Eric Keto & George B. Field, 2005, ApJ, 635, 1151

15 Pages·2005·0.56 MB·English

TheAstrophysicalJournal,635:1151–1165,2005December20 A
#2005.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A.
DARK CLOUD CORES AND GRAVITATIONAL DECOUPLING FROM TURBULENT FLOWS
Eric Keto1 and George Field1
Received2005February25;accepted2005August25
ABSTRACT
Wetestthehypothesisthatthestarlesscoresmaybegravitationallyboundcloudssupportedlargelybythermal
pressurebycomparingobservedmolecularlinespectratotheoreticalspectraproducedbyasimulationthatincludes
hydrodynamics,radiativecooling,variablemolecularabundance,andradiativetransferinasimpleone-dimensional
model.Theresultssuggestthatthestarlesscorescanbedividedintotwocategories:stablestarlesscoresthatarein
approximate equilibrium and will not evolve to form protostars, and unstable prestellar cores that are proceeding
towardgravitationalcollapseandtheformationofprotostars.Thestarlesscoresmightbeformedfromtheinterstellar
mediumasobjectsatthelowerendoftheinertialcascadeofinterstellarturbulence.Inaddition,weidentifyathermal
instabilityinthestarlesscores.Underparticularconditionsofdensityandmass,acoremaybeunstabletoexpansion
if the density is just above the critical density for the collisional coupling of the gas and dust so that as the core
expands,thegas-dustcouplingthatcoolsthegasisreducedandthegaswarms,furtherdrivingtheexpansion.
Subject headings: ISM: individual (L1544, B68, L1517B) — ISM: molecules — radiative transfer—
stars: formation
Online material: color figures
1. INTRODUCTION clouds, and star formation takes place within these collapsing
clouds within one crossing time (Elmegreen 2000). The ineffi-
Observations ofmolecular clouds show a power-law depen-
ciencyorslowrateofstarformationisthenattributedtothein-
dence between the size (length and mass) scales of the clouds
efficiencyintheformationofthesegravitationallyboundcloudsin
andthevelocitydispersionwithinthecloudsthatextendsfrom
theturbulentinterstellarmedium.
thelargestgiantmolecularcloudsdowntothescaleatwhichthe
Whilethetheoryoftheturbulentinterstellarmediumdoesnot
turbulentvelocitiesbecomesubsonic(Larson1981;Leungetal.
predictanymolecularcloudsinequilibrium,theobservedprop-
1982;Myers1983;Sandersetal.1985;Dameetal.1986;Fuller
ertiesofthesmallmolecularcloudsatthelowendofthecloud
&Myers1992).Thespectrummatchesthatexpectedinaturbu-
massspectrumknownasstarlesscoresareremarkablywellde-
lent cascade, suggesting that the clouds are hierarchical struc-
scribedasdiscretedynamicalunitsofstable,boundgaswithan
turesinasupersonicturbulentflow(MacLow&Klessen2004;
internalbalanceofforces.Thesestarlesscoresaredenseregions
Elmegreen&Scalo2004;Scalo&Elmegreen2004).Thepower-
(n (cid:1)104–106cm(cid:2)3)indarkcloudswithlinearscalesoftenths
lawrelationshipsbetweenthemass,length,andvelocitydisper- H2
ofpcandtotalmassesofafewsolarmasses.Thestarlesscores
sionalsofittherelationofvirialequilibrium,(cid:1)2(cid:1)2GM/L,asif
v contain no infrared sources above the sensitivity level of the
the cloudsweregravitationally bound structures. Ina picture of
Infrared Astronomical Satellite (IRAS; about 0.1 L at the dis-
theinterstellarmediumdominatedbyturbulence,therelationship (cid:3)
tanceofTaurus)andthusarethoughttobesitesofpossiblefu-
ofapparentvirialequilibriumreflectsadynamicequipartitionasa
tureratherthancurrentstarformation(Myersetal.1983;Myers
resultofthecouplingbetweenkineticandpotentialenergiesinthe
&Benson1983;Benson&Myers1989;Beichmanetal.1986;
turbulentflowratherthanastaticequilibriumwithingravitation-
Ward-Thompson et al. 1994; Tafalla et al. 1998; Lee & Myers
allyboundclouds(Larson1981;Ballesteros-Paredesetal.1999;
1999;Leeetal.2001).
Klessenetal.2005).Whilegravitymaybelargelyresponsiblefor
Observations of dust column density in the starless cores
generatingthesupersonicflowsthatformtheclouds,theclouds
appearapproximatelyasexpectedforcoresinhydrostaticequi-
themselvesaretheresultofcompressionduetoinertialforcesin
librium. The observed density profiles are characterized by a
theturbulenthydrodynamics.Cloudsareformedwhereconverg-
core-envelopestructurewithaninnerregionofweaklydecreas-
ingstreamsintheturbulencecreatezonesofhigh-densitygas,but
ing density and a surrounding envelope with a steeper density
thesecloudsformedbycompressionmayjustaseasilydissipate
gradient (Ward-Thompson et al. 1999; Bacmann et al. 2000;
astheflowschangedirectionandvelocityonacrossingtimescale.
Shirleyetal.2000,2002;Evansetal.2001;Youngetal.2003;
In the inertial range of the interstellar turbulence, there are no
Ketoetal.2004).Detailedobservationsofsomeindividualdark
cloudswithaninternalequilibriumbetweenself-gravityandthe
clouds show density profiles that appear to match within a few
supporting forces of thermal, turbulent, or magnetic pressure.
percentthoseofpressure-confined,hydrostaticspheres(Bonnor-
Gravitationally bound clouds may form if the compression is
Ebertspheres;Bonnor1956;Alvesetal.2001;Tafallaetal.2004).
strongenoughtoboostthelocalgasdensitybeyondgravitational
Whiletheobservedmorphologiesofthestarlesscoresarenotal-
instability.However,thesecloudswillbeshort-lived,collapsing
wayssphericalandaspectratiosof2:1arecommon,theobserved
toformstarswithinafree-falltime.Inthetheoryoftheturbulent
densityprofilesdonotdeviatesignificantlyfromthoseexpected
interstellar medium, these are the only gravitationally bound
innear-equilibrium.
Radio-frequency spectral line observations that reveal the
1 Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge, gas velocities within the cores confirm this near-equilibrium
MA02138. state.Theobservedspectrallinesofallthestarlesscoresarequite
1151
1152 KETO & FIELD Vol. 635
narrowwithnowingsandnearlythermalwidths,anindication Our research thus aims to reduce the ambiguity of the com-
thatanygasvelocitiesareafewtenthsofthesoundspeedorless parison by considering information that could be derived from
(Zhou et al. 1994; Wang et al. 1995; Gregersen et al. 1997; molecularlineobservationsofthestarlesscoresinadditiontothe
Launhardtetal.1998;Gregersen&Evans2000;Leeetal.1999, densitystructurederivedfromobservationsofthedustcontinuum.
