Revealing Gravitational Collapse in Serpens G3-G6 Molecular Cloud using Velocity Gradients
DD RAFT VERSION F EBRUARY
15, 2021Typeset using L A TEX preprint2 style in AASTeX63
Revealing Gravitational Collapse in Serpens G3-G6 Molecular Cloud using Velocity Gradients Y UE H U ,
1, 2
A. L
AZARIAN ,
2, 3
AND S NE ˇ ZANA S TANIMIROVI ´ C Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA Department of Astronomy, University of Wisconsin-Madison, Madison, WI 53706, USA Center for Computation Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010
ABSTRACTThe relative role of turbulence, magnetic fields, self-gravity in star formation is a subject of intensive debate.We present IRAM 30m telescope observations of the CO (1-0) emission in the Serpens G3-G6 molecularcloud and apply to the data a set of statistical methods. Those include the probability density functions (PDFs)of column density and the Velocity Gradients Technique (VGT). We combine our data with the Planck 353 GHzpolarized dust emission observations, Hershel H column density. We suggest that the Serpens G3-G6 southclump is undergoing a gravitational collapse. Our analysis reveals that the gravitational collapse happens atvolume density n HI ≥ cm − . We estimate the plane-of-the-sky magnetic field strength of approximately120 µG using the traditional Davis-Chandrasekhar-Fermi method and 100 µG using a new technique proposedin Lazarian et al. (2020). We find the Serpens G3-G6 south clump’s total magnetic field energy significantly sur-passes kinetic energy and gravitational energy. We conclude that the gravitational collapse could be successfullytriggered in a supersonic and sub-Alfv´enic cloud. Keywords:
Interstellar medium (847); Interstellar magnetic fields (845); Interstellar dynamics (839) INTRODUCTIONStar formation in molecular clouds is regulated by a combi-nation of MHD turbulence, magnetic fields, and self-gravityover scales ranging from tens of parsecs to (cid:28) . pc (Jokipii1966; Shu 1977, 1992; Shu et al. 1994; Kennicutt 1998b; Mc-Kee & Ostriker 2007; Li & Henning 2011; Hull et al. 2013;Caprioli & Spitkovsky 2014; Andersson et al. 2015; Kenni-cutt 1998a). A number of observational and numerical stud-ies suggest that the interaction of turbulent supersonic flowsand magnetic fields produce high-density fluctuations whichserve as the nurseries for new stars (Nakamura et al. 1999;Tilley & Pudritz 2003; McKee & Ostriker 2007; Federrath2013; Shimajiri et al. 2019; Hu et al. 2020b). However, thebalance between the three is still obscured. In particular, it ischallenging to identify the gravitational collapse, i.e., wheregravity takes over.To get insight into self-gravity in molecular clouds, the col-umn density Probability Density Functions (PDFs) are pop-ularly used. In non-gravitational collapsing isothermal su-personic environments, the PDFs appear as log-normal dis-tribution with its width controlled by the sonic Mach num-ber (Vazquez-Semadeni et al. 1995; Klessen 2000; Robert-son & Kravtsov 2008; Kritsuk et al. 2011; Collins et al.2012; Padoan et al. 2017; Burkhart 2018). Self-gravity,further, introduces a power-law tail at high-density range [email protected]; [email protected] (Vazquez-Semadeni et al. 1995; Collins et al. 2012; Robert-son & Kravtsov 2008; Burkhart 2018; K¨ortgen et al. 2019).The transition from log-normal PDFs to power-law PDFs canreveal the density threshold, above which the gas becomesgravitational collapsing.The Velocity Gradients Technique (VGT; Gonz´alez-Casanova & Lazarian 2017; Yuen & Lazarian 2017a; Lazar-ian & Yuen 2018a; Hu et al. 2018, and reference therein) is anovel approach to study the magnetic fields, turbulence, andself-gravity in the interstellar medium (ISM). VGT is rootedin the MHD turbulence theory (Goldreich & Sridhar 1995)and the turbulent reconnection theory (Lazarian & Vishniac1999). These theories revealed the anisotropic nature of tur-bulent eddies, i.e., the eddies are elongating along with thelocal magnetic fields. The magnetic field direction, there-fore, is parallel to the eddies’ semi-major axis. The gradientsof velocity fluctuations, which are perpendicular to the semi-major axis, play the role of probing the magnetic fields (Cho& Vishniac 2000; Maron & Goldreich 2001; Cho et al. 2002).Since subsonic velocity and density fluctuations exhibit sim-ilar statistical properties, the density gradients (or intensitygradients) are also perpendicular to the magnetic fields (Yuen& Lazarian 2017b; Hu et al. 2019c). The above considerationis the basis on which the VGT was developed.However, this perpendicular relative orientation betweenthe gradients and the magnetic fields can be broken by self-gravity. In the case of gravitational collapse, the gravitationalforce pulls the plasma in the direction parallel to the mag-netic field producing the most significant acceleration. Be- a r X i v : . [ a s t r o - ph . GA ] F e b H U , L AZARIAN & S
TANIMIROVI ´ C cause gradients are pointing to maximum changes, the veloc-ity gradients are thus dominated by the gravitational acceler-ation, being parallel to the magnetic fields (Yuen & Lazarian2017b; Hu et al. 2020b, 2019a). For intensity gradients, theconsideration is similar. The most significant accumulationof material happens in the direction of gravitational collapseso that the intensity gradients are parallel to the magneticfields. This phenomenon has been observed in the molec-ular clouds Serpens (Hu et al. 2019a), G34.43+00.24 (Tanget al. 2019), and NGC 1333 (Hu & Lazarian 2021). Thisparticular reaction with respect to self-gravity enables VGTto reveal regions of gravitational collapse and quiescent ar-eas where turbulent motions, thermal pressure, and magneticsupport dominate over gravitational energy.The Serpens cloud, which is famous for its high star forma-tion rate (Cambr´esy 1999) and high surface density of youngstellar objects (YSOs; Eiroa et al. 2008), serves as an excel-lent object to study the gravitational collapse. In particular,the Serpens G3-G6 region (Cohen & Kuhi 1979) is relativelyisolated and does not have nearby H II regions, or obviouslarge scale gas shocks, both of which can cause changes inthe alignment between magnetic fields and velocity gradi-ents. Our study, therefore, focuses on the Serpens G3-G6 re-gion. The CO (2–1) and CO (2–1) emission lines of thisregion were previously observed with the Arizona Radio Ob-servatory Heinrich Hertz Submillimeter Telescope (Burleighet al. 2013) with angular resolution of ≈ (cid:48)(cid:48) . Our new CO (1–0) emission line data obtained with the IRAM 30-m tele-scope complement previous observations and have higher an-gular resolution ( ≈ . (cid:48)(cid:48) ). These data, including the Her-schel H column density image (Andr´e et al. 2010) and thePlanck 353 GHz dust polarization data (Planck Collabora-tion et al. 2020), were used in this study to resolve kine-matic, magnetic, and gravitational properties for Serpens atscales ≤ pc. The kinematic property is directly measuredfrom the emission lines’ width, while we use two differentapproaches to estimate either magnetic or gravitational prop-erties. The magnetic field strength was estimated by boththe Davis-Chandrasekhar-Fermi