Measuring Turbulence with Young Stars in the Orion Complex
DDraft version Tuesday 12 th January, 2021
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Measuring Turbulence with Young Stars in the Orion Complex
Trung Ha, Yuan Li,
1, 2
Siyao Xu, ∗ Marina Kounkel, and Hui Li
5, 6, ∗ Department of Physics, University of North Texas, Denton, TX 76203, USA Department of Astronomy, University of California, Berkeley, CA 94720, USA Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA Department of Physics and Astronomy, Western Washington University, 516 High St, Bellingham, WA 98225, USA Department of Physics, Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA02139, USA Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA
ABSTRACTStars form in molecular clouds in the interstellar medium (ISM) with a turbulent kinematic state.Newborn stars therefore should retain the turbulent kinematics of their natal clouds. Gaia DR2 andAPOGEE-2 surveys in combination provide three-dimensional (3D) positions and 3D velocities ofyoung stars in the Orion Molecular Cloud Complex. Using the full 6D measurements, we compute thevelocity structure functions (VSFs) of the stars in six different groups within the Orion Complex. Wefind that the motions of stars in all diffuse groups exhibit strong characteristics of turbulence. Theirfirst-order VSFs have a power-law exponent ranging from ∼ . − . INTRODUCTIONThe interstellar medium (ISM) is turbulent. Morespecifically, the cold dense star-forming molecular cloudsare turbulent. For example, Larson (1981) finds a powerlaw relation between velocity dispersion and cloud sizewith an exponent of 0.38, similar to the Kolmogorov lawfor incompressible turbulence (Kolmogorov 1941). Tur-bulence plays an important role in regulating the clouddynamics and star formation (Mac Low & Klessen 2004;McKee & Ostriker 2007; Hennebelle & Falgarone 2012),from cloud scales all the way down to the scales muchsmaller than molecular cores (Hull et al. 2017).The ubiquitous interstellar turbulence induces veloc-ity fluctuations on all scales in both diffuse interstel-lar phases and dense star-forming gas. Statistical mea-surements of velocity fluctuations in the multi-phase [email protected] ∗ NASA Hubble Fellow
ISM can be done using the Velocity Channel Analy-sis (VCA), the Velocity Coordinate Spectrum (VCS)(Lazarian & Pogosyan 2000, 2006), the Delta-variancetechnique (Stutzki et al. 1998; Ossenkopf & Mac Low2002), or the Principal Component Analysis (Heyer &Brunt 2004; Roman-Duval et al. 2011) based on spectro-scopic data. These analyses generally reveal power-lawspectra of turbulent velocities (Lazarian 2009a; Chep-urnov et al. 2010).Many processes can drive turbulence in the ISM, suchas gravity, shear, and various forms of stellar feedbackincluding supernova explosions (Krumholz & Burkhart2016; Padoan et al. 2016). Turbulent motions shapethe filamentary density structures in molecular clouds(e.g., Federrath 2016; Xu et al. 2019). Dense cores andprotostars subsequently form in dense filaments (Andr´e2017).Besides gas tracers that are commonly used to probeturbulence in the volume filling ISM, turbulent velocitiescan also be sampled using point sources such as molecu- a r X i v : . [ a s t r o - ph . GA ] J a n lar cores formed at the density peaks generated in highlysupersonic turbulence. For example, Qian et al. (2012)developed statistical measurements of core-to-core ve-locity dispersion over a range of length scales , i.e., thecore velocity dispersion technique. Qian et al. (2018)find signatures for a turbulent power-law spectrum inthe Taurus Molecular Cloud, and demonstrate that thestatistics of turbulent velocities are imprinted in the ve-locities of dense cores (Xu 2020).Because prestellar cores are nurseries of stars, new-born stars should naturally inherit the velocities of coresand hold the fossil information on the turbulent veloci-ties of their natal clouds, as long as dynamical interac-tions of young stars have not erased that information.In this work, we carry out the first statistical analysisof star-to-star velocity structure function (VSF) acrossthe Orion complex over a range of length scales, from ∼
200 pc down to < DATA PROCESSING2.1.
