Energy of oscillation: The role of the negative energy mode in overstable nonradial oscillations of rotating stars

We discuss the energies of adiabatic nonradial oscillations of nonrotating and uniformly rotating stars. For the sake of simplicity, we neglect the Eulerian perturbation of the gravitational potential (Cowling approximation) and the horizontal component of angular velocity of rotation ("traditional approximation"). If a dispersion relation for an eigenmode of nonradial oscillations is given as script D sign(ω) = 0, with ω being the angular frequency of the oscillation, the energy of the oscillation (kinetic plus potential energies) is shown to be proportional to ω + ∂ + script D sign/∂ + ω. This leads to the interpretation that the resonance between two positive energy modes yields an avoided crossing, while the resonance between one positive energy mode and one negative energy mode yields an overstable mode. We find that the oscillation energies of p- and g-modes of nonrotating stars and g-modes of rotating stars are definitely positive, while the energy of an oscillatory convective mode of rotating stars is positive or negative, depending on the ratio of the frequencies of rotation to oscillation, Ω/ω. For a negative energy oscillation, its amplitude grows when energy is extracted from rather than fed into the oscillation. When a resonance occurs between a negative energy mode and a positive energy mode, energy flows from the negative energy mode to the positive energy mode forming an overstable oscillation. This mechanism interprets the previously found overstable g-mode oscillations coupled with an oscillatory convective mode in a rotating massive main-sequence star, because the oscillatory convective mode is found to have negative energy.