Measuring Stellar Long Surface Rotation Periods by Combining Multi-Sector/Cycles Long Cadence Data from TESS D.B. PALAKKATHARAPPIL¹, M. GILLES¹ ², R.A. GARCÍA¹, A. HAMY³, A. PRIN³, S.N. BRETON⁴, S. MATHUR⁵ & A.R.G. SANTOS⁶ 1. Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, F-91191, Gif-sur-Yvette, France, 2. ESTACA Paris-Saclay, Montigny-le-Bretonneux 3. École Centrale-Supélec, Univesité Paris-Saclay 4. INAF - Osservatorio Astrofisico di Catania 5. IAC - Instituto de Astrofisica de Canarias and ULL 6. Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP Introduction The evolution of surface magnetic features on stars, such as spots and faculae, produce modulations in light curves. These modulations allow us to measure the surface rotation period, Prot, of stars from Main Sequence (e.g. McQuillan et al. 2014, Santos et al. 2019, 2021) to Red Giant stars (Ceillier et al. 2017). Nevertheless, the effectiveness of determining Prot heavily depends on the length of the observation period. Previously, apart from the long observation campaigns of TESS (Ricker et al. 2015), it was not possible to measure Prot longer than about 13 days, half of TESS's 27-day sector observation period. TESS has completed two all-sky observations; combining this multi-sector data allows us to probe rotation periods beyond the typical limit of around 13 days. Measuring the rotation period from combined multi-sector data is challenging due to inter-sector gaps (~13 days), sector gaps (over years), and phase shifts in the light curve while combining long-gapped data. In this work, we evaluate the limits of going up to ~20 days from at least two continuous sectors in multiple campaigns, minimum possible multi-sector/cycle configuration. We assess the best strategies to extract Prot and provide yields based on Kepler (Borucki et al. 2010) data observed as TESS cycles. Identification of bad Sectors & Background correction PYTADACS Calibration TESS QLP Light Curves The MIT Quick Look Pipeline (QLP, Huang et al., 2020) generates light curves for approximately 19 million stars from FFIs. The 'SAP_FLUX' data product from QLP provides normalized best aperture photometric flux that is corrected for background without detrending the signal, ensuring that periodic features are preserved for accurate rotational period analysis. Additionally, the availability of pre-processed light curves eliminates the need to download Full Frame Images (FFIs) and compute light curves from scratch. However, several calibration strategies should be done to obtain precise rotation periods, as explained below. In a single sector, signiﬁcant ﬂux ﬂuctuations can occur (Fig. 1, Sector 26), likely resulting from background calibration issues within the QLP. Such variations signiﬁcantly compromise the accuracy of the Prot calculations, whether using the Autocorrelation Function (ACF) or the Global Wavelet Power Spectrum. It is crucial to identify these aﬀected sectors and either remove or correct them for more reliable Prot estimation. The Fig. 2 on the right illustrates the estimation of an accurate period post- correction. Proper stitching of Light curves TESS data include inter-sector gaps of approximately 6 hours to 1 day around every ~13 days and sector gaps of 1-2 days. QLP flux is normalized by sector, which limits analysis to shorter rotation periods. To derive longer rotation periods, it is essential to properly stitch light curves across these gaps. As depicted in Fig. 3, without adequate stitching, the estimated rotation period will be biased. A stitching routine utilizing Bayesian inference, adapted from the TASOC pipeline (Handberg & Lund 2016, Handberg et al., 2021), has been implemented in PYTADACS (soon available through git, García et al. in prep.). This approach allows for the detection of longer periods, as illustrated in Fig. 4. Rotation Period Estimation After the correction of sectors and stitching of the light curve, a filter is appli