Surface trapping and leakage of low-frequency g-modes in rotating early-type stars – I. Qualitative analysis

A qualitative study of the surface trapping of low-frequency non-radial g-modes in rotating early-type stars is undertaken within the Cowling, adiabatic and traditional approximations. A dispersion relation describing the local character of waves in a rotating star is derived; this dispersion relation is then used to construct propagation diagrams for a 7M⊙ stellar model, which show the location and extent of wave trapping zones inside the star. It is demonstrated that, at frequencies below a cut-off, waves cannot be fully trapped within the star, and will leak through the surface. Expressions for the cut-off frequency are derived in both the non-rotating and rotating cases; it is found from these expressions that the cut-off frequency increases with the rotation rate for all but prograde sectoral modes. While waves below the cut-off cannot be reflected at the stellar surface, the presence of a sub-surface convective region in the stellar model, due to He ii ionization, means that they can become partially trapped within the star. The energy leakage associated with such waves, which are assigned the moniker virtual modes due to their discrete eigenfrequencies, means that stability analyses which disregard their existence (by assuming perfect reflection at the stellar surface) may be in error. The results are of possible relevance to the 53 Per and SPB classes of variable star, which exhibit pulsation frequencies of the same order of magnitude as the cutoff frequencies found for the stellar model. It is suggested that observations either of an upper limit on variability periods (corresponding to the cut-off), or of line-profile variations due to virtual modes, may permit asteroseismological studies of the outer layers of these systems.