Seismic measurement of the locations of the base of convection zone and helium ionization zone for stars in the {\it Kepler} seismic LEGACY sample

1.5 MB·

by  Kuldeep Verma

Draftversion February1,2017
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
SEISMIC MEASUREMENT OF THE LOCATIONS OF THE BASE OF CONVECTION ZONE AND HELIUM
IONIZATION ZONE FOR STARS IN THE KEPLER SEISMIC LEGACY SAMPLE
Kuldeep Verma1,2,3, Keyuri Raodeo4, H. M. Antia2, Anwesh Mazumdar4, Sarbani Basu5, Mikkel N. Lund6,1,
V´ıctor Silva Aguirre1
Draft versionFebruary 1, 2017
ABSTRACT
7
Acoustic glitches are regionsinside a star where the soundspeed orits derivatives changeabruptly.
1
These leave a small characteristic oscillatory signature in the stellar oscillation frequencies. With
0
the precision achieved by Kepler seismic data, it is now possible to extract these small amplitude
2
oscillatory signatures, and infer the locations of the glitches. We perform glitch analysis for all the
n
66 stars in the Kepler seismic LEGACY sample to derive the locations of the base of the envelope
a
convection zone and the helium ionization zone. The signature from helium ionization zone is found
J
toberobustforallstarsinthesample,whereastheconvectionzonesignatureisfoundtobeweakand
1 problematic,particularlyforrelativelymassivestarswithlargeerrorbarsontheoscillationfrequencies.
3
Wedemonstratethattheheliumglitchsignaturecanbeusedtoconstrainthepropertiesofthehelium

ionization layers and the helium abundance.
]
R Subject headings: stars: fundamental parameters — stars: interiors — stars: oscillations — stars:
S solar-type
.
h
p 1. INTRODUCTION tion observed in CoRoT data to determine the loca-
- tion of the helium ionization zone in a red giant. Sim-
o Anaccurateunderstandingofstellarstructureandevo-
ilarly, Mazumdar et al. (2012) used CoRoT data for
r lution is of paramount importance to astrophysics, and
t HD49933 to determine the acoustic depths of the he-
s physics in general. Indeed, the properties of stars are
lium ionization zone and base of the envelope convec-
a usedto inferthe natureofthe associatedexoplanetsand
[ to learn the history of the Milky Way. Seismic data tion zone (see also, Mazumdar et al. 2011; Roxburgh
from CoRoT (Baglin 2006; Baglin et al. 2009) and Ke- 2011). Mazumdar et al. (2014) performed a more ex-
1 tensive study using data from about a year of obser-
pler (Borucki et al. 2009; Koch et al. 2010) space mis-
v
vations by Kepler for 19 stars to determine the acous-
7 sions have revolutionized our understanding of stellar
tic depths of both glitches. In this work, we extend
8 structure. The precise set of observed oscillation fre-
the study to 66 stars using Kepler data, covering up
9 quenciesarebeing usedto testthe varioushypothesesof
to 3.5 years of observations. Apart from location, the
8 stellar physics.
oscillatory signal from the helium ionization zone was
0 It hadbeen proposedthat the signaturesof the acous-
shown to be sensitive to the envelope helium abun-
. tic glitches in the oscillation frequencies of distant stars
1 dance (Basu et al. 2004; Monteiro & Thompson 2005;
could be used to determine the depths of the base of
0 Houdek & Gough2007). Verma et al.(2014b)haveused
the envelopeconvectionzoneandheliumionizationzone
7 oscillation frequencies from Kepler to determine the en-
(Monteiro et al. 2000; Mazumdar & Antia 2001; Gough
1 velope helium abundance of a binary system, 16 Cyg A
2002; Roxburgh & Vorontsov 2003). The amplitude of
:
v the glitch signature is a few orders of magnitude smaller & B.
i than the background smooth component, and a set of The independent measurement of the locations of
X
the base of the convection zone and helium ioniza-
precise oscillation frequencies are required in a suffi-
r tion zone can be used to constrain the stellar prop-
cientlylargefrequencyrangetousethistechnique. With
a
erties better. The fundamental stellar parameters
the availability of the high quality seismic data from
are not only useful in the context of stellar evolu-
CoRoT and Kepler space missions, it has become possi-
tion but also in the studies of exoplanets (see, e.g.,
ble to apply this technique to distant stars. Miglio et al.
Nutzman et al. 2011; Gilliland et al. 2013; Liu et al.
(2010) used the modulation of the frequency separa-
2014; Silva Aguirre et al. 2015), stellar populations
(Chaplin et al. 2011), and galactic archeology (see, e.g.,
[email protected]
1Stellar Astrophysics Centre, Department of Physics and Miglio et al. 2013; Casagrande et al. 2014, 2016). The
Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 stellar ages are particularly important, and cannot be
AarhusC,Denmark determined directly. The standard technique to deter-
2Tata Institute of Fundamental Research, Homi Bhabha
mine stellar age compares the observed surface proper-
Road,Mumbai400005,India
3CentreforExcellenceinBasicSciences,UniversityofMum- ties of the star with the corresponding quantities from
bai,Kalina,Mumbai400098, India the stellar evolution models. This approach is effec-
4Homi Bhabha Centre for Science Education, TIFR, V. N. tive in determining the ages of stellar clusters, pro-
PuravMarg,Mankhurd,Mumbai400088, India
5AstronomyDepartment, YaleUniversity,P.O.Box208101, vided the observed sample includes stars at different
NewHaven,CT065208101, USA evolutionary stages, including those beyond the main-
6SchoolofPhysicsandAstronomy,UniversityofBirmingham, sequence. But for the isolated field stars, we need ad-
Edgbaston, Birmingham,B152TT,UK
2 Verma et al.
ditional seismic constraints to determine the ages re- 0.4
liably. For instance, the observed oscillation frequen-
8
0.2
cies or their appropriate combinations along with spec-
troscopic data are used to obtain the best-fit stellar 0.0
6
model(see,e.g.,Mathur et al.2012;Chaplin et al.2014; x)
Mcenettclya,lfSeielvtaaAl.gu20ir1r4e;etSiallv.a(A20g1u3i)rrehaevteaol.bt2a0in15ed). steRllaer- L (L)⊙4 --00..42 Fe/H] (de
ages to about 10% accuracy using seismic data in addi- [
-0.6
tionto spectroscopicobservations. The processofdeter-
2
mining the best-fit stellar model typically involves min- -0.8
imization of a cost function, which is a highly nonlin-
ear function and may have multiple minima (see, e.g., 0 -1.0
6600 6300 6000 5700 5400 5100
Aerts et al. 2010), and in some cases two or more min- Teff (K)
ima may be close in terms of the quality of the fits. The
additional information from the acoustic glitches can be Fig.1.— Hertzsprung-Russell diagram showing the parameter
used to resolve these near degeneracies. space covered by the LEGACY sample. The luminosity of stars
was obtained from the best-fit model. The colors represent the
The locations of the acoustic glitches can also be used
observedsurfacemetallicityofthestar.
to constrain the input physics of the stellar evolution
models. This was demonstrated by Mazumdar (2005) ysis and stellar model fitting are discussed in Section 5,
using synthetic data for a CoRoT target star. The evo- Section6demonstratestheimportanceoftheglitchanal-
lutionary models require the heavy element abundance, ysisinstellarmodelfitting,andfinallywesummarizethe
Z, of the star, which is generally derived from the ob- conclusions of this study in Section 7.
served [Fe/H] assuming the relative abundances to be
2. SPECTROSCOPICANDSEISMICDATA
similar to the solar abundances. The recent revision of
the solar heavy element abundances using 3D hydrody- NASA’s Kepler space mission has observed solar-like
namic model for the solar atmosphere (Asplund et al. oscillationsinoverhundredSun-likemain-sequencestars.
