Super-Earths with solid surfaces such CoRot-7b and Kepler-10b are expected to be extremely hot. It has been suggested that they must be atmosphere-free and that a lava ocean is present on their hot dayside. Here, we use several dedicated thermal models to study how observations with NIRSPEC on the JWST could further confirm and constrain, or reject the atmosphere-free lava ocean planet model for very hot super-Earths. Using CoRoT-7b as a working case, we explore the consequences on the phase-curve of a non tidal-locked rotation, with/without an atmosphere, and for different values of the albedo. We simulate future observations of the reflected light and thermal emission from CoRoT-7b with NIRSPEC-JWST and look for detectable signatures, such as time lag, as well as the possibility to retrieve the latitudinal surface temperature distribution. We demonstrate that we should be able to constrain several parameters after observations of two orbits (42 h) with a broad range of wavelengths: i)The Bond albedo is retrieved to within +/- 0.03 in most cases. ii) The lag effect allows us to retrieve the rotation period within 3 hours for a planet, with rotation half the orbital period. iii) Any spin period shorter than a limit in the range 30 - 800 h, depending on the thickness of the thermal layer in the soil, would be detected. iv) The presence of a thick gray atmosphere with a pressure of one bar, and a specific opacity higher than 1.0E-5 m-2 kg-1 is detectable. v) With spectra up to 4.5 mu, the latitudinal temperature profile can be retrieved to within 30 K with a risk of a totally wrong solution in 5 % of the cases. We conclude that it should thus be possible to distinguish the reference situation from other cases. Surface temperature map and the albedo brings important constraints on the nature or the physical state of the soil of hot super-Earths.
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