Turbulence and star formation efficiency in molecular clouds: solenoidal versus compressive motions in Orion B

0.84 MB·

by  Jan H. Orkisz

Astronomy&Astrophysicsmanuscriptno.turbulence-in-OrionB c ESO2017
(cid:13)
January5,2017
Turbulence and star formation efficiency in molecular clouds:
⋆
solenoidal versus compressive motions in OrionB
JanH.Orkisz1,2,3,Je´roˆmePety2,3,Maryvonne Gerin3,EmericBron4,8,VivianaV.Guzma´n6,7,Se´bastienBardeau2,
JavierR.Goicoechea8,PierreGratier5,FranckLePetit4,Franc¸oisLevrier3,HarveyLiszt9,KarinO¨berg6,Nicolas
10 4 11 12
Peretto ,EvelyneRoueff ,AlbrechtSievers ,andPascalTremblin
1 Univ.GrenobleAlpes,IRAM,F-38000Grenoble,France
7 2 IRAM,300ruedelaPiscine,F-38406SaintMartind’He`res,France
1 3 LERMA,Observatoire deParis,PSLResearchUniversity,CNRS,SorbonneUniversite´s,UPMCUniv. Paris06,E´colenormale
0 supe´rieure,F-75005,Paris,France
2 4 LERMA,ObservatoiredeParis,PSLResearchUniversity,CNRS,SorbonneUniversite´s,UPMCUniv.Paris06,F-92190,Meudon,
France
n
5 Laboratoired’astrophysiquedeBordeaux,Univ.Bordeaux,CNRS,B18N,alle´eGeoffroySaint-Hilaire,33615Pessac,France
a
6 Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA,02138,USA
J
7 JointALMAObservatory(JAO),AlonsodeCordova3107Vitacura,SantiagodeChile,Chile
4 8 ICMM,ConsejoSuperiordeInvestigacionesCientificas(CSIC).E-28049.Madrid,Spain
9 NationalRadioAstronomyObservatory,520EdgemontRoad,Charlottesville,VA,22903,USA
] 10 SchoolofPhysicsandAstronomy,CardiffUniversity,Queen’sbuildings,CardiffCF243AA,UK
A
11 IRAM,AvenidaDivinaPastora,7,Nu´cleoCentral,E-18012Granada,Espan˜a
G 12 MaisondelaSimulation,CEA-CNRS-INRIA-UPS-UVSQ,USR3441,Centred’e´tudedeSaclay,F-91191Gif-Sur-Yvette,France
.
h
p
- ABSTRACT
o
r Context.Thenatureofturbulenceinmolecularcloudsisoneofthekeyparametersthatcontrolstarformationefficiency:compressive
t
s motions,asopposedtosolenoidalmotions,cantriggerthecollapseofcores,ormarktheexpansionofHiiregions.
a Aims.Wetrytoobservationallyderivethefractionsofmomentumdensity(ρv)containedinthesolenoidalandcompressivemodesof
[
turbulenceintheOrionBmolecularcloudandrelatethesefractionstothestarformationefficiencyinthecloud.
1 Methods.TheimplementationofastatisticalmethoddevelopedbyBrunt&Federrath(2014),appliedtoa13CO(J=1 0)datacube
−
obtainedwiththeIRAM-30mtelescope,allowsustoretrieve3-dimensionalquantitiesfromtheprojectedquantitiesprovidedbythe
v
observations,yieldinganestimateofthecompressiveversussolenoidalratioinvariousregionsofthecloud.
2
6 Results.DespitetheOrionB molecular cloud beinghighly supersonic (meanMach number 6), thefractions of motion ineach
∼
modedivergesignificantlyfromequipartition.Thecloud’smotionsareonaveragemostlysolenoidal(excess> 8%withrespectto
9
equipartition), which is consistent with its low star formation rate. On the other hand, the motions around the main star-forming
0
regions(NGC2023andNGC2024)provetobestronglycompressive.
0
. Conclusions.Wehavesuccessfullyappliedtoobservationaldataamethodthatwassofaronlytestedonsimulations,andhaveshown
1 thattherecanbeastrongintra-cloudvariabilityofthecompressiveandsolenoidalfractions,thesefractionsbeinginturnrelatedto
0 thestarformationefficiency.Thisopensanewpossibilityforstar-formationdiagnosticsingalacticmolecularclouds.
7
1 Keywords.Turbulence-Methods:statistical-ISM:clouds-ISM:kinematicsanddynamics-Radiolines:ISM-ISM:individual
: object:OrionB
v
i
X
1. Introduction as the major mechanism that shapes the clouds: their fractal
r
a geometry is related to the propertiesof their turbulentvelocity
The evolution of molecular clouds is controlled by a com-
field (Pety&Falgarone 2000; Federrathetal. 2009). The
plex interplay of large-scale phenomena and microphysics:
dissipation time scale for the turbulent energy of a molecular
chemistry and interaction of the matter with the surrounding
cloudisshorter( 1Myr,MacLow1999)thantheageofsuch
far-UV and cosmic-ray radiation control the thermodynamic ∼
clouds ( 20 30Myr, Larson 1981). Hence, a continuous
state of the gas and its coupling to the magnetic field. The ∼ −
energyinjectionmustexist(Hennebelle&Falgarone2012,and
medium is highly turbulent, with Reynolds numbers reaching
references therein). The proposed injection mechanisms may
107 and magnetic Reynolds numbers reaching 104 (Draine
be either external, for instance Galactic shear or nearby super-
2011). Magneto-hydrodynamic (MHD) turbulence is one of
novae explosions (Kim&Ostriker 2015), or internal, like the
the main counter-actions to gravity (Hennebelle&Falgarone expansionof Hiiregionsandmolecularoutflowsoftherecently
2012; Federrath&Klessen 2012; Padoanetal. 2014), as well
formed stars (Hennebelle&Falgarone 2012). The nature of
turbulence, and notably its solenoidal or compressive forcing,
⋆ Based on observations carried out at the IRAM-30m single-
also playsa keyrole in the star formationefficiency(hereafter,
dish telescope. IRAM is supported by INSU/CNRS (France), MPG
SFE) of molecular clouds (Federrath&Klessen 2012, 2013).
(Germany)andIGN(Spain).
1
JanH.Orkiszetal.:TurbulenceandstarformationefficiencyinOrionB
In particular, compressive motions bear the mark of various
phenomena related more or less directly to the formation of
stars: infall on filaments, collapsing dense cores, expansion
around young stars... As a result, we can expect that a more
“compressive” cloud is a more “active” cloud, probably more
likely to be formingstars, as proposedby Federrath&Klessen
(2012).
