Reliability of NH3 as the temperature probe of cold cloud cores

0.24 MB·English

by  M. Juvela

Astronomy&Astrophysicsmanuscriptno.juvela˙nh3 (cid:13)c ESO2012
January17,2012
Reliability of NH as the temperature probe of cold cloud cores
3
M.Juvela1,J.Harju2,1,N.Ysard1,T.Lunttila1,
1 DepartmentofPhysics,P.O.Box64,FI-00014,UniversityofHelsinki,Finland,mika.juvela@helsinki.fi
2 FinnishCentreforAstronomywithESO(FINCA)
ReceivedSeptember15,1996;acceptedMarch16,1997
ABSTRACT
2
Context. Thetemperatureisacentralparameteraffectingthechemicalandphysicalpropertiesofdensecoresofinterstellarclouds
1
andtheirpotentialevolutiontowardsstarformation.Thechemistryandthedustpropertiesaretemperaturedependentand,therefore,
0
interpretationofanyobservationrequirestheknowledgeofthetemperatureanditsvariations.Directmeasurementofthegaskinetic
2
temperatureispossiblewithmolecularlinespectroscopy,theammoniamolecule,NH ,beingthemostcommonlyusedtracer.
3
n Aims. Wewanttodeterminetheaccuracyofthetemperatureestimatesderivedfromammoniaspectra.Thenormalinterpretationof
a NH observationsassumesthatallthehyperfinelinecomponentsaretracingthesamevolumeofgas.However,inthecaseofstrong
3
J temperaturegradientstheymaybesensitivetodifferentlayersandthiscouldcauseerrorsintheopticaldepthandgastemperature

5 estimates.
1 Methods. Weexamineaseriesofsphericallysymmetriccloudmodels,1.0and0.5M⊙ Bonnor-Ebertspheres,withdifferentradial
temperatureprofiles.WecalculatesyntheticNH3 spectraandcomparethederivedcolumndensitiesandtemperaturestotheactual
] valuesinthemodels.
A Results. Forhighsignal-to-noiseobservations,theestimatedgaskinetictemperaturesarewithin∼0.3Koftherealmassaveraged
G temperatureandthecolumndensitiesarecorrecttowithin∼10%.WhentheS/Nratioofthe(2,2)spectrumdecreasesbelow10,the
temperatureerrorsareoftheorderof1Kbutwithoutasignificantbias.Onlywhenthedensityofthemodelsisincreasedbyafactor
h. ofafew,theresultsbegintoshowsignificantbiasbecauseofthesaturationofthe(1,1)maingroup.
p Conclusions. Theammoniaspectraarefoundtobeareliabletraceroftherealmassaveragedgastemperature.Becausetheradial
- temperatureprofilesofthecoresarenotwellconstrained,thecentraltemperaturecouldstillbedifferentfromthisvalue.Ifthecores
o areopticallyverythick,therearenolongerguaranteesoftheaccuracyoftheestimates.
r
t Keywords. ISM:clouds–ISM:molecules–Radiolines:ISM–Stars:formation–radiativetransfer
s
a
[

1. Introduction haveahyperfinestructure.Forexamplethe(1,1)spectrumcon-
1
v tains18componentsfromwhichatleastfiveseparategroupsof
2 The dense molecular cloud cores are of particular interest be- linesareusuallydiscernible.Theintensityratioofthehyperfine
6 cause their collapse can lead to the formation of new stars. To componentsprovidesaconvenienttooltoestimatethelineopti-
0 understandthestar formationprocess,one needsto understand caldepths.Because ofthese properties,ammoniais a common
3 thephysicalandchemicalpropertiesofthecores.Thetempera- tool for gas temperature measurements, especially in the case
1. tureofthecores,especiallyasafunctionofposition,isdifficult of dense cores (Ho&Townes 1983). The high critical density
0 to measure and yet it affects everything from core stability to (∼2·103cm−3forthe(1,1)transitionascalculatedfromthera-
2 chemistryanddustevolution. tio of the spontaneousand collisional de-excitationrates at the
1 Dustandgasbecomethermallycoupledonlyathighdensi- temperature of 15K) guaranteesthat one is measuring temper-
: ties,n(H )∼105cm−3andabove(Goldsmith2001).Becausethe ature of dense gas. On the other hand, NH is resilient against
v 2 3
continuum surface brightness is strongly affected by the warm depletionandtheabundanceisobservedtodroponlyinthevery
i
X outercloudlayers,observationsofthedustemissionarenotideal densestandcoldestcores(Bergin&Tafalla2007).