2001, 2004a; Alves et al. 2001; Keto et al. 2004). The cores Sincewemodelonlyindividualcloudsinquasi-equilibriumrather
contain a variety of gas motions: for example, contraction, ex- thantheentirelargerscaleturbulentcascade,wecanincludein
pansion,orbothwithindifferentregionsofthecore;buttheob- ourmodelsphysicssuchastheradiativeequilibriumofthedust
servationthatthegasvelocitiesareallsubsonicimpliesthatthe and gas and the depletion of molecules from the gas phase by
inertial forces are small and that there is near balance between freezingontodustgrainsthat,withthepresentcomputingpower,
thethermalandgravitationalforces. couldnotbefullyincorporatedintoathree-dimensionalhydro-
Astateofnear-equilibriumasindicatedbyboththedustand dynamic simulation. Finally, our models can be run at much
spectral line observations would allow the starless cores to be higherspatialresolution.Thelevelofdetailprovidedbythead-
long-lived,apparentlyatoddswiththetransiencerequiredofthe ditional physics and resolution improves the predictive power
cloudsinthetheoryoftheturbulentinterstellarmedium.How- ofourmodelsandreducestheambiguityofthecomparison,al-
ever, there might be no conflict between the two concepts of lowingustoimproveontheresultsofBallesteros-Paredesetal.
clouds as either transient entities or as pressure-supported dy- (2003).
namicalunitsifthelargercloudsarewithinandthestarlesscores The results of our comparison suggest that the observations
are below the inertial range of the supersonic turbulence. The ofthestarlesscoresarewellexplainedbymodelsincloseequi-
idea that the starless cores with their narrow line widths are librium. For a given external pressure, the equilibrium may be
cloudsatthesubsonicbottomoftheturbulentcascadehasbeen stable or unstable to gravitational collapse, depending on the
discussedintheliteraturesinceLarson’sdescriptionofthesize– massofthecore.Itseemsreasonabletosupposethattherewillbe
line width relationship (Larson 1981; Padoan 1995; Goodman adistributionintheinitialmassesofthecoresbecausethecores
et al. 1998; Va´zquez-Semadeni et al. 2003). What has evolved are formed at the subsonic scale at the bottom of the turbulent
since the earliest description of the interstellar medium as a cascadethatitselffollowsapower-lawdistributioninsizescale
turbulentcascadeistheconceptofthelargerscalecloudsinthe intheinertialregime.Coresthataremoremassiveanddenseat
inertial range of supersonic turbulence as purely transient phe- the time offormation (n >105 cm(cid:2)3 and a few M ) will be
H2 (cid:3)
nomenaentirelyoutofequilibrium.Thequestionnowiswhether ultimately unstable togravitational collapse. Lessdenseexam-
allcloudsintheinterstellarmedium,includingthestarlesscores, ples may be in stable equilibrium, but stability does not imply
aretransitory. thatthecoresmustbestatic.Forexample,stablecoresmayos-
We propose to test the hypothesis that the starless cores are cillateinsizeanddensityaroundanequilibriummean.Whilethe
clouds in quasi-equilibrium by comparing the predicted char- unstablecoresfollowaprogressiontowardgreaterdensitiesand
acteristicsofmodelnear-equilibriumcoresagainstobservations. infall velocities, there is no evolutionary sequence linking the
Ifthemodeledcoresprovideanadequatedescriptionoftheob- stabletotheunstablecores.Thelessdensestablecoreswillnot
servations,thenwewouldconcludethatquasi-equilibriumstruc- ofthemselvesprogresstowardgreaterdensityandultimatecol-
turescananddoexistintheinterstellarmedium.Giventhatthe lapse,butmaypersistindefinitelyuntilexternalconditionschange.
interstellar medium is dominated on larger scales by the struc- Inthispicturetherearetwoclassesofcores,andsomeofthein-
turesofsupersonicturbulence,thenthestarlesscoresmustbethe efficiencyorslowrateofstarformationisduetotheformationof
members at the bottom of the hierarchy, with scales below the asignificantfractionofthepopulationofcoresthatarestableand
sonicscale. notinclinedtoevolvetoformstars.
Asimilarapproachofcomparingmodelcloudsintheturbu-
lent hierarchy with observations was discussed in Ballesteros- 2. PHYSICS OF THE STARLESS CORES
Paredes et al. (2003). That research examined the density
2.1. Hydrostatic and Radiative Equilibrium
structures of a number ofclouds produced in a numerical sim-
ulationofinterstellarsupersonicturbulenceandcomparedtheir Given the power-law relationship (cid:1) (cid:1)r1/3 between the tur-
v
structures to the density profiles of pressure-supported, self- bulentvelocity,(cid:1) ,andthecloudlengthscale,r,thesizescale
v
gravitatingclouds.Whilenoneofthemodelcloudswereinequi- where (cid:1) isthe sound speed defines the fragmentation scale of
v
librium, a number of them had density structures that matched theturbulentflow(Larson1981;Padoan1995;Goodmanetal.
thoseexpectedofcloudsinequilibrium.Thus,onewouldcon- 1998; Va´zquez-Semadeni et al. 2003). Below this length scale,
clude that observations of the density structure of clouds is an theturbulentvelocitiesaresubsonic,andtheturbulentenergyis
ambiguous test of their state of equilibrium. While that com- lessthanthethermalenergy.Thefactthatthethermalpressure
parisonwasmotivatedbyobservationsofthedustcontinuumin dominates the dynamics immediately leads to a qualitative de-
starless cores, an observation that typically provides only the scription of the structure of subsonic cores that matches the
densitystructure,therearemanyobservationsofmolecularlines observed properties of the starless cores. Subsonic turbulence
instarlesscoresthatprovideinformationonthevelocitystruc- cannot produce substructure within the core, and therefore the
ture,thetemperature,andthechemistryofthecores.Withthis density profile must be smooth and monotonic. Moreover, the
additional information, the comparison is less ambiguous. For density gradients must be characteristic of thermal pressure
example, the clouds in Figure 9 of Ballesteros-Paredes et al. support,becausetheinertialforcesaresmallunlessthecoresare
(2003)wouldneverbeconfusedwithequilibriumstructuresbe- infree-fallcollapse.Thus,theremustbeapproximateforcebal-
cause ofthesupersonicvelocitieswithinthe model clouds. The ance,withthermalpressureratherthanturbulentpressureasthe
dynamicsofcloudswithsupersonicvelocitiesaredominatedby dominantsupportagainstgravity.
inertialforcesandnottheirinternalpressure.Therefore,‘‘obser- Ifoneimaginesaboundarytothecore,thisleadstoadescrip-
vations’’ofthesemodelcloudsthatincludedspectroscopywould tion of the starless cores as hydrostatically supported spheres
indicatethattheycannotbeinequilibrium. confinedbyanexternalpressure.Ifthegasisisothermal,these
No. 2, 2005 DARK CLOUD CORES AND GRAVITATIONAL DECOUPLING 1153
spheresaregenerallyreferredtoasBonnor-Ebert(BE)spheres areconventionallycharacterizedbythecentraldensity,butwe
describedbythesolutionoftheLane-Emdenequation,truncated usethetemperatureattheboundarybecausetheboundarytem-
atsomeradius,withanexternalpressurethatissetequaltothe perature,unlikethecentraltemperature,isnearlyindependent
internal pressure, P¼nkT, at the truncation radius (Bonnor ofthemassanddensityofthesphere.Alinearscalingconstant,
1956).Becausethecoresareembeddedinlargerscalemolecular a,relatedtotheratioofthermaltogravitationalenergy,isused
clouds,theyarecontinuouswiththesurroundingmoleculargas, tonondimensionalizethelengthandmass:
withnosharpboundariesinpressure,temperature,ordensity.In
particular,thenotionofthestarlesscoresasBEspheresdoesnot a¼(cid:1) kTR (cid:2)1=2; ð6Þ
require or imply a hot, rarefied interstellar medium around the Gm4(cid:3)(cid:2)
c
cores. Because the velocity dispersion in the molecular clouds
increaseswiththelengthscale,thereissomescalearoundacore (cid:5)¼r=a; ð7Þ
wherethe turbulence becomes supersonic, thermal pressure no M(r)
longerdominates,andthedynamicsaredominatedbytheiner- z¼ : ð8Þ
4(cid:3)a3(cid:2)
tialforces.Thisscaledefinestheboundaryofthecore.However, c
thisdefinitionismoreconceptualthanprecise,andinpractice,
From the equations above, one derives two coupled ordinary
theboundariesinourmodelsarechosenmoresimply;forexam-
differentialequations,whichwesolvewithasecond-orderRunge-
ple,tosetthetotalmassofthecore.
