Pulsation models for the roAp star HD 134214

# Pulsation models for the roAp star HD 134214

## Abstract

Precise time-series photometry with the MOST satellite has led to identification of 10 pulsation frequencies in the rapidly oscillating Ap (roAp) star HD 134214. We have fitted the observed frequencies with theoretical frequencies of axisymmetric modes in a grid of stellar models with dipole magnetic fields. We find that, among models with a standard composition of and with suppressed convection, eigenfrequencies of a model with and a polar magnetic field strength of 4.1kG agree best with the observed frequencies. We identify the observed pulsation frequency with the largest amplitude as a deformed dipole () mode, and the four next-largest-amplitude frequencies as deformed modes. These modes have a radial quasi-node in the outermost atmospheric layers (). Although the model frequencies agree roughly with observed ones, they are all above the acoustic cut-off frequency for the model atmosphere and hence are predicted to be damped. The excitation mechanism for the pulsations of HD 134214 is not clear, but further investigation of these modes may be a probe of the atmospheric structure in this magnetic chemically peculiar star.

## 1Introduction

HD 134214 (HI Lib) is a cool Ap (CP2) star with a spectral type of F2 Sr Eu Cr [38]. [25] and [26] discovered light variations in this star with a period of 5.65 min, confirming HD 134214 as a member of class of rapidly oscillating Ap (roAp) stars. The roAp class, established by [28], now consists of about 40 known members [32]. The 5.65-min period of HD 134214 is the shortest known among the roAp stars. Only one frequency was detected in the first observations, in later groundbased photometry [27] and in 10 hours of photometric monitoring by the MOST satellite in 2006 [10].

The monoperiodicity of HD 134214 was rejected after 2 hours of time-resolved spectroscopy by [33] who detected 5 additional frequencies in the radial velocities of some lines of rare earth elements. Soon after, 8.8 h of spectroscopy by [31] led to the identification of 2 more frequencies. Furthermore, new photometry they obtained revealed a second significant frequency in light. [31] found that the photometric amplitude ratio of the secondary to the principal frequencies () is similar to the ratio in radial velocity amplitudes. Recently, [18] detected 10 independent oscillation frequencies in HD 134214 from a 26-d (2008 April - May) photometric time series obtained by MOST, confirming all 8 previously reported frequencies. HD 134214 is now one of the richest multi-periodic roAp stars known, comparable to HD 24712 (HR 1217) [29], HD 101065 (Przybylski’s star) [37], and HD 60435 [35].

The pulsations observed in roAp stars are p-modes of low degree and high radial order, affected by globally coherent magnetic fields of strengths  kG. The periods range from 5.6 to 21 min. The amplitudes of many roAp pulsation modes are modulated with the star’s rotation period. This has been explained by the Oblique Pulsator Model proposed by [28], in which the pulsation is axisymmetric with respect to the dipole magnetic axis, which is itself obliquely inclined to the stellar rotation axis by an angle . However, no amplitude modulation has been observed in HD 134214 in any of the photometric or spectroscopic data obtained to date. This may indicate that the magnetic obliquity is small (), or that the rotation axis is nearly coincident with the line of sight () [26].

How does the strong magnetic field of an roAp star affect its pulsations? Lorentz forces on the moving plasma in the stellar atmosphere mean that the angular dependence of the eigenfunction of a nonradial mode cannot be described by a single spherical harmonic. In addition, p-mode (acoustic) oscillations couple with magnetic oscillations in the outer layers where the Alfvén velocity is comparable to the sound speed . Due to this coupling, part of the acoustic oscillation energy is converted to magnetic oscillations which propagate inward as slow waves. These slow waves are expected to be dissipated deep in the interior of the star because of their very short spatial wavelengths [39]; i.e., magnetic coupling damps the p-mode oscillations. Including these effects, the properties of adiabatic p-mode oscillations in the presence of dipole magnetic fields were first investigated by [16], [9], [15], and [44]. The results from different methods qualitatively agree with each other [43]. The oscillation frequency of a p mode generally increases as the strength of the magnetic field increases, because the phase velocity of magneto-acoustic waves increases. The gradual frequency increase is interrupted occasionally by an abrupt fall of a few tens of Hz, when the damping due to slow-wave dissipation reaches a peak [15].

