Uncertainties in the solar photospheric oxygen abundance

0.66 MB·

by  M. Cubas Armas

Astronomy&Astrophysicsmanuscriptno.main (cid:13)cESO2017
January25,2017
Uncertainties in the solar photospheric oxygen abundance
M.CubasArmas1,2,A.AsensioRamos1,2,andH.Socas-Navarro1,2
1 InstitutodeAstrofísicadeCanarias(IAC),AvdaVíaLácteaS/N,38200LaLaguna,Tenerife,Spain
e-mail:[email protected]
2 DepartamentodeAstrofísica,UniversidaddeLaLaguna,38205LaLaguna,Tenerife,Spain
Received;accepted
ABSTRACT
7
1 Aims.Thepurposeofthisworkistobetterunderstandtheconfidencelimitsofthephotosphericsolaroxygenabundancederivedfrom
0 three-dimensionalmodelsusingtheforbidden[OI]lineat6300Å,includingcorrelationswithotherparametersinvolved.
2 Methods.Weworkedwithathree-dimensionalempiricalmodelandtwosolarintensityatlases.WeemployedBayesianinferenceas
n atooltodeterminethemostprobablevalueforthesolaroxygenabundancegiventhemodelchosen.Weconsideredanumberoferror
a sources,suchasuncertaintiesinthecontinuumderivation,inthewavelengthcalibrationandintheabundance/strengthofNi.
J Results.Ourresultsshowscorrelationsbetweentheeffectsofseveralparametersemployedinthederivation.TheBayesiananalysis
providesrobustconfidencelimitstakingintoaccountallofthesefactorsinarigorousmanner.Weobtainthat,giventheempiricalthree-
4
dimensionalmodelandtheatlasobservationsemployedhere,themostprobablevalueforthesolaroxygenabundanceislog((cid:15) ) =
2 O
8.86±0.04.However,wenotethatthisuncertaintydoesnotconsiderpossiblesourcesofsystematicerrorsduetothemodelchoice.

] Keywords. Sun:abundances–Sun:atmosphere–Sun:photosphere–Methods:statistical
R
S
.
h1. Introduction A key factor in the abundance derivation is the solar atmo-
p sphere model used. It seems natural that a three-dimensional
-Thesolarchemicalcompositionisstillunderdebate,beingpar- model should be preferred over a one-dimensional one. How-
oticularlyimportantthecaseoftheoxygen(thethirdmostabun-
ever, when the three-dimensional model is a numerical simula-
r
dantelementintheSunafterhydrogenandhelium).Oneofthe
t tion and the one-dimensional model has been obtained empir-
sabundancesettraditionallyacceptedconsidersanoxygenabun-
a ically by fitting observations, it is not clear that the former is
dancesoflog((cid:15) )=8.93±0.04(Grevesseetal.1984),intheas-
[ O betterthanthelattertofitobservations(Ayres2008).
tronomical usual scale referred to hydrogen (where log((cid:15) ) =
O
1log(N /N )+12).Thisabundancewaslaterdownwardrevised
X H
vto log((cid:15)O) = 8.83±0.06 (Grevesse & Sauval 1998). With this Derivingelementabundancesisfarfromtrivial.Theresults
9
abundance set, there is excellent agreement between solar in- are not determined solely by the solar atmosphere model used.
0
teriormodelspredictionsandthehelioseismology.However,us- Theyarealsodependentofthesolaratlasadoptedtofitthemodel
8
6ingmorerecentthree-dimensionaltheoreticalmodels,lowerso- and even some details on the calibration (choice of the contin-
0larmetallicitieshavebeenobtained.Forexample,Asplundetal. uum level, spectrum rectification, wavelength calibration, etc).
.(2004)reachedlog((cid:15)o)=8.66±0.05,whichmayspoiltheagree- Recently,somestudieshadnotedthedifferencesbetweentheso-
1
ment with the helioseismology (see Basu & Antia 2008). Note laratlases(e.g.,Caffauetal.2008,2009;Doerretal.2016)and
0
however that a recent study suggests that we might be under- theneedtousemorethanoneintheanalysis.Anotherobvious
7
1estimating interior opacities which could restore the agreement factoristhevalueofthelinestrengthemployed(parameterized
:withhelioseismology(Baileyetal.2015).Giventhatoxygenisa intermsoflog(gf)),whichhavebeendiscussedinsomeworks
vveryimportantelementforstellarinteriormodels,adoptingsuch (e.g.,Storey&Zeippen2000;Johanssonetal.2003).
