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Observational Constraints on Models of Beta Hydri

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Observational Constraints on Models of Beta Hydri Observational Constraints on Models of β Hydri João Fernandes Observatório Astronómico da Universidade de Coimbra, Coimbra, Portugal Mário J.P.F.G. Monteiro Departamento de Matemática Aplicada, Faculdade de Ciências da Universidade do Porto, and Centro de Astrofísica da Universidade do Porto, Porto, Portugal 2002 September 25 Abstract. A review of the models for the star β Hydri is done by adjusting the evolutionary sequences to the most recent bolometric, spectroscopic and astrometric data. The dependence of the solution on some of the relevant modelling parameters is analysed and the degeneracy of the solution using the HR Diagram analysis is discussed. We briefly describe how the seismic study of the star can help to clarify some of the degeneracy of the model solutions. Keywords: β Hydri; Stellar Evolution; Oscillations 1. Observations of β Hydri β Hydri (HD 2151, HR 98, HIP 2021) is a G2 IV star, in the neighbourhood of the Sun, belonging to the old disc of the Galaxy. This evolved star is fre- quently seen as the distant future of the Sun (Dravins et al. 1993). Recently, Bedding et al. (2001) and Carrier et al. (2001) confirmed the presence of solar-type oscillations in β Hydri. In Table I we present the available data on β Hydri. Table I. Observational data for β Hydri. Observable Value Source L=L 3:51±0:11 Blackwell et al. (1998);DiBenedetto (1998) Teff (K) 5774±60 DiBenedetto (1998) Z=X 0:019±0:003 Favata et al. (1997);Castro et al. (1999) ∆ν (µHz) 56:2±2:0 Bedding et al. (2001) 2. Models and HR Diagram Analysis The stellar evolution calculations were computed with the CESAM code ver- sion 3. (Morel 1997). Details on the physics of these models can be found in Lebreton et al. (1999). We construct our reference model, hereafter S0, by tuning the parameters in order to reproduce the observations on the HRD. © 2002 Kluwer Academic Publishers. Printed in the Netherlands. 514 Fernandes & Monteiro Figure 1. Mass-Luminosity degeneracy - evolutionary tracks for the test models S0 (full line), S1 (dashed line) and S2 (dotted line) - see Table II. The observed values (given in Table I) are indicated by the box. Table II. Models of β Hydri calculated for different sets of parameters. All models have α=1:4, αov=0:25 and Z=0:014. The frequency separations are given in µHz. The evolutionary tracks for these models are shown in Fig. 1. Mod M=M Y t (Gyr) R=R L=L Teff (K) ∆ν¯ ∆ν¯r S0 1.10 0.27 6.820 1.899 3.540 5751 55.47 55.47 S1 1.05 0.30 6.414 1.878 3.529 5778 55.17 54.27 S2 1.15 0.23 7.125 1.883 3.477 5749 57.45 56.74 Our reference model has the following characteristics: Mass, M=1:1 M , Helium, Y=0:27, Metals, Z=0:014, MLT parameter, α=1:4 and overshoot- ing, αov=0:25. This model reproduces the observed quantities (L, Teff) to as good as 1% for an age of 6.82 Gyr. In Table II we describe other models of β Hydri having the observed luminosity and effective temperature (within the error bars), but using different combinations of the modelling parameters. The knowledge of the HRD position - L and Teff - is not sufficient to define all stellar parameters. In fact, any point in the HRD depends on stellar mass, helium, metallicity, age and some free parameters on the input physics of stellar models. For a single star, other than the Sun and some binary systems (Fernandes et al. 1998), there are far more unknowns than observables. In Observational Constraints on Models of β Hydri 515 the case of β Hydri we find a correlation between helium abundance and mass. Equation (1) quantifies this, reproducing the observed luminosity of β Hydri to within 1% , for the range of mass between 1.0M and 1.15M . The corresponding values of Z=X also fall within the error bars of the observed values. M Y(±0:01) = 1:00 − 0:67 : (1) M With this relation we can estimate the mass of β Hydri, finding that MHRD 0:04 = 1:10 ± ; (2) M 0:07 where the error in Z=X is also included. 