4 Asteroseismology of RXJ 2117+3412

4.1 Period spacing

In the asymptotic limit, non-radial g-modes are equally spaced in period. Two main methods can be used for detecting regularly spaced periods: the Kolmogorov-Smirnov (K-S: Kawaler 1988) and inverse variance (IV: O'Donoghue 1994) significance tests. In principle, these methods should be applied to the periods of the central (m=0) modes, as they are not affected by rotation (to first order), unlike the $m=\pm 1$ modes. Among the 48 significant modes detected in RXJ 2117+3412, one finds only two triplets and eight doublets. Before performing the mode identification, one does not know which component of the doublets is the central m=0 one, nor the m value of the remaining single modes. Neither of the proposed tests will distinguish between the period interval between adjacent modes and the rotational splitting. To minimize this effect and get a clearer test, one first selects a limited range in the period distribution. Considering that all multiplets are found at frequency lower than 1500 $\mu$Hz, and that the modes at higher frequency are sparsely distributed up to 4340 $\mu$Hz, are all of low amplitude, and that most of them result from linear combinations of lower frequency modes, the tests were performed in a frequency domain restricted to the interval 600 $\mu$Hz-1500 $\mu$Hz ($\approx$1600 s- 700 s periods). This interval contains 42 modes. For the triplets, the tests use the central mode period. For the doublets we can proceed in two different ways: either we use both observed modes or we use only the average of the two periods for each doublet. In the latter case, the K-S test shows a double minimum - typical for a K-S test - with the first minimum at 21.0 s and the second minimum at 22.1 s. The inverse variance test shows a single, obvious maximum at 21.9 s. Figure 10 shows both the K-S and the inverse variance tests. The agreement between the two methods is quite good, but the inverse variance test does not suffer the ambiguous double peak seen in the K-S test. However, if those tests are useful to detect the expected equidistant pattern in the period distribution, they do not provide a good enough estimate of the period spacing since they assign equal weights to all the periods. The determination of the period spacing, $\Delta P$, can be refined by the weighted linear least-squares fit. The weight assigned to each period is inversely proportional to the square of its uncertainty. The uncertainties are defined in the following way: for single mode, the frequency uncertainty equals to the average frequency splitting found in the triplets (this is the real uncertainty of the frequency of the m=0 mode). For the doublets, we take the average frequency of the two components affected by an uncertainty equal to half their frequency difference. For the triplets, the m=0 mode is identified and the uncertainty is the formal error given by the non-linear least-squares fit to the time series data. Our best linear weighted fit gives a preliminary period spacing $\Delta P = 21.540$  $\rm s \pm 0.030$ s. The period spacing derived here is slightly larger than the 20.5 s derived in the discovery paper (Vauclair et al. 1993). Close inspection of the K-S test shown in their paper reveals the double minimum, with the other much shallower minimum at 22 s. Considering the much cleaner determination of the period spacing deduced from the weighted least-squares fit in this paper, it is not worth discussing that particular point any further. The main difference between the 1992 discovery analysis and the present one is that we do not find any evidence of the $\ell =2$ modes in the present data from the period spacing tests. If a substantial number of $\ell =2$ modes were present in the power spectrum, their signature in the period spacing tests would be a secondary minimum (maximum) at 21.540 s/$\sqrt{3}$ $\approx$12.43 s in the K-S (IV) tests. Inspection of Fig. 10 does not show any evidence for such a $\ell =2$ modes signature.

Figure 10: The results of tests for period spacings. Upper panel: Kolmogorov-Smirnov (K-S) test; lower panel: Inverse Variance (IV) test. The K-S test shows a double minimum for period spacings $\Delta P \approx$ 21 s and $\Delta P \approx 22$ s. The IV test shows a single maximum for a period spacing $\Delta P \approx 22$ s. The single peak points to a single $\ell$ value being present, which we show to be $\ell =1$. We see no evidence of $\ell =2$ modes, which should have a peak near 12.4 s.

4.2 Mode identification

For the mode identification, we proceed by iteration as follows. We assume that all the modes seen in RXJ 2117+3412 have the same value of $\ell$. This assumption relies on the equidistant period spacing, discussed above (Sect. 4.1), and on the similar rotational splitting found in the multiplets. It is not possible to give an absolute identification for the k order of the modes. Only differential k can be asserted. The reference mode is chosen at 945.156 $\mu$Hz, the central m=0 component of a triplet, whose korder is some unknown k₀. We will attempt now to identify the relative order $\Delta k= k-k_{0}$, as well as the azimuthal number m for as many modes as possible. Note that the convention chosen for the sign of m is different from the one used in classical textbooks: here m=-1 is associated with the low frequency component of a triplet (retrograde mode) while m=+1 corresponds to the high frequency component (prograde mode). This sign convention is the same as in Unno et al. (1989) and Winget et al. (1991, 1994). The identification procedure starts by considering modes for which the m=0 frequency is well determined, either from the two complete triplets, (945 $\mu$Hz, 1101 $\mu$Hz), or from the three doublets whose components are separated by twice the rotational splitting (978/988 $\mu$Hz, 1045/1055 $\mu$Hz and 1179/1190 $\mu$Hz). In this latter case the m=0 frequency is obtained by averaging the two components, assuming that the triplets are symmetric. These five m=0 modes give a unique determination of the period spacing, either by applying the inverse variance test (O'Donoghue 1994), which yields a period spacing of $\Delta P= 21.508~\rm s \pm 0.173$ s (HWHM), or a linear unweighted least-squares fit which yields $\Delta P=21.506~\rm s\pm 0.099$ s. The inverse variance procedure also yields the values of the relative radial order of the modes, $\Delta k$; they are listed in the fourth column of Table 9. We are not able to assign an absolute k value, because there are no models of suitable quality at that high a luminosity in the literature.

