Dissecting the molecular structure of the Orion B cloud: Insight from Principal Component Analysis

4.1 MB·

by  Pierre Gratier

Astronomy&Astrophysicsmanuscriptno.pca-in-OrionB˙arXiv (cid:13)c ESO2017
January17,2017
Dissecting the molecular structure of the OrionB cloud:
Insight from Principal Component Analysis(cid:63)
PierreGratier1,EmericBron2,3,MaryvonneGerin4,Je´roˆmePety5,4,VivianaV.Guzman6,JanOrkisz5,4,Se´bastien
Bardeau5,JavierR.Goicoechea2,FranckLePetit3,HarveyLiszt7,KarinO¨berg6,NicolasPeretto8,EvelyneRoueff3,
AlbrechSievers5,andPascalTremblin9
1 Laboratoired’astrophysiquedeBordeaux,Univ.Bordeaux,CNRS,B18N,alle´eGeoffroySaint-Hilaire,33615Pessac,France
2 ICMM,ConsejoSuperiordeInvestigacionesCientificas(CSIC).E-28049.Madrid,Spain.
3 LERMA,ObservatoiredeParis,PSLResearchUniversity,CNRS,SorbonneUniversite´s,UPMCUniv.Paris06,F-92190,Meudon,
France
7
4 LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universite´s, UPMC Univ. Paris 06, E´cole normale
1
supe´rieure,F-75005,Paris,France
0
5 IRAM,300ruedelaPiscine,38406SaintMartind’He`res,France
2
6 Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA,02138,USA.

n 7 NationalRadioAstronomyObservatory,520EdgemontRoad,Charlottesville,VA,22903,USA
a 8 SchoolofPhysicsandAstronomy,CardiffUniversity,Queen’sbuildings,CardiffCF243AA,UK.
J 9 MaisondelaSimulation,CEA-CNRS-INRIA-UPS-UVSQ,USR3441,Centred’e´tudedeSaclay,F-91191Gif-Sur-Yvette,France.

6
1

] ABSTRACT
A
Context.Thecombinationofwidebandreceiversandspectrometerscurrentlyavailablein(sub-)millimeterobservatoriesdeliverwide-
G
fieldhyperspectralimagingoftheinterstellarmedium.Tensofspectrallinescanbeobservedoverdegreewidefieldsinaboutfifty
. hours.Thiswealthofdatacallsforrestatingthephysicalquestionsabouttheinterstellarmediuminstatisticalterms.
h
Aims.Weaimatgaininginformationonthephysicalstructureoftheinterstellarmediumfromastatisticalanalysisofmanylines
p
fromdifferentspeciesoveralargefieldofview,withoutrequiringdetailedradiativetransferorastrochemicalmodeling.
-
o Methods.Wecoupledanonlinearrescalingofthedatawithoneofthesimplestmultivariateanalysismethods,namelythePrincipal
r ComponentAnalysis,todecomposetheobservedsignalintocomponentsthatweinterpretfirstqualitativelyandthenquantitatively
t
s basedonourdeepknowledgeoftheobservedregionandoftheastrochemistryatplay.
a Results.Weidentify3principalcomponents,linearcompositionsoflinebrightnesstemperatures,thatarecorrelatedatvariouslevels
[ withthecolumndensity,thevolumedensityandtheUVradiationfield.
Conclusions.Whensamplingasufficientlydiversemixtureofphysicalparameters,itispossibletodecomposethemolecularemission
1
inordertogainphysicalinsightontheobservedinterstellarmedium.Thisopensanewavenueforfuturestudiesoftheinterstellar
v
medium.
5
0 Keywords.ISM:molecules,ISM:clouds,ISM:photon-dominatedregion(PDR),Object:OrionB,Method:statistical
2
4
0
1. Introduction self: astrochemical modeling (Le Bourlot et al. 2012; Agu´ndez
.
1 &Wakelam2013).Furthercomplexityariseswhenconsidering
0 Molecularcloudshaveacomplexstructure,withfilamentshost- radiativetransfertoderivelineintensitiesfromthelocalchemi-
7 ingdensecoresandimmersedinalowdensitydiffuseenvelope. calcompositionandphysicalstructure.
1 Large scale dust continuum maps obtained with Herschel have
The last few years have seen the installation of new
: provided a breakthrough, by showing the tight relationship be-
v wideband receivers and spectrometer at millimeter and sub-
tween the filaments and the dense cores. These maps however
i millimeterradiotelescopes.Withtheseinstruments,linesurveys
X donotprovideinformationonthegasdynamicsoritschemical
of several GHz bandwidth and several tens of thousands of
r composition. Furthermore the relationship between the submil-
a limeterdustemissionandthegascolumndensityisaffectedby spectral channels are the new default mode of observations.
Combined with wide field imaging capabilities both for single
thedusttemperatureandpossiblevariationsofthedustemissiv-
dishandinterferometers,hyperspectralimagingisnowroutinely
ity. Molecular line emission maps provide alternative means to
carriedoutwiththeseinstruments.
studymolecularcloudstructureandrelateittotheflowkinemat-
Theanalysisandinterpretationoftheselargedatasets,con-
ics.Molecularlineemissionislinkedtotheunderlyingphysical
sistingofthousandsofspatialpositionsandtensofthousandsof
propertiesoftheISM,suchasdensity,gasanddusttemperatures,
spectral channels, will benefit from the use of statistical tools.
