Detection of l=4 and l=5 modes in 12 years of solar VIRGO-SPM data --- Tests on Kepler observations of 16 Cyg A and B

12 MB·

by  Mikkel Nørup Lund

DRAFTVERSIONJANUARY28,2014
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
DETECTIONOF(cid:96) =4AND(cid:96) =5MODESIN12YEARSOFSOLARVIRGO-SPMDATA
—TESTSONKEPLEROBSERVATIONSOF16CYGAANDB—
MIKKELNØRUPLUND1(cid:70),HANSKJELDSEN1,JØRGENCHRISTENSEN-DALSGAARD1,
RASMUSHANDBERG2,1,ANDVICTORSILVAAGUIRRE1
1StellarAstrophysicsCentre,DepartmentofPhysicsandAstronomy,AarhusUniversity,
NyMunkegade120,DK-8000AarhusC,Denmark
2SchoolofPhysicsandAstronomy,UniversityofBirmingham,Edgbaston,Birmingham,B152TT,UK
DraftversionJanuary28,2014
4 Abstract
1
We present the detection of (cid:96) = 4 and (cid:96) = 5 modes in power spectra of the Sun, constructed from
0
12 yr full-disk VIRGO-SPM data sets. A method for enhancing the detectability of these modes in as-
2
teroseismic targets is presented and applied to Kepler data of the two solar analogues 16 Cyg A and
n B. For these targets we see indications of a signal from (cid:96) = 4 modes, while nothing is yet seen for
a (cid:96) = 5 modes. We further simulate the power spectra of these stars and from this we estimate that
J
it should indeed be possible to see such indications of (cid:96) = 4 modes at the present length of the data

7 sets. In the simulation process we briefly look into the apparent misfit between observed and calculated
2 mode visibilities. We predict that firm detections of at least (cid:96) = 4 should be possible in any case at

the end of the Kepler mission. For (cid:96) = 5 we do not predict any firm detections from Kepler data.
]
R
S Keywords: asteroseismology — methods: data analysis — stars: individual (16 Cyg A; 16 Cyg B) — stars:
oscillations—stars: solar-type—Sun: oscillations
.
h
p
- 1. INTRODUCTION as experienced for distant stars. Owing to the difference in
o
thestellarnoisepropertiesbetweenvelocityandphotometric
r Stars observed for the purpose of asteroseismology are
t measured in full-disk integrated light. In such observations measurements,thehighest(cid:96)=4modes(hexadecapole)have
s
been seen in e.g. BiSON (Chaplin et al. 1996a) and GOLF
a there will be an unavoidable geometrical cancellation effect
[ between bright and dark patches on the stellar surface from data (Roca Corte´s et al. 1998) for a long time - with the de-
tection of these already predicted by Christensen-Dalsgaard
the standing harmonic oscillations excited in the star. The
1 & Gough (1989). In full irradiance observations of the Sun
NASAKeplersatellite,dedicatedtofindingplanetsusingthe
v on the other hand only modes up to (cid:96) = 3 have so far been
transit method (Koch et al. 2010), delivers very high-quality
3 directlyobserved(see,e.g.,Garc´ıa2009).
photometric data which are ideal for asteroseismic studies
0
Itisclearthatthedetectionofhigher-degreemodeswould
0 (Gilliland et al. 2010b). However, even with such exquisite
beofgreatimportanceforstellarmodelingwiththeextracon-
7 data it has so far only been possible to detect modes of de-
straintsadded.Itwouldtoagreaterextentbecomepossibleto
. gree (cid:96) = 3 (octupole) in two main sequence (MS) stars,
1 performstellarstructureinversions(e.g.,Basuetal.1997),al-
16 Cyg A and B (Metcalfe et al. 2012), while detections in
0 beitnotwithveryhighprecision.Intheidealcasewherethese
subgiants and red giants have been seen for some time (see,
4 higher degree modes could actually be resolved, a wealth of
e.g., Bedding et al. 2010; Huber et al. 2010; Mosser et al.
1 information could be extracted on for instance the rotational
2012b).
: propertiesofthestar(e.g.,Gizon&Solanki2004).
v Turning to our own star, the Sun has been studied exten-
Inthisstudy,wetakeacloserlookatthephotometricdata
i sively using helioseismology. Here the surface can be re-
X from VIRGO as these data sets hint at what might be possi-
solvedwherebytheconcernofcancellationeffectsisavoided
r and very high degree modes can be studied. As mentioned ble with long time series from the Kepler satellite, and the
a results of this will be presented in Section 2. In Section 3,
above, we do not have this luxury when studying other stars
we describe the method we propose to use for other stars in
usingasteroseismology; hereinsteadthegloballow-(cid:96)modes
thesearchforthesehigherdegreemodes.WetestinSection4
can be studied. To learn about other stars from our Sun, the
thismethodontwoofthemostpromisingtargetsintheKepler
unresolved surface has been studied in so-called Sun-as-a-
field,viz.16CygAandB,withresultspresentedinSection5.
star observations, and this has been ongoing for more than
Furthermore,wesimulatethepowerspectraofthesetwotar-
a decade with velocity observations from ground (e.g., Bi-
gets,describedinSection6,andpresenttheresultsfromthese
SON1;seeChaplinetal.1996b),andspace(e.g.,GOLF2;see
in Section 7. In Section 8, we test the signal from (cid:96) = 4 in
Gabrieletal.1995)andspace-bornphotometricobservations
the solar data when using data lengths corresponding to the
(e.g.,VIRGO3;seeFro¨hlichetal.1995;Frohlichetal.1997).
amountofdataavailablefor16CygAandB.Finally,wewill
Theseobservations suffer fromthesame cancellationeffects
discussourfindingsinSection9.
(cid:70)[email protected]
1BirminghamSolarOscillationNetwork.
2GlobalOscillationsatLowFrequencies.
3VariabilityofSolarIrradianceandGravityOscillations.
2 LUNDETAL.
and (cid:96) = 5 modes that is seen. The strong peak ∼2777µHz,
whichcoincideswiththefrequencyofan(cid:96) = 5mode,isnot
0.07 ofstellarnaturebutisanartefact,havinghalfthefrequencyof
thesignalfromtheDAS.Theobservedfrequenciesalsomatch
0.06 up with values reported from e.g. LOI5 (Appourchaux et al.
1995;Appourchaux&VirgoTeam1998),BiSON(Broomhall
0.05 etal.2009)(upto(cid:96) = 4), MDI6 (see, e.g.,Larson&Schou
]
2m 2008), BiSON+LOWL (Basu et al. 1997), and GONG7 (for
pp0.04 (cid:96)=5)(see,e.g.,Kommetal.2000). SeeFigure3foranex-
[er ample of the correspondence between the red SPM-VIRGO
w
o0.03 data and the frequency estimates from the above references
P
fortwoofthecentralorders(n = 19, 20). These(cid:96) = 4and
0.02 (cid:96) = 5modeshavenotpreviouslybeenreportedonthebasis
of full-disk photometry data, and all of the comparison fre-
0.01 quency estimates stem from either observations done in ve-
locity or from photometry where the solar surface is semi-
0.00 resolved.
