Seismic Analysis of 70 Ophiuchi A: a New Quantity Proposed New

New Astronomy 13 (2008) 541–548

Contents lists available at ScienceDirect

New Astronomy

journal homepage: www.elsevier.com/locate/newast

Seismic analysis of 70 Ophiuchi A: A new quantity proposed

Y.K. Tang a,c,*, S.L. Bi b,a, N. Gai a,c a National Astronomical Observatories/Yunnan Observatory, Chinese Academy of Sciences, Kunming 650011, PR China b Department of Astronomy, Beijing Normal University, Beijing 100875, PR China c Graduate School of the Chinese Academy of Sciences, Beijing 100039, PR China article info abstract

Article history: The basic intent of this paper is to model 70 Ophiuchi A using the latest asteroseismic observations as Received 16 July 2007 complementary constraints and to determine the fundamental parameters of the star. Additionally, we Received in revised form 26 February 2008 propose a new quantity to lift the degeneracy between the initial chemical composition and stellar Accepted 26 February 2008 age. Using the Yale stellar evolution code (YREC7), we construct a series of stellar evolutionary tracks Available online 2 March 2008 for the mass range M = 0.85–0.93 M with different composition Y (0.26–0.30) and Z (0.017–0.023). Communicated by W. Soon i i Along these tracks, we select a grid of stellar model candidates that fall within the error box in the HR diagram to calculate the theoretical frequencies, the large- and small-frequency separations using Guen- Keywords: ther’s stellar pulsation code. Following the asymptotic formula of stellar p-modes, we deﬁne a quantity Stars: oscillations Stars: evolution r01 which is correlated with stellar age. Also, we test it by theoretical adiabatic frequencies of many mod- Stars: individual: 70 Ophiuchi A els. Many detailed models of 70 Ophiuchi A are listed in Table 3. By combining all non-asteroseismic observations available for 70 Ophiuchi A with these seismological data, we think that Model 60, Model 125 and Model 126, listed in Table 3, are the optimum models presently. Meanwhile, we predict that

the radius of this star is about 0.860–0.865 R and the age is about 6.8–7.0 Gyr with mass 0.89–0.90

1. Introduction Recently, Carrier and Eggenberger (2006) detected solar-like oscillations on the K0 V star 70 Ophiuchi A (HD 165341), and iden- The solar 5-min oscillations have led to a wealth of information tified some possible existing frequencies. They obtained the large about the internal structure of the Sun. These results stimulated separation Dm = 161.7 ± 0.3 lHz by observation over six nights with various attempts to detect solar-like oscillations for a handful of HARPS. The spectroscopic visual binary system 70 Ophiuchi is one solar-type stars. Individual p-mode frequencies have been identi- of our nearest neighbors (5 pc) and is among the first discovered fied for a few stars: a Cen A (Bouchy and Carrier, 2002; Bedding binary stars. It was observed first by Herschel in 1779. So 70 et al., 2004), a Cen B (Carrier and Bourban, 2003; Kjeldsen et al., Ophiuchi A is famous as the primary of a visual and spectroscopic 2005), l Arae (Bouchy et al., 2005), HD 49933 (Mosser et al., binary system in the solar neighborhood. Although many observa- 2005), b Vir (Martic´ et al., 2004a; Carrier et al., 2005b), Procyon A tion data have been obtained since 1779, the theoretical analysis of (Martic´ et al., 2004b; Eggenberger et al., 2004a), g Bootis (Kjeldsen 70 Ophiuchi A has only been made by Fernandes et al. (1998).Bya et al., 2003; Carrier et al., 2005a), b Hyi (Bedding et al., 2001; Car- calibration method which takes into account the simultaneous rier et al., 2001) and d Eri (Carrier et al., 2003b). Based on these evolution of the two members of the binary system, they analyzed asteroseismic data, numerous theoretical analyses have been per- 70 Ophiuchi A by means of standard evolutionary stellar models formed in order to determine precise global stellar parameters using the CESAM code (Morel, 1997) without microscopic diffu- and to test the various complicating physical effects on the stellar sion. They found that the metallicity of 70 Ophiuchi A is very close structure and evolutionary theory (Thévenin et al., 2002; Eggen- to the solar one, the values of the mixing-length parameter a and berger et al., 2004b, 2005; Kervella et al., 2004; Miglio and Mont- helium abundance Y are near the Sun. They thought that the star albán, 2005; Provost et al., 2004, 2006). is younger than the Sun and 3 ± 2 Gyr is probably a limit considering the age versus stellar rotation relation with its rotation velocity (vsini 16 km s1). * Corresponding author. Address: National Astronomical Observatories/Yunnan The aim of our paper is to present the model which can be con- Observatory, Chinese Academy of Sciences, Kunming 650011, PR China. Fax: +86 strained by these seismology data. The observational constraints 3920154. E-mail addresses: [email protected], [email protected] (Y.K. Tang), bisl@ available for 70 Ophiuchi A are summarized in Section 2, while bnu.edu.cn (S.L. Bi). the numerical calculations are presented in Section 3. The seismic

analyses are carried out and a new quantity r01 as an indication of large separation of 161.7 lHz, the solution Dm = 172.2 lHz cannot stellar age is proposed in Section 4. Finally, the discussion and con- be ruled out deﬁnitely. We refer to these two groups of results in clusions are given in Section 5. the paper and make analyses in Sections 4 and 5.

