The nature of strange modes in classical variable stars

Strange modes have been found in the radial spectrum of many luminous stars, such as PAGB stars. The strange modes are characterized by having small amplitudes in the interior of the envelope, and egregious periods and growth-rates. It has been common belief that the strange modes are a result of strong nonadiabaticity. Recently, and perhaps surprisingly, such modes have also been found in classical Cepheid models, even though these are weakly nonadiabatic stars. Here we show that in fact there is nothing strange about these modes and that they must exist even in the adiabatic limit. They are essentially acoustic surface modes.

By means of a simple change of variables and {\sl without approximation}, the adiabatic linear pulsation equation for the radial displacement is reduced to a Schr\"odinger like equation in which the radial coordinate is the local sound traversal time. In this formulation, the narrow hydrogen partial ionization region is seen to act as a potential barrier, separating the star into two regions. Modes can be trapped either in the inner or in the surface region. Coupling through the barrier gives rise to resonances between the inner and surface regions. The strange modes are those in which the ratio of inner to surface amplitude is at a minimum. The potential problem formulation shows that strange modes exist in the adiabatic limit. As a function of the stellar parameters the relative location of the barrier changes, and this gives rise to the phenomenon of {\sl avoided level crossings} along a sequence of models.

The appearance of strange modes and the associated level crossings can be exhibited with an {\sl analytically} solvable toy model when the potential barrier is approximated by a delta function. In the full nonadiabatic models the same trapping mechanism remains responsible for the appearance of strange modes. The unusual growth-rates are seen to also be a consequence of the relative minimum of the inner amplitude for these modes. Again the behavior of the nonadiabatic modes can be well mimicked by a simple analytical toy model.

The strange modes can be linearly unstable to the left of the fundamental and first overtone blue edges. Hydrodynamical calculations show that the strange limit cycle pulsations (a) are extremely superficial as the linear eigenvectors already indicate, in fact they have negligible amplitudes interior to the partial hydrogen ionization front, and \th (b) the pulsations have surface radial velocities in the 0.1 -- 1.0 km/s range, but extremely small photospheric velocities, and luminosity variations in the milli-magnitude range. \th These modes are therefore expected to be difficult to observe. }