The GAPS Programme at TNG XXXVI. Measurement of the Rossiter-McLaughlin Effect and Revising the Physical and Orbital Parameters of the HAT-P-15, HAT-P-17, HAT-P-21, HAT-P-26, HAT-P-29 Eccentric Planetary Systems

Aim: We aim to refine the orbital and physical parameters and determine the sky-projected planet orbital obliquity of five eccentric transiting planetary systems: HAT-P-15, HAT-P-17, HAT-P-21, HAT-P-26, and HAT-P-29. Each of the systems hosts a hot Jupiter, except for HAT-P-26 which hosts a Neptune-mass planet. Methods: We observed transit events of these planets with the HARPS-N spectrograph, obtaining high-precision radial velocity measurements that allow us to measure the Rossiter-McLaughlin effect for each of the target systems. We used these new HARPS-N spectra and archival data, including those from Gaia, to better characterise the stellar atmospheric parameters. The photometric parameters for four of the hot Jupiters were recalculated using 17 new transit light curves, obtained with an array of medium-class telescopes, and data from the TESS space telescope. HATNet time-series photometric data were checked for the signatures of rotation periods of the target stars and their spin axis inclination. Results: From the analysis of the Rossiter-McLaughlin effect, we derived a sky-projected obliquity of 13, -26.3, -0.7, -26 degree for HAT-P-15b, HAT-P-17b, HAT-P-21b and HAT-P-29b, respectively. Due to the quality of the data, we were not able to well constrain the sky-projected obliquity for HAT-P-26b, although a prograde orbit is favoured. The stellar activity of HAT-P-21 indicates a rotation period of 15.88 days, which allowed us to determine its true misalignment angle (25 degree). Our new analysis of the physical parameters of the five exoplanetary systems returned values compatible with those existing in the literature. Using TESS and the available transit light curves, we reviewed the orbital ephemeris for the five systems and confirmed that the HAT-P-26 system shows transit timing variations, which may tentatively be attributed to the presence of a third body.