Kuttaalgorithm,thatdefinethedensityofasphereinhydrostatic
Whether a core in equilibrium can be long-lived depends on
equilibriumforthetemperatureprofile,(cid:4),determinedbyradiative
itsdynamicalstabilityagainstgravitationalcollapse.Thegravita-
equilibrium:
tionalstabilityofanisothermalsphereisdescribedbyBonnor’s
criterionthatthepressureattheboundaryofthecoreshouldin- dz
¼(cid:5)2(cid:1); ð9Þ
creaseasthevolumeofthecoreisreduced.Similartothecaseof
d(cid:5)
anisothermalsphere,thestaticstabilityofanonisothermalsphere,
(cid:1) (cid:2)
which might include somenonthermal internal pressure, can be d(cid:1) 1 (cid:1) d(cid:4)
¼(cid:2) z þ(cid:1) : ð10Þ
determinedbythesignofthechangeinboundarypressureasthe d(cid:5) (cid:4) (cid:5)2 d(cid:5)
volumeofthecoreisreduced.Becausethetemperaturevariation
inthestarlesscoresvariesbyonlyseveraldegreesaroundanav- Closely following several previous papers on the radiative
eragetemperatureofabout10K,astabilityanalysisofthenon- equilibriumofthedarkcloudcores,wecalculatetheradiative
isothermalmodelsshowsaresultthatissimilartotheisothermal equilibriumofthegasassumingthatthegasisheatedbycosmic
case,althoughthecriticalvaluesaresomewhatdifferent. rays, cooled by molecular line radiation, and either heated or
Inx3,wedescribeourstaticmodelforstarlesscoresasspheres cooled by collisional coupling with the dust, depending on
in hydrostatic and radiative equilibrium. This model serves as whetherthedustishotterorcoolerthanthegas(Larson1973,
thestartingpointforthecalculationsofthehydrodynamicevolu- 1985;Evansetal.2001;Shirleyetal.2002;Zucconietal.2001;
tion, but it also provides a reasonable approximation to the in- Stamatellos&Whitworth2003;Gonc¸alvesetal.2004).Forthe
ternalstructureofastarlesscoreatanypointinitsevolution.This dust,
followsbecausethecoolingtimeofacoreismuchshorterthan
its dynamical time and because the cores are always in near- (cid:1)ISR(cid:2)(cid:2)d ¼0; ð11Þ
equilibrium,unlesstheyareingravitationalfreefall.Thiswillbe
justifiedlaterwiththeresultsoftheevolutionarycalculations. where (cid:1) is the rate at which dust grains are heated by the
ISR
external interstellar radiation field and (cid:2) is the rate at which
d
3. MODELS OF HYDROSTATIC SPHERES grainscoolbyblackbodyradiationmodifiedbythedustemis-
IN RADIATIVE EQUILIBRIUM sivity.Theequilibriumgastemperatureisdefinedsimilarlyas
thetemperatureatwhichthenetgascoolingrateLiszero,
Thehydrostaticequilibriumisdefinedbytheusualequations
fortheforcebalancebetweenpressureandgravity,themassof
L¼(cid:1) (cid:2)(cid:2) (cid:2)(cid:2) ; ð12Þ
CR line gd
thesphere,andtheequationofstate(Chandrasekhar1939):
dP GM(r)(cid:2) where(cid:1)CRistherateofheatingbycosmicrays(eq.[13]),(cid:2)line
¼ ; ð1Þ istherateofcoolingbymolecularlineradiation(eq.[14]),and
dr r2
(cid:2) istherateofenergytransferbetweenthegasandthedustby
gd
dM
¼4(cid:3)r2(cid:2); ð2Þ collisions(eq.[15]).Inequation(11),thelackofacorrespond-
dr ing term for the collisional coupling is an approximation re-
flecting the much higher rate of energy transfer by the dust
P¼kT(cid:2)=m; ð3Þ
throughabsorptionandemissionofradiation(Goldsmith2001;
Gonc¸alvesetal.2004).Thisallowsthedusttemperaturetobe
where m is the average mass per molecule, including helium
calculatedindependentlyofthegastemperature.Inequilibrium,
andheavierelements.Theseequationsarenondimensionalized
thetotalradiativegascoolingrateL¼0,butduringthehydro-
bydefining
dynamicevolution,wheretheradiativegascoolingrateenters
(cid:4)¼T=T ; ð4Þ into the equation for the change in internal energy of the gas
R
(eq.[20]),thisrateisnotgenerallyzero,becausethegastem-
(cid:1)¼(cid:2)=(cid:2) ; ð5Þ peratureisusuallydifferentfromitsvalueinradiativeequilib-
c
riumbecauseofcompressionorexpansionofthegas.
where(cid:2) andT areareferencedensityandtemperature.Weuse The dust temperature at each point in the cloud is calcu-
c R
the densityatthe centerof the sphere because the BEspheres lated by assuming radiative equilibrium between the incoming
1154 KETO & FIELD Vol. 635
Fig. 1.—Curve of volume and boundary pressure of a 5 M sphere in Fig.2.—SameasFig.1,exceptthatthesphereisisothermalat10K.[See
(cid:3)
hydrostaticequilibriumandradiativeequilibriumparameterizedasafunction theelectroniceditionoftheJournalforacolorversionofthisfigure.]
ofthegasdensityinthecenterofthesphere.Thelogarithmofthegasdensity
ismarkedatpointsalongthecurve.Thespheresaresupportedagainstgravity
bythethermalpressureoftemperaturesrangingfrom7to14K.Thespheres throughtransitionsoflessabundantspeciessuchas13CO.Thus,
withcentraldensities,volumes,andboundarypressuresthataretotherightof
while the optical depth of one transition itself is a sensitive
themaximumpressureareinstableequilibrium.Thosespheresonthecurveto
theleftofthepressuremaximumareinunstableequilibrium.[Seetheelec- functionoftheassumedvelocitygradient,thetotalcoolingrate
troniceditionoftheJournalforacolorversionofthisfigure.] isalesssensitivefunction,sinceoverawiderangeofdensityand
velocity gradient there are always some transitions of optical
depthunity.
