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Are C-rich ultra iron poor stars also He-rich?

(abridged) The three most iron poor stars presently known ([Fe/H] equal to -5.96, -5.4 and -4.75) are carbon-rich, they are called C-Rich Ultra-Metal Poor Stars (CRUMPS). The origin of their peculiar surface abundances is not understood. We propose a synthetic view of the different models so far proposed to explain the peculiar abundances observed at the surface of the CRUMP stars. We deduce some expected trends based on nucleosynthetic arguments and look for signatures allowing to discriminate among models. We discuss the conditions for having CRUMP stars which are He-rich, i.e. with a mass fraction of helium greater than 0.30 and up to 0.60. We discuss the chemical composition of stars made of interstellar medium mixed with wind material of very metal poor massive stars, with wind plus supernova ejecta and with material extracted from the envelope of early AGB stars. Rotating and non-rotating models are considered. CRUMP stars formed from wind material of massive stars mixed with small amounts of pristine interstellar medium are He-rich (helium mass fraction between 0.30 and 0.60), Li-depleted and present low $^{12}$C/$^{13}$C ratios (inferior to 10 in number). Such He-rich stars, if discovered, would confirm that the most metal poor CRUMPs formed from essentially pure wind/envelope material. They would provide the most direct way to probe the nucleosynthetic outputs of the first generations of stars. We show that rotation is a key ingredient to explain the abundance patterns of CRUMPS stars.