2. Observational constraints 3. Stellar models

2.1. Non-asteroseismic observation constraints We will construct a grid of stellar evolutionary models by the Yale stellar evolution code (YREC; Guenther et al., 1992) with The mass of this star was investigated by Batten et al. (1984), microscopic diffusion. The initial zero-age main sequence (ZAMS) Heintz (1988), Fernandes et al. (1998) and Pourbaix (2000). In this model used for 70 Ophiuchi A was created from pre-main sequence paper, we adopt the value of the mass deduced from Fernandes evolution calculations. In these computations, we do not consider et al. (1998). The effective temperature was determined by Gray rotation and magnetic ﬁeld effects. These models are computed and Johanson (1991). So far, the metallicity values obtained by using OPAL equation of state tables EOS2001 (Rogers and Nayfo- observation are [Fe/H] = 0.05 (Peterson, 1978) and [Fe/H] = 0.00 nov, 2002), the opacities interpolated between OPAL GN93 (Igle- (Perrin et al., 1975). We choose [Fe/H] = 0.0 ± 0.1 as a representa- sias and Rogers, 1996) and low temperature tables (Alexander tive value according to Fernandes et al. (1998). and Ferguson, 1994). Using the standard mixing-length theory, The mass fraction of heavy-elements, Z, was derived assuming we set a = 1.7 for all models, close to the value which is required log[Z/X] [Fe/H] + log[Z/X] and [Z/X] = 0.0230 (Grevesse and to reproduce the solar radius under the same physical assumptions Sauval, 1998), for the solar mixture. So we can deduce the [Z/ and stellar evolution code. Meanwhile, it must be emphasized that

X]s = 0.0183–0.0290. there are still a number of uncertainties in our analyses, foremost All non-asteroseismic observational constraints are listed in Ta- among which is the still open question of the mixing-length theory ble 1. responsible for the stellar model. The nuclear reaction rates have been updated according to Bahcall et al. (1995). The Krishna- 2.2. Asteroseismic constraints Swamy Atmosphere T–s relation is used for this solar-like star (Guenther and Demarque, 2000). Also, we consider the microscopic Solar-like oscillations of 70 Ophiuchi A have been detected by diffusion effect, by using the diffusion coefficients of Thoul et al. Carrier and Eggenberger (2006) with the HARPS spectrograph. (1994). Since 70 Ophiuchi A, like a Cen B, is less massive than the Fourteen individual modes are identified with amplitudes in the Sun, the mass contained in its convective zone is much larger range 11–14 cm s1. Although they listed two groups of frequen- and, therefore, the effect of microscopic diffusion is much smaller cies by mode identification (see Table 2 in Carrier and Eggenberger, (Miglio and Montalbán, 2005; Morel and Baglin, 1999). However, 2006), one group of frequencies with an average large separation it is necessary to consider this effect as a physical process in stellar Dm = 161.7 lHz was suggested to be more reliable than the other modeling (see Provost et al., 2005; Provost et al., 2006). with an average large separation Dm = 172.2 lHz. The star 70 In general, the determination of parameters (M,t,Yi,Zi) fitting Ophiuchi A is very similar to a Cen B with the same spectral type the observational constraints needs two steps. The first step is to and similar large separation, which has a mean small separation construct a grid of models with position in the HR diagram in of 10 lHz. It is thought that the small separation should be similar. agreement with the observational values of the luminosity, the By inspecting the results of the mode identification, they note that effective temperature and the surface metallicity. The principal the value of the small separation coming from the identification constraints deduced from non-asteroseismic observation are listed with the large separation of 172.2 lHz is significantly different in Table 1. The error box, which is composed of observational effec- from 10 lHz. If the large separation is 172.2 lHz, the small separa- tive temperature and luminosity, represents the possible position tion will be lower than 6.5 lHz in the frequency range 3–4.5 mHz. of 70 Ophiuchi A in the HR diagram (see Fig. 1a). According to Although this identification is less reliable than the one with a the results of Fernandes et al. (1998), we list the parameter space

of mass M, the initial heavy-element abundance Zi and the initial helium abundance Yi in Table 2. Since the microscopic diffusion is included in our paper, we give the wider parameter space of ini- Table 1 tial heavy-element abundance Zi than the range of Zi of Fernandes Non-asteroseismic observational data of 70 Ophiuchi A et al. (1998).