Xiadramaticrevisionswouldhaveimplicationsoverabroadrange
oftopicsinastrophysics.
r
a Further works have not yet converged to a satisfactory res- In this paper, we present a study of the oxygen abun-
olution of the issue. On one hand, some studies found a rel- danceinthesolarphotospherewiththenoveltythatweapplied
atively low oxygen abundance (e.g., Socas-Navarro & Norton Bayesianinferencetoproperlydisentangletheeffectsoftherel-
2007;Pereiraetal.2009;Grevesseetal.2010);but,ontheother evantparametersinvolved.Inordertodothat,weusedathree-
hand,highoxygenabundancehasalsobeenobtained(e.g.,Cen- dimensional empirical model of the solar atmosphere (Socas-
teno&Socas-Navarro2008;Ayres2008;Socas-Navarro2015b). Navarro 2011, 2015b) to fit the forbidden [O i] line at 630.03
Perhapsanintermediatevalue,suchaslog((cid:15) )≈8.75(e.g.,Caf- nmobservedintwodifferentsolaratlases.InSection2,wede-
O
fau et al. 2008, 2015) might be able to satisfy all of the con- scribethesolarmodelused,thesynthesesmadeandthesolarat-
straintsiftheuncertaintieshadbeenunderestimatedthusfar(i.e., lasobservations.AbriefsummaryoftheBayesianinferenceand
themodelatmosphereistypicallyassumedtobeperfectexcept atablewithourpriorsaregiveninSection3.Then,wepresent
in Socas-Navarro 2011). In conclusion, the oxygen abundance theresultsinSection4and,finally,someconclusionsinSection
issueremainsopenandlargelycontroversial. 5.
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A&Aproofs:manuscriptno.main
2. Modelatmosphereandobservationaldata of72%and28%respectivelyforeachisotope(Johanssonetal.
(2003);seeTable1).
2.1. Solarmodelusedandsynthesis
For the synthesis with NICOLE , we chose a wavelength
range from 630.0 to 630.3 nm, including also the Fe i lines at
The solar atmosphere model that is used in this work was de-
630.15and630.25nm.WeintroducedtheFeilinesbecausethe
rivedbySocas-Navarro(2011,2015a)basedonobservationsof
[Oi]lineisformedinthefarwingofthe630.15nmFeilineand
thespectro-polarimeter(SP;Litesetal.2001)ofthesolaropti-
weneedtoincludethiseffectinoursynthesis,sinceitaffectsthe
caltelescopeonboardtheHinodesatellite(Kosugietal.2007).
continuumestimate.
Theobservationshaveafieldofviewveryclosetothediskcen-
We introduce an additional simplification in the forward
terandthewavelengthrangecoversfrom6300.89to6303.27Å.
modeling, given that the model of Socas-Navarro (2011) has
ThespectrumateachpixelwasinvertedusingthecodeNICOLE
200 × 200 spatial pixels in the field of view. Synthesizing the
(Socas-Navarro et al. 2015) to determine a column with the
wavelength range in each one of the 1000 models in our grid
height stratification of temperature, line-of-sight velocity and
would be very demanding computationally. For this reason, we
magnetic field vector. The reader is referred to the papers cited
carriedoutaMontecarloselectionofasubsetof1000pixels,as
aboveformoredetailsonthemodel.
a good compromise between accuracy and computational time,
Ouraimistocomparethesyntheticlineprofileswiththose
that produce an average profile that is “equivalent” to the aver-
observed in different atlases of the solar spectrum. The spatial
ageprofileofthewholesnapshot.By“equivalent”wemeanthat
resolutionoftheseobservationsisextremelypoor(oftheorder the differences between the average intensity profile of the full
oftensofarcsec).Thus,wemimickthisresolutionbyaveraging
mapandthatofthesubsetissmallerthan0.10%intheoxygen
all profiles synthesized in all pixels of the snapshot. As a con-
line.
sequence,theforwardmodelinthisthree-dimensionalsnapshot
In summary, we constructed a grid of 1000 models where
turns out to be very time consuming. Given that the Bayesian
each model has 1000 pixels. For each one of them, we synthe-
analysis that we explain in the following section requires the
sized the spectral lines of interest using NICOLE, taking into
evaluation of several tens of thousand of forward models, it is
account the atomic information compiled in Table 1. For the
convenient to pre-compute a grid of models and carry out the
sakeofreproducibility,thetablealsoincludesthereferencesfor
Bayesianinferenceusingasimulatorthatjustinterpolatesonthe the log(gf) values. The log(gf) value for the Fe i at 630.25 nm
gridofmodels(O’Hagan2006).Thisgreatlyacceleratesthein-
wastakenfromSocas-Navarro(2011).Theradiative,Starkand
ferencebyseveralordersofmagnitude.
van der Waals damping parameters are also provided for each
The precomputed database is built using a cartesian grid in line except the Fe lines. For those, we give instead the α and
allparametersofthemodel.Themodelparameterscanbesep- σ damping parameters using the method of Anstee & O’Mara
aratedintwodifferentclasses:slowparameters,thatrequirethe
(1995)obtainedwiththecodeofBarklemetal.(1998).
