Mode visibilities in radial velocity and photometric Sun-as-a-star helioseismic observations

0.46 MB·English

by  David Salabert

Astronomy&Astrophysicsmanuscriptno.15946 (cid:13)c ESO2011
January25,2011
Mode visibilities in radial velocity and photometric Sun-as-a-star
helioseismic observations
D.Salabert1,2,3⋆,J.Ballot4,andR.A.Garc´ıa5
1 InstitutodeAstrof´ısicadeCanarias,E-38200LaLaguna,Tenerife,Spain
2 DepartamentodeAstrof´ısica,UniversidaddeLaLaguna,E-38206LaLaguna,Tenerife.Spain
3 Universite´deNiceSophia-Antipolis,CNRS,ObservatoiredelaCoˆtedAzur,BP4229,06304NiceCedex4,France
4 Laboratoired’AstrophysiquedeToulouse-Tarbes,Universite´deToulouse,CNRS,31400Toulouse,France
5 LaboratoireAIM,CEA/DSM-CNRS,Universite´Paris7Diderot,IRFU/SAp,CentredeSaclay,91191Gif-sur-Yvette,France
1
1 Receivedxxxxxx;acceptedxxxxxx
0
2 ABSTRACT

n Weanalyze more than 5000 days of high-quality Sun-as-a-star, radial velocity GOLF and photometric VIRGO/SPMhelioseismic
a observations to extract precise estimates of the visibilities of the low-degree p modes and the m-height ratios of the l = 2 and 3
J
multipletsinthesolaracousticspectrum.ThemodevisibilitiesareshowntobelargerduringtheGOLFred-wingconfigurationthan

4 duringtheblue-wingconfiguration,andtodecreaseasthewavelengthoftheVIRGO/SPMchannelsincreases.Wealsoshowthatthe
2 modevisibilitiesareindependentofthesolaractivitycycleandremainconstantoverallwithtime,butthatneverthelesstheyfollow
short-term fluctuations on a time scale of a few months. The l = 1 mode visibility also increases significantly toward the end of

] theyear1999.Comparisonswiththeoreticalpredictionsareprovided.Eventhoughthereisqualitativeagreement,somesignificant
R
discrepanciesappear,especiallyforthel=3modes.Thelimbdarkeningalonecannotexplaintherelativevisibilitiesofmodes.These
S preciseestimatesshouldbeusedasreferencesfortheextractionofthep-modeparametersforanyfutureinvestigationusingtheGOLF
. andVIRGO/SPMobservations.
h
p Keywords.Methods:dataanalysis–Sun:helioseismology
-
o
r
st 1. Introduction the l = 2 and 3 multiplets (the so-called m-height ratio) dur-
a ingthepeak-fittingprocedurewhenestimatingthep-modechar-
[ The potential of helio- and asteroseismology to provide in- acteristics (e.g., Salabertetal., 2004), while the acoustic pow-