method (Davis 1951; Chan-drasekhar & Fermi 1953) based on the polarization measure-ment and a new approach based on the values of sonic Machnumber M S and Alfv´en Mach number M A (Lazarian et al.2020). Also, we employ the PDFs and VGT to confirm thegravitational collapsing nature of the Serpens G3-G6 cloud.The paper is organized as follows. In § 2, we give the de-tails of the observational data used in this work, includingdata reduction. In § 3, we describe the methodology and al-gorithms implemented in VGT. In § 4, we present our resultsof identifying the gravitational collapsing regions on SerpensG3-G6 clump and we analysis the dynamics of the clump.We give discussion in § 5 and conclusion in § 6. OBSERVATIONS AND DATA REDUCTION2.1. CO (1–0) emission line We obtained a new CO (J = 1–0) fully sampled map ofSerpens using the IRAM 30m telescope (Carter et al. 2012).The observations were obtained in July 2020 using 16 h of telescope time under average summer weather conditions (6mm median water vapor). We covered a field of view (FoV) (cid:48) × (cid:48) . The CO (J = 1–0) emission was observed us-ing the EMIR receiver and the VESPA spectrometer usinga bandwidth of 60 MHz at 0.092 MHz resolution ( ≈ − ). The half-power beamwidth (HPBW) at 110.201354GHz is ≈ . (cid:48)(cid:48) .We used the on-the-fly scanning strategy with a dump timeof 0.7 s and a scanning speed of (cid:48)(cid:48) /s to ensure a sampling ofthree dumps per beam along the scanning direction with thescanning direction reversed after each raster line (i.e., zigzagscanning mode). We covered the full FoV ( ≈ (cid:48)(cid:48) × (cid:48)(cid:48) size. In total, we employed about 13 minper tile.Data reduction was carried out using theGILDAS1/CLASS software . The data were first calibratedto the T A scale and were then corrected for atmospheric ab-sorption and spillover losses using the chopper-wheel method(Penzias & Burrus 1973). A polynomic baseline of secondorder was subtracted from each spectrum, avoiding velocitieswith molecular emission. The spectra were then gridded intoa data cube through convolution with a Gaussian kernel ofFWHM ∼ / of the IRAM-30m telescope beamwidth atthe rest line frequency. The typical (1 σ ) RMS noise levelachieved in the map is 0.33 K per 212 m s − velocity chan-nel. A large table of the individual spectra was made and thespectra were finally combined to obtain a regularly griddedposition-position-velocity data cube, setting the pixel size to5 (cid:48)(cid:48) . Here we convert the measured antenna temperature T A to brightness temperature T b through T b = (F eff / B eff )T A ,where F eff = 0 . is the forward efficiency of the IRAM 30m telescope and B eff = 0 . is the main beam efficiency at110.201354 GHz (Pety et al. 2017).2.2. CO (2–1) and CO (2–1) emission lines The CO (2–1) and CO (2–1) emission lines wereobserved with the Heinrich Hertz Submillimeter Telescope(Burleigh et al. 2013) while the H column density data isobtained from the Herschel Gould Belt Survey (Andr´e et al.2010). Each line was measured with 256 filters of 0.25 MHzbandwidth, giving a total spectral coverage of 41 km s − at aresolution of 0.33 km s − .The angular resolution of the emission liens is (cid:48)(cid:48) (0.04pc) with a sensitivity of 0.12 K RMS noise per pixel in onespectral channel (Burleigh et al. 2013). The radial velocity ofthe bulk of the emission ranges from about -1 to +18 km s − DENTIFYING THE G RAVITATIONAL C OLLAPSE IN S ERPENS WITH V ELOCITY G RADIENTS CO (2-1) and from +2 to +13 km s − for CO (2-1)(Burleigh et al. 2013). We select the emissions within theseranges for our analysis.2.3. Polarized dust emission
To trace the magnetic field orientation in the POS, weuse the Planck 353 GHz polarized dust signal data from thePlanck 3rd Public Data Release (DR3) 2018 of High Fre-quency Instrument (Planck Collaboration et al. 2020) .The Planck observations defines the polarization angle φ and polarization fraction p through Stokes parameter maps I,Q, and U: φ = 12 arctan( − U, Q ) p = (cid:112) Q + U /I (1)where − U converts the angle from HEALPix convention toIAU convention and the two-argument function arctan isused to account for the π periodicity. To increase the signal-to-noise ratio, we smooth all maps from nominal angular res-olution (cid:48) up to a resolution of (cid:48) using a Gaussian kernel.The magnetic field angle is inferred from φ B = φ + π/ . METHODOLOGY: THE VELOCITY GRADIENTSTECHNIQUE3.1.
Theoretical consideration
The Velocity Gradients Technique (VGT; Gonz´alez-Casanova & Lazarian 2017; Yuen & Lazarian 2017a; Lazar-ian & Yuen 2018a; Hu et al. 2018) is the main analysistool in the work. It is theoretically rooted in the advancedmagnetohydrodynamic (MHD) turbulence theory (Goldreich& Sridhar 1995, henceforth GS95), including the conceptof fast turbulent reconnection theory (Lazarian & Vishniac1999, henceforth LV99). These theories explained that tur-bulent eddies are anisotropic (see GS95) so that their semi-major-axis is elongating along the local magnetic fields (seeLV99). This was numerically demonstrated, by Cho & Vish-niac (2000) and Maron & Goldreich (2001). In particular,LV99 derived the anisotropy relation for the eddies in localreference frame: l (cid:107) (cid:39) L inj ( l ⊥ L inj ) M − / (2)where l ⊥ and l (cid:107) are the perpendicular and parallel size ofeddies in respect to the local magnetic field. M A is the Alf´enMach number and L inj is the injection scale of turbulence.The corresponding scaling of velocity fluctuation v l at scale l is then: v l (cid:39) v inj ( l ⊥ L inj ) M / (3)where v inj is the injection velocity. Explicitly, as theanisotropic relation indicates l ⊥ (cid:28) l (cid:107) , the velocity gradients scale as (Yuen & Lazarian 2020a): ∇ v l ∝ v l l ⊥ (cid:39) v inj L inj ( l ⊥ L inj ) − M A (4)which means that the smallest resolved eddies induce thelargest gradients. The gradients are perpendicular to the localmagnetic fields.3.2. Principal component analysis
As we discussed above, the velocity fluctuation, i.e., thevelocity variance, is related to the eddy’s size along the LOS.Large velocity variance eddies give the most significant con-tribution to the velocity gradients. To get the largest velocityvariance, Hu et al. (2018) proposed the Principal ComponentAnalysis (PCA) to pre-process the spectroscopic PPV cubes.PCA treats the observed brightness temperature T b ( x, y, v ) in PPV cube as the probability density function of threerandom variables x , y , v . By splitting the PPV cube into n v velocity channels along the LOS, the covariance matrix S ( v i , v j ) and its corresponding eigenvalue equation are de-fined as: S ( v i , v j ) ∝ (cid:90) dxdyT b ( x, y, v i ) T b ( x, y, v j ) − (cid:90) dxdyT b ( x, y, v i ) (cid:90) dxdyT b ( x, y, v j ) (5) S · u = λ u (6)here i, j = 1 , , ..., n v and λ is the eigenvalues associatedwith the eigenvector u . The equation gives totally n v eigen-values. Each eigenvalue λ i is proportional to the squared ve-locity variance v i along the LOS. The velocity channel cor-responding to the largest eigenvalue, therefore, has the mostsignificant contribution to the velocity gradients.In the frame of VGT, the PPV cube can be pre-processedby PCA in two different ways. The first one is treating theeigenvalues λ as the weighting coefficients for each velocitychannel (Hu et al. 2018). By integrating the weighted veloc-ity channels along the LOS, we obtained the velocity centroidmap C(x,y) : C ( x, y ) = (cid:82) dvT b ( x, y, v ) · v · λ ( v ) (cid:82) dvT b ( x, y, v ) (7)Large eigenvalues enhance the significance of their corre-sponding velocity channels, while small eigenvalues give asuppression.Alternatively, instead of weighting velocity channels, thePPV cube can be projected into a new orthogonal basis con-structed by n v eigenvectors (Hu et al. 2020a). Its corre-sponding eigenvalue constrains the length of each eigenvec-tor. Since the new axes are oriented along the direction ofmaximum variance, the PPV cube on a new basis exhibitsthe largest velocity variance. The projection of is operatedby weighting channel T b ( x, y, v j ) with the corresponding H U , L AZARIAN & S
