For a moderately to fast rotating star this straightforward interpretation is altered, as the modes group according to both $\ell$ and m for these stars, and for fast rotators the dominant role may be played by m in connection with the inclination angle. The reason is that gravity darkening becomes as important as limb darkening.

In the case of BN Cnc we can expect the m dependence to be at least as significant as the $\ell$ dependence for the amplitude ratio of a given mode. As a consequence the $R_{\alpha}$ observable only provides an indicator of the degree of surface structure of the observed modes. It can still be used to divide the observed modes in certain classes, containing only a subset of the total modes.

A sensible approach to Fig. 10 is to identify the two modes f₂ and f₅ as modes with small spatial structure. If there are radial modes among the observed set, these two modes are thus prime candidates. This can be concluded without reference to any other evidence. It actually turns out to be consistent with the results of Hernández et al. (1998) that there can be at most two radial modes in this star. They found a small set of solutions that were within the uncertainty of the parameters for the cluster and the star, taking rotation into account. Earlier, ignoring rotation, the f₅ and f₄ modes were assigned n=6 and $5, \ell =0$ by Perez Hernandez et al. (1995). The analysis, including effects of rotation, is not able to discriminate against any of the choices (f₂,f₅) and (f₄,f₅) as sets of radial modes. Even (f₂,f₁) is an acceptable pair giving a frequency ratio within the possible parameter space.

The f₆ mode would need to have high spatial structure, meaning that it cannot be radial and most likely can be assigned to $\ell =2, m=0,\pm2$ , since according to Frandsen (2000) only such a mode can achieve such high values of R. Higher $\ell$ values are not likely, as the mode is observed photometrically.

As a check, using only the NOT data, $R_\beta$ was found and confirmed the low ratio of the f₂ and f₅ modes, and the high ratio of the f₆ mode, see Fig. 11.

$\begin{figure} \par\includegraphics[width=6.1cm,clip]{H2906F11.ps} \end{figure}$

Figure 11: Amplitude ratios of H $\alpha$ and H $\beta$ relative to photometry. The dashed line show $R\alpha = R\beta$ . Generally there is good agreement between the two ratios, despite the higher noise in H $\beta$ .

The remaining modes are difficult to classify. If one accepts that f₂ and f₅ are radial, then they have to be non-radial modes. If we assume that f₂ corresponds to n=5 and f₅ to n=6, it is possible that f₄, f₁, and f₃ are $\ell =1$ belonging to consecutive orders, e.g. n = 5,6,7. Additional observations are needed to sort this out.

Table 4: The identifications from this work and comparison with previous suggestions. Amplitudes are given in promille for $\Lambda ^{{\rm H}\alpha }$ and in mmag for the photometry of Paper I.
mode Amplitude from Mode ID

$\Lambda ^{{\rm H}\alpha }$ Paper I this work previous^a

f₁ 2.34 3.01 $\ell =1$

f₂ 1.28 2.52 $n=5, \ell =0$

f₃ 1.69 2.42 $\ell =1$

f₄ 1.84 2.32 $\ell =1$ $n=5, \ell =0$

f₅ 1.00 2.13 $n=6, \ell =0$ $n=6, \ell =0$

f₆ 1.29 0.63 $\ell =2-3$

**Table 4:** The identifications from this work and comparison with previous suggestions. Amplitudes are given in promille for $\Lambda ^{{\rm H}\alpha }$ and in mmag for the photometry of Paper I.
mode	Amplitude from	Mode ID
	$\Lambda ^{{\rm H}\alpha }$	Paper I	this work	previous^a
f₁	2.34	3.01	$\ell =1$
f₂	1.28	2.52	$n=5, \ell =0$
f₃	1.69	2.42	$\ell =1$
f₄	1.84	2.32	$\ell =1$	$n=5, \ell =0$
f₅	1.00	2.13	$n=6, \ell =0$	$n=6, \ell =0$
f₆	1.29	0.63	$\ell =2-3$

^a Perez Hernandez et al. 1995, but see text.