UVradiationfield,andcosmicrayionizationrate.Butthesere-
PrincipalComponentAnalysis(PCA)isoneofthemostwidely
lationships are complex and their detailed study is a field in it-
used multivariate analysis method. It has been used to study
(cid:63) Based on observations carried out at the IRAM-30m single- the Interstellar Medium (ISM) using molecular emission maps
dish telescope. IRAM is supported by INSU/CNRS (France), MPG (Ungerechts et al. 1997; Neufeld et al. 2007; Lo et al. 2009;
(Germany)andIGN(Spain). Melnicketal.2011;Jonesetal.2012).
1
PierreGratieretal.:DissectingthemolecularstructureoftheOrionBcloud:InsightfromPrincipalComponentAnalysis
Table 1. Properties of the observed spectral lines. The last six columns show the statistics of the data before and after asinh
reparametrization.
Originaldata Afterasinhreparametrization
Molecule Transitions Frequency Noise Min. Median Max. Std. Min. Median Max. Std.
(MHz) (K) (K) (K) (K) (K) (K) (K) (K) (K)
12CO J=1→0 115271.202 0.09 −0.39 13.40 57.11 10.18 −0.37 2.39 3.32 0.91
13CO J=1→0 110201.354 0.04 −0.19 1.38 36.43 3.27 −0.19 0.97 3.03 0.74
CS J=2→1 97980.953 0.06 −0.36 0.06 15.53 0.48 −0.35 0.06 2.48 0.23
HCN J=1→0,F=2→1 88631.848 0.10 −0.58 0.15 10.32 0.39 −0.52 0.15 2.22 0.25
HCO+ J=1→0 89188.525 0.09 −0.45 0.26 8.07 0.47 −0.42 0.25 2.07 0.30
SO N=3→2,J=2→1 99299.870 0.06 −0.43 0.04 6.46 0.24 −0.40 0.04 1.92 0.17
CN N=1→0,J=3/2→1/2,F=5/2→3/2 113490.970 0.09 −0.58 0.10 6.33 0.27 −0.52 0.09 1.91 0.20
HNC J=1→0 90663.568 0.08 −0.49 0.07 6.01 0.27 −0.45 0.07 1.88 0.19
CCH N=1→0,J=3/2→1/2,F=2→1 87316.898 0.12 −0.62 0.08 5.72 0.22 −0.55 0.08 1.85 0.18
C18O J=1→0 109782.173 0.06 −0.30 0.06 5.55 0.42 −0.29 0.06 1.83 0.26
N2H+ J=1→0,F1=2→1,F=3→2 93173.764 0.08 −0.44 0.00 4.53 0.13 −0.41 0.00 1.70 0.10
CH3OH J=2→1,K=0→0,(A+) 96741.375 0.06 −0.34 0.01 2.24 0.08 −0.32 0.01 1.26 0.08
In this paper, we address the following question: can PCA telescope at almost the same time, as the observed bandwidth
provide a method to study the underlying physics of the ISM wascoveredinonlytwofrequencytunings.Thenoiseforeach
when applied to a large dataset of molecular emission, without map,alongwiththeminimum,median,maximum,andvariance
performingeitherradiativetransferorastrochemicalmodeling? values are listed in Table 1. The noise is computed by fitting a
Thearticleisdividedasfollows.Section2presentsthedata gaussian function to the negative part of the histogram of pixel
usedinthisstudy.Section3describesthestatisticalmethodused brightnesses.Thisenablestocomputethenoisewithoutneeding
in this paper and its implementation. Results are presented in tomaskouttheemission.
Sect 4 first by analyzing the output of the PCA, and further
by comparing theses outputs with independent maps of physi-
3. PrincipalComponentAnalysis
cal conditions in OrionB in Sect. 5. The last section discusses
theseresults.
3.1. Principle
We use the following standard statistical terms: the dataset is
2. Data composed of samples each described by individual features. In
our case, each spatial pixel is a sample and each line intensity
The data used in this paper is selected from the ORION-B
isafeature(thefulldatasetthuscorrespondstoadatamatrixof
project (PI: J. Pety), which aims at mapping with the IRAM-
132300samplestimes12features).
30mtelescopealargefractionoftheSouth-Westernedgeofthe
PrincipalComponentAnalysis(PCA)isawidelyusedmulti-
OrionB molecular cloud over a field of view of 1.5 square de-
dimensionalanalysistechnique(Jolliffe2002),whichcanbede-
greesinthefull3mmatmosphericwindowat200kHzspectral
finedinseveralmathematicallyequivalentways.Itaimsatfind-
resolution. Pety et al. (2016) describe in detail the data acqui-
inganeworthogonalbasisofthefeaturespace(whoseaxesare
sition and reduction strategies. Table 1 lists the 12 transitions
calledprincipalcomponents,orPCs),sothatforeachk,thepro-
selectedinthispaperfromthealreadyobservedfrequencyrange
jectionontothehyperplanedefinedbythefirstkaxesisoptimal
(from 84 to 116GHz), based on the inspection of the full data
inthesensethatitpreservesmostofthevarianceofthedataset
cube.
(orequivalentlythattheerrorcausedbythisprojectionismini-
For each line, we focus on emission coming from a lim-
mal).PCAthusdefinessuccessiveapproximationsofthedataset
ited1.5kms−1-velocityrangecenteredonthepeakvelocity(i.e.,
byhyperplanesofincreasingdimension.
