Asymptotic and measured large frequency separations

0.49 MB·English

by  B. Mosser

Astronomy & Astrophysics manuscript no. mesure11 (cid:13)c ESO 2013
January 3, 2013
Asymptotic and measured large frequency separations
B. Mosser1, E. Michel1, K. Belkacem1, M.J. Goupil1, A. Baglin1, C. Barban1, J. Provost2, R. Samadi1, M.
Auvergne1, and C. Catala1
1 LESIA, CNRS, Universit´e Pierre et Marie Curie, Universit´e Denis Diderot, Observatoire de Paris, 92195 Meudon
cedex,France; e-mail: [email protected]
2 Universit´ede Nice-Sophia Antipolis, CNRS UMR 7293, Observatoire de la Cˆote d’Azur,Laboratoire J.L. Lagrange,
3
BP 4229, 06304 Nice Cedex 04, France
1
0 Preprint online version: January 3, 2013
2
n ABSTRACT
a
J Context. With thespace-bornemissions CoRoT andKepler, alarge amount of asteroseismic data is nowavailable and
has led to a variety of work. So-called global oscillation parameters are inferred to characterize the large sets of stars,
2
perform ensemble asteroseismology, and derive scaling relations. The mean large separation is such a key parameter,

easily deduced from the radial-frequency differences in the observed oscillation spectrum and closely related to the
]
R mean stellar density.It is therefore crucial to measure it with thehighest accuracy in orderto obtain themost precise
asteroseismic indices.
S
Aims.Astheconditionsofmeasurementofthelargeseparationdonotcoincidewithitstheoreticaldefinition,werevisit
.
h the asymptotic expressions used for analyzing the observed oscillation spectra. Then, we examine the consequence of
p thedifference between theobserved and asymptotic valuesof themean large separation.
- Methods.Theanalysisisfocusedonradialmodes.Weuseseriesofradial-modefrequenciesinpublishedanalysesofstars
o withsolar-likeoscillationstocomparetheasymptoticandobservationalvaluesofthelargeseparation.Thiscomparison
r relies on the properuse of thesecond-order asymptotic expansion.
t
s Results. We propose a simple formulation to correct the observed value of the large separation and then derive its
a asymptotic counterpart. The measurement of the curvature of the radial ridges in the ´echelle diagram provides the
[
correcting factor. Weprovethat, apart from glitches duetostellar structurediscontinuities, theasymptoticexpansion

is valid from main-sequence stars to red giants. Our model shows that the asymptotic offset is close to 1/4, as in the
2
theoretical development,for low-mass, main-sequence stars, subgiants and red giants.
v
Conclusions. High-quality solar-like oscillation spectra derived from precise photometric measurements are definitely
7
better described with the second-order asymptotic expansion. The second-order term is responsible for the curvature
8
observed in the ´echelle diagrams used for analyzing the oscillation spectra, and this curvature is responsible for the
6
differencebetween theobserved and asymptoticvalues of thelarge separation. Taking it intoaccount yields a revision
1
of the scaling relations, which provides more accurate asteroseismic estimates of the stellar mass and radius. After
.
2 correction of the bias (6% for the stellar radius and 3% for the mass), the performance of the calibrated relation is
1 about 4% and8% for estimating, respectively,thestellar radius and thestellar mass for masses less than 1.3M⊙; the
2 accuracy is twice as bad for higher mass stars and red giants.
1
: Key words.Stars: oscillations - Stars: interiors- Methods: data analysis - Methods: analytical
v
i
X
1. Introduction scaling reations governing the amplitude of the oscillation
r
a signal(e.g.,Stello et al.2011; Mosser et al.2012a). Incase
The amount of asteroseismic data provided by the space-
an oscillationspectrum is determined with a low signal-to-
bornemissionsCoRoTandKepler hasgivenrisetoensem-
noise ratio, the mean large separation is the single param- 20
bleasteroseismology.Withhundredsofstarsobservedfrom
eter that can be precisely measured (e.g., Bedding et al.
5 the main sequence to the red giantbranch,it is possible to
2001; Mosser et al. 2009; Gaulme et al. 2010).
study evolutionary sequences and to derive seismic indices
fromglobalseismicobservationalparameters.These global The definition of the large separation relies on the
parameters can be measured prior to any complete deter- asymptotic theory, valid for large values of the eigenfre-
minationoftheindividualmodefrequenciesandareableto quencies, corresponding to large values of the radial order. 25
10 provide global information on the oscillation spectra (e.g., However,itsmeasurementisderivedfromthelargestpeaks
Michel et al.2008).Forinstance,themeanlargeseparation seenintheoscillationspectruminthefrequencyrangesur-
∆ν between consecutive radial-mode frequencies and the rounding νmax, thus in conditions that do not directly cor-
frequency ν of maximum oscillating signal are widely respond to the asymptotic relation. This difference can be
max
used to provide estimates of the stellar mass and radius taken into account in comparisons with models for a spe- 30
15 (e.g., Kallinger et al. 2010; Mosser et al. 2010). Almost all cific star,but not in the considerationof large sets of stars
otherscalingrelationsmakeuseof∆ν,as,forinstance,the for statistical studies.
In asteroseismology, ground-based observations with a
Send offprint requests to: B. Mosser limited frequency resolution have led to the use of sim-
1
B. Mosser et al.: Asymptotic and measured large frequency separations
Fig.1.E´chellediagramoftheradialmodesofthestarKIC Fig.2.Variationofthe largeseparationsν −ν as a
n+1,0 n,0
9139163 (from Appourchaux et al. 2012). The red dashed functionofν forthestarHD49933(fromBenomar et al.
n,0
line indicates the quadratic fit that mimics the curvature. 2009). The red dashed line indicates a linear fit.