TANIMIROVI ´ C eigenvector element u ij , in which the corresponding eigen-channel I i ( x, y ) is: I i ( x, y ) = n v (cid:88) j u ij · T b ( x, y, v j ) (8)This step produces a total of n v eigen-channels in the eigen-vectors space. Generally, both the weighting and projectingapproaches are figuring out the maximum velocity variance,which is the most important in velocity gradients’ calcula-tion. In this work, we adopt the projection approach to con-struct the pseudo-Stokes-parameters; see the discussion be-low.Note here the channel number n v should be sufficientlylarge so that the channel width ∆ v satisfies ∆ v < (cid:112) δ ( v ) ,where (cid:112) δ ( v ) is the 1D velocity dispersion. We call thechannel, which satisfies the criteria as a thin velocity chan-nel. Due to the velocity caustic effect, which comes from thenon-linear mapping from real space to PPV space, the thinvelocity channel’s intensity fluctuation is mainly induced byvelocity fluctuation instead of density fluctuation (Lazarian& Pogosyan 2000). The thin velocity channels, therefore,record velocity information, which is used to calculate ve-locity gradients. 3.3. Sub-block averaging
The PPV cubes are pre-processed by PCA to extract themost crucial velocity components resulting in n v eigen-channels. Each eigen-channel is convolved with 3 × G x and G y to calculate pixelized gradient map ψ ig ( x, y ) : (cid:53) x I i ( x, y ) = G x ∗ I i ( x, y ) (cid:53) y I i ( x, y ) = G y ∗ I i ( x, y ) ψ ig ( x, y ) = tan − (cid:18) (cid:53) y I i ( x, y ) (cid:53) x I i ( x, y ) (cid:19) , (9)where (cid:53) x I i ( x, y ) and (cid:53) y I i ( x, y ) are the x and y compo-nents of gradient respectively, and ∗ denotes the convolution.Note that the turbulent eddy discussed above is a statisticalconcept. A single gradient, therefore, does not necessarilycorrelate to the magnetic field. The orthogonal relative orien-tation between velocity gradients and the magnetic field ap-pears only when the sampling is statistically sufficient. Thestatistical sampling procedure is proposed by Yuen & Lazar-ian (2017a), i.e., the sub-block averaging method. Firstly,the sub-block averaging method takes all gradients orienta-tion within a sub-block of interest and then plots the corre-sponding histogram. A Gaussian fitting is then applied to thehistogram. The Gaussian distribution’s expectation value is G x = − − − , G y = − − −
10 0 0+1 +2 +1 the statistically most probable gradient’s orientation withinthat sub-block. Incidentally, the sub-block averaging alsopartially suppress the RMS noise in the spectroscopic data.The selection of sub-block size, which controls the num-ber of sampling points, is crucial. Hu et al. (2020b) later im-proved the sub-block averaging method to be adaptive. Thesub-block centers were selected continuously, locating at theposition (i, j), i, j = 1, 2, 3, etc. All gradients pixels withinthe rectangular boundaries [i-d/2, i+d/2] and [j-d/2, j+d/2] areselected to do averaging, where d is the sub-block size. Wevary the sub-block size and check its corresponding fittingerrors within the 95% confidence level. When the fitting er-ror reaches its minimum value, the corresponding sub-blocksize is the optimal selection. We refer to this procedure asthe adaptive sub-block (ASB) averaging method (Hu et al.2020b). 3.4.
Pseudo-Stokes-parameters
By repeating the gradient’s calculation and ASB for eacheigen-channel, we obtain totally n v eigen-gradient maps ψ igs ( x, y ) with i = 1 , , ..., n v . In analogy to the Stokes pa-rameters of polarization, the pseudo Q g and U g of gradient-inferred magnetic fields are defined as: Q g ( x, y ) = nv (cid:88) i =1 I i ( x, y ) cos(2 ψ igs ( x, y )) U g ( x, y ) = nv (cid:88) i =1 I i ( x, y ) sin(2 ψ igs ( x, y )) ψ g = 12 tan − ( U g Q g ) (10)The pseudo polarization angle ψ g is then defined correspond-ingly. Similar to the Planck polarization, ψ B = ψ g + π/ gives the POS magnetic field orientation.3.5. Alignment measure
The relative alignment between magnetic fields orienta-tion inferred from Planck polarization φ B and velocity gra-dient ψ B is quantified by the Alignment Measure (AM,Gonz´alez-Casanova & Lazarian 2017): AM = 2( (cid:104) cos θ r (cid:105) −
12 ) (11)where θ r = | φ B − ψ B | and (cid:104) ... (cid:105) denotes the average within aregion of interests. The value of AM spans from -1 to 1. AM= 1 mean φ B and ψ B are parallel, while AM = -1 indicates φ B and ψ B are perpendicular. The standard deviation dividedby the sample size’s square root gives the uncertainty σ AM .3.6. Double-peak histogram
The double-peak histogram achieves the detection of thegravitational collapsing region. Unlike the gradients ψ B indiffuse regions, the gravitational collapse flips the gradientsby 90 ◦ being ψ B + π/ . The histogram of gradients’ anglecan be used to extract this change. In the diffuse region, the DENTIFYING THE G RAVITATIONAL C OLLAPSE IN S ERPENS WITH V ELOCITY G RADIENTS Figure 1.
Map of H column density obtained from the HerschelGould Belt Survey. The YSOs are identified by Harvey et al. (2007). histogram exhibits a single Gaussian peak locates at ψ B . Thesingle peak of the histogram becomes ψ B + π/ in gravita-tional collapsing regions. However, in transitional regions,which cover, for instance, half diffuse material and half grav-itational collapsing, the histogram is therefore expected toshow two peak values of ψ B and ψ B + π/ . We call this typeof histogram the double-peak histogram (DPH).The DPH works as follows. Every single pixel of ψ B isdefined as the center of a sub-block to draw the histogram.An envelope, which is a smooth curve outlining the extremes,is adopted for the histogram to suppress noise and the effectfrom insufficient bins. Any term whose histogram weight isless than the mean weight value of the envelope is masked.After masking, we work out the peak value of each consecu-tive envelope. Once more than one peak values appear, andthe maximum difference is within the range 90 ◦ ± σ g , where σ g is the total standard deviation, the center of this secondsub-block is labeled as the boundary of a gravitational col-lapsing region. The details of the algorithm are presented inHu et al. (2020b). RESULTS4.1.
