A transit timing variation observed for the long-period extremely low density exoplanet HIP 41378f.

HIP 41378 f is a temperate $9.2\pm0.1 R_{\oplus}$ planet with period of 542.08 days and an extremely low density of $0.09\pm0.02$ g cm$^{-3}$. It transits the bright star HIP 41378 (V=8.93), making it an exciting target for atmospheric characterization including transmission spectroscopy. HIP 41378 was monitored photometrically between the dates of 2019 November 19 and November 28. We detected a transit of HIP 41378 f with NGTS, just the third transit ever detected for this planet, which confirms the orbital period. This is also the first ground-based detection of a transit of HIP 41378 f. Additional ground-based photometry was also obtained and used to constrain the time of the transit. The transit was measured to occur 1.50 hours earlier than predicted. We use an analytic transit timing variation (TTV) model to show the observed TTV can be explained by interactions between HIP 41378 e and HIP 41378 f. Using our TTV model, we predict the epochs of future transits of HIP 41378 f, with derived transit centres of T$_{C,4} = 2459355.087^{+0.031}_{-0.022}$ (May 2021) and T$_{C,5} = 2459897.078^{+0.114}_{-0.060}$ (Nov 2022).