35 plified and incomplete forms of the asymptotic expansion. wherenisthe p-moderadialorder,ℓisthe angulardegree,
With CoRoT and Kepler data, these simplified forms are ∆ν is the large separation, and ε is a constant term. The
still in use, but observed uncertainties on frequencies are terms∆ν andεarediscussedlater;the dimensionlessterm 80
much reduced. As for the Sun, one may question using the d0 is relatedto the gradientof soundspeedintegratedover
asymptotic expansion, since the quality of the data usu- the stellar interior; d1 has a complex form.
40 ally allows one to go beyond this approximate relation.
Moreover,modelinghasshownthatacousticglitchesdueto
2.2. The asymptotic expansion used in practice
discontinuities or important gradients in the stellar struc-
turemayhampertheuseoftheasymptoticexpansion(e.g., Whenreadingtheabundantliteratureonasteroseismicob-
Audard & Provost1994). servations, the most common forms of the asymptotic ex- 85
45 In this work, we aim to use the asymptotic relation in pansion used for interpreting observed low-degree oscilla-
aproperwaytoderivethe genericpropertiesofa solar-like tion spectra (e.g., Mosser et al. 1991; Bedding et al. 2001;
oscillationspectrum.Itseemsthereforenecessarytorevisit Bouchy et al.2005,forground-basedobservations)aresim-
the different forms of asymptotic expansion since ∆ν is in- ilar to the approximate form
troduced by the asymptotic relation. We first show that
50 it is necessary to use the asymptotic expansion including ℓ
ν ≃ n+ +ε ∆ν −ℓ(ℓ+1)D . (2) 90
n,ℓ 0 0 0
its second-order term, without simplification compared to 2
(cid:18) (cid:19)
the theoretical expansion. Then, we investigate the conse-
quence of measuring the large separation at νmax and not Compared to Eq. (1), the contribution of d1 and the vari-
inasymptoticconditions.Therelationbetweenthetwoval- ation of the denominator varying as νn,ℓ are both omitted.
55 ues, ∆ν for the asymptotic value and ∆ν for the ob- This omission derives from the fact that ground-based ob-
as obs
served one, is developed in Section 2. The analysis based servationshaveatoocoarsefrequencyresolution.Equation
onradial-modefrequenciesfoundintheliteratureiscarried (2) is still in use for space-borne observations that provide 95
out in Section 3 to quantify the relation between the ob- a much better frequency resolution (e.g., Campante et al.
served and asymptotic parameters, under the assumption 2011;White et al.2012;Corsaro et al.2012).Thelargesep-
60 thatthe second-orderasymptoticexpansionis validfor de- aration∆ν0 andtheoffsetε0 aresupposedtoplaytheroles
scribing the radial oscillation spectra. We verify that this of ∆ν and ε in Eq. (1): this is not strictly exact, as shown
hypothesis is valid when we consider two regimes, depend- later. 100
ingonthestellarevolutionarystatus.WediscussinSection Observationally, the terms ∆ν0 and D0 can be derived
4theconsequencesoftherelationbetweentheobservedand from the mean frequency differences
65 asymptotic parameters. We also use the comparison of the
∆ν = hν −ν i, (3)
seismic and modeled values of the stellar mass and radius 0 n+1,ℓ n,ℓ
torevisethescalingrelationsprovidingM andRestimates. 1
D0 = h (νn,ℓ−νn−1,ℓ+2)i, (4)
4ℓ+6
2. Asymptotic relation versus observed parameters wherethe squarebracketsrepresentthe meanvaluesinthe
observedfrequencyrange.Forradialmodes,theasymptotic
2.1. The original asymptotic expression expansion reduces to 105
70 The oscillation pattern of low-degree oscillation pressure ν =(n+ε ) ∆ν . (5)
n,0 0 0
modes can be described by a second-order relation (Eqs.
65-74 of Tassoul1980). This approximate relation is called Clearly, this form cannot account for an accurate descrip-
asymptotic, since its derivation is strictly valid only for tion of the radial-mode pattern, since the ´echelle diagram
largeradialorders.The developmentof the eigenfrequency representationshowsannoticeablecurvatureformoststars
75 ν proposed by Tassoul includes a second-order term, with solar-like oscillations at all evolutionary stages. This 110
n,ℓ
namely, a contribution in 1/ν : curvature, always with the same concavity sign, is partic-
n,ℓ
ularly visible in red giant oscillations (e.g., Mosser et al.
ℓ ∆ν2 2011; Kallinger et al. 2012), which show solar-like oscil-
ν = n+ +ε ∆ν−[ℓ(ℓ+1)d +d ] (1)
n,ℓ 2 0 1 ν lations at low radial order (e.g., De Ridder et al. 2009;
(cid:18) (cid:19) n,ℓ
2
B. Mosser et al.: Asymptotic and measured large frequency separations
115 Bedding et al. 2010a). Figure 1 shows a typical example
of the non-negligible curvature in the ´echelle diagram of a
main-sequence star with many radial orders. The concav-
ity corresponds equivalently to a positive gradient of the
large separation (Fig. 2). This gradient is not reproduced
120 by Eq. (5), stressing that this form of the asymptotic ex-
pansion is not adequate for reporting the globalproperties
oftheradialmodesobservedwithenoughprecision.Hence,
it is necessary to revisit the use of the asymptotic expres-
sionforinterpreting observationsandto better accountfor
125 the second-order term.