r todeterminethegastemperatureatthecentreofthedensecores. The analysis of NH3 is based on the assumption that the
a Thewidthsofmolecularlinescansometimesbeusedtoderive (J,K) transitions and their different hyperfine components are
stringent limits on the gas temperature (e.g., Harjuetal. 2008) alltracingthesamephysicalregion.Becauseofthedifferentop-
butthevalueofthismethodislimitedbythepresenceofturbu- tical depths, the weaker hyperfine components could be more
lentvelocitycomponent. sensitivetodeeperlayersinthecloud.Theeffectshouldbecome
In observationsofmostmolecules,it isdifficultto separate noticeable if the main lines are moderately optically thick and
the effects of density, opticaldepth, and temperature.The situ- thesourcehasstrongtemperaturegradients.
ationisdifferentforammonia,NH ,whoserotationalenergyis Thetemperaturestructureofquiescentcorescan,in princi-
3
characterisedbythequantumnumbers(J,K).Becausethedipole ple, be calculated from the balance of the various heating and
transitions between the different K ladders are forbidden,their cooling mechanisms (e.g. Goldsmith 2001; Gallietal. 2002).
relativepopulationsdependonlyoncollisionsandthusmeasure The gas within the cores is heated by the cosmic rays and the
thegaskinetictemperature.Eachrotationalstate(J,K)isfurther external stellar radiation field and it is cooled down by the
dividedtoinversiondoublets.Theinversiontransitions(1,1)and line radiation from a number of atomic and molecular species.
(2,2) are at 23.7GHz and 23.72GHz respectively,and are eas- Depending on the temperature difference between the gas and
ily observable from the ground. The inversion transitions also the dust, the gas-dust collisions can result in either heating or
1
M.Juvelaetal.:ReliabilityofNH asthetemperatureprobeofcoldcloudcores
3
coolingofthegas.Juvela&Ysard(2011)discussedtherangeof Table1.Themaincoremodels.
possibletemperatureprofilesindensecores,emphasisingtheun-
certaintiesassociatedwithmanyofthecontributingfactors(e.g., Model Modification T rangesofthe0.5
kin
theexternalradiationfield,thegas-dustcoupling,andtheabun- and1.0M⊙models(K)
dancesofthespeciesresponsibleforthemaincoolinglines).In Default – 5.3-9.3,4.6-8.4
thispaperweusethemodelsofJuvela&Ysard(2011)toexam- χ χ=(0.1−1.0) χdefault 5.3-11.0,4.8-11.0
Λ Λ =(0.3-1.0) Λdefault 6.4-10.0,5.3-8.5
ine the reliability of the ammonia temperature estimates in the g,d g,d g,d
presenceofradialtemperaturevariations. σv σv=(0.1-1.0) σdvefault 5.3-10.9,4.8-11.4
v 0-1.0kms−1 5.3-9.1,4.8-8.3
The content of the paper is the following. In Sect. 2 we radial
PEH includeΓ(PEH)for 9.2-14.7,9.2-19.1
present the methods concerning the core models, the calcula-
ζ=3·1017s−1
tionofsyntheticammoniaspectra,andthederivationoftemper-
atureandcolumndensityestimates.Theresultsarepresentedin
Sect. 3, where we compare the derived estimates to the corre-
spondingactualvaluesinthemodels,inparticularstudyingthe
effects of observational noise and increased column densities.
wherethemodelcloudisassumedtobeilluminatedbyanunat-
ThefinalconclusionsarepresentedinSect.5.
tenuatedinterstellarradiationfield(Mathisetal. 1983).Forthe
detailsofthecalculationsoftheT profiles,seeJuvela&Ysard
kin
2. Methods (2011).
Inadditiontotheabovedescribedmodels,weconsidercases
2.1.Coremodels
wherethekinetictemperaturesarethesamebuttheturbulentline
We use the cloud models discussed by Juvela&Ysard (2011). width is decreased by a factor of ten in the NH3 calculations.
These are 0.5M⊙ and 1.0M⊙ Bonnor-Ebert spheres (ξ=6.5, In particular, in the central part of the σV-model the linewidth
T =10K) for which the gastemperatureswere calculated un- then becomes practically thermal. We use the Juvela&Ysard
kin
derdifferenthypotheses.Thisresultedinasetofradialtemper- (2011) results only as examples of the possible shapes of the
atureprofileswithvaryingshapes,thetemperaturebeingofthe temperature profiles and the purpose of the present paper is to
orderof10K.Whilethedusttemperaturedecreasesinwardsin probetheireffectontheanalysisofammoniaspectraatageneral
starlesscores,thesituationcanbedifferentforthegastempera- level.
turebecauseofthe efficientmolecularline coolingin the outer
part.Inthemodels,thegastemperatureprofilesweremostlyin-
2.2.Syntheticspectra
creasingtowardsthecentre.Whenthephotoelectricheatingwas
included,thesituationisreversedandthesurfacetemperatureis
Foreachmodel,syntheticpara-NH (1,1)and(2,2)spectrawere
raisedcloseto20K. 3
obtained with the help of radiative transfer calculations. The
ThecalculationsfollowGoldsmith(2001)apartfromthefact
non-LTEradiativetransferproblemissolvedwithaMonteCarlo
that the cooling rates due to line emission are obtained with a
program (Juvela 1997). The the model clouds are divided into
Monte Carlo radiative transfer program. The cooling rates are
100shells.Togetbetterresolutionintheouterregionswherethe
calculated directly for 12CO, 13CO, C18O, C, CS, and o-H O.