1000 2000 3000 4000 5000 6000
Frequency [µHz] We can further visualize the power excess from the (cid:96) = 4
and (cid:96) = 5 modes in the well-known e´chelle diagram (Grec
Figure1. PowerspectraofVIRGO-SPMdata, smoothedwitha1.8µHz
etal.1983),wherethepowerspectrumisfirstdividedinseg-
boxcarfilter. Thecoloroftherespectivepowerspectracorrespondstocolor
oftheSPM-filterused.Asseenthepowerlevelsareclearlyhighestintheblue ments corresponding to the so-called large separation given
band,followbythegreenbandandlastlytheredband.Inordertomakethe by the frequency difference between modes of same degree
distinctionbetweenthepowerspectraeasiercircleshavebeenaddedonthe
and consecutive radial order. Subsequently these segments
peaksofthecentral(cid:96)=0,1modesfortheredandthegreenpowerspectra.
SeealsoFigure12forthepositioninwavelengthofthedifferentfilters. arestackedontopofeachother. Inpractice,thefrequencyis
plottedagainstitsmodulotothelargeseparation-wehaveon
2. SOLARANALYSIS
the ordinate plotted the mid frequency of the respective seg-
2.1. Data ments. Inordertoobtainanicerepresentationoftheridges,
itiscustomarytoaddanarbitraryvaluetothefrequenciesbe-
ForthesolaranalysisweusedthedatafromFro¨hlich(2009)
foretakingthemodulo,therebyallowingashiftoftheridges
with the corrections described therein. More specifically we
on the abscissa (see, e.g., Bedding 2011). In Figure 4, we
use 12 yr data sets from the three Sun photometers (SPM)
of the VIRGO instrument on-board the ESA/NASA SoHO4 showthee´chellediagramofthepowerspectrumfromthered
band,herealsowiththeModelSfrequenciesover-plotted. In
spacecraft. Themidwavelengthsofthesephotometersareat
Figure5,wegivethesamee´chellediagram,thoughinanar-
402nm(blue),500nm(green),and862nm(red).Thepower
rowerfrequency range. To makeclearer the excessfrom the
spectra computed from the corrected time series are shown
(cid:96) = 4 and (cid:96) = 5 modes, the power spectrum has been di-
in Figure 1. These were calculated using a sine-wave fitting
videdbyalinearrepresentationofthestellarbackground,and
method(see,e.g.,Kjeldsen1992;Frandsenetal.1995), nor-
thecolorshavebeentruncatedsuchthatthemaximumcorre-
malized according to the amplitude-scaled version of Parse-
spondsroughlytothehighest(cid:96)=3mode.
val’s theorem (see Kjeldsen & Frandsen 1992), in which a
sine wave of peak amplitude, A, will have a corresponding
peakinthepowerspectrumofA2. 3. ENHANCINGTHEDETECTABILITY
The peak seen at 5555µHz is an artefact and stems from Tomakeeasierthedetectionofthe(cid:96)=4and(cid:96)=5modes
theDataAcquisitionSystem(DAS)ofVIRGOwhichhavea in stars other than the Sun, we propose here a simple three
corresponding reference period of 3 minutes (Jime´nez et al. step method that should accomplish this. Firstly, the power
2005). spectrum is stretched, this is described in Sections 3.1-3.2.
Secondly, the power spectrum is smoothed in Section 3.3.
2.2. (cid:96)=4and(cid:96)=5Modes Thirdly,thestretched,smoothedpowerspectrumiscollapsed
in Section 3.4. The final spectrum we tentatively name the
In Figure 2, we show a zoom-in around the frequency of
SC-spectrum(Straightened-Collapsed).
maximumpower,ν ,ofthepowerspectraaftera1.8µHz
max
binninghasbeenapplied.
Inthisfigure,wehaveindicatedfrequenciescomputedfrom 3.1. ParameterizationofthePowerSpectrum
the solar structure model Model S (Christensen-Dalsgaard Inthefirststep,weparameterizethepowerspectruminor-
etal.1996). Thesefrequencieshavebeencorrectedfornear- dertoaccountforthepossibledeparturesfromtheasymptotic
surfaceeffectsasdescribedbyChristensen-Dalsgaard(2012, descriptionoftheconstituentmodes. Themotivationforthis
Equation(14)). stepis,thatintheendweneedtocorrectforsuchdeparturesin
Fromamerevisualinspectionofthisfigure,itisquiteclear ordertofacilitatetheco-addition(collapsing)ofpowerfrom
that the prominent peaks in the power spectrum agree with many radial orders of a specific degree . This is inspired by
thefrequenciesfromModelS.Thisisofcoursetobeexpected astepintheOctavecode(Hekkeretal.2010),wherethefre-
consideringthatModelSiscalibratedtomatchtheSun.How- quency scale of the power spectrum is modified to account
ever, the match also includes many of the Model S (cid:96) = 4 for departures in the asymptotic description and thereby ob-
(hexadecapole) and (cid:96) = 5 (dotriacontapole; Ellis & Gough
1988) modes, confirming that it is indeed signal from (cid:96) = 4 5TheLuminosityOscillationsImager(onSoHO).
6MichelsonDopplerImager(onSoHO).
4SolarandHeliosphericObservatory. 7GlobalOscillationNetworkGroup.
SEARCHFOR(cid:96)=4AND(cid:96)=5MODES 3
ℓ=0
0.0007
ℓ=1
0.0006
ℓ=2
0.0005 ℓ=3
0.0004 ℓ=4
ℓ=5
0.0003
0.0002 RED SPM
0.0001
0.0025
]0.0020
2m
p
p0.0015
[r
e
w0.0010
o
P
0.0005 GREEN SPM
0.006
0.005
0.004
0.003
0.002
0.001 BLUE SPM
2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800
Frequency [µHz]
Figure2. Zoom-inonthepowerspectraoftheVIRGO-SPMdata,binnedinsegmentscovering1.8µHz.Thecoloroftherespectivepowerspectracorresponds
tocoloroftherespectivetheSPM-filterused. Notethedifferentverticalaxes. ThepositionsofsurfacecorrectedModelSfrequencieshavebeenindicatedby
verticallines,showingaclearcorrespondencetotheobservedpeaksinthepowerspectra.Thelegendgivesthecolorsandlinestylesusedforthedifferentmode
degrees.
