NASA Technical Reports Server (NTRS) 20120013322: All-Sky Observational Evidence for An Inverse Correlation Between Dust Temperature and Emissivity Spectral Index

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Mon.Not.R.Astron.Soc.000,1–17(2011) Printed4January2012 (MNLATEXstylefilev2.2)
All-sky Observational Evidence for An Inverse Correlation between
Dust Temperature and Emissivity Spectral Index
1(cid:2) 2 1
Z. Liang, D. J. Fixsen and B. Gold
1DepartmentofPhysics&Astronomy,TheJohnsHopkinsUniversity,3400N.CharlesSt.,Baltimore,MD,21218,USA
2UniversityofMaryland,GoddardSpaceFlightCenter,MD,20771,USA
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4January2012
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c
e ABSTRACT
D
We show that a one-component variable-emissivity-spectral-index model (the free-α
0
model)providesmorephysicallymotivatedestimatesofdusttemperatureattheGalacticpo-
3
lar caps than one- or two-componentfixed-emissivity-spectral-indexmodels (fixed-α mod-

] els)forinterstellardustthermalemissionatfar-infraredandmillimeterwavelengths.Forthe
A comparisonwehavefitall-skyone-componentdustmodelswithfixedorvariableemissivity
G spectralindextoanewandimprovedversionofthe210-channeldustspectrafromtheCOBE-
FIRAS,the100−240μmmapsfromtheCOBE-DIRBEandthe94GHzdustmapfromthe
.
h WMAP.Thebestmodel,thefree-αmodel,iswellconstrainedbydataat60−3000GHzover
p 86percentofthetotalskyarea.Itpredictsdusttemperature(Tdust)tobe13.7−22.7(±1.3)K,
- the emissivity spectral index (α) to be 1.2−3.1 (±0.3) and the optical depth (τ) to range
o
r 0.6−46×10−5witha23percentuncertainty.Usingtheseestimates,wepresentall-skyev-
t idenceforaninversecorrelationbetweentheemissivityspectralindexanddusttemperature,
s
a whichfitstherelationα=1/(δ+ω·Tdust)withδ =−0.510±0.011andω =0.059±0.001.
[ This best modelwill be usefulto cosmic microwavebackgroundexperimentsfor removing

foregrounddustcontaminationanditcanserveasanall-skyextended-frequencyreferencefor
1
v futurehigherresolutiondustmodels.
0
Keywords: dust,extinction–infrared:ISM–submillimetre:ISM–Galaxy:general–meth-
6
ods:dataanalysis–technique:spectroscopic.
0
0
.
1
0
2 1 INTRODUCTION ter (FIRAS, FIRASExplanatorySupplement (1997)) instrument
1 onboardtheCosmicBackgroundExplorer(COBE,Mather1982)
Anaccuratemodelofthermaldustemissionatthefar-infraredand
: satellite to constrain dust models with emissivity proportional to
iv millimetrewavelengths isimportant for cosmic microwave back- ν2. They found that dust emission is best described by a three-
X ground (CMB)anisotropy studiesbecause ithelpstoremove one componentdustmodel:awarm(16−21K)andacold(4−7K)
of the three major diffused foreground contaminants. In the last
r componentthatarepresenteverywhereinthesky,andaninterme-
a decade,experimentssuchastheWilkinsonMicrowaveAnisotropy diate temperature (10−14 K) component that exists only at the
Probe(WMAP,Bennettetal.2003a)havepreciselymeasuredthe
InnerGalaxy.In1996,Boulangeretal.independentlyderivedan-
angularvariationsinCMBsignalinordertounderstandtheglobal
other set of dust spectra using the FIRAS measurements and fit
geometryandexpansionoftheuniverse.However,studyingvaria- it a one-component ν2 emissivity dust model. They find that the
tionsthatare10−5 thestrengthoftheprincipalsignalisdifficult,
averagespectrum of dust associated withHI gashasanaveraged
and the removal of contaminating signals in thedata needs to be temperatureof17.5±0.2K.In1998,Lagacheetal.usedDIRBE
doneaccurately.Fortheseexperiments,adusttemplate,suchasone
bands at 100, 140 and 240 μm to decompose FIRAS spectra at
extrapolatedfromtheFinkbeiner,Davis&Schlegel(1999)(FDS) |b| > 10◦ intoacirrusandacoldcomponent. For61percentof
studyofinterstellardustinthefar-infraredhasbeenusedtoremove
theskywherethecoldemissionisnegligible,theyfoundthatthe
thermal dust contribution from sky measurements (Bennettetal.
cirrushadameantemperatureof17.5Kwithadispersionof2.5K.
2003b;Hinshawetal.2007;Goldetal.2009).
For the 3.4 per cent sky where both cirrus and cold components
Among major efforts to derive an all-sky dust model from arepresent,thetwocomponentsarebothassumedtofollowaν2
observational data, Reachetal. (1995) use dust spectra derived
emissivitylaw,withthecirruscomponentfoundtohaveatemper-
frommeasurementsoftheFarInfraredAbsoluteSpectrophotome- atureof17.8±1.2K,andthecoldcomponentwithatemperature
of 15 ± 0.8 K. A widely used dust model in CMB studies was
(cid:2) E-mail:[email protected] obtainedbyFinkbeineretal.(1999).Theirbestmodel(Model#8)
(cid:2)c 2011RAS
2 Z. Liang, D.J. FixsenandB. Gold
totheFIRASdustspectraconsistoftwodustcomponents:acold newsetofFIRASdustspectraandtounifycalibrationsofthedata
component following a ν1.67 emissivity law with temperature at sets.InSection4,wepresentresultsandanalysisfromfittingone-
7.7−13.1K,andawarmcomponentfollowingaν2.70emissivity componentdustmodelswithfixedandvariableemissivityspectral
lawwithtemperatureat13.6−21.2K.Adecadelater,thePlanck indextothedata.InSection5,wecomparetemperaturepredictions
Collaboration (PlanckCollaboration 2011a) used the Planck-HFI of the free-α model with those of the fixed-α models and show
(350μm–2mm)andIRAS100μmdatatoderiveall-skydusttem- thatonlythefree-αmodel givesphysicallymotivatedpredictions
peratureandopticaldepthmapsusingaone-componentmodelwith ofdusttemperatureattheGalacticpolarcaps.Wealsodiscussthe
emissivity proportional to ν1.8. They found that themedian tem- implicationsofthefree-αmodelontheinversecorrelationbetween
perature of the sky at 10◦ above and below the Galactic plane is emissivityspectralindexanddusttemperature.Conclusionsalong
17.7K,seealsoLiang(2011). withsuggestionsforhowtouseourresultsarepresentedinSection
Thislistofresultshighlightsthediversefindingsinthestudy 6.
of thermal dust emission at far-infrared and millimetre wave-
lengths. It also shows that the derived dust properties depend as
much onthefittingmethodand thefunctional formof themodel
2 OBSERVATIONS
asonthedata. Withadded new andmoresensitivedatafromthe
WMAP,wenowcanconstrainmodelparameterswithmuchgreater Thefollowingsectionsreviewtheinstrumentsandthedatasetswe
accuracy. usedinthefollowinganalysis.
