ) and external ( T ext ) turbulentenergy, assuming that molecular clouds are confined by the exter-nal pressure rather than completely isolated. Note that all energieshere are measured per unit length.The internal kinetic energy T int is calculated as: T int = 12 M line σ , int , (4)where M line is the line mass along the filament, which can beobtained in Fig. 3 of Paper I. While σ tot , int measured from COand C O (2-1) are similar, the one from CO (2-1) was finallyadopted in the calculation since CO emission tends to tracelarger-scale gas. The external turbulent energy T ext can be ex-pressed as a function of the external pressure P ext : T ext = k P ext πR , (5)where k is the Boltzmann constant and R fil is the filament ra-dius, ∼ . pc as measured in Paper I. A conservative value of P ext = 5 × K cm − is assumed, which is in the range be-tween K cm − for the general ISM (Chromey, Elmegreen &Elmegreen 1989) and K cm − for several molecular clouds as-sociated with HI complexes (Boulares & Cox 1990).The gravitational potential energy W is derived from the linemass of the filament: W = − M G, (6)where G is the gravitational constant. The magnetic energy M iswritten as: M = B µ πR , (7)where B is the magnetic field strength and µ is the permeability offree space. B is estimated following the empirical linear relation-ship between the field strength and gas column density. This rela-tionship was summarized by Crutcher (2012) from Zeeman mea-surements in the form: B = C × N H − µG, (8) MNRAS000 , 000–000 (0000) arge-scale periodic velocity oscillation in G350.54 + − E n g e r y ( e r g p c − ) × C2C3C4C5C6C7AC7BC8 W T net M T net + M (a) G350.5-N 0 . . . . . − − E n g e r y ( e r g p c − ) × C1 W T net M T net + M (b) G350.5-S Figure 5.
Virial analysis through the comparisons between turbulent ( T ), magnetic energies ( M ) and gravitational potential ( W ) for G350.5-N (in panel a),and for G350.5-S (in panel b). Several peaks are related to dense cores (also see Fig. 3) as indicated by vertical lines. The comparison shows that the gravitydominates over both turbulence and magnetic fields, and is a major driver to the global fragmentation of G350.5 from clouds to dense cores. where C is a constant, and N H is the column density of atomic hy-drogen gas. This constant was given to be 3.8 by Crutcher (2012)under the assumption of magnetic critical condition and a sphericalshape of clouds, and improved to be 1.9 by Li et al. (2014) as-suming a sheet-like shape. The mean column density of the cloudG350.5, N H = 8 × cm − (Paper I), indicates an average B-field of µ G. Even though this estimate of the B-field strength isvery indirect, it can give us some insight into the role of the B-fieldversus gravity (see below).Figure 5 presents the comparison between turbulent ( T ), grav-itational ( W ) and magnetic ( M ) energies per unit length along themain structure of the two filaments G350.5-N, and G350.5-S. It canbe seen that W (black curve) is greater than the sum of T net and M (light grey curve) along the main structure of both G350.5-Nand G350.5-S with respective mean ratios of W/ ( T net + M ) =2 . ± . , and . ± . . This trend is in particular evident for thedense cores which correspond to the peaks in the distributions (seeFig. 5). This suggests that gravity dominates over both turbulenceand magnetic fields, and is a major driver to the global fragmenta-tion of G350.5 from clouds to dense cores. This is consistent withour previous analysis (Paper I) showing that G350.5 could have un-dergone radial collapse and fragmentation into distinct small-scaledense cores. Moreover, one can see that only turbulence withoutthe aid of the B-field is not able to counteract gravity, suggestingthe critical role of B-field in regulating molecular clouds againstgravity. This is in agreement with existing observations showingthat magnetic fields are important in filamentary clouds (e.g., Li etal. 2013, 2014, 2015; Stutz & Gould 2016; Contreras et al. 2016).Future more accurate B-field strength measurements will providedirect constraints on the role of the magnetic fields in G350.5. As mentioned in Sect. 3.2, a large-scale periodic velocity oscilla-tion is found along the