10.5kms−1) of the main velocity component along the line of
This is equivalent to the diagonalization of the covariance
sight.Averagingthethree0.5kms−1velocitychannelsallowsus
matrix, so that the principal components are naturally uncorre-
to get a consistent dataset from the radiative transfer and kine-
lated.Itcanbethoughtofasfindingtheprincipalaxesofinertia
maticsviewpoints.Inparticular,weavoidtheneedtodisentan-
gle1)theeffectsofhyperfinestructuresofsomelines,and2)the ofthecloudofsamplesabouttheirmeaninfeaturespace,andis
thus a way to analyze the covariance structure of the data. The
complexvelocitystructureofthesource(Orkiszetal.subm.).
principalcomponentsareorderedbydecreasingprojectedvari-
The observed field of view covers 0.81deg×1.10deg and
ance.AsaresultPC1istheaxisoflargestvarianceinthedata.
containstheHorseheadnebula,andtheHiiregionsNGC2023,
PC2isthentheaxisoflargestvarianceatconstantPC1(orthog-
NGC2024, IC434, and IC435. The angular resolution ranges
onaltoPC1)andsoforth.Neglectingtheaxesoflowestvariance
from22.5to30.5(cid:48)(cid:48).The12resultingmapshaveacommonpixel
thenallowstodefinealow-dimensionalhyperplaneinwhichthe
sizeof9(cid:48)(cid:48)thatcorrespondstoaNyquistsamplingforthehighest
dataset is approximately embedded. An important property to
frequencylineobserved(12CO(1–0)at115.27GHz).Themaps
keepinmindisthelinearityofPCA(itdefineslow-dimensional
thuscontains315×420pixels.Atadistanceof∼400pc(Menten
hyperplanes,andnotgenerallow-dimensionalhypersurfaces).
etal.2007),themapsgiveusaccesstophysicalscalesbetween
Acommonvariant1 intheapplicationofPCAistonormal-
∼50mpcand10pc.
ize the variations of the dataset around the mean by the stan-
Figure1showsthe12mapsoftheresultingbrightnesstem-
dard deviation of each features, before applying the PCA. This
perature multiplied by an ad-hoc factor in order that they can
amounts to diagonalizing the correlation matrix instead of the
share the same color look-up table, even though the intrinsic
covariancematrix.Thecorrelation-basedvariantallowstoavoid
brightnesstemperaturesofthedifferentlinesdifferbymorethan
one order of magnitude. The relative calibration of the differ- 1 ItactuallygoesbacktooneofthetwoearliestdescriptionsofPCA:
entlinesisexcellentbecausetheywereobservedwiththesame Hotelling(1933)
2
PierreGratieretal.:DissectingthemolecularstructureoftheOrionBcloud:InsightfromPrincipalComponentAnalysis
12CO 13CO 2 CS 5 HCN 10
× × ×
HCO+ 10 SO 10 CN 10 HNC 10
× × × ×
CCH 10 C18O 15 N H+ 15 CH OH 15
2 3
× × × ×
40
30
20
)
(0
y 10
δ
0
10
−
20
− 40 30 20 10 0
δx( )
0
0 6 12 18 24 30 36 42 48
T (K)
mb
Fig.1.MapsofmolecularemissioninKelvinmainbeamtemperatures.
having one feature dominating the variance, and is indicated if moment, elemental abundances,...) that are not relevant for our
therelativescalesofthefeaturesarenotrelevantforthepurpose analysisofthechemicalvariationsacrossthemap,weusehere
of the analysis. As the relative intensity scales of the different thecorrelation-basedversionofPCA.
molecules used here are largely affected by properties (dipole
3
PierreGratieretal.:DissectingthemolecularstructureoftheOrionBcloud:InsightfromPrincipalComponentAnalysis
of the N H+(1−0) is Gaussian to first order, the histogram of
101 101 2
n 13CO(1−0) exhibits heavy tails similar to power laws. As a
o
ncti 100 100 result, extreme intensity values might dominate the covariance
utionfu10−1 10−1 slotrwuecrtuirneteonfsitthyevdaaltuae,sh.idingthevariationsatthemorecommon
Distrib1100−−32 10−2 brigFhrtnoemssthteemppheyrasitcuarel visieawlspooidnet,sitraakbilneg. PtCheAloisgaariltihnmearotfecthhe-
0 10 20 30 40 2 1 0 1 2 3 niquewhichdecomposethedataasasumofuncorrelatedcom-
− −
101 101 ponents.Applyingittothelogarithmofthedataallowsadecom-
nfunction101−001 100 pttuoorstehiteiinountneadrsmearslypoirnofgdruartcaitdooisa,ftifpvarecotdtorurascn,tssafneadrndathnpudoscwdheeesrmclraiicwbaesls,etmffheeocrdtesa.taaTdsaatkpriutnecgd-
utio10−2 10−1 thelogarithmsofthedataistheequivalentinastrochemistryto
Distrib1100−−43 10−2 dstouidnigesc.oPloert-ycoeltoarl.m(a2g0n1i6tu)dsehodwiagtrhaamt tahnealliynseisininteogpratitceadlborrigUhVt-
0 10 20 30 40 2 1 0 1 2 3 nesstemperaturesofourdatasetistofirstordercorrelatedtothe
− −
Brightnesstemperature(K) Reparametrizedbrightnesstemperature(K)
columndensityofmatteralongthelineofsight.Weexpectthis
aspecttoappearinourPCAanalysis,andsecondorderchemi-
Fig.2.Effectoftheasinhrenormalizationontheintensitydistri-
calvariationsaroundthistrend,whichwouldberevealedbyline
butions.Leftcolumn:beforerenormalization,rightcolumn:after
ratios, are thus better described as multiplicative –rather than
renormalization,Toprow: N H+,bottomrow: 13CO.