Figure 11: Illustration of the mode identification procedure. The figure shows how the observed modes are identified for the cases $\Delta k= -3$ and -4. The two large dots represent known periods of m=0 modes. The two straight lines are at $\pm$2 s from the linear least-squares fit to the m=0 modes identified so far. We assume all the m=0 modes to lie within this range. This generously allows for mode trapping which proves to be much smaller ($\leq$1.3 s). The small dots represent the observed modes which need identification. At $\Delta k= -3$, there is a doublet with periods of 994.387 s and 989.569 s. The mode with 994.387 s period must be the m=0 component, since it is the one whose period lies between the two lines, with the other mode being the prograde m=+1 component. For $\Delta k=-4$, we see only one mode, which must be the retrograde (m=-1) mode, since its period falls outside and above the $\pm$2 s band. The calculated period of the unobserved m=0 mode is 972.247 s.

Starting from this point, new m=0 modes are identified in a step by step procedure, using linear interpolation/extrapolation. We allow for up to 2 s departures from the linear trend to account for possible mode trapping effects. Later, we will show that this is consistent with the actual trapping cycle, whose maximum amplitude is $\sim$1.3 s. An illustration of the mode identification procedure is given in Fig. 11 for the case of the $\Delta k= -3$ and -4 modes. From the straight line fit to the first five modes, we find that the m=0 mode for $\Delta k= -3$ should appear at $994.2\pm 2$ s or in the frequency interval 1003.8 to 1007.9 $\mu$Hz. This estimate allows a unique identification of the m=0 component within the doublet at 1005/1010 $\mu$Hz as the 1005.645 $\mu$Hz mode. We can now include this mode into linear regression and continue the procedure. We then estimate the frequency range of the m=0 mode for $\Delta k=-4$, which is 1025.9 to 1030.2 $\mu$Hz. It is clear that the mode observed at 1023.594 $\mu$Hz must be an m=-1 component of the triplet. The m=0 component is not seen, but its frequency can be estimated from the rotational splitting. We include the new m=0 frequency into the linear regression and continue the procedure until we have determined the relative radial order ($\Delta k$) and azimuthal order (m) for as many of the observed modes as is possible. At each intermediate step, a linear fit to m=0 modes is redetermined.

We notice that the rotational splitting changes with period (see Fig. 13), so we must interpolate the value for the $\Delta k=-4$, -8 modes. For $\Delta k$ smaller than -11, the behaviour of rotational splitting is not known, and we assume a constant splitting of $\Delta f= 5.332$ $\mu$Hz for these modes. While this is an arbitrary assumption, it has a negligible effect on the trapping cycle parameters (Sect. 4.3) that we determine below. We stop the mode identification at frequencies lower than 780 $\mu$Hz, because below this limit the spacing between adjacent mmodes and between different overtone modes become comparable and we can no longer tell them apart. Also, we refrained from determining periods of unobserved m=0 modes for $\Delta k$ larger than 0, because the variation of rotational frequency splitting with $\Delta k$ is not well established here because there are not enough multiplets detected at these low frequencies. In addition, at low frequencies, any error in the interpolated/extrapolated value of $\Delta f$ would translate in relatively large error of the mode's period and affect our mode trapping results. Our procedure shows that some modes which could have been interpreted as multiplets by considering only their frequency difference in Table 6 cannot be so. That is the case for the modes at 789 $\mu$Hz and 793 $\mu$Hz, which cannot correspond to the same $\Delta k$ value, and for the modes at 830, 836 and 840 $\mu$Hz, which must be a combination of a single mode and a doublet of the successive k mode. Our proposed mode identification is summarized in Table 9.

4.3 Mode trapping

A linear least-squares fit to the 20 m=0 identified or inferred modes, leads to an average period spacing $\Delta P=21.639$  $\rm s \pm 0.021$ s. Figure 12 shows the residuals of this linear fit. They prove to be significantly different from zero and vary quasiperiodically, which previous experience (cf. Winget et al. 1991) indicates is due to mode trapping. Knowing that there is a signature of mode trapping superimposed on the asymptotic equidistant period distribution, one can repeat the fitting procedure to get the trapping cycle parameters by adding a sine function to the linear fit with a function of the form:

$$P(\Delta k)= a + \Delta P \times \Delta k + A \times \sin( \frac{2\pi \Delta k}{T_{k}} + c)$$

where $\Delta P$ is the refined value of the period spacing (in s), A is the semiamplitude of the trapping cycle (in s) and T_k is the length of the trapping cycle (trapping period) expressed in number of modes. This new fitting yields $\Delta P$, A and T_k simultaneously. One derives $\Delta P= 21.640$  $\rm s\pm 0.012$ s, A= 0.71  $\rm s\pm 0.11$ s and $T_{k}= 3.799 \pm 0.042$(which translates in a period of the trapping cycle $P_{\rm tc}=82.21~\rm s\pm 0.91$ s). The mode $\Delta k=4$ (period 1146 s) does not satisfactorily fit the trapping cycle. Repeating the fitting procedure with this mode excluded leads to: $\Delta P= 21.618~\rm s \pm 0.008$ s, $A= 0.823~\rm s\pm 0.078$ s and $T_{k}= 3.880 \pm 0.026$ ( $P_{\rm tc}= 83.88~\rm s\pm 0.57$ s). These latter values will be used in the following discussion but either solution would lead to very similar results. The semi-amplitude of the mode trapping is comfortably smaller than the 2 s allowance accepted in the mode identification procedure (Sect. 4.2). Our implicit assumption that the trapping cycle is strictly periodic, i.e., can be fitted with a single sine function, is satisfied closely enough, as witnessed by the small errors of both the semiamplitude and the trapping period. This implies that either there is only one chemical composition interface, presumably between the He-rich envelope and the C/O core, or the beating between the trappings induced by this interface and those induced by another potential C/O interface is too small to be detected (see also Fig. 14).

 

Table 9: Period list and mode identification in RXJ 2117+3412.