sights into the interiors of the Sun and other distant stars is ers of the l = 1, 2, and 3 modes relative to the l = 0 modes
1
unique. Measurements of the properties of the normal modes
v are allowed to vary (the so-called mode visibility). However,
of solar oscillations have contributed greatly to our knowl-
4 in asteroseismology, the mode visibilities are fixed to theoret-
edge of both the internal structure (e.g. Basuetal., 1996;
4 icalvalues(Christensen-Dalsgaard&Gough,1982;Palle´etal.,
5 Antia&Chitre,1998;Couvidatetal.,2003)anddynamics(e.g. 1989a)becauseofthelowersignal-to-noiseratio(SNR)oftypi-
4 Thompsonetal.,1996;Chaplinetal.,1999;Garc´ıaetal.,2004) caldatasetsandshortertimeseries,andthem-heightratiosare
1. of the Sun’s convective and radiative layers as far down to its expressed as a function of the inclination of the rotation axis
core(e.g.Mathuretal.,2007;Garc´ıaetal.,2007;Mathuretal.,
0 only (Appourchauxetal., 2008; Garc´ıaetal., 2009). Although,
2008). Moreover, strong constraints on the stellar properties
1 the acoustic frequencies (e.g., Woodard&Noyes, 1985), pow-
1 havealsobeenobtainedwithseismicanalysisfrombothground- ers, and damping rates (e.g., Palle´etal., 1990) vary with the
: based (e.g. Bazotetal., 2005) and space-based (e.g. Bruntt, magnetic activity cycle of the Sun and the first observation of
v
2009;Christensen-Dalsgaardetal.,2010;Metcalfeetal.,2010)
similar variations of the p-mode parameters in a solar-like star
i
X observations. wasrecentlyreportedbyGarc´ıaetal. (2010), themodevisibil-
r However, our knowledge of the properties of the solar and ities are supposed to be independent of both solar and stellar
a stellar interiorsdependsonourability to correctlymeasurethe magneticactivity.Fro¨hlichetal.(1997)observedvariationswith
resonant modes of oscillations, which are commonly modeled theheightinthesolaratmosphereintheintensitydatacollected
asLorentzianprofilesintheacousticpowerspectrum.Themode by the Variability of Solar IRradiance and Gravity Oscillation
parameters(amplitude,linewidth,frequency,rotationalsplitting, (VIRGO;Fro¨hlichetal.,1995)atthebeginningoftheSolarand
and asymmetry) are then extracted by fitting the model to the HeliosphericObservatory(SoHO)mission.However,itremains
data using maximum-likelihoodfunctions (Appourchauxetal., to be verified if these values are different in the radial veloc-
1998). The rotation of a star lifts the degeneracy of the os- ityGlobalOscillationsatLowFrequency(GOLF;Gabrieletal.,
cillation modes and splits the eigenfrequencies into 2l + 1 m- 1995)measurementsbetweentheblue-andred-wingobserving
components, as −l ≤ m ≤ +l, where l is the angular degree periods.
and m the azimuthal order. In Sun-as-a-star helioseismology, The space-based Convection, Rotation, and plane-
only the l+|m| even componentsare visible and it is common tary Transits (CoRoT, Micheletal., 2008) and Kepler
practice to fix the height ratios between the m-components of (Boruckietal., 2009) missions are currently providing in
large quantities high-quality photometric measurements. After
⋆ [email protected] several years of Kepler observationsof the same asteroseismic
1
D.Salabertetal.:ModevisibilitiesinSun-as-a-starhelioseismicobservations
targets,theSNRwillbehighenoughtomeasurethevisibilities approximation(Ledoux, 1951), thusthe mode frequencyis ex-
and m-height ratios in a wide range of solar-like stars in the pressedasν =ν −mδν ,whereν isthecentralfrequency
l,m,n l,n l,n l,n
Hertzsprung-Russell diagram at different evolution stages andδν istherotationalsplitting.Wenotethatweassumedthe
l,n
(Beddingetal., 2010; Chaplinetal., 2010). The forthcoming samelinewidthforthecomponentsofagivenmultiplet.Because
observations from the radial velocity Stellar Observations of their close proximity in frequency,modes are fitted in pairs
NetworkGroup(SONG, Grundahletal., 2007) andthe photo- (i.e., l = 2 with 0, and l = 3 with 1). While each mode pa-
metric PLAnetary Transits and Oscillations of stars (PLATO1) rameterwithina pairofmodesarefree,thepeakasymmetryis
instruments will also provide extremely precise measurements set to be the same within pairs of modes. When visible in the
of stellar oscillations in the very near future. Simultaneous powerspectra,thel=4and5modeswereincludedinthefitted
radial velocity and photometric observationswill be extremely model. An additive value B is added to the fitted profile repre-
useful in improvingour understanding and modeling of stellar sentingaconstantbackgroundnoiseinthefittedwindow.Since
atmospheres. SoHOobservestheSunequatorwards,onlythel+|m|evencom-
ponentsare visible in Sun-as-a-star observationsof GOLF and
VIRGO/SPM.Toreducethenumberoffreeparametersandsta-
2. Dataandanalysis bilize the peak-fitting procedure, it is common practice to fix
theheightratiosbetweenthevisiblem-componentsforboththe
Weanalyzedsimultaneousobservationscollectedbythespace-
l = 2 and l = 3 multiplets to their estimated values. To derive
based, Sun-as-a-star Global Oscillations at Low Frequency
observationalestimates ofthe m-heightratios, the m = ±2 and
(GOLF;Gabrieletal.,1995)andVariabilityofSolarIRradiance
m = 0 components of the l = 2 multiplets, and the m = ±3
and Gravity Oscillation (VIRGO; Fro¨hlichetal., 1995) instru-
and m = ±1 components of the l = 3 multiplets were fit-
mentsonboardtheSolarandHeliosphericObservatory(SoHO)
tedusingindependentheightsH ,assumingthatcomponents
spacecraft.TheGOLFinstrumentisaresonantscatteringspec- l,m,n
with opposite azimuthal order m have the same heights, i.e.,
trophotometer measuring the Doppler wavelength shift – inte-
H = H . Finally, the mode parameters were extracted
grated over the solar surface – in the D and D Fraunhofer l,−m,n l,+m,n
1 2 by maximizing the likelihood function by assuming the power
sodium lines at 589.6and 589.0nm, respectively.The VIRGO
spectrumstatisticsisgivenbyaχ2withtwodegreesoffreedom.
instrument is composed of three Sun photometers (SPM) at
Thenaturallogarithmsofthemodeheight,linewidth,andback-
402 nm (blue channel), 500 nm (green channel), and 862 nm
groundnoisewerevariedresultinginnormaldistributions.The
(redchannel).Atotalof5021daysofradialvelocityGOLFand
formal uncertaintiesin each parameter were then derived from
intensity VIRGO/SPM observations starting on 1996 April 11
the inverseHessian matrix.The blue and redperiodsof GOLF
and ending on 2010 January 8 were analyzed, with respective
werealsoanalyzedseparately,aswellasthemeanpowerspec-
dutycyclesof95.4%and94.7%.TheGOLFvelocitytimeseries
trumofthethreeVIRGO/SPMs(blue,green,andredchannels).
were obtained following Garc´ıaetal. (2005) and calibrated as
describedinJime´nez-Reyesetal.(2003).Moreover,inthecon-
text of this paper, it is important to remember that the GOLF 3. Modevisibilities
signalhasasmallintensitypollutionofaround14%(Palle´etal.,
1999).Fortechnicalreasons,GOLFhasbeenobservingonlyone Because ofaveragingeffectsoverthe solardisk withSun-as-a-
sideofthesodiumdoublet:inthebluewingfromApril111996 starobservations,theobservedrelativeamplitudesofthemodes
untilJune251998andagainfromNovember182002untilnow; aremodifiedbyvisibilityfactorsVl.Itispossibletorecoverthe
and,inthemeanwhile,betweenSeptember1998untilNovember visibilities by assuming that the intrinsic amplitudes of modes
182002,themeasurementswereobtainedfromtheredwingof with close frequencies are similar. By denoting Al,n to be the
the sodium doublet, which originates higherup in the solar at- observedamplitudeofamode,wefindthatVl/V0 ≈ Al,n/Al=0,n′
mosphere (Garc´ıaetal., 2005). In addition, two extended gaps whenthemodes(l,n)and(l=0,n′)haveclosefrequencies.
arepresentintheanalyzedtimeseriesbecauseofthetemporary Thus,wecanmeasurethevisibilitiesinthespectrumas
loss of the SoHO spacecraft. The first gap of ∼100 days hap- (V /V )2 = P /P , (2)
penedduringthesummerof1998andwasduetoafailureinthe 1 0 1,n 0,n
gyroscopes.Thesecondoneofaperiodof∼1monthinJanuary (V /V )2 = P /P , (3)
1999 occurred while new software was being uploaded to the 2 0 2,n−1 0,n
spacecraft. and
The power spectrum of each time series was fitted to esti-
matethemodeparametersofthel = 0,1,2,and3modesusing (V3/V0)2 = P3,n−1/P0,n, (4)
amulti-stepiterativemethod(Salabertetal.,2007).Eachmode
whereP isthemodepower,directlylinkedtothesquareofthe
componentofradialordernandangulardegreelwithazimuthal l,n
modeamplitude,definedasP ∝ H Γ ,whiletheheightH
ordermwasparameterizedwithanasymmetricLorentzianpro- l,n l,n l,n l,n
ofagivenmultiplet(l,n)isdefinedasthesumoftheheightsof
file(Nigam&Kosovichev,1998),as
itsm-components,as:
Ll,m,n(ν)= Hl,m,n(1+α1l,n+xl,xm2,n)2+α2l,n, (1) Hl,n =m=+lHl,m,n. (5)
l,m,n mX=−l
where xl,m,n = 2(ν−νl,m,n)/Γl,n, and νl,m,n, Γl,n, and Hl,m,n rep- Since Γl,n is assumed to be a smoothly varying function also
resentthe modefrequency,the linewidth,and the heightof the of the frequency, we could directly use the ratios of the mode
spectraldensity,respectively.Thepeakasymmetryisdescribed heights Hl,n to measure the visibility. We decided to use the
by the parameterα . Therotationis treatedusing a first-order mode powers instead because, as demonstrated by Ballotetal.
l,n
(2008),theycanbeaccuratelymeasuredevenwhentheindivid-
1 http://sci.esa.int/plato ualm-componentswithinagivenmultipletstartblendingaround
2
D.Salabertetal.:ModevisibilitiesinSun-as-a-starhelioseismicobservations
3 3
Raw Corrected
2.5 2.5
2 2
0 0
P P
/ 1.5 / 1.5

