Very unusual light phenomena have been observed inside superconducting niobium cavities under an electrical excitation equivalent t o a n a v erage acceleration of 4 MV m.Of particular interest is the observation of what appear to be objects executing elliptical orbits. A small particle model that yields a cylindrical harmonic oscillator potential well inside such a c a vity is developed and appears to o er considerable promise as an analytical framework in which to understand these observations; the harmonic oscillator potential well has elliptical orbits as general solutions. However, explicit calculations indicate that with the cavity excitation parameters associated with this data, it is highly unlikely that a small object orbiting in such a potential well can bemade of any known material. De ciencies of the model are discussed, and avenues for further study, both theoretical and experimental, are indicated. PACS: 46.90 +s; 52.80 Mg