Inthiswork,weproposetomeasureforthefirsttime(toour
knowledge)therelativefractionsofmomentuminthesolenoidal
andcompressiblemodesofturbulenceinamolecularcloud,fol-
lowingamethoddevisedandtestedonnumericalsimulationsby
Bruntetal.(2010)andBrunt&Federrath(2014).Ourgoalisto
obtainaquantitativeestimationofthesefractionsfromobserva-
tionaldata,andtocomparetheirratiowiththestarformationef-
ficiency,derivedfromindependentdata.Weinvestigatewhether
differentfractionsofcompressiveandsolenoidalmotionsmight
providea diagnosticforthe variationofthe star formationeffi-
ciencyamongmolecularclouds.Adifferentproportionofcom-
pressiveforcingmightbethereasonwhysomemolecularclouds
form stars at a high rate while others do not (Federrathetal.
2010;Federrath&Klessen2012;Renaudetal.2014).
Ourobjectofstudyisalargeregionofanearbygiantmolec-
ularcloud,namelythesouth-westernedgeoftheOrionBcloud Fig.1.H columndensitymapofthe south-westernpartofthe
2
(Barnard33orLynds1630).OrionBisrelativelyclosetous,ata OrionB giant molecular cloud, derived from Herschel Gould
typicaldistanceof 400pc(Mentenetal.2007;Schlaflyetal. Belt Survey observations (Andre´etal. 2010; Schneideretal.
∼
2014),sothataspatialresolutionof25 correspondsto0.05pc 2013).ThefieldobservedbytheOrionBcollaborationandused
′′
or104AUinthecloud.ThetotalmassofOrionBisestimatedto forthisworkisoverlaidinred.Thebluesquaresmarkthenebu-
be7 104M (Lombardietal.2014),andtheaverageincident laeNGC2024,NGC2023andtheHorsehead,andthepinkstar
FUV×radiatio⊙n field is G 45 (Petyetal. accepted). OrionB symbolsmarkthe starsAlnitak,HD38087andσOri(Northto
0
∼
is located in the Orion GMC complex (Krameretal. 1996; South).
Rippleetal. 2013), east of the famous OrionBelt. Alnitak,
the eastmost of the three belt stars shines in the foregroundof
the cloud. The south-western edge of the cloud represents an efficiencyinOrionB.IntheAppendix,wepresentourcomputa-
ideal laboratory to study star formation, and features several tionoftheMachnumbermapinthecloud.
remarkable regions. First, the cloud is illuminated by the
massive star σOri that creates an Hii region, the emission
nebula IC434, bounded on its eastern side by an ionization 2. Observations
front. Silhouetted against this bright background,a dark cloud
2.1.TheOrionBprojectdataset
can be seen: the famous Horsehead nebula. HD38087 also
creates a small Hii region, IC435. Still embedded in OrionB, The OrionB project (PI: J. Pety) has already mapped with the
thestar-formingregionNGC2024,knownastheFlameNebula, IRAM-30m telescope the south-western edge of the OrionB
hostsseveralmassive O-typeyoungstellar objects,which have molecular cloud over a field of view of 1.5 square degrees in
created compact Hii regions inside the cloud. NGC2024 lies the full frequencyrange from 84 to 116GHz at 200kHz spec-
justeastoftheAlnitakstar,andiscrossedbyafilamentwhich tralresolution(Petyetal.accepted).TheredrectangleinFig.1
isseeninabsorptioninvisiblelight,andinemissionintheradio showsthefieldofviewobserveduptonow,overanH column
2
range.ThereflectionnebulaNGC2023isa quietercounterpart densitymapproducedbytheHerschelGouldBeltSurveycon-
ofNGC2024,hostingyoungB-type stars. Itlies North-Eastof sortium(Andre´etal.2010;Schneideretal.2013).Themapped
the Horsehead nebula. The rest of the cloud contains extended areacovers56 98arcminutes(about6 11parsecsattheas-
× ×
and quieter areas with strong filamentary structures (Fig. 1). sumeddistanceofOrionB,400pc)insize.
Thisareahasbeenextensivelyobservedinthe3mmrangewith Sofar,theobservationsprovidedabout250000spectraover
the IRAM-30m telescope (PI: J. Pety). This survey has led to a 32GHz bandwidth, yielding a position-position-frequency
a series of papers (Liszt&Pety (2016); Petyetal. (accepted); cubeof370 650 160000pixels,eachpixelcovering9 9
′′ ′′
Gratieretal.(subm.))towhichthisarticlebelongs. 0.5kms 1 (a×tNyq×uistsampling).Thereduceddatasetam×ount×s
−
to100GBofdata.Thedatareductionisdescribedindetailsin
The paper is organised as follows. We briefly describe in Petyetal.(accepted).
Section2theobservationsbytheORION-B(OustandingRadio This dataset is unique for its extensive coverage, both spa-
ImagingofOrioNB)collaboration,andthedataweusehere.In tially and spectrally, at a typical resolution of 25 and a me-
′′
Section 3, we present the concepts and equations of the statis- dian noise of 0.1K [T ] per channel over a 32GHz spectral
mb
tical method and the details of its implementation, from noise range. From the spatial point of view, the survey gives us ac-
filtering to the computation of power spectra and the estima- cess to a large range of scales in the molecular cloud, from
tionofvelocity-densitycorrelations.Theresultsaredescribedin 50mpc to 10pc. For the analysis of turbulence properties (see
Section4,anddiscussedinSection5withaspecialemphasison Sect. 4.3), it means that we are able to study a large fraction
therelationoftheturbulencepropertieswith thestarformation ofthe inertialrangeof theturbulence,witha potentialviewon
2
JanH.Orkiszetal.:TurbulenceandstarformationefficiencyinOrionB
the injection scale. The dissipation scale, on the other hand, is 3. Derivingtherelativefractionofsolenoidal
of the order of a milli-parsec (Hennebelle&Falgarone 2012; motionsfromaposition-position-velocitycube
Miville-Descheˆnesetal. 2016), and, at a distance of 400pc,
is only accessible using millimetre interferometers, and out of In order to measure the fraction of the solenoidal and com-
reachfortheIRAM-30mtelescope. pressive turbulencemodes, we apply the method developedby
Bruntetal. (2010) and Brunt&Federrath (2014). In this sec-
From the spectral point of view, having such a large band-
tion, we first recall the method, its assumptions, and the way
width observed in one go allowed us to image over 20 chem-
weimplementedit.
ical species (Petyetal. accepted), among which those listed in
Tab. 1. As opposed to several small-bandwidth mappings, the
spectral lines in this survey are observed in the same condi-
3.1.Descriptionofthemethod
tionsandarewellinter-calibrated,whichgivesanunprecedented
spectralaccuracyforsuchalargefieldofview. 3.1.1. Principlesandassumptions
Thekeypointofthismethodisthefactthattheobjectswe ob-
2.2.The13COspectraldatacube servearefundamentally3-dimensional(e.g.,amolecularcloud),
but the observer only has access to a 2-dimensionalprojection
Mostoftheworkpresentedherewasperformedonthe13CO(J =
along the line of sight of that object. Bruntetal. (2010) devel-
1 0)datacube,whichcoversavelocityrangeof40kms 1cen-
− oped a method to retrieve properties of the 3-dimensional ob-
tre−d around the source systemic velocity of 10.5kms 1 and a
− ject that we are interested in, and which corresponds to a 3-
rest frequency of 110.201354GHz. The datacube presents an
dimensional field F , via the properties of the 2-dimensional
3D
RMSnoiseofσ=0.17K,andamediansignal-to-noiseratioof
observational F , which is a projection of F along the z
2D 3D
T /σ=7.9.