2.3. Radial modes depicted by second-order asymptotic
expansion
We have chosen to restrict the analysis to radial modes.
Weexpressthesecond-ordertermvaryinginν−1 ofEq.(1)
130 with a contribution in n−1:
A
as
ν = n+ε + ∆ν . (6)
n,0 as as
n
(cid:18) (cid:19)
Fig.3. E´chelle diagrams of the radial modes of a typi-
Compared to Eq. (1), we added the subscript as to the
cal red-clump giant, comparing the asymptotic expansion
different terms in order to make clear that, contrary to
(Eq. (6), blue triangles) and the development describing
Eq. (2), we respect the asymptotic condition. We replaced
the curvature (Eq. 9, red diamonds). Top: diagram based
135 the second-order term in 1/ν by a term in 1/n instead of
on ∆ν observed at ν ; the dashed line indicates the
1/(n+ε ),sincethecontributionofε inthedenominator obs max
as as
vertical asymptotic line at ν ; the dot-dashed line indi-
can be considered as a third-order term in n. max
catesthe asymptoticlineathighfrequency.Forclarity,the
Thelargeseparation∆ν isrelatedtothestellaracous-
as ridge has been duplicatedmodulo ∆ν . Bottom: diagram
tic diameter by obs
based on ∆ν ; the dot-dashed line indicates the vertical
as
−1 asymptotic line at high frequency.
R dr
140 ∆ν = 2 , (7)
as
c
Z0 ! Mosser et al.(2011)haveproposedincludingthe curvature 165
of the radial ridge with the expression
where c is the sound speed. We note that ∆ν , different
0
from the asymptotic value ∆νas, cannot directly provide ν = n+ε + αobs [n−n ]2 ∆ν , (9)
the integral value of 1/c. We also note that the offset ε n,0 obs max obs
as 2
has a fixed value in the original work of Tassoul: (cid:16) (cid:17)
where ∆ν is the observed large separation, measured in
obs
1 a wide frequency range aroundthe frequency νmax of max-
145 ε = . (8)
as,Tassoul 4 imum oscillation amplitude, αobs is the curvature term, 170
and ε is the offset. We have also introduced the di-
obs
Intheliterature,onealsofindsthatε =1/4+a(ν),where mensionless value of ν , defined by n = ν /∆ν .
as max max max obs
a(ν) is determined by the properties of the near-surface Similar fits have already been proposed for the oscilla-
region (Christensen-Dalsgaard& Perez Hernandez 1992). tionspectraoftheSun(Christensen-Dalsgaard& Frandsen
The second-order expansion is valid for large radial or- 1983; Grec et al. 1983; Scherrer et al. 1983), of αCenA 175
150 dersonly,whenthe second-ordertermA /nis small.This (Bedding et al. 2004), αCenB (Kjeldsen et al. 2005),
as
meansthatthelargeseparation∆ν correspondstothefre- HD203608 (Mosser et al. 2008b), Procyon (Mosser et al.
as
quency difference between radial modes at high frequency 2008a),andHD46375(Gaulme et al.2010).Infact,sucha
only, but not at ν . We note that the second-order term fit mimics a second-ordertermandprovidesa linear gradi-
max
can account for the curvature of the radial ridge in the ent in large separation: 180
155 ´echelle diagram, with a positive value of A for reproduc-
ing the sign of the observed concavity. as νn+1,0−νn−1,0 =(1+α [n−n ])∆ν . (10)
obs max obs
2
The introduction of the curvature may be considered as
2.4. Taking into account the curvature
anempiricalformofthe asymptoticrelation.It reproduces
In practice, the large separation is necessarily obtained the second-order term of the asymptotic expansion, which
from the radial modes with the largest amplitudes ob- has been neglected in Eq. (2), with an unequivocal corre- 185
160 served in the oscillation pattern around ν (e.g., spondence.Itreliesonaglobaldescriptionoftheoscillation
max
Mosser & Appourchaux 2009). In order to reconcile ob- spectrum,whichconsidersthatthemeanvaluesoftheseis-
servations at ν and asymptotic expansion at large fre- mic parameters are determined in a large frequency range
max
quency,wemustfirstconsiderthecurvatureoftheridge.To around ν (Mosser & Appourchaux 2009). Such a de-
max
enhance the qualityof the fit ofradialmodes inredgiants, scription has shown interesting properties when compared 190
3
B. Mosser et al.: Asymptotic and measured large frequency separations
toalocalonethatprovidesthelargeseparationfromalim- 3. Data and analysis
ited frequency range only around ν (Verner et al. 2011;
max 3.1. Observations: main-sequence stars and subgiants 235
Hekker et al. 2012).
It is straightforward to make the link between both Published data allowedus to construct a table of observed
195 asymptotic and observed descriptions of the radial oscil- values of αobs and εobs as a function of the observed large
lation pattern with a second-order development in (n − separation ∆νobs (Table 1, with 94 stars). We considered
n )/n ofthe asymptotic expression.Fromthe identi- observationsofsubgiantsandmain-sequencestarsobserved
max max
ficationofthedifferentordersinEqs.(6)and(9)(constant, byCoRoTorbyKepler.Wealsomadeuseofground-based 240
varying in n and in n2), we then get observations and solar data. All references are given in the
caption of Table 1.