2009) are known to be inconsistent with the helioseis- TheLEGACYsampleconsistsof66main-sequencestars
mic constraints (Basu & Antia 2008; Gough 2013, and observed in short cadence mode for at least 12 months
references therein), while the earlier solarabundance ta- (most of the stars have a time series of approximately
bles of Grevesse & Sauval (1998), obtained using 1D so- 3 years), and spans a large range in metallicity. Fig-
lar atmospheric model, provide better agreement. Re- ure1showsthelocationsofthestarsintheHertzsprung-
cent measurements of iron opacity in a condition similar Russell diagram. We used the spectroscopic and seis-
to the solar interior shows that the iron opacity used mic data from Lund et al. (2016), and refer the reader
in stellar models are significantly low (see, Bailey et al. to that paper for the details on the target selection,
2015). This may partly resolve the issue of solar abun- compilation of the spectroscopic data, and the com-
danceproblem,butnotcompletely. Itwouldbeinterest- putation of the stellar oscillation frequencies (see also,
ingtoseeiftheasteroseismicdatacantelluswhetherthe Silva Aguirre et al. 2016).
problem is with opacities, or solar abundances, or with
both (see, e.g., Mazumdar et al. 2010). The position of 3. FITTINGTECHNIQUES
the base of the convection zone is very sensitive to the Acoustic glitches inside a star are regions where the
opacityofthestellarmaterial,whichalsodependsonthe sound speed or its derivatives show an abrupt variation
abundance of the heavy elements, and can throw some on length scales shorter than the typical wavelengths of
light on the above issues. the acoustic modes. Such glitches introduce an oscilla-
Lund et al. (2016) have recently determined the oscil- tory component, δν , in the frequencies of stellar oscil-
g
lation frequencies of 66 main-sequence stars for which lations as a function of the radial order, n, of the form,
there are more than one year of Kepler data. This δν (ν)∝sin(4πτ ν+ψ ), whereτ is the acousticdepth
g g g g
sample is known as Kepler seismic LEGACY sample. of the glitch (see, Gough & Thompson 1988; Vorontsov
Silva Aguirre et al.(2016)haveusedthesesetoffrequen- 1988;Gough1990),andψ isthephaseoftheoscillatory
g
cies to derive the properties of all stars in the sample. signal.
They found the masses, radii, and the ages with average ThetwomainsourcesofacousticglitchesinaSun-like
uncertainties of about 4%, 2%, and 10%, respectively. main-sequence star are the base of the envelope convec-
The long duration of the observations ensures sufficient tion zone (CZ) where the second derivative of the sound
precisionto study the acoustic glitches inthese stars. In speed,c,isdiscontinuous,andthehelium(He)ionization
this work, we use the above set of oscillation frequen- zone where the first adiabatic index, Γ , varies rapidly.
1
cies for all the 66 stars to estimate the acoustic depths Both of these glitches lie deep inside the star where the
of the base of the convection zone and helium ionization non-adiabatic effects are weak, and the glitch signatures
zone. We extend the work of Silva Aguirre et al. (2016) are not significantly affected by the poorly understood
by including the additional observables obtained from near-surfacelayers. Theboundaryofthecoreconvection
the glitch analysis to our stellar model fitting. zone does not contribute a significant oscillatory signal
Therestofthepaperisorganizedasfollows: Section2 as this is aliased to a signal with a very small acoustic
describes the spectroscopic and seismic data used in the depth (Mazumdar & Antia 2001), and that cannot be
study, the techniques to fit the glitch signature are de- distinguished from the background smooth component
scribed in Section 3, Section 4 presents the procedure of the frequency. It was customarily believed that the
to get the best-fit model, the results of the glitch anal- oscillatory signal from the helium ionization zone arises
Glitch analysis of stars in Kepler seismic LEGACY sample 3
fromthedipinΓ -profileinthesecondheliumionization with a related to the amplitude, τ being the acous-
1 c CZ
zone. Thefittedacousticdepthoftheheliumsignalwas, tic depth of the base of the convection zone, and ψ
CZ
however,foundtobesignificantlysmallerthanthedepth being the phase of the signal. The third term is the os-
oftheHeiiionizationzone,andHoudek & Gough(2007) cillatorycontributioncoming fromthe helium ionization
attributed this difference to the neglect of the acoustic zone with a related to the amplitude, c related to the
h 2
cut-off frequency in the phase function and the signal width of Γ -peak between the He i and He ii ionization
1
from the He i ionization zone. Broomhall et al. (2014) zones, τ being the acoustic depth of the Γ -peak, and
He 1
found that the fitted acoustic depth of the helium signal ψ being the phase. This function contains a total of
He
in the models of the red-giants agrees with the acoustic 22 free parameters when fitting l = 0, 1, and 2 modes
depth of the peak in Γ -profile between the He i and He (5×3 = 15 polynomial coefficients a (l), a , τ , ψ ,
1 i c CZ CZ
ii ionization zones. Verma et al. (2014a) did a detailed a , c , τ , ψ ).
