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Up: The mode trapping properties


   
3 Results

For our template model we have computed g-modes with $\ell = 1, 2$ and 3 (we do this because geometric cancellation effects grow progressively for larger $\ell$ in non-radial oscillations; see Dziembowski 1977), with periods in the range of 50 s $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}P_k
\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}$ 1300 s. Let us quote that for mode calculations we have employed up to 5000 mesh points. For all of our pulsation calculations, the relative difference between Pk and $P_k^{\rm V}$remains lower than 10-3. This gives an indication of the accuracy of our calculations.

We begin by examining Figs. 2 to 4, the upper panels of which show the logarithm of the oscillation kinetic energy of modes with, respectively, $\ell = 1, 2$ and 3 in terms of computed periods. Middle panels depict the values for the forward period spacing $\Delta P_k$ ( $\equiv P_{k+1} - P_k$) together with the asymptotic value $\Delta P_{\rm A}$ as given by dotted lines[*]. Finally, in the bottom panel of these figures we depict the $C_{\ell ,k}$ values as well as the asymptotic values (dotted lines) that these coefficients adopt for high overtones, that is $C_{\ell,k} \approx 1 / \ell (\ell + 1)$ (Brickhill 1975). An inspection of plots reveals some interesting characteristics. To begin with, the quantities plotted exhibit two clearly different trends. Indeed, for $P_k \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}500 {-} 600$ s and irrespective of the value of $\ell$, the distribution of oscillation kinetic energy is quite smooth. Note that the $\log(E_{\rm kin})_k$ values of adjacent modes are quite similar, which is in contrast with the situation found for lower periods. On the other hand, the period spacing diagrams show appreciable departures of $\Delta P_k$ from the asymptotic prediction (Eq. (3)) for $P_k \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}} 500 {-} 600$ s. As well known, this is due mostly to the presence of chemical abundance transitions in DA white dwarfs. In contrast, for higher periods the $\Delta P_k$of the modes tend to $\Delta P_{\rm A}$. Also, note that the $C_{\ell ,k}$ values tend to the asymptotic value for $P_k \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}500 {-} 600$ s.

 \begin{figure}
\par\includegraphics[width=13.2cm,clip]{H3435F2.eps}
\end{figure} Figure 2: The logarithm of the oscillation kinetic energy, forward period spacing and first order rotation splitting coefficient (upper, middle and lower panels, respectively) for modes with $\ell = 1$ as a function of computed periods. Dotted lines show the asymptotic behaviour for $\Delta P_k$ and $C_{\ell ,k}$. The values of $(E_{\rm kin})_k$ correspond to the normalization $y_1= \delta r / r= 1$ at r= R*.


 \begin{figure}
\par\includegraphics[width=13.2cm,clip]{H3435F3.eps}
\end{figure} Figure 3: Same as Fig. 2, but for $\ell = 2$.


 \begin{figure}
\par\includegraphics[width=13.2cm,clip]{H3435F4.eps}
\end{figure} Figure 4: Same as Fig. 2, but for $\ell = 3$.

An important aspect of the present study is related to the mode trapping and confinement properties of our models. For the present analysis we shall employ the weight functions, WF. We elect WFbecause this function gives the relative contribution of the different regions in the star to the period formation (Kawaler et al. 1985; Brassard et al. 1992a,b). We want to mention that we have also carefully examined the density of kinetic energy (the integrand of Eq. (1)) for each computed mode. For our purposes here, this quantity gives us basically the same information that provided by WF. We show in Fig. 5 to 7 the WF for all of the computed modes corresponding to $\ell = 1$. In addition, we include in each plot of these figures the Ledoux term in arbitrary units (dotted lines) in order to make easier the location of the chemical transition regions of the model. In the interests of a proper interpretation of these figures, we suggest the reader to see also Fig. 2. For low periods, a variety of behaviour is encountered. For instance, the g1 mode is characterized by a WF corresponding to the well known mode trapping phenomenon, that is, g1 is formed in the very outer layers irrespective of the details of the deeper chemical profile, as previously reported by previous studies (see Brassard et al. 1992a,b). In contrast, it is the helium-carbon-oxygen transition that mostly contributes to the formation of the g2 mode, whilst the hydrogen-helium transition plays a minor role. This mode would be representative of the "confined modes'' according to Brassard et al. (1992ab). WF for modes g3 and g4 is qualitatively similar to that of g1, except that they are not exclusively formed in the hydrogen-rich envelope, but also in the helium-carbon-oxygen interface. On the other hand, the high-density zone underlying the helium-carbon-oxygen transition plays a major role in the formation of mode g5. From Eq. (1) is clear that the $(E_{\rm kin})_k$ values are proportional to the integral of the squared eigenfunctions, weighted by $\rho$. As a result, the g5 mode is characterized by a high oscillation kinetic energy value (see Fig. 2). Note that the helium-carbon-oxygen transition region also contributes to the formation of modes g6 and g10. The g10 mode is particularly interesting, because it is formed over a wide range of the stellar interior, thus being also a high kinetic energy mode. The WFs corresponding to remaining modes do not differ appreciably amongst them. They exhibit contributions mainly from the outer layers of the model, though they also show small amplitudes in deeper regions. Note that for all of the modes shown in Figs. 5 to 7 there is a strong contribution to WFs from the hydrogen-helium transition region. This indicate that, as found in previous studies, this chemical interface plays a fundamental role in the period formation of modes. We want to mention that we have elected for this analysis the dipolar modes ($\ell = 1$) for brevity; the results for $\ell= 2, 3$are qualitatively similar to those of $\ell = 1$.

