HD 203608, a quiet asteroseismic target in the old galactic disk
B. Mosser, S. Deheuvels, E. Michel, F. Thevenin, M.A. Dupret, R. Samadi, C. Barban, M.J. Goupil
aa r X i v : . [ a s t r o - ph ] A p r Astronomy & Astrophysics manuscript no. h8 c (cid:13)
ESO 2018October 26, 2018
HD 203608, a quiet asteroseismic target in theold galactic disk ⋆ ⋆⋆
B. Mosser , S. Deheuvels , E. Michel , F. Th´evenin , M.A. Dupret , R. Samadi ,C. Barban , and M.J. Goupil LESIA, CNRS, Universit´e Pierre et Marie Curie, Universit´e Denis Diderot,Observatoire de Paris, 92195 Meudon cedex, Francee-mail: [email protected] Laboratoire Cassiop´ee, Universit´e de Nice Sophia Antipolis, Observatoire de la Cˆoted’Azur, CNRS, BP 4229, 06304 Nice Cedex 4, FranceSubmitted: April 2007
ABSTRACT
Context.
A short observing run with the spectrometer harps at the ESO 3.6-m tele-scope was conducted in order to continue exploring the asteroseismic properties of Ftype stars. In fact, Doppler observations of F type on the main sequence are demandingand remain currently limited to a single case (HD 49933). Comparison with photomet-ric results obtained with the CoRoT mission on similar stars will be possible with anenhanced set of observations.
Aims.
We selected the 4th magnitude F8V star HD 203608, in order to investigate theoscillating properties of a low-metallicity star of the old galactic disk.
Methods.
A 5-night asteroseismic observation program has been conducted in August2006 with harps . Spectra were reduced with the on-line data reduction software pro-vided by the instrument. A new statistical approach has been developed for extractingthe significant peaks in the Fourier domain.
Results.
The oscillation spectrum shows a significant excess power in the frequencyrange [1.5, 3.0 mHz]. It exhibits a large spacing about 120.4 µ Hz at 2.5 mHz. Variationsof the large spacing with frequency are clearly identified, which require an adaptedasymptotic development. The modes identification is based on the unambiguous sig-nature of 15 modes with ℓ = 0 and 1. Conclusions.
This observation shows the potential diagnostic of asteroseismic con-straints. Including them in the stellar modeling enhances significantly the precisionon the physical parameters of HD 203608, resulting in a much more precise position inthe HR diagram. The age of the star is now determined in the range 7 . ± .
07 Gyr.
Key words. techniques: radial velocities – stars: evolution– stars: oscillations B. Mosser et al.: Asteroseismic study of HD 203608
1. Introduction
New stable spectrometers dedicated to very precise radial velocity measurements have per-mitted rapid progress in observing solar-like oscillations in solar like stars (see for exampleBedding & Kjeldsen 2006 for a recent review). The questions raised with the observationof the CoRoT target HD49933 (Mosser et al. 2005), an active F5V star, led us to continuethe observations of F stars. We therefore planned to measure, identify and characterize thesolar-like oscillations of another low-metallicity F type dwarf star, with a 5-night run.The selected target HD 203608 (HIP 105858, HR 8181, γ Pav) is a F8V star whichbelongs to the group of the Vega-like stars with an infrared excess attributed to thepresence of circumstellar dust warmed by the central star. Its age is estimated to rangebetween 6.5 to 14.5 Gyr (Bryden et al. 2006), according to previous work includingLachaume et al. (1999), based on different estimators: comparison to theoretical isochrones,rotational velocity, strength of chromospheric calcium emission lines, stellar metallicity, andspace velocity. Its metallicity [Fe/H]= − v sin i = 2 . ± . − ,Reiners & Schmitt 2003) which is excellent for Doppler asteroseismology, especially for aF star. On the other hand, with broader lines than G stars, F stars are demanding targetsfor ground-based seismic Doppler observations. As a result, HD 203608 is an exciting starto compare with a similar star, the active F5V star HD 49933, a main target of the satelliteCoRoT already studied by spectrovelocimetry (Mosser et al. 2005).Section 2 reports the current status on the physical parameters of HD 203608.Observations are presented in Section 3, with the analysis of the time series and of theactivity signal. The seismic analysis, based on the unambiguous detection of ℓ = 0 and1 modes is exposed in Section 4. Asymptotic parameters are extracted from the Fourierspectrum, then individual eigenfrequencies and amplitudes. The modeling of HD 203608 ispresented in Sect. 5. Section 6 is devoted to conclusions.