2 additive–factors.
4.0 3.2.2. Impactofnoise
3.5 Thepresenceofnoisehowevercausesthepossibilityofnegative
valuesinpixelswheresomelinesareundetected.Thelogarithm
3.0
transform cannot be applied to these negative noise values. In
addition,asthelogarithmstretchesthelowestvaluescompared
2.5
tothelargestones,itwillalsotendtostretchthepositivenoise
y2.0 valuesoftheundetectedpixels,thusgivingthemmoreweightin
thecovarianceofthedata.
1.5
There may be two different reasons for a non-detection: ei-
1.0 ther the measurement is not sensitive enough to detect the line
y=aasinh(x/a)
ortheregionjustdoesnotcontainthespeciesthatemitstheline.
0.5 y=x ThelattercasehappensinparticularinHiiregions(e.g.,IC434),
y=aln(2x/a) where 12CO(1–0) is photo-dissociated by the far UV photons.
0.0
0 1 2 3 4 5 6 Inthisparticularcase,wecouldremovefromourdatasetallthe
x samples (pixels) where no 12CO(1–0) is detected. This just as-
sumesthatno 12CO(1–0)detectionsathighsensitivityimplythe
Fig.3.Plotofthe asinhfunction(solidline)showingtheasymp-
toteswhen x → 0(dashdottedline)and x → +∞(dottedline). absenceofmoleculargas.However,thismethodisnotgeneric.
Theparametera(herea=1)istracedwithathinverticalline For instance, N2H+(1–0) is only detected in dense cores (Pety
et al. 2016), and restricting ourselves to these regions would
drasticallylimitthescopeofourstudy.Ifwewishtousethisim-
portanttracerofthemoleculargaswhilestillcoveringtherange
In this work, weused the PCA implementation available in
of different chemical regimes present in our full map, we thus
the Python package scikit-learn (Pedregosa et al. 2011),
need to find an alternative. Moreover, in the 3mm band, radio
whichusesasingularvaluedecompositiontocomputetheprin-
recombinationlines,whichemitinHiiregions,couldinprinci-
cipalcomponentaxes.
plebeaddedtoourdatasettostudytheformationofmolecular
gas.
3.2. Reparametrizationofinputdata Addingathresholdingstepbeforetakingthelogarithmwill
onlyworsenthescopeofundefinedvalues.WhiletherearePCA
3.2.1. Ontheneedofareparametrization
methods(seee.g.Ilin&Raiko2010)thatcantakemissingdat-
As seen in Table 1, some of the tracers have large dynamical apoints into account, they rely on the fact that these missing
ranges(2ordersofmagnitudefor 13CO(1–0)and 12CO(1–0)). points have the same statistics than the measured points (i.e.,
Figure 2 shows the histogram of the brightnesses temper- when a value is missing, it is independent of the actual value).
atures of two lines with contrasting behavior in our dataset, This is clearly not the case here as the missing values (unde-
namely 13CO(1−0), and N H+(1−0). As the dynamic range tected lines) are missing because they are below the sensitivity
2
is large both in intensity and number of pixels per bin, these threshold.Thesearecalledcensoredvaluesinstatistics.Wethus
histograms use the Bayesian Blocks algorithm (Scargle et al. searchforafunctionthatislineararound0(inthenoisedomi-
2013, using here the Python implementation from AstroML, nateddomain),andwhichisasymptoticallyequaltoalogarithm
seeVanderplasetal.2012;Ivezic´ etal.2014),whichadaptsthe forlargevalues(comparedtothenoiselevel).Theinversehyper-
bin width to the underlying distribution. While the histogram bolicsinusfunction, asinh(x),fulfillstheseconditions.Wethus
4
PierreGratieretal.:DissectingthemolecularstructureoftheOrionBcloud:InsightfromPrincipalComponentAnalysis
12CO 13CO CS HCN
HCO+ SO CN HNC
CCH C18O N H+ CH OH
2 3
40
30
20
)
(0
y 10
δ
0
10
−
20
− 40 30 20 10 0
δx( )
0
0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
−
Fig.4.Mapsofmolecularemissionafterreparametrizationbytheasinhfunction.
usedthefollowingfunctiontoreparametrizethedatabeforeap- where the parameter a is the typical value for which the func-
plyingPCA tionbehaviorchangesfromalineartoalogarithmicregime(see
Fig.3).
T(x)=aasinh(x/a), (1)
5
PierreGratieretal.:DissectingthemolecularstructureoftheOrionBcloud:InsightfromPrincipalComponentAnalysis
Thefirstprincipalcomponentexplainsthemajority(60%)of
100
thetotalcorrelationpresentintheoriginaldataset.Thusalarge
part of the variations in the dataset occur along a single axis
(i.e. all lines are strongly correlated to each other). The second
80
principalcomponentaccountsforabout10%ofthecorrelation.
