Modeling ɛ Eri and Asteroseismic Tests of Element Diffusion

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4 594 N. Gai et al Basic Equation for Element Diffusion Most studies on chemical diffusion use either the method of Chapman - Enskog (Chapman & Cowling 1970) or the method of Burgers (1969) for deriving the transport properties from the Boltzmann equation. In our work the element diffusion is described using the equations of Burgers (1969) with the diffusion velocity coefficients from Thoul et al. (1994). A single model includes both helium and metal element diffusion. The change rate of the element mass fractions due to the diffusion is now written as (Thoul et al. 1994; Bahcall & Loeb 1990): X s t = 1 [r 2 X s T 5/2 ξ s (r)] ρr 2, (2) r where the partial derivatives are evaluated in the local rest frame of the mass shell inside the star, ρ, r, T are the local values of density, radius, temperature, and X s is the mass fraction of the element s. The diffusion velocity for species s is defined by and the function ξ s (r) has the expression, ξ s (r) =A p (s) ln p r w s = T 5/2 ξ s, (3) ρ + A T (s) ln T r + c e,2 A c (s) ln C c r. (4) In Equation (4), A p, A T, A c correspond to the gravitational setting, thermal diffusion and concentration gradient diffusion, respectively, and are functions of the mass only. The suffix c refers to all species other than helium Treatment of Diffusion in the Evolutionary Code The Yale stellar evolution code has been modified to include the effects of He diffusion from gravitational settling and thermal diffusion (Bahcall & Pinsonneault 1992a) using the method of Bahcall & Loeb (1990) and with the metal diffusion added in The method by which element diffusion is treated in a standard stellar evolution code has been presented by Bahcall & Pinsonneault (1992a) in detail. In the diffusion subroutine given by Bahcall & Pinsonneault (1992a), the main task is to solve the diffusion equation and calculate the change of the element abundance by diffusion. Equation (2) can be written in the following form, where D 1, D 2 are the diffusion coefficients. For helium diffusion, { D 1 (Y )= and for metal diffusion, dx dt = 1 [ d ρr 2 dr (D 1)+ d ( dr D 2 (Y )= { D 1 (Z) = D 2 (Z) = 5/2 Fgy r2 T ln Λ Fgy r2 T 5/2 ln Λ A X c, 5/2 Fgz r2 T ln Λ Fgz r2 T 5/2 ln Λ A Z c. D 2 dx dr d ln p dr X (A X p + A X T ), d ln p dr Z (A Z p + AZ T ), )], (5) The radius, temperature and density are in the nondimensional units defined by r = r /R, T = T /10 7 K, ρ = ρ /100 gcm 3 and t = t /10 13 yr, respectively (Bahcall & Pinsonneault 1992a; Chaboyer et al. 1992). In Equations (6) and (7), F gy and F gz are adjustable multiplicative factors of the helium and metal diffusion coefficients. In our work, we set both these two factors to 1.0. The exact numerical solution of A p, A T, A c are calculated by an export subroutine developed by Thoul (1994). } } (6) (7)

