Inferring mode inertias in evolved solar-like stars

0.52 MB·

by  O. Benomar

Draftversion January22,2014
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
ASTEROSEISMOLOGY OF EVOLVED STARS WITH KEPLER:A NEW WAY TO CONSTRAIN STELLAR
INTERIORS USING MODE INERTIAS
O. Benomar1,6,7, K. Belkacem2, T.R. Bedding1,6, D. Stello1,6, M.P. Di Mauro4, R. Ventura5 , B. Mosser2, M.J.
Goupil2, R. Samadi2, R.A. Garcia3
Draft version January 22, 2014
ABSTRACT
4 Asteroseismology of evolved solar-like stars is experiencing a growing interest due to the wealth
1 of observational data from space-borne instruments such as the CoRoT and Kepler spacecraft. In
0 particular, the recent detection of mixed modes, which probe both the innermost and uppermost
2 layersofstars,pavesthewayforinferringthe internalstructureofstarsalongtheirevolutionthrough
n the subgiant and red giant phases. Mixed modes can also place stringent constraints on the physics
a of such stars and on their global properties (mass, age, etc...). Here, using two Kepler stars (KIC
J 4351319 and KIC 6442183), we demonstrate that measurements of mixed mode characteristics allow
us to estimate the mode inertias, providing a new and additional diagnostics on the mode trapping
1
2 andsubsequentlyonthe internalstructureofevolvedstars. We howeverstressthatthe accuracymay
be sensitive to non-adiabatic effects.

] Subject headings: stars: oscillations, stars: interiors, methods: data analysis
R
S
. 1. INTRODUCTION strain the evolutionary status of evolved low-mass stars
h (Bedding et al.2011;Mosser et al.2011a). Thestrength
Asteroseismology, with the help of the space-borne
p of mode mixing depends mainly on the trapping of the
missions such as CoRoT (e.g. Baglin et al. 2006a,b;
- modeandthereforeontheevanescentregionbetweenthe
o Michel et al. 2008) and Kepler (e.g. Borucki et al. 2010)
inner and outer resonant cavities of stars.
r gives us access to the internal structure of stars. This is
t Todate,asteroseismologyhasgenerallyusedthemode
s madepossiblethroughthedetectionofoscillationmodes
a propagating throughout the stars (Appourchaux et al. frequencies to probe stellar interiors. In this Letter, we
[ show that we can also use mode inertias. Theoretical
2008; Metcalfe et al. 2012; Gizon et al. 2013).
While formainsequencestarsweobserveonly accous- calculations (e.g. Dupret et al. 2009) show that inertias
1
of mixed modes vary with frequency in a way that is
v tic modes, subgiants and red giants also show a rich
related to the efficiency of the trapping, which changes
2 spectrumofso-calledmixedmodes (e.g.Beck et al.2011;
significantlyasthestarevolves. Moreover,Samadi et al.
5 Bedding et al.2011; Mosser et al.2011a;Benomar et al.
(2008)showedthatmodeinertiadependsstronglyontur-
1 2013). They have a dual nature, behaving as acoustic
bulent pressure. Hence, mode inertias can provide strin-
5 modesinthestellarenvelopeandasgravitymodesinthe
gent constraints on mode trapping, reveal stellar interi-
. core. They are thereforedetectable at the stellar surface
1 ors and potentially provide a new insight on the physics
while providing information on the innermost regions.
0 of stars. Furthermore, based on an asymptotic analysis,
This aspect of mixed modes has motivated extensive
4 Goupil et al. (2013) proposed a way to observationally
theoreticalwork (e.g. Dziembowski 1971; Scuflaire 1974;
1
derive the ratio between the inertia in the g cavity and
: Osaki 1975; Aizenman et al. 1977; Dziembowski et al.
v 2001; Christensen-Dalsgaard2004; Dupret et al. 2009). the total inertia of a mixed mode.