2. Stellar parameters
The atmospheric parameters of HD 203608 have been discussed in several studies(see Cayrel de Strobel et al. 1997). It results a range of 5929 K < T eff < Send offprint requests to : B. Mosser ⋆ Based on observations obtained with the harps ´echelle spectrometer mounted on the 3.6-mtelescope at ESO-La Silla Observatory (Chile), programme 077.D-0720 ⋆⋆ Data corresponding to Fig. 1, Fig. 2 and Table 3 are available in electronicform at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/. Mosser et al.: Asteroseismic study of HD 203608 3
Jonsell et al. (2005) and del Peloso et al. (2005) adopted a similar value: T eff = 6070 Kwith an uncertainty of ±
100 K. But this star is a Vega-like one which exhibits infraredexcess (Mannings & Barlow 1998). The flux absorption and reemission in the infrared isdifficult to estimate, and its effective temperature may be underestimated.The absolute bolometric magnitude was determined using the apparent V magnitude(Table 1) combined with its Hipparcos parallax ( π = 107 . ± .
19 mas, van Leeuwen 2007).The bolometric correction used is from Flower (1996). The resulting luminosity is
L/L ⊙ =1 . ± .
13 (Table 2).
The atmospheric stellar abundance published in the recent literature converges to the value − g is in error. IT99 having adopted T eff = 6072K. Like Jonsell et al. (2005), we can use the NLTE values proposed by IT99 for [Fe/H]: − g value is deduced from the Hipparcos distance. This parameter has alarge uncertainty ( ≃ − .
60 dex for Fe. In most of metal-poorstars the alpha elements present an enhancement compared to iron element abundance,up to 0.4 dex. Idiart & Th´evenin (2000) have computed NLTE abundances for Ca and Mgelements in HD 203608. For that star, Ca appears to follow Fe, contrary to Mg whichis slightly enhanced compared to Fe. In the LTE abundance analyses, Th´evenin (1998)did not found any strong enhancement of the alpha elements Ca, Mg, Si, and only aslight enhancement for the oxygen element: [O/Fe] ≃ f = − . ± .
10 with respect to the solarmetallicity (Grevesse & Noels 1993). Because this star belongs to the old disk, we shalladopt for the calibration an initial abundance Y i =0.25. The accurate Hipparcos parallax determination may help to constrain the adopted T eff value with the estimate of the radius derived from surface brightness relations, as al-ready done for several asteroseismic studies (Kervella et al. 2004, Th´evenin et al. 2005).Values of the radius derived from the magnitudes in different infrared bands are given inTable 1. Creevey et al. (2007) have demonstrated the advantage to constrain the stellarmodeling with asteroseismic data with an independent interferometric measure of the ra- B. Mosser et al.: Asteroseismic study of HD 203608
Table 1.
Color magnitudes of HD 203608, and inferred radii.
Band Magnitude
R/R ⊙ V 4.22 0.96 ± ± ± ± ± ± Table 2.
Physical parameters of HD 203608 T eff ( K ) 6070 ± f − . ± . L/L ⊙ ± ± g (cm s − ) 4.30 ± v sin i ( km s − ) 2.4 ± Fig. 1.
Time series of the signal intensity. Night 4 suffered from an unusually large seeing(up to 3.5”). The efficiency of the collected signal is highly sensitive to the sky opacity andto the seeing.
JD - 2453950-6-4-20246 v ( m / s ) Fig. 2.
Time series of the radial velocity of HD 203608 measured with the pipeline reductionof harps (unfiltered data). The signal is free of any important low-frequency component. . Mosser et al.: Asteroseismic study of HD 203608 5
Fig. 3.