Observable Value Source By adjusting three parameters M, Yi and Zi listed in Table 2,we can obtain many evolutionary tracks passing through the error box Mass, M/M 0.89 ± 0.04 (1) Effective temperature, Teff (K) 5322 ± 20 (2) in the HR diagram. Now we consider a function which describes Luminosity, log(L/L) 0.29 ± 0.03 (1) the agreement between the observations and the theoretical Metallicity, [Fe/H]s 0.0 ± 0.1 (1) results: Surface heavy-element abundance, [Z/X]s 0.02365 ± 0.00535 (3) ! 2 X3 theo obs References – (1) Fernandes et al. (1998), (2) Gray and Johanson (1991), (3) this C C v2 i i ; paper. obs ð1Þ i¼1 rCi

theo where C represent the following quantities: L/L, Teff and [Z/X]s, C represent the theoretical values and Cobs represent the observa- Table 2 obs tional values listed in Table 1. The vector rCi contains the errors Input parameters for model tracks on these observations which are also given in Table 1. Variable Minimum value Maximum value d As Fernandes et al. (1998) has pointed out that the age of 70 Ophiuchi A is 3 ± 2 Gyr, it is reasonable for us to choose the evolu- Mass, M/M 0.85 0.93 0.01 Initial heavy-element abundance, Zi 0.017 0.023 0.001 tionary tracks passing through the error box within 8 Gyr. We se- Initial helium abundance, Yi 0.26 0.30 0.01 lect 44 evolutionary tracks passing through error box as our Note – The value d deﬁnes the increment between minimum and maximum possible candidates to go on with our investigations. Fig. 1a gives 2 parameter values used to create the model array. 44 evolutionary tracks, and Fig. 1b presents v as a function of Y.K. Tang et al. / New Astronomy 13 (2008) 541–548 543

Fig. 1. Evolutionary tracks in the HR diagram and v2 as a function of age for 70 Ophiuchi A. (a) The selected individual stellar evolutionary tracks (44 in total). (b) v2 values 2 calculated for 70 Ophiuchi A observational data using different Zi, Yi and mass, plotted as a function of age. v refers to non-asteroseismological observables as denoted by Eq. (1). the age correspondingly. It is well-known that v2 is smaller, the they are seen as a peak in the Fourier transform of the power spec- more competitive is the candidate. Fig. 1b shows that models with trum and they are mostly uncontaminated by composition effects, v2 smaller than 1 have ages between 3 Gyr and 7 Gyr. From Fig. 1a, these large separations provide an efficient way to constrain a stel- we find that the upper section of the error box is empty. The reason lar model. It is also important to remember that the theoretical fre- for the empty upper section of the error box is related to the range quencies calculated in our paper should not be expected to match of initial parameters, like mass, initial composition and specially the observed frequencies of Carrier and Eggenberger (2006) per- the mixing-length parameter. We think that the future interfero- fectly. We think that there are three reasons for this. Firstly, our the- metric measurement of the radius could reduce the domain of oretical models do not match the mass and radius of the star the possible position in the HR diagram (e.g., Provost et al., 2006). precisely. Secondly, the uncertainty in calculating the sound speed The second step is to determine the optimum model using the in the outer layers of the models becomes significant and non-adi- asteroseismic measurements. We will select a grid of models along abatic effects become important. Thirdly, at high frequencies, the these 44 tracks shown in Fig. 1a to calculate the low-lp-modes fre- effect of the convection-oscillation interactions is larger; the quencies. We list the representative models extracted from every description of convection is an open problem. Although the differ- tracks in Table 3. ences between the theoretical frequencies and the observed fre- The detailed pulsation analysis is described in the next section. quencies could result in a significant effect on the large separations, we think that the effect is small due to the large sepa- 4. Pulsation analysis rations corresponding to differences between frequencies of modes with the same angular degree l and consecutive radial order n. 4.1. Selecting the optimum model Therefore, in our paper, we think that the matching the observable large separations is the important criterion to select the optimum Using Guenther’s pulsation code (Guenther, 1994), we calculate model. In Table 3, we find that the average large separations of the adiabatic low-lp-mode frequencies of the selected models. We Model 60, Model 125 and Model 126 are 161.7, 161.92 and define the large separations Dm and small separations dm in the 161.68 lHz, in good agreement with the mean value derived from usual way (Tassoul, 1980): Carrier and Eggenberger (2006). So we can tentatively say that these models may be the best fit models. In Fig. 2, we plot the observa- Dm m m ð2Þ n;l n;l n1;l tional results for the large separations and the errors. Also we plot and the large separation as a function of frequency for Model 54 in Fig. 2a, Model 60 in Fig. 2b, Model 125 in Fig. 2c and Model 126 dm m m ; ð3Þ n;l n;l n1;lþ2 in Fig. 2d. We clearly find that the theoretical large separations of where n is the radial order, l is the degree, and m is the frequency. the Model 60, Model 125 and Model 126 are consistent with the Because the expected acoustic cutoff has a limit, we only calculate observations. Model 54, as the representative of many non-fit mod- the mean large- and small separations by averaging over n = 10– els, is not consistent with the observational large separations. 30 (see Murphy et al., 2004). Within these 44 tracks, we list 129 Therefore, we have sufficient reasons to say that Model 60, Model models in Table 3. hDmli represents the mean of large separations 125 and Model 126 are really the best fit models. Meanwhile, we Dmn,l for n = 10–30. The frequency range corresponds to about can predict that the radius of star is 0.860–0.865 R and the age 2000–6000 lHz. Additionally, hDmi represents the mean of hDmli for is about 6.8–7.0 Gyr with mass 0.89–0.90 M presently. l = 0–3. In the same way, hdm02i and hdm13i represent the mean of Once the asteroseismic observation can confirm the large sepa- dmn,0 and dmn,1 for n = 10–30, respectively. So far, we only know the rations to be 161.7 ± 0.3 lHz and the theory Model 60, Model 125, large separations and the fourteen individual modes of the star Model 126 are considered as the best models, we can predict that based on the asteroseismic data of Carrier and Eggenberger the mean small separation hdm02i is about 10.29–10.48 lHz and the