l l
P P
1 1
0.5 0.5
0 0
1500 2000 2500 3000 3500 4000 1500 2000 2500 3000 3500 4000
Frequency (µHz) Frequency (µHz)
Fig.1.Raw(left)andcorrected(right)modevisibilities(V/V )2ofl=1(bluedots),l=2(redsquares),andl=3(blackdiamonds)
l 0
relativetol=0asafunctionoffrequencyinthecaseoftheradialvelocityGOLF(solidlines)andintensityVIRGO/SPM(dashed
lines)observations.
3100µHzbecauseoftheirincreasinglinewidthswithfrequency.
Althoughconsistentvaluesareobtainedbetweenthetwodefini- 2
GOLF
tions,thecomputationofthevisibilitiesdefinedastheratiosof 1.8 VIRGO/SPM
themodeheightsislimitedtofrequencieslowerthan3100µHz,
1.6
as shown by Salabertetal. (2011). The l = 1, 2, and 3 mode
visibilities calculated as in Eqs. 2, 3, and 4 are represented in 1.4
Fig.1asafunctionoffrequency,forbothradialvelocityGOLF 2) 1.2
and intensity VIRGO/SPM (i.e., the mean power spectrum of V0
1
thethreeSPMsonVIRGO)observations.Theserawmodevis- /