1e−4 1e−4
ℓ=4 ℓ=4
2.6 ℓ=5 ℓ=5
2.4
2.4
] ]2.2
2m 2m
p p
p p
[er 2.2 [er 2.0
w w
o o
P P
2.0 1.8
1.8 1.6
2990 3000 3010 3020 3030 3040 3050 3060 3130 3140 3150 3160 3170 3180 3190
Frequency [µHz] Frequency [µHz]
Figure3. Zoom-insontwocentralorders(n=19,20)ofthepowerspectrumfromVIRGOdataoftheredSPM-filter.A1.8µHzboxcarsmoothinghasbeen
applied.Theverticallines(twodrawnforeachdegree)showtherangeinthemodefrequencyestimates(includingerrorbars)fromLOI(Appourchaux&Virgo
Team1998),BiSON(Broomhalletal.2009)(only(cid:96)=4),MDI(Larson&Schou2008),BiSON+LOWL(Basuetal.1997),andGONG(only(cid:96)=5)(Komm
etal.2000).Asseen,thecorrespondencebetweentheseestimatesandtheVIRGOdataisunequivocal.
tainahighersignalinthepower-spectrum-of-power-spectrum Bedding&Kjeldsen2010).
(PS⊗PS). It is well-known from both theory and observations (see,
We start from the following version of the asymptotic fre- e.g., Mosser et al. 2011; Kallinger et al. 2012; Mosser et al.
quencyrelation(Tassoul1980), whichtoagoodapproxima- 2013) that Equation 3.1 is in fact only an approximate de-
tion is applicable to acoustic modes of high radial order n, scription as ∆ν, D , and (cid:15) all have small dependencies on
0
andlowangulardegree(cid:96): both frequency and degree. There exist departures from the
asymptoticdescriptiononbothsmallandlargescales. Small-
ν =∆ν(n+(cid:96)/2+(cid:15))−(cid:96)((cid:96)+1)D . (3.1)
n(cid:96) 0 scaleoscillationsinthelargeseparationoriginate,e.g.,from
Inthisequation,∆ν isthelargeseparation,(cid:15)isadimension- sharp changes in the sound-speed profile (one source being
lessoffsetsensitivetothesurfacelayers,andD isaquantity theHe II ionizationzone;Houdek&Gough2007)-wewill
0
sensitivetothesound-speedgradientnearthecoreofthestar not account for this in the following. The larger scale de-
(see, e.g.,Scherreretal.1983;Christensen-Dalsgaard1993; partures manifest themselves as overall curvatures or tilts of
4 LUNDETAL.
asymptoticrelation(seealsoMosseretal.2011,2013):
ℓ=4 ℓ=2 ℓ=0 ℓ=5 ℓ=3 ℓ=1 ν˜n(cid:96) =∆νpivot(n+(cid:96)/2+(cid:15))−(cid:96)((cid:96)+1)D0 (3.2)
4000 dD
−(cid:96)((cid:96)+1) 0(n−n )
dn pivot
3500
Hz] +(n−n )2d∆ν/dn.
[µcy 3000 pivot 2
uen We have denoted this model frequency by ν˜n(cid:96) to indicate
q thatthisisapredictedvalueonly. ∆ν denotesthevalue
e pivot
Fr2500 of the large separation at n = n (various formulations
pivot
exist for finding ∆ν , see, e.g., Kallinger et al. 2012). Here
0
2000 npivot (which is not constrained to be of integer value) can
be seenasthe pivotpoint forthe variationsin boththe large
separationandD -thefrequencyatwhichn∼n isgen-
1500 0 pivot
0 20 Fr4e0quency6 0mod 13850.19 [µH1z0]0 120 erally coinciding with the frequency of maximum power of
themodes,ν . InthedescriptionofEquation3.2,wehave
max
Figure4. E´chellediagramofthe1.8µHzsmoothedpowerspectrum, red assumedthatthevariousparametersarefrequency-dependent
SPMband.TheModelSfrequenciesindicatedareofdegrees;(cid:96)=0(blue),
ℓ=4 ℓ=2 ℓ=0 ℓ=5 ℓ=3 ℓ=1 onlyandneglectedanypotentialdependenciesonangularde-
(cid:96) = 1(red),(cid:96) = 2(green),(cid:96) = 3(magenta),(cid:96) = 4(cyan),and(cid:96) = 5
(yellow).Thegrayscalegoesfromwhite(lowpower)toblack(highpower), gree. In the following, we will not touch upon the physical
andisgivenonalogarithmicscale. meaning behind the frequency dependencies in ∆ν and D
0
but merely use them in our parameterization of the frequen-
ciesinthepowerspectrum.
The interested reader is referred for instance to Tassoul
(1980), Houdek & Gough (2007), and most recently Mosser
3600 et al. (2013) and references therein for more on the physics
behindthedepartures.
Hz]3400 3.2. ModifyingtheFrequencyScale
µ
[y Themodificationorstraighteningofthefrequencyscalein
c
n3200 thepowerspectrumcomesdowntothreesteps:
e
u
q
e
Fr 1. FittingoftheparameterizationgivenbyEquation3.2.
3000
Thefittingofthemodesisbestillustratedinthee´chelle
diagram described in Section 2.2, and we will do so
2800 fromnowonward.
0 20 40 60 80 100 120 If only observed frequencies are available, e.g.
Frequency mod 135.19 [µHz] from peak-bagging (e.g. Appourchaux 2003),
Figure5. Zoom-inonthee´chellediagraminFigure4. Toenhancethevis- the above parameterization is simply fitted to
ibilityofthe(cid:96)=4,and(cid:96)=5modesthepowerspectrumhasfirstbeendi- these frequencies, with the free parameters being
videdbyalinearfunctionapproximatingthestellarbackground,furthermore
Ψ={d∆ν/dn,D ,dD /dn,(cid:15),n }. Note that
thecolorshavebeentruncatedsuchthatthemaximumpowerlevel(black) 0 0 pivot
correspondsroughlytothehighest(cid:96)=3modes.Thepowerexcessfromthe even if the identification of the modes with respect to
soughtformodesclearlylinesupwiththeModelSfrequencies-theconven- theradialorderiswrongagoodfitcanstillbeobtained
tionrelatingcolortodegreeisthesameasadoptedinFigure4,asistheorder as n is a free parameter. The large separation
pivot
ofthedegreeoftheridges.
∆ν is not included in the optimization as this can be
determined relatively easily, and any small deviations
from∆ν atn canbeaccountedforby(cid:15).
pivot pivot
Ifontheotherhandastellarmodelwithcomputedfre-
quencies is available, and which after applying a sur-
facecorrection(e.g.,Kjeldsenetal.2008a)matchesthe
the ridges in the e´chelle diagram. With the larger scale de-
observed data well, one can apply the fitting to these
partures in mind, Hekker et al. (2010) described one way
modeled frequencies. Here it is then possible to use
to find the variation in the large separation as a function of
alsothemodelcalculatedfrequenciesforthe(cid:96)=4and
the radial order, d∆ν/dn (see also Mosser & Appourchaux
(cid:96)=5modesinthefitting.