entire filament. This oscillation feature maybe related to kinematics of either core formation or a large-scaleoscillation (e.g., wave-like perturbations triggered by material ac-cretion flows onto the filament or a standing wave Stutz, Gonzalez-Lobos, & Gould (2018)). In what follows, we will make an attemptto investigate the nature of the observed large-scale velocity oscil- lation in G350.5-N. Since G350.5-S is disconnected to G350.5-Nin Fig. 3 and G350.5-S itself has insufficient detection in the diffuseregion, the nature of the velocity field of G350.5-S requires detailedinvestigation with sensitive and higher-resolution observations. Tobetter visualize the velocity oscillation along G350.5-N, we makethe position-velocity (PV) diagram shown in Fig. 6a. The colorscale is the main-beam temperature with at least σ ( σ = 0 . K)detection within − . to − . km s − .To highlight the velocity fluctuation feature, we calculate theaverage velocities weighted by the main-beam temperature, asshown in cyan crosses. In Fig. 6a, the velocity oscillation behavesperiodically along the filament. Particularly, it can be fit with asine function with a wavelength of ∼ . pc and an amplitude of ∼ . km s − . In Paper I, nine identified cores are found to be distributed al-most periodically along the entire filamentary cloud G350.5 andthe average projected separations between them are measured tobe ∼ . pc. This separation is consistent with the prediction( ∼ . pc in Paper I) by the “sausage” ( gravitational ) instability(Chandrasekhar & Fermi 1953; Nagasawa 1987), and comparableto the wavelength of the periodic velocity signature ( ∼ . pc,see above) inferred from the periodic velocity distribution as well.Therefore, the periodic velocity fluctuation may be associated withthe kinematics of filament fragmentation into cores through gravi-tational instability (GI hereafter).Actually, this periodic-type velocity fluctuation was alreadypredicted in the analytic models dedicated to GI-induced core for-mation in a uniform, incompressible filament threaded by a purelypoloidal magnetic field (e.g., Chandrasekhar & Fermi 1953). Insimulations, the GI is generally represented with redistributions ofthe gas in the filament via motions that have a dominant velocitycomponent parallel to the cylinder axis due to longitudinal gravita-tional contraction at least during the first stages of evolution (e.g.,Nakamura, Hanawa, & Nakano 1993; Fiege & Pudritz 2000b). Thegas redistribution processes can be summarized in the schematicmodel of core formation as seen in Fig. 7a, where the motions ofcore-forming gas converge towards the core center along the fila-ment, leading to the density enhancements peaking at a position of MNRAS , 000–000 (0000)
H.L. Liu et al. − − − − V ( k m s − ) Fit curveBlue-shifted fitObservedIntensity-weighted velocity . . . . . . T mb (K) (a) PV diagram d (pc)24 N H ( c m − ) C2C3C4C5C6C7aC7bC8
Cubic interp.Nearest-neighbour interp. (b) N H along the filament Figure 6. (a-b:) Velocity and column density distributions extracted along the spine of the filament G350.5-N. (a): Position-velocity diagram of C O (2-1).The detections greater than σ ( σ = 0 . K) are shown in grey-scale within the velocity channels from − . to − . km s − . The cyan crosses are theweighted average velocities by brightness temperatures, and fitted with a sine function with a wavelength of ∼ . pc and an amplitude of ∼ . km s − .Literally, the blue color curves stand for the blue-shifted velocities with respect to the systematic velocity of − . km s − , and the red ones for the red-shiftedvelocities. (b:) Density distribution. Density peaks related to dense cores are indicated, i.e., C2 to C8, which are extracted from Fig. 3. The velocity and columndensity distributions do not show a good one-to-one correspondence between one another. vanishing velocity. Assuming that both density and velocity pertur-bations (oscillations) are sinusoidal (periodic), a λ/ shift betweenthem can be expected (Hacar & Tafalla 2011 for more details). Thesimilar pattern of velocity oscillation and its association with thedensity distribution have been reported in Taurus/L1517 (Hacar &Tafalla 2011).Comparing with the