5 Modeling ɛ Eri and Asteroseismic Tests of Element Diffusion 595 Table 2 Model Parameters of ɛ Eridani Model A1 A2 A3 B1 B2 B3 C1 C2 C3 M/M =0.83 M/M =0.83 M/M =0.83 Z i =0.012 Z i =0.013 Z i =0.014 Y i = Y i = Y i = Dif. None. Y.Dif Y & Z.Dif None. Y.Dif Y & Z.Dif None. Y.Dif Y & Z.Dif Age(Gyr) L/L R/R T eff (K) r cz/r τ cz (s) τ 0 (s) τ cz(s)/τ 0 (s) Δν In Equation (5) the partial time derivative is evaluated at constant mass shell of the star. The diffusion equation is solved with zero hydrodynamic velocity of the stellar plasma. In the right hand of Equation (5), the first term depends on the first spatial derivatives of the element mass fraction and the second term on the second spatial derivatives of the element abundance. The first term is solved explicitly using the two-step Lax-Wendroff technique (Press et al. 1986) by neglecting the second derivatives of the element abundance. We then use this trial solution as the initial abundance to determine the second derivatives by a fully implicit method (Bahcall & Pinsonneault 1992a) and so obtain the final updated element abundance. For the helium diffusion, we only consider the diffusion of 4 He and ignore that of 3 He (Loeb et al. 1989). The time rate of change of the 4 He mass fraction Y is equal in magnitude and opposite in sign to the rate of change of the hydrogen mass fraction. The diffusion of all metal elements is assumed to proceed at the same rate as fully ionized iron. Because of the diffusion of metal elements, the radiative opacity has to be changed after each time step for each spherical shell. We calculate the effect of metal element diffusion on the opacity by computing the total metal element abundance at each model radius Z(r), and then interpolating for the opacity between opacity tables with different total metal element abundances (Bahcall et al. 1995). The diffusion subroutine carries out the diffusion calculations using data supplied by other parts of the Yale code, in which the thermal structure and the element abundances are calculated. It is assumed that the amount of diffusion within a given time step is too small to affect significantly the changes in the thermal structure, abundances, and nuclear reaction rates that are calculated elsewhere. After the diffusion computation, we update the element abundance and then calculate the thermal structure. In the diffusion subroutine, the element diffusion is treated only in the radiation region. The characteristic time for the element to diffuse through a solar radius under solar conditions is of the order of yr (Bahcall et al. 1995), much larger than the age of the sun or ɛ Eri. However, helioseismic inferences have demonstrated the significance of element diffusion (Cox et al. 1989; Bahcall & Loeb 1990; Bahcall & Pinsonneault 1992a; Proffitt 1994). 3.3 Method of Computation and Results of Analysis Evolutionary Tracks and Candidates for Pulsation In order to reproduce observational constraints of ɛ Eri, we have computed a grid of evolutionary tracks for six values of M/M : 0.80, 0.81, 0.82, 0.83, 0.84 and 0.85, and for values of the initial metallicity Z i : 0.011, 0.012, and From a series of evolutionary tracks computed, we selected those models that land within the observational constraints in log T eff, log L/L and log R/R in the H-R diagram. Then we further chose the models on the basis of the constraint of the observed [Z/X]. From the computation alone, we note that the age of ɛ Eri is likely to be in the range Gyr. Here we consider the age ( 1 Gyr) as a property of the star considered in SD 89. According to the age estimated from observation, as given in the introduction, we

6 596 N. Gai et al. Fig. 1 (a) Evolutionary tracks (12 in total) in the H-R diagram starting from the ZAMS. The solid lines mark the box delimited by the observed luminosity and effective temperature. The two dash dot lines mark the region delimited by the interferometric radius. (b) Enlargement of the error box of panel (a). The models A3, B3 and C3 are denoted by the filled circle, triangle and pentacle, respectively. constrain the age of models to the range of Gyr. We have 12 tracks falling within the observational error box and these are plotted in Figure 1(a). To deduce the set of parameters that best agrees with the observations, we performed a χ 2 minimization as described in Eggenberger (2005). The χ 2 function is defined as χ 2 4 ( C theo i Ci obs ) 2, (8) σ i=1 where the vector Ci obs (L/L, T eff, R/R, [Z/H] s ). The vector Ci obs and the observational error vector σ are given in Table 1. The χ 2 minimization yields three models, A3, B3 and C3, and these are presented in Figure 1(b) and Table Calibration of Models with and without Diffusion Based on the selected models A3, B3 and C3, we calibrated three groups of models with mass M/M = 0.83 and different initial chemical compositions. The models of the same group have the same initial chemical compositions, but with different types of element diffusion. See Table 2. We use the stellar pulsation code of Guenther (1994) to calculate the eigenfrequencies and the large spacings of the models listed in Table 2. For solar-like stars, the eigenfrequencies ν n,l of oscillation modes, characterized by the radial order n at harmonic degree l, satisfy the simplified asymptotic relation (Tassoul 1980): ( ν n,l =Δν n + l 2 + α + 1 ) + ɛ n,l. (9) 4