Period (s)	Frequency ($\mu$Hz)	m	$\Delta k$
1267.360	789.042	+1	+10
1259.790	793.783	-1	+9
1203.792	830.708	+1	+7
1196.076	836.067	-1
1189.956	840.367:	0	+6
1174.422	851.483	-1	+5
1146.346	872.337	0	+4
1124.117	889.587	0	+3
1117.568	894.800	+1
1103.292	906.378	0	+2
1084.927	921.721	-1	+1
1063.193	940.563	-1
1058.026	945.156	0	0
1052.732	949.909	+1
1043.261	958.533	-1
1038.118	963.282	0	-1
1021.582	978.874	-1
(1016.467)	(983.800)	0	-2
1011.403	988.726:	+1
994.387	1005.645	0	-3
989.569	1010.541	+1
976.950	1023.594	-1
(972.247)	(1028.545)	0	-4
956.306	1045.690	-1
(951.750)	(1050.697)	0	-5
947.236	1055.703	+1
911.816	1096.712	-1
907.489	1101.942	0	-7
903.160	1107.224	+1

 

Table 9: continued.

Period (s)	Frequency ($\mu$Hz)	m	$\Delta k$
889.880	1123.747:	-1
(885.736)	(1129.004)	0	-8
847.490	1179.955	-1
(843.692)	(1185.267)	0	-10
839.928	1190.578:	+1
824.749	1212.490	-1
821.145	1217.812	0	-11
802.918	1245.457	-1
(799.495)	(1250.789)	0	-12
(778.921)	(1283.828)	0	-13
775.699	1289.160	+1
760.424	1315.055	-1
(757.354)	(1320.387)	0	-14
733.948	1362.495	0	-15
715.696	1397.242	-1
(712.975)	(1402.574)	0	-16
694.831	1439.198	-1
(692.267)	(1444.530)	0	-17

Notes. Periods and frequencies in parenthesis are for the m=0 components which are not observed and whose values are calculated on the basis of the known rotational splitting (see text for details). The chosen convention for m is: m=-1 for retrograde modes and m=+1 for prograde modes.

Note that there is no correlation between mode trapping and mode amplitude, where trapped modes are those defining the minima in Fig. 12. There are 6 such minima. The corresponding modes have periods ($\Delta k$) of 733.97 s (-15), 799.49 s (-12) and/or 821.15 s (-11), 885.74 s (-8) and/or 907.49 s (-7), 972.25 s (-4), 1058.03 s (0) and 1124.11 s (+3). Looking at the trapped mode amplitudes during different runs (Table 6), we see that trapped modes can have either low or high amplitudes. Mode trapping and amplitude were also found to be uncorrelated in PG 1159-035 (Winget et al. 1991) and in the DBV GD 358 (Winget et al. 1994). Clearly, the amplitude of a mode is not simply governed by its linear growth rate. The period spacing found here for RXJ 2117+3412 is remarkably similar to the ones found in other PNNV and GW Vir stars: $\Delta P= 22.3$ s in NGC 1501 (Bond et al. 1996), 21.5 s in PG 1159-035 (Winget et al. 1991), 21.6 s in PG 2131+066 (Kawaler et al. 1995) and 21.1 s in PG 0122+200, (Vauclair et al. 2001). We do not understand what mechanism is forcing these pulsators - that have different masses and luminosities - to display the same period spacing. O'Brien (2000) suggests that an interplay between the driving zone depth and the maximum allowed pulsation period, as a function of $T_{{\rm eff}}$ and the total mass, can explain the tendency for higher mass pre-white dwarfs to pulsate at cooler $T_{{\rm eff}}$ than lower mass ones. If higher mass GW Vir stars pulsate at cooler temperature than low mass ones, then the average period spacings could be similar for all of the GW Vir stars. However, it is still unclear how this preserves the almost constant $\Delta P$ observed over a factor of almost 1000 in luminosity. Clearly, the "numerology'' is telling us something about the nature of the GW Vir and PNNV stars, but we do not yet understand it.

4.4 Rotation rate

Since the multiplets in the combined list of frequencies are $\ell =1$ modes split by rotation, one may derive the corresponding rotation rate of RXJ 2117+3412. In the slow, solid body rotation limit we have $\Omega\ll\sigma_{\ell,k}$ , where $\Omega$ is the angular rotation frequency and $\sigma_{\ell,k}$ is the non-rotating angular pulsation frequency for a mode of degree $\ell$ and of order k. In this limit, the frequencies for a rotating star are given by

$$\sigma_{\ell,k,m}= \sigma_{\ell,k} + m(1- C_{\ell,k})\Omega + o(\Omega^{2})$$

where $C_{\ell,k}$ takes a simple form in the asymptotic limit of high order gravity modes, which applies to pre-white dwarfs, as discussed by Winget et al. (1991). In that case $C_{\ell,k} \simeq 1/\ell(\ell+1)$as shown by Brickhill (1975). In the above expression for the frequencies, solid body rotation is assumed. Any differential rotation present in the star would result in an additional term $C^{'}_{\ell,k,m}$ to $C_{\ell,k}$ which depends on k, and so is different for different modes. Assuming all the modes to be $\ell =1$ modes, we derive an average rotation period for RXJ 2117+3412, using the relation

$$P_{\rm rot}= 1/2\Delta\bar{f}$$

where $\Delta$$\bar{f}$ is the mean rotational splitting. From the observed triplets and doublets (10 cases), one finds an average rotational splitting of $\Delta \bar{f}= 4.998 \pm0.23~\mu$Hz, from which one derives a mean rotational period of 1.16  $\rm d\pm 0.05$ d, where the uncertainty is derived from the average deviation of the rotational splitting to the mean value. This is a conservative overestimate of the uncertainty since it does not take into account the variation of the rotational splitting with period discussed below.

Figure 12: A plot showing observational evidence of mode trapping. The residuals of the period distribution relative to the average period spacing ( $\Delta P= 21.618$ s) for the 20 m=0 modes are plotted as a function of the period. The best fit with a sine wave gives a trapping cycle of $3.88 \pm 0.026$ mode (i.e. 83.88  $\rm s \pm 0.57$ s) with a semi-amplitude $A= 0.823~\rm s\pm 0.078$ s. The modes defining the minima on this plot are trapped modes, while those defining the maxima are nontrapped modes. Filled circles represent the m=0 modes, which are either observed or whose frequencies are determined by interpolation between the observed m=-1 and m=+1 components of the triplets. Open circles represent the m=0 modes whose frequencies have been inferred from the observed singlets and the known rotational splitting.