A second motivation for our work is to understand whether
dust optical properties (Draine&Lee 1984) are the same at far-
infrared and millimetre wavelengths from the perspective of em- 2.1 COBEDIRBE
piricalmodelfitting.Thatfar-infrareddustemissivityfollowsaν2
The DIRBE instrument was a cryogenically cooled 10-band ab-
power law has been widely accepted (see list above), yet the va-
solute photometer designed tomeasure thespectrum and angular
lidity of such extrapolation has not been proved by theory, lab-
distributionofthediffuseinfraredbackground. Ithada0◦.7beam
oratory experiment or empirical model fitting. In fact, laboratory
andcoveredthewavelengthrangefrom1.25to240μm.Duringits
measurementsofAgladzeetal.(1996)andMennellaetal.(1998)
lifetime, the DIRBE achieved a sensitivity of 10−9 W m−2 sr−1
foundthatemissivityofamorphoussilicateandcarbongrainsdif-
feredfromaν2powerlawandhadasignificanttemperaturedepen- at most wavelengths (Boggessetal. 1992; Silverbergetal. 1993;
DIRBEExplanatorySupplement1998).
dence.Thisinconsistencybetweenobservationandpracticemeans
Weusethe1997“Pass3b”Zodi-SubtractedMissionAverage
thatourunderstandingofdustemissioninthefar-infraredandmil-
(ZSMA)Mapsatbands100,140and240μm.Thesemapsmeasure
limetreisincomplete. Inthiswork, weattempttofillthisgapby
theGalacticandextragalacticdiffuseinfraredemission,andhave
first deriving best-fitting dust models with emissivity spectral in-
beencalibratedtoremovezodiacallight(zodi).Theyareavailable
dexfixedatdifferentvaluesandasavariable,andthencomparing
at the LegacyArchive for Microwave Background Data Analysis
thequality-of-fitofthesemodels.Basedonourfindings,weargue
(LAMBDA)1.
thatdustemissivitydiffersinthefar-infraredandmillimetrefrom
theoptical.
Athirdreasonforour workistodemonstrateaspatialaver-
2.2 COBEFIRAS
aging technique to increase signal-to-noise of spectra. At low in-
tensityregionsdataoftencomewithlargeuncertainty.Ifsuchdata The FIRAS instrument was a polarizing Michelson interfer-
areuseddirectlytoconstraintamodel,resultsarehighlyuncertain ometer designed to precisely measure the difference between
parameters. At times this problem is treated with averaging data the CMB and a blackbody spectrum. The FIRAS had a 7◦
within a predefined sky region. This approach has the disadvan- beam and covered the frequency range from 1 − 97 cm−1 at
tageofusingapresupposeddustdistributioninthederivationofa 0.45 cm−1 resolution (Boggessetal. 1992; Fixsenetal. 1994a;
solution whilefiguring out the distributionispart of the research FIRASExplanatorySupplement1997).
question.Herewemakenoassumptionofthedustdistributionbut WederiveanewsetofdustspectralmapsfromtheDestriped
insteadusethesignal-to-noiseofthedatatodeterminetheamount Sky Spectra of the Pass 4 final data release. The procedures are
ofspatialaveragingneededforthedata.Theresultsarehigherspa- describedinSection3.The2106063-pixelmapscomprisethemain
tialresolutionforregionswithgoodsignal-to-noiseandlessaver- bodyofspectralinformationweuseinmodelfitting.
agingfortheoriginaldataset.Thistechniquecouldbeappliedto SixtypesofuncertaintieshavebeencharacterizedbytheFI-
similarproblemsinotherareasofresearch. RASTeam.Fixsenetal.(1994b);FIRASExplanatorySupplement
Finally,regardingthemanyempiricalmodelswenowknow, (1997);Matheretal.(1999)provideextensiveinstructionsonhow
e.g. the ones listed above, one cannot help but ask: How do we to treat each type of uncertainty. Since we build models that re-
test the validity of these models? Beside having good constraint spondtobothspectralandspatialvariationsofdustemission,the
onmodel parameters, aretherephysicallymotivatedtestswecan following analysis includes all six FIRAS uncertainties: detector
usetoverifypredictions ofthesemodels? Here,wepropose one: noise (D), emissivity gain uncertainties (PEP),bolometer param-
to compare dust temperature distribution with the distribution of eter gain uncertainties (JCJ), internal calibrator temperature er-
dustheatingsourceattheGalacticpoles.Weconductanindepen- rors (PUP), absolute temperature errors (PTP), and destriper er-
dentandcomprehensivetestonall-skyone-componentmodels,and rors (β). Section 7.10 of FIRASExplanatorySupplement (1997)
showthatallbuttheone-componentfree-αmodelfailthistest. provides very helpful instructions on how to assemble the co-
Thestructureofourmanuscriptisasfollows.InSection2,we variance matrix. For example, the D and β matrices vary only
review observations by the COBEsatellite’s DIRBE and FIRAS
experiments and the WMAPsatellite that are used in our model
construction. InSection3wedetailprocedures takentodeducea 1 TheLAMBDAWebsiteishttp://www.lambda.gsfc.nasa.gov/
(cid:2)c 2011RAS,MNRAS000,1–17
All-skyObservationalEvidenceforA TemperatureDependent EmissivitySpectralIndex 3
Table1.SpectralCoverageofDIRBE,FIRAS&WMAP
λ 1/λ ν
(μm) (cm−1) (GHz)
DIRBE 100 100 2998
140 71 42
240 42 1249
FIRAS 103−4407 2−97 68−2911
WMAP 3189 3 94
among pixels, while the PEP, JCJ, PUP and PTP matrices dif-
ferfordifferent frequencies. Interestedreadersarereferredtothe
FIRASExplanatorySupplement(1997)fordetails.