Xi Oscillations are very sensitive diagnostics that can be MotivatedbythelatestKepler observations,andbased
on the mode fitting results from Benomar et al. (2013),
used to constrainthe stellar structure and to give an in-
r we show that in conjunction with frequencies, mode in-
a sight to internal properties of stars, such as the extent
ertias could help us to infer precisely the cavities of os-
of the convective zone (Roxburgh 2009). In particular,
cillations. We describe in Section 2 a method for in-
mixed modes provide valuable information on the core
ferring mode inertias from the measurements of mode
structure which, for example, allows us to better con-
amplitudes and linewidths. In Section 3 we apply the
1Sydney Institute for Astronomy (SIfA), School of Physics, proceduretotwostarsobservedbyKepler, KIC6442183
UniversityofSydney, NSW2006,Australia and KIC 4351319,and compare these results with mod-
2LESIA,ObservatoiredeParis,CNRSUMR8109,Universit´e els. We present diagnostics of the stellar structure of
ParisDiderot,5placeJ.Janssen,92195Meudon,France
3Laboratoire AIM, CEA/DSM-CNRS-Universit´e Paris evolved stars, derived from the measurements of inertia
Diderot; IRFU/SAp, Centre de Saclay, 91191 Gif-sur-Yvette in Section 4.
Cedex, France
4 INAF-IAPSIstitutodiAstrofisicaePlanetologiaSpaziali 2. INFERENCEOFMODEINERTIARATIOS
ViadelFossodelCavaliere100,00133,Roma,Italy 2.1. Preliminary definitions
5INAF-Astrophyscial Observatory of Catania, Via S. Sofia
78,95123, Catania,Italy Inthis sectionweshowhowto useobservedmodeam-
6Stellar Astrophysics Centre, Department of Physics and plitudesandlinewidthstomeasuremodeinertiaforstars
Astronomy, Aarhus University, Ny Munkegade 120, DK-8000
showing mixed modes. Let us consider a stochastically-
AarhusC,Denmark
7Department of Astronomy, The University of Tokyo, driven mode. If this mode is resolved, i.e. if the dura-
113-033, Japan tion of the observations is significantly longer than the
2 Benomar et al.
mode lifetime, it has a Lorentzian profile in the power Equation(5)deservessomecomments. Firstly,itlinks
spectrum (e.g. Appourchaux et al. 2012). In contrast, the observables H and Γ with the quantities P and M,
if the duration of the observations is shorter than the which are only derived from modelling. Secondly, be-
mode lifetime, the mode will have an unresolved sinc- cause we require the frequencies ν and ν to be close
0 1
squared profile (such as expected for solar g modes, see to each other (say |ν −ν | < ∆ν/2), the shapes of the
0 1
Belkacem et al. 2009). Thus, it is only possible to mea- eigenfunctions in the uppermost layers are very similar.
sure linewidths for resolved modes and we restrict our As shown by Dupret et al. (2009), this implies that the
discussion to this case. driving are the same:
The height H of a mode is given by (e.g.
P M ≃P M . (6)
Libbrecht 1988; Chaplin et al. 1998; Baudin et al. 2005; 0 0 1 1
Chaplin et al. 2005; Belkacem et al. 2006)
Equation 6 means that, at similar frequencies and with
P similar mode shapes in the super-adiabatic layers, the
H = , (1) work done by the driving source on the modes is the
2η2M
same. Theoretical computations on evolved stars span-
where P is the excitation rate, η is the damping rate, ning from the early-subgiantphase to the top of the red
and M is the mode mass, which is proportional to the giantbranchshowthatthisapproximationisaccurateto
mode inertia I (e.g. Goldreich et al. 1994): better than one percent in the observedfrequency range
(nearν )butfailsatverylowfrequencies,wheremode
M= I , with I = M ξ~2 dm. (2) amplitumdaexsaretoosmallto be observed. Nevertheless,a
2 thorough investigation of the accuracy of Equation 6 is
ξ~(R) Z0 (cid:12) (cid:12) desirable.