scenario of GI-induced core formationmodels, we do not observe a λ/ ( ∼ . pc) phase shift betweenthe velocity and density distributions (see Fig. 6). In addition, thepredicted vanishing of velocity (see above) is not observed in mostof the density enhancements cores (i.e., C2, C5, C6, C7a,b, andC8), and instead the velocity extremes coincide spatially with thedensity enhancements. Note that the above-mentioned models arerather idealised, and only designed for a straight filament. For ex-ample, if a filament were randomly kinked/curved on small scales,neither the systematic phase shift between the velocity and densitydistributions or the vanishing of velocity at the position of coreswould be expected in the core formation process (see scenario 1in Fig. 12 of Henshaw et al. 2014). However, we believe that thepossibility of a randomly kinked structure is very low in G350.5-N since (random) small-scale kinks would not maintain a large-scalecoherent, periodic oscillation. On the other hand, a regular but os-cillating geometry driven by some physical mechanism could bepossible (e.g., Gritschneder, Heigl, & Burkert 2017, see model 2 inFig. 7, and Sect. 4.2 for more discussions).Moreover, upon inspection of Fig. 3, we can group all densecores into six main mass-accumulation clumps, i.e., C1 [red], C2[red], C3+4 [red], C5+6 [red+blue], C7a+b [red+blue], and C8[red]. They are almost periodically separated, which is demon-strated to be a result of filament fragmentation through GI (seeabove, and Paper I). Three of these clumps might have undergonefurther fragmentation due to GI or Jeans collapse (e.g., Kainulainenet al. 2017), leading to the observed multiplicity (i.e., from clumpsto cores; C3+4 to C3 and C4, C5+6 to C5 and C6, and C7a+b toC7a and C7b). Such clump-scale fragmentation may influence thesmall-scale (inter-core) velocity distribution within the clumps. Asa result, red and blue-shifted velocities would be expected on eitherside of dense cores in a straight filament. Assuming that G350.5-Nis straight to some extent (see above), no red and blue-shifted ve-locities appearing on either side of the dense cores within two of MNRAS000
Cubic interp.Nearest-neighbour interp. (b) N H along the filament Figure 6. (a-b:) Velocity and column density distributions extracted along the spine of the filament G350.5-N. (a): Position-velocity diagram of C O (2-1).The detections greater than σ ( σ = 0 . K) are shown in grey-scale within the velocity channels from − . to − . km s − . The cyan crosses are theweighted average velocities by brightness temperatures, and fitted with a sine function with a wavelength of ∼ . pc and an amplitude of ∼ . km s − .Literally, the blue color curves stand for the blue-shifted velocities with respect to the systematic velocity of − . km s − , and the red ones for the red-shiftedvelocities. (b:) Density distribution. Density peaks related to dense cores are indicated, i.e., C2 to C8, which are extracted from Fig. 3. The velocity and columndensity distributions do not show a good one-to-one correspondence between one another. vanishing velocity. Assuming that both density and velocity pertur-bations (oscillations) are sinusoidal (periodic), a λ/ shift betweenthem can be expected (Hacar & Tafalla 2011 for more details). Thesimilar pattern of velocity oscillation and its association with thedensity distribution have been reported in Taurus/L1517 (Hacar &Tafalla 2011).Comparing with the scenario of GI-induced core formationmodels, we do not observe a λ/ ( ∼ . pc) phase shift betweenthe velocity and density distributions (see Fig. 6). In addition, thepredicted vanishing of velocity (see above) is not observed in mostof the density enhancements cores (i.e., C2, C5, C6, C7a,b, andC8), and instead the velocity extremes coincide spatially with thedensity enhancements. Note that the above-mentioned models arerather idealised, and only designed for a straight filament. For ex-ample, if a filament were randomly kinked/curved on small scales,neither the systematic phase shift between the velocity and densitydistributions or the vanishing of velocity at the position of coreswould be expected in the core formation process (see scenario 1in Fig. 12 of Henshaw et al. 2014). However, we believe that thepossibility of a randomly kinked structure is very low in G350.5-N since (random) small-scale kinks would not maintain a large-scalecoherent, periodic oscillation. On the other hand, a regular but os-cillating geometry driven by some physical mechanism could bepossible (e.g., Gritschneder, Heigl, & Burkert 2017, see model 2 inFig. 