%) Itissignificantlylessthanthefirstcomponent,butmorethanany
(
n
atio 60 othercomponents.PC3,4and5correspondtosimilaramounts
el of correlations (around 5% each) and PC6 slightly less (3.3%).
corr PC1 to 6 collectively explain more than 90% of the correlation
d
ne 40 inourdataset.TheremainingPCshavesimilarlowamountsof
plai explainedcorrelation(from2%forPC7to0.9%forPC11).
x
E
20
4.2. Discussionoftheprincipalcomponents
ThePCsdefinedbyouranalysisrepresentnewaxesinthefea-
0 2 4 6 8 10 12 ture space (the full 12 PCs are simply a rotation of the initial
Numberofcomponents basis of the feature space), deduced from the data itself. They
can thus be expressed in terms of the original axes, as a linear
Fig.5. Percentage of the explained correlation as a function of combination of the (transformed) line intensities. Figure 6 dis-
thenumberofcomponentsinthePrincipalComponentAnalysis, plays the quantitative contribution of each initial feature (line)
dottedline:cumulativepercentage. toeachPC.Analternativeviewoftherelationshipbetweenthe
PCs and the line intensities, namely the correlation wheels, is
presentedinFig 7.
The only free parameter in the method is the threshold a.
Eachsample(pixelbrightness)canthenbeprojectedonthe
Appendix A discusses our choice, i.e., a = 8×0.08 = 0.64K
new axes, providing new coordinates commonly called com-
equal to 8 times the median noise of the dataset. In short, we
ponent scores. The PCA method considers the pixels as inde-
selectthevalueofathatmaximizesthecorrelationsofthefirst
pendent samples, and thus ignores the spatial structure of the
3principalcomponentswithindependentknownmeasurements
molecularemission.Itisneverthelesspossibletoreconstructthe
ofthecolumndensity,volumedensity,andUVillumination(see
maps of the component scores. Figure 8 shows these projected
Section5).Thisappendixalsodemonstratesthatourresultsare
maps. The chosen color look-up table emphasizes that positive
quiteinsensitivetotheexactvalueofa.
(red-colored)andnegative(blue-colored)valuesoftheprojected
TherightcolumnofFig.2showstheresultoftheasinhtrans-
maps–correspondingtovariationsaboveandbelowtheaverage
formation on the intensity distributions of two transitions rep- alongtheconsideredaxis–clearlyextractadifferentspatialpat-
resentativeofbright(13CO(1–0))andweak(N H+(1–0))aver-
2 ternperprincipalcomponent.
aged lines. In the case of the bright lines, the dynamic range is
The first principal component is a linear combination of all
drasticallyreducedwiththeheavytailbeingtransformedintoa
tracers,withsimilarpositiveweightsforalllines(Fig.6).Itthus
secondpeakinthedistributionbutwithnovaluesabove3.Inthe
describes correlated variations of all molecules, and these ac-
case of N H+ the distributions before and after reparametriza-
2 count for most of the variations in the dataset. It is a natural
tionareverysimilar.Figure4showsthe12mapsofthemolec-
consequence of the fact that all lines are well correlated (posi-
ularemissionafterreparameterizationbytheasinhfunction,but
tively) to each other. Pety et al. (2016) show that the emission
before the normalization step of the PCA analysis. The bright-
of all lines is correlated to first order with the column density
nesstemperaturesofallthemapshavebeencompressedbetween
ofmatteralongthelineofsight.Thiscomponentisthusproba-
about-0.5and3.5,withlowsignal-to-noisebrightnesstempera-
blyrelatedtothetotalcolumndensity,whoseincreasecausesto
turesbetween-0.5and0.5beingmostlyuntransformed.
firstorderanincreaseinalllines(inthelinearapproximationof
PCA,non lineareffectssuch assaturationofthe 12COlineare
4. Results not captured). The corresponding component map (Fig. 8) in-
deedresemblesamapofcolumndensity.Thisrelationbetween
The PCA method exposes the correlations between the line PC1andthetotalcolumndensityofmatterwillbeinvestigated
brightness temperatures. The derived PCs give the main axes morequantitativelyinSect.5.NotethatasPC1hasonlypositive
of correlated variations in the data set. As such, it does not di- coefficientsforalllines,orthogonalityensuresthatallotherPCs
rectlyyieldphysicalinformationunderlyingthedataset.Inthis willrepresentcontrastsbetweendifferentlines.
section, we describe the results of the application of this sta- The second principal component represents the axis of
tistical method to our dataset and we start to discuss their pos- largestvariationatconstantPC1(orthogonaltoPC1).Thisaxis
sible physic interpretation based on our a priori astrochemical ofvariationsisdominatedbypositivecontributionsof N H+and
2
knowledge.ThepossiblerelationsbetweenthesePCsandphys- CH OH,andnegativecontributionof 12COand 13CO.Thefirst
3
icalvariablesareinvestigatedinalatersection. two tracers are chemically associated with dense and cold re-
gions of the interstellar medium. For instance, N H+ is easily
2
destroyedbyCO,itcanthusonlybeabundantinthegasphase
4.1. Correlationfractionexplainedbythedifferentprincipal
when CO has been depleted on the grain surfaces (Pety et al.
components
2016). The component map shows strong positive values high-
Figure 5 shows the percentage of the correlation explained by lighting known dense cores, including the clumps in the head
eachPC(asafunctionoftheprincipalcomponentnumber)along andintheneckoftheHorsehead(Ward-Thompsonetal.2006).