h 2 He He
study of glitch signals from various ionization zones of To determine the parameters of Eq. (1), we perform a
helium in main-sequence stellar models to find that the regularizedleast-squares fit by minimizing the function,
fitted acoustic depth alwaysmatches that of the peak in
Γ1 between the two ionization zones. Their attempt to χ2 = νn,l−f(n,l) 2+λ2 d3Pl(n) 2, (2)
fitthesignaturesfrombothionizationzonesofheliumre- g (cid:20) σ (cid:21) (cid:20) dn3 (cid:21)
sultedina significantlybetter fit forthe solaroscillation Xn,l n,l Xn,l
frequencies, but the fit was again to the usual peak be- whereσ isthequoteduncertaintyontheobservedν
n,l n,l
tweenthetwoheliumionizationzones,andapeakabove andλistheregularizationparameter. Notethatwehave
the hydrogen ionization zone. More significantly, the in- used a third derivative regularization instead of the sec-
clusionofthesecondglitchdidnotaffecttheparameters ondderivativeusedinVerma et al.(2014b,a). The third
for the helium glitch (the glitch between the two helium derivative regularizationmarginally improves the stabil-
ionization zones). Hence in this work, we fit the glitch ityofthefit. Theregularizationparameterisdetermined
signature from the helium ionization zone only. in the same way as in Verma et al. (2014b) for the solar
The amplitudes of the oscillatory signature from the oscillation frequencies, and the same value is used for
acoustic glitches are approximatelythree or more orders stars in the Kepler LEGACY sample. The cost function
of magnitude smaller than the backgroundsmooth com- χ2 defined in Eq. (2) is a nonlinear function of the fit-
g
ponent, which makes it hard to extract them from the
ting parameters, hence the fit may not converge to the
stellaroscillationfrequencies. Therearetwopopularap-
global minimum, particularly if the initial guess is not
proaches to extract the glitch signatures: the first at-
closeenough. Wesearchfortheglobalminimuminasub-
tempts to fit the oscillation frequencies directly, while
spaceoftheparameterspacebyrepeatingthefittingpro-
the second tries to fit the second differences of the oscil-
cess for 100 sets of randomly choseninitial guesses. The
lationfrequencies. Weusebothfittingmethods,Method
fit with minimum value of the standard chi-square (first
AandMethod Basdescribedbelow,toderivethe glitch
term in Eq. 2) among 100 trials is accepted as the best
parameters. The two independent methods can be used
fit. We generate 1,000 realizations of the observed os-
to assess the associated systematic uncertainties in the
cillation frequencies assuming the uncertainties on them
estimated glitch properties.
areuncorrelatedandnormallydistributed. Wefitallthe
realizations to get the distributions of the fitted param-
3.1. Fitting frequencies directly (Method A)
eters. The median of the distribution is accepted as the
There are again two different approaches to extract parameter value while the 16th and 84th percentiles of
the glitch signatures from the stellar oscillation fre- the distributiongivethe negativeandpositiveerrorbars.
quencies: the first removes the smooth component The oscillatory signature from the base of the convec-
from the frequencies as a function of radial order, n, tion zone is typically weak with the amplitude of the
and fits the residual (see, e.g., Monteiro et al. 1994; order of the errors on the oscillation frequencies. Con-
Monteiro & Thompson 1998; Monteiro et al. 2000), sequently, the distribution of the fitted parameters may
while the second approach fits the smooth compo- have multiple peaks. We use only those realizations for
nent and the glitch signals simultaneously (see, e.g., whichthefittedacousticdepthfallsinthedominantpeak
Verma et al. 2014b,a). We have used the second ap- tocalculatethemediananderrorestimates. Theparam-
proach in this work as described below. eters associated with the CZ signature may not be cor-
We model the smooth component of the oscillation rect due to the problem of aliasing (Mazumdar & Antia
frequency using a l-dependent fourth degree polynomial 2001),inwhichcasethefitted acousticdepthisfoundto
in the radial order, n, where l is the harmonic degree. bethecomplementofτ ,i.e.,τ =T −τ (whereT is
CZ 0 CZ 0
Thefunctionalformsoftheglitchsignaturesareadapted theacousticradiusofthestar). Inspiteoftheproblemof
fromHoudek & Gough(2007). We fitthe oscillationfre- aliasing,theglitchanalysisisusefulasitcanrestrictτ
CZ
quency, νn,l, to the function, fromitsinfinitepossiblevaluestotwonumbers,viz.,τCZ
ac and T0−τCZ. In some cases, particularly for relatively
f(n,l)=Pl(n)+ν2 sin(4πτCZν+ψCZ) massive stars with large errorbars on the oscillation fre-
quencies, there may not be any well defined peak in the
+ahνe−c2ν2sin(4πτHeν+ψHe), (1) distribution of the fitted parameters for the convection
zone signal, as the values may be spread over a wide
where P (n) = 4 a (l)ni is the contribution of the
l i=0 i interval.
smooth component with a (l) being the coefficients of
P i
the polynomial. The second term is the oscillatory con- 3.2. Fitting second differences (Method B)
tribution coming from the base of the convection zone
4 Verma et al.
This is a well known method for extracting the signa- Ferguson et al. (2005). The metallicity mixtures from
tures of the acoustic glitches from the observed stellar Grevesse & Sauval (1998) was used. We used reaction
oscillationfrequencies(see,e.g.,Gough1990;Basu et al. rates from NACRE (Angulo et al. 1999) for all reac-
1994,2004). Theoscillationfrequenciesfollowcloselythe tions except 14N(p,γ)15O and 12C(α,γ)16O, for which
asymptotic expression of Tassoul (1980), which predom- updated reaction rates from Imbriani et al. (2005) and
inantly depends linearly on the radial order for a given Kunz et al. (2002) were used. Convection was modeled
degree. Hence taking the second difference of the oscil- using the standard mixing-length theory (Cox & Giuli
lation frequency with respect to n, 1968). An exponential overshoot (Herwig 2000) was in-
δ2νn,l :=νn−1,l−2νn,l+νn+1,l, (3) dcliuffduesdionforofsthaerlisuwmitahnmdahsesaevsygerleeamteernttshawnas1.i1n0coMrp⊙o.raTthede
reducesthebackgroundsmoothcomponentsignificantly. for stars of masses less than 1.35 M⊙ using the prescrip-
This, however,complicates slightly the fitting procedure tion of Thoul et al. (1994). For higher mass stars, the
becausethedifferenceshavecorrelatederrorbars,andthe diffusion prescription clearly overestimates the settling
covariance matrix has to be used in the definition of the ofheliumandheavyelementsintheenvelope,andhence
chi-square to be minimized. was not used. The adiabatic oscillation frequencies were
We fit the second differences of the oscillation calculatedusingtheAdiabaticPulsationcode(ADIPLS;
frequencies to the following function adapted from Christensen-Dalsgaard2008).
Houdek & Gough (2007), We constructed models independently for each star in
the LEGACY sample on a mesh of stellar parameters—
δ2ν =a +a ν+ b0 sin(4πντ +φ ) the mass M, initial helium abundance Yi, initial metal-
0 1 ν2 CZ CZ licity [Fe/H]i, mixing-length αMLT, and the overshoot
+c νe−c2ν2sin(4πντ +φ ), (4) parameter αOV. We generated 1,000 to 2,000 randomly
0 He He distributed mesh points for an individual star in a rea-
wherethefirsttwotermstakecareoftheresidualsmooth sonable subspace of the parameter space (we start with
component left after the second differences are calcu- achosensubspacewith 1,000meshpoints,andextendit
lated; the thirdtermis the oscillatorycontributioncom- uniformly if the best-fit model falls near the edge). The
ing from the base of the convection zone with b related models corresponding to every mesh point were evolved
0
to the amplitude, τ being the acoustic depth of the until the track enters in a box formed by the 4σ uncer-
CZ
baseoftheconvectionzone,andφCZ beingthephase;the taintiesintheobservedeffectivetemperatureTeff,surface
fourth term is the oscillatory contribution coming from metallicity [Fe/H], and average large frequency separa-
the helium ionization zone with c related to the ampli- tionh∆νi. Wefittedthesurfacecorrectedmodelfrequen-
0
tude,c relatedtothewidthofΓ -peakbetweentheHei cies (Kjeldsen et al. 2008) to the observed ones to break
2 1
and He ii ionization zones, τ being the acoustic depth the degeneracy inside the box, and accept the best-fit
He
of the Γ -peak, and φ being the phase. The above model as a representative model of the concerned star.