interstellar radiation field that heats the dust and the infrared
Energyistransferredbetweenthegasanddustbycollisional
emission that cools the dust (Zucconi et al. 2001; Gonc¸alves
coupling. If we assume a Maxwellian distribution of random
et al. 2004). Our calculation uses the parameterization of the
velocitiesinthegas(Burke&Hollenbach1983)andagrainsize
interstellar radiation field of Black (1994) and the parameteri-
distributionthatweassumefollowsthe(cid:2)3.5powerofthegrain
zationofthedustopacitiesofOssenkopf&Henning(1994)that
radiusfrom0.03to0.3(cid:8)m(Kru¨gel&Siebenmorgen1994)and
arederivedanddescribedinZucconietal.(2001).Theincoming
agas-to-dustmassratioof100,
interstellar radiation is attenuated by the overlying molecular
gas,andthecalculationoftheincomingradiationfieldinvolves
theopticaldepthsfromeachpointinthecoretothesurfaceofthe (cid:2)gd ¼10(cid:2)33n2H2T1=2(Tg(cid:2)Td) ergs cm(cid:2)3 s(cid:2)1: ð15Þ
coredefinedbythetruncationradiusforthenonisothermalBE
sphereandaveragedoverraysatallangles.Becauseofthelow ThisisessentiallythesamerateaswasusedbyZucconietal.
temperatures in the cores, the dust radiates primarily in the far (2001), Goldsmith (2001), and Gonc¸alves et al. (2004).
infrared,andthecoreisassumedtobetransparenttothislonger Anytwoofthethreeparametersofcentraldensity,totalmass,
wavelengthradiation. orexternalpressurealongwithatemperatureprofileissufficient
FollowingFalgarone&Puget(1985)orGoldsmith(2001),we to specify the solution of the Lane-Emden equation for hydro-
settherateofenergytransferfromcosmicraysintothegasas static equilibrium. The temperature profile may then be deter-
mined for a given density profile by solving for the radiative
(cid:1) ¼10(cid:2)27n ergs cm(cid:2)3 s(cid:2)1: ð13Þ
CR H2 equilibrium of the gas and dust. A few alternating solutions of
the hydrostatic and radiative equilibrium equations result in a
The molecular line cooling follows Goldsmith (2001) and
nonisothermalspherecharacterizedbyacentraldensityandtotal
uses the parameterized cooling functions for standard abun-
massthatisinequilibriumwithanexternal pressure andinter-
dancesthatarederivedinthatstudy:
stellarradiationfield.
(cid:2) ¼(cid:6)(T =10 K)(cid:7) ergs cm(cid:2)3 s(cid:2)1; ð14Þ
line g 3.1. Results of the Static Modeling
wheretheparameters(cid:6)and(cid:7)aregiveninTable2ofGoldsmith The stability of the nonisothermal cores is summarized in
(2001). Figure1andcomparedtothestabilityofanisothermalcorein
The parameterization is based on the large velocity gradient Figure2.Ineachfigure,theequilibriaonthecurvetotheleftof
approximation,assumingavelocitygradientof0.5or1.0kms(cid:2)1 themaximumequilibriumpressureareunstabletogravitational
pc(cid:2)1.Inthemodelcoresinourstudy,thevelocitygradientsare collapse.Becausethecentraldensitycorrelateswiththeexternal
almostneverthishigh.However,thelargenumberofmolecules pressureandinverselywiththevolume,thestabilityofasphere
withdifferentabundancesandtransitionsmakestheparameter- canalsobedefinedintermsofitscentraldensity.Thecurvesin
izationinsensitivetotheexactvelocitygradient.Molecularline Figures1and2canbethoughtofascurvesofequilibriumpres-
radiationescapesthecloud,coolingthegas,throughthosetran- sureandvolumeparameterizedbydensity.Somerepresentative
sitions that have optical depths of approximately unity. As the central densities areplottedalong thecurves.Acomparisonof
densityincreasesandthecorebecomesopticallythickinalower the curves shows that the critical boundary pressure or central
quantumtransitionofamoreabundantspeciessuchas12CO (1 densityatwhichtheequilibriumbecomesunstableisdifferentin
0),theradiationescapesthroughhighertransitionsof12COand thetwocases,butthecharacterofthecurvesisthesame.
No. 2, 2005 DARK CLOUD CORES AND GRAVITATIONAL DECOUPLING 1155
Fig.3.—SameasFig.1,exceptthatthegashasadditionalinternalenergyequal Fig.5.—SameasFig.4,exceptthatthecentraldensityofthesphereis106cm(cid:2)3.
toone-quarterofthethermalenergy.Thisadditionalenergycouldcomefromsmall- Inthisspherethegastemperatureinitiallyincreasesinwardasthemolecularline
scale turbulence or magnetic fields. With this additional internal pressure, the radiationthatcoolsthegasbecomesineffectivebecauseoftheincreasingoptical
cloudsareabletosupportahighermaximumcentralgasdensityandboundary depth.Asthegasdensityincreasestothecriticaldensityforgas-dustcoupling,the
pressure.[SeetheelectroniceditionoftheJournalforacolorversionofthisfigure.] gastemperatureapproachesthedusttemperature.Atthehighgasdensityofthis
sphere,thesphereisunstabletogravitationalcollapse.
If the gas has additional internal energy, such as might be
providedbysmall-scalesubsonicturbulenceormagneticfields, usuallydefinedastheradiusofthecentralregionoftheobserved
thecriticaldensitiesarehigher.ThestabilitydiagraminFigure3 coreinsidewhichthedensityisapproximatelyconstantandout-
shows the effect of additional internal energy equal to one- side which the density falls as a power law. For example, the
quarterofthethermalenergyofthegas.Theadditionalnonther- densityprofilemaybedescribedasn(cid:1)1/(1þr(cid:6))(Tafallaetal.
malenergyimprovesthestabilityofspheresovertheirthermal 2004) or more simply in terms of two zones, an inner zone of
counterparts. constantdensitywithinaradiusr andasurroundingpower-law
0
Thegasanddusttemperaturesandgasdensityforequilibrium envelope (Ward-Thompson et al. 1994). For cores with total
sphereswithcentraldensitiesof104and106cm(cid:2)3areshownin massesofseveralM ,thewidthoftheconstant-densityregionis
(cid:3)
Figures 4 and 5. These two densities lie on either side of the about2500AUinthecaseofunstableprestellarcoresandabout
criticaldensityforgravitationalinstabilityforacorewithatotal twicethator5000AUinthestablestarlesscores.
massof5M(cid:3).Thefiguresillustratethedifferencesinstructure The gastemperatureinastablesphere with lower density is
predictedforstableandunstablecores.Themostsignificantdif- approximatelyconstantat10K.Althoughthedusttemperature
ferencesinthetwocasesareinthedegreeofcentralcondensa- decreasesatsmallerradiibecausetheoverlyingcloudshieldsthe
tionandinthetemperature. dustinthecenterfromexternalstarlight,thegasanddustarenot
Thedensityprofileoftheunstablecoreshasasmallerradial collisionallycoupled,andthedusttemperaturehaslittleeffecton
widthorhigherdegreeofcentralconcentrationthanthatofthe thegas.Thegascoolsprimarilybymolecularlineradiation,and
stable counterparts. In the observational literature, the width is theincreasingopticaldepthtolineradiationcausesaslightin-
wardriseintemperature.