We have considered the radial eigenfrequencies, calcu-
n α
max obs
∆νas = ∆νobs 1+ 2 , (11) lated local large spacings νn+1,0−νn,0, and derived ∆νobs
and α from a linear fit corresponding to the linear gra- 245
α (cid:16) n3 (cid:17) obs
Aas = obs maxα , (12) dientgivenbyEq.(10).Thetermεobs isthenderivedfrom
2 1+n obs Eq. (9). With the help of Eqs. (11), (12) and (13), we an-
max
2 alyzed the differences between the observables ∆ν , ε ,
obs obs
ε = εobs−n2maxαobs. (13) αobs and their asymptotic counterparts∆νas, εas, and Aas.
as αobs Wehavechosentoexpressthevariationwiththeparameter 250
1+n
max 2 nmax, rather than ∆νobs or νmax. Subgiants have typically
n ≥15, and main-sequence stars n ≥18.
max max
200 When considering that the ridge curvature is smallenough
(nmaxαobs/2≪1), Aas and εas become 3.2. Observations: red giants
We also considered observations of stars on the red giant
α
Aas ≃ o2bs n3max, (14) branch that have been modelled. They were analyzed ex- 255
actly as the less-evolved stars. However, this limited set
n α
ε ≃ ε 1− max obs −n2 α . (15) of stars cannot represent the diversity of the thousands
as obs 2 max obs
of red giants already analyzed in both the CoRoT and
(cid:16) (cid:17)
Kepler fields(e.g.,Hekker et al.2009;Bedding et al.2010a;
These developments provide a reasonable agreement with Mosser et al. 2010; Stello et al. 2010). We note, for in- 260
the previousexact correspondencebetween the asymptotic stance, that their masses are higher than the mean value
and observed forms. of the largest sets. Furthermore, because the number of
205 The difference between the observed and asymptotic excited radial modes is much more limited than for main-
values of ε includes a systematic offset in addition to the sequence stars (Mosser et al. 2010), the observed seismic
rescaling term (1−n α /2). This comes from the fact parameters suffer from a large spread. Therefore, we also 265
max obs
that the measurement of ε significantly depends on the madeuseofthemeanrelationfoundforthecurvatureofthe
obs
measurementof ∆ν : a relative change η in the measure- red giant radial oscillation pattern (Mosser et al. 2012b)
obs
210 ment of the large separation translates into an absolute
α =0.015 ∆ν−0.32. (16)
changeoftheorderof−nmaxη.Thisindicatesthatthemea- obs,RG obs
surement of ε is difficult since it includes large uncer-
obs This relation, when expressed as a function of n and
max
tainties related to all effects that affect the measurement taking into account the scaling relation ∆ν ∝ ν0.75 (e.g., 270
of the large separation, such as structure discontinuities Hekker et al.2011),givesα =0.09n−0.96.Tmhaexexpo-
215 or significant gradients of composition (Miglio et al. 2010; obs,RG max
nent of this scaling relation is close to −1, so that for the
Mazumdar et al. 2012).
following study we simply consider the fit
α =2a n−1 , (17)
obs,RG RG max
2.5. E´chelle diagrams
witha =0.038±0.002.Redgiantshavetypicallyn ≤ 275
RG max
The´echellediagramsinFig.3comparetheradialoscillation 15.
patterns folded with ∆ν or ∆ν . In practice, the fold-
obs as
220 ing is naturally based on the observations of quasi-vertical
3.3. Second-order term A and curvature α
as obs
ridgesand provides∆ν observedat ν (Fig. 3 top). A
obs max
folding basedon∆ν wouldnotshowany verticalridgein We first examined the curvature α as a function of
as obs
theobserveddomain(Fig.3bottom).InFig.3wehavealso n (Fig. 4), since this term governsthe relation between
max
compared the asymptotic spectrum based on the parame- the asymptotic and observed values of the large separa- 280
225 ters ∆ν , A , and ε to the observed spectrum, obeying tion (Eq. (11)). A large spread is observed because of the
as as as
Eq. (9). Perfect agreement is naturally met for ν ≃ ν . acoustic glitches caused by structure discontinuities (e.g.,
max
Differencesvaryasα (n−n )3.Theyremainlimitedto Miglio et al. 2010). Typical uncertainties of α are about
obs max obs
a small fractionof the large separation,evenfor the orders 20%. We are interested in the mean variation of the ob-
far from n , so that the agreement of the simplified ex- served and asymptotic parameters,so that the glitches are 285
max
230 pressionissatisfactoryinthefrequencyrangewheremodes firstneglectedandlaterconsideredinSection4.1.Thelarge
haveappreciableamplitudes. Comparisonofthe´echelledi- spreadofthedataimpliesthat,asiswellknown,theasymp-
agrams illustrates that the observable ∆ν significantly toticexpansioncannotpreciselyrelateallofthe featuresof
obs
differsfromthephysically-groundedasymptoticvalue∆ν . a solar-likeoscillationspectrum.However,this spreaddoes
as
4
B. Mosser et al.: Asymptotic and measured large frequency separations
Fig.4. Curvature 103 α as a function of n = Fig.6. Same as Fig. 4, for the relative difference of the
obs max
ν /∆ν. The thick line corresponds to the fit in n−1 es- large separations, equivalent to n α /2, as a function
max max max obs
tablished for red giants, and the dotted line to its extrap- of n .
max
olation towards larger n . The dashed line provides an
max
acceptable fit in n−2 valid for main-sequence stars. Error
max
barsindicatethetypical1-σ uncertaintiesinthreedifferent
domains. The color code of the symbols provides an esti- if a global fit in n−1.5 should reconcile the two regimes, we 305
max
mate of the stellar mass; the colors of the lines correspond keepthe tworegimessince wealsohaveto considerthe fits
to the different regimes. oftheotherasymptoticparameters,especiallyεobs.Wealso
noteagradientinmass:low-massstarshavesystematically
lower α than high-mass stars.Masses were derived from
obs
the seismic estimates when modeled masses are not avail- 310
able (Table 1). At this stage, it is however impossible to
take this mass dependence into consideration.
