The $δ$ Scuti stars of the Cep--Her Complex. I: Pulsator fraction, rotation, asteroseismic large spacings, and the $ν_{\rm max}$ relation

Abstract: We identify delta Scuti pulsators amongst members of the recently-discovered Cep--Her Complex using light curves from the Transiting Exoplanet Survey Satellite (TESS). We use Gaia colours and magnitudes to isolate a subsample of provisional Cep--Her members that are located in a narrow band on the colour--magnitude diagram compatible with the zero-age main sequence. The $\delta$ Sct pulsator fraction amongst these stars peaks at 100% and we describe a trend of higher pulsator fractions for younger stellar associations. We use four methods to measure the frequency of maximum amplitude or power, $\nu_{\rm max}$, to minimise methodological bias and we demonstrate their sound performance. The $\nu_{\rm max}$ measurements display a correlation with effective temperature, but with scatter that is too large for the relation to be useful. We find two ridges in the $\nu_{\rm max}$--$T_{\rm eff}$ diagram, one of which appears to be the result of rapid rotation causing stars to pulsate in low-order modes. We measure the $\nu_{\rm max}$ values of $\delta$ Sct stars in four other clusters or associations of similar age (Trumpler 10, the Pleiades, NGC 2516, and Praesepe) and find similar behaviour with $T_{\rm eff}$. Using \'echelle diagrams we measure the asteroseismic large spacing, $\Delta\nu$, for 70 stars, and find a correlation between $\Delta\nu$, rotation, and luminosity that allows rapid rotators seen at low inclinations to be distinguished from slow rotators. We find that rapid rotators are more likely than slow rotators to pulsate, but they do so with less regular pulsation patterns. We also investigate the reliability of Gaia's vbroad measurement for A-type stars, finding that it is mostly accurate but underestimates $v\sin i$ for slow rotators ($v\sin i < 50$ km.s$^{-1}$) by 10--15%.