4 Non-radial pulsation model

4.1 The Geneva photometric data

The method of photometric amplitudes as described by Watson (1988) allows us to derive the degree $\ell$ of the pulsation mode from multicolour photometry by comparing the observed amplitudes at some wavelengths (i.e. those of the central wavelengths of the passbands of the used photometric system) with the theoretical amplitudes calculated for several values of $\ell$ and a free parameter S taking into account non-adiabatics effects ( $S\in [0,1]$, 0: fully non-adiabatic, 1: adiabatic). We note that the parameter S is the same as the parameter R of Stamford & Watson (1981). As the theoretical amplitudes are proportional to an unknown wavelength-independent function, ratios of the amplitudes are considered in order to eliminate this function. In Table 3 we give the amplitudes obtained with a sine fit for the separate Geneva filters, together with the amplitude ratios with respect to the U-filter. We refer to Heynderickx et al. (1994) for a full description of the method we used in this work and to Briquet et al. (2001) for a brief description. In general the mode identification by this method is successful for $\beta$ Cep stars (see e.g. Heynderickx et al. 1994) and SPBs (De Cat 2001). For HD131120 we tested $\ell$ from 0 to 7 and we choose the degree $\ell$ using a discriminant $\eta_{\ell}(S)$. This discriminant is the square root of the sum of squares of the differences between the observed and theoretical amplitude ratios divided by 7. For each $\ell$ we determine the value of S for which the discriminant $\eta_{\ell}(S)$ is minimal. Then we choose the mode for which the discriminant attains the lowest value. The minima of the discriminant $\eta _{\ell }$ are given in the left columns of Table 4. The best solutions are $\ell = 1, 2, 4$ and 6. They all have very similar values for the discriminant and so are equivalent in quality. Moreover the derived amplitude ratios have a large uncertainty (see Table 3). This observational uncertainty is larger than the difference between the competing pulsational models. We then conclude that the photometric data do not allow us to determine the degree $\ell$.

 

 

Table 3: Amplitudes of the least-squares sine fits to the Geneva data in the different filters, computed together with their standard errors using the SAS-software package. Ratios of the amplitudes with respect to the U-filter are also listed.

Filter	Amplitude	Ratio
U	0.0183 $\pm$ 0.0013	1
B1	0.0145 $\pm$ 0.0010	0.792 $\pm$ 0.077
B	0.0125 $\pm$ 0.0009	0.685 $\pm$ 0.069
B2	0.0123 $\pm$ 0.0010	0.672 $\pm$ 0.072
V1	0.0103 $\pm$ 0.0008	0.566 $\pm$ 0.060
V	0.0109 $\pm$ 0.0008	0.599 $\pm$ 0.058
G	0.0111 $\pm$ 0.0008	0.608 $\pm$ 0.061

4.2 The spectroscopic data

4.2.1 Mode identification by the moment method

Figure 2: Phase diagram of the first three moments of the SiII 4128 Å line. We show the observed values (dots), the fit using 0.6374 c/d and its first harmonic (solid line), and the best spot model (dashed line).

Figure 3: Phase diagram of the equivalent width of the SiII 4128 Å line, which is expressed in Å. The comparison with the best spot model is represented in dashed line.

The moment method is the most objective criterion to identify modes of non-radial pulsators with a slow rotation. The identification of modes is performed by comparing the amplitudes of the first three observed moments of a line profile $\langle v^1\rangle$, $\langle v^2\rangle$ and $\langle v^3\rangle$ (see Aerts et al. 1992) with theoretically calculated ones by means of a discriminant. For a full description of the method we refer to Aerts (1996).