As a consequence of Eq. (14), we find that the second-
order asymptotic term A scales as n2 for red giants
as max
and as n for less-evolvedstars (Fig. 5). We note, again, 315
max
alargespreadofthevalues,whichisrelatedtotheacoustic
glitches.
3.4. Large separations ∆ν and ∆ν
as obs
Theasymptoticandobservedvaluesofthelargeseparations
ofthe setofstarsareclearlydistinct(Fig.6).Accordingto 320
Eq. (11), the correction from ∆ν to ∆ν has the same
obs as
relativeuncertaintyasα .Therelativedifferencebetween
Fig.5. Same as Fig. 4, for the second-order asymptotic obs
∆ν and∆ν increaseswhenn decreasesandreaches
termA asafunctionofn .ThefitsofA inthediffer- as obs max
as max as a constant maximum value of about 4% in the red giant
ent regimes are derived from the fits of α in Fig. 4 and
obs regime, in agreement with Eq. (17). Their difference is re- 325
the relation provided by Eq. (12).
duced at high frequency for subgiants and main-sequence
stars, as in the solar case, where high radial orders are ob-
served,butstilloftheorderto2%.Thisrelativedifference,
290 not invalidate the analysis of the mean evolution of the
evenif small,depends onthe frequency.This canbe repre-
observed and asymptotic parameters with frequency.
sented, for the different regimes, by the fit 330
Thefitofα derivedfromredgiants,validwhenn
obs max
is in the range [7, 15], does not hold for less-evolved stars
with larger nmax (Fig. 4). It could reproduce part of the ∆νas =(1+ζ) ∆νobs, (19)
295 observedcurvature of subgiants but yields too large values
inthemain-sequencedomain.Inordertofitmain-sequence with
starsandsubgiants,itseemsnecessarytomodifytheexpo-
nent of the relation α (n ). When restricted to main- 0.57
obs max
ζ = (main-sequence regime: n ≥15), (20)
sequence stars, the fit of αobs(nmax) provides an exponent nmax max
300 ofabout−2±0.3.Wethuschosetofixtheexponenttothe
ζ = 0.038 (red giant regime: n ≤15). (21)
max
integer value −2:
α =2a n−2 , (18) The consequence of these relations is examined in
obs,MS MS max
Section 4.2. As for the curvature, the justification of the
with a =0.57±0.02. Having a different fit compared to two different regimes is based on the analysis of the offset 335
MS
theredgiantcase(Eq.(16))isjustifiedinSection3.5:even ε .
obs
5
B. Mosser et al.: Asymptotic and measured large frequency separations
Fig.7. Same as Fig. 4, for the observed(diamonds) and asymptotic (triangles) offsets. Both parameters are fitted, with
dotted lines in the red giant regime and dashed lines in the main-sequence regime; thicker lines indicate the domain
of validity of the fits. The triple-dot-dashed line represents the Tassoul value ε = 1/4, and the dot-dashed line is the
as
model of ε (varying with log∆ν in the red giant regime, and constant for less-evolvedstars).
obs obs
Fig.8. Same as Fig. 7, with a color code depending on the effective temperature.
6
B. Mosser et al.: Asymptotic and measured large frequency separations
3.5. Observed and asymptotic offsets 4. Discussion
The offsets ε and ε are plotted in Fig. 7. The fit of We explore here some consequences of fitting the observed
obs as
ε , initially given by Mosser et al. (2011) for red giants oscillationspectrawiththeexactasymptoticrelation.First, 395
obs
340 andupdatedbyCorsaro et al.(2012),isprolongedtomain- we examine the possible limitations of this relation as de-
sequence stars with a nearly constant fit at ε ≃ 1.4. finedbyTassoul(1980).Sincetheobservedandasymptotic
obs,MS
We basedthis fit onlow-massGstarsin orderto avoidthe values of the largeseparationdiffer,directly extractingthe
morecomplexspectraofFstarsthatcanbeaffectedbythe stellar radius or mass from the measured value ∆νobs in-
HD 49933 misidentification syndrome (Appourchaux et al. duces a non-negligible bias. Thus, we revisit the scaling 400
345 2008;Benomar et al.2009).The spreadofε islargesince relations which provide estimates of the stellar mass and
as
the propagationofthe uncertaintiesfromα toε yields radius. In a next step, we investigate the meaning of the
obs as
a large uncertainty: δεas ≃2δαobs/αobs ≃0.4. term εobs. We finally discuss the consequence of having
ε exactly equal to 1/4. This assumption would make the
Wenoteinparticularthatthetwodifferentregimesseen as
asymptoticexpansionusefulforanalysingacousticglitches. 405
for α correspond to the different variations of ε with
obs obs Demonstrating that radial modes are in all cases based on
350 stellar evolution. The comparison between ε and ε al-
as obs ε ≡1/4 will require some modeling, which is beyond the
lows us to derive significant features. The regime where as
scope of this paper.
