HD 35843: A Sun-like star hosting a long period sub-Neptune and inner super-Earth

We report the discovery and confirmation of two planets orbiting the metal-poor Sun-like star, HD 35843 (TOI 4189). HD 35843 c is a temperate sub-Neptune transiting planet with an orbital period of 46.96 days that was first identified by Planet Hunters TESS. We combine data from TESS and follow-up observations to rule out false-positive scenarios and validate the planet. We then use ESPRESSO radial velocities to confirm the planetary nature and characterize the planet's mass and orbit. Further analysis of these RVs reveals the presence of an additional planet, HD 35843 b, with a period of 9.90 days and a minimum mass of $5.84\pm0.84$ $M_{\oplus}$. For HD 35843 c, a joint photometric and spectroscopic analysis yields a radius of $2.54 \pm 0.08 R_{\oplus}$, a mass of $11.32 \pm 1.60 M_{\oplus}$, and an orbital eccentricity of $e = 0.15\pm0.07$. With a bulk density of $3.80 \pm 0.70$ g/cm$^3$, the planet might be rocky with a substantial H$_2$ atmosphere or it might be a ``water world". With an equilibrium temperature of $\sim$480 K, HD 35843 c is among the coolest $\sim 5\%$ of planets discovered by TESS. Combined with the host star's relative brightness (V= 9.4), HD 35843 c is a promising target for atmospheric characterization that will probe this sparse population of temperate sub-Neptunes.