(2006). Guenther (1998) pointed that the large separations are most radius of the star is about 0.860–0.865 R. Direct measurements of easily identiﬁable characteristics in the p-mode spectrum. Because stellar diameters from interferometric observations should provide 544 Y.K. Tang et al. / New Astronomy 13 (2008) 541–548

l=0 l=0 l=1 l=1 a l=2 b l=2

l=0 l=0 l=1 l=1 c l=2 d l=2

Fig. 2. Large-frequency separations vs. frequency for the Models 54, 60, 125 and 126 (in Table 3). The observable large separations Dm versus frequency for p-modes of degree l =0(d), l =1(N) and l =2(j) are obtained from Carrier and Eggenberger (2006), which correspond to the average large separation of 161.7 lHz. Open symbols correspond to pffiffiffi large separation averages taken between non-successive modes and vice versa. All individual errors are fixed to 2 1:1 lHz (half resolution). an independent check for asteroseismic predictions such as Kervel- 4.2. Asymptotic formula and frequency analysis la et al. (2003a); Kervella et al. (2003b). In order to compare the theoretical p-mode frequencies de- 4.2.1. Large separations and small separations duced from the models in Table 3 with the observational frequen- It is well-known from asymptotic theory that the large separa- cies provided by Carrier and Eggenberger (2006), we plot the tions are mainly sensitive to the stellar radius (Tassoul, 1980; echelle diagram of every model and find that no model can fit Christensen-Dalsgaard, 1984). More precisely, the asymptotic observational frequencies. For the exact values of the frequencies, behavior of Dm is expected to scale with (M/R3)1/2, where M is the considering the above three reasons, a linear shift of a few lHz be- mass of the star and R is its radius. Murphy et al. (2004) find that tween theoretical and observational frequencies is perfectly a degeneracy in predicted radius occurs for models of different acceptable. Taking it into account, we define the mean value of mass. Here, the degeneracy means that the hDmi changes with ra- the difference between the theoretical and observational frequen- dius and mass (see Fig. 4 in Murphy et al., 2004). In order to lift cies (e.g., Eggenberger et al., 2004b, 2005): the degeneracy, Fernandes and Monteiro (2003) and Murphy et al. (2004) assumed homology to compare theoretical models 1 XN hD i mtheo mobs ; ð4Þ by introducing a ‘‘reduced” radius, such as m N i i i¼1 3=2 hDmri¼hDmn;liðR=RÞ : ð5Þ where N is the number of observable frequencies (N = 14).

Taking into account the systematic difference hDmi between the- Here, we name the quantity hDmri ‘‘reduced” large separation. We oretical and observable frequencies, we plot the differences be- draw the ‘‘reduced” large separation hDmri versus mass in Fig. 5 tween calculated and observed frequencies in Fig. 3 and the and list the values of hDmri for each model in Table 3. From Fig. 5, echelle diagram in Fig. 4. The observable frequencies correspond we ﬁnd that the degeneracy was lifted approximately. It is easily to the average large separation of 161.7 lHz in these ﬁgures. seen that the values of the hDmri are relatively consistent with each