Data Acquisition
We obtain the locations, parallax distances, andproper motions of stars in the Orion Molecular CloudComplex from Gaia’s second data release (Gaia Collab-oration et al. 2018). The Gaia catalog includes five-parameter astrometric data from over 1.3 billion stars.The line-of-sight velocities of the stars were observed inthe near infrared during the Apache Point ObservatoryGalactic Evolution Experiment 2 (APOGEE-2). TheAPOGEE spectrograph is mounted on the 2.5 m SloanFoundation Telescope of the Sloan Digital Sky Survey(SDSS), and is used to collect high resolution spectralimages of over 400,000 stars in the Milky Way (Gunnet al. 2006; Blanton et al. 2017). In total, 8891 starsfrom APOGEE-2, and 16754 stars from Gaia DR2 wereobserved towards the Orion Complex, as specified by
Figure 1.
Foreground: each dot represents the projectedposition of a star in the Orion Complex that we include inour analysis. The star’s color corresponds to its main group.Background: greyscale optical image of the Orion MolecularCloud Complex, courtesy of Rogelio Bernal Andreo.
Kounkel et al. (2018). Combining these two surveys pro-vides six-dimensional information of a subset of stars.Kounkel et al. (2018) developed a hierarchical cluster-ing algorithm to assess membership of the Orion Com-plex and its structure based on the position of stars inthe phase space (position+velocity). They identify fivemain groups inside the Complex, namely Orion A, OrionB, Orion C, Orion D, and λ Ori. We require all stars inour sample to be a part of one of these groups, exclud-ing all of the stars that could not be clustered together(i.e., field stars, or other members of Orion that mayhave peculiar velocity - such as spectroscopic binariesor runaways). Furthermore, we require a complete in-formation on six-dimensional position and velocity foreach star, i.e, a star needs to have both astrometric so-lutions from Gaia DR2 as well as radial velocities fromAPOGEE. Through this, we obtain a sample of 1439stars.We note that the Orion Nebula Cluster (ONC) is apart of the Orion A molecular cloud, however, it is veryunique, being a singularly most massive young cluster inthe solar neighborhood. The environment that is foundin the ONC is different from the more diffuse environ-ment in the southern part of Orion A. Thus, for thepurpose of our analysis, we separate ONC and the restof the Orion A. Throughout the paper, Orion A refersonly to the “tail” of the cloud (L1641 and L1647), ex-cluding stars in ONC. The reasoning behind this splitis discussed in Section 3. The term “group” in this pa-per refers to all “stellar systems” that are spatially andkinematically associated, while “cluster” only refers toopen clusters such as the ONC.Table 1 lists the number of stars and the mean age ofeach main group and the ONC. We plot the projectedpositions of the stars in our analysis over a greyscaleimage of the Orion Complex in Figure 1.2.2.
Method of analysis
To test our hypothesis that young star clusters “re-member” their local gas kinematics at the time of birth,we compute the VSFs for all identified groups, as wellas the VSF of the combined Orion Complex catalog,both with and without the ONC. We first convert theradial velocity and proper motion of stars into velocityin Cartesian coordinate, in the Local Standard of Rest(LSR). Figure 2 shows the 3D velocity information ofstars used in this study in each cloud.For each cloud, we compute the first-order VSF andits uncertainties. The VSF for a sample of stars can becalculated in the following way: for each pair of stars,we first record the physical separation (cid:96) of the pair andcompute the magnitude of the vector velocity difference δ(cid:126)v as the velocity difference | δv | . We then compute theaverage of the velocity differences (cid:104)| δv |(cid:105) within logarith-mic bins of (cid:96) .The VSF is related to the kinetic energy power spec-trum of turbulence. It describes how velocity