One should keep in mind that the frequency separation within multiplets may deviate from uniformity for different reasons. There are at least four physical processes that could affect the frequencies: i) non-linearities resulting from resonant coupling between components of multiplets, ii) mode trapping, iii) structural changes in the wave propagation cavities, and iv) magnetic field. In case i), slight changes in the frequency of multiplets components are expected even in the case of modest non-linearity in the pulsations. The non-linearities result in both amplitude and frequency variations for selected modes as described in Goupil et al. (1998). In their application to the case of the DBV GD358, however, they find that the frequency splitting is changed by the non-linear effects by no more than 2%. In case ii), mode trapping also introduces small variations in the frequency shift due to rotational splitting (Kawaler et al. 1999). Since the radial structure of the mode is affected by trapping, so is the rotational kernel. If rotation is non-uniform, it will affect the splitting constant. In their discussion of the rotational splitting in PG 1159-035, Kawaler et al. (1999) show that the effect of mode trapping may change the frequency separation of the rotationally split components by 2.5% in the period range 400  ${\rm s}\leq P \leq 800$ s. They also show that the rotation rate inside PG 1159-035 decreases with increasing radius. The variations could be much larger would the gradient of the rotation curve be steeper.

Figure 13: Rotational splitting as a function of periods. The frequency splitting of the 10 multiplets (2 triplets and 8 doublets) is plotted as a function of the m=0 mode period. With the exception of the mode at 1124 s (open circle), a clear trend is present for the other modes (filled circles): the rotational splitting decreases with increasing period.

In case iii), changes in the structure of the outer layers modify the properties of the propagation regions. These changes would modify the trapping cycle and indirectly affect the rotational splitting in case of non-uniform rotation as discussed in case ii). There are evidences for spectroscopic and photometric variations in RXJ 2117+3412 (Feibelman 1999) as mentioned above which are not unexpected in a mass losing star. However, there is not enough data to determine whether 1) the mass loss rate could also be time dependent and whether 2) there is a correlation between spectroscopic and photometric variations. Qualitatively, one may expect that any variation in the mass loss rate should affect the chemical composition of the outer He-rich layer, where both the excitation and the propagation of the modes takes place. This should result in variations for both the frequencies and the amplitudes of the modes. Such effects need to be quantitatively estimated. Finally, in case iv), a magnetic field would produce an asymmetry in the frequency shifts of the m=+1 and -1 components relative to the m=0 component, since the additional frequency shift induced by the magnetic field is proportional to m².

As a consequence, any variations in the frequency splitting measured between multiplets may be due to a combination of at least these four effects and deriving any information on a potential differential rotation profile is a speculative task. Figure 13 shows the variation of the rotational splitting as a function of period for all multiplets observed in RXJ 2117+3412. Only secure detections are considered (marginal detections are rejected). Rotational splittings derived from 1994 WET data are preferred because these data have the best window function. For data from either the 1992 WET or the 1993 data set, splittings showing the smallest formal error were preferred. Figure 13 shows a clear trend of decreasing rotational splitting with increasing period. Only one mode does not fit the trend: it is the 1124.117 s mode, seen only during the 1992 WET. Several explanations can account for this discrepancy. We note from Table 6 that for the few cases where we have frequency splitting information for the same mode in different years, that the 1992 WET data have the largest splitting for unknown reasons. We suspect that the systematically larger frequency splittings in the 1992 data affect the 1124.117 s mode, although we cannot rule out the possibility that there is an incompletely corrected 2d alias present. The observed rotational splitting changes by a factor of 1.14 in the period interval 821 s to 1058 s, covering 11 k orders. The factor is 1.24 if the longest period mode (at 1189 s) is included, covering 17 k orders. This variation of $\Delta f$ is 13-14 times greater than the non-differentially rotating case considered by Kawaler et al. (1999) who find $a \approx 2$% change in the rotational splitting between 400 s and 800 s, covering 20 k orders, in their model of PG 1159-035. The conservative conclusion is that the rotational splitting in RXJ 2117+3412 is inconsistent with solid-body rotation. Kawaler et al. (1999) show that a rotation law that decreases or increases outwards may have similar signatures in a $\Delta f$-Period diagram. They also show that mode trapping affects the rotational splitting and that it is the phase shift between the trapping seen in the period spacing ($\Delta P$-Period diagram) and the one seen in the rotational splitting ($\Delta f$-Period diagram) which contains the pertinent information on the rotation velocity law. We do not see any such trapping cycle in the $\Delta f$-Period diagram of RXJ 2117+3412, which is in agreement with the weak trapping indicated by the small amplitude of the trapping cycle, and this precludes any further statement on the internal rotation profile of RXJ 2117+3412. The observed trend of the rotational splitting in RXJ 2117+3412 is surprisingly smooth. One would have expected a rather complex internal rotation law, if one considers that the star is i) still contracting towards the white dwarf cooling sequence with a short time scale ($\approx$ $10^{4}{-}10^{5}\,{{\rm yr}}$) and ii) is loosing mass at a rate of a $\approx$ $10^{-7}~M_\odot~{{\rm yr}}^{-1}$(Koesterke et al. 1998; Koesterke & Werner 1998).