Fig. 3a, b, c and d correspond to the Model 54 with hDmi = mass. It is successful using the hDmri instead of large separations to 30.059 lHz, Model 60 with hDmi = 63.75 lHz, Model 125 with indicate the stellar mass. hDmi = 70.252 lHz, and Model 126 with hDmi = 63.24 lHz, respec- The small separations, like the large separations, will be visible tively. Fig. 4a, b, c and d show the echelle diagram of the Model as peaks in the Fourier transform of the power spectrum. At the 54, Model 60, Model 125 and Model 126 respectively. For p-modes earlier stage, Christensen-Dalsgaard (1984) proposed that the cal- in the asymptotic theory (n l), the large separations are nearly culation of small separations could put a constraint on the age the constant; meanwhile the so-called ‘‘echelle diagrams” present star. Subsequently, Ulrich (1986) realized that only if the composi- the frequencies in ordinates, and the same frequencies modulo tion of the star is known completely can one use the small separa- the average large separation in abscissa. So the asymptotic theory tions to correctly identify a stellar age. This point has been predicts an approximated vertical line for given degree. In this illustrated in Murphy et al. (2004). Thus, the various chemical com- case, Fig. 4b, c and d show that the theoretical frequencies of Model positions create a degeneracy in age determination (see Murphy 60, Model 125, Model 126 can ﬁt the observable frequencies with et al., 2004). Namely, the small separations dm change with the ini-

161.7 lHz. Meanwhile, we ﬁnd that the systematic differences hDmi tial composition and age. In the next section, we will discuss this are larger than the results of a Cen B obtained by Eggenberger et al. problem and propose a quantity which may be correlated with (2004b). It is interesting to analyze the difference in future work. stellar age. Y.K. Tang et al. / New Astronomy 13 (2008) 541–548 545

a b

c d

Fig. 3. Differences between calculated and observable frequencies for the Model 54, 60, 125 and 126 in Table 3. The systematic shifts hDmi for the four models are 30.059, 63.75, 70.252 and 63.24 lHz, respectively. The observable frequencies correspond to the average large separation of 161.7 lHz (see text for details).

c d

Fig. 4. Echelle diagrams for the Models 54, 60, 125 and 126 (in Table 3), which average large separations hDmi are 166.22, 161.7, 161.92 and 161.68 lHz, respectively. The systematic shift hDmi, which have been applied to the theoretical frequencies, are 30.059, 63.75, 70.252 and 63.24 lHz, for the four models, respectively. Open symbols refer to the theoretical frequencies, and ﬁlled symbols to the observable frequencies. Circles are used for l = 0 modes, triangles for l = 1 modes, and squares for l = 2 modes. The observable frequencies correspond to the average large separation of 161.7 lHz (see text for details).

4.2.2. A new quantity proposed because of their small amplitude. So far, we only obtain knowledge At the present time, we can know the stellar internal structure of the stellar interior from the limited modes (l = 0, 1, 2, 3) which and understand the stellar evolution from oscillation frequencies. can be observed. Many authors proposed some quantity as diag- Thus asteroseismology provides a window to ‘‘see” the interior of nostic purposes to probe the stellar internal and put constraints star. But the observation of solar-like oscillations is very difﬁcult on the model parameters (Christensen-Dalsgaard, 1984, 1988, 546 Y.K. Tang et al. / New Astronomy 13 (2008) 541–548

tions of the equilibrium model. It should be noted that the classical asymptotic theory of Tassoul (1980), although providing good results at the ﬁrst order in frequency, does not represent with accu- racy the p-mode spectrum of the stars considered. Several authors (e.g., Gabriel, 1989; Audard and Provost, 1994; Roxburgh and Vorontsov, 2000a,b, 2001) have discussed the difﬁculties of the asymptotic theory, particularly for evolved models with rapid variation in the sound speed in the core. Using Eq. (2) and the asymptotic formula (10), the large separation can be written as follows (Gough and Novotny, 1990):