We calculated the first three observed moments of a line profile $\langle v^1\rangle$, $\langle v^2\rangle$ and $\langle v^3\rangle$ (see Aerts et al. 1992). The first moment is equivalent to the radial velocity, for which we only found the frequency f=0.6374 c/d. The linear pulsation theory predicts that $\langle v^2\rangle$ varies with both f and 2f while $\langle v^3\rangle$ varies with f, 2f and 3f. In order to compare the frequencies found in the observed moments to the theoretical predictions we performed a frequency analysis on the observed moments. In the second moment, it was not possible to determine a frequency and in the third moment we found only the frequency f=0.6374 c/d. A phase diagram for the frequency f=0.6374 c/d for $\langle v^1\rangle$, $\langle v^2\rangle$ and $\langle v^3\rangle$ is shown in Fig.2. It is clear that the second moment does not vary with f nor with 2f. Such a situation does not correspond to linear pulsation theory as described above. We also computed the moment of order zero, which is the equivalent width of the line. The same frequency of 0.6374 c/d is present in it and a phase diagram is shown in Fig.3. We point out that the relative EW variation is about 10%. Such a large value is not encountered for confirmed SPBs (see De Cat 2001).

The discriminant is a function of the differences between the observed and theoretically calculated amplitudes of the first three moments. For each set of wavenumbers ($\ell$,m) we determine the values of $v_{\rm p}$, i, $v_{\Omega }$ and $\sigma$ for which the discriminant $\Gamma_{\ell}^m(v_{\rm p},i,v_{\Omega},\sigma)$ is minimal. Then we chose the mode $(\ell ,m)$ for which the discriminant attains the lowest value. The outcome of the mode identification with the discriminant is listed in the right columns of Table 4 for the best solutions in parameter space. We tested $\ell$ from 0 to 6 because the discriminant is only able to correctly identify modes with low to moderate degree. The other velocity parameters were varied in the interval [0.1;2]km s^-1 with a step 0.1 km s^-1 for the amplitude of the radial part of the pulsation velocity $v_{\rm p}$, $[1^{\circ};90^{\circ}]$ with a step $1^{\circ}$ for the inclination angle i, [40;70]km s^-1 with a step 1km s^-1 for the projected rotational velocity $v_{\Omega }$ and [1;20]km s^-1 with a step 1km s^-1 for the intrinsic line-profile width $\sigma$. The more probable mode is $(\ell,m)=(3,0)$. However, there are other candidates of almost equal probability, as can be seen in Table 4. We point out that the moment method is not able to distinguish the sign of the azimuthal number m.

  

Table 4: Left: the different minima of the discriminant $\eta _{\ell }$ of the method of photometric amplitudes together with the most likely value of the free parameter S taking into account non-adiabatic effects. Right: the different minima of the discriminant $\Gamma _{\ell }^m$ of the moment method for the SiII 4128 Å line for the best solutions. $v_{\rm p}$ is the amplitude of the radial part of the pulsation velocity, expressed in km s^-1; i is the inclination angle; $v_{\Omega }$ is the projected rotational velocity, expressed in km s^-1 and $\sigma$ is the intrinsic line-profile width, also expressed in km s^-1.

$$\begin{tabular}{ccc\vert cccccccc} \hline% $\ell$ & $\eta_{\e... ...dots &\vdots &\vdots &\vdots &\vdots & \\ \hline \end{tabular}$$

Unfortunately no confidence intervals for the minima of the discriminant and the corresponding velocity parameters $v_{\rm p},\ i, \ v_{\Omega}$ and $\sigma$ can be determined. Consequently, we generated theoretical line-profile variations for the modes for which the discriminant attains the lowest value (see Table 4) in order to choose the mode which gives the best fit compared to the observed line-profile variations. We define the "best fit model'' as the one which has the smallest standard deviation in the intensity over all profiles $\Sigma \equiv \frac{1}{N} \sum_{j=1}^{N} \sqrt{\frac{1}{n_j} \sum_{i=1}^{n_j} \left ( I_{i,{\rm obs}}^j - I_{i,{\rm th}}^j \right )^2}$, with N the number of spectra and n_j the number of wavelength pixels in the spectra j. We found that $\Sigma$ has about the same value for the most likely modes listed in Table 4, which is 0.004 and we are then again not able to determine the most likely mode from the discriminant.