In the absence of any consistent physical interpretation of the rotational splitting variation, we can only conclude that the average rotation period is $\approx$1.1 d, and that the frequency splitting is not consistent with solid body rotation. The average rotation period for RXJ 2117+3412 is within the range of values derived from asteroseismology for other pre-white dwarfs. The PNNV NGC 1501 has a rotation period of 1.17 d (Bond et al. 1996). Among other pulsating PG 1159 stars having rotation periods derived from rotational splitting PG 1159-035 has a period of 1.38 d (Winget et al. 1991), PG 2131+066: 5.07 h (Kawaler et al. 1995) and PG 0122+200: 1.61 d (O'Brien et al. 1996, 1998; Vauclair et al. 2001). This trend persists with the cooler DBV and DAV white dwarfs, which have rotation periods ranging from 9 to 58 hours (Bradley 2001). Spruit (1998) argues that such rotation periods around one day for white dwarfs can be expected if some small non-axisymmetries occur in the mass loss process along the AGB evolutionary phase. In the case of RXJ 2117+3412, which is still losing mass in its present pre-white dwarf phase (Werner et al. 1996; Koesterke et al. 1998; Koesterke & Werner 1998), the complex structure of its low surface brightness planetary nebula suggests such a non-axisymmetrical mass loss.

4.5 Magnetic field

The existence of a magnetic field would also lift the degeneracy of the modes by splitting a mode of degree $\ell$ into $\ell$+1 components. Since the cumulative power spectrum of RXJ 2117+3412 does show some triplets, the fine structure must at least be dominated by rotational splitting as the magnetic field alone would only produce doublets for $\ell =1$ modes. However, if a weak magnetic field is superimposed on the rotation, its effect would be to shift each component relative to its non magnetic frequency, with the shift in frequency proportional to m²B², where B is the strength of the magnetic field. Both the $m=\pm 1$components are equally shifted by the magnetic field to higher frequencies. The m=0 component is also shifted to higher frequency by a smaller amount (see Unno et al. 1989; Jones et al. 1989). As a result, a frequency asymmetry in the triplets could be the signature of such magnetic field. Unfortunately, there are only two true triplets in the power spectrum of RXJ 2117+3412 to search for such an asymmetry. Considering these two triplets, one does find that the differences between their prograde and retrograde mode frequency splitting is within the formal uncertainties in the frequency measurement with $\sigma= 0.2$ $\mu$Hz. The corresponding upper limit of the magnetic field, obtained by scaling the results of Jones et al. (1989) for $\ell =1$ modes (their Fig. 1) is of the order of $B\leq500$ G. As this is taken from the calculations for a pure carbon white dwarf model by Jones et al. (1989), it can only be an approximate value when scaled to RXJ 2117+3412.

4.6 Mass of the He-rich outer layers

Mode trapping is interpreted as the signature of chemical stratification in the star. Such a stratification is induced by the previous history of nucleosynthesis within the star and the gravitational settling combined with diffusion acting in a strong gravitational field. The effect of mode trapping on the frequency of the pulsation modes has been studied in detail for the pre-white dwarf pulsators (Kawaler & Bradley 1994), although for luminosities lower than that of RXJ 2117+3412. The trapping cycle observed in RXJ 21117+3412, folded by the trapping phase, is shown in Fig. 14. It does not show evidence of a double peaked structure; this absence suggests that we detect probably only one chemical composition transition zone between the He-rich outer layers and the C/O core. The amplitude of the trapping cycle (A) depends on the gradient of the mean molecular weight through the transition zone and on the thickness of the He-rich layer, while the period of the trapping cycle depends mainly on the thickness of the He-rich outer layer. The thickness of the He-rich outer layer in RXJ 2117+3412 could be precisely determined only through the calculation of realistic models, which are not yet available. The best we can do now is to use the results published by Kawaler & Bradley (1994) and extrapolate them to the range of parameters of RXJ 2117+3412. The extrapolation may not be too bad, since at least the average period spacing is only weakly dependent on luminosity. The trapping period depends on the thickness of the He-rich envelope at fixed $T_{{\rm eff}}$, as shown in Fig. 3 of Kawaler & Bradley. As can be inferred from this figure, the logarithm of the outer layer fractional mass is related to the trapping period through a very tight linear relation. Similarly, at fixed mass of the He-rich outer layer, the trapping period depends on log ( $T_{{\rm eff}}$), as shown in their Fig. 4. Again, the trapping period as a function of log ( $T_{{\rm eff}}$) is accurately fitted by a linear relation. We combine these two relations to construct an interpolation formula representing the models of Kawaler & Bradley:

$$T_{k}= -1.633 \log\left(q_{Y}\right) + 27.65 \log\left(T_{{\rm eff}}\right)+ \rm const.$$

where q_Y is the fractional mass of the He-rich envelope. Using this formula, we can now estimate the thickness of the He-rich outer layer in RXJ 2117+3412 relative to PG 1159-035. To do this, we first redetermine the trapping cycle in PG 1159-035 in the same way as we have done here for RXJ 2117+3412, because our method differs slightly from the way it is done for PG 1159-035 by Winget et al. (1991). Here, the mode trapping is derived from the residuals of the period distribution relative to the average period spacing, while Winget et al. (1991) derive the mode trapping from the forward difference [i.e. $P(k+1)-P(k)~{\rm vs.}~P(k)$] diagram. With the definition adopted here, the redetermined trapping cycle in PG 1159-035 is $T_{k}=3.752 \pm 0.039$, which translates to a trapping period of $P_{\rm tc}= 80.60~{\rm s} \pm 0.84$ s through a period spacing of $\Delta P= 21.483~{\rm s} \pm 0.040$ s, with a semiamplitude of A= 1.59  $\rm s \pm 0.32$ s.

Figure 14: Folded trapping cycle. The residuals of the period distribution (same as in Fig. 12) but plotted vs. trapping cycle phase. The mode $\Delta k=+4$ does not fit the trapping cycle satistactorily and has been omitted. The cycle is repeated twice for a better visibility. The plot shows a single well defined maximum and minimum, indicating that we see only one trapping interface in the star, presumably between the He-rich outer layer and the C/O core.