[14] modified the method of [15] to make it possible to treat quadrupole magnetic fields, while [42] extended the method of [44] to nonadiabatic oscillations. Furthermore, [8] and [7] found that pulsation axis does not necessary align with the magnetic axis if rotational effects are taken into account. Theoretical pulsation analyses including magnetic effects have been applied to several roAp stars to fit observed frequencies; for example, Equ [19], 10 Aql [21], HD 101065 [37], HD 24712 [45], and three roAp stars recently found by the Kepler satellite [5].

In this paper, we compare oscillation frequencies of HD 134214 obtained by [18] from 2008 MOST photometry to theoretical frequencies of axisymmetric low-degree modes calculated by the method of [42], including the effect of dipole magnetic fields.

## 2Fundamental parameters of HD 134214

In the literature, we note two recent sets of estimates of luminosity and effective temperature: and by [22], and and by [24]. In both cases, the Hipparcos parallax was used to derive luminosity, and photometric indices of the Strömgren and the Geneva system to obtain . Also, [40] obtained () and from Strömgren photometry.

The possible locations of HD 134214 in the HR diagram (HRD) set by these estimates are shown in Fig. ?; filled and open circles with error bars are for the [24] and [22] estimates, respectively, and the range of obtained by [40] is shown near the bottom of the diagram. Also shown in this figure are the positions of other roAp stars (crosses) adopted mainly from [4] and [24]. This figure indicates that HD 134214 is a relatively cool roAp star and its location in the HRD is similar to that of the prototypical roAp star HD 24712 (= HR 1217). Not only their positions in the HRD are similar, but also their pulsational properties [18]. We will return to this similarity later in the paper.

[34] obtained a mean surface magnetic field of 3.1 kG from the Zeeman splitting pattern, but found no measurable longitudinal fields in HD 134214. They also found a slight modulation of the mean field strength with a period of days, although [1] found no long-term variation in the mean light. If the periodicity indicates the rotation period of HD 134214, it corresponds to an equatorial velocity of km s. Combined with the projected rotation velocity  km s obtained by [40], we find the inclination angle to be as small as . Such a nearly pole-on condition in HD 134214 was also inferred from the absence of pulsation amplitude modulations (and no rotational frequency splittings) by [27].

## 3Stellar and pulsation models

We have generated unperturbed stellar models for our pulsation analysis from evolution models calculated by a Henyey-type code for a mass range of , using OPAL opacity tables [23] supplemented with low-temperature tables by [2]. Some of the calculated evolutionary tracks are shown in Fig. ?.

For the outermost atmospheric layers we have employed the standard relation given in [47]. In some models, helium is assumed to be depleted above the first He ionization zone (appropriate for the stratified atmosphere of an Ap star) in the same way as described by [45]. We have suppressed envelope convection in majority of the models, assuming that a strong magnetic field should inhibit convective instability in the envelope. For those models where we did not suppress envelope convection, we used the mixing-length theory formulated by [20] with a mixing length of 1.5 pressure scale heights. Core convection is always included, but with no overshooting. Table 1 lists the assumed parameters adopted for each series of evolutionary models, in which ‘Y’ and ‘N’ mean ‘included’ and ‘not included’, respectively.

We have calculated nonadiabatic frequencies for axisymmetric p-modes in the presence of dipole magnetic fields (but without rotation) using the method described by [42] and [44], where perturbations of magnetic fields are included by adopting the ideal MHD approximation. The pulsation axis is assumed to be aligned with the magnetic axis. Frequencies are obtained for field strengths at intervals of 0.1 kG in the range (where = polar field strength). Since the angular dependence of the eigenfunction of a pulsation mode in the presence of a magnetic field deviates from a single spherical harmonic, we express it by a sum of terms proportional to Legendre functions (or , where is co-latitude measured from the pulsation axis) with for an even mode, or for an odd mode; i.e., the kinetic energy of a pulsation mode is distributed among components with different degrees . To designate the angular dependence of a mode we use which is equal to the degree of the component having the largest fraction of kinetic energy among the expanded terms of the mode. In most cases, we have employed twelve values to express an eigenfunction. When the expansion did not converge sufficiently after 12 terms, up to 14 Legendre functions were used.