recomputation of the synthesis in the whole snapshot because
theyaffecttheradiativetransfer,andfastparameters,thatcanbe
applied directly to the profiles with simple manipulations. The 2.2. Observationaldata
firsttwoslowparametersaretheoxygenandnickelabundances
Inthisstudywealsointendtoassesstheinfluenceofthechoice
(notethatthe[Oi]featureanalyzedherehasaNiiblend).The
of solar observation on the abundance determination. We used
thirdslowparameterisanenhancementfactorforthevelocities
two different intensity atlases of the solar photosphere which
inthelowerlayersofthesnapshot.Thisparameterisintroduced
have a good signal-to-noise (S/N) ratio and have been widely
because the model of Socas-Navarro (2011) is constructed fit-
used in previous works (e.g. Caffau et al. 2008, 2009; Ayres
ting the strong Fe i lines at 630.1 and 630.2 nm, and it misses
2008;Maiorcaetal.2009).
someofthedynamicsinthelowerlayersoftheatmosphere,pre-
The first one is the disk center intensity atlas of the solar
cisely where the oxygen line forms. Fortunately, the nearby Sc
iiline,verysimilarinstrengthandformationheighttothe[Oi] spectrum from 3000 Å to 10000 Å by Delbouille et al. (1973).
ThespectrumwasobtainedattheInternationalScientificStation
line,isanexcellentcalibrationtoolforthismissingturbulence,
oftheJungfraujochandmaybedownloadedfromtheBASS2000
which we parameterize as a multiplicative enhancement factor
web server1. The second atlas that we considered was the Kitt
applied on the velocities of the lower layers. For more details,
Peak intensity FTS (Fourier Transform Spectrometer) atlas by
seethediscussioninSocas-Navarro(2011).Thefastparameters
Neckel & Labs (1984). This atlas was produced with the FTS
are: two for a linear correction to the continuum, a global ve-
instrument at the McMath telescope and spans the wavelength
locityshiftthattakesintoaccountimprecisioninthewavelength
calibration (this correction is important because a degeneration rangefrom3290to12510Å.
existsbetweentheratioofNiandOabundances,thisisexplained
inmoredetailedinsect4.2below)andtheuncertaintyofthefit.
3. Bayesiananalysis
Toendupwithawellsampleddatabase,wetooktenpossi-
blevaluesforeachoneoftheslowparameters,resultingagridof We employthe formalism ofBayesian inference (e.g., Gregory
1000models.Theoxygenandnickelabundanceswerevariedbe- 2005) to quantify the uncertainty in our conclusions derived
tweentypicalvaluesreportedintheliterature(seee.g.,Grevesse from the use of a prescribed model, a bunch of free parame-
etal.1984;Anders&Grevesse1989;Grevesse&Sauval1998; ters and a set of priors (information known a priori). The aim
Asplundetal.2009).Theirminimumandmaximumvaluesare oftheBayesiananalysisistocomputetheposteriorprobability
showninTable2togetherwiththosefortheenhancementfactor. distributionassociatedwithallmodelparametersbytakinginto
It is important to stress that we considered the nickel abun- accounttheinformationprovidedbytheobservationsandallour
danceintheanalysisbecausethespectralfeatureat630.0nmis a-prioriinformation.Itreliesontwofundamentaltools.Thefirst
actuallytheresultofablendbetweenthe[Oi]lineat630.03nm one is the Bayes theorem, that describes very simply how the
andaNiilineat630.03nm.Weconsideredthetwomajoriso-
topes58Niiand60Nii,withalog(gf)valueof-2.11andafraction 1 http://bass2000.obspm.fr/solar_spect.php
Articlenumber,page2of6
M.CubasArmas etal.:Uncertaintiesinthesolarphotosphericoxygenabundance
Ion Wavelength[nm] Ex.Pot.[eV] log(gf) Conf.(lower) Conf.(Upper) γ γ γ σ α
rad Stark Waals
[Oi] 630.0304 0.000 -9.717a 3P 1D 0.0 0.05 1.00 ... ...
2 2
Nii 630.0335 4.266 -2.253b 3D 3P 2.63 0.054 1.82 ... ...
1 0
Nii 630.0355 4.266 -2.663b 3D 3P 2.63 0.054 1.82 ... ...
1 0
Scii 630.0678 1.507 -1.898c 3P 3D 2.30 0.05 1.30 ... ...