Lomb-Scargle periodogram, and inset of the window function. The time series usedfor this spectrum excludes the noisiest value; night 4 is completely filtered out with such atreatment.dius. However, the infrared excess and the presence of the disk around the star make thisestimate inaccurate, as shown for the Vega-like star τ Ceti. Its diameter measured withinterferometry has been first used to constrain its age and mass (Di Folco et al. 2004); butrecently, Di Folco et al. (2007) have measured the K band excess due to hot dust exozodi-acal disk, and then found a smaller diameter. Therefore, we prefer not to use in this studythe HD 203608 diameter as a fixed constraint.The mass of HD 203608 has been estimated by Jonsell et al. (2005) using evolutionarytracks inversion and assuming an age of 14 Gyr :
M/M ⊙ = 0 .
88. No uncertainty on themass has been proposed by the authors. As for the radius, we cannot consider the stellarmass as a constraint. On the other hand, one of the objective of these observations is toshow the capability of asteroseismology to derive accurate values of these parameters.
3. Observations
About 47.6 hours observations were obtained between August 5 and 10, 2006, representing2504 individual measurements (Table 3). The exposure time was 33 s, giving one mea-surement each 68.5 s, and a Nyquist frequency about 7.3 mHz, much above the predictedoscillation cutoff frequency. For 4 nights out of 5, the observation duration at this winterperiod was greater than 10.3 hours per night. The fourth night was affected by strongperturbations, with an extremely bad seeing (up to 3.5”) and many cirrus. Both effectsyielded a low intensity signal (Fig. 1): most of the photons do not enter the 1” fibre whenthe seeing is too much degraded.The mean SNR in the ´echelle spectrum at 550 nm was typically better than 250 atairmass less than 2. The noise level finally derived from the high frequency part of theFourier spectrum is about 5.1 cm s − , corresponding to a high frequency noise in the timeseries about 1.6 m s − and to a noise equivalent velocity of about 1.0 m s − / √ mHz.The time series show that HD 203608 does not in fact present any signature of ac-tivity (Fig. 2), contrary to HD 49933 that showed a strong low-frequency modulation(Mosser et al. 2005). This difference can be due to the higher stellar age. It can be also B. Mosser et al.: Asteroseismic study of HD 203608
Table 3.
Journal of radial velocity measurements. The 2nd night was split in two parts.∆ T represents the length of observation each night. The dispersion σ RV is derived fromthe high frequency noise recorded in the power spectrum of each night. Date Number of ∆
T σ RV Aug. 06 spectra (hr) (m s − )05 583 11.0 1.506 358 +65 6.9 + 1.2 1.7 + 1.707 570 10.9 2.608 377 7.2 3.009 551 10.4 2.2 Table 4.
Different treatments of the time series, with different filter levels (arbitrary unit,similar to the y-axis in Fig. 1). The dispersion σ ν is directly measured in the high frequencypart of the power spectrum; σ t = σ ν p n c / n c the number of points in the time series. cut η n c σ t σ ν level % (m s − ) (cm s − )0 45 2054 2.0 5.70.1 42 2373 1.8 5.30.2 35 1960 1.6 5.10.3 28 1580 1.4 5.10.4 21 1165 1.4 5.70.5 14 811 1.3 6.2 related to a low value of the stellar inclination, accounting for the low v sin i value. Thequasi pole-on observations should then hamper any significant signature of the rotationalmodulation.