In the unstable core with high density, the increased optical
depthtotheexternalstarlightreducestheheatingofthedust.Be-
causethedustiscooledbyopticallythinfar-infraredradiation,
thecoolingrateremainsconstant,andthedustbecomescolder
toward the center. At central densities >106 cm(cid:2)3, the extinc-
tion through the core causes the dust temperature to decrease
from17Kattheboundarytobelow8Katthecenter.
Theunstablecorehasahighenoughdensitythatthegasbe-
comesopticallythicktomolecularlineradiationandinaddition
beginstocollisionally coupletothedust.Towardthecenterof
thecore,atdensities>105cm(cid:2)3,thegasanddustarecollision-
ally wellcoupledand thegasiscooled predominantly through
thedust.Inthedensestpartoftheunstablecore,wherethecou-
plingismostefficient,thegastemperatureapproachesthedust
temperature, around 8 K. Outside of the center, the collisional
coupling and hence the gas cooling is reduced as the density
decreases,andthegastemperatureclimbstoabout13K.Atstill
Fig.4.—Dustandgastemperatureandgasdensityasafunctionofradiusfora larger radii the optical depth tothe surface ofthe corebecomes
sphereinradiativeandhydrostaticequilibriumwithacentraldensityof104cm(cid:2)3. lowenoughthatthegascancoolefficientlythroughmolecularline
Thelogarithmofthegasdensityisshownbythelowerline.Thetwoupperlines
radiation,andthetemperaturefallstojustbelow10K.Thecriti-
representthedusttemperature(dashedline)andthegastemperature.Theordinate
axismaybereadaseitherthelogarithmofthedensityincm(cid:2)3orthetemperaturein caldensityofthenonisothermalcoresinradiativeequilibriumis
unitsofK.Thesphereisinstablegravitationalequilibrium. above that of the 10 K isothermal core even though the central
1156 KETO & FIELD Vol. 635
temperatureislowerinthenonisothermalcore.Thisisbecausethe stablestarlessandunstableprestellar,mayhavecoherentinter-
coresinradiativeequilibriumhaveahigheraveragetemperature. nalvelocities. The stable starless cores canexhibit avarietyof
Itiscoincidentaltotheconditionofradiativeequilibriumthat oscillatorymotions,expansionandcontraction,thatdonotresult
thedensitiesatwhichthecoresdevelopastronglyvaryingtem- inthecollapseordissipationofthecore.Theunstableprestellar
peratureprofilearealsothedensitiesatwhichthecoresbecome cores have inward velocities that ultimately evolve to gravita-
unstabletogravitationalcollapse.Nonetheless,thiscircumstance tionalcollapseatfree-fallvelocities.Thetwotypesofcorescan
resultsinausefuldivisionofthepopulationofthestarlesscores be distinguished by their spectral line profiles and widths. The
intotwoclasses.Inthefirstcategoryarecoresthataretrue‘‘star- spectrallinesofunstableprestellarcorestendtoshowstrongly
less’’ cores in that they will not of their own contract to form asymmetric profiles split by self-absorption. Also, the spectral
protostars.Inthesecondcategoryare‘‘pre-protostellar’’or‘‘pre- linewidthsofvolatilemolecularspeciesarebroader intheun-
stellar’’ cores that have not yet formed a protostar, but are un- stableprestellarcoresthantheyareinthestablestarlesscores.
stable to collapse of the core and the formation of a protostar
4. HYDRODYNAMIC MODEL
withinafree-falltime.Wecombinetheobservationalandtheo-
reticaldescriptionsandrefertothefirstcategoryasstablestarless To model the evolution ofthestarless cores,we use a simple
coresandthesecondasunstableprestellarcores.Intheabsence one-dimensionalhydrodynamiccodewithafirst-orderLagrangian
of changing external conditions, there is no evolutionary path discretization and Richtmyer–von Neumann pseudoviscosity.
linking the two categories of cores. Stable cores do not evolve Whilethismethodisnotassophisticatedasthepiecewisepara-
tobecomeunstable. bolicmethod(PPM;Colella&Woodward1984)usedinoneof
thepreviousdynamicalstudiesofBEspheres(Foster&Chevalier
3.2.Comparison to Observations
1993),theimprovementincomputerspeedallowsourhydrody-
Examplesofwell-studiedstablestarlesscoresincludeB68and namiccodetoberunwithafactorof10moregridpointsthanin
L1517B. Well-known cores in the unstable prestellar category that previous study, and this allows better spatial resolution de-
includeL1544andL1521F.Estimatesofthecentraldensityinthe spitetheinherentsmoothinginthepseudoviscositymethod.Also,
stablestarlesscoreL1517Bare2:2;105cm(cid:2)3fromdust(Tafalla inourstudytheresolutionisnotasimportantasintheFoster&
et al. 2004) and 105:1(cid:4)0:3 cm(cid:2)3 from the molecular line N H+ Chevalierstudy.First,ourinvestigationconcernsonlythestability
2
(Ketoetal.2004).ThewidthofthedensityprofileinL1517Bis ofstarlesscores.Thus,wefollowtheevolutionofcoresonlytoa
3500(0.025pc)fromdust(Tafallaetal.2004)and0:022þ0:008 pc scale of AU, which is sufficient to determine the stability and
(cid:2)0:016
from N H+. In the unstable prestellar core L1521F, the central evolutionarypathofthecore.Forthetypicalconditionsinstarless
2
density is estimated at 106 cm(cid:2)3, with a width of 0.017 pc for cores,mostofthevelocitiesaresubsoniconthisscale.Incontrast,
the central region of the core (Crapsi et al. 2004). The density the previous calculations of Foster & Chevalier also sought to
structure of L1544 appears similar, with a central density of characterize the asymptotic behavior of the flow at the time of
105:7(cid:4)0:4cm(cid:2)3andawidthof0:004(cid:4)0:002pcfromtheobser- formationofapointsourceattheflowcenter.Nearagravitational
vationsofKetoetal.(2004)and106cm(cid:2)3withawidthof2000 pointsource,theflowcanacceleratetoarbitrarilyhighMachnum-
(0.015pc)fromtheobservationsofTafallaetal.(1998).Ingen- bers.Incontrast,todeterminethefateofacore,oursimulations
eral,thestablestarlesscoreshavelowerdensitieswithbroader neednotfollowevolutionmuchpastthepointwhentheflowhas
densityprofilesthantheunstableprestellarcores. accelerated to more than unity Mach number. Supersonic infall
The theoretically predicted decrease in dust temperature has velocitiesimplythatthecoreisinfree-fallcollapse,anditsfate
been observed in several cores. Infrared observations show a issealed.
dust temperature profile that decreases from about 15 K at the Thetypicalcoresizeinourmodelsis0.2pc.Thegridisinitially
edge of the core to 8 K in the center (Ward-Thompson et al. linear with approximately 5000 points, yielding a resolution of
2002;Paganietal.2003,2004). about10AU.Thesoundspeedisabout0.2km(cid:2)1,sotheCourant
Molecularlineobservationsofcoresthoughttobestableand condition implies a maximum time step of about 250 yr. The
not currently evolving toward star formation, such as L1517B simulationsarerunwithatimestepof1/10themaximumindi-
and B68, show nearly constant gas temperatures with little catedbytheCourantcondition,initiallyabout25yr.Attheendof
variation across the core. Observations of L1517B in NH in- the evolution of an unstable core, the typical maximum spatial
3
dicateagastemperatureof10K(Tafallaetal.2004),whileob- resolutioninthecenterofthecoreislessthan1AU,withatime
servations of N H+ indicate an average temperature of 9þ7 K resolutionoflessthan1yr.Aswillbediscussedbelow,thelim-
2 (cid:2)2
(Ketoetal.2004).TemperatureestimatesofB68rangefrom10 iting resolution in these simulations is not that of the hydrody-
to16K(Alvesetal.2001;Ladaetal.2003). namics,butoftheradiativetransfer.