2 mn1;l mn;l Dmn;l ¼ mn;l mn1;l ¼ m0 ½Alðl þ 1ÞBm0 : ð11Þ mn;l mn1;l

Taking the first order of mn,l for the n l approximately, we can obtain the result like Gough and Novotny (1990) and Eq. (11) becomes "# 1 Alðl þ 1ÞB Dm m m : n;l 0 1 l l 0 ð12Þ ðn þ 2 þ Þðn 1 þ 2 þ Þ Fig. 5. The ‘‘reduced” large separation vs. mass for each of the 129 stellar models. Using the same approximate method, we can obtain the expression of small separation 1993; Ulrich, 1986, 1988; Gough, 1987, 1990, 2003). For p-modes ð4l þ 6Þm0A of solar-like stars, the usual frequency separations are the large dmn;l ¼ dml;lþ2 : ð13Þ n þ l þ separation defined by Eq. (2) and the small separation defined by 2 Eq. (3). Additionally, Roxburgh (1993) and Roxburgh and Voront- As the small separations are rather sensitive to composition and sov (2003) considered the following separations: therefore to the structure of the core, especially the extreme sensi- tivity of the stellar core density stratification to several parameters 1 d01ðnÞ¼ ðmn1;0 4mn1;1 þ 6mn;0 4mn;1 þ mnþ1;0Þ; ð6Þ (Guenther and Demarque, 2000; Morel et al., 2000), we define an- 8 1 other quantity regarding the ratio of average small separation adja- d ðnÞ¼ ðm 4m þ 6m 4m þ m Þ; ð7Þ 10 8 n1;1 n1;0 n;1 nþ1;0 nþ1;1 cent in l and defined the ratios d of small to large separations as follows: hdm0;2i ij r01 ¼ : ð14Þ hdm1;3i dm02ðnÞ dm13ðnÞ d02ðnÞ¼ ; d13ðnÞ¼ ; Dm1ðnÞ Dm0ðn þ 1Þ Using Eq. (14), we calculate the values of r01 and list it in Table 3. ð8Þ Based on the results of numerical calculations, we plot the ratio dm01ðnÞ dm10ðnÞ d01ðnÞ¼ ; d10ðnÞ¼ : r versus age in Fig. 6. Fortunately, we find that the ratio r is Dm1ðnÞ Dm0ðn þ 1Þ 01 01 tightly correlated with age and decreases monotonously with age. The ratios dij of small to large separations are independent of the We think that this most likely is due to the perturbation to the grav- structure of the outer layers of a star, and therefore provide a diag- itational potential, neglected in the asymptotic relation (10), which nostic of the stellar interior alone. affects modes of the lowest degrees most strongly and which prob- In addition, Gough (1990), Monteiro and Thompson (1998), ably increases with evolution due to the increasing central density. Vauclair and Théado (2004), Houdek and Gough (2007a) gave the These effects are most important for modes of the lowest degrees second differences D2ml(n): which penetrate most deeply and hence affect dm0,2 more than

D2mlðnÞ¼mnþ1;l þ mn1;l 2mn;l: ð9Þ dm1,3, leading to the dependence of r01 on age.

The second differences D2ml(n) can be used to reveal the variation of the first adiabatic exponent c1 dependent on the influence of the ionization of helium on the low-degree acoustic oscillation frequencies in the model of solar-type stars. Recently, Houdek and Gough (2007b) stated that the second differences can provide a measure of helium abundance and hence precisely lift the degeneracy between composition and age. Summarizing the above character separation, we find that the investigation of lifting the degeneracy between the chemical compositions and the age is interesting. We begin with our investigation from a well-known asymptotic formula.

The asymptotic formula for the frequency mn,l of a stellar p-mode of order n and degree l was given by Tassoul (1980) l m ’ n þ þ m ½Alðl þ 1ÞBm2m1; ð10Þ n;l 2 0 0 n;l where the characteristic m0 is related to the run of sound travel time across the stellar diameter; A is a measure of the sound speed gra- dient and most sensitive to conditions in the stellar core (see Gough and Novotny, 1990; Gough, 2003; Christensen-Dalsgaard, 1993; Fig. 6. The ratio of small separations adjacent in l vs. age for each of 129 stellar Guenther and Brown, 2004), and B are constants which are func- models. Y.K. Tang et al. / New Astronomy 13 (2008) 541–548 547