4. The seismic signature
Computing the power spectrum of the complete time series requires a careful treatmentof the noisiest data. Different ways have been proposed, as the weights introduced in theLomb Scargle analysis by Butler et al. (2004). In fact, this method makes the noise uniformin the time series and minimizes the noise level in the power spectrum. As a consequence, itmodifies the signal and yields a degraded window function, which increases the interferencesin the power spectrum between the signal, the aliases and the noise.On the other hand, eliminating noisy data requires a criterion for defining a threshold,and translates immediately in a degradation of the duty cycle. Optimizing simultaneouslythe signal-to-noise ratio (SNR) and the duty cycle is therefore necessary. Choosing theoptimum threshold for the lowest noise level is possible, but it does not insure the highestefficiency for detecting eigenmodes, since highly unpredictable interferences occur betweenthe signal, its aliases and the noise. . Mosser et al.: Asteroseismic study of HD 203608 7
Fig. 4. ´Echelle diagram, with all peaks with a false alarm probability less than 10%, col-lected for 10 different treatments (with increasing cutoff in SNR, hence decreasing valueof the duty cycle). The size of the symbol is representative of the mode amplitude. A darksymbol indicates a peak present in many treatments, whereas a light grey symbol corre-sponds to a peak only present in one treatment. The strong peaks at 2.3 mHz with anabscissa around 40 µ Hz emerge only in the noisiest data set, and therefore are not selected.We have endorsed this impossibility of an a-priori criterion for the best solution, andproposed to circumvent it: instead of performing only one single power spectrum, we havecalculated many, each one corresponding to a different threshold level in the time series(Table 4). Thresholds were based on the signal intensity (Fig. 1). In each case, the LS pe-riodogram was computed and the highest peaks were selected according to the analysis inAppourchaux (2004) that gives a test for detecting peaks of short-lived p modes embeddedin a power spectrum of noise. Simulations have confirmed that the single treatment cor-responding to the optimum solution is far from providing all significant peaks. The peakswith a false alarm probability lower than 10% are plotted in an ´echelle diagram based on alarge splitting of 120.5 µ Hz (Fig. 4). The procedure makes then possible the identificationof the major ridges in the ´echelle diagram. The most confident peaks in these ridges arethen identified (Fig. 5); those selected have the minimum false alarm probabilities obtainedin each periodogram.
The ´echelle diagram exhibits clearly the regular pattern constructed by the asymptotic be-havior of low-degree high-frequency pressure modes. Clear ridges appear in the frequencyrange [2.3 - 2.8 mHz]. The multiple calculations of periodograms, as many as thresholdvalues were fixed, allowed us to put in evidence more peaks than obtained with the cal-culation of a single spectrum. For 17 peaks (including the aliases) finally detected in therange [2.3 - 2.8 mHz] with a false alarm probability less than 10%, 12 at most were presentin a single power spectrum.The ridges in the ´echelle-diagram show a curvature in the range [2.3 - 2.8 mHz]. Thismay be interpreted as a significant second order term to the asymptotic law, or as the
B. Mosser et al.: Asteroseismic study of HD 203608
Fig. 5. ´Echelle diagram, selecting the major peaks detected in the two major ridges iden-tified in Fig. 4 (black symbols), completed with peaks with a larger false alarm probability(up to 25%, grey symbols) but following the asymptotic pattern, and excluding the peakspresent in the neighbor alias of the most prominent ridges.
Fig. 6.
Collapsogram of the ´echelle diagram, with all identified large spacings rescaled onone single value ∆ ν eq , for modes identified in the range [2.2, 2.9 mHz]. Modes ℓ = 0 or 1can be identified, as well as their alias (with a prime). The small spacing derived from thefrequency difference δν allows us to identify the signature of ℓ = 2 modes.signature of a modulation in the spectrum due to an important density gradient inside thestar. In order to account for it, we propose a fit of the frequency pattern varying as: ν n,ℓ = ν n + n ℓ ∆ ν − ℓ ( ℓ + 1) D + n ℓ C (1)with n ℓ = n − n + ℓ D measures the small spacing, as usually done in most of the previousworks reporting single-site asteroseismic observations. C represents the variation factorof the large spacing with the radial order, and defines the local curvature in the ´echellediagram. It corresponds to a global linear increase of the large spacing