2009; Roxburgh 2009). This variation in the large separa-
tion was used in Campante et al. (2010), where it was intro- Withthisfirststep,weobtainestimatesfortheparame-
ducedintheasymptoticrelationasatermquadraticinn-we tersenteringEquation3.2.
willfollowthesameprocedureforthisterm. Thesmallsep-
2. Estimatefrequenciesforatargeteddegree.
aration δν ≡ν −ν =6D , and in turn D , is
02 n(cid:96)=0 n−1(cid:96)=2 0 0
fortheSunfoundtodecreasealmostlinearlywithfrequency The reason for selecting a specific degree (the ”tar-
(Elsworthetal.1990). Withthisinmind,weintroducetoour geted” degree) is that as we have included a variation
modified asymptotic relation a term dD/dn which is linear in D in our parameterization, we cannot modify the
0
in n. We end up with the following modified version of the frequencyscale(asinHekkeretal.2010)suchthatall
SEARCHFOR(cid:96)=4AND(cid:96)=5MODES 5
degrees fulfil Equation 3.1. The reason for this is the
factor of (cid:96)((cid:96)+1) on the term describing the variation
dependingonD0,makingthisterm(cid:96)-dependent. νnℓ - Measured
From the estimated parameters (Ψ) found above one ˜νnℓ - Estimated from fit
can now use Equation 3.2 to estimate the frequencies Interpolation "I" of ˜νnℓ
νs˜tna(cid:96)ncoefabesptheceifi(cid:96)c=or4taorgre(cid:96)te=dd5egmreoed.eTs.hiOsnceouslhdofuoldribne- ∆ν0ν)old
d I(
awarethatsmallerrorsin(cid:15)andD cancauseabigoff- o
0 m
set in the estimate of, e.g., the (cid:96) = 4 or (cid:96) = 5 mode y δν
frequencies.ThebiggestissueisD0duetothefactorof enc | |
(cid:96)((cid:96)+1)onthisparameter;furthermore,thisvalueisone u
q
ooffdthifefemreonsttddeigffireceulatroenneesetdoedc.onInstcraoinntraasstm,(cid:15)an(aynmdo∆dνes FreI(ν)0
0
forthatmatter)canbefairlywellconstrainedbymodes
of the same degree. This issue is mainly a concern if
thefitoftheparameterizationinstep(1)ismadetoob-
servedfrequenciesonly,e.g.,ofpeak-baggedmodesup ν0 νold
to(cid:96)=2,andthedesireistoidentifypowerfrommodes Frequency
ofahigherdegree,e.g. (cid:96) = 4or(cid:96) = 5. Theconcern
Figure6. Illustrationoftheconceptatthebaseofthestraighteningproce-
becomesirrelevantifawell-matchingmodelexists. dure.Thefigurecanbeseenasaflippede´chellediagram,withthefrequency
astheindependentvariableandthemoduloofthefrequencywiththelarge
3. Correct for deviation from Equation 3.1 for a specific separationasthedependentvariable.Theopenthick-edgecirclesgive”mea-
sured”frequencies,meaningfrequenciesobtainedfromeitherpeak-bagging
degree.
orfromastellarmodel. Equation3.2isfittedtothese. Openthin-edgecir-
Wenowwishtotakeoutthedependenciesonfrequency clesgivethe”estimated”frequencies,i.e.,theonesfoundwiththebestfitof
Equation3.2foragivendegree. Thesolidblacklinegivestheinterpolation
in the modes spacings. This is to get a power spec-
betweentheseestimatedfrequencies.Thereferencefrequencyν0definesthe
trum that for modes of a targeted degree follow Equa- correction,δν,thatshouldbeappliedtoavalueνoldonthefrequencyscale
tion 3.1 more strictly, i.e. with equidistance between inordertoobtainapowerspectrumthatforaspecifictargeteddegreefollows
modefrequencies-equivalenttoastraightridgeinthe Equation3.1moreclosely.
e´chellediagram. Thisisaccomplishedbychangingthe the interpolation to the frequencies estimated from the ob-
frequency scale in the power spectrum using frequen- tained fit of Equation 3.2. The dashed black lines give the
ciescomputedinstep(2)forthemodesofthetargeted behaviorofthefitafterthestraighteningprocedure, withthe
degree. straightened Model S frequencies now given as circles. The
straightening is targeted at the (cid:96) = 4 modes, and they now
An interpolation is made between the computed mode
forma verticalline. Asseen, the positiononthe abscissa of
frequencies from step (2) as the independent variable
thisverticallineisgivenbythevalueofthefitatthechosen
and their modulo with the large separation as the de-
referencefrequencyν . Thereferencefrequencyisgivenby
pendent variable - this would correspond to a flipped 0
the horizontal dashed black line, and were here set equal to
e´chellediagram,seeFigure6. Wedenotetheobtained
ν =3150µHz. Intherightpanelthesameprocedureis
interpolationby”I”. max,(cid:12)
followed,buthereonlythemodeshaving(cid:96)=0−2wereused.
Thisinterpolationnowfollowstheridgeinthee´chelle Theuseoflessmodesleadstoafit(step1)withapoorlyde-
diagramfromthecomputedfrequenciesofthetargeted terminedvalueofD (mainly),whichinturnresultsinabad
0
degree. We can now for every frequency (νold) in the estimationof(cid:96)=3−5modes(step2). Whenusingthebadly
frequencyscaleofthepowerspectrumcomputeanew estimatedvaluesof(cid:96)=4modesinthestraightening(step3),
modifiedfrequency(νnew)byaddingavalueδν: an offset between the actual (straightened Model S frequen-
cies)andpredicted(straightenedestimatedfrequencies)value
ν =ν +δν. (3.3)
new old ontheabscissaofthestraightened(cid:96)=4modesisseen.
This added value is obtained from the interpolation as
(seeFigure6) 3.3. Smoothing
For every mode of degree (cid:96), there are 2(cid:96) + 1 degenerate
δν =I(ν )−I(ν ). (3.4)
0 old m-componentsranginginvaluefrom−(cid:96)to+(cid:96),withmbeing
With this value of δν the reference frequency sets the theso-calledazimuthalorder.Thedegeneracyofthesemodes
position on the abscissa of the targeted ridge in the is lifted when the star rotates, with the m-components being
e´chelle diagram - by choice we set the reference fre- spreadoutinfrequencyandtherebytakingawaypowerfrom
quency equal to the value of ν . As we wish to co- the central frequency at m = 0. In the straightening proce-
max
addthesegmentedpowerspectrum,weneedintheend dure, itisthepositionofthiscentralm = 0componentthat
tomapthemodifiedpowerspectrumontoafrequency isfound.