Integrated intensity map and H column density map Fig. 1 presents H column density map for the Serpens G3-G6 clump. The H column density structures are elongatedand filamentary. Here we plot the distribution of young stel-lar objects (YSOs) within this clump. The YSOs are identi-fied by Harvey et al. (2007). The evolutionary stage of a YSOcan be classified as (in order of youngest to oldest): Class I,flat, Class II, and Class III (Lada 1987; Andre & Montmerle1994; Greene et al. 1994). We find Class I and flat YSOsconcentrate on the south dense clump core and the north-eastfilamentary tail. Class II YSOs’ mainly locate at the north Note this sub-block is different from the ASB, which is used to determinethe mean gradients’ orientation. The second sub-block implemented in theDPH is used to extract the change of gradients. The second sub-blocksize can be different from the one for the ASB. In this work, we adopt thesecond block size × pixels, which are statistically sufficient Hu et al.(2020b). filamentary tail, while Class III only occupies a small frac-tion. YSOs’ high surface density at the dense clump core andthe north tail indicates these two regions are actively formingstars.The CO (1-0) emission line observed with the IRAM30-m telescope zooms into the south dense clump with aresolution of . (cid:48)(cid:48) ( ≈ . pc, see Fig. 2). The radialvelocity of the emission ranges from about 2.6 to 12.0 kms − . The spectral line appears to have two apparent peaks: 2.22 K at 7.65 km s − and 0.58 K at 4.9 km s − . Af-ter fitting a double Gaussian profile to the integrated spectralline shown in Figure 2, we obtain the 1D velocity disper-sion σ v, D = 1 . ± . km s − for the dominating emissionfeature. Fig. 2 also shows the integrated intensity maps forthe south dense clump core of Serpens G3-G6. The inte-gration of CO (1-0) considers pixels where the brightnesstemperature is larger than 0.9 K, which is about three timesthe RMS noise level. The CO (1-0) emission lines covera wider . ◦ × . ◦ area. The clump’s CO (1-0) is stillfilamentary, spanning from north to south.Fig. 3 shows representative velocity channels from the CO (1-0) emission line data, averaged over 636 m s − .The intensity structures seen in velocity channels do not ap-pear filamentary. In particular, two distinct dense structuresappear at 7.0 km s − and 7.6 km s − . These structures sug-gest that the velocity caustic effect is significant, i.e., velocityfluctuations dominate the thin velocity channels (Lazarian &Pogosyan 2000).4.2. gravitational collapsing regions identified from PDFs The column density PDFs are widely used to study tur-bulence and self-gravity in ISM. The PDFs in gravitationalcollapsing isothermal supersonic turbulence follow a hybridof log-normal distribution P LN ( s ) in low-intensity rangeand power-law distribution P P L ( s ) in high-intensity range(Vazquez-Semadeni et al. 1995; Robertson & Kravtsov 2008;Collins et al. 2012; Burkhart 2018; K¨ortgen et al. 2019): P LN ( s ) = D (cid:112) πσ s e − ( s − s σ s , s < S t P P L ( s ) = DCe ks , s > S t (12)where s = ln ( N/N ) is the logarithm of the normalized col-umn density N . S t is transitional density between the P N and P L . s = − σ s is the mean logarithmic density. Thestandard deviation σ s for magnetized turbulence is related tothe sonic Mach number M S , driving parameter b , and com-pressibility β (Molina et al. 2012): σ s = log(1 + b M S2 ββ + 1 ) (13)The turbulence driving parameter b = 1 / for purelysolenoidal driving, while b = 1 for purely compressive driv-ing. For a natural mixture of solenoidal and compressivedriving, b is ≈ . (Federrath & Banerjee 2015; Federrathet al. 2016). Assuming P NL ( s ) + P P L ( s ) is normalized, H U , L AZARIAN & S
TANIMIROVI ´ C Figure 2.
Map of integrated brightness temperature of CO J = 1–0 (top) over velocity range 2.6 to 12.0 km s − and the mean brightnesstemperature spectra (bottom) averaged over the region. The emission line is observed with the IRAM 30m telescope. The contour indicates amean intensity value of 28.68 K km/s. Figure 3. CO J = 1–0 spectral channel maps averaged over 636 m s − and spaced 212 m s − apart. Mean LSR velocity is labeled in upperright. continuous, and differentiable, the coefficient D and C are(Burkhart et al. 2017; Burkhart 2018): D = ( Ce S t k − k + 12 + 12 erf( 2 S t + σ s s √ σ s )) C = e . k +1) kσ s σ s √ π (14)In Fig. 4, we plot the H column density PDFs for the en-tire Serpens G3-G6 clump. The PDFs are log-normal withdispersion σ s ≈ . ± . till S t ≈ . , which indi-cates the gas is gravitational collapsing when its density islarger than N = e S t N ≈ . × cm − . This col-umn density threshold reveals that the south dense clump isgravitational collapsing. The characteristic slope k of thatpower-law is -2.40 ± . , which is related to the cloud mean free-fall time, the magnetic fields, and the efficiency of feed-back (Girichidis et al. 2014; Burkhart et al. 2017; Guszejnovet al. 2018). Also, we zoom into the south dense region (seethe red box in Fig. 4), which corresponding to the measured CO (1-0). The south dense clump’s PDFs are still the com-bination of P LN and P P L . The transition density S t ≈ . and the characteristic slope k = − . ± . identified thesame gravitational collapsing regions as before.4.3. Magnetic fields morphology traced by VGT
In Fig. 5, we present the results of the VGT analysis us-ing the CO (2-1) and CO (2-1) emission lines data sets.We use the emission within the velocity ranges [+1, +13] kms − for calculation following the recipe presented in Sec. 3.Pixels, where brightness temperature is less than three timesthe RMS noise level (0.4 K), are blanked out. We average DENTIFYING THE G RAVITATIONAL C OLLAPSE IN S ERPENS WITH V ELOCITY G RADIENTS Figure 4. Top:
Map of of H column density. Cyan areas indicatethe gravitational collapsing regions identified from the PDFs. Mid-dle:
The PDFs of entire column density map.
Bottom:
The PDFsof south dense clump (see the red box on the top map). σ s gives thedispersion of the log-normal distribution. S t is the transition densitythreshold and k is the slope of the reference power-law line. the gradients over × pixels and smooth the gradientsmap ψ g (see Eq.10) with a Gaussian filter with kernel width1, i.e., over 5 pixels.We also compare the magnetic field inferred from thePlanck 353 GHz polarized dust signal data (FWHM ≈ (cid:48) ).We re-grid the Planck polarization further to achieve thesame pixel size as emission lines. The polarization vectoris also averaged over × pixels to match the gradientsmap. We find the resulting gradients of CO (2-1) have Figure 5. Top:
The POS magnetic fields inferred from VGT (redsegments) using CO (2-1) emission line and Planck 353 GHz po-larization (blue segments). Bottom:
The POS magnetic fields in-ferred from VGT (red segments) using CO (2-1) emission lineand Planck 353 GHz polarization (blue segments). Cyan outlinesindicate AM < good agreement with the Planck polarization, showing AM= 0.69 ± . . Several anti-alignment vectors appear in thesouth dense region and image boundary. It is likely that thedust polarization and the CO (2-1) emission probe differ-ent spatial regions. The optically thick tracer CO samplesthe outskirt diffuse region of the cloud with volume den-sity n ≈ cm − , while dust polarization likely tracesdenser regions. The theory of Radiative Torque (RAT) align-ment (see Lazarian & Hoang 2007; Andersson et al. 2015for a review) predicts that dust grains can remain alignedat high densities, especially in the presence of the embed-ded stars. This is in agreement with numerical simulations(Bethell et al. 2007; Seifried et al. 2019). Therefore, we ex-pect that for n HI ≥ cm − grains are aligned in the re-gions that we study. In addition, the low resolution ( (cid:48) ) ofthe Planck data also contributes to the misalignment. TheVGT measurements from CO (2-1) gives less alignment(AM = 0.41 ± . ) with the Planck polarization. In particu-lar, the gradients become perpendicular to the magnetic fieldin the south dense region. As the south dense region is iden-tified to be gravitational collapsing by the PDFs (see § 4.2),this change the gradients’ direction suggests that presence ofa gravitational collapse(see § 3). Since the overall magneticfields horizontally cross the clump, while the gradients are H U , L AZARIAN & S
TANIMIROVI ´ C Figure 6.