with the radial order n such as:∆ ν ( n ) = ∆ ν ( n ) + C n ℓ (3) . Mosser et al.: Asteroseismic study of HD 203608 9 In order to account for the curvature and the irregularities in the ´echelle diagram, webuilt a rectified collapsogram, as in Mosser et al. (2008). This rectified collapsogram of the´echelle diagram (Fig. 6), corrected from the variation with frequency of the large spacingsbetween modes of same degree, puts in evidence the signature of ℓ =2 modes. Their locationcompared to radial modes is in agreement with the small spacing derived from the frequencydifference δν = ν n, − ( ν n, + ν n − , ) / ℓ =0 and 1 modes. We may thenpropose an identification of eigenmodes with degrees ℓ = 0, 1 and 2, from which we canderive the local values of the large spacing (Fig. 7).The large spacing is about 120.3 ± µ Hz at 2.6 mHz, matching the value 119 µ Hzexpected from the scaling based on the square root of the mean density p G M/R . Thevariation factor C , around 0 . +0 . − . µ Hz in the range [2.3 - 2.8 mHz], gives a measure of thevariation of the large spacing with the radial order n . Variations of the small spacings with frequency show a more complicated pattern thanexpected from simple asymptotics (Fig. 8). The decrease with frequency of the small fre-quency differences is so pronounced that, instead of the development introduced with Eq 1,we prefer a development closer to the asymptotic form (Tassoul 1980): ν n,ℓ = ν n + n ℓ ∆ ν − ℓ ( ℓ + 1) n + ℓ/ A + n ℓ C (4)The characteristic frequency A introduced in this asymptotic development, around24 ± µ Hz, corresponds at 2.6 mHz to a classical small spacing value D about1.2 ± µ Hz. The correction introduced with the 1 /n term appears efficient, but is notsufficient to account for the variation of the small frequency difference.The evolved star HD 203608 may present a core mainly composed of helium, responsiblefor a significant central decrease of the sound speed. Such a core has a strong influence onthe second order asymptotic term, through the integral term d c/r , that the classical Tassouldevelopment cannot take into account. The rapid density and sound speed variations maybe modeled as a discontinuity. The asymptotic form in that case (Provost et al. 1993, seetheir Eq 4 .1) introduces many parameters for describing the modulation of the smallspacings. With eigenfrequencies identified over a limited frequency range, it has little senseto try to fit all of them, so that we prefer to linearize the development in the form: ν n,ℓ = ν n + n ℓ ∆ ν − ℓ ( ℓ + 1) n + ℓ/ P ℓ + Q ℓ n ℓ ] + n ℓ C (5)It yields small spacings such as: δν ≃ n [ P + Q n ℓ ] − C (6) δν = 6 n [ P + Q n ℓ ] (7)with δν = ν n, − ν n − , . The parameters introduced by Eq 5 are summarized in Tab. 5.Compared to the A factor introduced by Eq 4, the relative errors on the parameters P and P are sensitively reduced. The identification of individual eigenmodes is finally givenin Table 6. We note that the location of ℓ = 2 modes is possibly influenced by aliases of Fig. 7.
Variation of the large spacing with frequency, for the degrees ℓ = 0 (squares), 1(circles) and 2 (diamonds). The dispersion with respect to the linear fit (dashed line) iscompatible with the frequency uncertainty around 1.2 µ Hz derived from Libbrecht 1992,that yields twice this uncertainty on each frequency difference.
Fig. 8.
Variation of the small spacings ( n + ℓ/ δν / n + ℓ/ δν / ℓ = 1 (circles) and 2 (diamonds) modes, compared to ℓ = 0.the ℓ = 0 modes, since the configuration of the time series yield reinforced signatures ofthe alias related to a 4-day periodicity. According to its mass and luminosity and following the power law given bySamadi et al. (2007), the maximum amplitudes of HD 203608 was supposed to be about30 cm s − . In order to estimate the maximum oscillation amplitude, we have constructedsynthetic time series, based on a theoretical low degree p-modes eigenfrequency pattern.The modes lifetimes are estimated from the eigenfrequency widths, between 1 and 4 µ HzFWHM (Houdek et al. 1999). Due to the short duration of the time series, possible largeuncertainties in the lifetimes estimate translate in very small uncertainty in the result. Themaximum amplitudes are assumed to follow a Gaussian distribution in frequency. The syn- . Mosser et al.: Asteroseismic study of HD 203608 11
Table 5.
Estimation of the asymptotic parameters, relying on the modes detected in thefrequency range [2.3, 2.8 mHz], and 3- σ error bars. Asymptotic parameters (at 2.6 mHz)with a Tassoul-like development∆ ν ± µ Hz C = d∆ ν/ d n . +0 . − . µ Hz A ± µ Hz D ≃ A /
20 1.2 ± µ HzSecond order terms P . ± . µ Hz P . ± . µ Hz Q − . ± . µ Hz Q − . ± . µ HzAmplitudes ν max v max ± − Table 6.