2 2
Fei 630.15012 3.654 -0.718c 5P 5D ... ... ... 834.4 0.243
2 2
Fei 630.24940 3.686 -1.13 5P 5D ... ... ... 850.2 0.239
1 0
Table1.Adoptedatomicparameters.γ ,γ andγ aretheradiative,theStarkandthevanderWaalsdampingparameters(units108rad
rad Stark Waals
s−1).Thelog(gf)aretakenfrom:aStorey&Zeippen(2000),bJohanssonetal.(2003),cVALDdatabase.
prior information is updated with the acquisition of new obser- Parameter Range Type
vations,totheposteriorinformation.Ifwedenotethemodelpa- log((cid:15) ) (8.55,9.20) Flat
o
rametersofinterestwiththevectorθ,andtheobservationswith log((cid:15) ) (5.84,6.36) Flat
Ni
D,theBayestheoremstatesthat v[km/s] (-1,1) Flat
p(θ|I)p(D|θ,I) Enhancementfactor (0.5,2.3) Flat
p(θ|D,I)= . (1) σ (0.0001,0.02) Jeffreys
p(D|I)
Cont.slope[I/I ˙Å−1] (-0.1,0.1) Flat
c
IntheBayestheorem, p(θ|I)isthepriorprobabilitydistribution Cont.yintercept[I/I ] (0.9,1.1) Flat
c
of θ, that encodes all the a-priori information we know about
Table2.Priorsselectedforeachparameterofouranalysis.
theparameters(i.e.,ifaquantityispositive,thatacertainregion
of the space of parameters is more probable than others, etc.).
p(D|θ,I)isthelikelihood,whichencodestheinformationabout
the model parameters that can be extracted from the observa- whichstatesthattheobservationsforthei-thwavelengthpoint,
tions. p(θ|D,I)istheposteriordistribution,thatdescribesevery- O(λi), can be modeled with I(θ,λi) plus some uncertainty that
thing that we know about the model parameters. p(D|I) is the haszeromeanandvarianceσ2.Followingthestandardapproach
model evidence, which is an unimportant constant in our case, (e.g.,Gregory2005),thelikelihoodhastheformofanuncorre-
because it does not depend on θ. Finally, I refers to any im- latedmultivariatenormaldistribution,whichreducesto:
portant context information that is necessary for the inference. (cid:89)n 1 (cid:34) −[O(λ)−I(θ,λ)]2(cid:35)
In our case, for instance, the specific radiative transfer model p(D|θ,I)= √ exp − i i , (4)
thatweuse.Thesecondfundamentaltoolisthemarginalization, 2πσ 2σ2
i=1
thatisusedtoobtaintheposteriordistributionforanyparameter
takingintoaccounttheuncertaintiesandcorrelationswithother wherenisthenumberofobservedwavelengths.
parameters: We sample the posterior distribution using the nested sam-
(cid:90) plingtecniquebySkilling(2004)2.
p(θ |D,I)= dθ ...dθ p(θ|D,I) (2)
1 2 n
4. Results
In this work we considered seven free parameters, that we
considernecessaryforexplainingalltheexpectedvariabilityof
4.1. Linefit
the line profiles: the oxygen abundance (which is the main pa-
rameterofinterest),theabundanceofnickel(necessarytorepro- Figure1showsourfitsofthe[Oi]andSciilinesinourmodel
ducethelineshapebecausetheoxygenlineofinterestisblended (black lines) with respect to the atlases (blue). The left panel
withaNiiline),aglobalwavelengthshifttoaccountforpossible uses the Neckel atlas and the right one uses the Delbouille at-
inaccuraciesinthewavelengthscale,anenhancementfactorfor las.Insteadofasinglefit,weprovideasamplingofmodelsthat
the dynamics that is described above, two free parameters that arecompatiblewiththeobservations.Theyareobtainedbysyn-
areusedtocorrectthecontinuumlevelanditsslope,andfinally thesizing models with parameters extracted from the posterior
a parameter σ that describes the (unknown) uncertainty of the distribution.Figure1clearlyshowsthatourmodelfitsverywell
observationsandalsoquantifiesthequalityofourmodeling. both atlases, even in very weak lines like these. However, the
In any Bayesian inference, it is crucial to define the priors. modelspresentsomediscrepancieswiththeobservationsinthe
In our case, we define the priors as flat (meaning there is no continuum region in between the two lines, where the models
preference)foralmostallparametersintherangesshowninTa- setthecontinuumtoahigherlevel.Giventhatwehavethisef-
ble 2. The ranges have been chosen taking into consideration fect in both atlases, this might be produced by some unknown
theresultsobtainedinpreviousstudies(seee.g.,Grevesseetal. absorption in this region. The lines of interest are very weak,
1984; Anders & Grevesse 1989; Grevesse & Sauval 1998; As- thus, this region could affect our abundance determination in-
plundetal.2009)buttryingnottodiscardpartsofthespaceof directly by setting the continuum level on a wrong level. We
parametersthatmightbecompatiblewiththeobservations.The tried to minimize this effect by introducing the Sc ii line in the
onlynon-flatprioristhatofσ,forwhichweadoptedaJeffreys analysis,whichthensetsamuchbettercontinuumlevelestima-
prior, p(σ) ∝ σ−1, because it is a scale parameter that can po- tion. We have plans to refine this approach in the future by us-
tentiallyhavevaluesspanningseveralordersofmagnitude(e.g., ingamodelingthatisabletoabsorbthesedeficiencies(similar
Gregory2005). to the nonparametric approach based on Gaussian Processes of
Forcomputingthelikelihood,weassumethefollowinggen- Czekalaetal.2015).