ibilities (left panel of Fig. 1) vary with frequency – especially Vl0.8
for the l = 1 mode – that is mainly due to the variation in the (
0.6
mode heights and linewidths with frequency. The heights and
linewidths vary sensitively with frequency over half the large 0.4
frequencyseparation(∼67µHz),i.e.,theseparationinfrequency 0.2
betweenthe l = 0 andl = 1modesofthe same radialordern.
Thus, the visibilities are biased and to take account of this ef- 00 1 2 3
fect, this frequencydependencewas corrected by interpolating Angular degree l
with a spline function the powers P of the l = 1, 2, and 3
l,n Fig.2. Mode visibilities (V/V )2 averagedover frequencyas a
modestothefrequenciesofthel=0mode.Thecorrectedmode l 0
functionofangulardegreelfortheradialvelocityGOLF(black
visibilities are represented in the right panel of Fig. 1. As the
diamonds, solid line) and intensity VIRGO/SPM (red squares,
correctedmodevisibilitiescalculatedusingthemodepowersdo
dashedline)measurements.
notdisplaysignificantvariationswith frequency,we cansafely
averagethemoverfrequencytoobtainestimatesofthetrueval-
ues.Wenotethatweusetheexpressionmodevisibilityhereto blue-wingconfiguration,thel=3modebeingforinstancetwice
referto thecorrectedmodevisibility.Figure2 showsthemode asvisible duringthered-wingperiodthanin the blue-wingpe-
visibilities(V1/V0)2, (V2/V0)2, and(V3/V0)2 averagedoverfre- riod. The mode visibilities are shown to decrease as the wave-
quency,using the errorsas weights,measuredfromGOLFand lengthoftheVIRGO/SPMchannelsincreases.
VIRGO/SPM observations as a function of the angular degree
l. The represented values are tabulated in Table 1. The GOLF
4. Heightratiosbetweenm-components
visibilities were averaged between 1800 and 3600 µHz for all
l values, and the VIRGO/SPM visibilities from 2200 to 3600
The heightratios β between the visible m-componentsof the
l,n
µHz for the l = 1 and 2 modes (and from 2500 µHz for the l = 2 and l = 3 multiplets in Sun-as-a-star observations are
l = 3 mode). The lower limits are defined by the SNR of the
defined,respectively,as
modes, the intensity measurements such as VIRGO/SPM hav-
inga highernoiselevelatlowfrequencyandless sensitivityto β2,n = Hl=2,m=0,n/Hl=2,m=±2,n, (6)
the modes. The mode visibilities in the blue-wingand the red-
and
wingGOLFconfigurations,aswellasthevisibilitiesmeasured
separately in the blue, green, and red VIRGO/SPM channels β = H /H . (7)
3,n l=3,m=±1,n l=3,m=±3,n
are also given in Table 1. Significant differences are observed
As the linewidths are forced to be the same between m-
depending on the observing wing configuration of GOLF and
componentsofagivenmultiplet(l,n)inthepeak-fittingproce-
the VIRGO/SPM channels analyzed. The mode visibilities are
dure,theratiosthusmeasuredusingtheheightsH areiden-
greaterduringtheGOLFred-wingconfigurationthanduringthe l,m,n
ticaltotheonescalculatedusingthepowersP .Theseratios
l,m,n
3
D.Salabertetal.:ModevisibilitiesinSun-as-a-starhelioseismicobservations
Table1.ModevisibilitiesintheradialvelocityGOLFandintensityVIRGO/SPMmeasurements.
(V/V )2 GOLF GOLF GOLF VIRGO/SPM VIRGO/SPM VIRGO/SPM VIRGO/SPM
l 0
Bluewing Redwing Blue Green Red
(V /V )2 1.730±0.017 1.707±0.023 1.752±0.028 1.547±0.016 1.578±0.016 1.540±0.016 1.354±0.014
1 0
(V /V )2 0.884±0.012 0.827±0.017 1.016±0.022 0.655±0.012 0.696±0.012 0.632±0.011 0.465±0.008
2 0
(V /V )2 0.167±0.002 0.138±0.003 0.259±0.006 0.109±0.003 0.122±0.003 0.094±0.002 0.063±0.002
3 0
Notes.Themodevisibilitiesintheblue-wingandthered-wingGOLFconfigurationsarealsogiven,aswellasthevisibilitiesmeasuredseparately
intheblue,green,andredVIRGO/SPMchannels.
Table3. Modeledvisibilitiesandm-heightratioscomputedfor
1.5
thethreeVIRGO/SPMchannels.
1
β2
0.5
Modevisibility& Blue Green Red
m-heightratio (403µm) (501µm) (863µm)
0
1500 2000 Freque2n5c0y0 (µHz) 3000 3500 (V1/V0)2 1.60 1.54 1.45
(V /V )2 0.66 0.57 0.45
1 2 0
(V /V )2 0.06 0.03 0.01
3 0
β 0.667
2
β30.5 β3 0.600
Notes.Them-heightratiosdonotdependonthechannel.
0
1500 2000 2500 3000 3500
Frequency (µHz)
Fig.3. m-height ratios βl of the l = 2 (top) and l = 3 modes where
(bottom) as a function of frequency extracted from the GOLF
1
radialvelocitymeasurements.
V = 2π(2l+1) W(µ)P(µ)µdµ, (9)
l Z l
p 0
Table 2.Heightratiosβ betweenthe m-componentsofthe l =
l and
2 and l = 3 modes measured in the radial velocity GOLF and
intensityVIRGO/SPMobservations. α2 = (l−|m|)![P|m|(cosi)]2, (10)
l,m (l+|m|)! l
β GOLF VIRGO/SPM
l where P denotethe Legendrepolynomials,Pm are the associ-
β 0.634±0.033 0.751±0.059 l l
2 ated Legendrefunctions,and i is the inclinationof the rotation
β 0.400±0.020 0.633±0.065
3 axis relative to the line of sight, which is i = 90◦ for the Sun
toaverygoodapproximation.ThefunctionW(µ)istheweight-
ingfunction,whichdependsonlyon the distanceto thelimb µ
β are represented in Fig. 3 as a function of frequency in the
l,n