Identified peaks, with P their confidence level (here complementary to the falsealarm probability), and inferred peaks, as the function of the assumed radial order n ′ .Frequency uncertainty is around 1.2 µ Hz, according to the estimated lifetime, the obser-vation duration and the SNR (Libbrecht 1992). The detections and identifications outsidethe range [2.3-2.8 mHz] are not as certain as within it. Confidence levels are given whenmodes are directly identified in the ´echelle spectrum. For ℓ = 2 modes, they cannot begiven, since those modes are just inferred assuming a regular ´echelle pattern. ℓ = 0 ℓ = 1 ℓ = 2 n ′ ν obs P ν obs P ν obs mHz % mHz % mHz17 2.305 95 2.36118 2.369 91 2.425 87 2.48019 2.488 79 2.545 94 2.60220 2.608 86 2.667 92 2.72221 2.728 94 2.791 93 2.84322 2.849 88 2.911 81 2.96723 2.973 71 3.032 8424 3.093 75 3.154 71 thetic time series are then calculated using the model of a stochastically excited, dampedharmonic oscillator (Anderson et al. 1990), and include also a white noise.The maximum amplitude is adjusted in order to obtain comparable energy per frequencybin in the synthetic and observed spectra (the reference observed spectrum has a cut level at0.2, what corresponds to the minimum dispersion σ ν , as shown in Table 4). A Monte-Carloapproach finally shows that the best agreement is for a signal with a maximum amplitudeabout 22 ± − rms, with a Gaussian envelope centered at 2.6 mHz and with a 1.2mHz FWHM. The simulation shows that the noise component, precisely determined due to the large oversampling of the time series, is 1.57 ± − , in agreement with the highfrequency noise directly determined in the spectrum.The observed maximum amplitude and the predicted scaling (33 ± − rms) agreeonly marginally. The difference may be related to the low metallicity of HD 203608;a low metallicity yields a thinner convective envelope, then possibly lower amplitudes.Preliminary 3-D simulations of the outer layers of a star with a metallic abundance 10times smaller than solar result in a mode excitation rate about 2 times smaller. Actually,low metallicity corresponds to a mean opacity smaller compared to the solar one. Then,in the super-adiabatic region, where convection is inefficient because of the optically thinatmosphere, the radiative flux is larger than in a medium with a solar metallicity. In thatcase, convection can be less vigorous for evacuating the same amount of energy, leadingto a weaker driving. Therefore, it seems coherent that p modes in a star with sub-solarmetallicity [Fe/H] ≃ − ν Ind, [Fe/H] ≃ − .
4, Carrier et al. 2007; µ Ara, [Fe/H] ≃ ≃ − .
37, Mosser et al. 2005) already showed weaker amplitudes than expected.
5. Modeling
Interior models taking into account the new asterosismic constraints were all computedusing the evolutionary code CESAM2k (Morel 1997). We used the OPAL equation of stateand the nuclear data from NACRE (Angulo et al. 1999). The boundary layers are describedusing a model of atmosphere derived from the Kurucz model adapted to an undermetallicstar (Kurucz 1997). For the chemical composition, we used the solar mixture from Grevesse& Noels (1993). The revised abundances from Asplund et al. (2005) suggest lower abun-dances of C, N, O, Ne and Ar. Many studies showed however that the standard solar modelsusing the new abundances of Asplund are in disagreement with the sound speed profile,the radius and the helium abundance of the convection zone (Guzik et al. 2005). Thesenew abundances need to be confirmed or infirmed by independent 3D NLTE line transferstudies of the oxygen element. Nevertheless, we also computed models with the abundancesof Asplund, and checked that they do not induce any significant change in the results. Theconvection follows the description of Canuto & Mazzitelli (1991) with a mixing length pa-rameter λ close to 1. A model of the Sun using this description of the convection led to λ ≃ .
94. This value was then adopted for this study. The stellar models were computedwith microscopic diffusion using the formalism developed by Burgers (1969).