erativemodel:
2 We use the implementation of the Python package nestle, which
O(λi)= I(θ,λi)+n(λi), (3) canbeobtainedfromhttp://kbarbary.github.io/nestle/.
Articlenumber,page3of6
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Models and FTS Models and FTS
1.00 1.00
0.99 0.99
0.98 0.98
0.97 0.97
I/Ic I/Ic
0.96 0.96
0.95 0.95
0.94 0.94
0.93 0.93
6300.2 6300.3 6300.4 6300.5 6300.6 6300.7 6300.8 6300.2 6300.3 6300.4 6300.5 6300.6 6300.7 6300.8
Wavelength [ ] Wavelength [ ]
Fig.1.Somerepresentativefitsofthe[Oi]line(left)andSciiline(right)areshowninblack.TheblueprofilecorrespondtotheNeckelsolaratlas
intheleftpanelandtotheDelbouilleatlasintherightpanel.
4.2. Marginalposteriordistributions canbecompensatedwithalowernickelabundanceandahigher
oxygenabundance.
The joint and marginal posterior distributions for the five main
All marginal posterior distributions have a Gaussian-like
parameters (oxygen and nickel abundances, velocity, enhance-
shape. Therefore, we summarize them in Table 3 by providing
ment factor and σ) are shown in Fig. 2. The blue distributions
themedianandtheuncertaintydefinedbythepercentiles16and
correspondtotheNeckelatlaswhilethegreendistributionscor-
84 (equivalent to the standard 1σ uncertainty in the Gaussian
respondtotheDelbouilleatlas.Wedidnotshowthedistribution
case). It is clear in the table that the most probable value for
for the two parameters of the continuum because they are nui-
theoxygenabundanceisthesame(downtotheseconddecimal)
sance parameters that are only needed to have a good fit of the
in both atlases. The value obtained from this study is, hence,
line,buttheydonotproviderelevantinformation. log((cid:15) ) = 8.86±0.01.Thisvalueiswithinthecategoryofhigh
O
For both atlases, the joint and marginal posterior distribu- solar oxygen abundances: lower than the 8.93 ± 0.02 value in
tionsshowaGaussian-likeshape,wherethosecorrespondingto Grevesseetal.(1984),butcompatiblewiththe8.83±0.06revi-
theNeckelatlasalwayspresentalargerwidth.Thiswidthindi- sioninGrevesse&Sauval(1998).
cates that the uncertainties in the abundances inferred from the
Neckel atlas are larger than those obtained from the Delbouille
atlas.Moreover,themarginaldistributionforσalsoshowsthat 4.3. Moreexperiments
this parameter is smaller for the case of the Delbouille atlas.
Inordertoquantifyhowmuchtheimpactofthecontinuumlevel
Thus, our model seems to provide more constrained model pa-
and the correlation between the oxygen and nickel abundances
rametersandbetterfitsfortheDelbouilleatlas.
are,wecarriedouttwomorestudies.Theyareintendedalsoto
It is encouraging that the marginal posterior for the oxygen checktherobustnessofourresults.
abundance is consistent in both atlases, with that inferred from
First, we carried out the same analysis as before but reduc-
theDelbouilleatlasbeingcontainedinsidetheuncertaintyasso- ingthewavelengthrangetojusttakeintoaccountthe[Oi]line.
ciated with the Neckel atlas. It is interesting that the maximum
Although we do not display the fits or the marginal posteriors
marginal a posteriori (the peak of the marginal posterior) is lo-
forsimplicity,thesummaryoftheresultsisdisplayedinTable4
catedatroughlythesamevalueforthetwocases. (seerowscorrespondinglabeledas"Only[Oi]"and"noprior").