with the cumulative explained correlation as a function of the The third principal component shows positive contribution
numberofprincipalcomponentskeptinthedecomposition. ofCCHandCNthatareknowntobesensitivetoUVillumina-
6
PierreGratieretal.:DissectingthemolecularstructureoftheOrionBcloud:InsightfromPrincipalComponentAnalysis
PC1 60.1% PC2 10.9% PC3 6.5% PC4 5.9%
1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5
0.0 0.0 0.0 0.0
0.5 0.5 0.5 0.5
− − − −
1.0 1.0 1.0 1.0
− OOSN+ ONCHO+ H − OOSN+ ONCHO+ H − OOSN+ ONCHO+ H − OOSN+ ONCHO+ H
12C13CCHCHCOSCHNCC18CNH2HO3 12C13CCHCHCOSCHNCC18CNH2HO3 12C13CCHCHCOSCHNCC18CNH2HO3 12C13CCHCHCOSCHNCC18CNH2HO3
C C C C
PC5 5.0% PC6 3.3% PC7 2.0% PC8 1.9%
1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5
0.0 0.0 0.0 0.0
0.5 0.5 0.5 0.5
− − − −
1.0 1.0 1.0 1.0
− OOSN+ ONCHO+ H − OOSN+ ONCHO+ H − OOSN+ ONCHO+ H − OOSN+ ONCHO+ H
12C13CCHCHCOSCHNCC18CNH2HO3 12C13CCHCHCOSCHNCC18CNH2HO3 12C13CCHCHCOSCHNCC18CNH2HO3 12C13CCHCHCOSCHNCC18CNH2HO3
C C C C
PC9 1.7% PC10 1.4% PC11 0.9% PC12 0.4%
1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5
0.0 0.0 0.0 0.0
0.5 0.5 0.5 0.5
− − − −
1.0 1.0 1.0 1.0
− OOSN+ ONCHO+ H − OOSN+ ONCHO+ H − OOSN+ ONCHO+ H − OOSN+ ONCHO+ H
12C13CCHCHCOSCHNCC18CNH2HO3 12C13CCHCHCOSCHNCC18CNH2HO3 12C13CCHCHCOSCHNCC18CNH2HO3 12C13CCHCHCOSCHNCC18CNH2HO3
C C C C
Fig.6.Barplotsshowingthecontributionofeachlineintensitytoeachprincipalcomponent(withthefractionofthetotalcorrelation
accountedforbyeachPCgivenasapercentage).Theuncertainties(standarddeviations)showninredareobtainedbybootstrapping
asdescribedinSect.4.3.
tion,andnegativecontributionof N H+, CH OHandoftheCO PC2(butwithnegativevalueswherePC2isverylarge),andthis
2 3
isotopologues, that trace gas shielded from the UV field. This PCcouldthustracethechemistryofmoderatelydensegas.
componentthusprobablytracesthechemicalspecificitiesofUV Thesixthprincipalcomponentshowsnegativecontributions
illuminatedgas.Thecomponentmapclearlyshowspositiveval- ofHCN,HCO+ andCN,whichcanalloriginateinphotochem-
uesattheeasternedgeofthecloud,illuminatedbyσOri,andin istry,andpositivecontributionsfromCCH, C18O 13COandSO,
thestar-formingregionNGC2024. which are usually associated with more shielded gas although
CCH can also be bright in UV illuminated regions. Its compo-
Thefourthprincipalcomponentisparticularinthefactthat
nentmapshowsawideblueregionaroundNGC2024,similarto
itsmapalmostonlyshowslarge(positiveornegative)valuesin
thelargewarmdustregionseeninthedusttemperaturemapof
theregionsoflargepositivevaluesofPC2.Itthushighlightfur-
the region (Schneider et al. 2013). It could thus also be related
ther chemical variations inside dense cores. This component is
totheradiationfield,buttraceadifferentaspectfromPC3,char-
completelydominatedbyoppositecontributionsof CH OHand
N H+ and thus traces variations in the ratio of these tw3o lines. acterizedbylowerCCHintensitiesrelativetotheotherlines.
2 The remaining components are more difficult to interpret,
The component map seems to highlight smaller size cores em-
but tend to describe opposite variations in pairs of lines that
beddedinsomeoftheclumpsrevealedbyPC2,andthusprob-
ablyhighlightsthechemistryofthedensestcores(larger N H+ varied together in previous PC. PC7, 9 and 10 display oppo-
2 sitevariationsinpairsoflineofthegroupHCN,HNC,CNand
to CH3OHratios). HCO+,whosevariationswerecorrelatedinthepreviousPCsin
The fifth principal component shows positive contributions whichtheyhadlargeweights(PC1and6).PC8showsanticorre-
ofsulfurspecies(CSandSO),and C18O,andnegativecontribu- latedvariationsofSOand C18O,anditscomponentmapshows
tionsof 12CO,CCH,and CH OH.Itslargepositivevalueshigh- a striking spatial pattern with negative values (high SO/C18O
3
lightlargerscaleregionsembeddingthedensecloudsshownby ratios) in the Horsehead, the molecular gas at the base of the
7
PierreGratieretal.:DissectingthemolecularstructureoftheOrionBcloud:InsightfromPrincipalComponentAnalysis
1.0 1.0 1.0
%) 0.5 N2H+ %) 0.5 CCH %) 0.5 CH3OH
C2(10.9 00..05 HHNCCO+ C3(6.5 00..05 12CO NC2HH3+OH C4(5.9 00..05 13CO CCH
P− 12CO P− P− N2H+
1.0 1.0 1.0
− − −
1.5 1.0 0.50.0 0.5 1.0 1.5 1.5 1.0 0.50.0 0.5 1.0 1.5 1.5 1.0 0.50.0 0.5 1.0 1.5
− − − − − − − − −