From the analysis performed above based on the weight functions, we can clearly appreciate that for $P_k \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}500 {-} 600$ s the outer layer of the model appreciably contribute to the WFs. This is expected, because, as well known, g-modes in white dwarfs are envelope modes. As mentioned, the WFs of high order modes are very similar, indicating that these modes have essentially the same characteristics. At this point, we could, in principle, classify these modes either like trapped or partially trapped in the outer envelope or like "normal'' modes (in the terminology of Brassard et al. 1992), that is, without enhanced or diminished oscillation kinetic energies as in the case of eigenmodes corresponding to chemically homogeneous stellar models. In fact, the curves $\log(E_{\rm kin})_k - P_k$ depicted in Figs. 2 to 4, in particular for periods exceeding 500 - 600 s, strongly resemble the kinetic energy distribution corresponding to a model in which there no exist chemical interfaces. With the aim of solving such an ambiguity, we have performed pulsation calculations arbitrarily setting B= 0 in the computation of the Brunt-Väisälä frequency. As mentioned, the modified Ledoux treatment employed in the computation of the Brunt-Väisälä frequency bears explicitly the effect from changes in chemical composition by means of the Ledoux term B. So, by forcing B=0 the effects of the chemical transitions are strongly minimized (but not completely eliminated; see inset of Fig. 1c) on the whole pulsational pattern. In this way we obtain an approximate chemically homogeneous white dwarf model (see Brassard et al. 1992b for a similar numerical experiment). The oscillation kinetic energy values resulting for this simulated "homogeneous'' model are shown in Fig. 8 with dotted lines. In the interests of a comparison, we show the results corresponding to our (full) template model with solid lines. It is clear from the figure that the distribution of $(E_{\rm kin})_k$ values for both sets of computations (and for each value of $\ell$) is very similar in the region of long periods. However, note that the curves corresponding to the modified model are shifted to higher energies (by $\approx $0.2 dex) as compared with the situation of the full model. We have carefully compared the WF for each mode of the full model with the corresponding mode of the "homogeneous'' model (i.e. modes which have closest period values although generally for different radial order k). We found that, for modes with periods exceeding $\approx $600 s, the WFs are almost identical in both cases at the regions above the hydrogen-helium transition. However, below this interface the WFs corresponding to the "homogeneous'' model show larger amplitudes as compared with the case of the full model. Thus, we can conclude that for the full model, all the modes corresponding to the long period region of the pulsational spectrum must be considered as partially trapped in the hydrogen-rich envelope. In others words, the chemical distribution at the hydrogen-helium transition has noticeable effect on each mode, but this effect is the same for all modes. This conclusion is reinforced by the fact that the first order rotation splitting coefficients ( $C_{\ell ,k}$) for the full model adopt higher values as compared with those corresponding to the "homogeneous'' model (not shown here for brevity), thus lying nearest to the asymptotic prediction. As found by Brassard et al. (1992a,b), it is an additional characteristic feature of trapped modes in the hydrogen-rich outer region of white dwarfs.