7, and Sect. 4.2 for more discussions).Moreover, upon inspection of Fig. 3, we can group all densecores into six main mass-accumulation clumps, i.e., C1 [red], C2[red], C3+4 [red], C5+6 [red+blue], C7a+b [red+blue], and C8[red]. They are almost periodically separated, which is demon-strated to be a result of filament fragmentation through GI (seeabove, and Paper I). Three of these clumps might have undergonefurther fragmentation due to GI or Jeans collapse (e.g., Kainulainenet al. 2017), leading to the observed multiplicity (i.e., from clumpsto cores; C3+4 to C3 and C4, C5+6 to C5 and C6, and C7a+b toC7a and C7b). Such clump-scale fragmentation may influence thesmall-scale (inter-core) velocity distribution within the clumps. Asa result, red and blue-shifted velocities would be expected on eitherside of dense cores in a straight filament. Assuming that G350.5-Nis straight to some extent (see above), no red and blue-shifted ve-locities appearing on either side of the dense cores within two of MNRAS000 , 000–000 (0000) arge-scale periodic velocity oscillation in G350.54 + Observer
Model 1 (core formation) Model 2 (oscilating filament)L (distance) L (distance)0-VV0 0-VV0(a) (b)
Figure 7. (a:) schematic representation of GI-driven core formation along a filament, which is adapted from Fig. 12 of Hacar & Tafalla (2011). Due to GI,the motions of core-forming gas converge towards the core centers where the density enhances but the relative motions become static with respect to thesystematic velocity. In this scenario, assuming that both density ( δρ ) and velocity ( δV ) follow sinusoidal distributions, a λ/ phase shift between them wouldbe expected. (b): schematic representation of a physical oscillating filament (e.g., induced by an MHD wave). The filament moves toward and away from anobserver. In this scenario, the correspondence between the velocity extremes (i.e., red/blue-shifted velocities) and the density enhancements is expected if thefilament (density) oscillates sinusoidally. The red and blue colors indicate the red, and blue-shifted velocities (motions) with respect to an observer. Note thatthe physical intensities of δρ and δV are ignored in the schematic demonstration. the three clumps (i.e., C5+6, and C7a+b), therefore, imply that theclump-scale fragmentation (if any) would not significantly affectthe large-scale periodic velocity pattern.In addition, the average amplitude of the velocity oscillationsis measured to be ∼ . km s − . This value is around three timesgreater than that observed in Taurus/L1517 (Hacar & Tafalla 2011).This difference could depend on the mass of dense cores and the in-clination of the filament. Actually, the average mass of dense coresin G350.5-N is ∼ M (cid:12) , around 10 times higher than that in Tau-rus/L1517. Assuming free-fall gas accretion onto dense cores alongthe filament, we roughly calculate the infall velocity via the relation V infall = (cid:112) GM ∗ cos θ/r , where G is the gravitational constant, r is the radius within which gas flows onto dense cores, adopted to be / λ (0.33 pc), and θ is the inclination angle of the filament withrespect to the line of sight. As a result, the average mass ∼ M (cid:12) yields V infall = ∼ . km s − for θ = 0 ◦ , and ∼ . km s − for θ = 45 ◦ , both of which are around 4 times greater than the ampli- tude of the velocity oscillation in G350.5-N. Note that these infallvelocities should be overestimated to be the accretion velocity ofgas onto dense cores since neither gas accretion is purely free-fallin reality nor the infall commences at an infinite radius as assumedin the above equation (e.g., V´azquez-Semadeni et al. 2019). Keep-ing this uncertainty in mind, and given core formation through GIbeing at work in G350.5 (see above, and Paper I), we suggest thatcore-forming gas motions induced by GI could contribute in partto the observed velocity oscillation. However, another mechanism(e.g., see Model 2 in Fig. 7, and Sect. 4.2) is still required to explainthe discrepancy between the observations and idealized GI-inducedcore-formation models, i.e., the lack of a constant phase shift ob-served between the velocity and density distributions. MNRAS , 000–000 (0000)