The above equation can be now rewritten as

$$\Delta \log\left(q_{Y}\right)= 16.93~\Delta \log\left(T_{{\rm eff}}\right) - 0.61~\Delta T_{k}$$

where the symbol $\Delta$ represents the difference of a given parameter between RXJ 2117+3412 and PG 1159-035. For RXJ 2117+3412, we adopt $T_{{\rm eff}}$= 170000 K (Werner et al. 1996; Rauch & Werner 1997) and for PG 1159-035 $T_{{\rm eff}}$= 140000 K (Dreizler & Heber 1998) and q_Y=0.0035 M_* (average of the two estimates of Kawaler & Bradley 1994). We find the thickness of the He-rich envelope in RXJ 2117+3412 to be 0.078 M_*. This result must be treated with caution, however, because it is based on an extrapolation. We recall here, that the temperature and luminosity of RXJ 2117+3412 are outside the range covered by the models of Kawaler & Bradley (1994). We can obtain a more conservative estimate as follows: 1) we assume that the increase of q_Y with $T_{{\rm eff}}$ continues outside the range covered by the models, 2) instead of a true temperature of RXJ 2117+3412, we use the highest  $T_{{\rm eff}}$ for which models exist log ( $T_{{\rm eff}})= 5.184$ in our interpolation formula. This approach avoids extrapolation and yields a lower limit for the He-rich envelope mass of RXJ 2117+3412, which is q_Y=0.013 M_*. We conclude that the He-rich outer layer of RXJ 2117+3412 is at least 3.7 times more massive than that of PG 1159-035, and possibly more than 20 times more massive.

Despite similar trapping periods, we derive vastly different envelope thickness. This is entirely due to the difference in $T_{{\rm eff}}$ between RXJ 2117+3412 and PG 1159-035. A comparison of the trapping amplitudes of PG 1159-035 and RXJ 2117+3412 shows that the amplitude of RXJ 2117+3412 is only half that of PG 1159-035. The smaller trapping amplitude of RXJ 2117+3412 could arise from two effects (see Kawaler & Bradley 1994, their Fig. 3). First, the trapping amplitude decreases with increasing He-rich layer mass fraction and second, it also decreases towards longer periods. Both effects are the result of the peak amplitude portion of the eigenfunction moving away from the composition gradient, which decreases the resonance effect of mode trapping. The observed periods of PG 1159-035 are in the range of 430 s to 840 s, while in RXJ 2117+3412 they are in the range of 690 s to 1190 s.

4.7 Mass of RXJ 2117+3412

The presently available evolutionary models for the transition between the planetary nebulae nuclei and the white dwarfs are not suitable for interpreting RXJ 2117+3412. While the 0.7 $M_{\odot}$ evolutionary sequence of Wood & Faulkner (1986) fits the location of RXJ 2117+3412 in the $\log g$- $\log T_{{\rm eff}}$ diagram, the pure He surface composition of their models does not adequatly represent the observed abundances of RXJ 2117+3412. The more recent calculations by Gautschy(1997), while using a composition more compatible with the surface abundances of PG 1159 stars, do not fit the high luminosity and temperature of RXJ 2117+3412, except by considering stellar models with masses in excess of 0.7 $M_{\odot}$, which disagrees with the mass derived from asteroseismology (see below). None of these models takes into account the chemical stratification induced by diffusion in the presence of mass loss. As a consequence, one can hardly use them to calculate theoretical period spacings which one could use for asteroseismological mass determination.

In Vauclair et al. (1993), the mass estimate was based on an interpolation formula which did not take into account the luminosity dependence of the period spacing (Winget et al. 1991). This formula, used to estimate the mass of PG 1159-035 is probably not appropriate for RXJ 2117+3412, which is two orders of magnitude more luminous than PG 1159-035. Kawaler & Bradley (1994) calculated the period spacings, $\Pi_{0}=\sqrt{\ell(\ell+1)}\Delta P$, for a grid of pre-white dwarf models of various masses, including the luminosity dependence (their Fig.  2). They derive an interpolation formula that is valid for luminosities between $1.6 \leq \log(L/L_{\odot})\leq 2.8$. Extrapolating this formula to the luminosity of RXJ 2117+3412 log( $L/L_{\odot})\approx 4.0$ is risky. Rather than using their interpolation formula, we extrapolate $\Pi_{0}$directly from their Fig.  2 by spline functions. Using the period spacing derived in Sect. 4.3, $\Delta P= 21.618$  ${\rm s}\pm 0.008$ s, and assuming that this period spacing is valid for asymptotic $\ell =1$ modes, we obtain $\Pi_{0}=30.573~{\rm s}\pm 0.011$ s. At the luminosity of RXJ 2117+3412 (see next section), such a period spacing corresponds to a model of $\approx$0.56 $M_{\odot}$. The formula of Winget et al. (1991) gives a mass of 0.58 $M_{\odot}$. The value obtained from Kawaler & Bradley's interpolation formula would be 0.52 $M_{\odot}$, and the range $0.06~M_{\odot}$ is a rough estimate of the uncertainty for this preliminary mass determination. We should stress that the uncertainty we quote is entirely due to the fact that we have to extrapolate the existing theoretical calculations and does not include any observational uncertainty in $\log g$ or $T_{{\rm eff}}$. A much more precise mass estimate (as good as for PG 1159-035) must await models covering the parameter range of RXJ 2117+3412. For now, our best value for the mass of RXJ 2117+3412 is $0.56^{+0.02}_{-0.04}~M_{\odot}$.

If we interpret the observed period spacing as corresponding to $\ell =2$ modes, the derived mass would be $\approx$0.32 $M_{\odot}$. Such a low mass would be in conflict with the spectroscopically determined log g= 6.0. It would imply that the progenitor of RXJ 2117+3412 was in a binary system, for which we do not see evidence, and that the star would have a helium core as a result of previous mass transfer in the binary system. Higher $\ell$ values would imply even lower mass estimates for RXJ 2117+3412, which would make the conflict even more severe. On these grounds, we conclude that the modes observed in RXJ 2117+3412 must be $\ell =1$ modes. This is in agreement with the $\ell$ value implied by the fine structure found in the power spectrum.