2 excitationconditionscanchangemorerapidly,thethicknessof
The rates of the last two species are multiplied by 10.0 and
the shellswas decreasedtowardsthe surface.Theradiusof the
2.0, respectively, to take into account the contribution of other
innermost sphere and the thickness of the outermost shell are,
species. We examine the following cases without the photo-
respectively,∼6%and0.7%oftheradiusofthemodelcloud.
electric heating: the default model (‘def’), the low abundance
model (χ), the weak gas-dust coupling model (Λ ), the small Theammoniaspectraarecalculatedfollowingtheprescrip-
gd
velocity dispersion model (σ ), and the large velocity gradient tion presented in Ketoetal. (2004). The non-LTE populations
v
model (v). The names refer to the temperature calculations in of the (J,K)states of para-NH3 are obtainedfromthe radiative
Juvela&Ysard(2011). transfer calculationswhere,within each state, an LTE distribu-
The def-model was obtained assuming σ =1.0kms−1 line tion is assumed between the hyperfine levels (i.e., distribution
v
according to the statistical weights of the transitions). In the
widths (Doppler width) and constant abundances in the calcu-
radiative transfer calculations, we take into account the hyper-
lation of the cooling lines, no large scale velocity field, and a
fine components of the (1,1) and (2,2) lines. These consist of
coupling between the gas and dust temperatures that followed
18and21components,respectively.Inthe finalsynthetic(1,1)
that presented in Goldsmith (2001). In the σ -model, the tur-
v
bulent linewidth changes as a step function from 1.0kms−1 in and (2,2) spectra, many of these components are blended to-
the outer part, outside the half radius, to 0.1kms−1 within the gether. The molecular parameters for the (J,K) lines are from
the Lamda database (Scho¨ieretal. 2005) and the frequencies
innercore.Thev-modelistheonlyonewithalargescaleveloc-
and relative intensities of the hyperfine lines were taken from
ityfield.Ithasradialinfallvelocitiesthatincreaselinearlyfrom
0kms−1 atthe surfaceto 1.0kms−1 inthecentre.Thenameof Kukolich(1967).
thelowabundancemodel(χ)referstothecalculationsofthegas Thesynthetic spectraare calculated fora pencilbeam (i.e.,
kinetictemperaturewheretheabundanceofthecoolingspecies forasingleline-of-sightthroughthecentreofthecore)andav-
was assumed to be lower at the centre of the core. This was eragedoveraGaussianbeamwiththeFWHMequaltothecore
implemented as a step function with a factor of ten difference radius.Becausetheexcitationofthemoleculesvariesalongthe
between the inner and outer parts. However, in the modelling lineofsightandthehyperfinecomponentshavedifferentoptical
oftheNH spectra,thefractionalabundanceofpara-NH isal- depths,theobservedintensityratiosbetweenhyperfinecompo-
3 3
ways kept at a constant value of 10−8 (e.g. Hotzeletal. 2001; nentscan varydependingon the opticaldepthand temperature
Tafallaetal. 2004). This appliesalso to the χ-model.The pho- structureofthemodel.Figure1showsexamplesofthesimulated
toelectric heating is included only in one model (model PEH) spectra.
2
M.Juvelaetal.:ReliabilityofNH asthetemperatureprobeofcoldcloudcores
3
6 a Def (σ=0.1) b σV(0.1-1.0) veryweak andburiedin the noise. In the presentpaperwe use
themethodbtodotheanalysis.
4 The rotational temperature, T , is defined with the
12
()K 2 Boltzmannequation
Ta
0 N(2,2) 5
= e−41.5/T12 .
N(1,1) 3
−2
5 c v, (σ=0.1) d v, (σ=1.0) The kinetic temperature, Tkin, is estimated using the three
4 level approximation (including the levels (J,K)=(1,1), (2,1)
3 and (2,2)) as outlined by Walmsley&Ungerechts (1983). The
()TKa 012 tceimenptserCat2u2r→e21d/eCp2e2n→d1e1ncweaosfetshtiemraatteido aodfotphteincgoltlhiesiocnoaelfficcoieeffints-
−1 of Danbyetal. (1988). The method is the same as used in
−2 Harjuetal. (1993). The resulting relationship between T and
12
−20 −10 0 10 20 −20 −10 0 10 20 T isveryclosetothatgivenintheAppendixBofTafallaetal.
kin
v(km/s) v(km/s) (2004).Incoldgas,thesumofthecolumndensitiesofthe(1,1)
Fig.1. Examples of simulated ammonia spectra. Each frame and (2,2) levels is almost equal to the total column density of
shows the (1,1) (the upper line) and the (2,2) spectral profiles. para-NH3 as the populationsof the higher rotational levels are
Fortheplotting,thelatterhavebeenscaledbyafactorof3and negligible.Wecalculatethesecolumndensityestimatesfromthe
shifted by -2K. The text in the upper right hand corner indi- syntheticobservationsandcomparewiththecorrespondingtrue
cates the model in question and, in parentheses, the turbulent valuesinthemodelclouds.
linewidthusedinthecalculationoftheNH spectra.