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) Using Equation 2 and the fact that ξ~ (R) ≃
Here, ξ~(R), R(cid:12), dm,(cid:12)M are the displacement vector, the ℓ=0
(cid:12) (cid:12)
stellarradius,themasselementandthemassofthestar, ξ~ℓ=1(R) , we can consider that mode mas(cid:12)(cid:12)s is pro(cid:12)(cid:12)por-
respectively. (cid:12) (cid:12)
(cid:12)tionalto(cid:12)mode inertia. Equation6canbe thenbe recast
Power spectrum analysis mostly relies on fitting (cid:12) (cid:12)
Lorentzian profiles to the observed modes. The usual (cid:12)as (cid:12)
M I H Γ
form is 1 ≃ 1 ≃ 0 0 . (7)
H M I H Γ
L(ν)= , (3) 0 0 r 1 1
1+4(ν−ν )2/Γ2
c This shows that one can derive the ratio of mode in-
such that Γ = η/π is the full-width-at-half-maximum ertias (or mode masses) between a radial mode and a
(hereafter called linewidth) and ν is the central fre- neighboringdipole mixed-mode fromthe observedmode
c
quency. The observables needed to estimate the mode heights and linewidths. Using Equation 4, the inertia
inertia are the height H and the linewidth Γ. Note that ratio can also be rewritten in terms of mode amplitudes
they are also related to the mode amplitude: and linewidths,
I A Γ
A2 =πHΓ /2. (4) 1 ≃ 0 0 . (8)
I A Γ
0 1r 1
2.2. Relation between mode inertia, amplitude and
Note that observed height and linewidth are anti-
linewidth
correlated (e.g. Baudin et al. 2005), which makes it dif-
We recently used Kepler data to accurately mea- ficult to reliably measure inertia using Equation 7 when
sure the Lorentzian profiles of oscillation modes using an algorithm for which the result could be sensi-
(Benomar et al. 2013). To go a step further, and guided tivetothe parametrisation(suchasMLE).Onthe other
by those results, we now consider the comparisonof two hand,amplitudesandlinewidthsareweaklycorrelatedso
neighboring modes: that using Equation 8 will provide more reliable results
than Equation 7.
1. a radial mode (ℓ=0), denoted by the subscript 0,
and 3. OBSERVATIONALDETERMINATIONOFTHEMODE
INERTIARATIOS
2. adipolemode(ℓ=1)thatexhibitsamixedcharac-
ter, i.e., pressure-like behavior in the envelope and 3.1. Method
gravity-like behavior in the innermost layers. The Here, we show how to infer mode-inertia ratios from
subscript 1 will be used for this mode. Kepler observations. The power spectra were prepared
usingmethodsdescribedbyGarc´ıa et al.(2011). Wewill
We choose modes that are resolved, allowing us to ap-
first consider the star KIC 6442183,which was analyzed
ply Equation 1 for computing their heights and assume
by Benomar et al. (2013) using Quarters 5 to 7 (nine
modes of neighbor frequency (ν and ν ).
0 1 months). It was selected due to its high signal-to-noise
From Equation 1 and the fact that Γ=η/π, the ratio
oscillation spectrum and because it does not show large
of mode heights can be written
rotationalsplittingsthatwouldresultinamorecomplex
H P M Γ 2 power spectrum. Its global seismic characteristics are
0 0 1 1
= . (5) given in Table 1, and show KIC 6442183 to be an early
H P M Γ
1 (cid:18) 1(cid:19)(cid:18) 0(cid:19)(cid:18) 0(cid:19) subgiant star of about one solar mass.