The histograms of relative angle between the magneticfield inferred from Planck polarization and VGT using CO (top)and CO (bottom). nearly vertical, the resolution effects do not change this con-clusion.We plot the histograms of the relative alignment betweenrotated gradients and the magnetic fields in Fig. 6. The his-togram based on the CO (2-1) data is more concentratedaround 0, which means that two vectors are parallel. The dis-tribution based on the CO (2-1) data, however, spreads allthe way to π/ , which means that two vectors are perpen-dicular. Previous studies reported that the VGT results of the CO emission are more accurate than the ones derived us-ing the CO emission when comparing with dust polariza-tion (Hu et al. 2019b; Alina et al. 2020) in non-gravitationalcollapsing clouds. Here we find that the VGT of CO givesbetter alignment. It is likely because the gravitational col-lapse in the Serpens G3-G6 south clump happens in densegas n HI ≥ cm − . The CO emission samples morediffuse regions so that its gradients are less affected by self-gravity.4.4. Gravitational collapsing regions identified from VGT
The comparison of VGT and Planck polarization directlyreveals the region undergoing gravitational collapse. How- ever, insufficient resolution in polarization measurementslimits this approach in small scale studies. For instance, ourhigh-resolution CO (1-0) can measure the pixelized gradi-ent up to (cid:48)(cid:48) , which is 120 times higher than Planck’s resolu-tion (cid:48) . Nevertheless, the double-peak histogram (see § 3)extends the VGT to identify the gravitational collapsing re-gion independent of polarization measurements.In Fig. 7, we present the gradients map and the identifiedgravitational collapsing regions using the CO (1-0) emis-sion line. In dense areas, we find the gradients flips their di-rection by 90 ◦ comparing with surrounding low-intensity re-gions. The DPH finds three separate gravitational collapsingregions. The largest gravitational collapsing region coversthe one identified by the PDFs (see Fig. 4) and covers partsof low-intensity regions. It is likely the VGT is sensitive togravity-induced inflows, which span to also low-intensity re-gions, while the PDFs are detecting the already formed cores.Also, the DPH cover partially diffuse regions so that the ac-tual gravitational collapsing area is slightly overestimated.Nevertheless, both the VGT and the PDFs reveal that the cen-tral clump is gravitational collapsing.The gravitational collapse usually shows a signature of in-falling motions. Explicitly, an infall signature in a spectralline presents itself in the form of a red-blue asymmetry, gen-erally with a diminished redshifted component (Walker et al.1994; Myers et al. 1996). To search for the signature, we in-tegrated the CO (J = 1-0) emission in a velocity range of+3.0 to +7.0 km s − for blueshifted gas, as the velocity ofbulk motion is 7.65 km s − (see § 4). Redshifted gas is inte-grated in a velocity range of +8.3 to +12.0 km s − . In Fig. 8,the blueshifted and redshifted gases are overlaid in integratedintensity map of CO (J = 1-0). We can see the clump isdominated by blueshifted gas. In the high-intensity region,blueshifted and redshifted gases are overlapped, which indi-cates an infalling motion along the LOS.4.5. The overall energy budget of the Serpens G3-G6 southclump
Through both PDFs and VGT, we confirm that the SerpensG3-G6 south clump is gravitational collapsing. Here we de-termine the dynamics of the clump, focusing on the energybalance. The measured and derived physical parameters arelisted in Tab. 1.The area A of the south clump is measure with CO (1-0). We include all pixels where their integrated brightnesstemperature is above the mean value (see Fig. 2). Adopt-ing 415 ±
25 pc as the distance (Dzib et al. 2010), we find A ≈ . (0.2) pc. The value in brackets indicates the uncer-tainty, which is the average of upper bound and lower bound.Assuming a simple spherical geometry, we have the effec-tive LOS distance L ≈ . pc. The 1D velocity dispersion σ v, D ≈ . (0.2) km s − is measured from the CO (1-0)emission line (see Fig. 2). σ v, D contains the contributionfrom both turbulence velocity and shear velocity. The po-larization dispersion σ p ≈ . (0.72) deg is obtained fromthe Planck 353 GHz polarization, see Fig. 9. The mean H column density N ≈ . . × cm − . DENTIFYING THE G RAVITATIONAL C OLLAPSE IN S ERPENS WITH V ELOCITY G RADIENTS Figure 7. Top: the VGT map obtained from CO (1-0) for the Serpens G3-G6 south clump. Bottom: the gravitational collapsing regions(red area) identified by VGT.
Figure 8.
Integrated intensity CO (J = 1-0) emission in Serpens G3-G6 south clump. Blueshifted gas is integrated in a velocity range of +3to +7 km s − . Contours start at 1.5 times of the mean intensity value (cid:104) I blue (cid:105) with a step 0.25 (cid:104) I blue (cid:105) . Redshifted gas is integrated in a velocityrange of +8 to +12 km s − . Contours start at 1.5 times of the mean intensity value (cid:104) I red (cid:105) with a step 0.25 (cid:104) I red (cid:105) . From these physical parameters, we derive the totalmass M = 440 . (20.6) M (cid:12) and volume mass den-sity ρ = 8 . . × − g cm − . Adopting f =0 . , the total magnetic field strength B is calculated fromthe Davis-Chandrasekhar-Fermi method (Davis 1951; Chan-drasekhar & Fermi 1953), giving B = f √ πρ σ v, D /σ p =119 . . µG . The corresponding total kinetic energy E K = 1 . . × erg, total gravitational energy E G = − . . × erg , and total magnetic field energy E B = 7 . . × erg. The ratio of kinetic energyand gravitational energy | E K /E G | ≈ . . In particular, the magnetic field energy to kinetic energy and gravitational en-ergy ratios are | E B /E K | ≈ . and | E B /E G | ≈ . ,respectively. The role of magnetic field in Serpens G3-G6south clump is much more significant than turbulence andself-gravity.Note that kinetic energy and magnetic field energy may beoverestimated, as the dispersion σ v, D considers both turbu-lence velocity and shear velocity. The ratio E B /E K aftersimplification equivalents to E B /E K = (16 f ) / (3 µ σ p ) ,which is independent of σ v, D . Similarly, we have | E B /E G | = (40 B ) / (3 π µ GN µ H m H ) . The overes-0 H U , L AZARIAN & S
TANIMIROVI ´ C Physical Parameter Symbol/Definition Value (uncertainty) ReferenceArea A 2.14 (0.2) pc Measured1D velocity dispersion σ v, D − MeasuredPolarization dispersion σ p column density N × cm − MeasuredEffective diameter
L = 2 (cid:112) A /π volume number density n = N /L cm − DerivedMass of an H atom m H × − g Ref.1Mean molecular weight µ H ρ = n µ H m H × − g cm − DerivedMass M = N µ H m H A M (cid:12) DerivedMagnetic field strength B = f √ πρ σ v, D /σ p µG DerivedKinetic energy E K = 3 Mσ v, D / (0 . × erg DerivedGravitational energy E G = − GM / (5 L ) -6.04 (1 . × erg DerivedMagnetic energy E B = B L / (8 π ) × erg DerivedViral parameter α vir = 5( √ σ v, D ) / ( πGL ρ ) t ff = (cid:112) π/ (32 Gρ ) (0 . Myr DerivedSound speed (isothermal) c s = (cid:112) k B T /µ p m H
188 m s − Ref.1 ( µ p = 2 . )Alfv´en speed v A = B/ √ πρ − Derived3D sonic Mach number M S = √ σ v, D /c s M A = √ σ v, D /v A β = 2(M A / M S ) Table 1.