3
3. Results
2.3.Analysisofthespectra 3.1.Thebasicmodels
To derive estimates of the column densities and gas kinetic Thederivedpara-NH columndensitiesandthegaskinetictem-
3
temperatures,wesubjectthesyntheticNH3(1,1)andNH3(2,2) peratures were compared to the correspondingtrue values that
spectra to standard analysis as described in Ho&Townes were calculated as the mass averaged quantities in the model
(1983), Walmsley&Ungerechts (1983), and Ungerechtsetal. clouds, averaged over the same beam as what was used in the
(1986).ThehyperfinestructureoftheNH3(1,1)lineisfittedus- calculationofthesyntheticammoniaspectra.
ing a χ2 routine assuming a gaussian velocity distribution and The main results are presented in Figs. 2– 4. The profiles
that the different hyperfine states are populated according to ofthekinetictemperatureT areshownassolidblackcurves.
kin
LTE. The fit results are used to calculate the optical thickness Theseareacriticalinputparameterforthesimulationoftheam-
profile τ(v). The peak value τpeak, and the brightness tempera- moniaspectra.Withoutthephotoelectricheating,Tkin increases
ture in the corresponding channel (usually coincident with the towardsthe cloudcentre.The centraltemperatureis furtherin-
intensitymaximumTB,peak)aresubstitutedintheantennaequa- creasedwhen,withinonehalfofthecloudradius,theabundance
tion to derive the excitation temperature, Tex, of the transition. of the cooling species is decreased (model χ) or the turbulent
Theintegratedopticalthickness, τ(v)dv,wherevistheradial linewidth is decreased. In model Λ , the gas-dust coupling1
g,d
velocity,andT givethecolumnRdensityintheuppertransition is weakened in the inner core, leading to a small decrease in
ex
level,N (1,1),fromtheformula the temperature because T > T . We do not claim that all
u dust gas
thesetemperatureprofilesarelikelytobefoundinstarlesscores.
8π 1
N (1,1) = F(T ,T ) τ(v)dv, However,forthepresentcasethemostimportantfactisthatthey
u λ311 Aul 11 ex Z probea wide rangeof Tkin profilesthatpotentiallycouldresult
inabiasintheT estimates.
kin
where Aul is the Einstein coefficient for spontaneousemission, Figure 2 shows the Tkin estimates for the 1M⊙ cloud. The
λ11 = c/ν11 is the wavelengthof the transition,T11 = hν11/kB, horizontal thin and thick lines indicate the true mass averaged
andthefunctionF(T0,T)isdefinedby temperaturesforasingleline-of-sightthroughthecentreofthe
modelcloud(thinline) orweightedwith a gaussianbeamwith
1
F(T ,T)≡ . a FWHM equal to cloud radius. The dashed lines are corre-
0 eT0/T −1 spondinglytheT estimatesderivedfromtheNH spectraob-
kin 3
served with a pencil beam or the larger gaussian beam. In the
Addingtogethertheupperandlowerlevelsoftheinversiontran-
frames also are given the true and estimated column densities.
sition,thetotalcolumndensityofammoniainthe(1,1)levelis
For comparison, Figs. 3 and 4 show the results for 0.5M⊙
N(1,1) = Nu (1+eT11/Tex). m(σod≤e0l.s1aknmdsf−o1r)1. M⊙ modelswithasmallerturbulentlinewidth
Thecolumndensityofthe(2,2)level,N(2,2),canbederived Thetemperatureestimates derivedfromthe simulatedNH3
usingtwoslightlydifferentmethods:a)byestimatingτandT spectraarecorrecttowithin0.5Kandtheerrorsofthecolumn
ex
fromthehyperfinefitasinthecaseof(1,1);orb)byassuming
1 InJuvela&Ysard(2011)thegas-dustcouplingΛ wascalculated
thatT isthesameasforthe(1,1)transitionandthatthe(2,2) g,d
ex usingtheformulasgiveninGoldsmith(2001).Thetrueheatexchange
line is optically thin. In the case b the integral τ(v)dv is esti-
ratemaybehigher(seeYoungetal.2004).Thiswouldnothavealarge
matedfromtheintegratedintensityofthe(2,2)mRaingroupusing effectonthetemperatureinthecorecentreswherethegasandthedust
theantennaequation.Usuallyonlythemethodbisfeasible for arewellcoupled.However,itwouldmodifysomewhattheshapeofthe
observationaldataofcoldcloudsbecausethe(2,2)satellitesare T profiles.