Note that this relation does not account for mode visi- We also considered KIC 4351319, a red giant star al-
bilities. These will be considered in Section 3.1. ready analyzed by Di Mauro et al. (2011) using Q3.1
Inferring mode inertias in evolved solar-like stars 3
Table 1
Mainparameters forKIC6442183 andKIC4351319. ModelAisforKIC6442183. ModelBisforKIC4351319andusesthesame
physicsasmodelA.References: (a)Molenda-Zakowiczetal.(2013),(b)DiMauroetal.(2011),(c)Benomaretal.(2013),(d)Thisstudy
KIC6442183 ModelA(d) KIC4351319 ModelB(b)
Teff(K) 5738±62(a) 5632 4700±50(b) 4752
logg 4.14±0.10(a) 4.01 3.30±0.10(b) 3.5
[Fe/H] −0.120±0.050(a) −0.036 0.23±0.15(b) 0.20
∆ν (µHz) 65.07±0.09(c) 65.21 24.38±0.1(d) 24.6
νmax (µHz) 1160±4(c) 380.7±4(d)
∆Π1 (sec) 325±18(c) 359 97.2(d) 97.8
M (M⊙) ∼0.94(c) 1.02 ∼1.26(d) 1.32
R(R⊙) ∼1.60(c) 1.65 ∼3.40(d) 3.39
V2 1.52 1.55
ℓ=1
data (one month). For this work, we re-analyzed the as well as their ´echelle diagrams. Vertical lines are half-
power spectrum using Q3 to Q14 data (three years) waybetweenconsecutiveℓ=0p-modefrequencies(ν )
p,0
andadoptingaMarkovChainMonte-Carlosampler(e.g. and so give good estimates for the ℓ = 1 pure p-mode
Benomar et al. 2009; Handberg & Campante 2011), al- frequencies (ν ).
p,1
lowingprecisemeasurementsofallmodeparameters. Ta- Forthe subgiant,the mixedmodesclosesttothe ℓ=1
ble 1 summarizes the global characteristics of this star. p-mode frequencies have the lowest mode-inertia ratios
Notethatthemeasuredamplitudesarenotbolometric (I /I remains close to unity), while for mixed modes
1 0
and must therefore be corrected for the Kepler spectral that depart significantly from the p modes, I /I in-
1 0
response. Moreover, geometric and limb darkening ef- creases up to about two.
fects must be considered. Consequently, Equation 8 is Intheredgiant,thedensityofgmodesisgreaterthan
modified to become the density of the p modes, allowing us to better trace
variations of the mode inertia. I /I ranges between 1.5
I A Γ 1 0
1 ≃V 0 0 , (9) and 10 because, in this regime, all modes are strongly
1
I0 A1 rΓ1 mixed (none are pure p modes).
These behaviorsaredue to the difference inevolution-
wherethecoefficientV (hereaftercalledvisibility)isde-
1
ary stage. As a star evolves from subgiant to red giant,
finedasV2 =Hth/HthwithHthandHthbeingtheoreti-
1 1 0 0 1 thedensitycontrastbetweenthecoreandtheenvelopin-
calheightsforℓ=0andℓ=1modes. TocomputeV ,we
1 creases,and so do the mode-inertia ratios (Dupret et al.
use the results of Ballot et al. (2011), which considered
2009). This is also consistent with theoretical compu-
the spectral response of the Kepler spacecraft and in-
tations of mode inertias in evolved solar-like stars (e.g.
clude limb darkening and geometric effects. Ballot et al.
Dupret et al. 2009; Montalb´an et al. 2010).
(2011), based on the work of Michel et al. (2009), com-
puted a grid of values for V as a function of T , log g 4. COMPARISONWITHSTELLARMODELS
1 eff
and[Fe/H].Byinterpolatingthisgrid,weobtainthevis- In this section, we show how inertias can provide new
ibilities for KIC 6442183 and KIC 4351319 reported in diagnostics on the internal structures of evolved stars.