Physical parameters of the Serpens G3-G6 clump. All physical parameters are derived for pixels that fall within the 28.68 K km/s(mean intensity value) CO (1-0) intensity contours drawn in Fig. 2. All uncertainties are the average values of the upper bound and lowerbound for each physical parameter. The gas temperature T is assumed to be 10 K (Draine & Lazarian 1998). References: (1) Kauffmann et al.(2008). Figure 9.
The histogram of magnetic field angle obtained fromPlanck 353 GHz polarization. The angle is measured in IAU con-vention. timated σ v, D leads to a stronger magnetic field and alsoa overestimated | E B /E G | . Nevertheless, here we have | E B /E G | ≈ . , it is unlikely that the velocity disper-sion is overestimated by one order of magnitude so that | E B /E G | ≈ . Therefore, we expect the estimated magneticfield is indeed stronger than turbulence and self-gravity. Thesignificant magnetic field energy suggests that the gravita-tional collapse can happen in a strong magnetic field environ-ment, as numerically suggested by Hu et al. (2020b). In addition, assuming the gas temperature T ∼ K(Draine & Lazarian 1998), we have the isothermal soundspeed c s = 188 m s − and Alf´en speed v A = 3 . . km s − . The corresponding sonic Mach number M S andAlf´en Mach number M A are therefore 10.1 (1.8) and 0.52(0.10), respectively. Also, we have the compressibility β =0 . . . Recall that we have the measure PDF disper-sion σ s = 0 . ± . (see Fig. 4). The turbulence drivingparameter b can be derived from Eq. 13: b = ( e σ s − β + 1 β (15) Note in calculating total magnetic field energy, we use only the POS mag-netic field component. The actual magnetic field energy can be more sig-nificant. For strongly magnetized media the reconnection diffusion, which is theconsequence of turbulent reconnection (LV99), is important (Lazarian2005; Lazarian et al. 2012; Santos-Lima et al. 2010, 2020).
DENTIFYING THE G RAVITATIONAL C OLLAPSE IN S ERPENS WITH V ELOCITY G RADIENTS b ≈ . , which sug-gest the turbulence in Serpens G3-G6 south clump is mainlydriven by compressive force. The study in Federrath &Klessen (2012) shows that the supernova-driven turbulenceis more effective in producing compressive motions and isexpected to be more prominent in regions of enhanced stel-lar feedback. On the other hand, solenoidal motions are ex-pected to be more prominent in quiescent regions with lowstar formation activity. Therefore, intense star formation andsupernova in this region may contribute to the compressivedriving force here.4.6. Measuring mean magnetic field strength
The Planck measurement gives an overall view of the mag-netic field. However, its resolution limits our scope to smallerscales. The dispersion of polarization measures only largescale magnetic fields, so that its value may be underesti-mated. The smooth filed lines on a large scale suggest thisunderestimation is not significant.In addition to the DCF method, we also estimated the M A directly from velocity gradients using the new approach pro-posed in Lazarian et al. (2018a). It was shown that the prop-erties of velocity gradients over the sub-block is a functionof M A . In particular, the dispersion of velocity gradients’orientation was shown to exhibit the power-law relation with M A and a different power-law relation was obtained for theso-called ”Top-to-Bottom” ratio of the thin channel velocitygradients (VChGs) distribution: M A ≈ . T v /B v ) − . ± . , M A ≤ A ≈ . T v /B v ) − . ± . , M A > (16)where T v denotes the maximum value of the fitted histogramof velocity gradient’s orientation, while B v is the minimumvalue. The analytical justification of these relations is pro-vided in Lazarian et al. (2020). Here we, however, use theempirically obtained dependencies (Hu et al. 2019c). Thisnew way of obtaining Alfv´en Mach number provides us a M A distribution map for the Serpens G3-G6 south clump, asshown in Fig. 10. We find the median value of M A is 0.62for the south clump. We also repeat the analysis for the CO (2-1) emission line and find the median value M A ≈ . .We note that the difference of the evaluating M A using thenew gradient approach in Lazarian et al. (2018a) and the tra-ditional DCF method is that the new technique gets the valueof M A over an individual sub-block. Therefore we get nota single value of M A , but a distribution of magnetization of M A over the cloud image. Naturally, the new way of measur-ing the magnetization is much more informative compared tomeasuring the dispersion of the projected magnetic field overthe cloud image. This difference was demonstrated earlier inHu et al. (2019a) where for a set of molecular clouds the po-larization provided the mean value of magnetization and thedistribution of velocity gradients provided the detailed mapsof magnetization with the averaged value in good agreementwith that obtained using polarization. Figure 10.
The histogram of M A estimated from the distributionof velocity gradients over a sub-block (Lazarian et al. 2018a). Themedian values of M A are 0.62 and 0.53 for CO (1 - 0) and CO (2 - 1), respectively. In the present study, we also have a similar situation. Theaveraged value of M A estimated by the new VGT approachis close to the value that is obtained by measuring the direc-tions of polarization. The latter is M A = 0 . ± . . Thiscorrespondence is important as the technique in (Lazarian etal. 2018a) and the DCF-type measurements of M A are verydifferent both in terms of the information employed and howthis information is processed. This increases our confidencein our results.Having M A in hand one can use a new technique of mea-suring magnetic strength in Lazarian et al. (2020). The tech-nique termed there MM2, as it uses the values of two Machnumbers, the sonic one M S and the Alfv´en one M A . Usingthe relation derived in Lazarian et al. (2020) one can evaluatethe POS magnetic field as: B = Ω c s (cid:112) πρ M S M − , (17)where Ω is a geometrical factor. By adopting Ω = 1 (i.e., themagnetic field perpendicular to the LOS), M A = 0 . (i.e.,measured by VGT), M S = 10 . , ρ = 8 . × − g cm − ,and c s = 188 m s − , we get B ≈ µG , which is close tothe one ( ≈ µG ) derived from the DCF method. DISCUSSION5.1.