It is not easy to estimate a mean error for theoretical frequencies, because uncertainty depends on the frequency and . When the convergence of expansion is reasonably good (i.e., the ratio of the kinetic energy of the last component to that of the main component is less than 10 percent), the differences between the frequencies obtained using 12 and 14 Legendre functions are less than Hz. But among the cases with a mediocre convergence with the above ratio being 10-30 percent, the differences in some cases are Hz (in rare cases the difference can be up to Hz). From these properties, we consider the mean error of the theoretical frequencies due to the truncation of the expansion to be Hz. In addition, we expect errors from using spherically symmetric equilibrium models despite strong magnetic fields. Although it is not clear how seriously the magnetic non-sphericity would affect the frequencies of high-order p-modes, we assume that the uncertainty in theoretical frequencies would be Hz.

Since the observed pulsation frequencies of HD 134214 are all above the acoustic cut-off frequency (as in the case of HD 24712), we have applied the running wave outer boundary condition described in [45]. In the pulsation analysis for models that include envelope convection, we have used a frozen convection approximation for the divergence of convective energy flux. The perturbation of radiative flux is treated in the same way as done by [45], adopting the [49] theory for the time-dependent Eddington approximation.

## 4Frequency fittings

Table 2 lists frequencies and photometric amplitudes obtained by [18] from MOST photometry, and radial velocity amplitudes of Nd III lines obtained by [31]. The first column contains the designations of frequencies by [18], and the second column lists the corresponding designations by [31]. No frequencies corresponding to and were found by [31]. The ranking of the photometric amplitudes is very different than that of the radial velocity amplitudes (except for ) which might be related to differences in mode properties and/or long-time variations in pulsation amplitudes [31]. The last column of the table indicates for each frequency identified in our best models (see below).

For the fittings to be discussed in this section, we adopt only the eight confirmed frequencies () common to both the photometric and spectroscopic analyses. We omit the two frequencies having lowest amplitudes ( and ), allowing for the possibility that they could be modes of higher degree (). We will discuss later the result of fits including all 10 frequencies.

For each model sequence listed in Table 1, we have looked for a model whose frequencies of low-degree () axisymmetric modes best fit the eight frequencies (). The “goodness” of a model fit is measured by the mean deviation (MD) from the eight frequencies, where we have adopted the same weight for all eight, because the uncertainties in observed frequency values (Hz) are much smaller than theoretical ones (Hz). The theoretical frequencies have been interpolated with respect to radius along each evolutionary path to find the best radius, but no interpolation is performed with respect to because frequency may change discontinuously as a function of . (Recall that model frequencies are calculated at intervals of 0.1 kG.)

Table 3 lists parameters of the best fitting models we have found for various model sequences; the mean deviations (MD) are given in the last column. The loci of these best fitting models in the HRD are shown by open squares in Fig. ?. The mean deviations of the best models for different masses vary only slightly. A model with best reproduces the eight observed frequencies at polar magnetic field strength  kG among the D sequences, in which a standard composition is adopted, envelope convection is suppressed, and helium above the first He ionisation zone is depleted. The required magnetic field strength is plausible, given the mean modulus (3.1 kG) for this star obtained spectroscopically [34]. Fig. ? compares the theoretical frequencies of the best model as a function of with the observed frequencies; the closest agreement occurs for  kG (vertical dashed line). Generally, pulsation frequencies increase as the magnetic field strength increases, roughly preserving the large separation. However, since the dependence of frequencies on differs slightly for different , the relative loci of sequences for different degrees differ considerably from the case with no magnetic field.

The slow increase in frequency with is interrupted occasionally by a sudden decrease of a few tens of Hz; this behaviour was first recognised by [15]. Fig. ? shows that, in such a transition range, two distorted dipole modes appear. This is why two dipole modes appear within a large frequency separation in the echelle diagram shown in Fig. ?. These two deformed dipole modes have by definition, but have different distributions of kinetic energy among components of , and hence have different angular dependences. Such additional modes are needed to explain the frequency spectrum of the roAp star HD 101065 [37].

The theoretical frequencies of the model with  kG are compared in detail as an echelle diagram in Fig. ?. In this diagram, frequencies are folded modulo Hz according to the large frequency separation obtained by [18] from their observed frequencies of HD 134214. The large separation of the model is consistent with the observed one, although the former slightly depends on degree due to the fact that the magnetic effects also depend on . We find reasonably good agreement between theoretical and observed frequencies. The mean deviation MD of this model (Hz) seems to be slightly larger than the uncertainty expected for theoretical frequencies. A non-dipole component of the magnetic field could account for the discrepancy.