Table 1 (see Mosser et al. 2012a, for a discussion about To this end, we first compare the observed and modeled
observed visibilities). valuesofI /I . Wethendiscussthepotentialdiagnostics
1 0
Using Equation 9, together with the mode linewidths on the stars internal structure.
and amplitudes measured from Kepler observations, it
is possible to derive the mode-inertia ratios. However, 4.1. Comparison between observed and modeled
one must also consider how to compute the ratio of mode-inertia ratios
amplitudes A0/A1 and of linewidths Γ0/Γ1. To de- The structure models of KIC 6442183 and KIC
termine these ratios, we first interpolated A0(ν0) and 4351319 were computed using the ASTEC evolution
Γ0(ν0) at ν1 between consecutive radial modes. This code (Christensen-Dalsgaard 2008b) and the associated
relies on the fact that amplitudes and linewidths of eigenfrequenciesandeigenfunctionswerecomputedusing
radial modes vary smoothly from mode to mode (e.g. ADIPLS oscilation code (Christensen-Dalsgaard2008a).
Christensen-Dalsgaard2004; Dupret et al. 2009). The input physics used for the modelling is described in
To robustly estimate the uncertainties,the calculation detail by Di Mauro et al. (2011). The selected models
ofthemode-inertiaratioinvolvedtheprobabilitydensity well reproduce spectroscopic constraints (effective tem-
functions of the parameters from the MCMC process. perature and surface gravity), and the set of all the ob-
The median of the probability density function defined served frequencies, as well as the large (∆ν) and small
themostlikelysolution,whileconfidenceintervalsat±σ separations (δ ). For KIC 4351319, we used the evolu-
02
were computed using the cumulative distribution func- tionary model previously computed by Di Mauro et al.
tion and used as the lower and upper uncertainties. (2011). Theglobalparametersofthemodelsarereported
in Table 1.
3.2. Results
InFig.2, the ratioI /I derivedfromthe modelling is
1 0
In Fig. 1 we show the observedmode-inertia ratios for comparedwithobservationsandweseeverygoodagree-
KIC6442183andKIC4351319asafunctionoffrequency, ment for most modes. This strengthens the validity of
4 Benomar et al.
Figure 1. Inertiaratiobetweenℓ=1andℓ=0modesmeasuredforKIC6442183(a)andKIC4351319(b)againstfrequency. Approximate
positionsofthepureℓ=1pmodes(νp,1)areindicatedbyverticaldashedlines. E´chellediagramsofobservedfrequenciesforKIC6442183
(c)andKIC4351319 (d)withℓ=0,1,2,3modesreportedinblack, blue,red,browndiamondsrespectively. Warmercolorsindicatehigh
ratio,coldcolorsindicatelowerratio.
our method to derive the ratios I /I from the observed frequency.
1 0
mode amplitudes and linewidths, because the models Basedontheasymptoticrelationforpmode,itiscon-
werenot chosentoreproducethe modeinertiasbutonly ventional to define the small separation δν = −ν +
01 p,1
to match mode frequencies. It also suggests that a more (ν +0.5∆ν). As shownin Fig.1, ℓ=1 modes with the
p,0
accuratemodelofthese starcouldbe derivedusingboth lowest inertia behave almost as pure p-modes. There-
frequencies and inertias. fore, the frequency position of the inertia minimum al-
lowsustoestimateδν . ForKIC4351319,wemeasured
01
4.1.1. Small separation and period spacing diagnostics using δν01 =−0.69±0.30 µHz by fitting a second order poly-
collapsed ´echelle diagrams nomialwithinthe range[−∆ν/4,∆ν/4]onthe collapsed
´echelle. Forcomparison,thesameapproachonmodelfre-
In the following, we focus on the red giant KIC
quenciesgivesδν =−0.53±0.24µHz (cf. Fig.3). Note
4351319,sincethevariationsinI /I aregreater. E´chelle 01
1 0 thatnegativevaluesarefoundinmostevolvedstars(e.g.
diagrams, either in frequency or in period help to visu-
Bedding et al. 2010; Mosser et al. 2011b; Corsaro et al.
alise these variations and provide diagnostics on mixed
2012).
modes (see Fig. 3 and Fig. 4).