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As mentioned in Sect. 4.1, there could be a mechanism responsiblefor the observed relation between the velocity and density distribu-tions (see Fig. 6). We conjecture that this mechanism might be theMHD-transverse wave propagating along the filament, which canalso produce a periodic velocity oscillation. Actually, this mech-anism was reported both in observations and in simulations (e.g.,Nakamura & Li 2008; Stutz & Gould 2016). For example, combin-ing both observed spatial and velocity undulations and the helicalB-field measurements in the Orion integral-shaped filament (ISF),Stutz & Gould (2016) suggested that repeated propagation of trans-verse waves through the filament are progressively digesting thematerial that formerly connected Orion A and B into stars in dis-crete episodes. In three-dimensional MHD simulations of star for-mation in turbulent, magnetized clouds, including feedback fromprotostellar outflows, Nakamura & Li (2008) mentioned that stellarfeedback like outflows can induce large-amplitude Alfv´en waves,which perturb the field lines in the envelope that thread other partsof the condensed sheet. The large Alfv´en speed in the diffuse enve-lope allows different parts of the sheet to interact with each otherquickly. Such interaction can spring up global, magnetically medi-ated oscillations for the condensed material (Nakamura & Li 2008).The appearance of a large-scale MHD-transverse wave along thefilament is understandable as long as there is a poloidal compo-nent of B-field, which is expected with generally helical and otherconfigurations of 3D magnetic fields (e.g., Heiles 1997; Fiege &Pudritz 2000a,b; Stutz & Gould 2016; Schleicher & Stutz 2018;Reissl et al. 2018)In the filament G350.5-N, the driving source of the large-scaleMHD-transverse wave could result from outflows of young stel-lar objects (Nakamura & Li 2008). In addition, the gas accretionflows onto the filament (see Paper I) could be an additional driv-ing source. In principle, a wave-like shape of spatial density dis-tribution could be expected as observed in the Orion-A ISF (Stutz& Gould 2016). However, it can not be recognized from Fig. 2.This could be because of projection effects. That is, the large-scaleMHD-transverse wave oscillates toward and away from us (seemodel 2 in Fig. 7) while propagating along the filament. As a result,the filament is projected to be a rather straight morphology on theplane-of-sky but the periodic velocity oscillation can be observedto blue and red-shifted with respect to the systemic velocity.To conclude, given the GI-induced core formation being atwork in G350.5-N (see Sect. 4.1), we suggest that the observed pe-riodic velocity oscillation may result from a combination of thecore-forming gas motions induced by GI and a periodic physicaloscillation driven by a MHD transverse wave. This combinationcan make λ/ shift, as expected between the velocity and densitydistributions in the core-formation models (see Model 1 of Fig. 7),disappear due to the mixture of the two different motions. Instead,the wave could cause the correspondence between the velocity ex-tremes and density enhancements (see Fig. 6) if the wave is strongerthan the core-forming gas motions in velocity amplitude. Despite ofno definitive interpretation, it is worthwhile to test the possibility ofthe large-scale transverse wave being at work in G350.5-N. There-fore, we call for future high sensitivity/resolution Zeeman, and dustpolarization measurements to infer the B-field strength and to con-strain the field morphology (see above). We have analysed the internal kinematics of the filament G350.5with our observations of CO, and C O (2-1) by APEX. COemission reveals two clouds with different velocities. The majorcloud G350.5 corresponds to − . to − km