We identify the principal frequency ( mHz) as a distorted dipole mode, while frequencies and are identified as modes. We note that the frequencies and , have similar amplitudes (next largest after the principal frequency) and, hence, appear to share the same degree. Mode identifications in the other best fitting models listed in Table 3 are similar to those by D165. They are listed in the last column of Table 2.

Comparing the best models of the D170 and H170 sequences, we see that helium depletion in the upper atmosphere does not affect the “goodness” of the best model. The magnetic field strength of the best fit from the helium-depleted model sequence (D170) is smaller than the model without helium depletion (H170). A stronger magnetic field is needed for the latter model to have the same magnetic effect, because the mean density of the model without helium depletion is larger than the helium-depleted model at the same and , due to the lower opacity and higher mean-molecular weight.

It can be seen in Table 3 that including convective energy transport in the envelope convection zone (H160C, H165C, and H170C) slightly improves the quality of the fits but only by adopting stronger magnetic fields. The improvement, however, seems too weak to claim the presence of efficient convection in the polar regions of HD 134214. The necessity of the stronger fields is due to higher mean density in the outer envelope as discussed above. In this case, the higher density is caused by, in addition to the homogeneous helium abundance in the envelope, a decrease in the temperature gradient brought by convective energy transport. Changing metallicity does not significantly change the quality of fits.

## 5Mode properties

In the best fitting models discussed above, the principal frequency corresponds to a deformed dipole mode, while match modes whose amplitudes are very much smaller than that of the principal one (see Table 2). One could speculate that the large amplitude difference might be a difference in visibility due to different angular dependences of the amplitudes on the stellar surface between modes of and . However, a glance at Fig. ? demonstrates that this is not the case. This figure shows the latitudinal dependences of flux perturbations of the dipole mode (solid line) and one of the modes, (short dashed line) for  kG. The latitudinal dependences are similar to each other, although without a magnetic field, a dipole mode should vary as , quite different from the expected dependence (). The strong magnetic field suppresses pulsation amplitudes at low latitudes because gas cannot cross field lines, so that large amplitudes are strongly confined to the polar regions.

The only appreciable difference between the two modes is the parity with respect to the equatorial plane; modes are odd and modes are even. For the principal frequency to be identified as a deformed dipole mode, the angle between the magnetic axis and line-of-sight should be small so that the contribution from the opposite hemisphere is small. Combining this fact with the requirement of a small angle between the rotation axis and the line of sight (Sec.Section 2) indicates the obliquity angle of HD 134214 must be small too. With a small angle between the line of sight and the pulsation axis, there is little difference in the visibility between and modes. This is contrary to the conjecture of [18] that the large amplitude difference between and other frequencies is due to a large difference in their visibilities.

Fig. ? also shows the latitudinal distributions of flux perturbation for and 3 modes which are fitted to and , respectively. In contrast to the cases of and 2, the flux amplitudes of the and 3 modes show wavy variations as a function of latitude with quasi-nodal lines which would reduce the visibility of the modes. (Note that the mode is significantly deformed from spherical symmetry due to the contributions from high components.) This is consistent with the fact that and have smallest amplitudes among the eight frequencies adopted here.

Fig. ? shows runs of phase delay and amplitude of radial velocity variations in the atmosphere of a model which has parameters close to those of the best (interpolated) model. In this figure, the pulsation axis (and hence the magnetic axis) is assumed to be aligned with the line of sight. The properties are not very sensitive to the obliquity of pulsation and magnetic axes; we see very similar model behaviours if we adopt, for example, an obliquity of instead of . Fig. ? reveals a quasi-node around for both modes, where is again the optical depth calculated with the Rosseland mean opacity. Above the quasi-node, the amplitude and phase delay increase outward, indicating the presence of outwardly running wave components. This is consistent with the fact that both frequencies are above the critical acoustic cut-off. We note that the quasi-node is not the “false node” discussed by [48], because it appears in the eigenfunction.