1 He
amplitudes of the oscillatory signatures in the second In this manner, we get an ensemble of approximately
differences may be converted to the corresponding am- 1,000 to 2,000 representative models (depending on the
plitudes in the oscillation frequencies by dividing them totalnumberofmeshpoints)foreachstar. Notethatthe
with4sin2(2πτ h∆νi)(see,Basu et al.1994),whereh∆νi numberofmodelsinthe ensembleisnotexactlysameas
g
the number of mesh points, because not all the tracks
is the average large frequency separation.
enter the box.
Wefittheseconddifferencesofthe oscillationfrequen-
We took two different approaches to get the best-
cies to the function defined in Eq. (4) to determine the
fit model from the above ensemble: the first approach
parameters a , a , b , τ , φ , c , c , τ , and φ .
0 1 0 CZ CZ 0 2 He He
usedtheconventionalspectroscopicandseismicdatabut
WeagainsearchfortheglobalminimumasinMethodA
did not use the glitch information (termed as ‘Seismic-
with100trialsontheinitialguesses. Theparameterval-
Fit1’), while the second used all the information includ-
ues andthe associatederrorbarsarecomputed using the
ing from the glitches (termed as ‘GlitchFit’). These two
distributionofthe parametersobtainedbyfitting 10,000
approachesareusedonlyfortheMESAmodels. InSeis-
realizations of the observed oscillation frequencies. The
micFit1, we defined a cost function,
CZsignaturehasthesamelimitationsasthosediscussed
in the context of Method A.
2
p −p
4. BEST-FITMODELS χ2seismic = (cid:18) modσ obs(cid:19) , (5)
Xp p
We modeled each star in three different ways using
variousmethodsandevolutionarycodes. Theapproaches
where p represents 6 observable quantities; the T ,
eff
are briefly described below.
[Fe/H], large frequency separation averaged over the
radial modes h∆νi , average two-point frequency ratio
4.1. MESA models hr i, and the five-0point ratios r (n ) and r (n +3)
02 01 0 01 0
WeusedtheModulesforExperimentsinStellarAstro- (n0 is suitably chosen radial order, and the choice of
physics code (MESA; Paxton et al. 2011, 2013) for stel- n0+3ismadetoavoidthecorrelationamongtheobserv-
lar modeling. This code can be used with various input ables). We refer the reader to Roxburgh & Vorontsov
physics and data tables. We used the OPAL equationof (2003) for the definition of the ratios. The χ2 was
seismic
state (Rogers & Nayfonov 2002), Opacity Project (OP) minimized over the model ensemble to get the best-
high-temperature opacities (Badnell et al. 2005; Seaton fit model, and the uncertainties on the fitted param-
2005)supplementedwithlow-temperatureopacitiesfrom eters were estimated from the envelope of the χ2
seismic
Glitch analysis of stars in Kepler seismic LEGACY sample 5
(∆χ2 = 1). The detailed results for the LEGACY (2005). We used the 2005 version of the OPAL equa-
seismic
sample obtained using this approach were already pre- tion of state (Rogers & Nayfonov 2002). All nuclear re-
sented in Silva Aguirre et al. (2016, see sections and re- action rates were from Adelberger et al. (1998) except
sults relevantto ‘V&A’). Here,we onlycomparesomeof for14N(p,γ)15O reaction,for whichwe usedthe updated
those results with the results obtained when using the rates of Formicola et al. (2004). A large subset of the
additional information from the glitch analysis (see the models included the diffusion of helium and other heavy
next paragraph). elementswith the diffusion coefficientsfromThoul et al.
In GlitchFit, we incorporated the information coming (1994). The coefficients, however, were changed with a
from the glitch analysis to our fitting pipeline. For this multiplicative factor that depended on the mass of the
purpose, we fitted the signatures of the acoustic glitches models to inhibit the complete depletion of helium and
inthefrequenciesofallthemodelsintheensembleusing heavy elements in the envelope convection zone. The
MethodAtofindthevariousparametersassociatedwith coefficients were unchanged for the models with masses
the base of the convection zone and helium ionization uptoof1.25M⊙,whileforhighermassesthecoefficients
zone, and defined a cost function, weremultipliedbyafactor,f =exp −(M−1.25)2 ,where
2(0.085)2
q −q 2 M is the mass in solar unit. h i
χ2 =χ2 + mod obs , (6)
glitch seismic (cid:18) σ (cid:19) The Yale Monte-Carlo Method (YMCM;
Xq q Silva Aguirre et al. 2015) was used to determine
where q represents 3 observable quantities; amplitude of the best-fit model (‘SeismicFit2’). This fitting method
heliumsignatureaveragedoverthefrequencyrangeA , also does not use the glitch information and have been
He
applied only to the YREC models. The reason for
widthoftheΓ -peak∆ = c /8π2(Houdek & Gough
1 He 2 using two different names (SeismicFit1 and Seismic-
2007), and the acoustic deptph of the Γ1-peak. Note that Fit2) is that the detailed optimization process and the
there is no ambiguity in comparing the acoustic depths
observables used are different in the two cases. For each
of the helium ionization zone as obtained by fitting the
star, we start with using the average large frequency
observed and model frequencies. A model representing
separationand frequency of maximum power alongwith
the star must have similar helium glitch as the star,and
the spectroscopic estimate of the effective temperature
should leavesimilar signature on the oscillationfrequen-
togetanestimateofthemass(M)andradius(R)ofthe
cies,andhencethefittedτ forbothmustbeclose. The
He star using the Yale Birmingham Grid-Based modeling
differencesarisewhenwetrytoassociatethefittedacous-
pipeline (Basu et al. 2010; Gai et al. 2011). Since each
tic depth to a layer in the helium ionization zone (see,
of the observables has an associated error, we created
e.g.,Broomhall et al. 2014; Verma et al. 2014a). We did
several realizations of M, R, T , and [Fe/H]. For each
not include the parameters associated with the CZ sig- eff
realization (M, R, T , [Fe/H]), we used YREC in an
nature in the definition of χ2 for the reasons that eff
glitch iterative mode to obtain a model of the given mass M
we discussed earlier in Section 3.1. We minimized the
and [Fe/H] that had the required R and T . This was
χ2 over the ensemble to get the best-fit model, and eff
glitch done in two different ways: in the first approach, we
theuncertaintiesonthefittedparameterswereestimated kept α fixed at different values and iterated over Y
MLT i
in the sameway as inSeismicFit1. The MESAmodel in to get the model; and in the second approach, we kept
thesubsequentsectionswouldalwaysrefertothebest-fit Y fixed at different values and varied α to get the
i MLT
model obtained in this manner, unless stated otherwise. required model.