Observations ofthe core L1544 indicate anaverage temper- ThehydrodynamicalequationsinLagrangianformare
atureof8.75K(Tafallaetal.1998)fromNH linesand11(cid:4)4K
3
(Keto et al. 2004) from N2H+, both consistent within the ob- Z t
servational uncertainties with the models, but the spectral line r(s; t)¼sþ v(s; t)dt; ð16Þ
observationsdonothavetheangularresolutionorthesensitivity 0
todefinethetemperatureprofileofthegaswiththedetailofthe 1 1 (cid:3)r(s; t)(cid:4)2@r
models. In general, temperature estimates of the stable starless ¼ ; ð17Þ
(cid:2)(r; t) (cid:2)(s; 0) s @s
coresareafewdegreeswarmerthanthoseoftheunstablepre-
stellar cores, consistent with the higher temperatures predicted P(r; t)¼(cid:2)(r; t)kT=m; ð18Þ
bythemodels.
Inordertounderstandthevelocitystructureofthecoresandto @v 1 (cid:3)r(s;t)(cid:4)2@(Pþq) GM(r)2
¼(cid:2) (cid:2) ; ð19Þ
interpretspectrallineobservations,itisnecessarytomodelthe @t (cid:2)(s; 0) s @s r
hydrodynamicevolutionofthecores.Inxx4and6,respectively,
@E Pþq@(cid:2) L
we develop the hydrodynamic model and present some exam- ¼ (cid:2) : ð20Þ
plesoftheevolutionofcores.Wefindthatbothclassesofcores, @t (cid:2) @t (cid:2)
No. 2, 2005 DARK CLOUD CORES AND GRAVITATIONAL DECOUPLING 1157
In these equations, s is the original position of each 5. RADIATIVE TRANSFER AND CHEMISTRY
Lagrangian cell, r(s, t) is the corresponding position at time
Afterthenumericalhydrodynamicevolutionofthecloudhas
t, M(r) is the mass contained within radius r, E is the inter-
beencompleted,thetemperature,density,andvelocityasafunc-
nal energy of the gas, L is the radiative gas cooling rate de-
tionoftimearetransferredtoanon-LTEaccelerated(cid:2)-iteration
fined in equation (12), and q is the Richtmyer–von Neumann
(ALI) radiative transfer code to determine model spectral line
pseudoviscosity
profiles for comparison with observations (Keto et al. 2004).
( Themodelcoresareassumedtobeat150pc,thedistancetothe
l2(cid:2)½@vðs; tÞ=@s(cid:5)2 when @v=@s<0;
q¼ ð21Þ Taurusstar-formingregion,andtobeobservedwithatelescope
0 when @v=@s>0; withabeamwidthof2000.Becausethecoreradiiareontheorder
of10beams,theresolutionisnotcriticalexceptinthecasesof
wherelisaconstantthatisapproximatelytheshocksmoothing the unstable cores that have rapidly increasing density and ve-
widthoverthegridspacing.Theseequationsarediscretizedas locitygradientsintheircenters.
inRichtmyer(1957). Tocomputethespectrallineprofiles,wealsoneedtoknowthe
In our calculations of the hydrodynamic evolution of the abundancesofthemolecules,inadditiontotheinformationpro-
cores,westartwithacoreinhydrostaticequilibriumandinra- vided by the hydrodynamical evolution. The gas-phase abun-
diativeequilibrium.Inthesubsequenthydrodynamicevolution, dancesofmoleculesarevariablewithinthecore,becauseinthe
thetemperatureofthegasiscalculatedateachtimestep,taking colddenseinteriorsofthedarkcloudcores,moleculesfreezeout
intoaccountthePdV workofcompressionorexpansionandthe ofthegasphaseontothesurfacesofdustgrainsatdifferentchar-
rateforradiativecooling,L(eq.[12]).Usingtheparameterized acteristicdensitiesaccordingtotheirdifferentvolatilities(Brown
approximationsforthegascooling,thechangeininternalenergy et al. 1988; Willacy & Williams 1993; Hasegawa et al. 1992;
maybecalculatedforeachgridcellusingonlythelocalvaluesof Hasegawa & Herbst 1993; Caselli et al. 1999, 2002a, 2002b,
densityandvelocity(compression)ofthegasandthetempera- 2002c;Berginetal.1995,2002;Tafallaetal.1998;Aikawaetal.
turesofthegasanddust.Thedusttemperatureitselfdependson 2001). We follow Bergin et al. (2001) and Tafalla et al. (1998,
theglobalstructureofthecore:theangle-averagedopticaldepth 2004)andapproximatethedependenceoftheabundanceonthe
tothesurfacefromeachradialpoint.However,becausethelarger densitynasanexponential:
scalestructureofthecorechangesrelativelyslowlycomparedto
thetimestepofthehydrodynamicevolution,setbytheCourant X(n)¼X0exp((cid:2)n=nd); ð23Þ
condition,theangleaveragingtodeterminethedusttemperature
neednotbecomputedateverytimestep.Inaddition,becauseof with parameters for CS of abundance X0 ¼10(cid:2)8 and density
the approximation that the dust temperature is set only by the nd ¼104 cm(cid:2)3. A more complete treatment of the molecular
radiative equilibrium of the dust and is independent of the gas abundancesofthestarlesscoresderivedfromdetailedchemical
temperature, the dust temperature must be a smoothly varying modelingisdiscussedinLeeetal.(2004b).Thatstudyconcludes
functioninthecore.Thus,thedusttemperaturecanbecomputed that a simple parameterized model of depletion, as adopted in
less frequently and on a coarser grid and interpolated onto the oursimulations,isareasonableapproximationtotheirmorede-
hydrodynamicgrid. tailedanalysis.
Theinitialcorealwayscontainssomeperturbationsthatcause
6.MODEL CORES
some initial movement of the gas in the core. These perturba-
tions,atafractionofapercentlevel,areprimarilyduetocoarse Weadoptasastandardmodelacorewithatotalmassof5M
(cid:3)
griddingofthedustequilibriumandareonthescaleofthecoarse andconsideritsevolutionunderavarietyofconditionsthatcould
grid,about1/20ofthesizeofthecore.Theperturbationsarise beexpectedinandaroundthedarkcloudcores.Foragivenmass,
becausetheinitialdustandgastemperaturesexactlymatchcon- thecentraldensitydeterminesthegravitationalstabilityofanon-
ditions of hydrostatic and radiative equilibrium on the coarse isothermal BE sphere. We model the evolution of cores with a
grid, but the interpolated dust temperatures and thus the inter- rangeofcentraldensitiesn ¼103 107 cm(cid:2)3centeredaround
H2
polatedgas temperatures donot exactly match the gastemper- the critical density for gravitational instability, approximately
aturerequiredforhydrostaticequilibriumbetweengridpoints.In n (cid:1)104 cm(cid:2)3foracoreof5M .