peak axis. To achieve this, they use the fact that the Fourier trans-
The13CO(J = 1 0)linewaschosenbecauseitoffersone formF˜ ofthe2-dimensionalfieldisproportionaltothek =0
2D z
of the highest signal-−to-noiseratios over the whole map, but it cut through the Fourier transform F˜ of the 3-dimensional
3D
does not feature as much saturation as the 12CO(J = 1 0) field. In short, F˜ (k ,k ) F˜ (k ,k ,k = 0). If these fields
2D x y 3D x y z
− ∝
line. Signal is present at a signal-to-noise ratio greater than 5 are isotropic, i.e., if they are functions of k = k alone, with
in the whole map, except in the Hii regions around σ Ori and k = (k ,k ,k ) the wave vector and k the wave n|u|mber, the 2-
x y z
HD38087wheremoleculargasisphotodissociated.Thebright- dimensionalfieldallowsustoreconstructaveragepropertiesof
estregionsaretheNGC2023nebula,thecenteroftheNGC2024 the3-dimensionalfieldthankstosymmetryarguments.
nebula,anditsnorthernedge(Fig.4,leftpanel). TheBrunt&Federrath(2014)methodwasdevelopedasan
13COisagoodtracerofmoleculargas,frommoderatelydif- applicationoftheBruntetal.(2010)methodtothecaseofvec-
fuseandtranslucentregions(A = 1 5mag)uptomoderately tor fields. For a vector field such as a velocity or momentum
V
denseandshieldedgas(104cm 3,A −=10mag).COhasasmall field,thedimensionalityreductionduetotheprojectionismade
− V
dipole moment (0.11D), hence the rotational lines have low worse by the fact that only one component of the vector (the
Einsteincoefficients(e.g.,Mangum&Shirley2015).Thisleads line-of-sight one) can be measured, thanks to the Doppler ef-
to relativelyeasy collisionalexcitation,excitationtemperatures fect. In that case, the next main tool to retrieve 3-dimensional
approaching the kinetic temperature, and moderate line opaci- propertiesistheHelmholtztheorem,whichallowstodecompose
tiesexceptforthemostabundantspecies,12CO.Theabundance anyvectorfieldinitsdivergence-free(solenoidal)andcurl-free
ratio13CO/12CO is equalto 13C/12Corabout1/60whenchem- (compressive)components,F andF .Thesecomponentsarere-
ical fractionation reactions, which are limited to the most dif- latedviaalocalorthogonality⊥inFou||rierspace,F˜ (k) F˜ (k).
fuse regions, are inefficient (Wilson&Rood 1994). Therefore, Solenoidal modes can be pictured as the modes o⊥f a t⊥urbu||lent
the13CO abundancerelativetoH2 remainsapproximatelycon- incompressiblefield,madeofverticesandeddies.Ontheother
stantatalevelof 2 10−6(Dickman1978)acrossmostofthe hand,compressivemodes,madeofcompressionandexpansion
cloudvolume.13C∼O s×tartstobedepletedondustgrainsincold motions, are more likely to be generatedby phenomenalinked
cores,buttheserepresentonlyasmallfractionofthemassanda tostarformation.
negligiblefractionofthevolumeoftheOrionBmolecularcloud The application of these methods implies several require-
(Kirketal.2016). ments on the studied dataset. As mentioned earlier, the statis-
Figure 2 showsthe 13CO(J = 1 0) signalintegratedover tical isotropy of the cloud is the first necessary point, and en-
−
4 contiguous velocity ranges. It showcases the complexity of ables the use of 2-dimensional averages as a means to esti-
thespectralstructureofthecloud,withprominentvariationsof mate3-dimensionalproperties.Itmeansthatthemethodcannot
thelineprofilewiththeposition,whichisaconsequenceofthe be applied to individualfilaments, or to clouds where a strong
strongturbulenceatplayinthemolecularcloud. anisotropyissuspected,e.g.,duetothepresenceofastrongmag-
Up to four spectral components appear along each line-of- neticfieldatlowMachnumbers.
sight within the field (Fig. 3). A main component is visible Second, the field is required to go smoothly to zero on its
around 10kms 1, and a secondary component at lower veloc- borders.This propertyis needed to ensure that the decomposi-
−
ity (about 5kms 1). Sometimes an extra component at higher tionofthefieldisunique,sincetheHelmholtzdecompositionis,
−
velocity (about 14kms 1), or secondary peaks around 5 and in theory,definedupto a vectorconstant.Itisalso a necessary
−
10kms 1appeartoo.Thefirsttwocomponentsarethemostsig- condition for good behaviours of the Fourier transform, since
−
nificantonesatthescaleofthewholecloud,beingtheonlyones actual observational data are not periodic fields, unlike hydro-
visibleinthemeanspectrumof13CO(J =1 0),andhaveaver- dynamicalsimulations(seediscussioninBruntetal.2010).This
agevelocitiesof 9.7kms 1 and4.9kms 1 r−espectively.Allthe impliesthatthestudiedfieldshouldbeboundedinspace,likefor
− −
components,despitebeingquitedistinctonthespectralaxis,are exampleagravitationallyboundcloud.Inthecasewherethesig-
howeverthoughttobepartoftheOrionBcloud(seediscussion nalextendsuptotheedgeoftheobservedfield,thedatasethas
inPetyetal.accepted). tobeapodized.
3
JanH.Orkiszetal.:TurbulenceandstarformationefficiencyinOrionB
Fig.2. Maps of the average brightness temperature of the 13CO(J = 1 0) line in four contiguous velocity ranges. The main-
beam temperature scale is indicated by the color bar on the right. The c−ontour shows the value of 8.9K kms 1 in the W map,
− 0
correspondingto0.43Kinthemeantemperaturemapintegratedoverthe-0.5–20.5kms 1velocityrange.Thesetofcoordinates
−
usedfortheobservationalcampaigntakestheHorseheadPDRasareferencepoint,andalignstheIC434PDRalongtheverticalaxis
(14 counter-clockwiserotationwithrespecttoequatorialcoordinates).Thenumberedsquaresinthefirstpanelshowthepositions
◦
ofthespectrapresentedinFig.3,fromlefttoright.
Fig.3.13CO(J = 1 0)spectraalongselectedlinesofsightsin theOrionBcloud,showingthediversityofvelocitycomponents
−
(uptofourperspectrum).Thecoordinatesofthevariouslinesofsightsaregiveninarc-minutesinourcustomsetofcoordinates
(δx,δy).Theaveragespectrumforthewholefield,normalizedtohavethesamepeaktemperature,issuperimposedwithadashed
lineforcomparison.ThepositionofthespectraonthemapareshownbywhitesquaresinthefirstpanelofFig.2.