scalewitharegularstepsize. For this reason, we apply a smoothing to the straightened
powerspectruminordertoboostthedetectability,asitisde-
InFigure7,themethodofstraighteninghasbeenappliedto sirabletomergethepowercontainedintherotationallysplit
ModelSfrequencies. IntheleftpanelthefitofEquation3.2 m-components. Theeffectfromthismergingofpowertothe
ismadetoModelSfrequencies(squares)withdegrees(cid:96)=0 central frequency will depend on the size of the rotational
(blue), (cid:96) = 1 (red), (cid:96) = 2 (green), (cid:96) = 3 (magenta), (cid:96) = 4 splitting compared to the mean mode width. If the splitting
(cyan), and (cid:96) = 5 (yellow). The solid black lines illustrate isverylargecomparedtothemodewidththesmoothingwill
6 LUNDETAL.
4500 4500
4000 4000
Hz] Hz]
µ µ
[y 3500 [y 3500
c c
n n
e e
u u
q q
e3000 e3000
r r
F F
2500 2500
2000 ℓ=5 ℓ=3 ℓ=1 ℓ=4 ℓ=2ℓ=0 2000 ℓ=5 ℓ=3 ℓ=1 ℓ=4 ℓ=2ℓ=0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency mod 135.19 [µHz] Frequency mod 135.19 [µHz]
Figure7. MethodofstraighteningappliedtoModelSfrequencies,targetingmodeswith(cid:96) = 4. IntheleftpanelthefitofEquation3.2ismadetoModelS
frequencies (squares) with degrees (cid:96) = 0 (blue), (cid:96) = 1 (red), (cid:96) = 2 (green), (cid:96) = 3 (magenta), (cid:96) = 4 (cyan), and (cid:96) = 5 (yellow). The solid black
lines following these frequencies illustrate the interpolation of the obtained fit. The dashed black lines give the behavior of the fit after the straightening
procedure,withthestraightenedModelSfrequenciesnowgivenascircles. Notethatthefrequenciesofthetargeted(cid:96) = 4modesformaverticalline. The
valueofν0 isindicatedbythedashedhorizontalblackline. Therightpanelshowsthesamefit,buthereonlymodesofdegree(cid:96) = 0−2wereincludedin
thefitofEquation3.2.Asseen,thefitisnownoticeablyworseandthepositionofthestraighteningridgenolongeragreeswiththepredictedposition(dashedline).
notmergemuchofthepowerfromdifferentm-components. 3.4. CollapsingthePowerSpectrum
However, the smoothing will in any case decrease the point-
Prior to this final step in the construction of the SC-
to-pointscatterinthepowerspectrumfromtheχ2 noise, al-
spectrum, we assume that the stellar noise background has
lowinganyunderlyingstructurestostandoutmoreclearly. In
beenfittedandcorrectedfor.
addition,thesmoothingtakesouttheimpactofsmallwiggles
Whencollapsingthepowerspectrum,wewillbeco-adding
that unavoidably will be present in the ridge for the degree
the segmented power spectrum. As the modes in the power
of interest, even after the straightening procedure. The wig-
spectrumhaveadecreasingsignal-to-backgroundratio(SBR)
glesfromforinstancetheHeIIionizationzonewillalsostill awayfromthepeakatν ,weweightthepowerasafunc-
bepresentinthestraightenedridgeaswedidnotaccountfor max
tionofthedistancefromthisfrequency. Thisisdoneinorder
these. Itisdifficulttodeterminethesmoothinglevelthatop-
tonotsimplyaddnoisetothecollapsedspectrumwhenusing
timizesavisualdetection,asthisindeedisaveryqualitative
frequenciesfarawayfromν .
measure, and the impact of a given smoothing level will de- max
The envelope of the p-modes is most often described by a
pend on the rotational splitting, mode width and inclination
Gaussianfunction, andwechosethisasourweightingfunc-
angle of the specific star. Also, a ”too high” smoothing can
tion
resultinasignificantmergingofpowerfromcomponentsof (cid:18) (ν−ν )2(cid:19)
different degrees, which is undesirable. To visualize the im- G(ν)=exp − max . (3.6)
pact of the rotational splitting we can look at the visibility 2σ2
env
within a multiplet, which depends on the stellar inclination
Mosser et al. (2012a) give the following relationship be-
throughthevisibilityfunctionE (i)givenby(Dziembowski
(cid:96)m
1977;Gizon&Solanki2003) tweenνmaxandthefullwidthathalf-maximum(FWHM),and
therebythespreadoftheGaussianenvelope:
((cid:96)−|m|)!(cid:104) (cid:105)2
E(cid:96)m(i)= ((cid:96)+|m|)! P(cid:96)|m|(cosi) , (3.5) FWHM ≈0.66νm0.a8x8 ⇒ (3.7)
0.66ν0.88
σ ≈ max . (3.8)
env (cid:112)
where P|m| is the associated Legendre function, and i is the 2 2ln(2)
(cid:96)
stellar inclination, giving the angle between the stellar ro-
Withthis,wecanconstructthenewweightedpowerspectrum
tation axis and the line of sight, such that a pole-on view
tocollapse:
corresponds to an inclination of i = 0◦. See, e.g., Ap-
pendix A of Handberg & Campante (2011) for the E(cid:96)m(i) Pnew(ν)=Pold(ν)G(ν). (3.9)
factorsupto(cid:96)=4.
The power spectrum which now has a modified frequency
The behavior of E (i) for (cid:96) = 4 and (cid:96) = 5 modes is
(cid:96)m scale and weighted power values is then collapsed to form
illustrated in the top panel of Figure 8 as a function of the
the SC-spectrum, and the steps taken should ensure that the
stellarinclination.Anequivalentrepresentationisgiveninthe
signalfromthedegreeofinterestshouldbebetterdefinedin
lowerpanelofFigure8, wherethevisibilitiesinrotationally
frequency and easier to identify in power. Notice, that the
splitmultiplesof(cid:96) = 4(left)and(cid:96) = 5(right)areplottedin
applicationofEquation3.3onthepowerspectrumresultsas
grayscaleasafunctionofinclination(see Gizon&Solanki
wanted in frequency equidistance of the modes for the tar-
2003). Herethespreadofpowerinfrequencyfromm = 0is
geted degree, but for modes of another degree it will gener-
quiteclear,especiallyathighinclinationangles.
ally have the opposite effect; equivalently, when the power
spectrumiscollapsedingeneralonlythetargeteddegreewill
SEARCHFOR(cid:96)=4AND(cid:96)=5MODES 7
0.40 m=0 modes experiencing the largest effect of the mode bumping
0.35 m=1 shouldbeleftout.
(i)m00..3205 mmm===234 froAmnoththeemr aosdpeesct(cid:96)to=b0e−aw5arweiollftiossthoamtetheextceonldlapbseedcopnotawmer-
=4,0.20 inated with power from even higher degree modes. As also
ℓE0.15 mentionedin,e.g.,Appourchaux&VirgoTeam(1998)(cid:96)=1
0.10
modesarepollutedby(cid:96) = 6and(cid:96) = 9modes, while(cid:96) = 7
0.05
modes fall in the proximity of the (cid:96) = 4 modes, (cid:96) = 8 near
0.00
m=0
0.35 m=1 the(cid:96)=5modes,andsoforth. Forthisreason,itwillfurther-
0.30 m=2 more be very unlikely to pick up isolated signals from, e.g.,
(i)=5,m00..2205 mmm===345 (cid:96)=If6hy−po8thmeosidseste.sting is desired on the collapsed spectrum,
ℓE0.15 intheformofforinstanceanH0 test,thelaststepofweight-
0.10 ing could be left out, and a binning of points rather than a
0.05 smoothingwouldbemoresuitable.
