Praesepe

   
2 The mode identification problem

The past years have shown considerable effort to establish reliable mode identifications for a number of $\delta$ Scuti stars, but in most cases we are left with some ambiguity. Purely photometric techniques have the advantage of greater efficiency, and ultimately the ability to observe fainter stars than is possible with spectroscopy. A secure mode identification seems only possible, when several techniques agree. Especially for rotating stars, there are too many free parameters that one method can give a unique answer.

The photometric modal discrimination relies on model atmosphere calculations (e.g. Garrido 2000), which suffer from our lack of understanding of the physical conditions in $\delta$ Scuti stars. The "regions of interest'' used in the photometric discrimination are not only partially overlapping, but also strongly dependent on model parameters.

Another method has been the calculation of thousands of models to try to reproduce the frequencies (e.g. Pamyatnykh et al. 1998 for the case of XX Pyx), but without any convincing results. In fact, no stellar model has yet provided a unique fit to any set of observed frequencies.

The detection of line-profile variations caused by non-radial pulsation modes has been made possible by detailed spectroscopic studies. High resolution spectroscopy combined with high time resolution provides the ultimate information and works for low as well as high degree modes. In this case, large telescopes with efficient spectrographs are needed. For a recent review consult Mantegazza (2000). A different approach is the moment method introduced by Balona and discussed by Aerts & Eyer (2000) where references are given. Both methods work well when there are only a small number of modes (i.e. for a star like $\rho$ Pup with one radial mode), but when trying to fit a number of closely spaced modes, these spectroscopic methods are less powerful and often leave a lot of ambiguity in the classification. None of them has ever been applied to $\delta$ Scuti stars as faint as BN Cnc.

Detailed studies of line profile variations have been carried out for $\beta$ Cephei stars (e.g. Telting & Schrijvers 1997 or Schrijvers et al. 1997). If the S/N of the spectra is sufficiently high, good indications of the mode type can be derived. This type of analysis is more difficult for $\delta$ Scuti stars, where the Doppler velocities are smaller.

A more efficient technique in terms of photon efficiency, which can work with a modest resolution spectrum, is the use of ratios between the amplitude of the variation of the equivalent width (EW) of the Balmer lines divided by a photometric amplitude (e.g. Dall 1998; Frandsen 2000). The EW can be measured with "FOSC'' (Faint Object Spectroscopic Camera) type instruments which are very efficient (>20%) and easy to use. One can replace the EW by line-indices which rely on the temperature sensitivity and spatial sensitivity of the equivalent widths of spectral lines. It is a promising new technique, which has been successfully applied to a number of pulsating stars of different classes; the $\delta$ Scuti star FG Vir (Viskum et al. 1997, 1998), the roAp star $\alpha$ Cir (Baldry et al. 1998, 1999) and the EC14026 star PG1605+072 (O'Toole et al. 2000). It was suggested as a way to observe oscillations in solar-type stars by Bedding & Kjeldsen (1996) and has been theoretically explored for early type stars by Balona (2000). The variation of the EW basically tracks the changes in temperature and gravity. In $\delta$ Scuti stars the temperature dominates, while in EC14026 stars the gravity changes contribute at a similar level.

The modest resolution of the "FOSC'' type instruments and the instrumental flexures caused by the mounting in the Cassegrain focus means that radial velocity measurements are difficult. This is the reason that radial velocities are not discussed in the present project, as the data from the other sites alone is not enough to derive a decent power spectrum of the radial velocity variations.

   
2.1 The line-indices

For calculating line-indices, the spectra do not need to be wavelength or continuum calibrated and thus consist of "counts per pixel'' instead of fluxes as function of wavelength:

$$F_\lambda \longrightarrow S_x.$$ (1)

The quantity x is a pixel number and the unit of S is counts. One can of course work with fully calibrated spectra without problems.

The general idea is to integrate the counts in the line, using a suitable weight function, $W_x^{\rm L}$, which we will call a filter-function or just filter. A broad filter $W_x^{\rm C}$ is used for the continuum.

First the centre of the line is defined by moving an integration weight filter across the line, and choosing the point of maximum total counts as the line centre. The line centre is thus the position that maximises the counts in the filter, see Fig. 1.

Figure 1: To find the line centre a filter is moved across the line and the position that maximises the sum is the line centre. The width of the filter can be chosen arbitrarily. The dashed line is the fit to the continuum.

With the line centre pinned the continuum C_x is calculated from two adjacent areas on each side of the line, which appears to have a well defined continuum between weak lines. This is done by rejection iterations, fitting a straight line to the continuum. Even if the continuum is not well defined e.g. because of a continuous blend of weak lines, it is still possible to obtain a continuum measure from this blend, only will this give a lower value for the continuum. The effect will be to diminish each value of the line index by the same amount, which will in turn tend to make the apparent oscillation amplitude insignificantly larger.