Theabovemethodsfindthebest-fitmodelintwosteps. We computed the oscillation frequencies for all the
In the first step, we fix the evolutionary stage for a set models, and defined a cost function,
of initial conditions (M, Y , [Fe/H] , α , α ) using
i i MLT OV
oscillation frequencies, and filter out reasonable models χ2 =χ2 +χ2 +χ2 +χ2 , (7)
total ν ratios Teff [Fe/H]
ofthestar. Theoscillationfrequenciesmonotonicallyde-
crease as a star evolves, and hence reasonably constrain where χ2 was calculated using the surface corrected
ν
the evolutionary stage for a given initial condition. The modelfrequencies(twotermformulationofBall & Gizon
model frequencies have systematic uncertainties due to 2014), while χ2 was obtained using the uncorrected
ratios
thesurfaceeffect,hencewedonotusetheminthesecond modelfrequencies(containstermscorrespondingtoboth,
step, instead use the quantities that are relativelyinsen- r and r ). The first two terms on the right hand side
02 01
sitive to the surface effect, viz., the frequency ratios. In are reduced chi-squares. Since the ratios are strongly
this manner, we use both the oscillation frequencies and correlated, the full error covariance matrix was used to
their combinations to get the best-fit model. Since the define χ2 . The best-fit model for a star was the one
ratios
methods preserve a set of reasonable models of the star, with the lowest value of χ2 .
total
we may plot chi-square as a function of different stellar
parameters, which gives additional useful information, 5. RESULTS
e.g., the possibility of the secondary solutions.
We fitted the signatures of the acoustic glitches in the
oscillationfrequenciesofallthe66starsusingbothmeth-
4.2. YREC models
ods described in Section 3. The quality of fit primarily
A second set of best-fit models were calculated us- depends on the mass of star. Figures 2 and 3 show re-
ing the Yale Rotating Stellar Evolution Code (YREC; spectively the fits obtained using Methods A and B for
Demarque et al. 2008). The models used OPAL high KIC 8760414, 6116048, and 10068307. These stars with
temperature opacities (Iglesias & Rogers 1996) supple- masses close to 0.80, 1.05, and 1.36 M⊙, respectively,
mentedwithlow-temperatureopacitiesofFerguson et al. wereselectedtoberepresentativeofsub-solar,near-solar,
6 Verma et al.
0.8 l=0 l=1 l=2 ons40
δν(µHz) -000...404 ntage of realizati231000
-0.8 KIC 8760414 Perce 0
1800 2100 2400 2700 3000 3300 0 1000 2000 3000 4000
1.6 ons60
µHz) 00..08 of realizati3405
δν( -0.8 ntage 15
-1.6 KIC 6116048 Perce 0
1500 1800 2100 2400 2700 0 1000 2000 3000 4000
ns60
1.2 o
µHz) 00..06 of realizati3405
δν( -0.6 ntage 15
-1.2 KIC 10068307 Perce 0
600 800 1000 1200 0 2000 4000 6000 8000
ν(µHz) Acoustic depths (s)
Fig.2.— Fits to the observed oscillation frequencies of three stars using Method A. The left panels show the oscillatory part of the
frequency, δν, obtained by subtracting the smooth part, Pl, from them. The different points represent the observed frequencies, and the
continuous lineshowsthebest-fittothem. Therightpanels showthedistributionoftheacousticdepths oftheCZglitch(redhistogram)
andtheHeglitch(bluehistogram). Thehorizontal bars showthe ranges ofinitialguesses usedforthese twofitting parameters tosearch
fortheglobalminimum.
1.8 l=0 l=1 l=2 ons50
2δν(µHz) --00011.....06622 KIC 8760414 Percentage of realizati423100000
1750 2000 2250 2500 2750 3000 0 1000 2000 3000 4000
ns60
2δν(µHz) -2011....0000 ntage of realizatio341055
-2.0 KIC 6116048 Perce 0
1500 1800 2100 2400 2700 0 1000 2000 3000 4000
2.0 ons60
2δν(µHz) -011...000 ntage of realizati341055
-2.0 KIC 10068307 Perce 0
600 800 1000 1200 1400 0 2000 4000 6000 8000
ν(µHz) Acoustic depths (s)
Fig. 3.— Fits to the second differences of the observed oscillation frequencies using Method B. The different points in the left panels
representtheobservedseconddifferences,andthecontinuouslineshowsthebest-fittothem. Therightpanelsshowthedistributionofthe
acousticdepthsoftheCZglitch(redhistogram)andtheHeglitch(bluehistogram). Thehorizontalbarsshowtherangesofinitialguesses
usedforthesetwofittingparameterstosearchfortheglobalminimum.
Glitch analysis of stars in Kepler seismic LEGACY sample 7
and super-solar mass stars. Note the small amplitude of photosphere), and c is the sound speed.
the helium signature in the fit for KIC 8760414. This The top panel of Figure 5 compares the different es-
is expected for the low-mass stars because the depres- timates of the acoustic depth of the base of the con-
sion in their Γ -profile in the second helium ionization vection zone. The acoustic depth obtained by fitting
1
zone is shallow (see, e.g., Verma et al. 2014a), hence the the best-fit model frequencies agrees quite well with the
amplitude of the peak between the He i and He ii ion- acousticdepthcalculatedusingthecorrespondingsound-
izationzonesis small, consequentlythe amplitude ofthe speedprofile. Thecalculationoftheacousticdepthusing
helium signature is small. For low-mass stars, the small sound-speedprofilerequiresthedefinitionoftheacoustic
amplitude makes it difficult to fit the He signature un- surface, which is uncertain. Balmforth & Gough (1990)
lesssufficiently lowradialordermodesareobserved. We havearguedthatthe acousticsurfaceofastarshouldbe
found that the fit to the He signature was robust for all defined ata radialdistance in the atmospherewhere the
stars in the LEGACY sample, giving rise to a sharply extrapolated c2 from the outer convection zone vanishes
peaked unimodal distribution of τ (see, Figures 2 and (see also, Lopes & Gough 2001). An uncertainty in the
He
3). Thefitto the CZsignaturewasalsogenerallyrobust definition of the acoustic surface introduces a fixed shift
for stars of sub-solar and solar masses, with only a few inthe acousticdepths calculatedusing sound-speedpro-
problematiccases. However,fittingCZsignaturewasdif- file. Here,we assumedthe acousticsurfaceto be the top
ficult for super-solar mass stars, particularly for stars of mostlayeroftheEddingtonatmosphere(τ =10−5). The
M > 1.20 M⊙, which gave rise to multiple peaks in the scaleddifferencesoflessthan0.03betweenthefittedand
distributionofτ . Suchstarsaregenerallyhot,andthe calculated acoustic depths suggest that the true acous-
CZ
envelopeconvectionexcitesmodeswithshorterlife-time, tic surface is not very far from the assumed layer. The
which leads to larger line-width of the modes and larger pointscorrespondingtothedifferencebetweentheacous-
errorbaronthe mode frequencies. For some problematic tic depths obtained by fitting the observed and best-fit
stars, most of the oscillation frequencies have errorbars model frequencies are more scattered. This is primarily
that are larger than the average amplitude of the CZ duetothefactthattheobservedfrequencieshaveassoci-
signature. ated observationaluncertainties, and the fit to the weak
We modeled each star in the LEGACY sample using CZ signature in the observed frequencies is more prone
the approachesdescribedinSection4to findthe best-fit to aliasing than the fit to the model frequencies.