kin
3
M.Juvelaetal.:ReliabilityofNH asthetemperatureprobeofcoldcloudcores
3
12 Ntrue=2.49 Nobs=2.54 Nobs=2.54 12 Ntrue=4.98 Nobs=5.07 Nobs=5.07
11 NCtrue=1.05 NCobs=1.24 NCobs=1.21 11 NCtrue=2.11 NCobs=2.39 NCobs=2.37
K)10 K)10
( 9 ( 9
T T
8 8
7 7
Default Default
6 (cid:0) 6 (cid:3)
12 Nobs=2.53 Nobs=2.28 12 Nobs=5.05 Nobs=4.76
11 NCobs=1.23 NCobs=1.07 11 NCobs=2.37 NCobs=2.16
K)10 K)10
( 9 ( 9
T T
8 8
7 7
6 (cid:1)g,d (cid:2)v 6 (cid:4)g,d (cid:5)v
12 Nobs=2.75 Nobs=2.53 12 Nobs=5.22 Nobs=5.05
11 NCobs=1.22 NCobs=1.21 11 NCobs=2.36 NCobs=2.36
K)10 N2obs=2.77 K)10 N2obs=5.50
( 9 ( 9
T T
8 8
7 7
6 v PEH(AV=0) 6 v PEH(AV=0)
0.00 0.02 0.04 0.00 0.02 0.04 0.00 0.01 0.02 0.00 0.01 0.02
R[pc] R[pc] R[pc] R[pc]
Fig.2. Theradialtemperatureprofiles(solidblackcurves),the Fig.3. AsFig.2butforthehalfsolarmasscore.
estimated temperatures (dashed red lines), and the mass aver-
agedtruetemperatures(blackhorizontallines)fortheonesolar
mass models. The thin lines correspondto the data on a single 12 Ntrue=2.49 Nobs=2.59 Nobs=2.57
line-of-sightthroughthecloudcentreandthethicklinestodata 11 NCtrue=1.05 NCobs=1.19 NCobs=1.18
averaged with a gaussian beam. In the right hand part of each K)10
( 9
frame, the difference between the estimated and the real mass T
8
averaged T has been scaled by a factor of ten for better vis-
kin 7
Default
ibility. The column densities derived from the observationsare 6 (cid:6)
indicatedineachframe(unitsof1014cm−2),thesubscriptCde-
12 Nobs=2.56 Nobs=2.58
notingthecasewithbeamconvolution.Thetruecolumndensi- 11 NCobs=1.17 NCobs=1.18
tiesaregiveninthefirstframe.Forthev-model,thedottedline K)10
and the lower text entry indicate the result from a fit with two ( 9
T
velocitycomponents(pencilbeamonly). 8
7
6 (cid:7)g,d (cid:8)v
12 N2obs=2.52 Nobs=2.50
density values are ∼10% or less. This applies to the observa- 11 NCobs=1.18 NCobs=1.13
tionsconductedwithapencilbeamandmostobservationswith K)10
( 9
a gaussian beam. When one includes the convolution with the T
8
largebeam,thetemperaturevaluesreflectmorethevaluesfound
7
in the outer part of the cloud. However,comparedto the beam 6 v PEH(AV=0)
averagedrealkinetictemperature,thevaluesappeartobebiased 0.00 0.02 0.04 0.00 0.02 0.04
towardstheT in thecloudcentre.Theerrorislargestforthe R[pc] R[pc]
kin
σ modelwhereitcanapproach1K(Fig.2).
v Fig.4. AsFig.2 butwith thesyntheticammoniaspectramod-
The spectra of the v-model are double peaked because of elled with a smaller, 0.1kms−1 turbulent linewidth. For the
the infall motion that at the centre leads to maximum line-of- modelv,thepencilbeamobservationsarefittedwithtwoveloc-
sight velocities of ±1kms−1 (see Fig. 1). The redshifted peak itycomponentsandtheconvolvedspectrawithasinglevelocity
is stronger than the blueshifted peak. This is opposite to the component.
expected behaviour in contracting clouds and is caused by the
fact that the temperature is decreasing towards the core cen-
tre within the innermost ∼20% of the cloud radius. When the
3.2.Theinfluenceofnoise
spectra are modelled as the sum of two components, the indi-
vidual features described with gaussians, the analysis recovers Theinfluenceofobservationalnoisewasexaminedbyanalysing
the correct solution with good accuracy. In the case of narrow a series ofspectra with progressivelyhighersignal-to-noisera-
lines(σ ≤0.1kms−1)asinglecomponentfittothetwovelocity tios. The S/N ratio is defined as the ratio between the peak
componentswouldresultinanerrorof∼1K.Ontheotherhand, intensity of the (2,2) spectrum and the rms noise per chan-
whenthatmodelisobservedwithalargerbeam,onlyasingleve- nel. The rms noise added to the (1,1) spectra was the same in
locitycomponentremainsvisible.InFig.4thebeamconvolved brightness temperature units. In addition to the default model
datawerefittedwithasinglegaussianandthepencilbeamob- (marked ‘def’), the models with the temperature distributions
servationswith two components.Inboth cases the temperature correspondingtothe‘σ ’and‘PEH’caseswereexamined,i.e.,
v
estimatesremainwithin∼0.2Kofthecorrectvalue. modelswith strong inward and outward temperaturegradients.