Tracing magnetic field direction with VGT
Measuring the magnetic field in molecular clouds is gen-erally difficult. The polarization measurement is one prac-tical approach to trace the magnetic fields. It is rooted inthe theory of Radiative Torque alignment (Lazarian & Hoang2007; Andersson et al. 2015), which predicts that the polar-ized dust thermal emission is perpendicular to the magneticfield and the polarized starlight is parallel to the magneticfields. Through the measured dispersion of dust polariza-tion’s directions and the information of spectral broadening,2 H U , L AZARIAN & S
TANIMIROVI ´ C one can estimate the magnetic field through the DCF method(Falceta-Gonc¸alves et al. 2008; Cho & Yoo 2016). However,the polarization gives only the integrated magnetic field mea-surement along the LOS.To achieve the local magnetic field measurement in molec-ular clouds, several techniques, for instance, the CorrelationFunction Analysis (CFA; Lazarian et al. 2002; Esquivel &Lazarian 2011), the Structure-Function Analysis (SFA; Huet al. 2020; Xu & Hu. 2021), and the more recent Veloc-ity Gradients Technique (VGT; Gonz´alez-Casanova & Lazar-ian 2017; Yuen & Lazarian 2017a; Lazarian & Yuen 2018a;Hu et al. 2018), the have been proposed. These techniquesare based on the anisotropy of MHD turbulence, i.e., tur-bulent eddies are elongating along their local magnetic fielddirection (Goldreich & Sridhar 1995; Lazarian & Vishniac1999). In particular for VGT, the velocity gradients of eddiesare perpendicular to their local magnetic fields so that thegradients reveal the direction of the magnetic fields. Com-pared with dust polarization, the VGT, used in this work, ex-hibits several advantages in tracing magnetic fields. Firstly,it employs the spectroscopic data to measure the molecularcloud’s local magnetic fields so that it gets rids of the con-tamination from the foreground (Lazarian & Yuen 2018a;Hu et al. 2020a; Lu et al. 2020). Using spectroscopic dataand VGT, one can achieve a higher resolution of the result-ing magnetic field map. Also, VGT can be applied to multi-ple emission lines to reveal how the magnetic field changeswith the variation of density (Hu et al. 2019b; Hu & Lazarian2021).5.2. Identify gravitational collapsing regions by VGT
Observationally identifying gravitational collapsing re-gions is notoriously challenging. The column density PDFsprovide one possible solution. In isothermal molecularclouds, the PDFs follow a log-normal distribution for non-gravitational collapsing gas. The self-gravity, however, in-troduces a power-law tail to the high-density range. Thispower-law tail statistically reveals the critical density thresh-old above which the gas becomes gravitational collapsing.In this work, we find the gravitational collapsing gas corre-sponds to density N H ≈ . × cm − in the SerpensG3-G6 clump.VGT provides an alternative way to identify the gravita-tional collapse. In turbulence dominated diffuse media, thevelocity gradients are perpendicular to the magnetic fields(Lazarian & Yuen 2018a). In the presence of gravitationalcollapse, the gravity-induced velocity acceleration changesthe velocity gradients by 90 ◦ being parallel to the magneticfields (Yuen & Lazarian 2017b; Hu et al. 2020b). This changereveals the gravitational collapsing regions. For instance,VGT confirms the G3-G6 south clump is gravitational col-lapsing here. In addition, utilizing different emission lines,VGT can tell us the volume density range in which the col-lapsing occurs. As shown in Fig. 5, the change of gradientsappears only in CO map, which means the collapse hap-pens at volume density n HI ≥ cm − . The application of VGT is not limited to nearby molec-ular clouds. Multiple molecular clouds and molecular fila-ments can be easily separated from spectroscopic PPV cubes.A complete survey of gravitational collapsing clouds andfilaments is achievable using abundant spectroscopic datasets, for instance, the JCMT (Liu et al. 2019), GAS (Kauff-mann et al. 2017), COMPLETE (Ridge et al. 2006), FCRAO(Young et al. 1995), ThrUMMS (Barnes et al. 2015), CHaMP(Yonekura et al. 2005), and MALT90 (Foster et al. 2011) sur-veys. 5.3. Comparison with density gradients
In addition to velocity gradients, column density gradientsprovide an alternative way to study the magnetic fields. Forinstance, Soler et al. (2013) used the histogram of relativeorientation (HRO) to characterize the relative orientation ofcolumn density gradients and the magnetic fields. The HROtechnique relies on the polarimetry measurement to get mag-netic field information.The HRO should not be confused with the Intensity Gradi-ent Technique (IGT) which is the outshoot of the VGT Hu etal. (2019c). The IGT can identify the direction of the mag-netic field in subsonic turbulence and and identify shocks insupersonic turbulence. Different from HRO, IGT is indepen-dent of polarimetry and keeps the spatial information of den-sity gradients by employing the sub-block averaging method,which was firstly implemented in VGT (Yuen & Lazarian2017a),Here we make a comparison of HRO and IGT using theHerschel and Planck data. A numerical comparison is pre-sented in Hu et al. (2019c). For HRO, the density gradientswere directly calculated for each pixel following the recipeproposed in Soler et al. (2013). The gradients were thensegmented according to their corresponding column densityvalue. After that, we draw the histogram of the relative an-gle θ between segmented gradients and the magnetic fieldinferred from Planck polarization. The histogram shape pa-rameter ζ = A c − A e , where A c is the area under the centralregion of the HRO curve ( π < θ < π ), and A e is the areain the extremes of the HRO ( < θ < π and π < θ < π ).As for IGT, the pixelized gradient map was further processedby the adaptive sub-block averaging method (see § 3). A sim-ilar segmentation was also implemented to IGT and we usethe AM (Eq. 11) to quantify the performance of IGT. Notehere we rotate the density gradients by 90 ◦ for IGT so thatboth positive AM and ζ indicate the non-rotated gradientsare perpendicular to the magnetic field while negative AMand ζ represent a parallel relative orientation.Fig. 11 presents the results of HRO and IGT for the Ser-pens south clump. We find both ζ and AM are positive when N H < . × cm − . ζ drops to approximately zero inthe range of . × < N H < . × cm − and For some settings we know that the turbulence is subsonic. This is thecase, for instance for clusters of galaxies (Hu et al. 2020c) where IGT wasrecently used to predict the Plane of Sky (POS) magnetic field directions.
DENTIFYING THE G RAVITATIONAL C OLLAPSE IN S ERPENS WITH V ELOCITY G RADIENTS Figure 11.
The correlation between column density N H and ζ or AM calculated from density gradients (red and blue) or velocitygradients (lime). Pink shadow indicate the range in which ζ of HROis close to zero. then drops to negative value further. The transition density . × cm − is smaller than . × cm − given bythe PDFs. The behavior of AM is similar to ζ , but the changeis more dramatic. It is contributed by the sub-block averag-ing method, which enhances the statistically most importantcomponents so that density gradients can be used to tracethe magnetic field in the diffuse region. The transition frompositive AM and ζ to negative AM and ζ is likely causedby self-gravity, as gravitational collapse also flips the densitygradients by 90 ◦ (Hu et al. 2020b). However, this change canbe caused also by shocks, which make the density gradientsmore difficult in identifying gravitational collapsing regionthan velocity gradients.By analogy to the HRO for column density gradients, Huet al. (2019c) introduce it to the study of thin channel velocitygradients, called V-HRO. Here we use the velocity gradientscalculated from CO (1-0) emission line, Herschel columndensity data, and Planck polarization data to implement theV-HRO. The velocity gradients are calculated per pixel with-out the sub-block averaging method applied and are also seg-mented based on their corresponding column density value.We find at low-density range ( N H < . × cm − ), ζ ofV-HRO is positive and is larger than ζ of HRO, which indi-cates that velocity gradients (un-rotated) have better orthog-onal alignment with the magnetic fields. At the high-densityrange, the ζ of V-HRO drops dramatically to be negative anda flip of velocity gradients’ direction occurs. As this changeof direction is insensitive to shocks and it can be amplifiedby the sub-block averaging method, we therefore can identifythe gravitational collapsing region through velocity gradients(Yuen & Lazarian 2017b; Hu et al. 2020b).5.4. The interaction of turbulence, magnetic field, andself-gravity in star formation