difference( | δv ij | ) relates to the physical separation ( (cid:96) ij ) for givenpairs of points in space ( i , j ). In a turbulent flow, ki-netic energy cascades down from large scales to smallscales, and the velocity differences become smaller to-ward smaller scales. If the turbulence is in-compressible(subsonic), we expect the first-order VSF to be | δv ij | ∼ (cid:96) / ij within the inertial range (Kolmogorov 1941). Forcompressible (supersonic) turbulence, | δv ij | ∼ (cid:96) / ij (e.g.,Federrath 2013).To estimate uncertainties of the VSF, we first per-form random sampling of the measurements of stars’properties based on the observed uncertainties, and ob-tain 1000 realizations for each star following a Gaussiandistribution. We only consider measurement uncertain-ties of the parallax distances, proper motions, and radialvelocities. We ignore the uncertainties in RA and Decbecause they are typically much smaller than the un-certainties of other quantities. For each group, we thenobtain 1000 VSFs, with each iteration excluding a ran- dom star within the group. This allows us to accountfor both measurement errors and uncertainties due tosmall sampling sizes in some bins of (cid:96) . We take themean and standard deviation of the computed VSFs tobe the group’s VSF and its corresponding error. Wechoose 1000 iterations to ensure accuracy even thoughthe results are already converged at 100 iterations.We also compute the second and third-order VSFs ofthe groups, and discuss them in the Appendix. RESULTSFigure 3 shows the first-order VSF of stars in all sixindividual groups in the Orion Complex. The last twopanels show the combined VSF of all groups without andwith the ONC, respectively. To guide the eye, we alsoplot in each panel the expected scaling for supersonicand subsonic turbulence, along with Larson’s law mea-sured for turbulence in molecular clouds in the MilkyWay (Larson 1981).The VSFs of all six individual groups except ONCexhibit a power-law scaling, characteristic of a turbulentflow. The amplitudes and slopes vary from group togroup. The best-fit VSF slopes are listed in Table 1.The Orion A group (top left panel of Figure 3) islocated just outside of Barnard’s Loop (the shell-likefeature to the left of Orion A-D in Figure 1, see alsoOchsendorf et al. (2015)), which is proposed to haveoriginated from a supernova that erupted ∼ σ Ori, adense cluster of age ∼ Figure 2.
Velocities of stars in the six clouds in our analysis in Cartesian coordinate. The color map denotes stars’ velocity inthe y direction, which closely resembles their radial velocity. All clouds show some level of randomness in the velocities of thestars.
Table 1.
Properties of molecular clouds in the Orion ComplexOrion A Orion B Orion C Orion D Lambda Ori ONCNumber of Stars a
117 51 162 354 170 585Median Age (Myr) 2.2 1.3 4.1 4.9 3.7 2.7Crossing time (Myr) 16.1 13.6 8.0 12.9 12.8 6.9 b Fitting Range c (pc) 10 - 90 8 - 60 9 - 50 15 - 58 10 - 32 5 - 40Slope of VSF 0 .
18 0 .
14 0 .
19 0 .
53 0 .
36 0 . dense clusters on the VSF in more detail later in thissection.Stars in the Orion D group exhibit a curious VSF(middle left panel), with a slope more in line with su-personic turbulence (slope 1 /
2) than Kolmogorov at dis-tance scale 10 < (cid:96) <
60 pc and absolute velocity differ-ence higher than that of Larson’s law. We attributethese properties of Orion D’s VSF to its position insidethe supernova bubble. Located close by and at the front of the bubble (in our line-of-sight) as it expands, motionof stars in the Orion D cloud is additionally driven bythe shock wave of the supernova, hence its turbulentkinetic energy is amplified.The last independent group we look at is λ Ori (cen-ter panel of Figure 3). In the 10 < (cid:96) <
32 pc range, theVSF of stars in λ Ori closely exhibits the 0.38 slope ofLarson’s law. The amplitude also exactly follows Lar-son’s relation (Larson 1981), and is higher than Orion
Figure 3.