case oftheGOLFobservations.Somemarginalvariationswith classicallydefinedasthecosineoftheangle MCO,where M is
frequencycan be seen in Fig. 3, but we assume that the height theconsideredpointatthesolarsurface,CthecenteroftheSun,
d
ratiosareindependentoffrequency,andthatanaverageoverthe and O the observer. Within this approximation,and by assum-
spectrumwillyieldanestimate ofthetruevalue.Them-height ing that there is an equipartition of the energy between modes
ratiosofthel=2andl=3multipletsaveragedoverfrequency, ofclosefrequencyandthatthemodewidthdependsonlyonthe
usingthe errorsasweights,aregiveninTable2 forGOLFand frequency,wefindthat
VIRGO/SPMobservations.Nosignificantdifferencesarefound
H/H = P/P =(V/V )2, and (11)
inthem-heightratiosbetweentheGOLFblueandredobserving l 0 l 0 l 0
wings,andalsobetweenthethreeVIRGO/SPMchannels. P /P =H /H =(α /α )2. (12)
l,m′ l,m l,m′ l,m l,m′ l,m
For photometric measurements such as VIRGO/SPM observa-
5. Modelingvisibilities
tions, the initial assumption – W depends on µ only – is veri-
5.1.Photometricobservations fiedbecausethecontributiondependsmainlyonthelimbdark-
ening. As a consequence, the amplitude ratios between the m-
As previously mentioned, the apparent amplitude value of a components are purely geometrical effects, and the mode visi-
modedependsonthewaythatthecontributionsofeachelement bilityV iscomputedbyassumingtheweightingfunctiontobe
l
ofthesolardisktothetotalsignalareaveraged.Whenthecontri- W(µ) = I (µ)/I(0), the limb-darkening profile at the observed
λ
butionofasolar-diskelementtothetotalfluxdependsonlyonits wavelength λ. We computed the expected visibility by consid-
distancetothelimb,themodevisibilityVlandthem-heightratio ering the spectral bands of the three VIRGO/SPM channels.
aredecoupled(e.g.,Gizon&Solanki,2003;Ballotetal.,2006). We used the limb-darkening law from Neckel&Labs (1994).
TheobservedamplitudeA(obs)ofamode(n,l,m)islinkedtothe ResultsaregiveninTable3.
n,l,m
intrinsicamplitudeA bymeansoftherelation We comparethese predictionsto observations(cf.Table1).
n,l,m
First, the variationswith color are correctlyreproducedfor the
A(obs) =α VA , (8) different modes. The computations recover the observations to
n,l,m l,m l n,l,m
4
D.Salabertetal.:ModevisibilitiesinSun-as-a-starhelioseismicobservations
withinafewpercentaccuracy,exceptforthel=3modes,which Table 4. Modeledvisibilities and m-heightratios in radial ve-
are systematically higher than expected. Nevertheless, we note locityforGOLFobservations.
that, even for l = 1 and 2 modes, half of the predictions have
discrepancieswith the observationsthat are largerthan 3 σ. In Modevisibility& GOLF GOLF
addition,the error barsin Table 1 are purelystatistical, and do m-heightratio Bluewing Redwing
not take into accountmethodologicalor systematic errors. The (V /V )2 1.84 1.86
1 0
errorbarsmightthenbeunderestimated.Thereareseveralpos- (V /V )2 1.08 1.13
2 0
sible explanationsofthese differences.First, thesystematically (V3/V0)2 0.27 0.30
larger amplitudes observed for the l = 3 modes could be due β2 0.59 0.58
to the intrinsic amplitudes being larger than expected. It can- β3 0.43 0.40
notbeexcluded,butishardlyjustifiablefromatheoreticalpoint
of view today. Secondly, the differences might come from the
modeledweightingfunctionbeingunabletoperfectlyreproduce
theconsideredwingsandisingoodagreementwiththeobserved
therealone.Thelimb-darkeninglawcomingdirectlyfromob-
ratio (see Table 2). For the visibility, the globaltrend is recov-
servationsisreliable.Byidentifyingthelimb-darkeningprofile
ered, but the difference between blue and red wings for GOLF
with the weightingfunction,we assume that the relative inten-
islargerthanexpected:theobservedvalueforthe bluewingis
sity fluctuations δI/I induced by a mode are the same every-
significantlysmallerthanexpected.
whereonthevisiblesolardisk.However,thelimbisdarkerbe-
We also performed complementary computations to take
cause the light comes mainly from higher layers of the photo-
into account the horizontal motions of the modes. Since
sphere.With an adiabaticapproximation,andknowingthatthe
the contribution of the horizontal motions increases when
temperatureT variesslowlyintheatmosphere,itisshownthat
the frequency decreases and the mode degree increases (e.g.
δI/I = 4δT/T ∝ ξ , where ξ is the displacement. Since the
r r Christensen-Dalsgaard, 2003), it generates a small variation
modeisevanescentintheatmosphere,
in the mode visibility with frequency by a few percent.
ρξ2 ∝exp − 1−ω2/ω2 1/2h/H , (13) Nevertheless,itisnegligiblecomparedtotheobservationerrors.
r c
(cid:20) (cid:16) (cid:17) (cid:21)
where h is the height in the atmosphere, H is the pressure
6. Temporalevolutionofthemodevisibilities
scale height, and ω is the cut-off frequency (see for instance
c
Christensen-Dalsgaard, 2003). As the photosphere is a thin As for the p-mode frequencies (e.g., Woodard&Noyes, 1985;
