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A carbon star is a star with more carbon than oxygen in its stellar atmosphere. The spectral class C (for C-type star) indicates a star with molecular carbon lines, indicating such a star. Their effective temperature can be anywhere from within the range of M-type stars to G-type stars. Carbon and oxygen in a star's atmosphere forms carbon monoxide, and if carbon is left over, other carbon compounds are formed, giving the star a red appearance (most stars, including the Sun, have more oxygen than carbon in their atmospheres). One situation where stars form more-than-usual carbon is thought to be asymptotic giant branch (AGB) stars with helium producing additional carbon through helium fusion. Another means is accretion from a binary companion. Carbon stars can emit carbon in their stellar wind, leading to carbon in cosmic dust. Some related terms:
(For each of these terms, whether it can be called a subclass of carbon star, versus some other star type that has some carbon is a matter of semantics: the term carbon star is probably often used specifically for certain well-known types/scenarios.)