model and corresponding oscillation frequencies. Recall The helium ionization zones are extended in depth.
that the approach GlitchFit involves fitting the signa- Traditionally, it has been assumed while modeling the
tures of the acoustic glitches in the model frequencies form of the He glitch signature that it arises from the
using Method A. We used the same set of modes for the He ii ionization zone (see, e.g., Monteiro & Thompson
models as used for the observations. Table 1 lists the 1998;Houdek & Gough2007),whichimpliesthatthefit-
acousticdepthsoftheCZandHeglitchandaverageam- ted acoustic depth should represent a layer in the He ii
plitudeoftheHeglitchforboththeobservedfrequencies ionization zone where Γ is minimum (‘dip’). Recently,
1
and best-fit model frequencies from GlitchFit, as well as Broomhall et al. (2014) and Verma et al. (2014a) found
the mass, radius, and the age for all stars. respectively in the red-giant and main-sequence stellar
Forthe sakeofa clearpresentation,weshowherethat models that the fitted acoustic depth corresponds more
the results obtained using Methods A and B agree very closely to a layer between the He i and He ii ioniza-
well, and then present the results obtained using only tion zones where Γ is maximum (‘peak’). The bottom
1
Method A in most of the subsequent sections. Figure 4 panelofFigure5 comparesthe different estimatesof the
shows the differences between the acoustic depths ob- acousticdepthoftheheliumionizationzone. Theacous-
tained using Methods A and B. We can see an excellent tic depths obtained by fitting the observed and best-fit
level of agreement between the results of the two meth- model frequencies are in good agreement, as expected.
ods. This, however, does not guarantee the accuracy of We confirm that the fitted acoustic depth matches with
the results. In fact, the acoustic depth ofthe baseof the the acoustic depth of the peak in Γ -profile.
1
convectionzoneisincorrectforsomestarsinthesample.
Insuchcases,bothmethodsgivesystematicallyincorrect 5.2. Ensemble study
τ , as also noted by Reese et al. (2016).
CZ
Figure 6 shows the acoustic depths of the CZ and He
5.1. Acoustic depths of the CZ and He glitches glitch for all the 66 stars in the sample. The results ob-
tained using the two methods look very similar, except
The glitches observed in Sun-like main-sequence stars
that the errorbars obtained using Method B is on aver-
are regions of only sharp variation in sound speed (not
age larger than Method A, particularly the errorbar on
discontinuities in sound speed), and are extended in
the acoustic depth of the helium ionization zone. This
depth,particularlytheglitcharisingfromtheheliumion-
may be expected because the errorbars on the second
ization zone. To compare the acoustic depths obtained
differences increase by about a factor 2.5 in comparison
using glitchanalysis with the acousticdepth of the layer
to the errorbars on the oscillation frequencies, while the
which causes the signature, we computed the acoustic
amplitude of the signature increases approximately by a
depths of layers using sound-speed profile of the best-fit
model, factor 4sin2(2πτgh∆νi) (Basu et al. 1994). The factor
R∗ dr 4sin2(2πτgh∆νi) is generallysmaller than2.5 for the He
τr = , (8) signature(≈ 1.8,1.2,and1.3forKIC 8760414,6116048,
Z c
r and 10068307, respectively), effectively reducing its sig-
where r is the radial distance of the layer, R∗ the ra- nificance in the second differences in comparison to the
dius of the star (to the acoustic surface and not to the frequencies. There is a clear correlationseen in Figure 6
8 Verma et al.
TABLE1
PhysicalparametersforallstarsintheKeplerseismicLEGACYsample.
GlitchparametersusingMethodA StellarparametersusingGlitchFit
KIC τCZ,obs τCZ,mod AHe,obs AHe,mod τHe,obs τHe,mod M R t
(s) (s) (µHz) (µHz) (s) (s) (M⊙) (R⊙) (Gyr)
1435467a 5307+−8901 2898 1.067+−00..007762 0.957 1112+−3375 1056 1.39±0.04 1.706±0.020 2.7±0.3
2837475a 2422+−6655 1884 1.895+−00..114445 1.609 905+−2267 843 1.50±0.04 1.659±0.020 1.7±0.2
3427720 2068+−9798 2289 0.590+−00..007795 0.633 817+−5467 668 1.15±0.03 1.130±0.015 2.4±0.2
3456181b 5259+−354014 4373 1.088+−00..017232 0.689 1689+−5662 1657 1.50±0.04 2.157±0.020 2.6±0.3
3632418 4149+−118583 4157 0.727+−00..003304 0.655 1498+−3355 1420 1.31±0.04 1.867±0.020 2.8±0.3
3656476 3710+−110276 3575 0.536+−00..007555 0.435 961+−4312 993 1.09±0.03 1.322±0.015 8.4±0.4
3735871 1988+−7775 2323 0.598+−00..008741 0.589 845+−6700 651 1.20±0.04 1.133±0.020 1.5±0.2
4914923 3444+−215688 3497 0.580+−00..003323 0.553 1060+−3322 1025 1.07±0.03 1.357±0.015 6.8±0.3
5184732 2810+−7905 3117 0.735+−00..008865 0.663 849+−3382 826 1.17±0.03 1.329±0.015 4.0±0.4
5773345 3352+−118493 4099 0.798+−00..006605 0.785 1505+−3382 1491 1.55±0.04 2.045±0.020 2.5±0.4
5950854 3827+−129314 3275 0.410+−00..107908 0.236 1681+−126040 1062 1.03±0.02 1.269±0.010 10.3±0.4
6106415 2707+−110191 2685 0.542+−00..003363 0.522 861+−3300 832 1.05±0.04 1.209±0.020 4.9±0.3
6116048 2761+−5640 2790 0.467+−00..002287 0.448 922+−2253 893 1.05±0.02 1.235±0.010 5.7±0.3
6225718 2207+−312688 2425 0.910+−00..004421 0.919 773+−1154 736 1.12±0.03 1.220±0.015 2.5±0.2
6508366a 3325+−113601 4688 0.936+−00..008707 0.904 1527+−6518 1495 1.37±0.04 2.105±0.020 2.3±0.2
6603624 3089+−7678 2976 0.370+−00..002233 0.389 875+−2268 818 1.05±0.02 1.167±0.010 8.1±0.3