Finally,fromapracticalviewpoint,Bruntetal.(2010)have The3-dimensionalquantityweinferis
shownthattheirmethodworksbestforfieldswithpowerspectra
that are not too steep. Steep power spectra give measurements R=σ2 /σ2, (1)
p p
thatareverysensitive tothe low spatialfrequencies,whichare ⊥
the ratio of the variance of the transverse (solenoidal) momen-
usuallyuncertainduetopoorstatistics.
tumtothevarianceofthetotalmomentum.Forshort,Rwillbe
Thecomplianceofourdatasetwiththeserequirementsisdis-
referredtoasthe“solenoidalfraction”intherestofthispaper.
cussedindetailinSect.5.1.
AccordingtoBrunt&Federrath(2014),athypersonicMach
numbers (M = v/c > 5) the solenoidal fraction does not
sound
3.1.2. Equationsandnotations dependany more on the type of forcing,but instead converges
towards R 2/3. Brunt&Federrath (2014) note that this be-
∼
The studied quantity is the momentum density field (hereafter haviour is different from what is observed by Federrathetal.
“momentum”),p=ρv,withρandvthevolumedensityandthe (2011),wherethesolenoidalfractionconvergestodifferentval-
velocity. uesdependingonthe typeofforcing,butthisisdueto thefact
4
JanH.Orkiszetal.:TurbulenceandstarformationefficiencyinOrionB
that Brunt&Federrath (2014) consider the momentum density The3-dimensionalvarianceofthevolumedensity, (ρ/ρ )2 ,can
0
h i
field, while Federrathetal. (2011) describe the energy density bederivedusingtheBruntetal.(2010)method.Theexponentǫ
field. isasmall,positiveconstant,whichcanbeobtainedastheexpo-
This value of R 2/3 can be simply explained in terms nentofthe σ2 vs. ρpowerlaw,i.e.,thetypicalvelocitydisper-
∼ v
of equipartition of momentum between the compressive and sioninthecloudasafunctionofvolumedensity.Ifthedensity
solenoidal modes (see, e.g., Federrathetal. 2008). A value of andvelocityfieldswereuncorrelated,g wouldbeequalto1.
21
Rlowerthan2/3thereforemeansthatthereismoremomentum
inthecompressivemodesoftheflow,andthatthecloudisthus
3.2.Implementation
morelikelytoformstars.TheinfluenceoftheMachnumberis
furtherdiscussedinSect.4.1. Actualdatasufferfromseverallimitationsthatneedtobedealt
The available observables are a position-position-velocity withinordertoapplythemethoddescribedpreviously.Thefield
cube, its velocity-weighted moments and the power spectra of of view, the angular resolution, and the sensitivity are limited.
thesemoments.Wemaketwomajorassumptionsaboutthisdat- Thissectiondescribeshowtheseissuesweredealtwith.
acube,namelythatthe13CO(J =1 0)lineisopticallythinand
that its emissivity only depends on−the 13CO volume density.
3.2.1. Noisefiltering
These assumptions are true within less than 20%, with the ex-
ceptionofafewlinesofsighttowardsthecenterofNGC2024,
Computation of line moments is sensitive to noise in the line
whicharemoresaturated,butrepresentabout2%ofthewhole
wings. It is well-known that masking the position-position-
field. Under these assumptions, the position-position-velocity
velocity cube where signal stays undetected improves the de-
cubecanbeseenasadensity-weightedvelocityfield:thespec-
termination of the centroid velocity and linewidth. To define
trum obtainedfor each line of sight results fromthe projection
the mask containing the pixels detected at high significance,
oftheemissionofthematterpresentalongthislineofsight,and
thesepixelsarefirstgroupedintocontinuousbrightnessislands
movingatvariousvelocities.
that are made of neighbouring pixels in the position-position-
The useful moments are the zero-th, first and second order
velocityspace,whosesignal-to-noiseratioislargerthan2.The
moments of the momentum field, W , W and W , which are
0 1 2 noise level, σ, is measured outside the studied velocity range
definedasfollowsinBrunt&Federrath(2014): ( 0.5, 20.5kms 1).Thelistofislandsisthensortedbydecreas-
−
−
ingtotalflux.Thefirstislandcontainsabout97.8%ofthetotal
W0 = I(v)dv, W1 = vI(v)dv, W2 = v2I(v)dv. (2) signal.Thefollowingonesaresmallsignalclumpsthatarespa-
Z Z Z
tiallyorspectrallyisolatedfromthismainblock,andafterabout
Thespectrallineintensity,I(v),mayhavecontributionsfrom afewhundredislandsweareleftwithsingle-cellislandswhich
variouspositionsalongthelineofsight.Givenourassumptions arejustnoisepeaks.
onemissivity,andassumingthatthenaturallinewidthisnegligi- While it is easy to visually assess that the first few islands
blecomparedtotheoverallvelocitydispersion,wecandescribe correspond to genuine signal, it is more complex to determine
thesemomentsinanalternativeway: the transition to pure noise, as a significant fraction of the to-
tallinefluxcouldbehiddeninpixelsoffaintbrightness,atlow
W ρ(z)dz, W v(z)ρ(z)dz, W v(z)2ρ(z)dz. (3) signal-to-noiseratio.We thusstudiedtheinfluenceofthenum-
0 1 2
∝Z ∝Z ∝Z ber of islands used on R, the solenoidal fraction in the studied
cube.Thisinfluenceismodest,mainlybecauseover97%ofthe
The solenoidal fraction can be written in terms of these obser-
signalislocatedinthefirstisland.Usinguptoabout80islands
vationalquantitiesasfollows(seeBrunt&Federrath(2014)for
yieldsaverystablevalueofRwithlessthan0.1%variation.A
details):
steepincreaseofRisobservedwhenweenterthenoisydomain.
R hW12i hW02i/hW0i2 g hW2i −1B. (4) Wethususedthe80brightestislandsforallothercalculations.
Th≈eAhaWn02diBf1ac+toArs(haWre02fiu/hnWct0ioi2ns−o1f)th"ep21ohwWe0ris#pectraofW and 3.2.2. Momentcomputation
0
W . After selecting the signal islands in the position-position-
1
velocity cube, the moments are integrated from 0.5 to
f(k) f(0) f (k)k2x+ky2 20.5kms 1.Thecalculationshavetobeperformedinthe−center-
A= (cid:16)PkxkPxkykPykfz(k) −(cid:17)−f(0) , B= PkxPkkyxPkkzy f⊥⊥(k) k2 ,(5) omfi-nminagssthf−eracmenetroofidrevfeelroecnicteyoofftthheecclolouuddi,nwthheicLhSiRmfprlaiemsed.eTtheirs-
(cid:16)P P (cid:17) P P center-of-massvelocityissimplygivenby
where f(k)and f (k)aretheangularaveragesofthepowerspec-
traW˜0(kx,ky)and⊥W˜1(kx,ky),respectively. Wobs
Brunt&Federrath (2014) introduce a statistical correction Vc = h 1 i, (8)
W
factor, g21, of order unity, which measures the correlationsbe- h 0i
tweenthevariationsofthedensityandthevelocityfields,andis where Wobs is the first momentfield in the observer’sframeof
1
definedas reference.W , on the otherhand,is notvelocity-weighted,and
0
thereforedoesnotdependontheframeofreference.Fortheob-
ρ2v2 / ρ2
g21 = h ρv2i/hρ i. (6) servedfieldofview,weobtainVc =9.16±0.90kms−1.