4
M.Juvelaetal.:ReliabilityofNH asthetemperatureprobeofcoldcloudcores
3
) 88..12 a. DMe=fa1u.0lt 34..502m)(cid:9) ) 4300 a. (cid:11)Mv=1.0 342)/kn
K 8.0 c K m(cid:12)
T(kin7.9 3.014(10 T(kin2100 12140c
7.8 Np (1
7.7 2.5 0 0 N
2.8 PEH(A =0)
14 b. (cid:10)Mv=1.0 2.62) 9.85 b. M=1.0V 3.0 )/kn
K) 12 cm(cid:9) K) 9.80 2.52m(cid:12)
T(kin10 22..2414(10 T(kin9.75 2.0140c
8 Np (1
2.0 9.70 N
1.5
6
10.0 c. PEH(AV=0) 1.0 10.0
9.9 M=1.0 4.02) kn
K) 9.8 3.5 cm(cid:9) Fig.6. Thegaskinetictemperature(plussymbols)andthecol-
T(kin 99..67 3.014(10 aurmenscdaelnedsitwyit(hcrfoascsteosr)swknheinntthheerdaenngseitoiefs1o-6f4th.Teh1e.0cMolsuumn nmdoedne-l
Np sitieshavebeennormalizedwithkn.Thesolidlinesindicatethe
9.5 2.5 truevaluesandthedashedlinesarethevaluesdeducedfromthe
syntheticobservations.Theframescorrespondtodifferenttem-
10.0 100.0
S/N peraturedistributionsas markedin the upperrighthandcorner
ofeachframe.
Fig.5. The gas kinetic temperature (plus symbols) and the
column density (crosses) estimated from spectra with different
theoriginalσ and PEH models(Fig.2) andare notchanged2
signal-to-noise ratios. For each S/N ratio, the figure shows re- v
Therefore, the changes will be only because of the increase of
sultsfromfiverealisationsandtheiraverageasthedashedlines.
the density and the column density. Figure 6 shows the result-
Thehorizontallinesindicatethetruevaluesinthemodels.The
ingT andN(para-NH )estimatesresultingfromobservations
three frames correspond to the default model (def’), the σ - kin 3
V
with a pencil beam towards the cloud centre. As expected, the
model,andthePEHmodel,respectively,ofthe1.0M cloud.
sun estimatesshowsome biaswhenthe columndensitiesare much
higherthanfor the original1M⊙ Bonnor-Ebertspheres.In the
PEH modeltheT estimatescontinuetobeveryaccurateand
kin
Both the M = 0.5M⊙ and M = 1.0M⊙ cloud models were evenin columndensitytheerrorrisesto ∼ 50%onlywhenthe
used,calculatingtheammoniaspectrawitheither1.0kms−1 or densityis some 50 timesthe originalvalues.In thismodel,the
0.1kms−1linewidths.Again,thenarrowerlinewidthisnotcon- realtemperatureisquiteconstantwithinthecentralregionsand
sistent with the original derivation of the temperature profiles apparentlytheopticaldepthsarestillnothighenoughsothatthe
(Juvela&Ysard 2011) but gives some insight to clouds with spectrawouldreacttothewarmbutlowdensitysurfacelayers.
higherlineopticaldepths. Thechangesweremoredramaticintheσ -modelwherethere-
v
Figure 5 shows the estimated kinetic temperaturesand col- sultswereessentiallywrongafterkn ∼4.Inthismodeltheoptical
umndensitiesinrelationtotheirtruevaluesinthemodelclouds. depthswerehighertostartwith,becauseofthenarrowlinewidth
Thespectrawerecalculatedusingapencilbeam.TheS/Ncovers atthecentre.
a rangefrom∼3 to 300in multiplicativesteps of 1.5.For each Figure 7 shows the actual spectra calculated for kn=1 and
S/N ratio we show the results from five realizationsof spectra, kn=32.Intheσv-model,themaingroupbecomesstronglyself-
thedashedlinescorrespondingtotheirmeanvalue.Theresults absorbed and, because the kinetic temperature is higher in the
fromthe M =0.5M⊙andtheσV=0.1kms−1werequalitatively centre than at the surface, the intensity of the satellite lines is
similar. higherthanforthemaingroup.Thisis,ofcourse,asituationthat
cannotbeaccountedforintheanalysisassumingahomogeneous
medium.