In the previous studies of BN Cnc (and other variables in Praesepe), a search for the correct frequency ratios based on model predictions has been used to select possible pairs of radial modes. Arentoft et al. (1998) calculated the frequencies for two radial orders for an evolutionary sequence of models and approximated these with analytical relations given in their Eq. (2).

For a given choice of radial modes, the luminosity L will be a function of the effective temperature $T_{\rm eff}$ . We present in Fig. 12 the $L-T_{\rm eff}$ relations for the choice of the radial modes f(n=5) and f(n=6) found in the present paper, as well as from Hernández et al. (1998)

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{H2906F12.ps} \end{figure}$

Figure 12: Luminosity-temperature relations for the Arentoft et al. models selecting radial modes f(n=5) and f(n=6). The dot-dashed lines show the relations for the (wrong) identification presented in that paper, while the upper lines are based on the present identification (full) and the Perez Hernández et al. identification (dashed).

Hernández et al. came very close to our proposed solution. They have had a difficult case due to the presence of close frequencies of modes with similar amplitudes. The effect is minor and using non-rotating models the luminosity found from the mode identification in this paper and Hernández et al. is the same: $L / L_{\odot} = 22$ for $T_{\rm eff} = 7750$ K.

It is clear that using frequency ratios as the criteria only gives a hint to what might be radial modes. The same can be said about the evidence coming from observations of $R_{\alpha}$ and $R_\beta$ . However, if the two criteria agree for a mode, the probability of the identification being correct is considerably increased.

One of the reasons that frequency ratios are not such reliable indicators is the large value of the rotational velocity $v \sin i$ in BN Cnc and other Praesepe variables. They all have values above 100 kms^-1 except KW 284. The modeling is difficult and rotation also affects radial modes due to second or higher order terms. For BN Cnc, calculations by Kjeldsen et al. (1998) give changes for n = 5 and n = 6 of the order 1 $\mu$ Hz. At the same time the frequency ratio f(n=6)/f(n=5) changes from 1.1434 for a non-rotating star to 1.1386 for the rotating star. The change in effective temperature with the inclination angle ( $\sin i$ ) amounts to 100 K and the change of the apparent luminosity can be up to 10%. As $\sin i$ can not be measured, there are plenty of models that fit in the available parameter space.

Using the amplitude ratios R presented in this paper eliminates a lot of this freedom and points to the radial modes as those with small R values.

Is there any theoretical support for the conclusions about mode types in BN Cnc? Balona (2000) has performed extensive calculations of the EW variations for a small set of stellar models with parameters corresponding to $\delta$ Scuti stars. Simpler calculations for the ratio R have been carried out by Hansen (1999) summarized by Frandsen (2000, Eq. (9)). In the first calculation a stellar disk integrated line profile is computed at each point in time and the EW found by integrating the line profile. Having the detailed shape of the line profile makes it possible to compute other observables like line profile moments. Complete account of the Doppler shifts is secured. The second calculation derives the change in flux and the change of the EW for each surface element and then calculates the average for the whole stellar disk. Doppler shifts do not enter in this case. One can describe this as filter calculations.

The way the results obtained by Balona (2000, Figs. 1-3) are displayed makes the comparison with observations and other calculations difficult. All unstable modes are included in the diagrams regardless of the amplitude. Thus modes, which will have very small photometric amplitude occur although they in practice are unobservable photometrically. The calculations by Hansen (1999) show the expected dependence on the azimuthal number m. The modes with low values $R_{\alpha}$ are found to be $\ell = 0$ or m = 0 modes. Depending on inclination higher degree modes can sometimes show odd amplitudes in photometry or EW due to strong cancellation during the surface integral and therefore one can sometimes see large variations of the $R_{\alpha}$ parameter.

The difference between the two approaches is in principle only the order in which the integrals are performed. There are a few additional assumptions in the filter calculations, which we do not expect to have any major influence on the main results.

The results from the detailed line profile studies by Balona (2000) and the filter calculation by Hansen (1999) and Frandsen (2000) should be compared in detail; the filter calculations are relatively easy and fast to perform and in most cases sufficient for the interpretation of the EW results, but a verification is needed.