ε does not change with ∆ν coincides with the regime
obs obs
wherethecurvatureevolveswithn−2 .Thecorrectionpro-
max
vided by Eq. (13) then mainly corresponds to the constant 4.1. Contribution of the glitches
355 term2a . As a consequence,for low-massmain-sequence
MS
Solar and stellar oscillation spectra show that the asymp- 410
and subgiant stars the asymptotic value is very close to
totic expansionis notenoughfor describing the low-degree
1/4 (Eq. (8), as found by Tassoul 1980). In the red-giant
regime, the curvature varying as n−1 provides a variable oscillationpattern.Acousticglitchesyieldsignificantmodu-
max lation(e.g.,Mazumdar et al. 2012,andreferencestherein).
correction,sothatε is closeto1/4alsoevenifε varies
as obs
Defining the global curvature is not an easy task, since
360 with ∆ν .
obs
the radial oscillation is modulated by the glitches. With 415
the asymptotic description of the signature of the glitch
3.6. Mass dependence proposedby Provostet al. (1993), the asymptotic modula-
tion adds a contribution δν to the eigenfrequency ν
n,0 n,0
For low-mass stars, the fact that we find a mean value defined by Eqs. (1) and (6), which can be written at first
hε i of about 1/4 suggests that the asymptotic expansion order: 420
as
is valid for describing solar-like oscillation spectra in a co-
δν n−n
365 herent way. This validity is also confirmed for red giants. n,0 =βsin2π g, (23)
For thosestars,wemay assumethat εas ≡1/4.Havingthe ∆νas Ng
observed value ε much larger than ε can be seen as
obs as
whereβ measuresthe amplitudeoftheglitch,n itsphase,
an artefact of the curvature α introduced by the use of g
obs and N its period. This period varies as the ratio of the
∆ν : g
obs
stellaracousticradiusdividedbythe acousticradiusatthe
discontinuity.Hence,deepglitchesinducelong-periodmod- 425
1 α ulation,whereasglitches inthe upper stellar envelopehave
370 ε ≃ 1+n obs +n2 α . (22)
obs 4 max 2 max obs short periods. The phase ng has no simple expression.The
(cid:16) (cid:17) observed large separations vary approximately as
However, we note in Fig. 7 a clear gradient of εas with ∆νn,0 2πβ n−ng
the stellar mass: high-mass stars have in general lower εas ∆ν ≃1+αobs(n−nmax)+ N cos2π N , (24)
thanlow-massstars,similartowhatisobservedforε .In obs g g
obs
massive stars, the curvature is more pronounced; εobs and according to the derivation of Eq. (23). In the literature, 430
375 εas arelowerthaninlow-massstars.As aconsequence,the we see typical values of Ng in the range 6 – 12, or even
asymptoticvalueεascannotcoincidewith1/4,asisapprox- larger if the cause of the glitch is located in deep layers.
imately the case for low-mass stars. We tried to reconcile
The amplitude of the modulation represents a few percent
thisdifferentbehaviorbytakingintoaccountamassdepen- of ∆ν , so that the modulation term 2πβ/N can greatly
obs g
dence in the fit of the curvature. In fact, fitting the higher exceedthecurvatureα .Thisexplainsthenoisyaspectof 435
obs
380 curvature of high-mass stars would translate into a larger
Figs. 4 to 7, which is due to larger spreads than the mean
correction, in absolute value, from εobs to εas, so that it curvature.Fromεas =1/4,oneshouldgetεobs,as ≃1.39for
cannot account for the difference. main-sequence stars.If the observedvalue ε differs from
obs
We aretherefore left with the conclusionthat,contrary the expected asymptotic observed ε , then one has to
obs,as
to low-mass stars, the asymptotic expansion is less satis- suppose that glitches explainthe difference. The departure 440
385 factoryfordescribingtheradial-modeoscillationspectraof from ε = 1/4 of main-sequence stars with a larger mass
as
high-massstars.Atthis stage,wemayimaginethatinfact than 1.2M⊙ is certainly related to the influence of their
some features are superimposed on the asymptotic spec- convective core. If the period is long enough compared to
trum,suchassignaturesofglitcheswithlongerperiodthan the number of observable modes, it can translate into an
theradial-orderrangewheretheradialmodesareobserved. apparent frequency offset, which is interpreted as an offset 445
390 Suchlong-periodglitchescanbe due to the convectivecore in ε . Similarly, glitches due to the high contrast density
obs
in main-sequence stars with a mass larger than 1.2M⊙; between the core and the envelope in red giants, with a
they are discussed in Section 4.1. different phase accordingto the evolutionarystatus, might
7
B. Mosser et al.: Asymptotic and measured large frequency separations
modifydifferentlythecurvatureoftheiroscillationspectra. ofthescalingrelations,takingintoaccounttheevolutionary 505
450 These hypotheses will be tested in a forthcoming work. status of the star (White et al. 2011; Miglio et al. 2012),
cluster properties (Miglio et al. 2012), or independent in-
terferometric measurements (Silva Aguirre et al. 2012).