3.3.Modelsofhighopticaldepth
Inthedefault1Msunmodelthepeakopticaldepthsareτ(1,1)=0.7 4. Discussion
andτ =0.013.Inthe0.5M cloudtheseincreaseto1.2and
(2,2) sun
0.04,respectively.When the turbulentlinewidthwas decreased Tafallaetal. (2004) examined the validity of the kinetic tem-
from1.0kms−1 to 0.1kms−1,the valueswerenaturallyhigher, perature estimates from NH3 using isothermal models. In the
4.3and0.33forthe(1,1)and(2,2)lines,respectively.However, present study, we calculate the mass averaged kinetic temper-
asthesatellite linescontinuetobemostlyopticallythin,thisis atures in model clouds with realistic temperature profiles. The
notabigproblemfortheanalysisofthespectra.
2 Weareinvestigatingtheeffectsofdifferenttemperatureprofilesin
To determine when and how the errors increase to signifi-
general.However,wenotethatforhighk thegaskinetictemperature
n
cantlevels,wetookthe1.0Msuncloudsandcalculatedaseriesof shouldapproachthedusttemperatureandTkinisbeexpectedtodecrease
modelswherethedensitiesweremultipliedwithconstantfactors towardsthecorecentre(seee.g.Crapsietal.2007).Thesituationcould
kn between 1 and 64. The temperature profiles are taken from changeonlyifthecoreisheatedinternally.
5
M.Juvelaetal.:ReliabilityofNH asthetemperatureprobeofcoldcloudcores
3
6 a σ, kn=1 b PEH, kn=1 bulent linewidth,σV = 0.1km−1s−1, completely wrong results
werealreadyencounteredwithdensitiesafewtimeshigherthat
4
those of an 1M Bonnor-Ebertsphere. If the fractionalabun-
sun
()K 2 dancewerelargerthanthe 10−8 assumedin ourmodels,obser-
Ta vationsofnormalhydrostaticcorescouldbeaffected.However,
0
in starless cores the values are not likely to be much higher
−2 (Hotzeletal.2001;Tafallaetal.2002;Crapsietal.2007).
Tafallaetal. (2004) found that the excitation temperatures
8 c σ, kn=32 d PEH, kn=32 of the (1,1) and (2,2) lines varied by a factor of several. This
6 variation depends on several factors, the cloud optical depth,
)K 4 the volume density, and the variation of the kinetic tempera-
(
Ta 2 ture.Figure8showsthesituationinthoseofour1.0M⊙models
0 wherethedifferenceintheradialkinetictemperatureprofilesis
the largest. In the σ model the kinetic temperature increases
−2 V
−20 −10 0 10 20 −20 −10 0 10 20 inwards and, in combination of higher volume density and en-
v(km/s) v(km/s) hancedphotontrapping,theexcitationtemperaturesatthecloud
centrearethreetimesashighasatthecloudboundary.Towards
Fig.7. Comparison of the original spectra from the one solar
the centre also the difference between the excitation tempera-
massσ andthePEH modelsandthespectraobtainedafterthe
v
turesofthe(1,1)and(2,2)transitionsbecomesmorenoticeable.
densitieshavebeenscaledbyafactorofk =32.Fortheplotting,
n
On theotherhand,in the PEH case theoutwardincreasingki-
the(2,2)spectrahavebeenscaledbyafactorof3(upperframes)
netictemperaturehelpstokeeptheexcitationconditionsalmost
or0.5(lowerframes)andshiftedby-2K.
constant.Therefore,itisnotsurprisingthatthekinetictempera-
tureestimatedfromtheobservedspectraremainedaccurateeven
values are compared to the kinetic temperature estimates de- whenthedensitywasincreased(Fig.6).
rived with a standard LTE method, using the ammonia spectra Ammoniawas foundto be a veryreliable tracer of the real
‘observed’fromthesemodels. mass averaged gas temperature within the modelled Bonnor-
The estimated kinetic temperatures were generally found Ebert spheres. The temperature at the centre of a cloud could
to be very accurate. In the case of model clouds with masses still differ from this number by several degrees. In our models
0.5M⊙ and 1M⊙, the difference between the temperature de- this difference was at most ∼1K. However, if the signal from
rivedfromtheNH spectraandtherealmass-averagedtemper- verycentrebecomesweakerbecauseofalowerabundancecom-
3
ature was of the order of 0.1K, only rarely approaching 1K. binedwithalowkineticandexcitationtemperatures,thediffer-
Similarly,theestimatedcolumndensitiesofpara-NH remained encecanbemoresignificant.