Star formation serves as a hot debate in modern astro-physics. The present two classes of star-formation theorydiffer in the role played by magnetic fields. Mestel & Spitzer(1956) firstly proposed the strong magnetic field model thatclouds, which are mainly regulated by the magnetic fields,are formed with sub-critical masses. The magnetic supportcan be removed if mass could move across field lines. Thisremoval is mainly conducted by neutral gas through the pro-cess of ambipolar diffusion (Mestel 1966; Spitzer 1968).The ambipolar diffusion increases the mass to flux ratio. Asa consequence, cloud envelopes are subcritical, and cloudcores are supercritical. It, therefore, leads to the low ef-ficiency of star formation. This explanation assumes fluxfreezing conditions of the magnetic field in the absence ofturbulence. However, the turbulence is ubiquitous in ISM(Armstrong et al. 1995; Chepurnov & Lazarian 2010; Xu &Zhang 2017; Zhang et al. 2020) and in the presence of tur-bulence the flux freezing does not hold anymore, as the pro-cess of fast turbulent magnetic reconnection breaks the mag-netic freezing (Lazarian & Vishniac 1999). Based on the the-ory of turbulent reconnection, Lazarian (2005); Lazarian etal. (2012); Lazarian (2014) identified reconnection diffusion(RD) as an essential process of removing magnetic flux. Con-sequently, the total magnetic field may be larger than its viralvalue, while the region can be still contracting. The recon-nection diffusion (RD) theory predictions were confirmed inSantos-Lima et al. (2010, 2012, 2013) for molecular cloudsand circumstellar disk settings. It increases the mass to fluxratio and allows the magnetic field to equalize inside and out-side the cloud, decreasing magnetic support. Unlike the am-bipolar diffusion, which depends on the ionization, the re-connection diffusion is only related to the turbulent eddies’scale and the turbulent velocities (Lazarian 2005). The weakmagnetic field model (Mac Low & Klessen 2004; Padoan &Nordlund 1999) suggests that clouds are formed at the inter-section of turbulent supersonic flows. This model requiressupercritical clouds so that the gravitational collapse can betriggered without removing magnetic support.Our analysis reveals that the Serpens G3-G6 south clumpis supersonic ( M S = 10 . ) and sub-Alfv´enic ( M A =0 . ). The ratio λ between the actual and critical mass-to-magnetic flux ratios M/ Φ is (Crutcher 2004): λ = 7 . × − N H /B ≈ . . λ < suggests the cloud is sub-critical. Also, the total magnetic field energy in the SerpensG3-G6 south clump significantly surpasses the sum of totalkinetic energy and gravitational energy (see Tab. 1), suggest-ing the magnetic field is relatively strong in this gravitationalcollapsing clump.The typical strong magnetic field model neglects the tur-bulence effect. Consequently, it is difficult to accumulateenough gas along the field to overcome the magnetic field(Mestel & Spitzer 1956). However, turbulent compression The quantitative tests of the theoretical predictions related to the violationof the flux freezing and reconnection diffusion in a turbulent fluid can befound in Eyink et al. (2013) and Santos-Lima et al. (2020) U , L AZARIAN & S
TANIMIROVI ´ C can contribute to material accumulation. Since the strongmagnetic fields provide pressures perpendicular to the fieldlines, the compression preferentially follows the magneticfields and accumulates material along the perpendicular di-rection. Once the collapse is triggered, the inflows also pre-dominately move along the field lines. Consequently, thegravitationally bound cloud will be thin oblate spheroids, aswe see in Fig. 2. Our results suggest compressive turbulencein the sub-critical Serpens G3-G6 south clump. It supportsthe turbulent, strong magnetic field model.Another observation of the Serpens south region’s mag-netic field suggests that the magnetic supercriticality sets invisual extinctions larger than 21 mag , which also indicatesthat gravitational collapse can be triggered in a strongly mag-netized environment (Pillai et al. 2020).5.5. Uncertainty and robustness
In this work, we use the DCF method to estimate the totalmagnetic field strength. The DCF method assumes that: (i)ISM turbulence is an isotropic superposition of linear small-amplitude Alfv´en waves; (ii). turbulence is homogeneousin the region studied; (iii). the compressibility and densityvariations of the media are negligible; (iv). the variationsof the magnetic field direction and the velocity fluctuationsarise from the same region in space. These assumptionscould raise uncertainties in our estimation. Nevertheless, thedirect measurement from Zeeman splitting reveals that theLOS magnetic field strength is B ≈ µG for H I volumedensity ≈ × cm − (Crutcher 2012). As the mag-netic field is amplified in gravitational collapsing region, weexpect the uncertainty in our estimated POS magnetic fieldstrength B ≈ µG from DCF method and B ≈ µG from MM2 method are not significant.The measured velocity dispersion includes the contributionfrom both turbulent velocity and shear velocity. In our cal-culation, we did not separate the two components. As dis-cussed in § 4.5, the ratio between kinetic energy and mag-netic energy is independent of the velocity dispersion. Theuncertainty here mainly comes from polarization measure-ment. The velocity dispersion gives contributions to the ratioof magnetic energy and gravitational energy. However, theoverestimated velocity dispersion is unlikely to compensatefor the difference of eight orders of magnitude.Also, we assume the gas temperature T (cid:39) K. This as-sumption introduces uncertainties to the calculation of soundspeed and sonic Mach number. The gas temperature can beroughly estimated from the optically thick CO (2-1) emis-sion line (Pineda et al. 2010; Kong et al. 2015): J ν ( T ) = hν/k B exp( hν/ ( k B T )) − T = hν k B [log(1 + hν /k B T b + J ν ( T bg ) )] − = 11 . . T b + 0 .
194 )] − (18) where h is the Planck constant, ν is the emission frequencyof CO (2-1), k B is the Boltzmann constant, T b is the peakintensity of CO (2-1) in units of K, J ν ( T ) is the effectiveradiation temperature, and T bg = 2.725 K is the temperatureof cosmic microwave background radiation. By using a rangeof observed T b values, we estimate T ranges from 3.7 K to14.3 K, with a median value 7.1 K. A specie at transitionlevel J = 1-0 is brighter than the one at J = 2-1, so it wouldcorrespond to higher temperature. We therefore, expect thatthe assumption T = 10 K does not change significantly ourconclusions. CONCLUSIONThe energy balance between turbulence, magnetic fields,and self-gravity is still obscured in the process of star for-mation. In this work, we target the Serpens G3-G6 clump tostudy its kinetic, magnetic, and gravitational properties. Weemploy several data sets, including CO isotopologs’ emis-sion lines from the HHT and IRAM telescopes, H columndensity data from the Herschel Gould Belt Survey, and dustpolarization data from Planck 353 GHz measurement. Ourmain discoveries are:1. Using the column density PDFs and the VGT method,we confirm that the Serpens G3-G6 south clump isgravitational collapsing, which is in agreement withthe observed high surface density of YSOs.2. The VGT method reveals that the gravitational col-lapse in the Serpens G3-G6 south clump occurs at vol-ume density n HI ≥ cm − .3. We confirm that the VGT method can trace the mag-netic fields and identify gravitational collapsing re-gions independently of polarization measurements.4. We use the traditional DCF method and the new MM2technique to calculate the plane-of-the-sky magneticfield strength of the Serpens G3-G6 south clump.The magnetic field strengths estimated using the DCFmethod and the MM2 method are approximately 120 µG and 100 µG , respectively.5. We find that the magnetic fields energy dominates theenergy budget of the supersonic Serpens G3-G6 southclump, and that the turbulence is mainly driven bycompressive forces.6. We conclude that the gravitational collapse can be suc-cessfully triggered in a supersonic and sub-Alf´enic en-vironment.ACKNOWLEDGEMENTSA.L. acknowledges the support of the NSF grant AST1816234 and NASA ATP AAH7546. Y.H. acknowledges thesupport of the NASA TCAN 144AAG1967. Flatiron Insti-tute is supported by the Simons Foundation. This work isbased on observations carried out under project number 115-19 with the IRAM 30m telescope. IRAM is supported by DENTIFYING THE G RAVITATIONAL C OLLAPSE IN S ERPENS WITH V ELOCITY G RADIENTS
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