This is not the same with the rest of parameters. The For completeness, we repeat in this table the results of the pre-
marginal posteriors corresponding to the Delbouille atlas tend vious section for an easier comparison (labeled as "Both line"
tobeshiftedtowardssmallervalues,exceptfortheenhancement and"noprior").Thus,modifyingtherangeofwavelengths,the
factor, where the opposite happens. In this last parameter, the mostprobableoxygenabundancechangesfromlog((cid:15) )=8.86,
O
marginal distributions even indicate that the inferred enhance- avaluecompatiblewithGrevesse&Sauval(1998),tolog((cid:15) )=
O
ment factors are not compatible (at least to 3 standard devia- 8.94, a value compatible with Grevesse et al. (1984). Because
tions). we are neglecting the wing of the Sc ii line and the inferred
If we look at the panels with the joint posteriors (showing abundanceisdifferent,weconcludethatagoodestimateofthe
thecorrelationbetweenpairsofparameters),weseeaclearcor- continuumiscrucialfortheestimationofabundances.However,
relationbetweentheabundancesofoxygenandnickel.Thiscor- this large oxygen abundance seems improbable at the light of
relation is a direct consequence of the fact that the [O i] line is the really low nickel abundance inferred which is incompati-
a blend: we can obtain a fit equally good if we take a model ble with all previous reports in the literature (that usually give
with a higher nickel abundance and a lower oxygen abundance log((cid:15)Ni)>6).
and viceversa. Furthermore, these two parameters (oxygen and Thesecondtestismotivatedbythelownickelabundanceof
nickel abundances) are correlated with the velocity shift. This thepreviousresults.Inthiscase,weredotheanalysisbutinclud-
correlationmakessenseforsmallvelocities,sinceawavelength ingadifferentpriorforthenickelabundance.Thispriorissetto
shiftdisplacesthelineandonecanstillobtainagoodfitbyap- a Gaussian distribution with mean µ = 6.17 and standard devi-
propriatelymodifyingtheabundances.Forexample,ablueshift ation of σ = 0.07, as reported in Scott et al. (2009). We inten-
Articlenumber,page4of6
M.CubasArmas etal.:Uncertaintiesinthesolarphotosphericoxygenabundance
)
o
ǫ
(
g
o
l
6.35
6.30
)
Ni 6.25
ǫ
( 6.20
g
lo 6.15
6.10
0.08
s] 0.06
/
m 0.04
v[k 0.02
0.00
2.10
or2.05
ct2.00
fa1.95
h.1.90
n1.85
E1.80
2.0
3] 1.8
−
0 1.6
1
[ 1.4
σ
1.2
8.80 8.82 8.84 8.86 8.88 8.90 6.15 6.20 6.25 6.30 0.00 0.02 0.04 0.06 0.08 0.101.801.851.901.952.002.052.10 1.2 1.4 1.6 1.8 2.0
log(ǫo) log(ǫNi) v[km/s] Enh. factor σ [10−3]
Fig.2.Histogramsandcorrelationsoftheposteriordistributionsofthemainfiveparameters(inblueforNeckelsolaratlasandgreenforDelbouille
atlas).Fromlefttoright,andtoptobottom,weshow:oxygenabundance,nickelabundance,velocity(inkm/s),enhancementfactorandσ.
Atlas log((cid:15) ) log((cid:15) ) v[km/s] Enh.factor σ
o Ni
Neckel 8.861±0.014 6.22±0.03 0.05±0.01 1.92±0.03 0.0015±0.0001
Delbouille 8.856±0.006 6.18±0.01 0.014±0.005 2.05±0.01 0.00120±0.00005
Table3.Parametersmedianvaluesanddeviationsforbothatlases.
log((cid:15) ) log((cid:15) ) v[km/s] Enh.factor σ
o Ni
priorNi 8.864±0.013 6.21±0.03 0.05±0.01 1.92±0.03 0.0015±0.0001
Neckel
noprior 8.861±0.014 6.22±0.03 0.05±0.01 1.92±0.03 0.0015±0.0001
Bothlines
priorNi 8.858±0.006 6.18±0.01 0.014±0.005 2.05±0.01 0.00120±0.00005
Delbouille
noprior 8.856±0.006 6.18±0.01 0.014±0.005 2.05±0.01 0.00120±0.00005
priorNi 8.910+0.015 6.02±0.05 0.17+0.03 1.70±0.04 0.00052±0.00005
Neckel −0.019 −0.04
noprior 8.940+0.007 5.90+0.06 0.23+0.01 1.74±0.04 0.00050±0.00005
Only[Oi] −0.012 −0.04 −0.02
priorNi 8.937+0.004 5.88+0.04 0.171+0.007 1.99±0.02 0.00036±0.00005
Delbouille −0.007 −0.03 −0.013
noprior 8.940±0.002 5.85+0.02 0.178+0.003 2.00±0.01 0.00036±0.00002
−0.01 −0.004
Table4.Comparisonoftheresultsforthedifferentexperiments.
