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We perform a homogeneous search for and analysis of optical occultations andphase variations of the most favorable ultra-short-period (USP) ($P<1$~d)sub-Neptunes ($R_{p}<4 R_{\oplus}$) observed by $\textit{Kepler}$ and K2, withthe aim of better understanding their nature. We first selected 16$\textit{Kepler}$ and K2 USP sub-Neptunes, based on the expected occultationsignal. We filtered out stellar variability in the $\textit{Kepler}$ lightcurves, using a sliding linear fitting and, when required, a more sophisticatedapproach based on Gaussian Process regression. We simultaneously modeled theprimary transit, secondary eclipse, and phase variations in a Bayesianframework, by using information from previous studies and knowledge of the Gaiaparallaxes. We confirm the optical secondary eclipses for Kepler-10b($13\sigma$), Kepler-78b ($9.5\sigma$), and K2-141b ($6.9\sigma$), withmarginal evidence for K2-312b ($2.2\sigma$). We report new detections forK2-106b ($3.3\sigma$), K2-131b (3.2$\sigma$), Kepler-407b ($3.0\sigma$), andhints for K2-229b (2.5$\sigma$). For all targets with the exception of K2-229band K2-312b, we also find phase curve variations with a confidence level higherthan $2\sigma$. Two USP planets, namely Kepler-10b and Kepler-78b, shownon-negligible nightside emission. This questions the scenario of magma-oceanworlds with inefficient heat redistribution to the night-side for both planets.Due to the youth of the Kepler-78 system and the small planetary orbitalseparation, the planet may still retain a collisional secondary atmospherecapable of conducting heat from the day to the night side. Instead, thepresence of an outgassing magma ocean on the dayside and the low high-energyirradiation of the old host star may have enabled Kepler-10b to build up andretain a recently-formed collisional secondary atmosphere.

Probing Kepler's hottest small planets via homogeneous search and analysis of optical secondary eclipses and phase variations