6679371 3913+−118623 2937 1.278+−00..019043 1.200 1375+−4584 1352 1.63±0.04 2.248±0.020 2.0±0.3
6933899 4395+−229076 4179 0.466+−00..002214 0.401 1499+−4412 1374 1.12±0.03 1.583±0.015 6.6±0.4
7103006a 2941+−216980 3804 0.876+−00..100808 0.644 1292+−7707 1407 1.35±0.04 1.903±0.020 2.4±0.3
7106245 2916+−127354 2236 0.542+−00..118210 0.397 835+−111251 793 0.93±0.02 1.100±0.010 7.1±0.3
7206837 2665+−119047 3097 1.113+−00..112115 0.894 976+−4430 985 1.36±0.04 1.588±0.020 3.1±0.4
7296438 3635+−221006 3630 0.488+−00..006687 0.463 1098+−7881 986 1.15±0.03 1.393±0.015 6.6±0.3
7510397a 4982+−115557 3999 0.602+−00..002285 0.535 1606+−3349 1469 1.27±0.04 1.821±0.020 3.3±0.3
7680114a 2697+−320547 3739 0.435+−00..006491 0.527 1205+−6616 1082 1.13±0.03 1.423±0.015 7.2±0.3
7771282 3792+−222391 3684 0.893+−00..113275 0.776 1375+−110143 1051 1.30±0.03 1.659±0.015 2.9±0.3
7871531 2117+−7549 2091 0.237+−00..003459 0.163 749+−5894 629 0.80±0.02 0.858±0.010 9.2±0.4
7940546 3673+−9953 4368 0.754+−00..003471 0.651 1551+−3378 1463 1.33±0.03 1.915±0.015 2.7±0.3
7970740 1955+−6759 1824 0.215+−00..004451 0.198 551+−3488 580 0.73±0.03 0.761±0.015 10.1±0.4
8006161 2264+−9823 2163 0.424+−00..005435 0.457 563+−2200 552 1.00±0.03 0.933±0.015 4.9±0.2
8150065a 3749+−115547 2779 0.707+−00..570067 0.629 1012+−337518 854 1.12±0.03 1.367±0.015 4.0±0.3
8179536 2363+−6515 2308 1.076+−00..111150 1.000 803+−5555 757 1.20±0.03 1.331±0.015 1.6±0.2
8228742 4521+−9934 4736 0.545+−00..003218 0.509 1578+−3388 1491 1.28±0.03 1.825±0.015 4.6±0.4
8379927 2219+−222029 2160 0.704+−00..002393 0.654 703+−2212 659 1.11±0.03 1.114±0.015 1.7±0.2
8394589b 1952+−110488 2373 0.793+−00..006558 0.683 750+−5404 769 1.08±0.04 1.178±0.020 3.9±0.3
8424992 2662+−321526 2636 0.412+−00..117186 0.179 1045+−125150 766 0.95±0.03 1.060±0.015 9.1±0.3
8694723 3281+−6690 3793 0.693+−00..003358 0.678 1245+−3310 1220 1.09±0.03 1.522±0.015 4.5±0.3
8760414 2455+−113217 2528 0.222+−00..002296 0.181 968+−3367 929 0.80±0.02 1.018±0.010 12.4±0.4
8938364 4044+−6669 3891 0.390+−00..002243 0.343 1243+−4359 1170 1.06±0.03 1.386±0.015 9.8±0.4
9025370b 1722+−110085 2321 0.322+−00..100535 0.244 810+−113474 643 1.02±0.03 1.021±0.015 5.0±0.3
9098294 3128+−123133 2833 0.449+−00..004416 0.343 837+−3206 856 0.97±0.02 1.145±0.010 8.0±0.3
9139151a 3239+−8789 2277 0.683+−00..006632 0.698 788+−3303 685 1.20±0.03 1.168±0.015 1.9±0.2
9139163a 2182+−4541 2318 1.335+−00..007637 1.138 865+−2222 889 1.34±0.04 1.542±0.020 2.2±0.3
9206432 2839+−6642 2351 1.489+−00..111070 1.005 892+−3394 845 1.38±0.04 1.513±0.020 2.0±0.3
9353712b 6305+−115645 4028 0.775+−00..100937 0.766 1612+−7619 1599 1.53±0.04 2.178±0.020 2.7±0.3
9410862 2760+−112056 2692 0.430+−00..005757 0.374 1065+−5583 838 1.03±0.03 1.178±0.015 6.4±0.3
9414417 4289+−115979 4665 0.768+−00..005585 0.694 1468+−4318 1396 1.29±0.03 1.865±0.015 3.5±0.3
9812850b 4671+−316936 4368 1.154+−00..019073 0.782 1291+−4441 1297 1.26±0.04 1.760±0.020 3.3±0.3
9955598 1940+−141157 2089 0.250+−00..612175 0.195 980+−356702 548 0.87±0.02 0.877±0.010 6.5±0.3
9965715b 2800+−217500 2318 0.907+−00..101911 0.843 825+−4553 738 1.07±0.04 1.269±0.020 3.2±0.3
10068307b 4000+−239188 5349 0.601+−00..003316 0.508 1813+−2371 1719 1.36±0.03 2.053±0.015 3.2±0.4
10079226 2960+−130712 2460 0.518+−00..100898 0.494 870+−110259 678 1.11±0.03 1.145±0.015 2.8±0.2
10162436 4316+−221513 4024 0.775+−00..003470 0.767 1579+−3397 1499 1.18±0.04 1.898±0.020 3.7±0.3
10454113 2189+−7944 2362 1.079+−00..007715 0.942 734+−1292 722 1.16±0.03 1.237±0.015 1.9±0.2
10516096 3646+−112002 3602 0.484+−00..003346 0.432 1167+−4412 1112 1.17±0.03 1.447±0.015 6.2±0.3
10644253b 1804+−6852 2229 0.866+−00..100931 0.907 675+−3462 616 1.12±0.03 1.105±0.015 1.1±0.1
10730618 3691+−18090 3947 1.207+−00..113328 0.701 1260+−5447 1235 1.23±0.04 1.729±0.020 3.1±0.3
10963065 2900+−118900 2716 0.659+−00..006555 0.646 793+−4470 826 1.08±0.03 1.228±0.015 4.1±0.3
11081729 2339+−230265 2802 1.022+−00..115359 0.994 923+−91921 740 1.25±0.04 1.416±0.020 2.2±0.3
11253226 1954+−9753 2018 2.016+−00..110159 1.965 843+−2232 774 1.43±0.04 1.610±0.020 1.9±0.3
11772920 1847+−316780 2042 0.337+−00..103911 0.152 524+−8932 573 0.78±0.03 0.829±0.015 9.5±0.5
12009504 2996+−91408 2992 0.722+−00..005588 0.687 938+−5306 885 1.14±0.03 1.391±0.015 3.6±0.3
12069127a 3902+−110183 4661 0.863+−00..019248 0.733 1745+−111037 1697 1.61±0.06 2.314±0.030 2.5±0.3
12069424 2839+−7733 3063 0.451+−00..004347 0.418 910+−3312 886 1.10±0.02 1.237±0.010 6.7±0.3
12069449b 1790+−5650 2724 0.480+−00..002233 0.457 767+−2107 768 1.00±0.03 1.102±0.015 6.9±0.3
12258514 3906+−211683 3864 0.548+−00..001281 0.526 1274+−2230 1195 1.22±0.03 1.586±0.015 4.3±0.3
12317678a 2545+−6782 3991 1.498+−00..008893 1.035 1115+−3320 1260 1.19±0.04 1.758±0.020 3.4±0.3
Note. —Thesymbolsintheheaderhaveusualmeaning. Thequantitieswithsubscript‘obs’areobtainedbyfittingtheobservedfrequencies,whilewith‘mod’arefound
byfittingthebest-fitmodelfrequenciesfromGlitchFit.
a
ThedistributionofτCZ havemultiplepeaks. ThereisasmallpeakclosetotheτCZ obtainedusingsound-speedprofileofthebest-fitmodel,buttheacousticdepth
correspondingtothedominantpeakanditscomplementarefarfromtheτCZ. Thismayhappenifthesignatureisnotsignificant,i.e.,theoscillationfrequencieshave
largeerrorbars.
b
ThefittedacousticdepthofthebaseoftheconvectionzoneisclosetothecomplementoftheτCZobtainedusingsound-speedprofileofthebest-fitmodel.