tionallyincreasethevalueofσtodecreasetheinformativeness 4.4. Abundanceandlog(gf)factor
ofthepriorandletthedatadrivetheresults.Theresultsofthis
newanalysisarealsoshowninTable4(withlabels"priorNi"). Weareawarethatanimportantsourceofuncertaintyinthedeter-
The results show that the inference only with the [O i] line are minationofabundancesaretheatomicparametersofthespectral
notveryreliablebecause,whenincludingtheGaussianpriorfor lines,specificallythevalueoflog(gf).Wehavetriedtousethe
thenickelabundance,themodificationintheoxygenandnickel mostrecentdeterminations,butitistruethataslightlydifferent
abundancesaregreaterthanthepreviouslyquoteduncertainties. value of log(gf) inside its uncertainty would produce different
This fundamentally means that the result is strongly dependent oxygenandnickelabundances.Therefore,onecanconsiderour
ontheprior. However,thisisnot thecasewhenusingthe[O i] determination of the abundance to be indeed an inference over
and Sc ii lines together, where we obtain results that are essen- theproductgf(cid:15) ,whichremainsvalidforweaklines.
O
tiallyinsensitivetotheprior.
The gf value used in this work (Johansson et al. 2003) is
accepted as the most accurate, but its uncertainty can go up to
Articlenumber,page5of6
A&Aproofs:manuscriptno.main
10%.Thisuncertaintytranslatesintolog(gf)=−9.717±0.043. References
Ifthisistakenintoaccount,itwouldinduceanuncertaintyinthe
estimatedoxygenabundanceoflog((cid:15) )=8.86±0.04. Anders,E.&Grevesse,N.1989,Geochim.Cosmochim.Acta,53,197
O Anstee,S.D.&O’Mara,B.J.1995,MNRAS,276,859
Asplund,M.,Grevesse,N.,Sauval,A.J.,AllendePrieto,C.,&Kiselman,D.
2004,A&A,417,751
5. Conclusions Asplund,M.,Grevesse,N.,Sauval,A.J.,&Scott,P.2009,ARA&A,47,481
Ayres,T.R.2008,ApJ,686,731
There are many parameters involved in the determination of Bailey,J.E.,Nagayama,T.,Loisel,G.P.,etal.2015,Nature,517,56
abundances.Toendupwithanaccuratedeterminationofabun- Barklem,P.S.,Anstee,S.D.,&O’Mara,B.J.1998,PASA,15,336
dances,itiscrucialtoputemphasisonappropriatelyconsidering Basu,S.&Antia,H.M.2008,Phys.Rep.,457,217
Caffau,E.,Ludwig,H.-G.,Steffen,M.,etal.2008,A&A,488,1031
allofthem.Thisisthemotivationofthisworkanditrepresents
Caffau,E.,Ludwig,H.-G.,Steffen,M.,etal.2015,A&A,579,A88
a first step towards a reliable determination of the solar abun- Caffau,E.,Maiorca,E.,Bonifacio,P.,etal.2009,A&A,498,877
dances.Severalconclusionscanbeextractedfromourwork: Centeno,R.&Socas-Navarro,H.2008,ApJ,682,L61
Czekala,I.,Andrews,S.M.,Mandel,K.S.,Hogg,D.W.,&Green,G.M.2015,
– Wehaveusedaveryflexiblegenerativemodelthatcontains ApJ,812,128
sevenparameters.Someoftheseparametersarenuisancepa- Delbouille,L.,Roland,G.,&Neven,L.1973,Atlasphotometriqueduspectre
solairede[lambda]3000a[lambda]10000
rametersthatareofnodiagnosticinterestbutwhicharenec-
Doerr, H.-P., Vitas, N., & Fabbian, D. 2016, ArXiv e-prints
essary for a good modeling of the spectral line. Including [arXiv:1604.03748]
these nuisance parameters and marginalizing over them is Gregory,P.C.2005,BayesianLogicalDataAnalysisforthePhysicalSciences
crucialforareliabledeterminationofabundances. (Cambridge:CambridgeUniversityPress)
Grevesse,N.,Asplund,M.,Sauval,A.J.,&Scott,P.2010,Ap&SS,328,179
– Including nearby spectral lines turns out to be very impor-
Grevesse,N.&Sauval,A.J.1998,SpaceSci.Rev.,85,161
tant, because they modify the continuum level. We have
Grevesse,N.,Sauval,A.J.,&vanDishoeck,E.F.1984,A&A,141,10
foundthatagoodcharacterizationofthecontinuumlevelin Johansson,S.,Litzén,U.,Lundberg,H.,&Zhang,Z.2003,ApJ,584,L107
theseweaklinesiscrucialfortheinferenceofabundances. Kosugi,T.,Matsuzaki,K.,Sakao,T.,etal.2007,Sol.Phys.,243,3
– Wehavefoundsomedifferenceswhenapplyingexactlythe Kupka,F.,Piskunov,N.,Ryabchikova,T.A.,Stempels,H.C.,&Weiss,W.W.
samemodelingtodifferentatlases.Therefore,wefinditmay 1999,A&AS,138,119
Kupka,F.G.,Ryabchikova,T.A.,Piskunov,N.E.,Stempels,H.C.,&Weiss,
be convenient to infer solar abundances using the largest W.W.2000,BalticAstronomy,9,590
panoply of observations. However, we find a reliable oxy- Lites, B. W., Elmore, D. F., Streander, K. V., et al. 2001, in Presented at the
gen abundance with just 0.003 dex difference between the SocietyofPhoto-OpticalInstrumentationEngineers(SPIE)Conference,Vol.
