Testing the forward approach in modelling beta Cephei pulsators: setting the stage

1.6 MB·English

by  A. Miglio

Comm. in Asteroseismology
Contribution tothe Proceedingsofthe 38thLIAC/HELAS-ESTA/BAG,2008
Testing the forward approach in modelling β Cephei pulsators:
setting the stage
9
0
0 A.Miglio,J.Montalba´n, andA.Thoul
2
Institutd’AstrophysiqueetdeG´eophysique
n Universit´edeLi`ege,All´eedu6Aouˆt17-B4000Li`ege-Belgique
a
J

4
Abstract
1

] The information on stellar parameters and on the stellar interior we can get by studying
R pulsating stars depends crucially on the available observational constraints: both seismic
constraints (precision and number of detected modes, identification, nature of the modes)
S
and “classical” observations (photospheric abundances, effective temperature, luminosity,
.
h surfacegravity). Weconsiderthecaseofβ Cepheipulsatorsand,withtheaimofestimating
p quantitatively how the available observational constraints determine the type and precision
- ofourinferences,wesetthestageforHare&Houndexercises. Inthiscontributionwepresent
o preliminary resultsforonesimplecase,whereweassumeas“observed” frequencies asubset
r offrequenciesofamodelandthenevaluateaseismicmeritfunctiononadenseandextensive
st gridofmodelsofB-typestars. Wealsocomparethebehaviourofχ2 surfacesobtainedwith
a andwithoutmodeidentification.
[

Session: Poster
1
v
2
7 Tools
0
2 In order to set the stage for Hare&Hound exercises, the following three main components
. need tobedefined:
1
0 • Theoreticalpredictions: Thegridofmodelsweuseisbetadat(Thirion&Thoul
9 2006). Stellarmodelsandadiabaticfrequencies arecomputed, respectively,withcles
0 (Scuflaire et al. 2008a) and losc (Scuflaire et al. 2008b). The masses considered in
: thegridspanthedomainbetween7.8and18.5M⊙,ametalmassfractionZ between
v
0.01 and 0.025, an initial hydrogen mass fraction X = 0.70, and four values of the
i
X overshootingparameter(αov)intherange0-0.25. Frequencies oflow-orderoscillation
modesofdegreeuptoℓ=2arecomputed formain-sequence models.
r
a
• Observational constraints: we consider only seismic constraints, i.e. a subset of
theoretical eigenfrequencies of amodel M0inthe grid. The effects of rotation on the
oscillation modes are not considered in this first step, thus all modes are assumed to
beaxisymmetric(m=0). Concerning thedegree of theobserved modes, weconsider
thecasewhereℓisunknown aswelltheonewhereℓisavailableasaconstraint.
TestingtheforwardapproachinmodellingβCepheipulsators:
2 settingthestage
ℓ ν (µHz)
0 57.78
Table 1: Theoretical oscillation frequen-
cies of model M0 considered as observa- 1 58.93
tional constraints, the uncertainty on the 1 80.29
frequenciesisassumedtobe0.1µHz. 2 39.46
• Merit function: Inorder tocompute ameritfunction ateachpoint ofthegrid,we
useadoubleoptimisationproceduresimilartotheoneextensivelyadoptedinsdBaster-
oseismology(seee.g. Brassardetal.2001 andCharpinetetal.2005). Foreachmodel
inthegridwefindthebestglobalmatchbetween “observed”andtheoretical frequen-
cies by using a standard χ2 merit function. Then we study the properties of good-fit
modelslooking atminima inthe χ2 asa function of thestellarparameters/properties
(e.g. location in an HR diagram, mass, central hydrogen mass fraction (Xc), mean
density,...).
First test
Weconsider asseismicconstraints 4oscillation modesof themodel M0(see Table1), with
frequenciesinthetypicaldomainofthepulsationmodesobservedinβCepheistars. Themain
parameters and properties defining M0 are: M = 10M⊙, Xc = 0.2, αOV = 0, Z = 0.02,
logTeff =4.34andlogL/L⊙=4.02.
Wethencomputethemeritfunctiononasub-gridofmodelsofbetadat,whereαOVand
Z arethe sameasinM0. The behaviour of χ2 formain-sequence models of different mass
and evolutionary status isshown inFig. 1. Inthe casewhere ℓ isgivenasan observational
constraintforallthemodes(rightpanels),theproperties andparametersoftheinputmodel
M0 are easily recovered due to the appearance of an isolated global minimum in the χ2
function. The increase ofthe number of χ2 minima inleftpanels(ℓ isunknown) compared
to right panels allows to estimate the loss of information when mode identification is not
available. Nevertheless, even if no mode identification is available, the frequencies of 4
modesallowtoconstraintheparameterspaceinregionsclosetoχ2 minima.
The results of this simple test show that the approach presented here is viable tool to
determinethenumberandprecisionofobservationaldatarequiredtoconstraintheproperties
ofβCepheipulsators. However,anextensiveandthoroughstudyisneededtoinvestigatethe
effect ofconsidering both additional uncertainties (e.g. onparameters such asovershooting
and initial chemical composition, on rotational splittings and identification of m) as well
as other constraints (e.g. luminosity, Teff, logg, photospheric abundances, non-adiabatic
constraints).
Acknowledgments. A.M.andJ.M.acknowledgefinancialsupportfromtheProdex-ESACon-
tractProdex8COROT(C90199). A.M.isaCharg´edeRecherches andA.T.isaChercheur
qualifi´e oftheFRS-FNRS.
References
Brassard,P.,Fontaine,G.,Bill`eres,M.,etal. 2001,ApJ,563,1013
Charpinet,S.,Fontaine,G.,Brassard,P.,etal. 2005,A&A,437,575
Scuflaire,R.,Th´eado,S.,Montalb´an,J.,etal. 2008a,ApSS,316,83
Scuflaire,R.,Montalb´an,J.,Th´eado,S.,etal. 2008b,ApSS,316,149
Thirion,A.&Thoul,A.2006,ESASP-1306,383
A.Miglio,J.Montalba´n,andA.Thoul 3
ℓ not known ℓ known
Figure1:Colour-codedχ2(logarithmicscale)inthreedifferentplanes:logTeff-logL/L⊙(upperpanels),
logTeff-meandensity(middlepanels),andmass-Xc(bottompanels). Whitedotsrepresentthepositionof
theinputmodelM0ineachparameterspace. Rightpanelsshowχ2computedincludingtheidentification
ofℓasaconstraint,whereasinleftpanelsℓisconsideredunknown(ℓ=0,1or2).