3
correcttowithin∼10%. Inthispaperwehaveassumedtheammoniaabundancetobe
Inoneofthemodelstheinfallvelocityproducedtwoveloc- constantasafunctionofthecloudradius(seee.g.Tafallaetal.
ity components whose presence, in the case of large turbulent 2006).Iftheabundancevaries,NH3islikelytoprovideaccurate
linewidthsσ = 1.0km−1, was notimmediatelyobviousfrom estimates of the average temperature weighted not only by the
V
thelineprofiles.Theanalysiscompletedusingasinglegaussian massbutalsobythefractionalabundance.Iftheammoniaabun-
componentstill led to only ∼1K error. A fit with two velocity dance increasestowardsthe centre (e.g.Tafallaetal. 2002) the
componentsresultedintwiceashigherrorsinboththetempera- resultwillbeabetterestimateofthecentralcoretemperature.In
tureandtheestimatedcolumndensity(Fig.2).Whentheturbu- coresdenserthanconsideredherethesituationmaybereversed
lentlinewidthwasreducedinthecloudcoresothatthevelocity as even NH3 becomes depleted in the very centre of the core
componentsbecame clearly visible in the (1,1)profiles, the er- (Aikawaetal.2005).Asanillustrationoftheassociateduncer-
rorofasinglecomponentfitincreasedto∼1.5Kandthecolumn tainties, we consideredthe σv modelwith 1.0M⊙, varyingthe
densitywasoverestimatedby20%(Fig.4). However,thistime abundancebetween10−7.5 and10−8.5 asalinearfunctionofthe
the two component fit did recover the correct results with less logarithmoftheradius.OneobtainsanestimateofTkin=10.49K
than10%errors.Thisincreasesthegeneralconfidenceinthere- when the abundance increases inwards and 10.59K when the
sultsofobtainedfromammoniaspectraobservedwithadequate abundance increases outwards. Both are within 0.25K of the
resolution.Itappearsunlikelythattheeffectofhiddenfeatures, valuefromthepreviousconstantabundancemodel.Thus,inthis
e.g.ofmultiplevelocitycomponents,wouldbeamplifiedinthe casetheeffectoftheabundancegradientsisnotverysignificant.
T andN estimates. However, we will return to this questions in future papers that
kin
When the signal-to-noise of the (2,2) spectra is decreased discussself-consistentmodelsofthechemistryandofthether-
below S/N∼10, the temperature errors increased in our tests to malbalance.
∼1K.Theresultswerenotsignificantlybiasedsothatevenfora
setofsourcesobservedatlowS/N,the meanpropertiesshould
5. Conclusions
be relatively trustworthy. The sensitivity to noise naturally de-
pendsonthewidthofthelines. We have calculated ammonia emission spectra using non-LTE
When the column density is increased much beyond that radiativetransfermodellingandcloudcoretemperatureprofiles
foundinourbasicmodels,the(1,1)becomesopticallythickand takenfromtheliterature.Wehavecarriedoutanalysisofthesim-
therearenolongerguaranteesoftheaccuracyoftheresults.In ulatedspectraandcomparedtheresultswiththetrueparameters
particular,thelinesmightnolongerprobetheconditionsatthe ofthemodelclouds.Thishasledtothefollowingconclusions:
centreofverydensecores.Wecarriedoutonetestwithanadhoc
scalingofthedensities.Thebehaviourwasfoundtodependvery – For the ∼1M⊙ hydrostaticclouds, the kinetic temperatures
muchonthelinewidths(affectingtheinitiallineopacity)andthe and column densities derived from the ammonia observa-
temperaturestructureofthecloud.Inthecasewithasmalltur- tionsareaccuratetobetterthan10%accuracy.
6
M.Juvelaetal.:ReliabilityofNH asthetemperatureprobeofcoldcloudcores
3
18
16
14
K)
( 12 (cid:13)v
Tex PEH(A =0)
T,kin10 v
8
6
4
0.00 0.01 0.02 0.03 0.04 0.05
R(pc)
Fig.8. Comparisonoftemperaturesinthe1.0M⊙cloudmodels
σ and peh.Shownaretheradialprofilesofthekinetictemper-
v
ature (solid lines) and the excitation temperatures of the (1,1)
(dashedline)and(2,2)(dash-dottedline).
– The presenceofseveralvelocitycomponentscanbe identi-
fiedfromthelineprofilesbeforetheyhaveasignificanteffect
ontheaccuracyoftheanalysis.
– The uncertainty of the kinetic temperature estimates in-
creasesto∼1Kwhenthesignal-to-noiseofthe(2,2)spectra
isdecreasedtoS/N∼10.
– In models of higher density, the behaviour depends the ki-
netic temperature profile of the cloud. Quite wrong results
can already be encountered with models a few times more
opaquethanthe1M Bonnor-Ebertspheres.
sun
– Apartfromthemostopaquecores,theammoniawasfound
tobeaveryreliabletracerofthemassaveragedkinetictem-
perature.Thetemperatureatthecentreofdensecorescould
stilldifferfromthisvaluebyseveraldegrees.
Acknowledgements. Theauthors acknowledge thesupportoftheAcademyof
FinlandGrantsNo.127015,250741,and132291.Theauthorsthankthereferee
Dr.NealEvansforhelpfulcomments,whichimprovedthepaper.
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