With the chosen filter function $W_x^{\rm L}$ for the line in question, the integration is carried out across the line, yielding the line-index;

$$\Lambda^{\rm L} = \sum \frac{C_x - S_x}{C_x} W_x^{\rm L}.$$ (2)

The filter $W_x^{\rm L}$ typically fades off to zero as the line wings approach the continuum. An example of the weights used for continuum and line-index is given in Fig. 2.

Figure 2: An example of calculation of a line-index of H$\alpha$. The continuum is found from the two broad filters on each side, while the line-index is calculated as the counts in the line weighted by the central filter. Continuum slope and slow curvature do not pose a problem.

Thus, a line-index in a given filter will be proportional to the equivalent width (EW) of the line. It differs from a EW measurement in that it does not necessarily count the total flux in the line. In this sense it resembles the Strömgren H$\beta$ index, just performed with software filters. The advantage is of course that any type and number of filters can be applied to any line in the data set.

The justification of the term "line index'' as an analogue to a colour index can be seen as follows: we define an index $m_{\rm L} - m_{\rm C}$ as the difference between a narrow and a broad flux band centred on the line, analogous to the $\beta$-index:

$$m_{\rm L} - m_{\rm C} = -1.086 \ln \frac{\sum S_x W_x^{\rm L}... ... L}}{\sum S_x W_x^{\rm C} /\sum W_X^{\rm C}} + {\rm constant}.$$ (3)

Now, assuming $C_x = C = \sum S_x W_x^{\rm C} / \sum W_x^{\rm C}$ (2) can be rewritten

$$\Lambda^{\rm L} = \sum W_x^{\rm L} - \frac{\sum S_x W_x^{\rm L}}{\sum S_x W_x^{\rm C}} \sum W_x^{\rm C},$$ (4)

which can be inserted in (3), where we get

$$m_{\rm L} - m_{\rm C} = -1.086 \ln \left( 1 - \frac{\Lambda^{\rm L}}{\sum W_x^{\rm L}} \right)\cdot$$ (5)

If we look at variations $\delta(m_{\rm L} - m_{\rm C})$ and $\delta\Lambda$ around the mean values we then get

$$\delta(m_{\rm L} - m_{\rm C}) = 1.086 \frac{\delta \Lambda^{\... ...da^{\rm L}/ \sum W_x^{\rm L})} \propto \delta \Lambda^{\rm L},$$ (6)

and we thus see that $\delta(m_{\rm L} - m_{\rm C})$ and $\delta\Lambda$ are directly proportional, showing that the line index can be thought of as a colour index calculated using software filters.

One may of course define any number of line-indices for any spectral line by varying the filter-function used, hereby gaining information from different parts of the line. The choice of filter shape is not critical. We have used a so-called super-Gaussian filter with exponent 8, i.e.:

$$W_x = \exp \left( - \left( \frac{x - x_0}{b} \right)^8 \right)\cdot$$ (7)

The software for calculating line-indices and for performing the time series analysis has been collected in a software package named Ix, described by Dall (2000b).

2.2 Mode identification from line-index and photometry

The mode identification proposed here relies on the investigation by Bedding et al. (1996) of the sensitivity of different absorption lines to pulsation modes with different $\ell$. For non-rotating stars they showed that the Balmer lines have sensitivity similar to radial velocity measurements because of the strong limb darkening in these lines. Photometry on the other hand has very weak centre-to-limb variation.

Thus photometry and the line-indices of Balmer lines show different response to spatial variations across the stellar disk (i.e. to the $\ell$ value), because the Balmer lines sample the centre of the disk stronger than the limb while photometry smears out spatial variations across the disk. It follows that the ratio R between the amplitudes of the Balmer line-index and the photometry should be an increasing function of $\ell$, so that modes with different $\ell$ should be grouped in an amplitude ratio diagram. This applies only to a slowly rotating star.

Some concern about the ability of the method to discriminate between modes of different spherical degree has been raised by Balona (2000), who used stellar models to investigate the sensitivity of R on $\ell$. He finds in addition a strong dependence on $\sin i$ and m. His results will be addressed later. Indeed, for rotating stars the picture becomes much more complicated (Kjeldsen et al. 1998): as rotation increases, gravitational darkening becomes more important. This effect causes a stronger sensitivity to the azimuthal number m, because the orientational degeneracy is lifted. So we get in turn a dependence on the inclination angle. As shown by Hansen (1999) and Frandsen (2000), the m dependence is as least as strong as the $\ell$ dependence, with the values of amplitude ratios between photometry and line-index varying strongly with $\sin i$. Thus, to disentangle the $\ell$ and m dependencies, knowledge of the inclination angle is essential, which in turn means that the rotation period and $v \sin i$ must be known independently.

Viskum et al. (1998) conducted a detailed analysis of FG Vir that resulted in a suggestion of mode identification for the 8 strongest modes. This suggestion was later found to agree with an independent mode identification by Breger et al. (1999). In FG Vir a long period variation in the H$\alpha$ and Fe I line-indices, found by Viskum et al. was attributed to stellar activity and thus gave directly the rotation period of the star and, with the observed value of $v \sin i$, also the inclination angle.

Praesepe