Also plotted in Fig. ? are data of radial velocity/phase measurements from H bisector and lines of some rare earth elements of HD 134214 adopted from [41], where the optical depth at each H line depth and the formation depths of the other lines are assumed to be the same as in the case of HD 24712. This figure shows that the observational data qualitatively agree with the model; amplitude and phase-delay increases outward in the outermost layers, which also agree with the spectroscopic analysis by [31]. However, the observed variation of phase delay in the outermost layers is considerably steeper than the model prediction. The discrepancy probably indicates that the sound speed (or temperature) in the upper atmosphere of HD 134214 is lower than in the model. This is consistent with the actual temperature gradient being steeper due to a strong blanketing effect which is not included in our models.

The radial velocity amplitudes measured by Nd II/III tend to be larger than those of H, while the amplitudes of Pr III lines tend to be smaller than those of H. These differences can be attributed to the difference in the distributions of the elements Nd and Pr on the stellar surface. If the depths of line formation are similar to HD 24712, as we have surmised, then Figs. ? and ? suggest that Nd is distributed at a higher pulsational latitude than Pr.

At the quasi-node, the pulsation phase of the principal frequency changes dramatically by about 1.8 radians. The presence of the phase shift at is supported observationally by the phase shift from the deep part of H phase to the phases of Y II (Fig. ?).

The runs of phase and amplitude in the atmosphere for shown in dashed lines in Fig. ? are very similar to those for the principal frequency (; solid lines). This agrees with the finding by [31] that the amplitude ratio of the two frequencies ( and ) from the radial velocity variations (weighted toward the outermost layers) is similar to that measured for the light variations (which occur in the photosphere).

The similarity in the latitudinal and depth dependences between the principal mode and the secondary mode suggests that the huge amplitude difference is caused by a large difference in the excitation strengths of yet unknown mechanism.

## 6Fitting all ten frequencies

In the model fits described above, we omitted the two frequencies ( and ) with the lowest amplitudes because they might be due to high-degree () modes for which no theoretical frequencies are available. Here we discuss fits performed for models in the D165 sequence assuming all ten observed frequencies are low-degree () modes. Fig. ? shows the echelle diagram for the best fitting model from the D165 series. The observed frequencies are best fitted with  kG, resulting in a mean deviation MD = Hz. Compared to the best fitting D165 model for only eight frequencies (Table Table 3), this model is slightly hotter, but still within the uncertainty of the photometric effective temperature (Fig. ?). Note that in Fig. ?, frequencies are folded with a separation of Hz, which is larger than what is used in Fig. ? (Hz) because of the higher . In this model, are identified as modes rather than . Comparing Fig. ? (10-frequency fit) with Fig. ? (8-frequency fit), the quality of fit is better in the latter case. This is circumstantial evidence supporting the argument that and are modes of degree equal to or higher than 4.

## 7Discussion

### 7.1Excitation mechanism

The predicted frequencies and large separations of the best fitting model of each series are roughly consistent with the observed values for HD 134214. However, there is one serious problem: all observed frequencies are above the acoustic cut-off frequencies of our models and all modes with appropriate frequencies are damped modes.

Pulsations of roAp stars are now generally considered to be excited by the kappa-mechanism operating in the hydrogen ionization zone [4]. Our models are located in the instability region of the HR Diagram obtained by [13]. It turns out that p-mode pulsations of order up to ( mHz) are actually excited by the kappa-mechanism in the best fitting D165 model ( kG). But none of these predicted pulsations have been detected.

[3] found that the cut-off frequency increases if the relations based on more physically accurate atmospheric models are used. For the range appropriate for HD 134214, however, even their models predict cut-off frequencies below about 2.7 mHz, smaller than the principal frequency of 2.95 mHz in HD 134214. Another way to increase the cut-off frequency is to introduce a temperature inversion in the atmosphere, as proposed by [17]. Unfortunately, the inversion necessary in this case must be a few thousand degrees. Even if we assume a reflective boundary condition, no modes having frequencies comparable to those of HD 134214 are excited. In addition, observations indicate the presence of outwardly running waves in the outermost layers (Fig. ?) do not support a reflective boundary. Therefore, the outer boundary is probably not the cause of the excitation discrepancy.