2.3 WMAP
TheWilkinsonMicrowaveAnisotropyProbewasdesignedtode-
termine the geometry, content and evolution of the universe by
measuring temperature anisotropy of the CMB radiation. It con-
sisted of two back-to-back offset Gregorian telescopes, and used
20 high electron mobility transistor (HEMT) based differential
radiometers to measure the brightness difference between two
lines of sight that were 141◦ apart. At five frequency bands: 23,
33, 41, 61 and 94 GHz, the WMAP made full sky measure-
ments, which were analyzed by the data processing pipeline and
form 13(cid:3) FWHM HEALPix2 pixelization maps. The spin mo-
tion of the observatory and its scanning strategy symmetrized
the WMAP beams. Beam sizes were estimated using square-
root of the beam solid angle. In order of increasing frequen- Figure1.ExamplesofthenewFIRASdustspectra.Plottedinredarethe
cies, they are: 0.88, 0.66, 0.51, 0.35 and 0.22◦ (Jarosiketal. new dustspectra andtheir errorestimates; inpurple arethedustspectra
2003; Pageetal. 2003; Hinshawetal. 2003; Jarosiketal. 2007; derivedbytheFIRASteam;indarkgreenisthetotalskyintensitymeasured
byFIRAS;inblueistheCMBmonopole;andinorangeistheCMBdipole.
WMAPFive-YearExplanatorySupplement 2008; Hinshawetal.
2009;Hilletal.2009).
We use the dust temperature map (at 94 GHz) derived from
applied to both signal and noise maps. Procedures on zodi zero-
the“basemodel”inWMAP’sFive-Yearforegroundmodelinganal-
pointcorrectionsandFIRASsystematicerrorsareappliedtonoise
ysis by Goldetal. (2009). In the same study, Gold et al. used
mapsonly.
different models to account for diffused foreground emission at
different WMAP bands, with nonthermal synchrotron, thermal
bremsstrahlung,andthermaldustasthestandardcomponentsand
testedthepossibleexistenceofsteepeningsynchrotronand/orspin- 3.1 DeducingFIRASdustspectralmaps
ningdust.Theirlikelihoodanalysisshowedthatbasicmodelwith
TheFIRASdust spectra onLAMBDA exhibit a“jump” between
just three main foreground components was sufficient to subtract
thelow-andhigh-banddataduetoaninconsistantCMBmonopole
outforegroundsfromskymapsathighGalacticlatitudes.
temperaturesubtraction.Inaddition,thePass4datawerereleased
withanearlycalibration,makingitnecessarythatwederiveanew
setofdustspectra.Inthefollowingwedescribeprocedurestosub-
3 DATAPREPARATION tractfromtheDestripedSkySpectraablackbodyspectrumforthe
CMB,adipole of the Earth’smotion withrespect totheCMB, a
Sincewewanttomodelbothspectralandspatialvariationsofdust zodimodel,andcontributionfromthecosmicinfraredbackground
emission, we unify different hardware constraints and calibration (CIB).ExamplesofthenewdustspectraareplottedinredinFig.
standards to ensure that different data sets are compared on an 1.Forcomparison,correspondingdustspectraprovidedbytheFI-
equal footing. InSection3.1wediscuss theFIRASdata, thepri- RASTeamareplottedinpurple.
maryforthisstudy.InSections3.2–3.7weexplainalignmentof
theDIRBEandWMAPdatawiththeFIRASdata.Theprocedures
onbeamdifferences,mapprojections,spatialresolutions,DIRBE-
3.1.1 CMBmonopoleanddipole
FIRASabsolutecalibrations,andtemperature-fluxconversionare
The CMB temperature has been extensively treated (see
Matheretal. 1990; Fixsenetal. 1994b; Matheretal. 1994;
2 For definition and applications of the HEALPix projection, Fixsenetal. 1996; Matheretal. 1999; Fixsen&Dwek 2002;
refer to Go´rski,Hivon&Wandelt (1999), Go´rskietal. (2005), Fixsen 2009). The appropreate correction for the Pass 4 data set
Calabretta&Roukema(2007)andhttp://healpix.jpl.nasa.gov. isa2.7278blackbodyspectrum.
(cid:2)c 2011RAS,MNRAS000,1–17
4 Z. Liang, D.J. FixsenandB. Gold
A WMAP-determined dipole (Hinshawetal. 2009) is re- frequency measured by FIRAS. The central portion of the beam
moved from the Destriped Spectra. Specifically, Tdipole = (θ < 3◦.5)isapproximatedbyatophatsinceanyslightazimuthal
3.355mKand(l,b)=(263◦.99,48◦.26).Higherordervariationsin asymmetryshould havebeensmoothed out bytherotationof the
theCMBtemperaturewereignoredbecausetheyareinsignificant instrumentalongitsownaxis(Matheretal.1993)duringitsoper-
forthisstudy. ation.
On the other hand, DIRBE was built with a goal to reject
straylighttomeasuretheabsolutespectrumandangulardistribu-
3.1.2 Zodi
tion of the CIB. This goal was met by using a series of optical
Zodi is the thermal emission and scattered light from interplan- elements and baffle protections, among which the last field stop
etary dust in our solar system. Kelsalletal. (1998) derived a setthe0◦.7×0◦.7instantaneousfieldofviewforallspectralbands
time-dependent parametric model for its emission, and the FI- (Silverbergetal.1993;DIRBEExplanatorySupplement1998).To
RAS Team extended those resultsto the entire FIRAS frequency construct theZSMAmaps, theDIRBETeamcalculatedthezodi-
coverage (FIRASExplanatorySupplement 1997; Fixsen&Dwek acal light intensity using the IPD model by Kelsalletal. (1998),
2002).DerivationofthezodimodelforFIRAShingesonthefact andsubtracteditofffromeachweeklymeasurement.Theremain-
thatFIRASmeasurements overlapwithDIRBEbandsat140and ing signal was averaged over time. In this way, the ZSMA maps
240 μm. Therefore, the DIRBE model predictions for these two preservetheoriginal0◦.7×0◦.7angularresolutionoftheskyobser-
bandswerecoaddedandfittedwithapowerlawemissivitymodel vation.
and extrapolated to the frequency coverage of FIRAS. The zodi Thedust map fromtheWMAPwasone of theproducts de-
modelusedhereisamongFIRASdataproductsonLAMBDA.For rived from Markov chain Monte Carlo fittingof temperature and
furtherdetailsofthemodelderivation,seeFixsen&Dwek(2002). polarization data (Goldetal. 2009). Since their analysis used the
band-averaged mapsthatweresmoothedbya1◦ Gaussianbeam,
thedustmaphasthesameangularresolution.