4.8 Luminosity and distance

Knowing the total mass of the star from asteroseismology on one hand and the surface gravity and effective temperature from spectroscopy on the other hand, it is straightforward to derive the luminosity and the distance of the star.

The best fit model atmosphere for RXJ 2117+3412 indicates a surface gravity log g=6.0 ^+0.3_-0.2 (Rauch & Werner 1997) and an effective temperature $T_{{\rm eff}}= 170\,000~{\rm K} \pm 10\,000~{\rm K}$ (Werner et al. 1996). The radius derived from the mass ( $0.56^{+0.02}_{-0.04}~M_{\odot}$) and gravity is:

$$\log\left(R/R_{\odot}\right) = -0.91^{+0.10}_{-0.15}.$$

From the radius and $T_{{\rm eff}}$, one derives a luminosity (assuming a spectral energy distribution similar to a blackbody):

$$\log\left(L/L_{\odot}\right) = 4.05^{+0.23}_{-0.32}.$$

The distance can be derived from the ratio of the flux in the V band as observed on Earth to the flux emitted at the stellar surface, as given by the best fit model atmosphere (Werner et al. 1996 and Werner, personal communication). For this estimate, one uses a magnitude m_V= 13.16 (Motch et al. 1993) for RXJ 2117+3412 and the flux predicted by the model integrated through the spectral response of the V filter. One obtains a distance:

$$D= 1130~^{+340}_{-350}\,{{\rm pc.}}$$

In estimating the error of the distance, one takes into account the uncertainty of the radius and the luminosity due to the uncertainty of the surface gravity and of  $T_{{\rm eff}}$. But the uncertainty of the model flux due to the uncertainty of $T_{{\rm eff}}$ was not taken into account as only the best fit model ( $T_{{\rm eff}}= 170\,000$ K, $\log g= 6.0$) was used. The derived distance is in good agreement with the distance determined by Motch et al. (1993), D=1400  ⁺⁷⁰⁰_-500 pc. The uncertainty of our result is smaller because of the better determination of the mass and atmospheric parameters. The remaining uncertainty is dominated by the relatively large error of log g. However, one still has to account for the interstellar absorption because of the proximity of RXJ 2117+3412 to the galactic plane. In Motch et al. (1993), the interstellar absorption was neglected on the argument that four stars in the direction of RXJ 2117+3412 do not show reddening significant enough to affect the distance estimate of RXJ 2117+3412. Three of these stars have good Hipparcos parallaxes (Perryman 1997): HD 202904 (HR 8146) with $\pi=3.62~{\rm mas}\pm 0.56$ mas is at $\approx$275 pc, HD 204403 (HR 8215) with $\pi= 1.84$  ${\rm mas}\pm 0.56$ mas is at $\approx$540 pc, and HD 207516 (HR 8338) with $\pi= 6.61$  ${\rm mas}\pm 0.61$ mas is at $\approx$150 pc. The fourth star (HD 203921) is not really useful since its parallax is not good enough ( $\pi=1.46~{\rm mas}\pm 1.13$ mas). The three stars are significantly closer than RXJ 2117+3412 and therefore cannot yield a reliable extinction estimate out to the distance of RXJ 2117+3412 itself. Therefore, we estimate the interstellar extinction towards RXJ 2117+3412 according to the model of Chen et al. (1998). Their interstellar extinction model improves the model of Arenou et al. (1992) by using a large sample of open clusters in the galactic plane. However, the interstellar extinction at low galactic latitude is patchy and its small scale structure still needs to be determined. The extinction derived by Chen et al. (1998) depends on the galactic longitude and varies non linearly with the distance. As distance and interstellar absorption are related, we must iterate to get the distance of RXJ 2117+3412. We stop the iterations as soon as the last two consecutive distance determinations differ by only $\approx$0.5$\%$, which is comparable to the precision of m_V ( $m_V=13.16\pm 0.01$). This puts RXJ 2117+3412 at a closer distance:

$$D= 760 ^{+230}_{- 235}\,{{\rm pc.}}$$

The interstellar absorption at the average distance of RXJ 2117+3412 is: $A_{{\rm v}}=0.86$ mag. We predict the parallax of RXJ 2117+3412 to be 1.32 mas ^-0.30_+0.59 mas.

Adopting this new distance estimate, and its associated uncertainty, the linear extent of the planetary nebula is:

$$L_{\rm PN}= 2.9\pm~0.9\,{{\rm pc.}}$$

The planetary nebula surrounding RXJ 2117+3412 is still the largest one known.

4.9 Secular evolution

We showed in Sect. 4.6 that the He-rich outer envelope in RXJ 2117+3412 could be at least 3.7 times more massive than in PG 1159-035, and could possibly be greater than 20 times more massive. Knowing the mass loss rate of RXJ 2117+3412, we can estimate how long it would take to RXJ 2117+3412 to loose most of its He-rich envelope so that it would become similar to PG 1159-035. This would give an order of magnitude estimate for the evolutionary time scale. Koesterke et al. (1998) derive a mass loss rate of $\log \dot{M}=-7.0 \left(M_{\odot}~{{\rm yr}}^{-1}\right)$, from the C IV line profiles. More recently, Koesterke & Werner (1998) obtained a more precise mass loss rate of $\log \dot{M}= -7.4 \left(M_{\odot}~{{\rm yr}}^{-1}\right)$from O VI line profiles, which we will use here. If we assume that RXJ 2117+3412 is a progenitor of PG 1159-035, and also assume that PG 1159-035 had the same internal structure as RXJ 2117+3412 at the same luminosity, then we can derive the age difference between the two stars, assuming a constant mass loss rate, as:

$$\Delta t= \Delta M/\dot{M}$$

where $\Delta M$ is the difference in the He-rich envelope mass between RXJ 2117+3412 and PG 1159-035. It would take $\approx$1.3$\times$10⁵ yr (or $\approx$1.1$\times$10⁶ yr) for RXJ 2117+3412 to loose enough mass to become similar to PG 1159-035 if the He-rich envelope is 3.7 times (or 22.3 times) more massive than the envelope of PG 1159-035. This is only a lower limit to the evolutionary time scale, since it is expected that the mass loss rate decreases with decreasing luminosity.