From Fig. 6, the values of r0,1 in Table 3 and the above discus- work was supported by The Ministry of Science and Technology of sion, we can conclude that this quantity r01 can lift the degeneracy the People’s republic of China through Grant 2007CB815406, and between the chemical compositions and age. The analysis was in- by NSFC Grants 10173021, 10433030, 10773003, and 10778601. spired by Fernandes and Monteiro (2003) and Murphy et al. (2004). So, we can obtain r which may indicate stellar age, if we consider 01 Appendix A. Supplementary data a frequency ratio. As illustrated in Fig. 6, the quantity r01 is tightly correlated with stellar age over a substantial range of the remain- Supplementary data associated with this article can be found, in ing parameters, including composition. At the same time, we need the online version, at doi:10.1016/j.newast.2008.02.002. to point out that the range of variation of this quantity is relatively modest, compared to likely observational errors. Also, it is clear that the present data for 70 Ophiuchi A are not adequate to evalu- References ate this quantity. We think that using this quantity to evaluate the stellar age will be convenient based on asteroseismic data which Alexander, D.R., Ferguson, J.W., 1994. ApJ 437, 879. Audard, N., Provost, J., 1994. A&A 282, 73. will be provided in the future. Bahcall, J.N., Pinsonneault, M.H., Wasserburg, G.J., 1995. RvMP 67, 781. Batten, A.H., Fletcher, J.M., Campbell, B., 1984. PASP 96, 903. Bedding, T.R., Butler, R.P., Kjeldsen, H., et al., 2001. ApJ 549, L105. 5. Discussion and conclusions Bedding, T.R., Butler, H., Kjeldsen, R.P., McCarthy, C., Marcy, G.W., O’Toole, S.J., Tinney, C.G., Wright, J.T., 2004. ApJ 614, 380. Bouchy, F., Carrier, F., 2002. A&A 390, 205. The models of 70 Ophiuchi A are obtained by fitting effective Bouchy, F., Bazot, M., Santos, N.C., Vauclair, S., Sosnowska, D., 2005. A&A 440, 609. temperature, luminosity, surface metallicity and asteroseismic Carrier, F., Bourban, G., 2003. A&A 406, L23. observations. Carrier, F., Eggenberger, P., 2006. A&A 450, 695. Carrier, F., Bouchy, F., Kienzle, F., Bedding, T.R., Kjeldsen, H., Butler, R.P., Baldry, I.K., We list a series of possible models in Table 3. So far, we think O’Toole, S.J., Tinney, C.G., Marcy, G.W., 2001. A&A 378, 142. that Model 60, Model 125 and Model 126 are the best models. Carrier, F., Bouchy, F., Eggenberger, P., 2003b. In: Thompson, M.J., Cunha, M.S., With the advance in observation, the precise asteroseismic data Monteiro, M.J.P.F.G. (Eds.), Asteroseismology Across the HR Diagram. Kluwer, p. P311. will provide more strict constraints on theoretical models. The con- Carrier, F., Eggenberger, P., Bouchy, F., 2005a. A&A 434, 1085. clusions of the paper are: Carrier, F., Eggenberger, P., DAlessandro, A., Weber, L., 2005b. NewA 10, 315. Christensen-Dalsgaard, J., 1984. In: Mangeney, A., Praderie, F. (Eds.), Space Research Prospects in Stellar Activity and Variability. Paris Observatory Press, Paris, p. (1) Using the latest asteroseismic observation, we try our best to P11. construct the best model of 70 Ophiuchi A. So far, we only Christensen-Dalsgaard, J., 1988. In: Christensen-Dalsgaard, J., Frandsen(Reidel), S. select the Model 60, Model 125 and Model 126, which can (Eds.), Advances in Helio and Asteroseismology. p. 295. Christensen-Dalsgaard, J., 1993. In: Bron, T.M. (Ed.), Seismic Investigation of the Sun be fit for the observation, as the optimum models. These and Stars. A.S.P. Conf. Ser., vol. 42. p. 347. models correspond to the radius of this star being about Eggenberger, P., Carrier, F., Bouchy, F., Blecha, A., 2004a. A&A 422, 247. Eggenberger, P., Charbonnel, C., Talon, S., Meynet, G., Maeder, A., Carrier, F., 0.860–0.865 R and the age about 6.8–7.0 Gyr with mass 0.89–0.90 M . Bourban, G., 2004b. A&A 417, 235. Eggenberger, P., Carrier, F., Bouchy, F., 2005. NewA 10, 195. (2) By calculating many theoretical models, we want to use the Fernandes, J., Monteiro, M.J.P.G., 2003. A&A 399, 243. theoretical frequencies to compare with observational fre- Fernandes, J., Lebreton, Y., Baglin, A., Morel, P., 1998. A&A 338, 455. quencies and help to definitely validate the observational Gabriel, M., 1989. A&A 226, 278. Gough, D.O., 1987. Nature 326, 257. data. Gough, D.O., 1990. In: Progress of Seismology of the Sun and Stars. Proceedings of (3) We obtain a new quantity r01 which can lift the degeneracy the Oji International Seminar Hakone. Lecturer Notes Physics, vol. 367. Springer between the initial compositions and the stellar age. By cal- Verlag, Japan; p. 283. Gough, D.O., 2003. Ap&SS 284, 165. culation, we prove that it can be valuable for the indication Gough, D.O., Novotny, E., 1990. SoPh 128, 143. of the stellar age. Gray, D.H., Johanson, H., 1991. PASP 103, 439. (4) An important point is that we test our stellar structure and Grevesse, N., Sauval, A.J., 1998. SSRv 85, 161. Guenther, D.B., 1994. ApJ 422, 400. evolution theory. Meanwhile we find the confrontation of Guenther, D.B., 1998. In: Korzennik, S., Wilson, A., (Eds.), Proc SOHO 6/GONG 98 observations and theoretical models. Actually we have Workshop. Structure and Dynamics of the Interior of the Sun and Sun-like Stars, neglected some important effects, such as rotation and mag- ESA SP-418. p. 375. Guenther, D.B., Brown, K.I.T., 2004. ApJ 600, 419. netic field, which can have an impact on the internal struc- Guenther, D.B., Demarque, P., 2000. ApJ 531, 503. ture and evolution. Thus, the theory models which we Guenther, D.B., Demarque, P., Kim, Y.-C., Pinsonneault, M.H., 1992. ApJ 387, 372. have constructed cannot fit observation perfectly. The Heintz, W.D., 1988. JRASC 82, 140. detailed comparisons of individual mode frequencies will Houdek, G., Gough, D.O., 2007a. MNRAS 375, 861. Houdek, G., Gough, D.O., 2007b. AIPC 984, 219. also require taking into account the effects of turbulence in Iglesias, C.A., Rogers, F.J., 1996. ApJ 464, 943. the outer convective unstable layers in the stellar models, Kervella, P., Thévenin, F., Morel, P., Bordé, P., Di Folco, E., 2003a. A&A 408, 681. which shift the observed frequencies. The parameterization Kervella, P., Thévenin, F., Ságransan, D., Berthomieu, G., Lopez, B., Morel, P., Provost, J., 2003b. A&A 404, 1087. of turbulence tested on the Sun by Li et al. (2002) can be Kervella, P., Thévenin, F., Morel, P., Berthomieu, G., Bordé, P., Provost, J., 2004. A&A applied to models for solar-like stars as well. This parame- 413, 251. terization can be extended to other stars by using the Kjeldsen, H., Bedding, T.R., Baldry, I.K., Bruntt, H., Butler, R.P., Fischer, D.A., Frandsen, S., Gates, E.L., Grundahl, F., Lang, K., Marcy, G.W., Misch, A., Vogt, S.S., 2003. AJ three-dimensional radiative hydrodynamic simulations of 126, 1483. Robinson et al. (2003), which are based on the same micro- Kjeldsen, H., Bedding, T.R., Butler, R.P., Christensen-Dalsgaard, J., Kiss, L.L., scopic physics and can readily be parameterized in the YREC McCarthy, C., Marcy, G.W., Tinney, C.G., Wright, J.T., 2005. ApJ 635, 1281. Li, L.H., Robinson, F.J., Demarque, P., Sofia, S., Guenther, D.B., 2002. ApJ 567, 1192. stellar evolution code. Martic´, M., Lebrun, J.-C., Appourchaux, T., Korzennik, S.G., 2004a. A&A 418, 295. Martic´, M., Lebrun, J.C., Appourchaux, T., Schmitt, J., 2004b. In: Danesy, D. (Ed.), SOHO 14/GONG 2004 Workshop, Helio- and Asteroseismology: Towards a Golden Future, ESA SP-559. p. 563. Acknowledgements Miglio, A., Montalbán, J., 2005. A&A 441, 615. Monteiro, M.J.P.F.G., Thompson, M.J., 1998. New eyes to see inside the sun and stars. We are grateful to anonymous referees for their constructive In: Deubner, F.L., Christensen-Dalsgaard, J., Kurtz, D.W. (Eds.), Proc. IAU Symp, suggestions and valuable remarks to improve the manuscript. This 185. Kluwer, Dordrecht, p. 317. Morel, P., 1997. A&AS 124, 597. 548 Y.K. Tang et al. / New Astronomy 13 (2008) 541–548