PC1(60.1%) PC2(10.9%) PC3(6.5%)
1.0 1.0 1.0
0%) 0.5 SO 3%) 0.5 CCH CSO18O 0%) 0.5 CN C18O
5. 0.0 3. 0.0 2. 0.0 CCH
5( N2H+ 6( CH3OH 7( HCN SO
PC−0.5 CCH CH3OH PC−0.5 CN PC−0.5 HNC
1.0 1.0 1.0
− − −
1.5 1.0 0.50.0 0.5 1.0 1.5 1.5 1.0 0.50.0 0.5 1.0 1.5 1.5 1.0 0.50.0 0.5 1.0 1.5
− − − − − − − − −
PC4(5.9%) PC5(5.0%) PC6(3.3%)
1.0 1.0 1.0
%) 0.5 %) 0.5 %) 0.5
C8(1.9 00..05 HNHCCNSO C18CON C9(1.7 00..05 SOCHNCNHCN18CO C10(1.4 00..05 HHCCCNO18+OSO HNC
P− P− P−
1.0 1.0 1.0
− − −
1.5 1.0 0.50.0 0.5 1.0 1.5 1.5 1.0 0.50.0 0.5 1.0 1.5 1.5 1.0 0.50.0 0.5 1.0 1.5
− − − − − − − − −
PC7(2.0%) PC8(1.9%) PC9(1.7%)
1.0 1.0
9%) 0.5 CS 4%) 0.5 logNH2
C11(0. 00..05 HCO+ C18O HCN C12(0. 00..05 C1138COO H12CCOCO+S llooggnUH/U¯
P− P−
1.0 1.0 (cid:0) (cid:1)
− −
1.5 1.0 0.50.0 0.5 1.0 1.5 1.5 1.0 0.50.0 0.5 1.0 1.5
− − − − − −
PC10(1.4%) PC11(0.9%)
Fig.7.Correlationwheels,showingtheinitiallineintensitiesasvectorhavingascoordinatestheircorrelationcoefficientstoeach
PC,representedintheplanesofsuccessivepairsofPCs.Uncertaintiesfromourbootstrapinganalysis(seeSect.4.3)arepresentedas
thinblackcontoursaroundthearrows’heads(isodensitycontourscontaining68%ofthedistribution).Alsorepresentedincolored
(cid:16) (cid:17)
arrowsarethecorrelationsofourindependentphysicalparameterswiththePCs(red:logN ,green:logn ,blue:log U/U¯ ).
H2 H
Horsehead, and the small scale clumps in NGC2024, and pos- lightedregion,thushavinglittleweightinthecorrelationmatrix.
itive values (low SO/C18O ratios) in a dense filament stretch- PC12iscompletelyconstrainedbyorthogonalitytotheprevious
ing away from NGC2024. PC11 is strongly dominated by CS, PCsandisthusonlyaconsequence.
and thus shows specific variations of CS, mostly uncorrelated
with the other lines (somewhat anticorrelated with C18O), and
thatwerenotdescribedbythepreviousPCs.Itscomponentmap 4.3. Studyingtheeffectofnoise
shows small scale spots of positive values, mostly surrounding
NGC2024andNGC2023.Thefactthatitappearssolateinthe Thenoisynatureofourdatacanhavetwokindsofeffects.Itcan
decomposition can be explained by the small size of the high- firstinducevariabilityinourresults(theresultswouldvaryfora
differentrealizationoftherandomnoise).Weverifiedthestabil-
8
PierreGratieretal.:DissectingthemolecularstructureoftheOrionBcloud:InsightfromPrincipalComponentAnalysis
PC1 60.1% PC2 10.9% PC3 6.5% PC4 5.9%
-14.3 0.0 14.3 -9.2 0.0 9.2 -3.0 0.0 3.0 -5.1 0.0 5.1
PC5 5.0% PC6 3.3% PC7 2.0% PC8 1.9%
-3.4 0.0 3.4 -1.9 0.0 1.9 -1.5 0.0 1.5 -1.8 0.0 1.8
PC9 1.7% PC10 1.4% PC11 0.9% PC12 0.4%
-1.4 0.0 1.4 -1.1 0.0 1.1 -1.6 0.0 1.6 -0.6 0.0 0.6
40
30
20
)
(0
y 10
δ
0
10
−
20
− 40 30 20 10 0
δx( )
0
Fig.8.Principalcomponentmaps.Thesemapsrepresentthevalueofeachobservedpixelwhentheyareprojectedinthespaceof
theprincipalcomponents.
9
PierreGratieretal.:DissectingthemolecularstructureoftheOrionBcloud:InsightfromPrincipalComponentAnalysis
ityofourresultsbyusingabootstrappingmethod.Bootstrapping definedyoungmassivestars(seediscussioninPetyetal.2016).
is a method of choice to compute uncertainties on an estima- This allowed us to infer a link between the first three princi-
tor (here the PCA components) when the distribution of esti- palcomponentsandphysicalquantitiessuchasthecolumnden-
matorvaluescannotbeassumedtofollowasimpledistribution sity,thevolumedensity,andUVillumination.Inthissection,we
(Feigelson & Babu 2012). The idea of bootstrapping is to use willquantitativelyassertthesepotentialrelationsbystudyingthe
a Monte Carlo method to create new resampled datasets of the correlationofeachcomponentmapwithasetofindependently
samesizeastheoriginaldatasetbysamplingwithreplacement measuredmapsofphysicalparameters.
fromtheoriginaldataset.Weconstruct5000suchbootstrapped
datasetsandrunthePCAalgorithmoneach.BecausethePCA
5.1. Independentmeasureofthephysicalquantities
is invariant through the change of sign of the PCs, we ensure
thatthesignsareconsistentbeforecomputingthedistributionof
The goal of this section is to find the principal component that
thePCcoefficientstoavoidoverestimatingtheuncertainty.The
is best associated to each of the physical parameters, not to as-
resultsarepresentedbothfortheeigenspectrainFig.6,andthe
sign an absolute physical meaning to some of the components.