 \begin{figure}
\par\includegraphics[width=11.3cm,clip]{H3435F5.eps}
\end{figure} Figure 5: The normalized weight function (solid lines) in terms of the outer mass fraction, for modes g1 to g8 with $\ell = 1$. In the interests of comparison, dotted lines depict the run of the Ledoux term (arbitrary units).


 \begin{figure}
\par\includegraphics[width=11.3cm,clip]{H3435F6.eps}
\end{figure} Figure 6: Same as Fig. 5, but for modes g9 to g16.


 \begin{figure}
\par\includegraphics[width=11.3cm,clip]{H3435F7.eps}
\end{figure} Figure 7: Same as Fig. 5, but for modes g17 to g24.


 \begin{figure}
\par\includegraphics[width=12.9cm,clip]{H3435F8.eps}
\end{figure} Figure 8: The logarithm of the oscillation kinetic energy for modes with $\ell = 1$, $\ell = 2$ and $\ell = 3$ (upper, middle and lower panel, respectively), as a function of computed periods. In the interests of clarity, symbols corresponding to eigenmodes have been omitted. Solid lines correspond to our template model, and dotted lines correspond to the "homogeneous'' model in which B= 0.


 \begin{figure}
\par\includegraphics[width=11.6cm,clip]{H3435F9.eps}
\end{figure} Figure 9: Upper panel: hydrogen abundance distribution at the hydrogen-helium interface as given by multicomponent, non-equilibrium diffusion (solid line) and diffusive equilibrium in the trace element approximation (thin line). Lower panel: the logarithm of the squared Brunt-Väisälä frequency for the both treatment of diffusion mentioned above. The inset shows the prediction for the Ledoux term. For details, see text.


 \begin{figure}
\par\includegraphics[width=13.2cm,clip]{H3435F10.eps}
\end{figure} Figure 10: The logarithm of the oscillation kinetic energy (upper panel) and period spacing (lower panel) for $\ell = 1$ (Fig.  a)), $\ell = 2$(Fig.  b)) and $\ell = 3$ (Fig.  c)) in terms of the computed periods, for the case of diffusive equilibrium in the trace element approach. As found in previous studies, this approximation gives rise to a kinetic energy pattern and spacing of consecutive periods in which trapping signatures are clearly noted. This is in contrast to the prediction of time dependent element diffusion given in Figs. 2 to 4.

An important finding of this work is the effect of chemical abundance distribution resulting from time dependent diffusion on the mode trapping properties in DA white dwarfs. In fact, as shown in Fig. 2 to 4, for periods exceeding $\approx $500-600 s, the distribution of $(E_{\rm kin})_k$ is smooth, and $\Delta P_k$ values tend to the asymptotic value. This is quite different from that found in previous studies. Our calculations reveal that the capability of mode filtering due to mode trapping effects virtually vanish for high periods when account is made of white dwarf models with diffusively evolving chemical stratifications (see Córsico et al. 2001). In order to make a detailed comparison of the predictions of our models with those found in previous studies we have carried out additional pulsational calculations by assuming diffusive equilibrium in the trace element approximation at the hydrogen-helium interface (see Tassoul et al. 1990). This treatment has been commonly invoked in most of the pulsation studies to model the composition transition regions. The resulting hydrogen chemical profile and the corresponding Ledoux term and Brunt-Väisälä frequency N are shown in Fig. 9, together with the predictions of time dependent element diffusion. The trace element assumption leads to an abrupt change in the slope of the chemical profile which is responsible for the pronounced peak in the Brunt-Väisälä frequency at $\log(1 -
M_r / M_*) \approx -4$. As can be clearly seen in Fig. 10 for $\ell = 1$ to 3, the diffusive equilibrium in the trace element approximation gives rise to an oscillation kinetic energy spectrum and period spacing distribution that are substantially different from those given by the full treatment of diffusion (see Figs. 2 to 4), particularly for high periods. The most outstanding feature depicted by Fig. 10 is the trapping signatures exhibited by certain modes both in the $\log(E_{\rm kin})_k$ and $\Delta P_k$ values. This is in agreement with other previous results (see Brassard et al. 1992b, particularly their Figs. 20a and 21a for the case of $M_{\rm H} = 10^{-4}~M_*$)[*]. As well known, trapped modes correspond to those modes which are characterized by minima in their oscillation kinetic energy values and local minima in the period spacing having the same k-value or differing by 1. For the purpose of illustration, we compare in Figs. 11 and 12 the predictions of equilibrium diffusion in the trace element approximation and time-dependent element diffusion, respectively, for WF corresponding to the modes g38 and g39 with $\ell = 2$. Clearly, in the case of diffusive equilibrium in the trace element approximation, mode g39corresponds to a trapped mode characterized by small values of the weight function below the hydrogen-helium transition, as compared with the adjacent, non-trapped mode g38. By contrast, such modes show very similar amplitudes of their WF when account is taken of a full diffusion treatment to model the composition transition regions (see Fig. 12). We would also like to comment on the fact that the diffusive equilibrium condition is far from being reached at the bottom of the hydrogen envelope of our model. In Althaus et al. (2002) we argued that the situation of diffusive equilibrium in the deep layers of a DAV white dwarf is not an appropriate one for describing the shape of the chemical composition at the hydrogen-helium transition zone. In fact, during the ZZ Ceti stage time-dependent diffusion modifies the spatial distribution of the elements, particularly at the chemical interfaces (see also Iben & MacDonald 1985). In addition, for the case of thick hydrogen envelopes, we have recently found that under the assumption of diffusive equilibrium, a white dwarf does not evolve along the cooling branch, but rather it experiences a hydrogen thermonuclear shell flash (see Córsico et al. 2002). This is so because if diffusion had plenty of time to evolve to an equilibrium situation then the tail of the hydrogen distribution would have been able to reach hot enough layers to be ignited in a flash fashion.

