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

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