The models are constrained using the following observational quantities : T eff , L/L ⊙ ,[Z/X] f , and asteroseismic global parameters derived from the asymptotic development.Temperature, luminosity and final metallicity are fixed in the code; all other parameters . Mosser et al.: Asteroseismic study of HD 203608 13 are free, including the mass and the age. The difference between the computed models andthe observations is quantified by the χ function: χ = N X i =1 p obs i − p mod i σ obs i ! (8)Best-fit models are found by minimizing this function. We then obtain an estimate of theparameters which cannot be measured by observations, such as the mass of the star, itsage, the initial abundance of helium Y i and the initial metallicity [Z/X] i .The value of the initial helium content cannot be measured, but it can be assessed usingthe relative helium to metal enrichment of the galaxy ∆ Y / ∆ Z through the relation: Y = ∆ Y ∆ Z Z + Y p (9)where Y p is the primordial helium content. Pagel & Portinari (1998) have shown that theenrichment was such that 2 ≤ ∆ Y / ∆ Z ≤
5. With a primordial helium abundance of Y p = 0 . The observed frequencies from the data analysis range from 2.3 mHz to 3.2 mHz. The largespacing derived from the observations is determined with an uncertainty of 0.5 µ Hz in thisfrequency range, for the modes ℓ = 0, 1 and 2. The first models had to fit the values of T eff , L , [Z/X] f , [Z/X] i and Y i , plus this single asteroseismic constraint: only models which fit themean large spacing with an accuracy better than 0.5 µ Hz were kept. The resulting positionof the star in the HR diagram is narrowed (Fig. 9) and the precision of the parameters ofthe star is improved (Table 7).
Taking into account the other asteroseismic parameters is required to constrain the modelsmore efficiently. We therefore use the development expressed by Eq 5, which allows usto take into account the variation of the large spacing with frequency, and the differentbehaviors of the small spacings δν and δν (Eqs 6 and 7). The observational asteroseismicconstraints for the stellar modeling are then: ∆ ν , C , P , Q , P and Q .Figure 10 shows for instance the evolution of the large spacing with frequency, bothfor the observations and for one of the best-fit models. These evolutions are apparently invery good agreement: the values of the large spacing at 2.6 mHz agree within 0.1% andthe slope of the variations of the large spacing within 10%. The only parameter whichshows a marginal agreement is the one describing the evolution of δν , i.e. Q , as canbe seen on Fig. 11. In fact, none of the models reaches the value of the slope derivedfrom the observations. The agreement in the case 2-0 simply derives from the fact thatthe perturbation due to a dense core depends mainly on the factor n + ℓ/
2, and does notaffect significantly the spacing δν . On the other hand, it strongly affects the spacing δν .Hence, a small discrepancy between the best fit model and the observation translates intoa large discrepancy on this small spacing. Fig. 9.
Position in the HR diagram of the models which fit the mean value of the largespacing. The inner frame represents the 1- σ limits of log( T eff ) and log( L/L ⊙ ). The dashedlines set the 1- σ limits of the large splitting. The dotted line represents the location in theHR diagram of the models having a mean value of δν = 6 . µ Hz, like the one derived fromthe observations. The solid line shows an example of an evolutionary track fitting ∆ ν . Itscharacteristics are: M = 0 . M ⊙ , age = 6 . Y i = 0 .
26, [Z/X] i = − .
38. The numbersalong the evolutionary track stand for the age of the star in Gyr.
Fig. 10.
Variation of the large spacing with frequency, for the ℓ =0 (squares), 1 (circles)and 2 (diamonds) modes. The full symbols and the dashed line stand for the observations.The open symbols and the solid line stand for a model fitting the observational constraints.This phenomenon mainly accounts for the impossibility to get very low values for χ (the minimum value is χ ≃ . χ ≃ .
3. The resulting mod- . Mosser et al.: Asteroseismic study of HD 203608 15
Fig. 11.
Variation of the second order terms with frequency, derived from the ℓ = 1 (circlesand dashed lines) and 2 (diamonds and dotted lines) modes, compared to ℓ = 0. The opensymbols represent the model and the full symbols, the observations. Agreement is betterin the case 2-0 than in the case 1-0, since the perturbation due to a dense core dependsmainly on the factor n + ℓ/ Table 7.
Physical and seismic parameters of HD 203608 derived from the modeling of thestar (all models calculated with the convection parameter λ = 0 . a are those which only take into account the mean value of the large spacing. The modelsof type b are those which use the fitted values of ∆ ν , C , [Z/X] i , P , Q , P and Q . Physical parameters models a models b T eff ±
100 K 6037 ±
19 K
L/L ⊙ ± ± f − . ± .
05 dex − . ± .