layer, we generally assume that ξ does not vary too much. Palle´ etal., 1989b; Chaplinetal., 2007; Salabertetal., 2009),
r
Nevertheless,becauseofthe strongdropindensityρ,itispos- the acoustic power and damping parameters of the low-degree
siblynotexactlytrue.Moreover,non-adiabiticeffectsareproba- acousticmodeshavebeenfoundtovarywithsolaractivity(e.g.,
blynotnegligible.SinceV modesappeartoberelativelymore Palle´ etal., 1990; Chaplinetal., 2000; Salabertetal., 2003;
l=3
sensitivetotheshapeofW(µ)closetothelimb,theobserveddis- Simonielloetal., 2010). However, it has never been demon-
crepenciescould be due to the approximatedphysicsof modes stratedwhetherthemodevisibilitiesdependornotontheSun’s
intheatmosphere. magnetic activity. To verify this, we analyzed the GOLF and
Concerningthem-heightratio,thevalueareclosetotheob- VIRGO/SPM observations by dividing the original time series
servedones(cf.Table2),evenif,forl = 2modes,theobserved into contiguous 365-day and 91.25-day non-independent sub-
centralcomponentismarginallylargerthanpredicted(∼ 1.4σ). series (shifted by 1/4) and fitting the power spectrum of each
Sincethistermisnormallyinsensitivetothephysicsofthesolar subseries. For the SoHO vacation in 1998, the subseries with
atmosphere,discrepenciescanbeduetodifferencesintheintrin- duty cycles less than 50% were removed from the analysis.
sic amplitudes of the axisymmetric component. Nevertheless, Themodeparameterswereextractedinthesamemannerasex-
GOLF does not have similar systematics (see next section), plainedinSec.2,withtheonlydifferencethattheheightratios
meaningthatthesedifferencesarenotconclusive. betweenthevisiblem-componentsofthel=2and3modeswere
fixedtothevaluesdeterminedinSec.4andgiveninTable2,in
ordertostabilizethefittingprocedureoftheseshorttimeseries.
5.2.GOLFobservations
The mode visibilities obtained as in Eqs. 2, 3, and 4 for each
ForGOLFobservations,theweightingfunctiondoesnotdepend powerspectrumwerecorrectedfortheirfrequencydependences
onlyonµ, andwe havetoperformcompletecomputationstak- byinterpolatingthe powersof thel = 1,2,and3 modesto the
ing into account the instrumental response, which depends on l = 0frequenciesasdescribedinSec.3,andtheirweightedav-
thepositionsintheNaD andD linesthatareobserved.Thus, eragesoverfrequencyobtained.
1 2
mode visibilities change according to the observed wing (blue
orred),butalsowiththeepochoftheobservations:accordingto
6.1.The365-daysubseries
the position in the SoHO orbit, the Doppler shift change and
the lines are probed at different heights in the wings. Details Figure4showsthetemporalevolutionofthel=1,2,and3mode
oftheinstrumentalresponsecanbefoundineitherGarciaetal. visibilitiesmeasuredinthenon-independent365-dayGOLFand
(1998)orUlrichetal.(2000).Wenumericallycomputedthein- VIRGO/SPM power spectra. The mode visibilities do not vary
tegrals over the whole disk for the different modes through a with solar activity, and remain constant overall over the years,
simulatedinstrumentalresponsetakingintoaccountasimplified within the uncertainties. However, a clear dependence on the
limb-darkening of the NaD lines, the solar rotation and differ- GOLF wing configurations,as indicatedby the verticaldashed
entialrotation,andthegravitationalreddening.Wefirstassume lines,canbediscerned,mainlyinthel=3mode.TheGOLFvis-
thatthemodesinducepurelyverticalmotions. ibilitiesincreaseduringthered-wingobservingperiodbetween
TheresultsofthesecomputationsarelistedinTable4.First, the years ∼ 1998 and ∼ 2002, which is consistent with the re-
we note that the m-height ratio does not depend noticeably on sultsinTable1.Ontheotherhand,theVIRGO/SPMvisibilities
5
D.Salabertetal.:ModevisibilitiesinSun-as-a-starhelioseismicobservations
donotexhibitthesechanges.However,theVIRGO/SPMvisibil- valuesthatwethencomparedtotheoreticalvaluescomputedus-
ityofthel = 1modehasarelativelylargeincreasecenteredon ing the instrumental responses of GOLF and VIRGO/SPM by
the year2000,whichis also presentin the GOLFdata in addi- taking into account limb-darkening functions. In general, the
tiontothelargervisibilitiesduringthered-wingoperation.The predicted values qualitatively agree with the observations. We
VIRGO/SPMl = 2modevisibilityincreasesoverthesamepe- also showed that the mode visibility is a function of the wing
riodoftimeappearingtofollowtheGOLFvisibility.Asthelevel configurationofGOLF,thel=1,2,and3modesbeingmorevis-
ofprecisionofthedata,theVIRGO/SPMl=3visibilitiesdonot ible duringthe red-wingperiodthanduringthe blue-wingone.
showanychangescomparabletotheonesobservedinl=1and The mode visibilities were also shown to depend on the wave-
2.TheweightedmeanvaluesovertimeoftheGOLFvisibilities length of the VIRGO/SPMs, as predicted. However, some dis-
are 1.747±0.011,0.862±0.008,and0.171±0.003forl = 1, crepancies between observationsand theory remain, especially