The R values for radial modes should, according to the filter calculations we have performed, be quite similar. This is consistent with the results from Fig. 10 and with the values observed for FG Vir (Viskum et al. 1998), where $R \sim 0.5$ and for $\rho$ Pup (Dall et al., unpublished), where $R \sim 0.43$ . The observations do not show the spread in R seen in Figs. 1-3 in Balona (2000), but this might be due to the inclusion of the more extended range of modes in Balona's figure.

More studies of the usefulness of the observations of amplitude ratios are needed. These ratios do seem to provide an additional source of information, not necessarily superior to other diagnostics, but providing independent confirmation of the type of the modes. For a slow rotator like FG Vir the $R_{\alpha}$ parameter has been shown to be an indicator of the $\ell$ value. For BN Cnc it can be used to identify candidates for radial modes, but the interpretation is more complicated and other observables, like line profile moments, are needed to complete an identification.

Table 5: Phase differences between the oscillations of $\Lambda _{\rm {H}\alpha }$ and photometry.
mode $\phi_{\Lambda (\rm {H}\alpha)} - \phi_{\rm {phot}}$

[degrees]

f₁ $55 \pm 5$

f₂ $136 \pm 8$

f₃ $-8 \pm 9$

f₄ $-10 \pm 6$

f₅ $31 \pm 12$

f₆ $100 \pm 12$

**Table 5:** Phase differences between the oscillations of $\Lambda _{\rm {H}\alpha }$ and photometry.
mode	$\phi_{\Lambda (\rm {H}\alpha)} - \phi_{\rm {phot}}$
	[degrees]
f₁	$55 \pm 5$
f₂	$136 \pm 8$
f₃	$-8 \pm 9$
f₄	$-10 \pm 6$
f₅	$31 \pm 12$
f₆	$100 \pm 12$

We have also investigated the phase differences between the oscillations in $\Lambda ^{{\rm H}\alpha }$ and photometry. They are listed in Table 5, and shown in Fig. 13. We show $R_{\alpha}$ as a function of the phase difference. This resembles the well known photometric mode discrimination diagram (e.g. Garrido 2000), where an amplitude ratio between a colour index and a photometric band is plotted as a function of their phase differences. The grouping of modes in such a diagram can then, if one has a good model, help to identify the modes.

Recently Paparo & Sterken (2000) discussed the mode-typing potential of a wide selection of amplitude ratios and phase differences in the Strömgren photometric system. Based on an empirical approach they identified some promising discriminators, A(b)/A(y) and $\phi_{b-y} - \phi_b$ , without trying to explain the physical reasons for the observed groupings of modes. With our Fig. 13 we could take a similar approach and try to identify a pattern based on the already suggested radial modes f₂ and f₅. Note that the range in phase difference here is much larger than any observed between photometric bands.

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{H2906F13.ps} \end{figure}$

Figure 13: Amplitude ratios as function of phase differences. The phases are given in degrees.

Unfortunately, we do not have calculations, that permit us to make any interpretation of the phase differences. Let it be sufficient to note that it provides additional information about the modes, yet waiting to be understood.

7.2 Spectroscopic requirements

The diversity of the instrumentation has proved problematic. While oscillation signatures were found in all data sets, the noise was in many cases very high. There can be several causes for this: telescope tracking/guiding inaccuracy which likely worsens in bad weather conditions, instrumental instabilities or data reduction noise.

However, a very basic limitation is of course simple photon statistics: it was required to keep exposure plus readout times below $\sim$ 500 s to get sufficient sampling. For some combinations of telescope, spectrograph and detector, this requirement means that the resulting S/N is quite low, and the weighting of these data is correspondingly low. In particular the data from OAO and McD suffered from this problem, which is disappointing as these sites are situated at key longitudes. Recently the instrumentation at OAO was updated causing an overall increase in efficiency.

Instrumental instabilities

At the NOT such movements can amount to several pixels during a night, while at VBO the maximum shift from the beginning of the observations to the end is only around 0.5 pixels making drift problems insignificant. Since we do not see indications of instabilities in the NOT data we conclude that instrumental drifts are unlikely to be significant sources of errors.

7 Discussion

7.1 What is new?

7.2 Spectroscopic requirements

Instrumental instabilities