The same problem exists for another multi-periodic roAp star, HD 24712 (HR 1217), as discussed by [45]. These two examples clearly indicate that an instability mechanism other than the kappa-mechanism must be operating, at least for the super-critical frequency pulsations. The mechanism must be intimately connected with the interaction with magnetic fields because no such high-frequency oscillations occur in A-F stars other than the roAp stars. The atmospheric layers with a super-adiabatic temperature gradient and a magnetic field are overstable in linear analyses with the Boussinesque approximation, where density perturbations are neglected except for buoyancy forces [11]. Based on the fact that the periods of overstability for Ap stars with reasonable magnetic field strengths are comparable to the period range seen in roAp stars, [46] predicted overstability to be the excitation mechanism for roAp stars. [12] discussed some physical implications of overstability. Weaknesses of this concept were identified by [4]: as the analyses are local, it is not certain whether a global mode is excited, and even if global modes are excited it is not certain whether such a mode has any significant amplitude above the thin convective layer. Further investigations of the roAp excitation mechanism are definitely needed.

### 7.2Similarities and dissimilarities to HD 24712

refs.:1=[18], 2=[29], 3=[36], 4=this paper, 5=[45]

We have already mentioned several times in this paper the similarity of HD 134214 to another roAp star HD 24712 (HR 1217), and Table 4 compares a few properties of these two stars. The loci in the HR Diagram of the best fitting models for the two stars are nearly identical. The observed frequency ranges are above the acoustic cut-off frequencies of the models in both cases, and they are predicted to be damped. Both stars have similar large frequency separations and near zero small separations, , which is difficult to reproduce with models. Our best model (shown in Fig. ?) gives Hz, which for HD 134214 is much larger than the observed value of Hz. The same difficulty is reported for HD 24712 by [45]. Our current investigation of HD 134214 makes it clear that the problems in modelling of HD 24712 are not unique to this star, and may be the same for other roAp stars with similar parameters.

For completeness, we note some dissimilarities between the two stars. HD 24712 exhibits clear rotational amplitude modulation and associated frequency splittings, while HD 134214 shows none. The amplitude of the principal frequency of HD 134214 (identified by us as an mode) is very much larger than the other ones and has been nearly constant for a long time [31]. On the other hand, HD 24712 has three main frequencies with comparable amplitudes; the relative amplitudes vary on a time scale of a few years [29].

## 8Conclusions

We have fitted pulsation models, including magnetic perturbation effects and assuming dipole magnetic geometries, to the frequency spectrum of the rich multi-periodic star HD 134214, which pulsates with the shortest periods known among the roAp stars. We have found that a better fitting is obtained when we fit 8 frequencies common to the MOST photometry [18] and radial velocity data obtained by [31], rather than fitting the 10 frequencies obtained by MOST. This might indicate that the two new frequencies obtained by [18] have high surface degrees of .

Among the best fitting models found in our grid for various input physics, we found that a model with a mass of , effective temperature , luminosity , and polar magnetic field strength  kG is the best match to the data. The mean deviation of the fit is Hz, which is slightly larger than the expected numerical errors for the model frequencies. Non-dipole components in the magnetic field of HD 134214 may be additional error sources.

The effective temperature and luminosity of the best model are consistent with those derived from multicolour photometry and the Hipparcos parallax. We note that the parameters are nearly the same as those of the best model for HD 24712 obtained by [45].

The principal frequency, with a photometric amplitude ten times larger than that of the other frequencies, is identified as a deformed dipole mode. The five frequencies with the next largest amplitudes are identified as modes. In our model, the radial velocity variations of these modes have a quasi-node around , where oscillation phase shifts by radians. The radial velocity amplitude increases rapidly above the node. These properties qualitatively agree with the data from radial velocity measurements of H bisector and lines of rare earth elements.

Although the model frequencies and amplitude distribution in the atmosphere are consistent with observations of HD 134214, the excitation mechanism is still unknown. The kappa-mechanism in the hydrogen ionization zone, e.g., appears to be too weak to excite oscillations in the observed frequency range, above the acoustic cut-off frequency.

Finally, we note that Gruberbauer (2011 in preparation) has recently developed a new probabilistic method for finding a best model in reproducing observed frequencies. One of the advantages of the method is its ability of taking into account the possibility of the presence of systematic errors, which we disregarded completely in this paper. Applying such a probabilistic method to various roAp stars including HD 134214 itself might reveal systematic errors in the model frequencies as a function of the frequency and/or magnetic field strength.

## Acknowledgments

We are grateful to Don Kurtz, the referee of this paper, for his helpful comments and suggestions. MG has received financial support from an NSERC Vanier scholarship. WWW was supported by the Austrian Research Fond (P22691-N16).

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