2,and3respectively.InthecaseofVIRGO/SPM,thevaluesare forthel=3modes.Thesedifferencesimplythatthelimbdark-
1.605±0.012, 0.641±0.008, and 0.089±0.004, respectively. ening alone is insufficient to explain the observed visibilities.
TheseresultsareinagreementwithTable1.Similarconclusions This has to be analyzed in depth to be able to make reliable
areobtainedwiththeindividualVIRGO/SPMchannels. predictions for other stars. Nevertheless, we note that the dis-
crepenciesidentifiedherearesmallerthantheerrorbarsinmode
amplitudesobtainedwithafewmonthsofCoRoTorKeplerob-
6.2.The91.25-daysubseries
servations.
Figure5showsthetemporalevolutionofthel = 1and2mode By analyzing 365-day and 91.25-day GOLF and
VIRGO/SPM subseries, we also demonstrated for the first
visibilities measured in the non-independent 91.25-day GOLF
andVIRGO/SPMpowerspectra.Wedonotshowtheresultsfor time that the mode visibilities are constant in time and do
the l = 3 mode as they have a much larger scatter in as short not vary with the solar magnetic activity. However, the mode
visibilities exhibit short-term temporal variations of about
time series as these, because of the mode’s smaller visibility.
six months that are common to the GOLF and VIRGO/SPM
Although,Fig.5confirmsthatthemodevisibilitiesareindepen-
observations. Some of these fluctuations increase significantly,
dent of the solar activity, it shows fluctuations over short peri-
andasignificantincreaseinthel=1modevisibilitywasfound
ods of time, of about 6 months. The same analysis performed
using different lengths of subseries and amounts of shift (1/2, attheendoftheyear1999.Theoriginofboththeseshort-period
1/3,and1/4)confirmsthatthissix-monthperiodis notan arti- fluctuations and this bump at the end of 1999 is unclear but
might be related to the stochastic excitation of the individual
factofthemethodology.Moreover,theanalysisoftheartificial
modesorthevariationsintheacousticpoweranddampingrate
SolarFLAGdata(Chaplinetal.,2006)doesnotdetectthisperi-
ofthesolaroscillationswithtime.
odicity.However,itsoriginremainsunclear.Someofthesefluc-
Inthefuture,theobservationsfromtheradialvelocitySONG
tuations have significant increases, common to the GOLF and
VIRGO/SPMmeasurements.Theenhancementofthel=1vis- andphotometricPLATOinstrumentswillprovideinvaluablein-
putstomodelthestellaratmospheres.Finally,thesenewlycon-
ibility observed in the 365-day subseries could be due to a se-
strained l = 1, 2, and 3 mode visibilities and m-height ratios
ries of higher-than-averagepeaks between 1999 and 2001 that
provided in this paper should be used as inputs to any investi-
are observed in the 91.25-dayvisibilities. Inspection of the in-
gationof the Sun usingthe radialvelocityGOLF andintensity
dividual mode powers seems to indicate a significant reduc-
tion in the l = 0 power during that period. Similar results are VIRGO/SPMobservations.
obtained with the individual VIRGO/SPM channels. Thus, to
Acknowledgements. TheauthorswishtothankCatherineRenaudandAntonio
check whether the origin of this increase in the mode visibil-
Jime´nez for the calibration and preparation of the GOLF and VIRGO/SPM
ities is genuine and unrelated to intrinsic changes in SoHO, datasets.TheGOLFandVIRGO/SPMinstrumentsonboardSoHOareacoop-
we performed in Appendix A a similar analysis using the in- erativeeffortofmanyindividuals,towhomweareindebted.SoHOisaproject
dependent observations collected by the ground-based, multi- ofaninternational collaboration between ESAand NASA.DS acknowledges
thesupportofthegrantPNAyA2007-62650fromtheSpanishNationalResearch
site network BirminghamSolar Oscillations Network (BiSON;
Plan.ThisworkwassupportedbytheCNES/GOLFgrantatSAp/CEA-Saclay.
Elsworthetal., 1995) and Global Oscillation Network Group
This work utilizes data obtained by the Global Oscillation Network Group
(GONG;Harveyetal.,1996).WeshowinAppendixBthatiden- (GONG) program, managed by the National Solar Observatory, which is op-
tical temporal fluctuations seen in Figs. 4 and 5, including the eratedbyAURA,Inc.underacooperativeagreementwiththeNationalScience
increase in the l = 1 visibility in 2000, are also present in the Foundation. Thedatawereacquired byinstruments operated bytheBigBear
Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory,
BiSONandGONGobservations.
Udaipur Solar Observatory, Instituto de Astrof´ısica de Canarias, and Cerro
TololoInteramericanObservatory.Thisworkalsoutilizesdatacollectedbythe
BirminghamSolarOscillations Network(BiSON),whichisfundedbytheUK
ScienceTechnologyandFacilitiesCouncil(STFC).Wethankthemembersofthe
7. Conclusions
BiSONteam,colleagues atthehostinstitutes,andallothers,pastandpresent,
whohavebeenassociatedwithBiSON.
We have analyzed more than 5000 days of high-quality helio-
seismicobservationscollectedbythespace-based,Sun-as-a-star,
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D.Salabertetal.:ModevisibilitiesinSun-as-a-starhelioseismicobservations

