Application

One might expect that using more than two wavebands should allow a better determination of $\ell$. There is however, no obvious graphical technique that can be used. What is needed is a rigourous method of deducing the spherical harmonic degree which does not require these plots and which gives the probability of correct identification. This was done by Fontaine et al. (1996) for pulsating white dwarfs, where all wavebands are in phase ( $\psi = 180^\circ$) and only the relative amplitudes are sensitive to $\ell$. Their method involves minimising a $\chi^2$ goodness of fit criterion between observed amplitudes and amplitudes predicted for a given value of $\ell$. The $\chi^2$ criterion is a summation over as many wavebands as desired. In a recent paper (Balona & Evers 1999) we generalise the procedure proposed by Fontaine et al. (1996) to the case where both amplitude and phase is used in determining the mode. Furthermore, instead of allowing the non-adiabatic parameters, R and $\psi$, to take on their full ranges, linear non-adiabatic models were used to compute these parameters. In terms of the usual two-colour diagnostic diagram, this is equivalent to shrinking the area occupied by a given value of $\ell$ down to a point, thereby improving mode discrimination. In the new method, two or more wavebands can be used. The most likely value of $\ell$ is the one which gives the smallest value of $\chi^2$. Standard statistical tables allow the probability of mode identification to be estimated. Let the observed magnitude variation with time be

$$\begin{eqnarray*} M_j & = & A_{j,{\rm obs}}\cos(\omega t + \phi_{j,{\rm obs}})\\ & = & a_{1j} \cos \omega t + a_{2j} \sin \omega t \end{eqnarray*}$$

for filter j. We introduce an unknown scale factor, q, and phase shift, $\gamma$, to bring the calculated light curve, m_j, into best agreement with the observed light:

$$\begin{eqnarray*} q m_j & = & q B_{j,{\rm cal}}\cos(\omega t + \phi_{j,{\rm cal... ... b_1 - q_2 b_2)\cos \omega t + (q_1 b_2 + q_2 b_1)\sin \omega t \end{eqnarray*}$$

Here $b_1 = B_{j,{\rm cal}} \cos \phi_{j,{\rm cal}}$, $b_2 = -B_{j,{\rm cal}} \sin \phi_{j,{\rm cal}}$, $q_1 = q \cos \gamma$, $q_2 = -q \sin \gamma$. We need to minimise

$$\sigma^2 \chi^2 = \sum_j (M_j - q m_j)^2$$

Integrating over a pulsation cycle and minimising the resulting sum with respect to q₁ and q₂, we obtain the result

$$\begin{eqnarray*} {2\sigma^2\chi^2}\over{N} & = & <a_1^2> + <a_2^2> - {{(<a_1... ...\\ & - & {{(<a_1b_2> - <a_2b_1>)^2}\over{<b_1^2> + <b_2^2>}}. \end{eqnarray*}$$

where $\sigma^2$ is the variance for each filter (assumed the same for all filters) and N is the number of filters. Angle brackets denote averages over the filters. To determine the spherical harmonic degree, $\ell$, we begin by estimating the approximate effective temperature and gravity of the star. From a number of linear, nonadiabatic models which could represent the star, we use the calculated values of R and $\psi$ for modes which are unstable and which have pulsation frequencies similar to the observed frequency. This allows b₁ and b₂ to be calculated for each filter, so that $\chi^2$ can be found. The search is continued over a range of models spanning the expected value of $\log T_{\rm eff}$ and $\log g$. In this way, we obtain values of $\ell, n$ and $\chi^2$ for each observed frequency. The most probable identification is the one which minimises $\chi^2$. From the value of $\chi^2$ and the number of degrees of freedom (2(N-2), where N is the number of filters) we determine the probability of this identification.