The velocity scale in the observer’s frame of reference is
h i h i
Brunt&Federrath(2014)showthatthismaybewrittenas shiftedbyVc,beforecomputingW1andW2inthecenter-of-mass
frameofreference:
ρ 2 −ǫ
g21 =* ρ0! + . (7) W1=Z(vobs−Vc)·I(vobs)dvobs, W2=Z(vobs−Vc)2·I(vobs)dvobs.(9)
5
JanH.Orkiszetal.:TurbulenceandstarformationefficiencyinOrionB
Fig.4.Top:mapsofthe13CO(J = 1 0)fieldmomentsinthecloud’sframeofreference.Bottom:mapsofthephysicalquantities
−
directly derived from each field, with the centroid velocity being simply W /W , and the FWHM velocity dispersion given by
1 0
2√2ln(2)√W /W –thenormalizationoftheFWHMcorrespondstothatofafieldwithpurelyGaussianlineprofiles.
2 0
TheresultingfieldsareshowninFig.4. (2015).We keep the regionaffectedby apodizationas smallas
possibletominimizesignalalteration.Anapodizationwidthof
about 5 % of the smallest dimension of the field (i.e., roughly
3.2.3. Apodization
25 pixels) was the smallest value that efficiently smoothed out
thehigh-frequencyartefacts.Thisisconsistentaswellwiththe
ComputingthepowerspectraoftheW andW fieldsrequiresto
0 1
widthdeterminedinMartinetal.(2015).
takethe Fouriertransformofthese fields.Althoughcalculating
theFFT of2-dimensionalfieldsisan easytask, severalnumer- Oncethefieldisapodized,itmustbemadesquaretofollow
ical artefacts must be taken care of. In particular,the observed the isotropy requirementsof the method. The square was built
area does not reach the edges of the OrionB molecular cloud bypaddingwithzerostherightside(west)oftheobservedarea,
in all directionsasillustrated in Fig.1. This sharptruncationof as this is the location of the Hii region associated with σOri,
the13COemissionwillcreateartefactsintheFouriertransform, i.e., no signal is detected past the western edge of the field of
due to the convolution of the true Fourier spectrum by a sinus view. After this, the Fourier transform is calculated using the
cardinalfunctionthat oscillates at high frequencies.Apodizing FFTimplementationofNumpy1.8(Cooley&Tukey1964).
thefield isrequiredtoavoidthisbehaviour.We havechosento Apodizationisalinearfilterofthedata,andthushaseffects
multiplytheintensityby1 cos(πx/w),wherexisthepixelcoor- onthepowerspectrumatallfrequencies.Whileapodizational-
−
dinateandwtheapodizationwidth.Thisfunctiongoesfrom0to lows us to “clean” the spectrum at high spatial frequencies, it
1overwpixels.ThisapodizationfunctionisusedinMartinetal. alsoaltersthespectrumatthelowestfrequencies.Forexample,
6
JanH.Orkiszetal.:TurbulenceandstarformationefficiencyinOrionB
f(0)–thevalueofthepowerspectrumatspatialfrequencyk=0 bytheinverseofthevariancesobtainedineachbinwhencom-
–isdirectlyproportionaltothespatialintegralofW2,therefore putingthepowerspectrum.
0
losingsomesignalduetotheapodizationwillreducethevalueof The choice of fitting the power spectrum with a modified
f(0).Themethodchosentokeepthegoodpartsoftheapodized powerlawimpliesthattheunderlyingphysicalprocessesshould
andnon-apodizedspectrawasthefollowing:wefirstcomputed produce a power law. This is indeed the case for the inertial
the FFT of both the apodized and non-apodizedfields, respec- rangeofscalesinKolmogorovturbulence,andcanbeappliedas
tively W˜ and W˜ , then mixed them smoothly around k = 0 welltoBurgersturbulence(see,e.g.,Federrath2013).However,
ap nap
using a narrow (about 10 Fourier-space pixels) 2-dimensional the power spectrum of turbulence starts to deviate from a
GaussianG (k).TheresultingFourierfieldis power law at scales where the energy is injected (low spatial
mix
frequencies)ordissipated(highspatialfrequencies).Thepower
W˜final =Gmix(k) W˜nap(k)+(1 Gmix(k)) W˜ap(k) (10) spectra computed from our dataset somewhat deviates from a
· − ·
power law at low spatial frequencies (see Fig. 5). The power
Thepowerspectrumthusbehavesasthenon-apodizedspectrum law can therefore only be fitted and then used above a given
atlowk,keepingthecorrectvalueofthefieldintegral f(0),and spatialfrequency.Thepower-lawrangestartsaround 5.5.At
′
astheapodizedspectrumathighk,freeofthespectralparasites high spatial frequencies,no deviationsfrom the powe∼r law are
createdbythesharpedgesofthemap. detected(thefitsrendertheobservationsverywell).Thismeans
that either the dissipation scale happens at lower angular scale
thantheresolutionoftheobservationsoritishiddeninthenoise.
3.2.4. Powerspectracomputationandfit
The apodizedand correctedFFT of the field needsto be trans- Once the fit has been performed, the final version of the
formedintoanangle-averagedpowerspectrum f(k)or f (k).It power spectra is built using both the fit result and the observa-
is simply done by binning the modulus of the spatial fr⊥equen- tionalangle-averagedpowerspectra,andusedforfurthercalcu-
cies, and averaging the points found in these radial bins. The lations.Thefinalpowerspectrumis a purepowerlaw (without
resultingdiscretefunctioncanthenbelinearlyinterpolatedinto thebeamandnoise)abovethe1/5.5arcmin 1 threshold,andis
−
acontinuousfunction.Acriticalelementtothisexerciseresides equaltothelinearlyinterpolatedangle-averagedpowerspectrum
in thesamplingofthespatialfrequencyaxis.On theonehand, belowthisthreshold.Inparticular,we enforce f (0) = 0,since
the resulting angle-averagedspectrum should be as detailed as weareworkinginthecloud’srestframe. ⊥
possible,butontheotherhand,alargernumberofbinscanlead
to emptybins, containingnosampled pointsat all. As a result,
3.3.Density-velocitycorrelations
we used a number of bins of S/1.45, with S the size in pixels
ofthesquarefield,sothatthesizeofabinintheFourierspace We used the information on the mean line profiles to estimate
correspondsto slightly more than the length of the diagonalof the slope ǫ of the relation between the velocity dispersion and
pixelsintheFourierspace.