3.1.3 Emissionlines The FIRAS beam is the lowest common angular resolution
achievable among all three data sets, so the higher resolution
TheFIRASdetected18molecularandatomiclinesemittedbyin-
DIRBE andWMAPmaps need to be convolved withthe FIRAS
terstellargas.SincetheFIRASfrequencyresolutionismuchlarger
beam to make them all 7◦ maps. Additionally, since Fixsenetal.
thanthewidthofeachoftheselines,eachlineprofileiseffectively
(1997b)foundthattheFIRASbeamwaselongatedinthescandi-
FIRAS’sinstrumentresponsetoadeltafunction.Amongthese18
rectionby2◦.4,thatpatternismatchedinthedegradedDIRBEand
detectedemissionlines,notallofthemhaveadiscerniblepresence
WMAPmaps by convolving those data with an effective FIRAS
overthefullsky.Mostnotablyarethe[CII]and[NII]lines,which
beam.
exhibitadistinctgradientofintensityfromthecentreoftheGalaxy
tohigherlatitudes.Otheremissionlines,thoughdetected,areweak
inmostoftheskyexceptattheInnerGalaxy.ByInnerGalaxywe
meantheinnerGalacticdiskabouthalfthedistancetotheedgeof
theGalaxy.
Since the derivation of FIRAS line intensity maps on
LAMBDAusedtheirGalacticdustspectra,theFIRASlineinten- 3.3 Mapprojectionandspatialresolution
sitymapscannotbeusedheretoremoveemissionlinecontribution.
Both DIRBE and FIRAS maps are organized in COBEquadri-
Instead,intensitiesof[CII]and[NII]emissionarefitaspartsofthe
lateralized spherical cube format (quad-cube, Chan&O’Neill
overallmodelinthefollowing.
1975, O’Neill&Laubscher 1976, White&Stemwedel 1992, and
Calabretta&Greisen2002).WhiletheDIRBEmapsareinquad-
3.1.4 Cosmicinfraredbackground cube resolution level 9 (res9, 19m. 43 per pixel), the FIRAS maps
areinquad-cube resolution level 6(res6, 2◦.59 per pixel). Differ-
TheisotropicCIBsignalwasremovedfromskyspectrausingre- entfromtheFIRASandtheDIRBEmaps,theWMAPdustmapis
sults from three studies: For DIRBE measurements at 140 and in HEALPix (Go´rski,Hivon&Wandelt 1999, Go´rskietal. 2005,
240 μm,theCIB isremoved at15 and13 nWm−2 sr−1 respec- and Calabretta&Roukema 2007) resolution level 6 (res6, 54m. 97
tively according to Hauseretal. (1998). For the DIRBE band at per pixel). One way to reconcile these different formatsand spa-
100 μm, the CIB is removed at 25 nW m−2 sr−1, as given in tial resolutions is to carry out analysis inCOBEquad-cube res6.
Finkbeineretal. (2000). Notice that the Finkbeiner prediction is Thisdecision is made toretain maximum amount of information
withintheupperandlowerlimitsestimatedbytheDIRBETeam. containedintheoriginaldatasetsandachievethehighestcommon
To remove the CIB from FIRAS sky spectra, the CIB model in resolutionpossible.
Fixsenetal.(1998)isused.
Asaresult,DIRBEmapswerere-binnedtores6;theWMAP
dust mapwasfirstconvertedintoaquad-cube res9mapand then
re-binnedtores6.DuringWMAP’sdustmapconversion,wetook
3.2 Beamdifference
thefollowingstepstoensurethatnoexcessiveartificialnoisewas
The three instruments that produced the data used in this study introducedtothefinalmap:WecomparedtheoriginalHEALPix-
haddifferentbeampatterns.Forexample,theFIRASusedaquasi- projectionmapwiththere-binnedquad-cube-projection map,and
opticalmultimodehornantennatocollectradiationfroma7◦field found that 98.6 per cent of the 49,152-coordinate pairs sampled
of view (Matheretal. 1986). The horn was designed in a trum- gavenodifferencebetweenthequad-cube andtheHEALPixval-
petbellshapetoreduceresponsetooff-axisradiation.Asaresult, ues.Whentherewasadifference,themaximumwas0.0059mK,
whenthebeamprofilewasmeasuredontheground andinflight, whichamountedtoa0.11percentnoiseincreasefortheoriginal
it was found to have very low sidelobes over the two decades of HEALPixmap.
(cid:2)c 2011RAS,MNRAS000,1–17
All-skyObservationalEvidenceforA TemperatureDependent EmissivitySpectralIndex 5
3.4 Gradientcorrection where ν is the effective frequency (93.5 GHz) of the dust map
(Goldetal. 2009; Jarosiketal. 2003), and k isBoltzmann’s con-
The FIRAS dust maps are based on coadding interferograms, so
stant.
theirvaluesaregenerallynot atthedefinedcentreofmappixels.
Thispositionaldifferencerequiresanadditionalcorrectionstepto
prepare the quad-cube res6 DIRBEand WMAP maps. Detailsof
thistechniquearedescribedinFixsenetal.(1997b).Insummary,a
second-degreesurfacefunctionisfittotheintensityandlocationin- 4 RESULTSANDANALYSIS
formationofapixelanditsimmediateneighborsinoneofthecon-
4.1 Overviewofmodelfittingstrategy
vertedmaps.Thisfunctionisthenusedtopredictemissionatthe
FIRASmeanpositionforthatparticularpixel.Overall,a5percent
The thermal emission of a dust grain is described by a modified
rmscorrectionisappliedtoeachoftheDIRBEmapsat100,140
blackbodyfunction:
and240μmandtotheWMAPdustmap.
I (ν)=τ (cid:3) ·B (T ),
dust ν ν dust
3.5 Colorcorrection whereBν(Tdust)istheblackbodyspectrumattemperatureTdust,
(cid:3)ν =(ν/ν0)αistheemissivitywithspectralindexα,andτ isthe
In accordance with the IRAS convention opticaldepthnormalizedtofrequencyν0 =900GHz.