This estimate of the evolutionary time scale would imply a rate of period change of the order of $\dot{P} \leq 2.4 \times 10^{-10}\,{{\rm s\,s}}^{-1}$ ( $2.9 \times 10^{-11}\,{{\rm s\,s}}^{-1}$) for the low He-rich envelope mass case (high He-rich envelope mass) for periods around 1000 s. This is comparable to the $\dot{P} = (+13.0 \pm 2.6) \times 10^{-11}\,{{\rm s\,s}}^{-1}$ measured for the 516 s mode in PG 1159-035 (Costa et al. 1999). Such a high rate of period change would probably be detectable in RXJ 2117+3412 if one could find an isolated mode with a stable amplitude. As shown in Fig. 12, mode trapping in RXJ 2117+3412 offers the potential to measure $\dot{P}$ for both trapped and nontrapped modes. In the case of the trapped modes the resultant $\dot{P}$ would be dominated by the evolutionary time scale of the outer layers, where the structure reflects the competition between contraction and mass loss. The $\dot{P}$ of the nontrapped modes would be dominated by the core cooling time scale. Unfortunatly, it will be difficult to get an unambiguous $\dot{P}$measurement for any mode of RXJ 2117+3412, because of the rich pulsation spectrum (which requires multisite data) and because of the large amplitude variations of the modes.

The mode at 1315.05 $\mu$Hz, whose amplitude remained almost constant in the data sets presented here, unfortunately takes part in a linear combination with the 653.987 $\mu$Hz to form the 1968.952 $\mu$Hz (plus some other higher order combinations). So it can not be used as a clean mode to measure $\dot{P}$. As this mode may be suspected to take part in a mode coupling, its frequency variation may not reflect global stellar evolution, but would rather reflect some properties of the mode coupling. It is worth mentioning however that, while both the 653 $\mu$Hz and 1968 $\mu$Hz peaks suffered large amplitude variations in the period covered by the data, as seen in Table 6, the 1315 $\mu$Hz amplitude varies by only 20% during the same time. If those three modes were coupled, it is difficult to understand how the amplitude of two components could vary so much while the third remains almost constant. So long as we cannot determine whether the 1315 $\mu$Hz mode is an eigenmode or coupled to other modes, it would be hazardous to infer any physical meaning from a $\dot{P}$ measurement for that particular mode. One must conservatively conclude that with the data presently available, there is no stable enough mode in a clean enough part of the power spectrum which we could identify as a potential candidate for a $\dot{P}$ measurement. But given the presently poor physical constraints on evolution in this part of the H-R diagram, attempting to measure a rate of period change is a worthy challenge.

The rapid evolution of RXJ 2117+3412 shows up in the mass loss. RXJ 2117+3412 has a measured mass loss rate of $\log \dot{M}= -7.4 \left(M_{\odot}~{{\rm yr}}^{-1}\right)$, which has interesting implications for the region where pulsation driving takes place. The excitation mechanism ($\kappa$ mechanism due to carbon and oxygen partial ionization) operates at the depth of $T\approx10^{6}~$K, which lies in the outer $\approx$ 10^-8 M_* mass fraction of the star (Bradley & Dziembowski 1996). The observed mass loss rate implies that the material in the driving region is renewed on a time scale considerably shorter than the evolutionary time scale ($\approx$50 days!). As a consequence, the chemical composition of the driving region could change on this time scale if there is compositional stratification of the outer layers. This outflow of mass through the driving region may affect the efficiency of the excitation mechanism, especially if the mass loss rate is time dependent. The amplitude variations observed in the star, including variations on time scale as short as that exhibited by the mode at 717 $\mu$Hz during the 1994 WET campaign, could be related to the effect of the mass loss on the effectiveness of the driving in this region.

4.10 Is there evidence for $\epsilon$ mechanism in RXJ 2117+3412?

The non-radial g-mode instability in pre-white dwarf stars is triggered by the $\kappa$ and $\gamma$ mechanism induced by the partial ionization of carbon and oxygen at $T\approx 10^{6}$ K, as first suggested by Starrfield et al. (1983, 1984) and confirmed by the subsequent analysis of Stanghellini et al. (1991). More recently, instability studies using models computed with the new He/C/O OPAL opacities (Iglesias & Rogers 1993) show a better agreement with the observed blue edge of the instability strip and put some constraints on the composition of the driving region (Bradley & Dziembowski 1996; Saio 1996; Gautschy 1997). However, Kawaler et al. (1986) had also anticipated that during the PN and pre-white dwarf evolutionary phases, the possibility of a remnant He-burning shell that could drive g-modes by the $\epsilon$-mechanism. As the He burning necessarily occurs at the bottom of the He-rich outer layers, the periods of these unstable g-modes are in the range of 70 s-200 s, corresponding to low k orders for $\ell =1$ modes. Saio (1996) and Gautschy (1997) also find g-modes triggered by the $\epsilon$-mechanism in some of their models, with typical periods between $\approx$110 s and $\approx$150 s. However, such short period g-modes excited by $\epsilon$-mechanism have not been found in the surveys of PNN conducted by Grauer et al. (1987) and by Hine & Nather (1987).

At the high frequency end of the list given in Table 6, one finds some peaks which could be candidates for such $\epsilon$-mechanism driven modes, since their periods are in the range 230-290 s. But, as discussed in Sect. 3.2, all the peaks with frequency above 2180 $\mu$Hz ( $\rm Periods \leq 460$ s) are the result of linear combinations of lower frequency modes; they are not independent modes. The highest frequency independent mode has a frequency of 2174.884  $\mu$Hz (period of 459.8 s). A careful scrutiny of the power spectrum at even higher frequencies (up to 12000 $\mu$Hz), where the highest noise peaks are at a 0.20 mma level, does not reveal any significant peak. We conclude that our data show no evidence for low-k order mode driven by the $\epsilon$-mechanism.