Resonant Mode Coupling in $\delta$ Scuti Stars

Delta Scuti ($\delta$ Sct) variables are intermediate mass stars that lie at the intersection of the main sequence and the instability strip on the Hertzsprung-Russel diagram. Various lines of evidence indicate that nonlinear mode interactions shape their oscillation spectra, including the particularly compelling detection of resonantly interacting mode triplets in the $\delta$ Sct star KIC 8054146. Motivated by these observations, we use the theory of three-mode coupling to study the strength and prevalence of nonlinear mode interactions in fourteen $\delta$ Sct models that span the instability strip. For each model, we calculate the frequency detunings and nonlinear coupling strengths of $\sim 10^4$ unique combinations of mode triplets. We find that all the models contain at least $\sim 100$ well-coupled triplets whose detunings and coupling strengths are consistent with the triplets identified in KIC 8054146. Our results suggest that resonant mode interactions can be significant in $\delta$ Sct stars and may explain why many exhibit rapid changes in amplitude and oscillation period.