(IRASExplanatorySupplement 1988), DIRBE photometric Inadditiontomeasuringthermaldustemission,theprepared
measurementswerereportedinMJysr−1atnominalwavelengths, FIRASspectraretaincontributionsfrom[CII]and[NII]emission,
assumingthesourcespectrumtobeν·Iν =constant.Sinceeach duetothelackofpreciseall-skytemplates.Asaresult,twoemis-
DIRBEbandhasamuchwiderbandwidththanaFIRASchannel, sionlinesaremodeledatthesametimewiththedust:
spectral shape could have changed enough that at the nominal
I (ν)=[C ] f (ν),
wavelengththerealintensityissignificantlydifferentfromthenor- [CII] II intensity [CII]
malizedintensity.Asaresult,weincludecolor correctionsinthe
and
overall model, i.e., model predictions are compared with DIRBE
mKeiassuthreemcoelnotrscuosrirnegcttihoenrfealcattoiorndeIfiνn,emdodaesl =KIν,DIRBE,where I[NII](ν)=[NII]intensity f[NII](ν),
(cid:2)
wheref(ν)isthesyntheticlineprofiledeterminedbyFIRASre-
(I /I ) ·R dν
K = (cid:2) ν ν0 actual ν . (1) sponsetoadelta-functionsignal.Together,thefullmodelhasthe
(ν /ν) ·R dν
0 quoted ν form
(cid:2)
Isnkythniosrmeqauliazteiodnt,oth(eIνin/tIeνn0s)itayctautalfreisqutheencsypeνc0i,fiacndinRtenνsiistyDoIfRBthEe Itotal =Idust+I[CII]+I[NII]. (2)
relativesystemresponseatfrequencyν.ThevaluesofRνaredoc- Eachfull model isfittothedata byminimizing athree-part
umentedinDIRBEExplanatorySupplement(1998)Section5.5. χ2,witheachpartcorrespondingtooneofthethreedatasets:
χ2 =χ2DIRBE+χ2FIRAS+χ2WMAP, (3)
3.6 DIRBEuncertainties
where
(cid:3)
The DIRBE photometric system was maintained to ∼ 1 per cent
accuracy by monitoring the internal stimulator during 10 months χ2instrument= (Iobs−Imdl)i (M−1)ij (Iobs−Imdl)j, (4)
of cryogenic operation and observing the bright stable celes- i,j
tial sources during normal sky scans. It was absolutely cal- Here,Iobsistheobservedspectralintensity,Imdlisthemodelpre-
ibrated against Sirius, NGC7027, Uranus and Jupiter. Among diction,andMisthecovariancematrixoftherespectivedataset.
different types of uncertainties identified by the DIRBE Team, Inthefollowingsections,wefitone-component dustmodels
those relevant to this work are standard deviations of intensity tospectraoffixed(Section4.2)anddifferent(Section4.3)sizesky
maps, detector gain and offset uncertainties, and zodi model un- regions.Intheformercase,spectraretainthe7◦angularsizeofFI-
certainties (Hauseretal. 1998; Kelsalletal. 1998; Arendtetal. RASpixels;inthelattercase,the7◦spectraareaveragedbyvarious
1998).Forbands8–10,respectively,thedetectorgainsare:0.135, amountstoincreasesignal-to-noiseofthefinalspectra.Chi-square
0.106 and 0.116 nW m−2 sr−1; detector offsets are 0.81, 5 and perdegreeoffreedomvalues,χ2dof,areusedtoassessthequality-
2 nW m−2 sr−1; and zodi model uncertainties are: 6, 2.3 and of-fitofamodeltoeachspectrum.Inparticular,theone-component
0.5nWm−2 sr−1 (Arendtetal.1998).Theprocessofre-binning fixed-αmodelshave214−4=210degreesoffreedom.Adopting
thehighresolutionDIRBEmapsintoFIRASresolutionaffectsonly a 10 per cent probability cut-off for acceptable models, it corre-
thestandarddeviationsoftheoriginalmaps.Thefinaluncertainty sponds to a χ2dof (cid:2) 1.13. The free-α model has 209 degrees of
isthequadraturesumoftheindividualnoisecomponents. freedomandits10percentprobabilitycutoffisχ2dof (cid:2)1.13.
3.7 Temperature-intensityconversion
4.2 Fittingspectraof7◦skyregions
ForegroundmapsoftheWMAPproductionarereportedinantenna
temperature,TA,inmK.Ontheotherhand,mapsproducedbythe Wefitone-componentmodelswithfixedαintherange1.4−2.6
DIRBEandtheFIRASTeamsarereportedinspectralintensity,Iν, at 0.1 increment to the spectrum at each 7◦ pixel. The fits have
in MJy sr−1. In the following analysis, the WMAPdust map is acceptablevalues(χ2dof (cid:2)1.13)overmostoftheskyexceptatthe
convertedintofluxdensityvaluesfollowingIν = 2 (ν/c)2 kTA, Galacticplane.
(cid:2)c 2011RAS,MNRAS000,1–17
6 Z. Liang, D.J. FixsenandB. Gold
Figure2.χ2 vs.Galactic latitude. Thevalueofχ2 isobtainedfrom Figure3.χ2 distributionsofthe7◦ fitsusingone-componentα = 2.0
dof dof dof
fittingone-componentα=2.0modelto7◦spectracovering98.68percent model.ThereddistributionincludesonlypixelsatGalacticlatitudes|b|>
areaofthefullskywhereFIRASdataareavailable. Thisplotshowsthat 10◦;thebluedistributionincludesall6063pixelsatallGalacticlatitudes.
mostfits at|b| (cid:3) 10◦ havea χ2 ∼ 1, andfits at |b| (cid:4) 10◦ have a Best-fitting parameters of the Gaussians are printed in respective colors.
dof
χ2 >1. Bothχ2 distributionsarewellapproximatedbyaGaussian,whichmeans
dof dof
thatthereisnoapparent systematicbiasinthefits.Thatthedistributions
centerat0.93meansthaterrorsinthedataareslightlyoverestimated,and
4.2.1 Quality-of-fitofmodels thewidthsofthedistributions areasexpected (0.1)foradistribution of
randomdatawith210degreesoffreedom.
Asanexample,Fig.2presentsχ2dof asafunctionofGalacticlat-
itudefortheα = 2.0model.Itshowsthatmostfitsat|b| (cid:3) 10◦
haveχ2dof ≈1,andfitsat|b|(cid:4)10◦haveaχ2dof (cid:4)1.Fig.3com- 17.5±0.26K,comparedtoTdust ∼ 18.5±0.28Katα = 1.8.
paresthedistributionofχ2dof at|b| (cid:3) 10◦ withthedistributionof Thisdifferenceintemperatureislargerthanthesumoftheirerrors.