0.00
0 10 20 30 40 50 60 70 80 90
Inclination [◦] 4. ANALYSISOF16CYGAANDB
We will now apply the method of the SC-spectrum to the
90 90 twosolaranalogues16CygA8(G1.5V)andB9(G3V).These
80 80 two very similar stars are in fact part of a hierarchal triple
system(16Cygni),withafaintMdwarfasthethirdcompo-
70 70 nent. The most striking difference between the two stars is
thatmeasurementsinradialvelocityshow16CygBtohosta
60 60
] ] Jupiter-massplanetinahighlyellipticalorbit(Cochranetal.
◦ ◦
on [50 on [50 1997). Thesecondmajordifferencebetweenthetwostarsis
ati ati theLi-abundance,whichforsomereasonseemstobereduced
clin40 clin40 in the A-component by a factor of ≥ 4.5 relative to the B-
n n
I I component(see,e.g.,Frieletal.1993). Becauseofthesolar
30 30
similarityandchemicalpeculiaritiesofthetwomaincompo-
20 20 nentsthissystemhasbeenstudiedbynumerousgroups, see,
e.g., Schuler et al. (2011), and references therein for exam-
10 10 plesofthechemicalabundanceanalysis. Withitspositionin
the sky (Cygnus - The Swan), the system is fortuitously ob-
0 0
−2 Fr−e1quen0cy [µH1z] 2 −2 Fr−e1quen0cy [µH1z] 2 served by the Kepler satellite. The Kepler magnitudes10 of
these stars, 5.864 (16 Cyg A) and 6.095 (16 Cyg B), make
Figure8. ToptwopanelsgivethefunctionalformofE(cid:96)=4,m(i)(first)and them some of the very brightest stars in the Kepler field-of-
E(cid:96)=5,m(i)(second). Them = 0component(blue)goestoavalueof1at view, and ideal for asteroseismic studies. Such an astero-
i = 0◦. Bottomtwopanelsrenderthesameinformation,onlyhereatop
viewoftherotationallysplitmultiplets(νs =0.37µHz)isshown,withthe seismicstudyhasrecentlybeenconductedbyMetcalfeetal.
visibilitiesofdifferentazimuthalcomponentsgivenbythegrayscale,ranging (2012, hereafter MC12), where both stars were peak-bagged
fromblack(highvisibility)towhite(lowvisibility)(Gizon&Solanki2003). andsubsequentlymodeled-wewilluseresultsanddatafrom
Theverticaldashedlinesshowthepositionsofthesplitm-components.
thisstudyinouranalysis.
haveamorewell-definedsignalpeakwhilethepeaksofother Thereasonforchoosingthesestarsasastartingcaseforthe
degreeswilltendtobesmearedouttosomeextend. search of (cid:96) = 4 and (cid:96) = 5 modes is that both of them have
Evenwiththeaboveweightening,modesfarfromνmaxwill alreadyshownclearlydetectableoctupole(cid:96) = 3modes. The
mainly contribute noise to the SC-spectrum. Therefore, one prospectsofhavingdetectablemodeswith(cid:96)>3yieldssome
should preferably only include a relatively small number of very strong constraints on the stellar models. This comes in
centralorders. Theoptimumnumberofovertonestoaddfor addition to the constraint on the allowed model differences
aspecifictargetwilldependonthe SBR asafunctionoffre- fromthebinarityofthesystem.
quencyforthatspecifictarget,inadditiontothelargesepara- As of now, it has not been possible to constrain the incli-
tion,andisthereforenoteasilygeneralized. nation angles of the stars using asteroseismology, but work
iscurrentlybeingdonetoresolvethisissue(G.Davies2013,
3.5. Remarks privatecommunication).
ThechoiceofmodesinthefittingofEquation3.2isimpor-
4.1. KeplerData
tant. As seen in the e´chelle diagram of the Sun in Figure 4
theridgeshaveingeneralan”S”-shapefromthevariationin For both stars, we used short cadence (SC, ∆t = 58.85s;
thelargeseparation.Thesectionoftheridgethatiswellfitted Gilliland et al. 2010b) data from quarters Q7-Q13, corre-
by a quadratic term in n is only the top part of this ”S” (or, sponding in the end to ∼643 days, all downloaded from the
equivalently,onlythelowerpart),i.e.,fromabout2400µHz KASOC webpage11. The stars were not observed in SC in
and up. As seen the curvature of this upper part of the ”S”
8KIC12069424,HR7503,HD186408.
is quite well centred around ν at around 3150µHz. So,
max 9KIC12069449,HR7504,HD186427.
usingmodesthatliefarawayfromν canresultinabadfit
max 10 KeplermagnitudesarenearlyequivalenttoRbandmagnitudes(Koch
ofEquation3.2. etal.2010).
Formoreevolvedstars,mixedmodesshouldasfaraspos- 11KeplerAsteroseismicScienceOperationsCenter:kasoc.phys.au.
sible be excluded from the fit to the modes - so, e.g., (cid:96) = 1 dk.
8 LUNDETAL.
Q0-Q5. Asbothstarsarehighlysaturated12 Thesedatahave 5. RESULTSONDETECTABILITYFROMBONAFIDEDATA
therefore not been included in our data sets. The data type
WeappliedthemethodpresentedinSection3totheKepler
used is the uncorrected simple aperture photometry (SAP).
dataof16CygAandB.Figure10givesthee´chellediagrams
Thecorrectionofthetimeserieswasdonebyhigh-passfilter-
of the two stars, with the same color convention for the dif-
ingindividualsub-quarters(∼1month)byaonedaymoving
ferentdegreesasusedinFigure7fortheSun. Inthefigure,
medianfilter.Baddatapoints(oroutliers)werethenremoved,
observed mode frequencies are given as white stars with as-
with a bad datum being identified as one having a point-to-
sociated errorbars. Model frequencies (see Section 6.1) are
pointfluxdifferencefallingoutside3σ -withσ foundasthe
given as circles, with the distinction that filled circles were
standarddeviationofthepoint-to-pointfluxdifferencesofthe
used in the fitting of Equation 3.2, while white circles were
entire time series (Garc´ıa et al. 2011). We did not estimate
left out - see Section 3.5 for the considerations made in the
this standard deviation (STD) directly but used instead the
selection of modes to include in the fit. In the plot, we have
morerobustmedian-absolute-deviation(MAD),andfromthis
indicated the radial order n of the (cid:96) = 0 modes. The fit to
estimatedtheSTDviathescalingSTD=1.4826×MAD13.