H2 (cid:3)
bothstableandunstableclouds,theperturbationsaresmoothedin Acorecouldbesubjecttoavariationinexternalpressurefrom
atimescaleof(cid:9)/c (cid:1)104yrbysmallchangesintemperatureand a nearby supernova, bipolar outflow, or the turbulence of the
s
density that quickly establish radiative and hydrostatic equilib- interstellarmedium.Achangeinexternalpressurecouldcause
rium.Inthegravitationallystableclouds,oncetheperturbations oscillationsinacore,andasufficientincreaseinexternalpres-
are smoothed out, no further evolution takes place. In gravita- sure could cause a previously stable core to collapse. Accord-
tionallyunstablecores,eventhoughthe initial perturbations are ingly, we model the evolution of some cores in response to
quickly smoothed, the coreisunabletoestablishanexactequi- changingexternalpressure.
librium,andafteralongtime,severalsoundcrossingtimesortens Thedarkcloudcoresmayhavesomecomponentofmagnetic
offree-falltimes,the corewillbegingravitationalcollapse.The or turbulent support in addition to thermal support. In a one-
lengthoftimeforthecoretobegincollapseissimplythetimere- dimensionalmodel,thisadditionalsupportcanbeapproximated
quiredforverysmallperturbationstoreachanyappreciablemag- by modifying the equation of state to include a nonthermal
nitude.Oncetheinwardvelocitiesreachevenafewhundredthsof pressure.Thisassumesthatthenonthermalormagneticenergy
thesoundspeed,thecorecollapsesinaboutafree-falltime,tA(cid:1) hasacharacteristicscalemuchsmallerthanthecoresothatthe
ðG(cid:2)Þ(cid:2)1/2(cid:1)105 yr. Were it not for these perturbations deriving effectisatleastapproximatelythesameasanisotropicpressure.
fromthecoarsegriddingoftheradiativetransfer,itwouldbenec- Thisassumptionfollowsfromestimatesthatshowthatthelarge-
essarytointroduceperturbationsintotheinitialstructuresothat scale magnetic fields in cores are too weak to affect their evo-
theunstablecoreswouldcollapsebeforepatiencewiththehydro- lution(Nakano1998).Thus,wemayassumethatthenonthermal
dynamicevolutionwasexhausted. pressureissmall-scaleandisotropic.Ifthenonthermalpressure
1158 KETO & FIELD Vol. 635
isinitiallyproportionaltodensity,thentheinitialstructureofthe configurationthatisbothgravitationallyandthermallyunstable.
core may be determined as before with the Lane-Emden equa- The initial expansion of this core caused by the thermal insta-
tion,butwithahighereffectivetemperaturethaninthecaseof bilityresultsinastablecoreundergoingdampedoscillationsof
support only through thermal pressure. The subsequent evolu- expansionandcontraction.
tion of the core depends on the equation of state of the non-
thermalpressure.Forexample,thecoresthatwemodeledwitha 6.1.Hydrodynamic Evolution of an Unstable
nonthermaladiabaticindexgreaterthan4/3behaveasexpected Pre-Protostellar Core
andnever collapse toapointsource,butformaninnercore of
Thetheoreticalevolutionofagravitationallyunstablecoreis
approximately constant density supported by nonthermal pres-
dynamically as expected from previous simulations: small nu-
sure.Asthecoreevolves,thesurroundingenvelopecontinuesto
merical perturbations grow for several crossing times (several
collapse onto this inner core, creating a steep density gradient
Myr)untiltheyaresufficienttotipthecoreoffitsinitialunstable
and shock at the boundary of the inner core. Such a boundary
equilibrium towardcollapse(Hunter1977;Foster&Chevalier
region is never seen in observations of dark cloud cores, sug-
1993).Sincethecoolingtimeisshortcomparedtothedynamical
gestingthatthemoleculargasinthedarkcloudcoresmusthavea
time, the temperature profile is set almost entirely by radiative
lower value of the adiabatic index. In an analysis of the virial
equilibrium.Figures6–8showthepropertiesofthecoreasthe
theorem,McKee&Zweibel(1995)suggestedthattheadiabatic
collapseproceeds.Alsoshownareaccompanyingmolecularline
index for magnetic turbulence should be between 3/2 and 1/2,
spectraaswouldbeobservedtowardthecenterofthecore.
dependingonthetimeandlengthscalesoftheturbulence.Apar-
Figures6–8showthattheapproximateforcebalancebetween
ticularlyconvenientchoiceisanadiabaticindexofunity.Thisof
pressureandgravitythatissetintheinitialconfigurationismain-
coursematchestheadiabaticindexofthethermalpressure,and
taineduntiltheinfallvelocitiesbegintoapproachthesoundspeed.
thedynamicsarethereforesimilartoathermallysupportedcore
Thus, the density structure continues to approximate that of a
withahighertemperature.However,thenonthermalpressurere-
pressure-supported BEsphere as the core evolves toward free-
mainsproportional todensity only,whilethethermalpressure,
fallcollapse.
being proportional to the temperature as well as the density,
Ourhydrodynamicsimulation,aswellasanalyticconsidera-
changesasthetemperatureischanging.
tions(Whitworth&Ward-Thompson2001),showthatgravita-
We ran about 30 different numerical simulations to explore
tional collapse starting from an equilibrium-configuration BE
differentcombinationsandrangesofparameters.Densitiescon-
sphereischaracterizedbyinwardlyincreasingvelocities.Ave-
sideredinourstudyincludethose
locityprofilewithvelocitiesthatincreaseinwardisnotuniqueto
1. expectedtobestable; thedensityprofileofunstableBEspheresbutisstilladiagnos-
2. nearthecriticaldensityofthenonisothermalBEsphere; tic inasmuch as other initial configurations such as the uniform
3. expectedtobeunstable. spheres considered in Jeans instability and the well-studied sin-
gularisothermalspheres(Shu1977)andsingularlogotropicspheres
Differentequationsofstateconsideredinthisstudyarethose
(McLaughlin&Pudritz1997)evolvetocollapsewithverydiffer-
with
entvelocityprofiles.Inwardlyincreasingvelocitiesareindicated
1. thermalpressureonly; inmolecularlineobservationsofthecoresL1544andL1521Fby
2. thermalpressurewithanonthermalpressureinitiallypro- the velocity gradient inferred from the details of the observed
portionaltodensity: spectrallineshapesandfromthewidthsofspectrallinesofvola-
tilemolecularspeciesthatincreasetowardthecentersofthecores.
a) anadiabaticindexofnonthermalpressureequaltounity; Figures6–8showhowthespectralprofileofN H+developsthe
b) anadiabaticindexofnonthermalpressureequalto4/3. 2
characteristic asymmetric split with a stronger blue peak as the
Differentexternalconditionsappliedtothecloudwere infallvelocityincreases.Thischaracteristicsplitisseenintheob-
servedN H+spectrumfromL1544;forexample,inFigure2of
1. constantexternalpressure; 2
Williamsetal.(1999)andinFigures4and5ofKetoetal.(2004).