χ2doffortheentiresky.Bothdistributionsarewellapproximatedby Similarly,thedifferenceinτ ofthetwoαmodelsislargerthanthe
aGaussian,anindicationthatthefitsdon’thaveasignificantsys- sumoftheirerrors.Onthecontrary,intheLSNcase,thedifference
tematicbias.Thewidthsofthedistributionsareasexpected(0.1) inthebest-fittingTdustandτ ofα=1.8andα=2.0modelsare
foradistributionofrandomdatawith210degreesoffreedom.That wellwithintheuncertaintiesoftherespectiveparameters.
thedistributionscenter at0.93 means that statisticalerrorsof the Theseresultsdemonstratethesensitivityofthefitstomeasure-
data are slightly overestimated by ∼ 7%, and that the χ2dof cut- menterrors.Theexistenceofmeasurementnoiseinevitablycauses
offisreallyat∼ 1.21withaprobabilityof< 10%. Becausethe ahighdegreeofdegeneracybetweentheemissivityspectralindex
uncertainties of the FIRAS data include some systematic effects, and the dust temperature in the fits. While it is difficult to break
we do not feel at liberty to reduce the uncertainty. Based on the this degeneracy, high signal-to-noise data help. Fitting data with
χ2dof (cid:2)1.13cut,themodelisagoodfittothedataover87percent highsignal-to-noiseresultsinwellconstrainedparameters,which
ofthefullskyarea,butisrejectedbythedataattheGalacticplane. means that the choice of an α model can cause statistically sig-
Readers interested in the best-fit parameters (Tdust, τ, [CII] nificantdifferencesinthepredictionsoftheseparameters.Onthe
and[NII]intensities),theiruncertaintiesandcorrelationsforeach otherhand,fittinglowsignal-to-noisespectraresultsinsmalldif-
of theaforementioned αmodels arereferred to theauthor’s PhD ferenceinχ2dof andlargeerrorbarsofthebest-fitparameters,and
thesis(Liang2011). soisnotpossibletodifferentiatemodelswithdifferentfixedvalues
ofα.ThisisdemonstratedinFig.6,whichshowsthatthe68and
95percentconfidencecontoursofaHSNfitenclosemuchsmaller
4.2.2 Dependenceofbest-fittingparametersonthe
regionsintheT-αspacethanthoseofaLSNfit.
signal-to-noiseofdata
Figs.7showsskymapsofαandTdustthatcorrespondtothe
Figs.4and5presentχ2dof,Tdustandτ ofthebest-fittingαmodels minimum-χ2dof modelamongallmodelstestedateachpixel.Over-
fortwospectra:onehashighsignal-to-noise(HSN)andisatalow all,thetwomapsarenoisy,whichisaresultofinsufficientsignal-
Galacticlatitude,andtheotheronehaslowsignal-to-noise(LSN) to-noise in the data that prevents setting tight constraints on the
andisatahighGalacticlatitude.Inbothcases,theχ2dofvs.αplots best-fittingparameters.Morespecifically,bothmapsshowgreater
show that models with a wide range of different α values can fit consistencyinvalueatlowlatitudesandmorefluctuationsaround
the data well. In the HSN case, a curve fit to χ2dof as a function theGalacticpoles.Thatconsistentvaluesappearintheregionsur-
of α is a concave up parabola, with the minimum χ2dof = 0.89 rounding the Inner Galaxy is reasonable because star formation
atα = 1.8. Thedifference betweenχ2dof atα = 1.8 andthat at takesplaceintheGalacticdiskandatthebulge,andstarformation
α=2.0isΔχ2dof =0.01.At210degreesoffreedom,thismeans isthemostimportantheatsourcefordust.Thatlargefluctuations
aΔχ2 of∼ 2.1,whichisa2-sigmadifference.IntheLSNcase, appearnearthepoles,ontheotherhand,hastodowithlowsignal-
thebest-fittingcurvetoχ2dof vs.αisamuchflatterparabolaover to-noisedataintheseregionscomparedtothosemeasuredatlower
1.4 (cid:2) α (cid:2) 2.3withtheminimumχ2dof = 0.80atα = 1.6.The latitudes.Thishappens because fewdust grainsexist athighlati-
differencebetweenχ2dof atα=1.8andα=2.0is∼0.002. tudes,andtheydonotemitasstronglyasthoseclosetotheGalactic
Althoughmodelswithdifferentαhaveonlyasmalldifference disk.
inχ2dof,thebest-fittingTdustandτ aredifferentsignificantlyinthe Sincelowsignal-to-noisedatacannotgiveadequateconstraint
HSNcase:Atα=2.0,thebest-fittingdusttemperatureisTdust ∼ tomodelparametersandexacerbatesthedegeneracybetweendust
(cid:2)c 2011RAS,MNRAS000,1–17
All-skyObservationalEvidenceforA TemperatureDependent EmissivitySpectralIndex 7
Figure 4. χ2dof, Tdust and τ as a function of α. Each data point on Figure 5. χ2dof , Tdust and τ as a function of α. Each data point on
the Tdust and τ plots is thebest-fitting value ofthe corresponding one- the Tdust and τ plots is the best-fitting value ofthe corresponding one-
component fixed-α model to the 7◦ spectrum measured in the direction component fixed-α model to the 7◦ spectrum measured in the direction
l = 63◦.78andb = −11◦.53.Thissetofplotsservesasanexampleof l=254◦.32andb=65◦.08.Thissetofplotsservesasanexampleoflow
high signal-to-noise fits. The green curve fits the best-fitting values as a signal-to-noisefits.Thegreencurvefitsthebest-fittingvaluesasafunction
function of α. Inparticular, the plot of χ2 shows that the model with ofα.Becauseofthesmalldifferenceinχ2 andlargeerrorsinthemodel
dof dof
α=1.8isabetterfittothedatathanthemodelwithα=2.0.Thesmall parameters,fitresultscannotdifferentiate modelsatdifferentfixedvalues
errorbarsofTdust andτ showthatthechoiceofanαmodelcancause ofα.
statisticallysignificantdifferenceinthepredictionsoftheseparameters.
theamountofspatialaveragingneedstobeadjustedaccordingto
thesignal-to-noiseofthedata.
temperatureandspectralindex,inordertoconstructthebestdust
Onewaytoincreasethesignal-to-noiseofhigh-latitudespec-
model,weneedtofindwaystoincreasethesignal-to-noiseofthe
tra and to best preserve intrinsic variations in the signal detected
data.