themodelfrequencieswasmadewiththestraighteningofthe
Thefilteringofbaddatapointswasperformediterativelyfour
(cid:96)=4ridgeinmind,andtheblacklinedepictstheobtainedfit
times. Inaddition,weusedthe”Quality”14 entryintheFITS (similarcurvesofcourseexistfortheotherdegrees,seeFig-
filesfortheSAPtoremovepointswithknownartefacts. The ure7).Thedashedcyanlinegivesthepositionoftheridgeaf-
finaltimeserieshaddutycycles15 of85.7%(16CygA)and terthestraightening,withthereferencefrequencyν (dashed
0
82.2%(16CygB).Withthesedutycyclesinmindthespec- blackline)setequaltothevalueofν obtainedfromfitting
max
tral windows for the two stars were checked, and we found Equation 4.1 to the power spectra, viz., 2101µHz (16 Cyg
no significant leakage of power into side-lobes. In any case, A) and 2552µHz (16 Cyg B). The regions between the two
thesmoothingofthepowerspectrumshouldensurethanany thin horizontal lines give the range chosen in the collapsing
potentiallyleakedpowerisstillaccountedfor. ofthestraightenedpowerspectra, i.e., the∼7ordersclosest
The power spectra (see Figure 9) were calculated in the toν .Thisregionlargelycoincideswiththeregionwherein
max
samemannerasforthesolardata,exceptfortheuseofstatis- (cid:96)=3modesarereadilyobserved.
ticalweightsinthecomputation. Weightswerecomputedas In Figure 11 the SC-spectra are shown, and rendered in a
wi = 1/σi2, withσi foundfroma3daywindowedMADof multitudeofdifferentamountsofappliedsmoothing. Thetop
thecorrectedtimeseries. twopanelsgivethefullSC-spectra,whilethebottomtwopan-
For the modeling of the stellar background signal, we use elsshowzoom-inversions. Thedashedcyanlineindicateas
asumofpowerlaws(Harvey1985),hereintheversionpro- inFigure10thepositionofthestraightened(cid:96) = 4ridge,and
posed by Karoff (2008) (see also Huber et al. 2009), and in therebygivetheexpectedpositionofthepossibleexcessfrom
addition, we add a Gaussian function to account for the ex- (cid:96) = 4 modes. The horizontal lines give the median value of
cessp-modepower: the SC-spectrum in the minimum and maximum smoothing
cases. Thesehavebeenaddedsimplytoestablishareference
B(ν)=(cid:88)3 4σi2τi (4.1) levelinthespectrum. Forbothstarsweseeariseinthecol-
1+(2πντ )2+(2πντ )4 lapsedpowerattheexpectedpositionofthe(cid:96)=4ridge,while
i i
i=0 noclearexcesspowerisseendirectlyinthepowerspectrafor
+A exp(cid:18)−(ν−νmax)2(cid:19)+B , theexpectedfrequenciesofindividualmodes.
2σ2 N We also applied the method targeting the (cid:96) = 5 ridge, but
g
herenonoticeablerisewasseeninthecollapsedpower. The
where BN is the white shot-noise component, σi is the rms (cid:96) = 5ridgeisinanycasemoredifficulttocaptureduetoits
intensityofthe ithnoisecomponent, τi isthecorresponding closeproximitytothestrong(cid:96)=0modes.
timescaleofthenoisecomponent,Aandσgaretheamplitude Modelfrequencieswereobtainedfromthestellarmodelde-
and spread, respectively, of the p-mode envelope. The three scribed in Section 6.1 below, and no other model was tested
backgroundcomponentsincluded,andshowninFigure9,ac- for the straightening. However, we note that the small sep-
countfortheactivity(magenta),granulation(green),andfac- aration δν seems to be correctly reproduced as the model
02
ulae (blue) signals, respectively. When correcting the power frequenciesmatchwellwiththeobservedvaluesforboththe
spectrum for the stellar background, we of course leave out (cid:96) = 0and(cid:96) = 2ridges. Fromthedefinitionofδν as6D
02 0
the Gaussian p-mode envelope (dashed white) from the full (seeSection3.1)thismeansthatD isapproximatelycorrect
0
fit(thickred). andthe(cid:96) = 4ridgeshouldthereforebelocatedatnearlythe
sameplaceinthee´chellediagramsforallthemodelsthatcor-
rectlyreproduceδν .
02
12 ThesaturationlimitforKepler isaboutKp ∼ 11.5(Gillilandetal. 6. SIMULATEDPOWERSPECTRA
2010a)largecustomaperturemasksareneededinordertocaptureasmuch In order to check if a potential excess power in the SC-
fluxaspossible. However,theSCobservationsmadeinpartsofQ6didnot
spectrum is in line with what can be expected for (cid:96) = 4 or
makeuseofcustommasksontheCCDwhichresultedinaratherpoordata
quality. (cid:96)=5fromthecurrentamountofobserveddata,wesimulated
13Themultiplicativefactorof1.4826converts(approximately)theMAD thepowerspectraof16CygAandB.Thisalsoenablesatest
foranormaldistributiontoaconsistentmeasureoftheSTD,andcanbefound on the impact of longer observing times on the detectability
as1/Φ−1(3/4),withΦ−1representingtheinversecumulativedistribution ofthe(cid:96)=4and(cid:96)=5modes,andtherebyhinttotheneeded
functionorquantilefunction.
amountofdataforasoliddetection. Furthermore,wecanin-
14ThebitvaluesofvariousknownartefactscanbefoundinFraquelli&
Thompson(2012). vestigatetheroleofthecurrentlyunknownstellarinclination
15 Percentageoffinalnumberofpointstothetotalexpectednumberof angles, and test the effect of applying different amounts of
pointsgiventhelengthandcadenceofthetimeseries. smoothing. In the following, the various aspects underlying
SEARCHFOR(cid:96)=4AND(cid:96)=5MODES 9
PA
PA
2m]10-1 PG 2m]10-1 PG
p p
p p
[r [r
e e
w w PF
o o
P PF P
10-2 10-2
PS PS
101 102 103 101 102 103
Frequency [µHz] Frequency [µHz]
Figure9. Powerspectra(black)of16CygA(left)andB(right)fromQ7-Q13data,smoothedwitha1.8µHzboxcarfilter.Theoptimumfittothebackground
(red) includes, besides the Gaussian envelope from p-modes, the signature from activity (PA; magenta), granulation (PG; green), faculae (PB; blue), and
white/shotnoise(PS;black).ThedashedwhitelineshowsthebackgroundfitwithouttheGaussianenvelope.