 \begin{figure}
\par\includegraphics[width=11.5cm,clip]{H3435F11.eps}
\end{figure} Figure 11: The normalized weight function for modes g38 and g39 with $\ell = 2$ corresponding to the stellar model in which thehydrogen-helium chemical transition has been treated in the diffusive equilibrium and trace element approximation. Note the lower amplitude of WF below the hydrogen-helium transition for mode with k= 39, which corresponds to a trapped one inthe hydrogen envelope.


 \begin{figure}
\par\includegraphics[width=11.5cm,clip]{H3435F12.eps}
\end{figure} Figure 12: Same of Fig. 11, but for the case in which the hydrogen-helium chemical transition has been computed assuming time dependent element diffusion.

To place some of the results of the foregoing paragraph in a more quantitative basis, we list in Tables 2 and 3 the values for Pk, $\Delta P_k$ and $\log(E_{\rm kin})_k$ for modes corresponding to $\ell=$ 1 and $\ell=$ 2, in the case of equilibrium diffusion and time dependent element diffusion. In Table 2, the "m'' corresponds to minima, and "M'' stands for maxima. We have labeled the minima of $\Delta P_k$ and the minima and maxima of $\log(E_{\rm kin})_k$, in correspondence with Fig. 10. Note that for the case with $\ell = 1$there is a direct correlation (indicated by arrows) between minima in $\log(E_{\rm kin})_k$ and $\Delta P_{k-1}$ for most of high order modes, whereas for the case with $\ell=$ 2 this correspondence is between minima in $\log(E_{\rm kin})_k$ and $\Delta P_k$. The modes with minima in kinetic energy are classified as trapped (T) ones. In contrast to the case of equilibrium diffusion, the results corresponding to the time dependent element diffusion treatment do not show clear minima or maxima in kinetic energy, as can be appreciated in Figs. 2 to 4 and Table 3. We have compared the periods of our model with those kindly provided by Bradley corresponding to his 0.560 $M_{\odot }$ white dwarf model, and we find that our periods are typically 6% shorter. In part, this difference is due to the somewhat smaller mass of the Bradley's model and the different input physics characterizing both stellar models.