The period-´echelle diagram (Fig.4) presents the fre-
Frequency-´echelle diagrams are produced by repre- quency ratios as a function of the period ν−1 mod ∆Π ,
senting the frequencies as a function of the difference 1 1
where ∆Π is the average period spacing of the ℓ = 1
ν −(ν +0.5∆ν), ∆ν being measured using the radial 1
1 p,0 g-mode frequencies (Bedding et al. 2011) . The maxima
frequencies. Notethatincontrasttoanormal´echelledi-
of mode inertias are separated by ∆Π in the period-
agram(Fig. 1 c,d), here we remove any curvature in the 1
´echelle diagram, so by collapsing vertically over the cor-
radial ridge by substracting the measured radial mode
rectvalueoftheperiod-spacing,themode-inertia-period
frequency at each order. Collapsing the ´echelle diagram
plot shows a repetitive pattern of peaks, spaced by ∆Π
vertically and plotting inertia on the ordinate (bottom 1
(the pattern period). This is therefore a new method to
panels) reveals clearly the variation of the inertia with
Inferring mode inertias in evolved solar-like stars 5
Figure 2. Measured inertia ratio between ℓ = 1 and ℓ = 0 modes against frequency (black diamonds) for KIC 4351319 (a) and KIC
6442183(b). Theoreticalinertiaratioaresuperimposed(coloredpointsconnectedbylines). Dashedlineindicatesthelowerlimitofinertia
ratio(purepmodes). Thehigheristheinertia,thestrongerthemixedcharacter ofthemode.
6 Benomar et al.
Figure 3. Frequency-´echelle(top)andcollapsedfrequency-´echellediagramforKIC4351319forobservations(left)andformodelB(right).
Theminimaofinertiaarespacedby∆ν. Thedistanceto0intheabscissameasuresδν01.
measure the period spacing in evolved stars, with a pre- givenbyκ (thus by the periodspacing∆Π )appearing
g 1
cisionofbetterthan1%. Thehigherthegmodedensity, as the argument of the cosine function in Equation 10.
the smaller the uncertainty. For KIC 4351319 we found This is due to the fact that the term cos(κ ) is almost
p
∆Π =97.2 sec. This period spacing is compatible with constant over the considered frequency range.
1
that derived by other methods such as the asymptotic Moreover, the minimum value of the inertia ratio is
approachby Mosser et al. (2012b), which applied to our determinedbytheratioκ /κ inEquation10,whichde-
g p
data gives ∆Π1 = 96.5 sec. We also computed the pe- pendsontheinverseofthesquaredfrequency(1/ν2). We
riodspacingofthemodel1ofDi Mauro et al.(2011)and also note that the right-hand side of this equation is di-
found a value of 97.4 sec, which is compatible with the rectlyproportionaltoq2,whichisrelatedtothestrength
value of 97.8 sec found by integrating the Brunt Va¨is¨al¨a of the coupling and characterizes the evanescent region.
frequency (e.g. Christensen-Dalsgaard 2012). Relative Therefore,Equation10andEquation11confimthatthe
uncertainties are ≃0.1 % here. ratio I /I can be used as a tool to derive the period
1 0
spacing and also to constrain the evanescent region be-
5. DISCUSSIONANDCONCLUSION tweenthepandg cavities. Thisisinagreementwiththe
In this Letter, we presented a method to measure results of Sect. 4.1.1.