Glitch analysis of stars in Kepler seismic LEGACY sample 9
0.30 0.06
Observation Observation
MESA model MESA model
0.20 0.04
0.10 0.02
T0 T0
Bτ)/CZ 0.00 Bτ)/He 0.00
A - CZ A - He
τ( τ(
-0.10 -0.02
-0.20 -0.04
-0.30 -0.06
40 60 80 100 120 140 160 180 40 60 80 100 120 140 160 180
<∆ν> (µHz) <∆ν> (µHz)
0 0
Fig.4.—Comparisonoftheacousticdepths(baseoftheconvectionzoneintheleftpanelandheliumionizationzoneintherightpanel)
obtainedusingMethodsAandBforallstarsintheLEGACYsample. Theopencircleswitherrorbarcorrespondtothefittotheobserved
frequencies,whilefilledcirclestothefittothebest-fitmodelfrequencies.
0.30
0.15
T0
/Z 0.00
C
τ
∆
-0.15
∆τ = τfit - τfit ∆τ = τfit - τfit
CZ obs MESA CZ obs YREC
-0.30
∆τ = τfit - τc ∆τ = τfit - τc
CZ MESA MESA CZ YREC YREC
0.06
0.03
T0
/He 0.00
τ
∆
-0.03
∆τ = τfit - τfit ∆τ = τfit - τfit
He obs MESA He obs YREC
-0.06
∆τ = τfit - τc ∆τ = τfit - τc
He MESA MESA He YREC YREC
40 60 80 100 120 140 160 40 60 80 100 120 140 160
<∆ν> (µHz) <∆ν> (µHz)
0 0
Fig. 5.— Comparison of the acoustic depths obtained using glitch analysis (Method A) and the sound-speed profiles (MESA/YREC).
Theopencircleswitherrorbarshowthescaleddifference between theacoustic depths obtained byfittingtheobserved(τofibts)andbest-fit
model (τfit ) frequencies, while the blue circles show the scaled difference between the acoustic depths obtained by fitting the
MESA/YREC
best-fit model frequencies and using the sound-speed profile (τMcESA/YREC). The τMcESA and τYcREC in the bottom panels represent the
acousticdepthofthepeakinfirstadiabaticindexbetweentheHeiandHeiiionizationzones.
10 Verma et al.
0.8 Method A Method A
6600
0.7
6400
0.6
T0
/Z
C
τ 0.5
6200
0.4
6000
0.3
Observation MESA model K)
0.8 Method B Method B T (eff
5800
0.7
5600
0.6
T0
/Z
C
τ 0.5
5400
0.4
5200
0.3
Observation MESA model
0.12 0.16 0.20 0.24 0.12 0.16 0.20 0.24
τ /T τ /T
He 0 He 0
Fig. 6.—Scaledacousticdepthsofthebaseoftheconvectionzoneandheliumionizationzoneasobtainedusingglitchanalysis. Apoint
inapanel represents astar. Thepoints witherrorbarinthe leftpanels wereobtained byfittingthe observedfrequencies, whilepoints in
therightpanelswerefoundbyfittingthebest-fitmodelfrequencies. Thecolorsrepresenttheeffective temperatureofthestar.
between the acoustic depths of the base of the convec- tor of the evolutionary stage of the star—it decreases as
tion zone and helium ionizationzone. The largerscatter star evolves on the main-sequence—and the dependence
seen in the left panels is mostly due to aliasing of the seen in the bottom panel can be largely understood in
CZ signature. The helium signature is typically strong, terms of its dependence on the age. The dependence of
and the determination of τ is reliable. The fitted τ the τ on composition is significantly weaker than on
He He CZ
togetherwiththeabovecorrelationmaybeusedtoselect themass,andcanbeseenonlyifthemassisconstrained
thecorrectsolutioninthecaseswherethedistributionof to a narrow range.
τ have multiple peaks. As one may expect, the figure The helium signature in the oscillation frequencies
CZ
also shows that the cooler stars have deeper convection of the Sun-like stars cannot only be used to derive
zones as well as deeper helium ionization layers. the location of the helium ionization zone but also
Figure7showsthescaledacousticdepthofthebaseof can be used to estimate the envelope helium abun-
the convectionzone obtained using Method A as a func- dance. The average amplitude of the helium signature
tion of mass, age, large frequency separation averaged dependsontheamountofheliumpresentinitsionization
over radial modes, and average two-point ratio. The fit zone (see, e.g., Basu et al. 2004; Monteiro & Thompson
to the model frequencies is not as much affected by the 2005; Houdek & Gough 2007), which may be calibrated
problem of aliasing as the observed frequencies, unveil- against the corresponding amplitudes of the helium sig-
ing the relationships between the acoustic depth of the naturesinthemodelfrequenciesofdifferentenvelopehe-
base of the convection zone and various stellar proper- lium abundance to estimate its abundance (Verma et al.
ties better. The results obtained using Method B look 2014b). Figure 8 shows the acoustic depth of the he-
very similar. As can be seen from the topmost panels, lium ionization zone as well as the average amplitude of
the acousticdepthofthe baseofthe convectionzonede- the helium signature as a function of M, t, h∆νi , and
0
creasesas the mass (hence the effective temperature)in- r . The acoustic depth of the helium ionization zone
02
creases. Thisisexpectedasthehotterstarshavesmaller decreases as the mass increases. This is again expected
opacity, and the radiation can transport the energy in astheheliumgetsionizedclosertothesurfaceforhotter
the larger part of the envelope. The acoustic depth of stars. The acoustic depth of the helium ionization zone
the base of the convection zone increases as a function increases as a function of the age and large frequency
of the age and averagelarge frequency separation, while separation, while it decreases with the two-point ratio.
it decreases with the two-point ratio. The age and large The averageamplitude of the helium signatureincreases
frequencyseparationdependonthemassofthestar,and with the mass, as was noted by Verma et al. (2014a).
the globaltrendseen in the correspondingpanels are re- This complicates the calibration involved in the helium
sultofthatdependence. Thetwo-pointratioisanindica- abundancedetermination. The amplitude decreasesasa