7 Modeling ɛ Eri and Asteroseismic Tests of Element Diffusion 597 Fig. 2 Stellar structures of the group (C) models in Table 2 with Y i = and Z i = Solid lines: standard homogeneous model C1 without diffusion; dashed lines: model C2 with pure helium diffusion; dash dot lines: model C3 with both helium diffusion and metal diffusion. (a) Helium profiles as a function of the acoustic depths in the models, i.e. the time for the acoustic waves to travel from the surface down to the considered layer; (b) Metal profiles show a clear gradient at the bottom of the convection zone; (c) Adiabatic exponent Γ 1 profiles show a clear feature at the place of the He II ionization zones; (d) Sound speed gradients in which the dips are clearly visible at the place of the He II ionization zones and the bottom of the convection zones; (e) Second differences of the oscillation frequencies as a function of the frequencies; (f) Fourier transform of panel (e). The large spacings are defined by the frequencies of the same harmonic degree l and the adjacent radial order n: Δν n,l ν n,l ν n 1,l. (10) We average the large spacings over the modes l =0,1,2,3andn = 10, 11,..., 30 to obtain the mean value. From the age of the star in Table 2, we can see that the diffusion could speed up the evolution of stars. The model with both helium and metal diffusion has the smallest age of all the models in the same group. The mean of the large spacings of models A3, B3 and C3 is about 194 ± 1 μhz. The pure helium diffusion hardly affects ν n,l, but the effect is relatively obvious after adding the metal diffusion. It also seems that the diffusion effect is increase alone with increase of the initial heavy element abundance (Y i, Z i ). 4 ASTEROSEISMIC TEST, HELIUM AND METAL DIFFUSION EFFECTS 4.1 Second Differences Tests, Helium and Metal Gradients Stellar acoustic p-modes with low l degree can propagate deeply inside the stars. However, as mentioned by Gough (1990), rapid variations of the sound speed inside a star lead to partial reflections of the sound waves. A conveniently and easily evaluated measure of this oscillatory component is the second differences with

9 Modeling ɛ Eri and Asteroseismic Tests of Element Diffusion 599 Fig. 3 Upper panel: small spacings d 02(n). Lower panel: frequency separation ratios r 02(n) for the models in group (C). 4.2 Tests of Internal Structure The small frequency spacings are given by d ll+2 (n) =ν n,l ν n 1,l+2. (13) Using the asymptotic theory of p modes it is shown (Christensen-Dalsgaard & Berthomieu 1991; Basu et al. 2007), d ll+2 (n) (4l +6) Δν R nl dc dr 4π 2 ν n,l 0 dr r, (14) where R is the radius of the star and Δν nl is the large spacing. Now, in the core the gradient of the sound speed is large and r is small, so the integral in Equation (14) is dominated by the conditions in the core. So the small spacings usually test the stellar core, but the small spacings are also slightly affected by effects near the surface. To reduce the effect of uncertainties in the latter, the frequency separation ratios are used (Roxburgh & Vorontsov 2003), r 02 (n) = d 02(n) Δ 1 (n), r 13(n) = d 13(n) Δ 0 (n +1). (15) Each panel of Figure 3 shows the small spacing d 02 and the frequency separation ratios r 02 for the different models of group (C) with different types of diffusion as labeled in the figure. We see that the pure helium diffusion hardly alters the internal structure in these young, low mass models, while metal diffusion makes obvious differences in the internal structure. We now explore the changes of the internal structure from the asteroseismic test by comparing the internal physical parameters of the models with and without diffusion. It is known that the sound speed

10 600 N. Gai et al. Fig. 4 From top down: Relative sound-speed difference δc/c, temperature difference δt/t and mean molecular weight δμ/μ, respectively, as function of radius, for the models of group (C). All differences are with respect to model C1 of no diffusion. The dashed lines for the model with pure He diffusion (eg., δμ/μ =(μ Y.Dif - μ None.Dif )/μ None.Dif ). The dash-dot lines for the model with both He and Z diffusion (e.g. δμ/μ =(μ Y &Z.Dif - μ None.Dif )/μ None.Dif ). depends on both the mean molecular weight and the temperature (Basu et al. 2007; Bi et al ), c 2 K BT μm u T μ. (16) In Figure 4, we show the radial profiles of δc/c, δt/t and δμ/μ. It is seen that the effect of pure helium diffusion on the three parameters is small and hardly alters the values of c, T and μ in the radiation interior, and any effect is concentrated in the convection zone. Due to the diffusion, helium drifts inward just below the convection zone, which makes μ in the envelope smaller. On the other hand, the increase of opacity in the envelope due to the settling of He leads to a lower temperature. However, helium diffusion reduces both the T and μ, and also the sound speed c slightly in the convective envelope. So the small spacings and separation ratios are not much affected by helium diffusion.

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