correlationwheelsinFig.7. Itisthereforenotnecessarytohaveabsolutevaluesoftheinde-
ThePCcoefficientsappearoverallverystable,thevariances pendentlymeasuredphysicalparametersmaps.Onlytherelative
beingcompletelynegligibleforPC1and2,andverysmallfor variationofeachphysicalparameterisrequiredtocomputethe
PC 3 to 6. Only PC 7 and 8 show significant variability, with correlationcoefficient.Figure9showsthedifferentmapsofthe
somecoefficientschangingsign.ThiscanbeunderstoodasPCA physical quantities that we will correlate with the first 3 prin-
results are particularly sensitive to noise when two PCs corre- cipalcomponents. Thissection describeshow thesemaps were
spondtoverycloseeigenvalues,andPC7and8havetheclosest obtained.
eigenvalueswithrespectively2%and1.9%ofthetotalcorrela-
tion. Indeed, PCs with equal eigenvalues are degenerate in the
sensethatanybasisofthesubspacetheydefinesatisfiesthedef- 5.1.1. Columndensity
inition of PCA. As a result, when eigenvalues are not exactly
The dust column density map is from the Hershel Gould Belt
equalbutveryclose,thenoisecanresultinarandomrotationof
Survey(PI:P.Andre)OrionBmap(Andre´etal.2010;Schneider
thisgroupofPCsinsidetheirsubspace.Ourresults,whichfocus
etal.2013)2.Thismapwasobtainedbyfittingthefarinfrared
onthefirstfewPCs,arethusunaffectedbynoisevariability.
Spectral Energy distribution by greybodies. We apply a loga-
Thesecondpossibleeffectofnoiseistobiastheresults.PCA
rithmic scaling to the data to reduce the dynamical range, the
isunbiasedifthenoiseisspherical(i.e.,hasequalvarianceinall
resultingmapsisplottedintheleftpanelofFig.9.
directions)inthefinaldatasetonwhichPCAisapplied(i.e.af-
terstandardizationinourcorrelation-basedvariant).Inthiscase,
thenoisecanonlyhidethelowestPCs(thatdescribevariations 5.1.2. Volumedensity
smallerthanthenoiselevel)andmakethemdegenerate.Inour
casehowever,thenoiselevelsonthedifferentmolecularlinesare Volume density is a difficult quantity to measure because one
needs both a mass estimate and an associated volume. Density
initiallyclosebutnotequal(variationsbyafactorof3atmost,
isthusdependentonthescalethatitiscomputedat.Weusethe
seeTable1).Thenon-linearreparametrizationkeepstheserela-
catalogofcoresidentifiedandcharacterizedinKirketal.(2016).
tive variances. Finally, the last normalization step (by the stan-
We compute masses from each cloud’s 850µm flux using their
dard deviation) gives final noise variances proportional to the
equation (3). To do this, we assume a common temperature of
ratios of noise standard deviation to total standard deviation of
the reparametrized intensities. These differ by up to a factor of 17Kforallclouds.Fromthismassandtheirobservedsizeesti-
mateswecomputeavolumedensityforeachofthedensecores
14.8betweenthelines,andpossiblebiasesmaybepresentinour
inourobservedfieldofview.Inthiscasecorrelationcannotbe
results, giving higher weight to the lines with the largest ratios
carried out over the full map but we correlate the density mea-
of noise variance to total variance. However, it wasn’t possible
sured for each core with the value of the principal components
withthePCAmethodtobothavoidgivinghigherweighttothe
measuredinthenearestpixel.Thedataisshownasascatterplot
brightestlines(whichledustousethecorrelation-basedPCA),
inthemiddlepanelofFig.9.
andatthesametimeensureequalnoiseonallvariables.Wenote
thatthepreviousPCAstudiesofmolecularcloudswerelesscon-
cerned by noise-induced bias as they only used lines that were
5.1.3. UVradiationfield
clearlydetectedinallpixels.Ourchoiceallowedustoperform
theprincipalcomponentanalysisonthefullregion,whichledto We compute the UV radiation field by using the fact that PAH
the identification of two PCs associated with dense core chem- emissivityisroughlyconstantperunitHandunitradiationfield
istry. (Draine & Li 2007). In practice, we use the WISE (Meisner &
Finkbeiner 2014) 12µm maps divided by the column density
clipped to a maximum value of 1022cm−2. We do not claim to
5. Correlationoftheprincipalcomponentmaps have an absolute value of the UV radiation field but a quantity
withindependentlymeasured physical that should be proportional to it. The quantity log(U/U¯) where
U¯ is the mean value of U is shown in the rightmost panel of
parametersmaps
Fig.9.
In the previous section, we combined two sources of informa- TheproperwaytocomputetheUVradiationfieldfromPAH
tion to interpret the main principal components: 1) astrochem- emission would be to divide by the volume density but as we
istryteachesusthatsomemoleculestracecertainphysicalcon- discussedinthepreviousparagraph,itisnotpossibletogetafull
ditions,2)OrionBisanextremelywellstudiedsource,implying map of volume density. We choose to use column density as a
that the spatial structure of the source is well known. For in-
stance,themolecularcloudisknowntobeilluminatedbywell- 2 http://www.herschel.fr/cea/gouldbelt/en/
10