K2 and Spitzer phase curves of the rocky ultra-short-period planet K2-141 b hint at a tenuous rock vapor atmosphere

K2-141 b is a transiting, small (1.5 R ⊕ ) ultra-short-period (USP) planet discovered by the Kepler space telescope orbiting a K-dwarf host star every 6.7 h. The planet's high surface temperature of more than 2000 K makes it an excellent target for thermal emission observations. Here we present 65 h of continuous photometric observations of K2-141 b collected with Spitzer's Infrared Array Camera (IRAC) Channel 2 at 4.5 μm spanning ten full orbits of the planet. We measured an infrared eclipse depth of ${f_{{{\rm{p}} \mathord{\left/ {\vphantom {{\rm{p}} {{{\rm{f}}_{\rm{*}}} \right. \kern-\nulldelimiterspace} {{{\rm{f}}_{\rm{*}}} = 142.9_{ - 39.0}^{38.5}$ ppm and a peak to trough amplitude variation of $A = 120.6_{ - 43.0}^{42.3}$ ppm. The best fit model to the Spitzer data shows no significant thermal hotspot offset, in contrast to the previously observed offset for the well-studied USP planet 55 Cnc e. We also jointly analyzed the new Spitzer observations with the photometry collected by Kepler during two separate K2 campaigns. We modeled the planetary emission with a range of toy models that include a reflective and a thermal contribution. With a two-temperature model, we measured a dayside temperature of ${T_{{\rm{p,d}}} = 2049_{ - 359}^{362}$ K and a night-side temperature that is consistent with zero ( T p,n < 1712 K at 2 σ ). Models with a steep dayside temperature gradient provide a better fit to the data than a uniform dayside temperature (ΔBIC = 22.2). We also found evidence for a nonzero geometric albedo ${A_{\rm{g}}} = 0.282_{ - 0.078}^{0.070}$. We also compared the data to a physically motivated, pseudo-2D rock vapor model and a 1D turbulent boundary layer model. Both models fit the data well. Notably, we found that the optical eclipse depth can be explained by thermal emission from a hot inversion layer, rather than reflected light. A thermal inversion may also be responsible for the deep optical eclipse observed for another USP, Kepler-10 b. Finally, we significantly improved the ephemerides for K2-141 b and c, which will facilitate further follow-up observations of this interesting system with state-of-the-art observatories such as James Webb Space Telescope.