s − while the otherone corresponds to − . to . km s − . Our analysis shows that thefilament G350.5 as a whole is supersonic and gravitationally bound.In addition, we find a large-scale periodic velocity oscillation alongthe filament G350.5-N with a wavelength of ∼ . pc and anamplitude of ∼ . km s − . Comparing with the gravitational-instability induced core formation models, we suggest that the ob-served periodic velocity oscillation may result from a combinationof the kinematics of gravitational instability-induced core forma-tion and a periodic physical oscillation driven by a MHD trans-verse wave. To test the latter, future high sensitivity, and resolutionZeeman, and dust polarization measurements toward G350.5 arerequired to infer the B-field strength and to constrain the field mor-phology Acknowledgments
We thank the anonymous referee for constructive comments thatimproved the quality of our paper. This work was in part sponsoredby the Chinese Academy of Sciences (CAS), through a grant to theCAS South America Center for Astronomy (CASSACA) in San-tiago, Chile. AS acknowledges funding through Fondecyt regular(project code 1180350), “Concurso Proyectos Internacionales deInvestigaci´on” (project code PII20150171), and Chilean Centro deExcelencia en Astrof´ısica y Tecnolog´ıas Afines (CATA) BASALgrant AFB-170002. J. Yuan is supported by the National NaturalScience Foundation of China through grants 11503035, 11573036.This research made use of Astropy, a community-developed corePython package for Astronomy (Astropy Collaboration, 2018).
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APPENDIX A: OPTICAL DEPTH
Following the radiative transfer equations with an assumption ofoptically thin C O (2-1) emission (e.g., Garden et al. 1991), wecalculated the optical depth ( τ ) of CO, and C O (2-1) for theselected positions in the main structure of G350.5 as below: T COmb T C Omb ≈ − e τ − e τ ≈ − e τ − e τ /R , (A1)where T mb is the main-beam temperature for the two species, andthe isotope ratio R = [ CO] / [C O] is adopted to be 7.7 fol-lowing the derivation in Yuan et al. (2016). Instead of the weak-emission component of CO (2-1), its main velocity componentmatching the C O (2-1) counterpart was taken into account in thepractical calculations. With τ estimated, we can obtain τ viathe relation τ = τ /R . The statistics of the estimated opticaldepths for both species is shown in Fig. A1. It can be seen that allof C O (2-1) emission in the main structure of G350.5 are opti-cally thin while CO (2-1) emission is optically thick with opticaldepths up to . . N u m b e r COC O Figure A1.
Histogram of the optical depth for CO (a) and C O (b).The majority of C O (2-1) emission appears to be optically thin while CO (2-1) emission suffers more optically thick effects.
APPENDIX B: EXCITATION TEMPERATURE
We further evaluated the excitation temperatures for the selectedpositions in the main structure of the filament following Liu et al.(2015): T mb = h ν k [ J ν ( T ex ) − J ν ( T bg ) ][1 − exp( − τ )] f, (B1)where J ν is defined as ν/ k T ) − , T mb is the main-beam tem-perature, T ex is the exciting temperature, and T bg = 2 . K is thecosmic background radiation; τ is the optical depth, the fractionof the telescope beam filled by emission f is assumed to be 1, Band µ are the rotational constant and the permanent dipole momentof molecules respectively. Given the optical depth (in Sect. A) and T COmb , we obtained the excitation temperature of CO (2-1) fromEq. B1. As a result, the average excitation temperature of CO (2-1) in the main structure of the filament is ± . K, which is ∼ Kless than the corresponding average dust temperature there derivedfrom the dust temperature map (Paper I). This difference may be re-lated to non full thermalization of CO (2-1) in the main structureof G350.5.
APPENDIX C: GAUSSIAN FITTING RESULTS
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Gaussian-fitting results of both CO and C O (2-1) for the selected 41 positions in G350.5-N (upper panel), and 6 positions in G350.5-S (bottompanel). The red and blue colors represent the observed spectra of CO and C O (2-1), respectively. The black stands for the fitting result.MNRAS000