the local density (see Eq. 7). Among the lines detected in the
Additional observational constraints (noise, beam shape, meanspectrum,wehaveselectedlineswithdifferentspatialdis-
etc.)affectthedeterminationofthepowerspectrum.Following tributions(Petyetal.accepted).Totracethelowdensitygas,we
Martinetal. (2015), the power spectra are fitted with a power haveselected 12CO(J = 1 0) and HCO+(J = 1 0) as these
law,modifiedtotakeintoaccountthesingle-dishbeamandthe linespresentveryextended−emissionandhavemod−erateexcita-
noise. In our case, the beam is modelled as the Fourier trans- tionrequirements(Petyetal. accepted;Liszt&Pety2016).We
formofaGaussianofFWHMequaltothecuberesolution,i.e., haveincluded13CO(J = 1 0) asourtracer ofthe bulkof the
23.5′′.Thiscorrespondstoabout2.61pixels.Theconvolutionin gas.Thesomewhatdensera−ndmoreshieldedgasiswelltraced
the image space corresponds to a multiplication in the Fourier byC18O(J = 1 0), whilewe haveselected N H+(J = 1 0)
2
spacethatmostlyaffectsthehighspatialfrequencies. for the dense co−res. For these five species, we determined−the
The noise is not a Gaussian white noise, because of inter- FWHM byfittinga Gaussianlineprofiletothemeanprofileof
pixel correlationsand systematics. We thereforeuse the power thewholemap.Onlythe10kms 1component,whichispresent
−
spectraof30signal-freechannelsandaveragethemtoobtaina forallfivespecies,wasusedforthisfit.ForN H+ weusedthe
2
template of the noise power spectrum. In this case, we use the HFS fit method in GILDAS/CLASS1, which makes use of the
fullynoisydatacube,notthe 80firstsignalislands,in orderto informationonthehyperfinestructure.
havethesamespatialcorrelationsinnoiseforeachchannel(with While these lines are emitted by gas over a wide range of
orwithoutsignal),which wouldnotbe thecase witha masked densities,thereisaminimumdensityunderwhichthelineisnot
datacube.Thenoisetemplateintendstoreproduceasystematic detected,becauseoflackofexcitationorbecausethemoleculeis
behaviour,butwecanonlyuseafinitenumberofchannelswith notpresentinlowdensitygas.Itisthisminimumdensitywhich
randomnoise.Thenoisetemplateisthereforesmoothedtomake correspondstovelocitydispersionoftheline.Toderivetheden-
thissystematicpatternstandoutmore.GiventhatW1is,justlike sitiesassociatedwiththelineemission,weadoptedthreediffer-
W0, a linear combination of the channel maps, the same noise entmethods.
templateisusedforbothpowerspectra. For the low density and extended emission tracers, we de-
The fit is performed in the log(k)-log(f) space, so that the rivedthevolumedensitybycomparingtheminimumgascolumn
straightlineofthepowerlawstandsoutmore.Thefittingfunc- densitywheretheemissionisdetectedandtheresultingsizeof
tionisthereforethelogarithmof theemissionregions.Thisleadstogasdensitiesofafewhundred
cm 3for12CO(J =1 0),HCO+(J =1 0)and13CO(J =1 0).
10(a·log(k)+b) G˜beam(k)2+N noise(k) (11) The−emissionof12CO−(J =1 0)isdom−inatedbythelowden−sity
· · regions(Petyetal.accepted)−.TheemissionofHCO+(J =1 0)
where G is the Gaussian beam, G˜ is its (Gaussian) −
beam beam
Fourier transform, and noise is our noise template. The fitted 1 Seehttp://www.iram.fr/IRAMFR/GILDAS/formoredetailson
parametersarea,bandN.Duringthefit,thedataareweighted theGILDASsoftware.
7
JanH.Orkiszetal.:TurbulenceandstarformationefficiencyinOrionB
Fig.5.Left:PowerlawfittingoftheW powerspectrum.Upperpanel:data(bluecrosses),fitresult(thicksolidline)plottedover
0
the fitted domain,powerlaw convolvedwith the Gaussian beam (dottedline) extrapolatedto all spatialfrequencies,noise model
(dashedline). Lower panel, residuals. The scale in the Fourier space is given in UV distance as for interferometricobservations,
whichallowstovisuallyrelatetheresolutionandthetelescopediameter.Right:Sameresults,exceptfortheW powerspectrum.
1
is also dominated by low to intermediate density regions, and Table 1. Spectral tracers used in this study, and observed with
comesfromtheweakexcitationregime(Liszt&Pety2016).In the IRAM-30m telescope. The linewidths are derived from
bothcases,opacitybroadeningisnotverysignificant.We must theseobservations,thetypicaldensitiesfromtheseobservations
however consider that the line widths for these tracers are up- (ref. 1) or other works (ref. 2: Hily-Blantetal. (2005), ref. 3:
per bounds, due to this effect of opacity broadening.For these Kirketal.(2016)).
molecules,wearealsolimitedbythesensitivityoftheobserva-
tions, so that the densities should be regarded as upper limits. Line Frequency FWHM log(n(H )) Ref.
2
Hily-Blantetal.(2005)analysedthestructureandkinematicsof (J=1 0) (GHz) (kms 1) (cm 3)
− −
−
theHorseheadnebulaandderivedthedensityoftheextendedre- 12CO 115.271202 4.08 0.04 2.17 0.3 1
giontracedbyC18Oas3 5 103cm 3.Wehavekeptthisvalue HCO+ 89.188525 3.91±0.08 2.34±0.3 1
as the typical density tra−ced×by C18O−. Hily-Blantetal. (2005) 13CO 110.201354 2.97±0.03 2.65±0.3 1
haveshownthathigherdensityregionsexistwithdensitiessig- C18O 109.782173 2.55±0.01 3.60±0.15 2
nificantlylargerthan104 cm−3.Asveryfewpixelsaredetected N2H+ 93.173764 1.79±±0.10 4.10±±0.3 3
in N H+ in the region of the Horsehead nebula, we have used
2
the catalogue of dense cores identified by Kirketal. (2016) in
theirSCUBA2mapoftheOrionBcomplex.Wehavefound55 3.4.Estimationoftheuncertainties
coresassociatedwithN H+(J = 1 0)emission.Thedensities
2
havebeenderivedusingtheextracte−dfluxesandeffectiveradii,a Thecomputationofuncertaintiesimpliescomputingtheuncer-
uniformdusttemperatureof20K,andassumingsphericalgeom- taintyofeachelementofEq.4.WestartfromtheaverageRMS
etryforallcores.Themeandensityis104.1 cm−3 withascatter noise level in the data cube, 0.17K [Tmb] . This allows us to
ofaboutafactoroftwo.ThetemperatureoftheN2H+coreswas computethenoiselevelfortheW0,W1 andW2 maps.Thecom-
notindividuallychecked,butit is likely to be lowerthan 20K. putationis straightforwardcomparedto the computationof the
Inturn,thisimpliesanevenhigherdensityofN2H+.Therefore, uncertainty of the centroid velocity and linewidth because W1,
thederiveddensityshouldberegardedasalowerlimit. andW2 arenotnormalizedbyW0, i.e.,theirnoise distributions
stay Gaussian, whatever the value of W . Due to the velocity
0
weighting,the absolute uncertaintyincreasessignificantly with
the momentorder.However,the relative uncertaintieson W ,
0
h i
W and W aresimilar,withvaluesrangingfromabout10%
1 2
h i h i
for the whole field to 5% for the deepest zooms on NGC2023
Table 1 presents the resulting data, which are illustrated in andNGC2024(seeFig.8).TheuncertaintiesonthesumsAand
Fig. 6. The slope α = ǫ is derived from a least square fit of Bwereexplicitlycomputedaccordingtotheerrorbarsdescribed
the variation of the FW−HM with the density. We derive ǫ = inSect.3.2.4.TherelativeuncertaintyonArangesfrom24%for
0.15 0.03.Thepossiblesystematicsonthedensitiestracedby the full field to 11 % for the deepest zooms, and stays around
12CO±(J =1 0),HCO+(J =1 0)andN H+(J =1 0),aswell 13%forB.
