A study of the rapid rotator $\zeta$ Aql: differential surface rotation?

We report new, extremely precise, photopolarimetry of the rapidly-rotating A0 main-sequence star $\zeta$ Aql, covering the wavelength range $\sim$400--900nm, which reveals a rotationally-induced signal. We model the polarimetry, together with the flux distribution and line profiles, in the framework of Roche geometry with $\omega$-model gravity darkening, to establish the stellar parameters. An additional constraint is provided by TESS photometry, which shows variability with a period, $P_{\rm phot}$, of 11.1 hr. Modelling based on solid-body surface rotation gives rotation periods, $P_{\rm rot}$, that are in only marginal agreement with this value. We compute new ESTER stellar-structure models to predict horizontal surface velocity fields, which depart from solid-body rotation at only the $\sim$2% level (consistent with a reasonably strong empirical upper limit on differential rotation derived from the line-profile analysis). These models bring the equatorial rotation period, $P_{\rm rot,e}$, into agreement with $P_{\rm phot}$, without requiring any 'fine tuning' (for the Gaia parallax). We confirm that surface abundances are significantly subsolar ($\mbox{[M/H]} \simeq -0.5$). The star's basic parameters are established with reasonably good precision: ${M = 2.53\pm0.16\,\mbox{M}_\odot}$, $\log{L/\mbox{L}_\odot} = 1.72\pm0.02$, $R_{\rm p} = 2.21\pm 0.02\,\mbox{R}_\odot$, $T_{\rm eff} = 9693 \pm 50~\mbox{K}$, $i = 85{^{+5}_{-7}}^\circ$, and $\omega/\omega_{\rm c} = 0.95\pm0.02$. Comparison with single-star, solar-abundance stellar-evolution models incorporating rotational effects shows excellent agreement (but somewhat poorer agreement for models at $\mbox{[M/H]} \simeq -0.4$).