4.2. Scaling relations revisited
At this stage, we suggest deriving the estimates of the
The importance of the measurements of ∆ν and ν is stellar mass and radius from a new set of equations based 510
obs max
emphasizedbytheirabilitytoproviderelevantestimatesof on ∆νas or ∆νobs:
the stellar mass and radius: −2 1/2
R ν ∆ν T
max eff
−2 1/2 = , (28)
455 Robs = νmax ∆νobs Teff , (25) R⊙ (cid:18) νref (cid:19) (cid:18)∆νref(cid:19) (cid:18)T⊙(cid:19)
R⊙ (cid:18)νmax,⊙(cid:19) (cid:18) ∆ν⊙ (cid:19) (cid:18)T⊙ (cid:19) M ν 3 ∆ν −4 T 3/2
max eff
= , (29)
Mobs = νmax 3 ∆νobs −4 Teff 3/2. (26) M⊙ (cid:18) νref (cid:19) (cid:18)∆νref(cid:19) (cid:18)T⊙(cid:19)
M⊙ (cid:18)νmax,⊙(cid:19) (cid:18) ∆ν⊙ (cid:19) (cid:18)T⊙(cid:19) withnewcalibratedreferencesνref =3104µHzand∆νref =
138.8µHz based on the comparison with the models of 515
The solar values chosen as references are not fixed uni-
Table 1. In case ∆ν is used, corrections provided by
formly in the literature. Usually, internal calibration is obs
Eq. (27) must be applied; no correction is needed with
ensured by the analysis of the solar low-degree oscilla-
the use of ∆ν . As expected from White et al. (2011), we
460 tion spectrum with the same tool used for the asteroseis- as
noted that the quality of the estimates decreased for effec-
mic spectra. Therefore, we used ∆ν⊙ = 135.5µHz and tivetemperatureshigherthan6500Korlowerthan5000K. 520
νmax,⊙ = 3050µHz. One can find significantly different We therefore limited the calibration to stars with a mass
values, e.g., ∆ν⊙ = 134.9µHz and νmax,⊙ = 3120µHz less than 1.3M⊙. The accuracy of the fit for these stars
(Kallinger et al. 2010). We will see later that this diver-
is about 8% for the mass and 4% for the radius. Scaling
465 sity is not an issue if coherence and proper calibration are
relationsarelessaccurateforFstarswithhighermass and
ensured.
effective temperatures and for red giants. For those stars, 525
These relations rely on the definition of the large sepa-
the performance of the scaling relations is degraded by a
ration, which scales as the square root of the mean stellar
factor 2.
density (Eddington 1917), and ν , which scales as the
max
470 cutoff acoustic frequency (Belkacem et al. 2011). From the
observed value ∆ν , one gets biased estimates R(∆ν ) 4.3. Surface effect?
obs obs
and M(∆ν ), since ∆ν is underestimated when com-
obs obs 4.3.1. Interpretation of ε
paredto∆ν .Takingintoaccountthecorrectionprovided obs
as
by Eq. (11), the corrected values are We note that the scaling of ε for red giants as 530
obs
a function of log∆ν implicitly or explicitly presented
475 R ≃(1−2ζ)R and M ≃(1−4ζ)M , (27)
as obs as obs by many authors (Huber et al. 2010; Mosser et al. 2011;
with ζ defined by Eqs. (20) or (21), depending on the Kallinger et al. 2012; Corsaro et al. 2012) can in fact be
regime. The amplitude of the correction ζ can be as high explained by the simple consequence of the difference be-
as3.8%,butone musttakeinto accountthe factthatscal- tween ∆ν and ∆ν . Apart from the large spread, we 535
obs as
ing relations are calibrated on the Sun, so that one has to derivedthatthemeanvalueofε ,indicatedbythedashed
as
480 deduce the solar correction ζ⊙ ≃ 2.6%. As a result, for line in Fig. 7, is nearly a constant.
subgiants and main-sequence stars, the systematic nega- Following White et al. (2012), we examined how ε
obs
tive correction reaches about 5% for the seismic estimate varies with effective temperature (Fig. 8). Unsurprisingly,
ofthe mass andabout2.5% forthe seismic estimate ofthe we see the same trend for subgiants and main-sequence 540
radius.Theabsolutecorrectionsaremaximumintheredgi- stars.Furthermore,thereisaclearindicationthatasimilar
485 ant regime. This justifies the correcting factors introduced gradientis presentin redgiants.However,there is no simi-
for deriving masses and radii for CoRoT red giants (Eqs. lar trend for ε . This suggeststhat the physicalreasonfor
as
(9)and(10)ofMosser et al.2010),obtainedbycomparison explainingthegradientofε withtemperatureis,infact,
obs
with the modeling of red giants chosen as reference. firstly related to the stellar mass and not to the effective 545
We checkedthe biasinanindependent wayby compar- temperature.
490 ing the stellar masses obtained by modeling (sublist of 43 The fact that ε is very different from ε = 1/4 and
obs as
starsinTable1)withthe seismicmasses.Theseseismices- varies with the large separation suggests that ε cannot
obs
timatesappearedtobesystematically7.5%largerandhad be seen solely as an offset relating the surface properties
to be corrected.The comparisonof the stellar radiiyielded (e.g., White et al. 2012). In other words, ε should not 550
obs
the same conclusion:the mean bias was of the order 2.5%. be interpreted as a surface parameter, since its properties
495 Furthermore, we noted that the discrepancy increased sig- are closely related to the way the large separation is de-
nificantly when the stellar radius increased, as expected termined. Its value is also severely affected by the glitches.
from Eqs. (27), (20) and (21), since an increasing radius Any modulation showing a large period (in radial order)
corresponds to a decreasing value of n . will translate into an additional offset that will be mixed 555
max
This implies that previous work based on the uncor- with ε .
obs
500 rected scaling relations might be improved. This concerns In parallel, the mass dependence in ε also indicates
as
ensemble asteroseismic results (e.g., Verner et al. 2011; thatitdependsonmorethansurfaceproperties.Examining
Huber et al.2011)orGalacticpopulationanalysisbasedon the exactdependence ofthe offsetε canbe done by iden-
as
distancescaling(Miglio et al.2009).Thisnecessarysystem- tifying it with the phase shift given by the eigenfrequency 560
aticcorrectionmustbepreparedbyanaccuratecalibration equation (Roxburgh & Vorontsov 2000, 2001).