2
asonthelin−ewidthsof12CO(J−= 1 0)andHCO+(−J = 1 0), For the overall relative uncertainty, one must not only take
− −
alltendtomakethepowerlawsteeper.Therefore,wekeepthis intoaccounttheerrorsoftheindividualterms,butalsothecor-
valueofǫ asanupperlimit. relationbetweenthedifferentvariables.Inourcase,thedifferent
8
JanH.Orkiszetal.:TurbulenceandstarformationefficiencyinOrionB
and they also provide the spatial and statistical distribution at
goodangularresolution.Inparticular,theMachnumberismuch
smaller thantheaveragein thestar-formingregionsNGC2023
andNGC2024,belowthehypersonicregime.
4.2.Powerspectra
When the whole field is considered, the fit yields an exponent
a = 2.83 0.02 for the W field, and a = 2.50 0.07
0 0 1
− ± − ±
for the W field. When zooming into specific regions of the
1
cloud (NGC2023 and NGC2024, see Fig. 8), the values of
theseexponentsrangefroma = 2.52 0.08(widestfield)to
0
− ±
a = 3.04 0.05(smallestfield),andfroma = 2.24 0.09
0 1
− ± − ±
to a = 2.81 0.06. While the indices of W power spectra
1 1
− ±
arerarelyreported,therearemanyvaluesofspectralindicesfor
integratedlineintensitymapsintheliterature,andoura values
0
fallwellintherangeofspectralindicesforobservationsofCO
Fig.6.VariationoftheFullWidthatHalfMaximum(FWHM)as emission, dust emission, Hi emission and absorption compiled
afunctionofthegasdensity.Eachpointreferstothe(J =1 0) by Hennebelle&Falgarone (2012): the values range from -2.5
−
spectrallineofadifferentmolecule.Theredlineshowstheleast to-3.2,withmostvaluesaround-2.7.
squaresfit,withaslopeα= ǫ = 0.15. Inorderto havea meaningfulresultforthe A and Bcoeffi-
− −
cients,thepowerlawslopeofthepowerspectramustfollowthe
two steepness requirements mentioned by Bruntetal. (2010).
variablesare stronglycorrelated,since they are all by-products
On the one hand, the spectra should not be too steep, so that
of the same data cube. Therefore,we chose to use the average
theweightofthelowspatialfrequencies,forwhichtheavailable
ofthe variousrelativeerrorsas a rule-of-thumbestimate ofthe
information is scarce, does not become too large in the A and
overallerror. This approachyielded an overallrelative error of
B sums. On the other hand, the slopes should be steep enough
13%forthewholefield,and8%forthedeepestzooms.Wekept
toavoiddivergenceofthesesumsatlargefrequencies.Aslope
thehighestvaluetoallowforasafetymargin.
of a = 3 is at the limit between these two contradictingcon-
This 13% relative error ∆R/R corresponds to the median −
straints, since the divergenceof the 3-dimensionalsum is only
noise-to-signal ratio in the field, σ/T , which is also of the
peak logarithmic.Inourcase,thesumsarefinite,duetothefiniteres-
orderof13%–buttestingwithsimulatednoisewhetherthisisa
olution of the observations, so that with slopes between -2.24
coincidenceornotisoutofthescopeofthisarticle.
and-3.04,thesumsdonotgrowtooquicklyanddonotgivetoo
muchweighttothelowspatialfrequencies.
4. Results
4.3.Turbulencemoderatio
Inthissection,wefirstbrieflypresentthederivationoftheMach
numberinthecloud,thenwecomparetheobtainedpowerspec- We have determined a relative error of about 13% on the cal-
tra with other results in molecular clouds, and finally, we give culationoftheratioRfromtheposition-position-velocitycube.
theresultsofourcomputationofthesolenoidalfractionRinthe The correction factor g is determined independently, and we
21
OrionBcloud. assume a range of possible values for g , the lower limit be-
21
ing given by our calculations of Sect. 3.3, and the upper limit
resulting fromthe minimumestimate of ǫ 0.05accordingto
4.1.Machnumber
≃
Brunt&Federrath(2014).
According to Brunt&Federrath (2014) and Federrathetal. For the entire 13CO field, we obtain the following range of
(2011), at hypersonic Mach numbers (M > 5) the ratio of values:
solenoidal and compressive modes does not depend any more
onthetypeofforcing.Itisthusimportanttoalsoderivethedis- 0.72+0.09<R <1+0.0 (12)
0.09 13CO 0.09
tributionoftheMachnumbertobeabletointerprettheresults. − −
Using themapsof thesoundspeedderivedfrom12CO(J = In order to gain a deeper understandingof the dynamicsat
1 0)anddusttemperaturemaps,twodifferentestimationsofthe stakeintheOrionBcloud,themethodwasalsoappliedtosev-
M−ach number, M and M , were computed (see Appendix eralsub-regionsofthe13COmap.
max exc
fordetails).FiguresA.2andA.3showtheirspatialdistributions First, to check the reliability of the method and the homo-
andcomparetheirhistograms.The shapesofthe histogramsof geneityofthefield,weappliedaslidingsquarewindow,whose
Mach numbers computed with T and T are very similar, side is equal to the smallest dimension of the mapped area
max exc
and both show a large tail at high Mach number. Table A.1 (Fig.7).Thisavoidszero-paddingthestudiedfield.Theresults
listsseveralcharacteristicvaluesofbothdistributions.Themost showthateventhoughthevaluesareingeneralsomewhatlower
probablevalue( 3.5)oftheMachnumberismuchsmallerthan for these sub-fieldsthan for the whole 13CO field, they remain
∼
themeanormedianvalues( 6),forbothdistributions. marginallycompatiblewiththisresult,withintheestimatedun-
∼
Schneideretal. (2013) estimate the average Mach number certainties.
