6 Discussion

It is interesting to investigate if the apparent frequencies of the two non-radial pulsations modes in the primary of $\psi ^2$ Ori can shed some light on the question whether these modes are powered by tidal forces or from within the star.

6.1 Pulsation modes powered by tidal forces?

In a binary star, the perturbing force due to its companion star is periodic. Depending on the proximity of the orbit and on its eccentricity this periodic force is more or less sinusoidal, and can be expanded in a Fourier series in terms of the orbital frequency $\Omega_{\rm orb}$ (see e.g. Ruymaekers 1992; Smeyers et al. 1998). The perturbing frequencies as experienced by a mass element in the frame corotating with the star can be described as

$$\sigma_{\rm cor}=j\Omega_{\rm orb}+m\Omega_{\rm rot}$$ (1)

with j a positive integer indicating harmonic j-1 of the orbital frequency, and with $\Omega_{\rm rot}$ the rotation frequency of the star. This means that for a non-sinusoidal perturbing force the star experiences a spectrum of frequencies of which some may coincide with an eigenfrequency such that resonances occur.

The apparent pulsation frequency of a resonance mode as seen by an observer depends on the azimuthal order m of the mode and the rotation frequency of the star

$$\sigma_{\rm obs}=\sigma_{\rm cor}-m\Omega_{\rm rot} =j\Omega_{\rm orb}$$ (2)

where the rotation frequency is positive by definition and where negative m values represent prograde pulsation modes. Hence, if the pulsations are tidally induced, the observed pulsation frequencies should be multiples of the orbital period (Eq. (2)).

This binary has a very accurately determined orbital period, leading to $\Omega_{\rm orb}=0.395889$c/d. Taking the HWHM of the main peak of the window function, 0.083c/d, as an estimate of the error in our frequency determinations, we find that the observed 10.73c/d pulsation frequency is consistent with an integer value of j in Eq. (2): $j = 27.1 \pm 0.2$. However, the strongest detected frequency, 10.48c/d, is inconsistent with an integer value of j: $j = 26.5 \pm 0.2$.

We conclude that given the observed frequencies it is unlikely that both detected pulsations in the primary of $\psi ^2$ Ori are due to tidal forcing. However, it is clear that we need much more precise determinations of the pulsation frequencies in order to provide a conclusive answer. More high S/N spectra taken on a long time base are needed in order to achieve this.

6.2 Pulsation modes excited internally?

Here we investigate if the observed pulsation frequencies are consistent with those expected for internally excited $\beta$ Cephei oscillations. Dziembowski & Pamyatnykh (1993) present the pulsation frequencies of modes with low and intermediate degree $\ell$ in an $M = 12~M_\odot$ star. They present dimensionless frequencies in the corotating frame of the star, and hence for comparison we need to transform the observed frequencies using Eq. (2), assuming m=-6and $\Omega_{\rm rot}=\Omega_{\rm orb}$. To transform to dimensionless frequencies we assume the radius and mass of the primary in $\psi ^2$ Ori to be $R=5.4~R_\odot$ and $M=13.9~M_\odot$ (for inclination $i = 60\hbox{$^\circ$ }$).

The result of the above estimation is that the observed pulsation frequencies correspond to the lower limit of the unstable p-mode regime. If the observed modes are not sectoral, and hence m > -6, the transformation of Eq. (2) shifts the observed frequencies further into the p-mode regime. A similar conclusion was drawn for the observed $\ell \sim 9$ pulsation mode in the early-B type star $\omega^1$ Sco (Telting & Schrijvers 1998).