Tess asteroseismology of the known planet host star $\lambda^2$ Fornacis

The Transiting Exoplanet Survey Satellite (TESS) is observing bright known planet-host stars across almost the entire sky. These stars have been subject to extensive ground-based observations, providing a large number of radial velocity (RV) measurements. In this work we use the new TESS photometric observations to characterize the star $\lambda^2$ Fornacis, and following this to update the parameters of the orbiting planet $\lambda^2$ For b. We measure the p-mode oscillation frequencies in $\lambda^2$ For, and in combination with non-seismic parameters estimate the stellar fundamental properties using stellar models. Using the revised stellar properties and a time series of archival RV data from the UCLES, HIRES and HARPS instruments spanning almost 20 years, we refit the orbit of $\lambda^2$ For b and search the RV residuals for remaining variability. We find that $\lambda^2$ For has a mass of $1.16\pm0.03$M$_\odot$ and a radius of $1.63\pm0.04$R$_\odot$, with an age of $6.3\pm0.9$Gyr. This and the updated RV measurements suggest a mass of $\lambda^2$ For b of $16.8^{+1.2}_{-1.3}$M$_\oplus$, which is $\sim5$M$_\oplus$ less than literature estimates. We also detect a periodicity at 33 days in the RV measurements, which is likely due to the rotation of the host star. While previous literature estimates of the properties of $\lambda^2$ are ambiguous, the asteroseismic measurements place the star firmly at the early stage of its subgiant evolutionary phase. Typically only short time series of photometric data are available from TESS, but by using asteroseismology it is still possible to provide tight constraints on the properties of bright stars that until now have only been observed from the ground. This prompts a reexamination of archival RV data from the past few decades to update the characteristics of the planet hosting systems observed by TESS for which asteroseismology is possible.