Probing<i>Kepler</i>’s hottest small planets via homogeneous search and analysis of optical secondary eclipses and phase variations

We perform a homogeneous search for and analysis of optical occultations phase variations the most favorable ultra-short-period (USP) ($P<1$~d) sub-Neptunes ($R_{p}<4 R_{\oplus}$) observed by $\textit{Kepler}$ K2, with aim better understanding their nature. first selected 16 K2 USP sub-Neptunes, based on expected occultation signal. filtered out stellar variability in light curves, using sliding linear fitting and, when required, more sophisticated approach Gaussian Process regression. simultaneously modeled primary transit, secondary eclipse, Bayesian framework, information from previous studies knowledge Gaia parallaxes. confirm eclipses Kepler-10b ($13\sigma$), Kepler-78b ($9.5\sigma$), K2-141b ($6.9\sigma$), marginal evidence K2-312b ($2.2\sigma$). report new detections K2-106b ($3.3\sigma$), K2-131b (3.2$\sigma$), Kepler-407b ($3.0\sigma$), hints K2-229b (2.5$\sigma$). For all targets exception K2-312b, we also find curve confidence level higher than $2\sigma$. Two planets, namely Kepler-78b, show non-negligible nightside emission. This questions scenario magma-ocean worlds inefficient heat redistribution to night-side both planets. Due youth Kepler-78 system small planetary orbital separation, planet may still retain collisional atmosphere capable conducting day night side. Instead, presence an outgassing magma ocean dayside low high-energy irradiation old host star have enabled build up recently-formed atmosphere.