3000 ℓ=3 ℓ=1 ℓ=4 ℓ=2 ℓ=0 ℓ=5 3500 ℓ=3 ℓ=1 ℓ=4 ℓ=2 ℓ=0 ℓ=5
26
25 26
24 25
Hz]2500 2223 Hz]3000 2234
[µcy 2210 [µcy 2221
en 19 en2500 20
qu2000 18 qu 19
e 17 e 18
r r
F 16 F 17
15 2000 16
14 15
1500
1500
0 20 40 60 80 100 0 20 40 60 80 100
Frequency mod 103.40 [µHz] Frequency mod 116.97 [µHz]
Figure10. MethodfromSection3appliedtoKeplerdataof16CygA(left)andB(right). Thegrayscaleinthesee´chellediagramsrangefromwhiteatlow
powertoblackathighpower. Observedmodefrequenciesaregivenaswhitestarswithassociatederrorbars. Modelfrequencies(seeSection6.1)aregivenas
circles,withfilledcirclesusedinthefittingofEquation3.2,whilewhitecircleswereleftout.Theblacklinefollowingthe(cid:96)=4ridgeillustratetheobtainedfit
toestimatedfrequencies. Theradialorder,n,ofthe(cid:96)=0modesisindicatedbythenumbers. Thetwohorizontalblacklinesgivetherangeinfrequencythat
wascollapsedinmakingtheSC-spectrum,whiletheblackdashedlineindicatethefrequencyofνmax. Theverticaldashedcyanlinegivesthepositionsofthe
(cid:96)=4ridgeafterthestraightening.
thesimulationswillbedescribed,towit,thestellarmodeling, and specific details of the modeling. Notice, that the final
the synthetic power spectrum, visibilities, and noise proper- AMP parameters quoted in MC12 were found after using a
ties. localized Levenberg-Marquardt optimization algorithm (uti-
lizing singular-value-decomposition) to adjust the stellar pa-
6.1. StellarModeling rameters from the values found by the GA, viz., the values
given in Table 1. For this reason the values in Table 1 will
Forthestellarmodelsneededtocalculatethepulsationfre-
differ slightly from the ones given in MC12. With the best-
quencies we use pre-existing models16 from the asteroseis-
fitting AMP models, we use the ”Aarhus adiabatic pulsation
mic modeling portal (AMP; Metcalfe et al. 2009; Woitaszek
package”(ADIPLS;Christensen-Dalsgaard2008)tocompute
et al. 2010) - see Table 1 for the parameters of the adopted
theoscillationfrequenciesincluding(cid:96) = 4and(cid:96) = 5modes.
models. ThemodelswereinAMPfoundbyoptimizing(us-
Finally, we apply an empirical correction for surface effects
ing a parallel genetic algorithm (GA), see Metcalfe & Char-
inordertohavemodelfrequenciesthatresembletheobserved
bonneau2003)thefitoftheobservedoscillationfrequencies
frequenciesof16CygAandBascloselyaspossible.Follow-
(published in MC12) and observational constraints to mod-
ingtheprescriptionbyKjeldsenetal.(2008a),thefrequency
eled oscillation frequencies. The spectroscopic constraints
correctionappliedtothemodelfrequenciesisgivenby
listedinTable1arefromRam´ırezetal.(2009). Wereferthe
reader to MC12 and references therein for the input physics
1616CygA:amp.phys.au.dk//browse/simulation/191 (cid:18)ν (n)(cid:19)b
16CygB:amp.phys.au.dk//browse/simulation/189. δν ≡ν (n)−ν (n)=a obs , (6.1)
obs model ν
0
10 LUNDETAL.
5.00 5.00
4.75 4.75
12
4.51 4.51
4.26 10 4.26
4.01 4.01
10
3.76 3.76
Collapsed power 68 12222333........7025702583837272µmoothing window [Hz] Collapsed power 468 12222333........7025702583837272µmoothing window [Hz]
4 1.54S 1.54S
1.29 1.29
2 01..7094 2 01..7094
0.55 0.55
0.30 0.30
0 20 40 60 80 100 20 40 60 80 100 120
Frequency mod 103.40 [µHz] Frequency mod 116.97 [µHz]
5.00 5.00
4.75 1.4 4.75
4.51 4.51
1.4 4.26 4.26
4.01 4.01
3.76 1.2 3.76
Collapsed power11..02 12222333........7025702583837272µmoothing window [Hz] Collapsed power01..80 12222333........7025702583837272µmoothing window [Hz]
0.8 S S
1.54 1.54
1.29 1.29
1.04 0.6 1.04
0.6 0.79 0.79
0.55 0.55
0.30 0.30
0 20 40 60 80 100 20 40 60 80 100
Frequency mod 103.40 [µHz] Frequency mod 116.97 [µHz]
Figure11. SC-spectrafor16CygA(left)andB(right)withdifferentamountsofappliedsmoothing. ThetoptwopanelsgivethefullSC-spectra,whilethe
bottomtwopanelsshowzoom-inversions. ThedashedcyanlineindicateasinFigure10thepositionofthestraightened(cid:96) = 4ridgeandhencetheexpected
positionofapossibleexcessfrom(cid:96)=4modes.ThehorizontallinesgivethemedianvaluesoftheSC-spectraintheminimumandmaximumsmoothingcases.
where ν is a reference frequency which we set equal to the
0 Table1
mean of the observed radial modes; b is set to the solar cal-
AMPModeling
ibrated value of 4.823 (Mathur et al. 2012). The value of a
isfoundusingEquation1017 inKjeldsenetal.(2008a). Note Parameter ModelA ModelB
that when using Equation 6.1 to correct model frequencies
Results
outside the range of observed modes one should be aware
thatthisparameterizationofthesurfacecorrectionisnotsuit- M(M(cid:12)) 1.10 1.07
R(R(cid:12)) 1.2361 1.1256
able at frequencies far from the observed values due to a
L(L(cid:12)) 1.5669 1.2616
frequency-range dependence in the exponent b as the fre-
τ(Gyr) 6.5425 5.8162
quencydifferencedoesnotexactlyfollowasimplepowerlaw αML 2.06 2.00
(Kjeldsenetal.2008a). Y0 0.2510 0.2430
Z 0.02239 0.02032
6.2. TheSyntheticLimitSpectrum Teff(K) 5814.01 5771.32
ConstraintsinOptimization
Even though it has not yet been possible to extract mean
rotation periods from the power spectra of 16 Cyg A and B, Teff(K) 5825±50 5750±50
[Fe/H] 0.096±0.026 0.052±0.021
we can with gyrochronology make a rough estimate for the
logg 4.33±0.07 4.34±0.07
rotationperiod. Themeanrotationperiodisfoundusingthe
expressionofBarnes(2007): L(L(cid:12))a 1.56±0.05 1.27±0.04
No.offittedmodes 46 41
P =tna[(B−V) −c]b, (6.2) SurfaceCorrectionParameters
rot 0
a(µHz) -3.100 -4.927
with parameters n = 0.519, a = 0.773, b = 0.601, and ν0(µHz) 2215.812 2569.344
c = 0.4. Here t is the stellar age in Myr, for which we use aSeeMC12andreferencesthereinforthecalculationoftheluminosity.
themodelvaluesgiveninTable1. Thevaluesfor(B−V)
0
are0.64(16CygA)and0.66(16CygB),bothadoptedfrom
17Usingr=1. Johnson&Morgan(1953). UsingEquation6.2onthemodel