fromdifferentskydirectionsistobasetheamountofspectralaver-
agingonsignal-to-noiseoftheaveragedspectrum.Startingwiththe
baselevel,whereapixel’sownspectrumisusedtofitamodel,if
4.3 Fittingaveragedspectraofdifferent-sizeskyregions thefitdoesnotgivewellconstrainedparametersduetoinadequate
signal-to-noise,theproceduregoesontofittheaverageoftheorig-
Takingaverageofthehigh-latitudespectrabasedonlatitudinalor
inal spectrum and its eight immediate neighbors. Thisprocess of
longitudinaldivisionsoftheskycantightentheconstraintonmodel
involvingmoreoftheadjacentspectratoformanewaveragegoes
parameterssinceitincreasesthesignal-to-noiseofthedata.How-
onuntilthederivedparametersaresufficientlyconstrained.Inthis
ever,suchdivisionsarebasedonourexpectationsofthedistribu-
way,resultsfromfitsdoneatthebaselevelhaveaspatialresolu-
tionofGalacticdust.Sinceourknowledgeisnotcomplete,thedi-
visionsarenotoptimal.Inourexperiments(Liang2011),χ2ofre- tionof6.71(cid:5)(cid:6)2.Atthenextlevel,resultshaveaspatialresolution
gionalfitsaremuchhigherthanχ2offitstotheindividual7◦spec- of60.37(cid:5)(cid:6)2,andsoon.
trathatcomprisetheregionalaverages.Sincelargerskyregionsin-
cχl2udvealduieffmereeanntstythpaetstohfedauvsetreamgiinsgsihoanssapcehcitervae,dthaessutefefipciiennctresiagsneailn- 4.3.1 Tdust/δTdustconstraintonfixed-αmodels
to-noiseratio,sospectralvariationbecomesstatisticallyimportant. All-skyone-componentfixed-αmodelswithαintherange1.4−
Inordertopreserveinformationonspectralvariationinthemodel, 2.6andlowerlimitofTdust/δTdustat5,10,20and40areobtained
(cid:2)c 2011RAS,MNRAS000,1–17
8 Z. Liang, D.J. FixsenandB. Gold
Figure6.68and95percentprobabilitycontoursintheTdust-αspacefor Figure7.SkymapsofαandTdust.Ateachpixelthevaluecomesfrom
ahighsignal-to-noisespectrum(upperplot,measuredinthedirectionl= the7◦ fitwiththeminimumχ2 amongallone-component fixed-αfits
63◦.78andb=−11◦.53)andalowsignal-to-noisespectrum(lowerplot, intherange1.4 (cid:2) α (cid:2) 2.3atd0o.f1increment.ThesemapsareMollweide
l = 254◦.32andb = 65◦.08).Inbothplots,thewhitecrossrepresents projectionsoftheGalaxyinGalacticcoordinates.Thecentreofeachmapis
location oftheminimumχ2;theblueareaisthe68percentconfidence theGalacticcentre.Theupperandlowerendsoftheminoraxisare+90◦
region;andthegreenareaisthe95percentconfidenceregion.Theseplots and−90◦ latitudes respectively, andtheleftandrightendsofthemajor
demonstratetheeffectofmeasurementnoiseonthedegeneracybetweenα axisrepresent+180◦ and−180◦ longitudesrespectively. Bothmapsare
andTdustinspectralmodelfitting.Forthehighsignal-to-noisespectrum noisy,particularlysoathighlatitudes.Pixelsthatcorrespondtofitswitha
(upperplot),the68percentconfidenceregionisat1.7 < α < 1.9and χ2lessthan10percentprobabilityaremaskedinwhite.Thegroupofblack
17.9K < Tdust < 19.5K;forthelowsignal-to-noisespectrum(lower pixelsthatslantfromthecentreoftheupperleftquadrant(NorthEcliptic
plot),the68percentconfidenceregionhasamuchwiderextent,at1.2< Pole,NEP)tothecentreofthelowerrightquadrant(SouthEclipticPole,
α<2.2and18.5K <Tdust<24.7K. SEP)arepositionswhereFIRASdidnotprovidedata.
separately.Ingeneral,one-component fixed-αmodelswithdiffer-
entlowerlimitsontheTdust/δTdustvaluescanfitmostspectraex- modelwithTdust/δTdust (cid:5) 5,only0.59percentofthetotalsky
ceptthoseattheGalacticplane.Amorerestrictedlowerlimitonthe arearequire fitswitha60.37 (cid:5)(cid:6)2 resolutioninstead of thedefault
Tdust/δTdustvaluesrequiresagreateramountofspatialaveraging 6.71(cid:5)(cid:6)2.Fora10percentconstraintonTdust,0.47percentareaof
whichstressesthedustmodelandcausesχ2dof aroundtheGalactic thefullskyrequirefitstobeat167.70(cid:5)(cid:6)2resolution,7.24percent
poles to increase in value. Theupper plot of Fig. 8 demonstrates at60.37(cid:5)(cid:6)2resolution,andtherestat6.71(cid:5)(cid:6)2resolution.Abalance
thisrelationbyplottingχ2dof asafunctionofGalacticlatitudefor between having adequate constraint on parameters, preserving as
thecaseofα=2.0.Notethatχ2dof ofhigh-latitude(|b|>60◦)fits manyvalidmodelsaspossible,andkeepingregionalsizeslowcan
movefrom0.7−1.0to0.8−1.3aslowerlimitonTdust/δTdust beachievedattheTdust/δTdust (cid:5)10level.
startswithnoneandincreasesto40. A plot of χ2dof distributions for 13 all-sky fixed-α mod-
Histogramsoftheall-skycollectionsofχ2doffordifferentlim- els with α in the range 1.4 − 2.6 at 0.1 increment and satisfy
itsonTdust/δTdust arepresentedinthelowerplotofFig.8.The Tdust/δTdust (cid:5)10ispresentedinFig.9.Thehigh-χ2tailsofthese
three histograms for Tdust/δTdust (cid:5) 5, 10 and 20, in the shape distributionsshow that modelswiththelargest(2.6)andsmallest
of a Gaussian, peak at 0.93, 0.94 and 0.96 respectively, and they (1.4)valuesofαhavemorefitswithlargeχ2dof.Wepresentaspe-
all have a width of 0.10. The histogram for Tdust/δTdust (cid:5) 40 cificcomparisonoftheχ2dof excessfortheseall-skymodelsinthe
peaks at 1.00, has a width of 0.12 and a thick tail in the range lowerplotofFig.9.Thisplotshowsthattheall-skyα=1.7model
1.2 < χ2dof < 1.4. This shows that the demand for a 40-times isthebestbecauseitcanfitthelargestamountofdata(87.6percent
lowerlimitonTdust/δTdusthasputtoomuchstressonthemodel. areaofthefullsky).Fig.10presentsskymapsofχ2dof,spatialres-
Imposing a more stringent limit on Tdust/δTdust leads to a olution,dusttemperature,opticaldepth,andthesignal-to-noiseof
variety of spatial resolutions in each all-sky collection of fits. A parametersfromfittingtheone-component α = 1.7model.Each
more stringent requirement on Tdust/δTdust means lower spatial of the 6063 fits presented there satisfies the Tdust/δTdust (cid:5) 10
resolutionsforfitsathighlatitudes.Forone-component α = 2.0 requirementwiththeleastamountofspatialaveraging.Inthepa-
(cid:2)c 2011RAS,MNRAS000,1–17
All-skyObservationalEvidenceforA TemperatureDependent EmissivitySpectralIndex 9
Figure9.Upper:χ2 distributionsofthebest-fittingone-componentfixed-
Figure8.χ2dof vs.Galacticlatitude(upper)andχ2dof distributions(lower) α models with αdionf the range 1.4 – 2.6 at 0.3 increment and satisfy
o5,ft1h0e,a2l0l-saknydo4n0e,-croemsppeocntievnetlyα. W=ith2.0mofirtestrheasttrsicattiivsefyTTdduusstt//δδTTdduussttre(cid:5)- Tbedufisttb/yδTadounset-co(cid:5)m1p0o.neLnotwmeor:dePlewrciethntfiagxeedaeremaisosfivtihtyesfpuellctsrkaylitnhdaetxc(aχn2not
quirements,χ2dof ofhigh-latitude(|b| > 60◦)fitsmigratefromtherange cutoffcorrespondsto10percentprobability).Thisplotshowsthatα=d1o.7f
0.7−1.0totherange0.8−1.3.