Morel, P., Baglin, A., 1999. A&A 345, 156. Roxburgh, I.W., 1993. In: Approurchaux, T. (Ed.), PRISMA, Report of Phase A Study, Morel, P., Provost, J., Lebreton, Y., Thévenin, F., Berthomieu, G., 2000. A&A 363, 675. ESA 93. p. 31. Mosser, B., Bouchy, F., Catala, C., Michel, E., Samadi, R., Thévenin, F., Eggenberger, P., Roxburgh, I.W., Vorontsov, S.V., 2000a. MNRAS 317, 141. Sosnowska, D., Moutou, C., Baglin, A., 2005. A&A 431, L13. Roxburgh, I.W., Vorontsov, S.V., 2000b. MNRAS 317, 151. Murphy, E.J., Demarque, P., Guenther, D.B., 2004. ApJ 605, 472. Roxburgh, I.W., Vorontsov, S.V., 2001. MNRAS 322, 85. Perrin, M.N., Cayrel, de S.G., Cayrel, R., 1975. A&A 39, 97. Roxburgh, I.W., Vorontsov, S.V., 2003. A&A 411, 215. Peterson, R., 1978. ApJ 224, 595. Tassoul, M., 1980. ApJS 43, 469. Pourbaix, D., 2000. A&AS 145, 215. Thévenin, F., Provost, J., Morel, P., Berthomieu, G., Bouchy, F., Carrier, F., 2002. A&A Provost, J., Martic, M., Berthomieu, G., 2004. ESA SP 559, 594. 392, 9. Provost, J., Berthomieu, G., Bigot, L., Morel, P., 2005. A&A 432, 225. Thoul, A.A., Bahcall, J.N., Loeb, A., 1994. ApJ 421, 828. Provost, J., Berthomieu, G., Martic´, M., Morel, P., 2006. A&A 460, 759. Ulrich, R.K., 1986. ApJ 306, L37. Robinson, F.J., Demarque, P., Li, L.H., Soﬁa, S., Kim, Y.-C., Chan, K.L., Guenther, D.B., Ulrich, R.K., 1988. IAUS 123, 299. 2003. MNRAS 340, 923. Vauclair, S., Théado, S., 2004. A&A 425, 179. Rogers, F.J., Nayfonov, A., 2002. ApJ 576, 1064.