mode-inertiaratios,aquantitywhichuntilnow,wasonly The ratioI1/I0 providesanalternativeto using mixed
derived from modelling. This diagnostic is based on the mode frequencies and gives essentially the same con-
detection and precise measurement of mixed-mode am- straints (see Equation 9 of Mosser et al. 2012b). How-
plitudes and linewidths. Using Kepler data, we demon- ever,we note thatthe relationshipofI1/I0 as afunction
strated that it is possible to measure precisely the mode of frequency to (i) the period spacing (∆Π1) and (ii)
inertias of subgiants and early red giants. These can be the properties of the inner-most layers characterised by
used to determine the period spacing and the coupling q is different than Equation 9 of Mosser et al. (2012b).
between the p and g cavities of mixed modes. Those Therefore,the two observationalconstraintsarecomple-
results were validated by comparing mode-inertia ratios mentary, potentially providing stronger constraints on
derived from the observations and from stellar and pul- the stellar structure. Further investigations are required
sational modelling of KIC 4351319and KIC 6442183. to evaluate the advantages of observed inertia as con-
Togoastepfurther,andtobetterevaluatewhichcon- strains on the stellar structure.
straints inertias may provide, it is useful to adopt an We stress that inertia ratios are sensitive to non-
asymptotic approach. For this, we follow the work of adiabacitic effects, not accounted by Ballot et al. (2011)
Goupil et al. (2013) based on the asymptotic formalism when computing visibilities (Toutain & Gouttebroze
ofShibahashi (1979) for mixed modes. This allows us to 1993). Mosser et al.(2012a)notedthatheightratioscan
write differby 10%fromadiabaticallycomputedvalues,possi-
2 bly biasing inertia ratios by 5%.
I 1 κ cosκ
1 ≈ 1+ q2 g p , (10) Our approachrequires well-resolvedmixed modes and
I 4 κ cosκ
0 ( (cid:18) p(cid:19)(cid:18) g(cid:19) ) well-identifed mode rotationnal splittings, which can
only be achievedwith long observations. This limitation
where q measures the coupling between the p- and g-
is severe for the more evolved stars because their modes
mode cavities, as introduced by Mosser et al. (2012b),
have the longest lifetimes. The observations used here,
and
spanning more than three years, resolve mixed modes
π ν π π down to ν ≃ 120 µHz. We expect to push this limit
κ ≈ , and κ ≈ − , (11) max
p ∆ν g ∆Π ν 2 even further with the latest Kepler observations (reach-
1
ing four years).
where ν is the frequency. In Equation 10 and Equa-
tion 11, the period of the oscillationin the ratio I /I is
1 0
Inferring mode inertias in evolved solar-like stars 7
Figure 4. Period-´echelle(top)andcollapsedperiod-´echellediagramforKIC4351319forobservations(left)andformodelB(right). The
maximaofinertiaarespacedby∆Π1.
Acknowledgments. ”KB, BM, MJG, and RS ac- Chaplin,W.J.,Houdek, G.,Elsworth,Y.,Gough, D.O.,Isaak,
knowledge the ANR (Agence Nationale de la Recherche, G.R.,&New,R.2005,MNRAS,360,859
France) program IDEE (n◦ ANR-12-BS05-0008) “Inter- Christensen-Dalsgaard,J.2004, Sol.Phys.,220,137
—.2008a, Ap&SS,316,113
action Des E´toiles et des Exoplan`etes” as well as finan- —.2008b, Ap&SS,316,13
cialsupportfrom”ProgrammeNationaldePhysiqueStel- —.2012, AstronomischeNachrichten,333,914
laire” (PNPS) of CNRS/INSU, France”. RAG acknowl- Corsaro,E.,etal.2012, ApJ,757,190
DiMauro,M.P.,etal.2011,MNRAS,415,3783
edge the support of the European Community’s Seventh
Dupret,M.-A.,etal.2009,A&A,506,57
Framework Programme (FP7/2007-2013) under grant
Dziembowski,W.A.1971, ActaAstron.,21,289
agreement no. 269194 (IRSES/ASK). Dziembowski,W.A.,Gough, D.O.,Houdek,G.,&Sienkiewicz,
R.2001,MNRAS,328,601
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