8
B. Mosser et al.: Asymptotic and measured large frequency separations
4.3.2. Contribution of the upper atmosphere
The uppermost part of the stellar atmosphere contributes
to the slightmodification of the oscillationspectrum, since
565 the levelofreflectionofa pressurewavedepends onits fre-
quency (e.g., Mosser et al.1994): the higher the frequency,
thehigherthelevelofreflection;thelargertheresonantcav-
ity, the smaller the apparent large separation. Hence, this
effect gives rise to apparent variation of the observed large
570 separations varying in the opposite direction compared to
the effect demonstrated in this work. This effect was not
considered in this work but must also be corrected for. It
canbe modeledwhenthe contributionofphotosphericlay-
ers is takeninto account.For subgiants andmain-sequence
575 stars, its magnitude is of about −0.04∆ν (Mathur et al.
2012), hence much smaller than the correction from ∆ν
obs
to ∆ν , which is of about +1.14∆ν (Eq. (18)). Fig.9. Variation of the large separation difference
as
∆ν − (∆ν ) as a function of ν for the star
n+1,0 n,0 as n,0
HD49933.
4.4. A generic asymptotic relation
We now make the assumption that ε is strictly equal to
as with a′ = a /(1 + a ). The relationship given by
580 1/4 for all solar-like oscillation patterns of low-mass stars RG RG RG
Eq.(17)ensuresthattherelativeimportanceofthesecond-
and explore the consequences.
order term saturates when n decreases. Its relative in-
max
fluence comparedto the radialorder is a′ , of about 4%.
RG
4.4.1. Subgiants and main-sequence stars Contrary to less-evolved stars, the second-order term for 605
red giants is in fact proportionalto ν2 /ν and is predom-
max
The different scalings between observed and asymptotic inantly governed by the surface gravity.
seismic parameters have indicated a dependence of the ob-
585 served curvature close to n−2 for subgiants and main-
max
4.4.3. Measuring the glitches
sequencestars.Asaresult,itispossibletowritetheasymp-
totic relation for radial modes (Eq. (6)) as
Thesediscrepanciestothegenericrelationsexpress,infact,
the varietyofstars.Ineachregime,Eqs.(30)and(33)pro- 610
n
max
ν | = n+ε +a ∆ν (30) videareferenceforthe oscillationpattern,andEq.(23)in-
n,0 MS as MS n as
dicatesthesmalldepartureduetothespecificinteriorprop-
(cid:16) 1 n (cid:17)
max erties of a star. Asteroseismic inversion could not operate
≃ n+ +0.57 ∆ν (31)
as
4 n if all oscillation spectra were degenerate and exactly simi-
(cid:18) (cid:19)
1/2 lar to the mean asymptotic spectrum depicted by Eq. (30) 615
≃ n+ 1 + 12.8 M R⊙ T⊙ ∆ν .(32) and (33)). Conversely, if one assumes that the asymptotic
as
4 n (cid:18)M⊙ R Teff(cid:19) ! relationprovidesareliablereferencecaseforagivenstellar
model represented by its large separation, then comparing
Thisunderlinesthelargesimilarityofstellarinteriors.With an observed spectrum to the mean spectrum expected at
such a development, the mean dimensionless value of the ∆ν may provide a way to determine the glitches. In other 620
590 second-ordertermisa ,sincetheration /niscloseto words,glitchesmay correspondto observedmodulationaf-
MS max
1 and does not vary with n . However, compared to the ter subtraction of the mean curvature. We show this in an
max
dominantterminn,its relativevaluedecreaseswhenn example(Fig.9),wherewesubtractedthemeanasymptotic
max
increases.Inabsolutevalue,thesecond-ordertermscalesas slopederivedfromEq.(18)fromthelocallargeseparations
∆ν ν /ν, so that the dimensionless term d of Eq. 1 is ν −ν . 625
as max 1 n+1,0 n,0
595 proportional to ν /∆ν , with a combined contribution
max as
of the stellar mean density (∆ν term) and acoustic cut-
as
off frequency (ν , hence ν contribution, Belkacem et al. 5. Conclusion
max c
(2011)).
We have addressed some consequences of the expected dif-
ference between the observedand asymptotic values of the
4.4.2. Red giants large separation. We derived the curvature of the radial-
mode oscillation pattern in a large set of solar-like oscilla- 630
600 For red giants, the asymptotic relation reduces to tion spectra from the variation in frequency of the spac-
ings between consecutive radial orders. We then proposed
ν | = n+ε +a′ n2max ∆ν (33) a simple model to represent the observed spectra with the
n,0 RG as RG n as exact asymptotic form. Despite the spread of the data, the
(cid:18) (cid:19)
ability of the Tassoul asymptotic expansion to account for 635
1 n2
≃ n+ +0.037 max ∆ν (34) thesolar-likeoscillationspectraisconfirmed,sincewehave
as
4 n
(cid:18) (cid:19) demonstrated coherence between the observedand asymp-
1 18.3 M R⊙ T⊙ totic parameters. Two regimes have been identified: one
≃ n+ + ∆ν , (35)
(cid:18) 4 n (cid:18)M⊙ R Teff(cid:19)(cid:19) as corresponds to the subgiants and main-sequence stars, the
9
B. Mosser et al.: Asymptotic and measured large frequency separations
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