03 dex Y i ± ± i − . ± .
05 dex − . ± .
03 dex
M/M ⊙ ± ± ± ± R/R ⊙ ± a ♭ models b ♯ ∆ ν . ± . µ Hz 120 . ± . µ Hz δν . ± . µ Hz 6 . ± . µ Hz C ± µ Hz P . ± . µ Hz P . ± . µ Hz Q − . ± . µ Hz Q − . ± . µ Hz ♭ mean value over the frequencyrange [2.3 - 3.2 mHz] ♯ at 2.6 mHz els are expectedly older than those including diffusion (about 8 Gyr old), and the effectivetemperature is about 100 K higher. Fig. 12.
Minimization of the χ function. The minimum value of χ is χ ≃ .
5. Thesolid line, the dashed line and the dotted line delimit the areas in the HR diagram at 1, 2or 3 σ where respectively χ ≤ χ + 1 , σ uncertainties previous to this asteroseismic run.Finally, our study yields to a star reaching the end of the main sequence. The hydrogenis almost entirely exhausted in the center: the mass fraction of remaining hydrogen inthe core is of about 13%. The best models show no trace of a convective core duringthe evolution sequence, except plausibly during the first billion of years. As explicited byTables 2 and 7, the precision on the stellar parameters has been significantly improved.Localization in the HR diagram is refined by a factor greater than 5 in temperature andabout 8 in luminosity. The error bar on the mass is now defined, as low as 0.01 M ⊙ . Theprecision on the age, is much better, and we note that the age of the star corresponds tothe low limit given by previous works (6.5 →
6. Conclusion
This single-site 5-day long asteroseismic run on HD 203608 has given a much more preciseview of this star of the old galactic disk. We have developed a method for extracting thepeaks with the lowest false alarm probability. This method proves to be efficient for single-site observation with rapidly varying photometric conditions. The performing of multipleperiodograms combined with a statistical test allows us to extract more peaks than with asingle treatment. Using a criterion combining the minimization of false alarm probabilitiesand ´echelle diagram analysis, we have identified 15 ℓ = 0 and ℓ = 1 eigenmodes in thespectrum of HD 203608, from which we have derived lower amplitudes modes (including ℓ = 2 modes) as well as the asymptotic parameters.Despite the very short duration of the run, yielding a limited precision for the identi-fied eigenfrequencies, the fitting of the spectrum has required a more precise developmentthan the usual second order term − ℓ ( ℓ + 1) D . We have shown that the Tassoul originalform with a second order term decreasing with frequency − ℓ ( ℓ + 1) / ( n + ℓ/ A mustbe preferred to the development − ℓ ( ℓ + 1) D often used for interpreting ground-basedobservations. Nevertheless, the asymptotic development cannot account for the precise os- . Mosser et al.: Asteroseismic study of HD 203608 17 cillation pattern of HD 203608: this star exhibits clearly a large spacing with a sensitivedependence on the radial order, and small spacings depending on the mode degree. Thisdependence observed in the data are confirmed in the modeling: the strong compositionand sound speed gradients in the small core mainly composed of helium are responsible ofthe modulation of the oscillation pattern. The eigenfrequency precision in our data set isnot accurate enough to give additional independent constraints on the core size; continuouslong-duration observations are required for such a task.Contrary to a similar F type dwarf target (HD 49933, Mosser et al. 2005), HD 203608does not exhibit any noticeable activity. This may be due to the geometric configurationof the observation, with a possible very low inclination axis. By now, HD 203608 presentsthe lowest metallicity among dwarfs observed in asteroseismology. Similarly to HD 49933,modes amplitudes are sensitively smaller than expected from the scaling law. Two effectsmay explain this: first, both stars are undermetallic; second, the scaling has not yet beencalibrated on dwarf F stars. Observations with the satellite CoRoT will help understandingthat behavior.The modeling of HD 203608 has been achieved with the evolution code CESAM2k(Morel 1997). Taking into account the asteroseismic constraints (large and small spacings)allows us to propose a much more precise description of this star. Error bars on the phys-ical parameters have been divided by a factor of 2 to 8, in the framework of the presentphysical description used in this work. The age we derive for HD 203608 is about 7.25 ± L , T eff and log g (4.356 ± References
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