2. increasingexternalpressure:
An increase in the line width of N D+ is observed toward the
2
a) externalpressureremainingbelowthecriticalpressure centers of the cores L1544 and L1521F (Caselli et al. 2002c;
ofthenonisothermalBEsphere; Crapsietal.2004,2005).Althoughtheobservationsoflinewidth
b) externalpressuresufficientlyhightocausecollapse. donot indicatethe direction ofthe velocity, the increase inline
widthisindicativeofanincreaseinthemagnitudeofthevelocity
Hydrodynamicmodelswerecomputedforcombinationsofall
towardthecenterofthecore.
the cases above. Despite the variety of initial conditions and
equationsofstate,thebehaviorandpropertiesofthestablestar-
6.2. Hydrodynamic Evolution of a Stable,
lesscoresarequitesimilartoeachother,asarethebehaviorand
Oscillating Starless Core
propertiesoftheunstableprestellarcores.Inxx6.1and6.2,we
discussindetailtwomodelstoillustratethecharacteristicsofthe Hydrodynamic simulations of cores in stable equilibrium
twoclassesofcores.Wemodeltheunstableprestellarcoreswith show no evolution unless the core is subjected to a perturba-
a hydrodynamic simulation that starts from an initial configu- tion.Whileobservationsofmanycoresdonotshowvelocities
rationinunstableequilibriumandevolvestowardfree-fallgrav- abovetheobservationaldetectionlimit,about1/10ofthesound
itational collapse. The stable starless cores are modeled with a speed,othercoresindicateexpandingorcontractingmotionsor
simulationthatstartsinstableequilibriumandisperturbedinto morecomplexcombinationsindicativeofnonsphericalperturba-
oscillatorycontractionandexpansionbyasuddenincreaseinthe tions.Whilesomeofthesecoresmaybeunstableandevolving
external pressure. Finally, in order to illustrate the thermal in- towardcollapseasdescribedinx6.1,othersmaybeoscillating
stability that is related to the critical density for gas-dust colli- aroundanequilibriummean.Thenonsphericalmorphologiesof
sionalcoupling,wediscusstheevolutionofacorewithaninitial many cores, often showing aspect ratios of 2:1, may indicate
No. 2, 2005 DARK CLOUD CORES AND GRAVITATIONAL DECOUPLING 1159
Fig.6.—Evolutionofacorestartingfromaninitialconfigurationthatisinequilibriumbutunstabletogravitationalcollapse.Inthissimulation,thecloudremains
staticforabout3Myr,about3soundcrossingtimes,whilesmallperturbationsoriginatingfromnumericalnoisebuilduptomovethecloudoffitsunstableequilibrium.
Oncethecloudbeginstocontract,thevelocitiesinthecenterofthecloudbecomesupersonicinafew105yrandthecoreistheninunsupportedfreefall.Topleft:Density
(logcm(cid:2)3)andgasanddusttemperatures(K),inthesameformatasFigs.4and5,andalsothevelocity(soundspeedtimes(cid:2)10)asafunctionoftheradius(pc).For
example,avelocityof10ontheleftaxisindicatesaninwardvelocityequaltothesoundspeed.Bottomleft:Gravitationalforce(solidgreenline),pressureforce(dashed
redline),andpressure(solidpurpleline),incgsunits.Thispanel(inthesubsequentfigures)showsthatthecloudremainsinapproximateforcebalanceuntilthecore
beginsfree-fallcollapse.Right:SpectrallinesofNH+(1–0)andNH+(3–2),astheywouldbeobservedinthemodelcore.Theradiativetransfersimulationassumes
2 2
thatthecoreisatadistanceof150pcandthatthetelescopebeamis2000.Eachpanelshowsthehyperfinelinesthatarewithin(cid:4)1kms(cid:2)1.Thesearetheinnerthree
hyperfinesinthecaseofNH+(1–0).
2
oscillations about an equilibrium mean that may be stable or asymmetricprofilewithastrongerbluepeakthatischaracteristic
unstable. of inward motion. In Figure 10, at 1.613 Myr, the wave has
Inourexamplemodelofastablestarless core,webeginthe reflectedoffthecoreandisheadedoutward.Atthistime,mostof
hydrodynamic simulation in hydrostatic and radiative equilib- the core is still moving inward, responding to the increased
rium.Thecorestartswithacentraldensityof2;103cm(cid:2)3,well externalpressure,andthespectrallineprofileofCSstillshows
tothestablesideoftheBonnorcriteriafornonisothermalspheres, an asymmetric line split characteristic of inward motion. By a
soanincreaseinexternalpressurebyafactorof1.5,heldconstant timeof3.705Myr(Fig.11),whenthecontractionofthecorehas
thereafter, pushes the core close to but not past the maximum overshot the equilibrium point, the whole core is expanding
pressureforstability.Theincreaseinpressurepropagatesintothe outward. The spectral line profiles of CS now show an asym-
core,eventuallyreflectingoffthecenter,whilethecoreasawhole metricsplitlinewithastrongerredpeakthatischaracteristicof
adjustsitsdensityprofiletoanewequilibriumwiththeincreased outwardmotion.ObservationsofCSspectrainB68showasym-
external pressure. An initial overshoot of the new equilibrium metricsplitspectraofbothsenses,inwardmotionsandoutward
resultsindampedoscillationsofcontractionandexpansion.The motions, at different positions in the cloud; see, for example,
oscillationsimplythatinertialforcesareimportantinthedynam- Figure6ofLadaetal.(2003).
ics,buttheinertialforcesarestillsmallcomparedtothepressure Model spectra of N H+ from the same hydrodynamic simu-
2
andgravitationalforces,sothecoreremainsinapproximateforce lation show only symmetric profiles with no asymmetric split-
balancethroughouttheevolution. ting.ThedifferencebetweentheCSspectraandtheN H+spectra
2
Figure9showsthestructureofthecoreandtheaccompanying relates to the differences in the velocity fields in the regions
spectra when the head of the velocity and pressure wave has whereeachofthemolecularlinesisgenerated.Inthedensecen-
advanced halfway into the cloud at elapsed time 1.003 Myr. ters of thestarless cores, the CS molecule isdepleted from the
Becauseofthelowdensity,thegascoolsefficientlythroughout gas phase by freezeout onto dust grains, but the more volatile
thevolumeofthecloudbymolecularlineradiation,andthegas N H+ molecule maintains a constant abundance. Because the
2
temperature isnearlyconstant. Thepredictedspectralline pro- linebrightnessforanysubthermallyexcitedmoleculescaleswith
files of an oscillating core are particularly interesting in com- thesquareofthedensity,theN H+lineisformedpredominantly
2
parisonwithobservedprofiles.TheCS(1–0)spectrumshowsan inthedensegasinthecenterofthecore,whereastheCSlineis
Fig.7.—SameasFig.6,butatanevolutionarytimeof3.22Myr.Asthegasdensityinthecenterincreases,thegasbecomescollisionallycoupledtothedustand
thegastemperatureapproachesthedusttemperature.
Fig.8.—SameasFig.6,butatanevolutionarytimeof3.37Myr.Asthevelocitygradientsteepens,withtheinwardvelocityapproachingthesoundspeedinthe
centerofthecloudandthevelocitiesstillnearzeroattheedge,thelow-velocitygasattheedgeofthecloudabsorbstheemissionfromthecenterofthecloud,
creatingasplitspectrum.TheprofileoftheNH+(3–2)lineisdominatedbynumeroushyperfinecomponents.
2
1160