 
Table 2: Pulsation properties for $\ell = 1, 2$ modes corresponding to the case of equilibrium diffusion.
$\ell = 1$ $\ell = 2$
$\Delta P_{\rm A}=$ 46.26 s $\Delta P_{\rm A}=$ 26.71 s
k Pk   $\Delta P_k$   $\log(E_{\rm kin})_k$ k Pk   $\Delta P_k$   $\log(E_{\rm kin})_k$
  [s]   [s]   [erg]   [s]   [s]   [erg]
1 126.99 T 76.38   45.84 m 1 73.39 T 48.02   45.84 m
2 203.37   73.91   46.97 M 2 121.40   39.77   46.85 M
3 277.28 T 26.02 m $\leftarrow$ 44.60 m 3 161.17 T 18.11 m $\leftarrow$ 44.52 m
4 303.30   40.79   44.61 m 4 179.28   39.51   44.61 M
5 344.09   40.65   44.93 M 5 218.80 T 14.17 m $\leftarrow$ 43.77 m
6 384.75   44.00   43.62 m 6 232.97   17.06   43.89 M
7 428.75   61.25   43.43 7 250.03   33.50   43.49
8 490.00 T 59.93   42.90 m 8 283.53   40.06   42.89
9 549.93   21.12 m $\swarrow$ 42.99 M 9 323.59 T 24.67   42.53 m
10 571.05   47.22   42.59 10 348.26   18.24 m $\swarrow$ 42.48 M
11 618.27   44.54 m   42.23 11 366.50   19.88   42.27
12 662.81 T 48.65 $\nwarrow$ 41.91 m 12 386.39   35.55   41.95
13 711.46   30.69 m   42.24 M 13 421.94 T 21.12 m $\leftarrow$ 41.76 m
14 742.15 T 46.66 $\nwarrow$ 41.75 m 14 443.06   30.37   41.96 M
15 788.81   42.28 m   41.98 M 15 473.43 T 19.88 m $\leftarrow$ 41.58 m
16 831.09 T 48.51 $\nwarrow$ 41.48 m 16 493.32   25.75   41.66 M
17 879.60   43.99 m   41.55 M 17 519.06   19.59 m   41.45
18 923.59 T 48.15 $\nwarrow$ 41.16 m 18 538.65   30.84   41.21
19 971.75   33.31 m   41.31 M 19 569.49 T 22.68 m $\leftarrow$ 41.10 m
20 1005.05 T 47.02 $\nwarrow$ 41.16 m 20 592.18   27.64   41.29 M
21 1052.07   42.38 m   41.32 M 21 619.82   19.68 m   41.22
22 1094.45 T 55.95 $\nwarrow$ 41.10 m 22 639.51   32.53   41.26 M
23 1150.40   32.44 m   41.22 M 23 672.04 T 24.88 m $\leftarrow$ 41.00 m
24 1182.83 T 43.15 $\nwarrow$ 41.09 m 24 696.92   27.97   41.12 M
25 1225.98   40.36 m   41.16 M 25 724.89 T 21.88 m $\leftarrow$ 40.93 m
26 1266.34 T 53.93 $\nwarrow$ 40.93 m 26 746.77   25.83   41.25 M
27 1320.27   38.87 m   41.11 M 27 772.60 T 23.68 m $\leftarrow$ 41.01 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 28 796.28   28.33   41.18 M
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 29 824.61 T 22.94 m $\leftarrow$ 41.03 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 30 847.55   27.53   41.17 M
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 31 875.08 T 21.89 m $\leftarrow$ 41.04 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 32 896.97   30.30   41.14 M
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 33 927.27 T 25.47 m $\leftarrow$ 40.98 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 34 952.74   27.78   41.22 M
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 35 980.52 T 23.58 m $\leftarrow$ 41.06 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 36 1004.10   26.44   41.32 M
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 37 1030.54 T 23.93 m $\leftarrow$ 41.12 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 38 1054.46   28.40   41.35 M
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 39 1082.87 T 23.20 m $\leftarrow$ 41.18 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 40 1106.07   26.52   41.48 M
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 41 1132.58 T 23.99 m $\leftarrow$ 41.27 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 42 1156.57   29.72   41.46 M
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 43 1186.29 T 25.32 m $\leftarrow$ 41.32 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 44 1211.61   27.85   41.57 M
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 45 1239.47 T 24.74 m $\leftarrow$ 41.41 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 46 1264.20   26.62   41.68 M
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 47 1290.82 T 25.45 m $\leftarrow$ 41.48 m
$\ldots$ $\ldots$   $\ldots$   $\ldots$ 48 1316.27   26.83   41.70 M