Left to right, top to bottom: first-order VSF of stars in the Orion A, Orion B, Orion C, Orion D, Lambda Ori,ONC, all Orion without ONC, and all Orion with ONC. The error bars are obtained through a random sampling analysis, andrepresent the uncertainties from both observational errors and uncertainties due to small sample size. As reference, we also plota black dot-dashed line of slope 1/3 for the Kolmogorov turbulence, a pink dashed line of slope 1/2 for supersonic turbulence,and a solid blue line for Larson’s velocity dispersion correlation for molecular clouds: σ ( km · s − ) = 1 . · L ( pc ) . (Larson 1981).All panels use 40 logarithmic bins of (cid:96) from 1 pc to 170 pc. A and B, but lower than C and D. This result can beunderstood given λ Ori’s history and isolated positionin the Orion Complex. The λ Orion bubble (the bubble-like structure surrounding λ Ori in Figure 1) is proposedto have been created by a supernova explosion ∼ ∼ λ Ori where the stellar age has awide spread over ∼ ∼ τ r ∼ ∼ < (cid:96) <
60 pc scalerange. This is because D has the highest number ofstars amongst the 5 groups (Table 1).When the ONC is included, on small scales (2 < (cid:96) <
20 pc), the VSF flattens because it is heavily influencedby the cluster due to its dominant number of stars inthe catalog (585 stars vs. a total of 1439 stars). In fact,this is why we separate the ONC from Orion A in ouranalysis. If we include ONC in Orion A, the resultingVSF is significantly flatter. All the other star groups alsocontain star clusters. For example, Orion C has σ Ori.We do not separate these clusters because they are muchsmaller than the ONC, and removing them degrades thestatistics of our analysis. We note, however, that theymay have contributed to the flatter VSF of Orion C andthe flattening of the VSF in the other groups on smallscales. We discuss other uncertainties in more detail inSection 4.The combined VSF shows a bump at ∼
60 pc, similarto Orion D, suggesting energy injection at that scale.The mean age of the stars in this study is ∼ . ∼
80 pc)and its age ( ∼ DISCUSSIONS4.1.
Biases and Uncertainties
Gas in molecular clouds such as the Orion Complexis turbulent by nature. The theoretical foundation ofour analysis is that newborn stars retain this turbulenceinformation. However, after the stars are born, they de-couple from the hydro-dynamical forces of the gas, and no longer follow the turbulent flow. In other words, starsare only perfect tracers of turbulence at birth. Since starformation happens over a period of time, all of the stargroups contain stars that are over a few Myr old. Wehave identified two mechanisms through which stars canlose their memory of the ISM turbulence, causing biasesin our analysis.The first mechanism is dynamical relaxation, which wediscussed in the analysis of the ONC in Section 3. Dueto open clusters’ high density, close stellar encountersare common, which alter the stars’ original orbits andlead to dynamical relaxation. The relaxation time isestimated as a function of the number of stars and thecrossing time of the cluster: t relax = t cross · N · ln N/ , (1)Where N is the total number of stars in the cluster and t cross is the crossing time, defined as the typical time fora star to travel a distance equal to the half-mass size ofthe cluster: t cross = 2 · R hm σ v . (2)Here, R hm is the radius within which half of the cluster’sstars reside, and σ v is the bulk-motion-removed charac-teristic velocity of the cluster’s stars. It is computed asthe standard deviation of the velocity distribution withrespect to the median σ v ≡ σ [ | v i − v median | ].After the stars are relaxed, they no longer hold theturbulent kinematics of gas at the time of their forma-tion. For loose groups of stars such as those we ana-lyze, the crossing times are significantly longer than thegroup’s current age (Table 1). Furthermore, since eachgroup possesses a number of stars in the order of N ∼ (eg. N = 2800 stars within the dense core of the ONCwas used by Hillenbrand & Hartmann (1998)), the esti-mated relaxation time is an order of magnitude longerthan the crossing time. Therefore, dynamical relaxationhas most likely not affected the shape of the VSF forthe five groups in the Orion Complex, except for theONC. This is in agreement with Da Rio et al. (2017)who find that young stars in Orion A are kinematicallyassociated with the molecular gas but the ONC appearsmore dynamically evolved and virialized. Kounkel et al.