2 Modes correlation

2.1 Scientific context

Several authors have studied the temporal behaviour and the statistical distribution of modes amplitude or power, using time-frequency analysis. The distribution has been found to follow mainly an exponential behaviour [Baudin et al. 1996,Foglizzo et al. 1998], in agreement with a Boltzman distribution due to a stochastic excitation [Kumar 1997]. Although important for further data analysis, this does not constrain the description of the excitation. Nevertheless, some discrepancies remain in the high amplitude peaks [Gavryusev & Gavryuseva 1997,Chaplin et al. 1997] which could have a non-stochastic behaviour, but the statistics are rather poor and deserve some strengthening.

Several tentatives have been made already to measure the correlation between modes [Foglizzo et al. 1998], by looking at the temporal behaviour of the mode power. The method consists in analysing the integral of the power in a given bandwidth, so it is intrinsically limited to the study of correlation between modes well separated in frequency (namely the l=0 modes). They have found no evidence for correlation in the GOLF signal.

As pointed out by the authors, no significative correlation between the power of the modes should be found even if there are excited by the same noise source (because they have different frequencies), unless the excitation itself were to have a temporal correlation longer that the life-time of the modes.

We checked by a simulation that this correlation is almost zero when the modes do not overlap and when the temporal correlation of the excitation is shorter than the life-time of the modes. Therefore, the result from Foglizzo et al. is not contradictory with an excitation of the modes by a common signal, but suggests that in this case, excitation by a common source requires a temporal correlation shorter that the life-time of the mode.

2.2 Interest of the present method

We have developed a completely different method, based on a parametric model, which allows us to extract the profile of the modes as with the lorentzian fit, but can also recover the characteristics of the excitation signal. Unlike the method described above, this allows us to study correlation in the excitation of modes which are very close in frequency, such as the modes split by rotation.

This can be very useful in order to improve our description of the coupling between convection and oscillations. Indeed, the free parameters in the model will have geometrical effects on non-radial modes, that could be addressed if the correlation in the signal exciting different modes with the same or different geometries could be measured.

It is also of fundamental importance for the measurement of the internal rotation: the frequencies separation between modes of the same degree and order (the rotational splitting) is measured with the assumption that the modes are uncorrelated. If this hypothesis is not correct, the value of the splitting could be affected by a bias, which could modify drastically the rotation rate (see  Sect. 4.1).