For our template model we have computed g-modes with
and
3 (we do this because geometric cancellation effects grow
progressively for larger
in non-radial oscillations; see
Dziembowski 1977), with periods in the range of 50 s
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
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,
and 3 in terms of computed periods.
Middle panels depict the values for the forward period spacing
(
)
together with the asymptotic value
as given by dotted lines
. Finally, in the bottom panel of these figures we depict
the
values as well as the asymptotic values (dotted
lines) that these coefficients adopt for high overtones, that is
(Brickhill 1975). An
inspection of plots reveals some interesting characteristics. To
begin with, the quantities plotted exhibit two clearly different
trends. Indeed, for
s and irrespective of the
value of
,
the distribution of oscillation kinetic energy is
quite smooth. Note that the
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
from the asymptotic
prediction (Eq. (3)) for
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
of the modes tend to
.
Also, note that the
values tend to the asymptotic value for
s.
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 .
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
values
are proportional to the integral of the squared eigenfunctions,
weighted by
.
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 (
)
for brevity; the results for
are qualitatively similar to those of
.
From the analysis performed above based on the weight functions, we
can clearly appreciate that for
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
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
values for both sets of computations (and for each value of
)
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
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
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
(
)
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.
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 500-600 s, the distribution of
is smooth, and
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
.
As can be clearly seen in Fig. 10 for
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
and
values. This is in agreement
with other previous results (see Brassard et al. 1992b, particularly
their Figs. 20a and 21a for the case of
)
. 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
.
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.
![]() |
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,
and
for modes corresponding to
1 and
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
and the minima and maxima of
,
in
correspondence with Fig. 10. Note that for the case with
there is a direct correlation (indicated by arrows) between minima in
and
for most of high order
modes, whereas for the case with
2 this correspondence is
between minima in
and
.
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
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.
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k | Pk |
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k | Pk |
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[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 |
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44.60 m | 3 | 161.17 | T | 18.11 m |
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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 |
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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 | ![]() |
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 | ![]() |
42.48 M | |||
11 | 618.27 | 44.54 m | 42.23 | 11 | 366.50 | 19.88 | 42.27 | ||||
12 | 662.81 | T | 48.65 | ![]() |
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 |
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41.76 m | ||
14 | 742.15 | T | 46.66 | ![]() |
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 |
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41.58 m | ||
16 | 831.09 | T | 48.51 | ![]() |
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 | ![]() |
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 |
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41.10 m | ||
20 | 1005.05 | T | 47.02 | ![]() |
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 | ![]() |
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 |
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41.00 m | ||
24 | 1182.83 | T | 43.15 | ![]() |
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 |
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40.93 m | ||
26 | 1266.34 | T | 53.93 | ![]() |
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 |
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41.01 m | ||
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28 | 796.28 | 28.33 | 41.18 M | ||||
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29 | 824.61 | T | 22.94 m |
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41.03 m | ||
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30 | 847.55 | 27.53 | 41.17 M | ||||
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31 | 875.08 | T | 21.89 m |
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41.04 m | ||
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32 | 896.97 | 30.30 | 41.14 M | ||||
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33 | 927.27 | T | 25.47 m |
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40.98 m | ||
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34 | 952.74 | 27.78 | 41.22 M | ||||
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35 | 980.52 | T | 23.58 m |
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41.06 m | ||
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36 | 1004.10 | 26.44 | 41.32 M | ||||
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37 | 1030.54 | T | 23.93 m |
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41.12 m | ||
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38 | 1054.46 | 28.40 | 41.35 M | ||||
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39 | 1082.87 | T | 23.20 m |
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41.18 m | ||
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40 | 1106.07 | 26.52 | 41.48 M | ||||
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41 | 1132.58 | T | 23.99 m |
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41.27 m | ||
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42 | 1156.57 | 29.72 | 41.46 M | ||||
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43 | 1186.29 | T | 25.32 m |
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41.32 m | ||
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44 | 1211.61 | 27.85 | 41.57 M | ||||
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45 | 1239.47 | T | 24.74 m |
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41.41 m | ||
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46 | 1264.20 | 26.62 | 41.68 M | ||||
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47 | 1290.82 | T | 25.45 m |
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41.48 m | ||
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48 | 1316.27 | 26.83 | 41.70 M |
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k | Pk |
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k | Pk |
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[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 |
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28 | 785.33 | 25.47 | 41.02 |
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29 | 810.80 | 25.67 | 41.05 |
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30 | 836.47 | 24.62 | 41.04 |
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31 | 861.09 | 24.42 | 41.04 |
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32 | 885.51 | 25.09 | 41.05 |
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33 | 910.60 | 27.09 | 41.02 |
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34 | 937.69 | 27.41 | 41.03 |
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35 | 965.11 | 25.02 | 41.09 |
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36 | 990.12 | 24.63 | 41.15 |
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37 | 1014.75 | 24.66 | 41.18 |
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38 | 1039.41 | 25.64 | 41.21 |
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39 | 1065.05 | 26.07 | 41.23 |
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40 | 1091.12 | 24.20 | 41.29 |
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41 | 1115.32 | 24.33 | 41.35 |
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42 | 1139.66 | 26.34 | 41.36 |
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43 | 1166.00 | 27.19 | 41.37 |
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44 | 1193.19 | 26.48 | 41.42 |
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45 | 1219.67 | 26.20 | 41.46 |
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46 | 1245.87 | 25.35 | 41.52 |
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47 | 1271.22 | 25.13 | 41.56 |
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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.
Copyright ESO 2002