modelscanfitthelargestamountofdata(87.6percentareaofthefullsky).
rametermaps,ifafithaslessthan10percentχ2probability(i.e.,
χ2dof > 1.13for210degreesoffreedom),thenitscorresponding centrearound0.95.Thismeansthatthereisnoapparentsystematic
pixel ismasked inwhite. Fitresultsfor other αmodels at differ- biasinthefitsandtheerrorestimatesforthedataisaboutright.
ent levels of constraint on Tdust/δTdust are provided in the first Theplotofχ2dof vs.Galacticlatitude,lowerpanelofFig.12,
author’sPhDthesis(Liang2011). showsthatatlatitudes|b| < 10◦,χ2 continuestobe(cid:4) 1asis
dof
thecaseoffittingall-skyone-component fixed-αmodels.Athigh
4.3.2 α/δαconstraintonafree-αmodel latitudes,χ2dof donotflareupwithincreasingconstraintonα/δα,
asopposetothatwhichhappenswhenTdust/δTdustrequirements
Weapplyasimilarstrategytoconstrainfitsthatuseafree-αmodel. areimposed on afixed-α model. It confirmsour expectation that
Insteadofthesignal-to-noiseofthedusttemperature,wenowuse thefree-αmodelismoreadeptatfittingvariousspectralshapes.
signal-to-noiseoftheemissivityspectralindextogaugetheamount Fig. 13 presents distributions of α and Tdust of one-
of spectral averaging. As an example, we present sky maps of componentfree-αfitsthatusenoconstraintonanyparameterand
thebest-fittingparametersandtheirsignal-to-noiseforthefree-α thosethatsatisfyα/δα(cid:5)5.0,6.7and10.0.Thecentresofthedis-
modelwithα/δα(cid:5)10.0inFig.11. tributionsare at 1.80, 1.83, 1.85 and 1.88, respectively. Thisplot
Notice that the current dust temperature map has more con- showsthattheα/δαrequirementhastheeffectofmovingαfrom
sistentvaluesathighlatitudesneartheGalacticpolesthanthatob- below1.5tohighervalues.Withevenamoderateamountofcon-
tainedfromthe7◦-pixelfitsinFig.7.TheuncertaintyofTdust is straintonα/δα,therangeofαvaluesquicklyreducestobetween
lessthan7.4percentasaresultoftheconstraintonα/δα,andthe 1and3,anindicationthatvaluesoutsideofthisrangearerarein
uncertaintyofτ hasamaximumof23.25percent.Thespatialres- nature.
olutionofthesefitsarepresentedinanall-skymapinFig.11and AlsoshowninFig.13,theTdust distributionsdonotpeakat
summarizedinTable2. asinglevalue. Instead, thereisarange ofmost popular tempera-
The upper panel in Fig. 12 compares χ2dof distributions of turesbetween17and20K.Comparedtotheunconstrained case,
fitsthatusenoconstraint onanyparameter andthosethatsatisfy theα/δαrequirementsmoothsoutthehigh-temperaturepointsand
α/δα(cid:5)5.0,6.7and10.0.Itshowsthattheshapeoftheχ2dof dis- effectivelyreplacesthemwithvaluesatorbelow20K.
tributionsresembleaGaussianandtheconstraineddistributionsall Theamountsofconstraintondusttemperature,opticaldepth
(cid:2)c 2011RAS,MNRAS000,1–17
10 Z. Liang, D.J. FixsenandB. Gold
Table2.Spatialresolutionoftheone-componentfree-αfitswithχ2 (cid:2)1.13
dof
Level Regionalsizeoffits Percentageofthefullskyataregionalaverage
((cid:4)(cid:5)2) α/δα(cid:5)5.0 α/δα(cid:5)6.7 α/δα(cid:5)10.0
1 6.71 40.01 30.63 22.09
2 60.43 23.14 24.53 22.51
3 167.86 11.87 11.07 10.60
4 329.00 4.61 8.59 6.41
5 543.86 3.53 3.48 5.53
6 812.44 2.18 2.78 4.62
7 1134.73 1.01 2.12 3.30
8 1510.73 1.92 2.59
9 1940.45 1.22 2.23
10 2423.88 2.08
11 2961.03 1.90
12 3551.89 1.53
13 4196.47 0.72
14 4894.76 0.23
Total 87.66 87.19 86.34
Table3.Parametersandtheiruncertaintiesoftheone-componentfree-αfitswithχ2 (cid:2)1.13
dof
Constraintonα/δα Tdust δTdust
(K) (K)
min max
(cid:5) 5.0 10.11 23.83 2.67
(cid:5) 6.7 10.12 22.69 1.95
(cid:5)10.0 13.69 22.69 1.26
Constraintonα/δα α δα
min max
(cid:5) 5.0 1.08 4.71 0.91
(cid:5) 6.7 1.08 4.80 0.71
(cid:5)10.0 1.24 3.13 0.31
Constraintonα/δα τ δτ/τ
×10−5 (percent)
min max
(cid:5) 5.0 0.33 46.15 62.83
(cid:5) 6.7 0.45 46.15 49.02
(cid:5)10.0 0.61 46.15 23.25
(cid:2)c 2011RAS,MNRAS000,1–17