(2018) also found consistent kinematics between youngstars and gas in the whole Orion Complex where the gashas not been dispersed.In addition to the ONC, there are other clusters withineach group: σ Ori in Orion C, 25 Ori in Orion D, λ Orionis Cluster in λ Ori, and two small clusters NGC2024 and NGC 2068 in the Orion B group. These clus-ters have much smaller physical sizes than the ONC andcontain only a small fraction of stars of the group. Thisproperty of the clusters can be visually confirmed fromFigure 1. Thus, their existence does not significantlyaffect our overall results. However, they may have con-tributed to the flattening of the VSFs on small scalesthat we observe in almost all individual groups.Another potential source of bias in our analysis comessimply from the drifting of stars from their original po-sitions. Assuming no strong stellar encounters, starswith higher velocity differences drift apart at a fasterrate than stars with smaller velocity differences, causingthe VSF to steepen with time. This steepening happensover roughly t cross .In our analysis, most groups are relatively young, withtheir ages much shorter than t cross (Table 1), so drift-ing should not have had time to erase turbulence-drivenvelocity differences. Although drifting may have steep-ened the VSF of Orion D, which has the oldest medianage and the steepest slope amongst all the groups an-alyzed here. We plan to further explore the effects ofrelaxation and drifting in our future works both with alarger observed sample and with numerical simulations(Li et al. 2019).There are also other sources of uncertainties in ouranalysis, such as binary systems. While the barycentershould follow the turbulent velocity of the natal cloud,binaries also have orbital motion contributing to theoverall velocity measurement. While most systems withvery large orbital speeds have been excluded from theanalysis through the initial hierarchical clustering, sys-tems with orbital speeds of only a few km s − likelyremain.False positives in hierarchical clustering, such as con-tamination from the field stars can also affect our results.Kounkel et al. (2018) estimated that false positive frac-tion for the clustering algorithm can be as high as ∼ (cid:96) .4.2. Comparison with other Methods to MeasureTurbulence
Most of the previous studies on turbulence in astro-physical environments rely on the observations of gas.Although only velocity statistics can directly reflect thedynamics of turbulence and turbulent energy cascade, itis the density statistics that is widely used for measuringturbulence as it is much more easily accessible to obser-vations. For example, the delta-variance method hasbeen used to analyze the structure of observed molecu- lar cloud images in the ISM (Stutzki et al. 1998). Thesame method and a modification of it (Ar´evalo et al.2012) have also been used to infer turbulence in the hotintra-cluster medium based on X-ray images (e.g., Zhu-ravleva et al. 2014).Velocity centroids can recover the turbulent veloc-ity spectra in subsonic turbulence (Esquivel & Lazarian2005). The velocity channel analysis (VCA) using chan-nel maps and velocity coordinate spectrum (VCS) us-ing the fluctuations measured along the velocity axis ofthe Position–Position Velocity (PPV) cubes can also beused to extract turbulent velocity spectra in supersonicturbulence (see Lazarian (2009b) for a review). In par-ticular, the VCS technique only requires measurementsalong a few directions to obtain a reliable velocity spec-trum (Chepurnov & Lazarian 2009). Spectroscopic datahas also been used in the Principal Component Analy-sis to extract the size-line width relation and recover thestructure function within molecular clouds (e.g., Heyer& Brunt 2004; Roman-Duval et al. 2011).Recently, statistical measurements of the velocities ofgas phase point sources have been developed for, e.g.,dense molecular cores embedded in the background tur-bulent flow (Qian et al. 2012) and filaments in the cen-ters of galaxy clusters (Li et al. 2020). These analysesuse the 1D line-of-sight velocities and 2D positions pro-jected on the plane of the sky. The bias due to the pro-jection effect depends on the thickness of the structure(Qian et al. 2015), which is usually unknown.Our analysis in spirit is similar to that of gas phasepoint sources, but instead of using projected informa-tion, we take advantage of the full 6D (3D velocity +3D position) information of stars. Thus our results donot suffer from the poorly constrained projection uncer-tainties. Observing 6D information of the gas, especiallythe motions in the plane of the sky, is prohibitively chal-lenging (although see O’Dell & Henney 2008). Thus ourmethod has a unique advantage, and can potentially beused to help better understand the biases in the analysesbased on gas.4.3.