4.2.2 Mode identification by line-profile fitting

Another way to identify modes from line profiles is to compare the observed line-profile variations with theoretically calculated ones. At present the best code which simulates line profiles for a rotating star undergoing NRP is Townsend's (1997) code, called BRUCE, which includes the Coriolis force. Unlike the previous ones, this code is valid for all the ratios $\Omega/\omega$ of the rotation and pulsation frequencies and not only for $\Omega/\omega$ smaller than unity. Up to now, in BRUCE, temperature variations are taken to be adiabatic and two extra parameters must be included to simulate non-adiabatic temperature effects.

We search for the parameters for which the calculated profiles best fit the observed profiles by considering a large grid of possible wavenumbers and parameters. In order to keep the computation time feasible we averaged out all the observed profiles in phase bins of 0.05 of the variability cycle and worked with these 20 averaged observed profiles. They are shown as dotted lines in Fig.4. The observed profiles are compared to the theoretical profiles and to their orthogonal symmetric profiles in order not to favour a sense of rotation. As a measure of the goodness of fit we use the standard deviation $\Sigma$ in the intensity averaged over all profiles. The most likely mode and parameters are those that minimize $\Sigma$.

First we consider only the velocity perturbation and we cover the parameter space by varying the free parameters in the following way: $\ell$ from 0 to 6, the projected rotational velocity $v_{\Omega }$ from 30 to 60 km s^-1 with a step 5 km s^-1, the angle of inclination i between the rotation axis and the line of sight i from 10$^{\circ}$ to 90$^{\circ}$ with a step 10$^{\circ}$, the amplitude of the radial part of the pulsation velocity $v_{\rm p}$ from 2 to 20 km s^-1 with a step 2 km s^-1 ($v_{\rm p}$ from 0.5 to 1.5 km s^-1 with a step 0.5 km s^-1 for modes with m=0), the intrinsic line-profile width $\sigma$ from 2.5 to 20 km s^-1 with a step 2.5 km s^-1, the initial phase of the mode $\phi$ from 0 to 0.95 period with a step 0.05 period. Then we consider adiabatic temperature variations and used theoretical intrinsic profiles kindly provided by Dr. T. Rivinius. These are constructed using the atmospheric codes ATLAS 9 and BHT (Baschek-Holweger-Traving, see Gummersbach et al. 1998) and fixing the microturbulence at 2 km s^-1. Finally we consider non-adiabatic temperature effects by introducing two extra parameters which are the non-adiabatic temperature perturbation scaling factor $\Delta T$ and the non-adiabatic temperature perturbation phase shift $\psi$. We take $\Delta T$ from 0.2 to 1 with a step 0.2 and $\psi$ from 0$^{\circ}$ to 360$^{\circ}$ with a step 45$^{\circ}$.

The parameters that give the smallest $\Sigma$ are:

$$\begin{eqnarray*}\left\{\begin{array}{llllll} \ell = 2,\\ m = 0,\\ v_{\Omega} ... ...24,\\ \Sigma = 0.0040~{\rm continuum~units}. \end{array}\right. \end{eqnarray*}$$

We note that the magnitude amplitude corresponding to this solution is about 0.02, which is compatible with the observed one. The fit for these parameters is shown in Fig.4. This solution leads to the same line-profile variations as the solution without the inclusion of temperature effects and is not able to reproduce the large observed equivalent width variation. The theoretical profiles were adjusted to have the same equivalent width as the observed profiles. We note that this solution is not very different than the others in Table 4 and leads to the same value of $\Sigma$. Thus we have shown that the pulsation hypothesis does not agree well with the data.

Figure 4: Observed line profiles of the SiII 4128 Å line (dots) averaged over phase bins of 0.05 and theoretical line profiles (full lines) for the NRP model with $(\ell ,m)$=(2,0).