AE Ursae Majoris -- a $\delta$ Scuti Star in the Hertzsprung Gap

We analyze the photometric data and spectroscopic data that collect on the $\delta$ Scuti star AE UMa. The fundamental and the first overtone frequencies are confirmed as $f_{0} = 11.62560$\,c\,d$^{-1}$ and $f_{1} = 15.03124$\,c\,d$^{-1}$, respectively, from the frequency content by analyzing of the 40 nights light curve spanning from 2009 to 2012. Additionally, another 37 frequencies are identified as either the harmonics or the linear combinations of the fundamental and the first overtone frequencies, among which 25 are newly detected. The rate of period change of the fundamental mode is determined as $(1/P_{0})(dP_{0}/dt) = 5.4(\pm 1.9) \times 10^{-9}$\,yr$^{-1}$ as revealed from the $O-C$ diagram based on the 84 newly determined times of maximum light combined with those derived from the literature. The spectroscopic data suggests that AE UMa is a population\,I $\delta$ Scuti star. With these physical properties, we perform theoretical explorations based on the stellar evolution code MESA on this target, considering that the variation of pulsation period is caused by secular evolutionary effects. We finally constraint the AE UMa with the physical parameters as: the mass of $1.805 \pm 0.055\ M_{\odot}$, the radius of $1.647 \pm 0.032 \times 10^{11}$\,cm, the luminosity of $1.381 \pm 0.048~(\log L/L_{\odot})$ and the age of $1.055 \pm 0.095 \times 10^{9}$\,yr. AE UMa can be the (Pop. I) $\delta$ Scuti star that locates just after the second turn-off of its evolutional track leaving the main sequence, a star in the phase of the Hertzsprung Gap with a helium core and a hydrogen-burning shell.