 
Table 3: Same as Table 2, but for the case of time-dependent element diffusion.
$\ell = 1$ $\ell = 2$
$\Delta P_{\rm A}=$ 45.83 s $\Delta P_{\rm A}=$ 26.30 s
k Pk $\Delta P_k$ $\log(E_{\rm kin})_k$ k Pk $\Delta P_k$ $\log(E_{\rm kin})_k$
  [s] [s] [erg]   [s] [s] [erg]
1 126.98 76.87 45.84 1 72.14 49.34 45.84
2 203.85 66.57 46.83 2 121.48 34.89 46.69
3 270.43 33.77 44.77 3 156.38 21.63 44.72
4 304.19 35.98 44.48 4 178.00 32.08 44.41
5 340.17 36.30 44.69 5 210.09 23.30 44.10
6 376.47 44.67 43.97 6 233.39 10.74 43.84
7 421.13 67.75 43.30 7 244.12 36.72 43.40
8 488.88 51.08 43.04 8 280.85 31.47 43.05
9 539.95 26.84 42.78 9 312.31 33.28 42.67
10 566.80 46.27 43.04 10 345.59 16.62 42.44
11 613.06 44.76 42.18 11 362.21 19.97 42.30
12 657.83 46.99 42.03 12 382.18 30.40 42.05
13 704.82 32.45 42.08 13 412.58 25.26 41.87
14 737.27 42.44 41.93 14 437.84 26.30 41.86
15 779.71 44.78 41.88 15 464.14 24.63 41.78
16 824.49 43.42 41.68 16 488.77 21.89 41.58
17 867.91 50.43 41.50 17 510.66 22.90 41.59
18 918.34 43.07 41.23 18 533.56 24.96 41.30
19 961.42 38.91 41.20 19 558.53 27.11 41.14
20 1000.33 40.24 41.28 20 585.64 24.25 41.17
21 1040.57 45.71 41.19 21 609.89 22.92 41.21
22 1086.28 50.32 41.19 22 632.80 26.05 41.27
23 1136.60 40.88 41.13 23 658.86 28.13 41.09
24 1177.48 36.91 41.17 24 686.98 27.00 41.00
25 1214.40 42.99 41.06 25 713.98 23.95 40.97
26 1257.38 46.02 40.96 26 737.94 22.71 41.01
27 1303.40 46.39 40.99 27 760.64 24.69 41.07
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 28 785.33 25.47 41.02
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 29 810.80 25.67 41.05
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 30 836.47 24.62 41.04
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 31 861.09 24.42 41.04
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 32 885.51 25.09 41.05
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 33 910.60 27.09 41.02
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 34 937.69 27.41 41.03
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 35 965.11 25.02 41.09
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 36 990.12 24.63 41.15
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 37 1014.75 24.66 41.18
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 38 1039.41 25.64 41.21
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 39 1065.05 26.07 41.23
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 40 1091.12 24.20 41.29
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 41 1115.32 24.33 41.35
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 42 1139.66 26.34 41.36
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 43 1166.00 27.19 41.37
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 44 1193.19 26.48 41.42
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 45 1219.67 26.20 41.46
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 46 1245.87 25.35 41.52
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 47 1271.22 25.13 41.56
$\ldots$ $\ldots$ $\ldots$ $\ldots$ 48 1296.35 25.98 41.58


On the basis of our results, we claim that the treatment of the chemical profile at the chemical transitions is a key ingredient in the computation of the g-spectrum. This is particularly true regarding the mode trapping properties, which are considerably altered when a physically sound treatment of the chemical evolution is incorporated in such calculations. This conclusion is valid at least for massive hydrogen envelopes as predicted by our full evolutionary calculations. To get a deeper insight into these aspects, we examine the node distribution of the eigenfunctions (Figs. 13 and 14). According to Brassard et al. (1992a), this is an useful diagnostic for mode trapping. Here, a mode is trapped above the hydrogen-helium interface when its eigenfunction y1 has a node just above of such an interface, and the corresponding node in y2 lies just below that interface. Note that this statement is clearly satisfied by our model with diffusive equilibrium in the trace element approximation, as shown in Fig. 13. In this figure, the vertical dotted line at r= 0.927 R* indicate the location of the pronounced peak in the Brunt-Väisälä frequency (see Fig. 9). However, the node distribution becomes markedly different in the sequence with non-equilibrium diffusion and does not seem possible to find a well defined interface in that case (see Fig. 14). Thus, it does not seem to be clear that the above-mentioned trapping rule can be directly applied to this case. In fact, because the hydrogen-helium interface becomes very smooth in our models, the peak in the Brunt-Väisälä frequency is not very pronounced. Accordingly, the capability of mechanical resonance of our model turns out to be weaker. This causes the node distribution in the eigenfunctions to be quite different from that corresponding to the diffusive equilibrium approach.


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