to be of 8, with approximately30 40%error,derivingthis Second,wesearchedforsystematicvariationsofthefraction
∼ −
value from Herschel dust temperature and CO linewidth, and of solenoidalmodeswhenzoominginto specific regionsof the
findoutthatOrionBhasthehighestMachnumberofthesetof map.In particular,signsof the solenoidalor compressiveforc-
studiedclouds.Ourresultsarecompatiblewiththismeanvalue, ingareexpectedtoappearmainlyinregionsoflowMachnum-
9
JanH.Orkiszetal.:TurbulenceandstarformationefficiencyinOrionB
theemission,mostofthesignalisinthecenter,whichminimizes
theeffectsofthenecessaryapodizationonthefinalresults(only
7.9% of lost signal for the deepest NGC2023 zoom, 7.6% for
NGC2024).
Asfarastheisotropycriterionisconcerned,the2Dprojec-
tionisquiteobviouslyisotropicinthecaseofthezooms,since
theconsideredfieldsaresquare,andmuchlessinthecaseofthe
wholefield,inwhichtheregionwithsignalhasanaspectratioof
about2:1.Thethirddimensionisunknown,andinanycasecan-
notmatchsimultaneouslythedimensionsofthewholefieldand
those of the deepest zooms. However, for the large diffuse re-
gionslike forthebright,compactregions,the dimensionalong
the line of sight is supposed to be of the order of the dimen-
sionsin the planeof the sky.In the case of a zoomon a bright
regionembeddedin a diffuse one, the signal, and thereforethe
dimensiononwhichitisemittedalongthelineofsight,isdom-
inatedbythebrightgas.Therefore,sincethezoomsarecentred
on a brightregion, the correspondingdatacube behaves almost
as if the bright region was isolated and isotropic. Besides, the
Fig.7. Solenoidal turbulence fraction for sliding square areas
non-angle-averagedpower spectra of W and W do not show
0 1
with a width equal to the one of the full map (56 arcminutes).
anyapparentanisotropyatanyscale(exceptforthewindowing
Theshadedareacorrespondstotheg uncertainty,whiletheer-
21 effects). If the power spectra are isotropic in two dimensions,
rorbarsshowtheexperimentaluncertaintiesduetoobservational
then statistically we can expect the third dimension to follow
noise.Theverticallinemarkstheequipartitionlimit.Themapon
thisisotropyaswell.Therefore,theisotropyrequirementseems
theleftpresentsthecentresofthesquareareasforwhichthecal-
tobefulfilledbyourdatasetandthemethodcanbeapplied.
culationswereperformed,superimposedonthe13CO(J =1 0)
Second,aswasmentionedbyBrunt&Federrath(2014),the
−
T map.
peak methodissensitivetovaluesofthepowerspectraatlargespatial
scales(lowfrequenciesofthepowerspectra)duetothecharac-
teristicsofthesumsintheparametersAandB.Wethereforehad
bers (see Sect. 4.1). The zooms were thus performed into the
tofindawaytoobtainasmoothandreliablefunctionthatwould
NGC2023 and NGC2024 star-forming regions (Fig. 8) where
representan angle-averagedpowerspectrumat all frequencies.
the Mach number lies between about 3 and 5 (see Fig. A.2),
Oncethepowerspectrawerebinnedandfitted,wehavetwover-
mostlybecausethespeedofsoundishigherinregionsofhigher
sionsofthespectra,eachwithitsflaws.Thebinned(“dataonly”)
gastemperature,butalsobecausethevelocitydispersionisabit
spectrumsuffersfromobservationaleffects(noiseandbeam)but
lower.
hasalsolargeruncertaintiesatlowspatialfrequencies.Thefitted
Moreover, these regions offer the advantage of presenting
spectrum, if extrapolated to all spatial frequencies (“fit only”),
a strongly localized emission. One of the requirements of the
can give unphysical results in the lowest frequencies because
Brunt&Federrathmethodistouseisolatedfields.Thisisclearly
they lie outside the power law validity range. For example, if
notthecaseanymorewhenzoomingintothesespecificregions,
thefieldhadbeenzero-paddedallaroundtocreateaverylarge
and apodization is more necessary than ever to ensure that the
square, such an extrapolation would give very high values of
signalfallssmoothlytozeroontheedgesofthefield.However,
thespectrumatlowfrequencies,whereasphysicallytheyshould
byusingfieldsforwhichmostofthesignalisconcentratednear
beverylow,since thefieldwouldonaveragebealmostempty.
thecenter,theeffectsofapodizationareminimized,allowingus
Theseflawsledustochoosethecompositeschemedescribedin
tostayascloseaspossibletotherequirementofanisolatedfield.
Sect. 3.2.4, where the fit resultis used only in the “inertialdo-
Thesmallerthefield,thelowerthevalueofR:thisindicatesan
main”,and the low frequencieskeep the angle-averagedpower
increasingproportionofcompressiveforcing.
spectrum as it is. Using a different version of the power spec-
tra leads to quite different final results, as illustrated in Fig. 9.
5. Discussion However, it is important to note that even though the absolute
value of the solenoidalfraction R varies, the relative variations
5.1.Compliancewiththemethod’sassumptions
areconsistentacrossscaleswhateverthepowerspectracomput-
In order to apply the Brunt & Federrath method to real obser- ingscheme.Thus,resultssuchastheunusuallyhighsolenoidal
vationaldata,wewereabletoovercomeseveraldifficultiesand fraction at the scale of the whole cloud, or the variations of R
sourcesofuncertainty. when zooming out of the star-forming regions, stay valid, and
First,thecomplianceofthedatasetwiththerequirementsof arefurtherdiscussedbelow.
themethodmustbechecked,namelytheisotropyofthestudied
cloud,andits isolation. Thewhole field andthe zoomspresent
5.2.Physicalinterpretation
twooppositesituations.Theisolationcriterioniswellmetinthe
caseofthewholefield:wehavealmostnosignaltotheWestand To our knowledge, this work is the first attempt at apply-
totheSouthofthefield,andverydiffuseregionstotheNorthand ing the Brunt & Federrath method to actual observationaldata
to the East (see Fig. 4, first column),so thatthere is almostno (Brunt&Federrath2014; Lomaxetal. 2015). The results need
need for apodizationin order to have the signal goingdown to tobecomparedwithwhatwasdonesofaronnumericalsimula-
zeroontheedgesofthefield.Forthezooms,ontheotherhand, tions.
wearewellintothecloud,so thatthereissignalallthewayto Both Schneideretal. (2013) and our calculations (see
theedgesofthefield.However,sincewestudylocalmaximaof Sect.4.1andAppendix)showthatweareinacontextofhighly
10