Comparison with Theories and OtherObservations
There has been a lot of previous work on turbulencein Orion based on the observations of emission lines ofthe ionized gas (Castaneda 1988; O’dell & Wen 1992;McLeod et al. 2016; Arthur et al. 2016). Most of themare focused on small scales ( < ∼
100 pc), we have com-pared our results with Larson’s Law in Section 3. In-dividual groups may show slightly higher or lower am-plitude than the mean Larson’s relation, with slightlysteeper or shallower slopes, but overall, our results are ingood agreement with Larson’s Law for molecular cloudsin the Milky Way. Notably, Larson (1981) also finds aslightly lower-than-average amplitude for Orion B, con-sistent with our findings here even though we are usingdifferent tracers of turbulence.Our results show that regions heavily influenced bysupernova explosions can exhibit a higher level of tur-bulence, and even a steeper VSF. This is consistentwith numerical simulations showing that supernovae candrive turbulence in molecular clouds (Padoan et al.2016). Additionally, the steepened slope of Orion D isconsistent with Federrath (2013) finding that supernovaexplosions excite more compressible than solenoidal ve-locity modes in the gas, which tends to steepen theVSFs, and thus exhibit the 1/2 power law of super-sonic turbulence. Observations have also shown the lo-cal effects of supernovae on turbulence in other systems(e.g., in the Large Magellanic Clouds (Szotkowski et al.2019)). CONCLUDING REMARKS AND FUTUREWORKWe have introduced a new method to study turbu-lence in the ISM using motions of young stars. Ourresults demonstrate that stars in young star groups re-tain the memory of turbulence of their natal molecu- lar clouds. Our analysis uses the full 6D informationof stars to trace turbulence, and does not suffer fromprojection uncertainties. Our results can be used to pro-vide independent constraints on the turbulent propertiesof molecular clouds in addition to analysis of gas kine-matics, and also shed light on studies of the dynamicalevolution of star clusters in the Galaxy (Kamdar et al.2019).We plan to apply our analysis to a much larger sampleof star groups in the future. Meanwhile, we will performand analyze tailored numerical simulations of star for-mation in molecular clouds to better understand biasesand uncertainties in our analysis, as well as constrainingsubgrid models for star formation (Li et al. 2019).ACKNOWLEDGMENTSWe would like to thank Anna McLeod and Yuan-Sen Ting for helpful discussions. This work was partlyperformed at the Aspen Center for Physics, which issupported by National Science Foundation grant PHY-1607611. HL is supported by NASA through theNASA Hubble Fellowship grant HST-HF2-51438.001-Aawarded by the Space Telescope Science Institute, whichis operated by the Association of Universities for Re-search in Astronomy, Incorporated, under NASA con-tract NAS5-26555. S.X. acknowledges the support pro-vided by NASA through the NASA Hubble Fellowshipgrant H , CO , were found in numerical simulations of molecular clouds, which are related to theirdifferent volume filling factors and spatial distribution (Bertram et al. 2015). The second and third order VSFs thatwe present in Figure 4 are generally consistent with the Kolmogorov scaling. This finding agrees with the measurementusing the velocities of dense cores in Taurus cloud (Qian et al. 2018). It suggests that both the dense cores and thestars used here sample the turbulence in the relatively diffuse region with a large volume filling factor, and thus theKolmogorov scaling is expected (Xu 2020). REFERENCES Andr´e, P. 2017, Comptes Rendus Geoscience, 349, 187,doi: 10.1016/j.crte.2017.07.002Ar´evalo, P., Churazov, E., Zhuravleva, I.,Hern´andez-Monteagudo, C., & Revnivtsev, M. 2012,MNRAS, 426, 1793,doi: 10.1111/j.1365-2966.2012.21789.x Arthur, S. J., Medina, S. N. X., & Henney, W. J. 2016,MNRAS, 463, 2864, doi: 10.1093/mnras/stw2165Bertram, E., Konstandin, L., Shetty, R., Glover, S. C. O.,& Klessen, R. S. 2015, MNRAS, 446, 3777,doi: 10.1093/mnras/stu2372
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