Amplitude modulation in $\delta$ Sct stars: statistics from an ensemble study of Kepler targets

We present the results of a search for amplitude modulation of pulsation mode frequencies in 983 $\delta$ Sct stars, which have effective temperatures between 6400 $\leq T_{\rm eff} \leq$ 10 000 K in the Kepler Input Catalogue and were continuously observed by the Kepler Space Telescope for 4 yr. We demonstrate the diversity in pulsational behaviour observed, in particular nonlinearity, which is predicted for $\delta$ Sct stars. We analyse and discuss examples of $\delta$ Sct stars with constant amplitudes and phases; those that exhibit amplitude modulation caused by beating of close-frequency pulsation modes; those that exhibit pure amplitude modulation (with no associated phase variation); those that exhibit phase modulation caused by binarity; and those that exhibit amplitude modulation caused by nonlinearity. Using models and examples of individual stars, we demonstrate that observations of the changes in amplitude and phase of pulsation modes can be used to distinguish among the different scenarios. We find that 603 $\delta$ Sct stars (61.3 per cent) exhibit at least one pulsation mode that varies significantly in amplitude over 4 yr. Conversely, many $\delta$ Sct stars have constant pulsation amplitudes so short-length observations can be used to determine precise frequencies, amplitudes and phases for the most coherent and periodic $\delta$ Sct stars. It is shown that amplitude modulation is not restricted to a small region on the HR diagram, therefore not necessarily dependent on stellar parameters such as $T_{\rm eff}$ or $\log\,g$. Our catalogue of 983 $\delta$ Sct stars will be useful for comparisons to similar stars observed by K2 and TESS, because the length of the 4-yr Kepler data set will not be surpassed for some time.