4498,Proc.SPIEVol.4498,p.73-83,UV/EUVandVisibleSpaceInstrumen-
studieswiththetwosolaratlases.
tationforAstronomyandSolarPhysics,ed.O.H.Siegmund,S.Fineschi,&
– A reliable independent determination of the spectral line
M.A.Gummin,73
atomicparametersisveryimportant.Otherwise,thisuncer- Maiorca,E.,Caffau,E.,Bonifacio,P.,etal.2009,PASA,26,345
taintypropagatesaccordinglyontotheinferredabundances. Neckel,H.&Labs,D.1984,Sol.Phys.,90,205
– Given the interdependencies among all model parameters O’Hagan,A.2006,ReliabilityEngineering&SystemSafety,91,1290
Pereira,T.M.D.,Asplund,M.,&Kiselman,D.2009,A&A,508,1403
and the uncertainty in the observations, one should pursue
Piskunov,N.E.,Kupka,F.,Ryabchikova,T.A.,Weiss,W.W.,&Jeffery,C.S.
afullyBayesianapproachtakingintoaccountalleffectssi-
1995,A&AS,112,525
multaneously.Ideally,suchanapproachshouldalsobecon- Ryabchikova,T.,Piskunov,N.,Kurucz,R.L.,etal.2015,Phys.Scr,90,054005
sideredforaggregatingallsourcesofuncertaintyandcheck- Ryabchikova,T.A.,Piskunov,N.E.,Kupka,F.,&Weiss,W.W.1997,Baltic
ing,throughameta-analysis,forthecompatibilityofallthe Astronomy,6,244
Scott,P.,Asplund,M.,Grevesse,N.,&Sauval,A.J.2009,ApJ,691,L119
inferredabundancesthatexistintheliterature.
Skilling,J.2004,inAmericanInstituteofPhysicsConferenceSeries,Vol.735,
AmericanInstituteofPhysicsConferenceSeries,ed.R.Fischer,R.Preuss,&
Despitethemanyfactorsthatwetakeintoaccount,wefound U.V.Toussaint,395–405
agoodagreementinourresultsfortheoxygenabundance.Thus, Socas-Navarro,H.2011,A&A,529,A37
wecanconcludethat,basedonthethree-dimensionalmodelthat Socas-Navarro, H. 2015a, NICOLE: NLTE Stokes Synthesis/Inversion Code,
AstrophysicsSourceCodeLibrary
weused,themostprobablevaluefortheoxygensolarabundance
Socas-Navarro,H.2015b,A&A,577,A25
is log((cid:15) ) = 8.86±0.04. This value is classified as a high so-
0 Socas-Navarro,H.,delaCruzRodríguez,J.,AsensioRamos,A.,TrujilloBueno,
lar oxygen abundance and it is compatible with the results of J.,&RuizCobo,B.2015,A&A,577,A7
Grevesse&Sauval(1998). Socas-Navarro,H.&Norton,A.A.2007,ApJ,660,L153
Finally,acaveatisinorder.Notethatallresultsgiveninthis Storey,P.J.&Zeippen,C.J.2000,MNRAS,312,813
paperareconditionedontheempiricalmodelofSocas-Navarro
(2011, 2015a) and the radiative transfer of NICOLE being cor-
rect.Ifthisisnotthecase,theconclusionsmightchange.Thead-
vantageoftheBayesianframeworkisthatitistransparent(every
probabilitydistribution inEq.(1) isconditionedonthe a-priori
informationI).
Acknowledgements. FinancialsupportbytheSpanishMinistryofEconomyand
Competitiveness through projects AYA2014-60476-P and Consolider-Ingenio
2010CSD2009-00038aregratefullyacknowledged.AARacknowledgesfinan-
cialsupportthroughtheRamónyCajalfellowship.MCAacknowledgesFun-
daciónlaCaixaforthefinancialsupportreceivedintheformofaPhDcontract.
ThisresearchhasmadeuseofNASA’sAstrophysicsDataSystemBibliographic
Services.ThisworkhasmadeuseoftheVALDdatabase,operatedatUppsala
University,theInstituteofAstronomyRASinMoscow,andtheUniversityof
Vienna.Weacknowledgethecommunityeffortdevotedtothedevelopmentof
thefollowingopen-sourcePythonpackagesthatwereusedinthiswork:numpy,
matplotlib,nestle,andseaborn.
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