3. Seismic Analysis The seismic information that can more easily be identified in the power spec- trum of the oscillations is the large frequency separation ∆ν. This value is expected to scale with (M=R3)1=2. Here we calculate the separation (Monteiro et al. 2002) as being its value at a fixed frequency after removing the periodic component present due to the Helium ionization zone (Monteiro and Thomp- son 1998). Considering that from the observations we can only estimate the radius, the best way to compare the models with the observed seismic data is by considering a “reduced” separation; 3=2 3=4 3 R L Teff;0 ∆ν¯r ≡ ∆ν¯ = ∆ν¯ (3) R0 L0 Teff where the subscript “0” indicates the values for a reference model (in here, this is model S0). For a pure homologous scaling of the separation, ∆ν¯r would be a function of stellar mass alone. It is not however a pure homologous scal- ing as the scaling factor will depend on the physics of the models, namely, the chemical composition and convection, among others (known or unknown). The present bolometric and spectroscopic observations give the determi- nation of the radius with an uncertainty of about 4% while, at present, the large frequency separation is not known to better than 4.0%. Consequently, the uncertainty on the value of the mass is still large, with the preliminary result giving that M∆ν¯ = 1:09 ± 0:22 : (4) M where we have used that for β Hydri, ∆ν¯r=55:2 µHz. If the large frequency separation is determined with a precision below 1% –which is expected to be achieved by the space missions in Asteroseismology– the result is dominated by the error in the Luminosity, which results on an 516 Fernandes & Monteiro uncertainty for the mass of about 14%. This is an independent estimate of the value determined with the HRD analysis, as discussed above, and therefore extremely valuable to raise some of the degeneracy we have found. 4. Conclusions In this work we have recalculated the best models for adjusting β Hydri in the HR Diagram. The modelling parameters which have been evaluated are the Helium abundance (Y ) and the stellar mass (M). The HRD analysis does not allow us to determine these parameters due to the strong degeneracy in (Y;M). Given the recent seismic observations of this star, and the planned missions in Asteroseismology, we have discussed the possibility of using seismic observations to remove the (Y;M) degeneracy (see also Fernandes & Monteiro 2003). It has been found that the large frequency separation ∆ν¯r can provide, at least partially, the necessary extra dimension to the HRD. Acknowledgements This work was supported in part by Fundação para a Ciência e a Tecnologia - Portugal, through projects PESO/P/PRO15128/1999 and ESO/FNU/43658/2001. References Bedding, T.R., Butler, R.P., Kjeldsen, H., et al.: 2001, ApJ 549, L105. Blackwell, D.E., Lynas-Gray, A.E.: 1998, A&A 129, 505. Carrier, F., Bouchy, F., Kienzle, F., et al., 2001, A&A 378, 142. Castro, S., Porto de Mello, G.F., da Silva, L.: 1999, MNRAS 305, 693. DiBenedetto, 1998, A&A 339, 858. Dravins, D., Lindegren, L., Nordlund, A., et al.: 1993, ApJ 403, 385. Favata, F., Micela, G., Sciortino, S.: 1997, A&A 323, 809. Fernandes, J., Monteiro, M.J.P.F.G.: 2003, A&A (in press). Fernandes, J., Lebreton, Y., Baglin, A., Morel, P.: 1998, A&A 338, 455. Lebreton, Y., Perrin, M.-N., Cayrel, R., Baglin, A., Fernandes, J.: 1999, A&A 350, 587. Monteiro, M.J.P.F.G., Thompson, M.J.: 1998, in: F.-L. Deubner, J. Christensen-Dalsgaard and D.W. Kurtz (eds.), New Eyes to See Inside the Sun and Stars, Kluwer, 317. Monteiro, M.J.P.F.G., Christensen-Dalsgaard, J.,Thompson, M.J.: 2002, in: F. Favata, I.W. Roxburgh and D. Galadì-Enríquez (eds.), Stellar Structure and Habitable Planet Finding, ESA SP-485, ESA Publications Division, 291. Morel, P., 1997, A&A 124, 597..
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