2
1.5
0
P

/

l
P 1
0.5
0

1996 1998 2000 2002 2004 2006 2008 2010
Date (year)
Fig.4. Temporalevolutionofthe l = 1 (blue),l = 2 (red),andl = 3 (black)modevisibilities measuredfromthe analysisof the
non-independent365-daypowerspectraofbothradialvelocityGOLF(opencircles)andintensityVIRGO/SPM(dots)observations.
Thevertical,dashedlines(fromlefttoright)separatetheperiodsofblue,red,andblue-wingGOLFoperations,respectively.
3
0
P 2

/

1
P 1
0
1998 2000 2002 2004 2006 2008 2010
0
P
1
/

2
P
0
1996 1998 2000 2002 2004 2006 2008 2010
Date (year)
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7
D.Salabertetal.:ModevisibilitiesinSun-as-a-starhelioseismicobservations
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1583 BiSON andGONGintegratedl = 1,2,and3modevisibilities,
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thepeak-fittingitself,suchasthesizeofthefittingwindows,or
thesmallmultipletfrequencyasymmetryatl = 2and3,which
AppendixA: ModevisibilitiesinBiSONandGONG
is not taken into account in our analysis. In the case of the in-
integratedobservations
tegrated GONG time series, no previously measured m-height
This analysis presented in these appendices was motivated by ratioswerefoundinthepublishedlitterature.Wenotethatsince
the observation of an increase in the temporal evolution of the the visibility of the l = 3 mode is close to being zero in the
visibilities, mainly for the l = 1 mode, at the end of the year GONGintegrateddata,them-heightratioforl=3couldnotbe
1999, seen in the Sun-as-a-star GOLF and VIRGO/SPM in- measuredaccurately.
struments onboard the SoHO spacecraft. We wish here and in
Appendix B to establish whether the origin of these variations
might come from SoHO itself and not from the Sun. We de- 2 Respective datasets are available from
cided then to perform the same analysis using two indepen- http://bison.ph.bham.ac.uk/data.php and
denthelioseismicdatasetscollectedbytheground-based,multi- http://gong.nso.edu/data/.
8
D.Salabertetal.:ModevisibilitiesinSun-as-a-starhelioseismicobservations
3 3
BiSON GONG integrated
2.5 2.5
2 2
P0 P0
/ 1.5 / 1.5

Pl Pl
1 1
0.5 0.5
0 0
1500 2000 2500 3000 3500 4000 1500 2000 2500 3000 3500 4000
Frequency (µHz) Frequency (µHz)
Fig.A.1.Modevisibilitiesofl = 1 (bluedots),l = 2 (redsquares),andl = 3 (blackdiamonds)relativetol = 0as a functionof
frequencyextractedfromtheanalysisoftheBiSON(left)andGONGintegrated(right)observations.
Fig.A.2.Temporalvariationsinthel=1modevisibilitiesextractedfromtheanalysisofthenon-independent365-day(leftcolumn)
and91.25-day(rightcolumn)powerspectraoftheGOLF,VIRGO/SPM,GONG,andBiSON(fromtoptobottom)instruments.The
verticaldashedlinescorrespondto1999October10.
AppendixB: Temporalevolutionofthel= 1mode
visibilityinseveralSun-as-a-starhelioseismic
instruments
Wecomparethetemporalevolutionofthelow-degreemodevis-
ibilitiesofthesolaroscillationsmeasuredwiththespaced-based
9
D.Salabertetal.:ModevisibilitiesinSun-as-a-starhelioseismicobservations
Table A.2. Height ratios β between the m-components of the
l
l = 2 and l = 3 modesmeasuredin the BiSON and integrated
GONGtimeseries.
β BiSON GONG
l
β 0.625±0.032 0.882±0.045
2
β 0.445±0.020 —a
3
Notes. (a) Owing to the very small visibility of the l = 3 mode in
the GONG integrated data, the m-height ratio for l = 3 could not be
accuratelymeasured.
GOLFandVIRGO/SPMinstrumentsonboardtheSoHOspace-
craft,tothevisibilitiesmeasuredwiththeground-based,multi-
site BiSON, and GONG networks.The BiSON andGONG in-
tegrated time series were divided into contiguous 365-day and
91.25-daynon-independentsubseries (shifted by 1/4) and ana-
lyzed in the same manner as explained in Sec. 6. Figure A.2
showsthetemporalevolutionofthevisibilityofthel = 1mode
extractedfromtheanalysisofthenon-independent365-dayand
91.25-dayGOLF,VIRGO/SPM,BiSON,andGONGintegrated
powerspectra.Thebumpofthel = 1modevisibilityattheend
oftheyear1999observedinthe365-daysubseries(seeFig. 4)
is present in all the analyzed independent datasets. As already
mentioned in Sec. 6, this bump is most likely the result of a
seriesofhigher-than-averagepeaksbetween1999and2001ob-
servedinthe91.25-dayvisibilities.Moreover,similarshort-term
variationsofabout6monthscanbeobservedinthe91.25-days
subseries of the GOLF, VIRGO/SPM, BiSON, and GONG in-
struments,someofthemexhibitingsignificantandcommonin-
creases.Thesameanalysisperformedusingdifferentlengthsof
subseries and overlapping factors confirms that this six-month
periodicityisnotanartifactofthemethodology.
10