Asteroseismological analysis of the polluted ZZ Ceti star G29-38 with TESS

G\,29$-$38 (TIC~422526868) is one of the brightest ($V=13.1$) and closest ($d = 17.51$\,pc) pulsating white dwarfs with a hydrogen-rich atmosphere (DAV/ZZ Ceti class). It was observed by the {\sl TESS} spacecraft in sectors 42 and 56. The atmosphere of G~29$-$38 is polluted by heavy elements that are expected to sink out of visible layers on short timescales. The photometric {\sl TESS} data set spans $\sim 51$ days in total, and from this, we identified 56 significant pulsation frequencies, that include rotational frequency multiplets. In addition, we identified 30 combination frequencies in each sector. The oscillation frequencies that we found are associated with $g$-mode pulsations, with periods spanning from $\sim$ 260 s to $\sim$ 1400 s. We identified %three distinct rotational frequency triplets with a mean separation $\delta \nu_{\ell=1}$ of 4.67 $\mu$Hz and a quintuplet with a mean separation $\delta \nu_{\ell=2}$ of 6.67 $\mu$Hz, from which we estimated a rotation period of about $1.35 \pm 0.1$ days. We determined a constant period spacing of 41.20~s for $\ell= 1$ modes and 22.58\,s for $\ell= 2$ modes. We performed period-to-period fit analyses and found an asteroseismological model with $M_{\star}/M_{\odot}=0.632 \pm 0.03$, $T_{\rm eff}=11\, 635\pm 178$ K, and $\log{g}=8.048\pm0.005$ (with a hydrogen envelope mass of $M_{\rm H}\sim 5.6\times 10^{-5}M_{\star}$), in good agreement with the values derived from spectroscopy. We obtained an asteroseismic distance of 17.54 pc, which is in excellent agreement with that provided by {\sl Gaia} (17.51 pc).