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<jats:title>Abstract</jats:title> <jats:p>Stellar rotation was proposed as a potential age diagnostic that is precise, simple, and applicable to a broad range of low-mass stars (<jats:inline-formula> <jats:tex-math> <?CDATA $\leqslant $?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn1.gif" xlink:type="simple" /> </jats:inline-formula>1 <jats:inline-formula> <jats:tex-math> <?CDATA ${M}_{\odot }$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn2.gif" xlink:type="simple" /> </jats:inline-formula>). Unfortunately, rotation period (<jats:inline-formula> <jats:tex-math> <?CDATA ${P}_{{\rm{rot}}}$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn3.gif" xlink:type="simple" /> </jats:inline-formula>) measurements of low-mass members of open clusters have undermined the idea that stars spin down with a common age dependence (i.e., <jats:inline-formula> <jats:tex-math> <?CDATA ${P}_{{\rm{rot}}}$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn4.gif" xlink:type="simple" /> </jats:inline-formula> <jats:inline-formula> <jats:tex-math> <?CDATA $\propto \ \sqrt{\mathrm{age}}$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn5.gif" xlink:type="simple" /> </jats:inline-formula>): K dwarfs appear to spin down more slowly than F and G dwarfs. Agüeros et al. interpreted data for the ≈1.4 Gyr-old cluster NGC 752 differently, proposing that after having converged onto a slow-rotating sequence in their first 600–700 Myr (by the age of Praesepe), K dwarf <jats:inline-formula> <jats:tex-math> <?CDATA ${P}_{{\rm{rot}}}$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn6.gif" xlink:type="simple" /> </jats:inline-formula> stall on that sequence for an extended period of time. We use data from <jats:italic>Gaia</jats:italic> DR2 to identify likely single-star members of the ≈1 Gyr-old cluster NGC 6811 with <jats:italic>Kepler</jats:italic> light curves. We measure <jats:inline-formula> <jats:tex-math> <?CDATA ${P}_{{\rm{rot}}}$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn7.gif" xlink:type="simple" /> </jats:inline-formula> for 171 members, more than doubling the sample relative to the existing catalog and extending the mass limit from <jats:inline-formula> <jats:tex-math> <?CDATA $\approx 0.8$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn8.gif" xlink:type="simple" /> </jats:inline-formula> to ≈0.6 <jats:inline-formula> <jats:tex-math> <?CDATA ${M}_{\odot }$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn9.gif" xlink:type="simple" /> </jats:inline-formula>. We then apply a gyrochronology formula calibrated with Praesepe and the Sun to 27 single G dwarfs in NGC 6811 to derive a precise gyrochronological age for the cluster of 1.04 ± 0.07 Gyr. However, when our new low-mass rotators are included, NGC 6811's color–<jats:inline-formula> <jats:tex-math> <?CDATA ${P}_{{\rm{rot}}}$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn10.gif" xlink:type="simple" /> </jats:inline-formula> sequence deviates away from the naive 1 Gyr projection down to <jats:inline-formula> <jats:tex-math> <?CDATA ${T}_{{\rm{eff}}}\approx 4295$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn11.gif" xlink:type="simple" /> </jats:inline-formula> K (K5V, 0.7 <jats:inline-formula> <jats:tex-math> <?CDATA ${M}_{\odot }$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn12.gif" xlink:type="simple" /> </jats:inline-formula>), where it clearly overlaps with Praesepe’s. Combining these data with <jats:inline-formula> <jats:tex-math> <?CDATA ${P}_{{\rm{rot}}}$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn13.gif" xlink:type="simple" /> </jats:inline-formula> for other clusters, we conclude that the assumption that mass and age are separable dependencies is invalid. Furthermore, the cluster data show definitively that stars experience a temporary epoch of reduced braking efficiency where <jats:inline-formula> <jats:tex-math> <?CDATA ${P}_{{\rm{rot}}}$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjab2393ieqn14.gif" xlink:type="simple" /> </jats:inline-formula> stall, and that the duration of this epoch lasts longer for lower-mass stars.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>In this paper, we present an analysis of the pulsating behavior of <jats:italic>Kepler</jats:italic> target KIC 10284901. The Fourier transform of the high-precision light curve reveals seven independent frequencies for its light variations. Among them, the first two frequencies are main pulsation modes: F0 = 18.994054(1) day<jats:sup>−1</jats:sup> and F1 = 24.335804(4) day<jats:sup>−1</jats:sup>; the ratio F0/F1 = 0.7805 classifies this star as a double-mode high-amplitude <jats:italic>δ</jats:italic> Scuti (HADS) star; another two frequencies, <jats:italic>f</jats:italic> <jats:sub> <jats:italic>m</jats:italic>1</jats:sub> = 0.4407 day<jats:sup>−1</jats:sup> and <jats:italic>f</jats:italic> <jats:sub> <jats:italic>m</jats:italic>2</jats:sub> = 0.8125 day<jats:sup>−1</jats:sup>, are detected directly, and the modulations of <jats:italic>f</jats:italic> <jats:sub> <jats:italic>m</jats:italic>1</jats:sub> and <jats:italic>f</jats:italic> <jats:sub> <jats:italic>m</jats:italic>2</jats:sub> to F0 and F1 modes (seen as quintuplet structures centered on these two modes in the frequency spectrum) are also found. This is the first detection of a double-modulation effect in the HADS stars. The features of the frequency pattern and the ratio (<jats:italic>f</jats:italic> <jats:sub> <jats:italic>m</jats:italic>1</jats:sub>/<jats:italic>f</jats:italic> <jats:sub> <jats:italic>m</jats:italic>2</jats:sub> ≈ 1:2), as well as the cyclic variation of amplitude of the two dominant pulsation modes, which seem to be similar to that in Blazhko RR Lyrae stars, indicate this modulation might be related to the Blazhko effect. A preliminary analysis suggests that KIC 10284901 is in the bottom of the HADS instability strip and situated in the main sequence.</jats:p>