Recently, several Type Ib supernovae (SNe; with the prototypical SN 2005E) have been shown to have atypical properties. These SNe are faint (absolute peak magnitude of similar to-15) and fast SNe that show unique composition. They are inferred to have low ejecta mass (a few tenths of a solar mass) and to be highly enriched in calcium, but poor in silicon elements and nickel. These SNe were therefore suggested to belong to a new class of calcium-rich faint SNe explosions. Their properties were proposed to be the result of helium detonations that may occur on helium accreting white dwarfs. In this paper, we theoretically study the scenario of helium detonations and focus on the results of detonations in accreted helium layers on low-mass carbon-oxygen (CO) cores. We present new results from one-dimensional simulations of such explosions, including their light curves and spectra. We find that when the density of the helium layer is low enough the helium detonation produces large amounts of intermediate elements, such as calcium and titanium, together with a large amount of unburnt helium. Alternatively, enough carbon enrichment of the accreted helium as a result of convective undershoot at the early stages of the runaway can avoid the production of iron group elements as the alpha particles are consumed avoiding iron production. Our results suggest that the properties of calcium-rich faint SNe could indeed be consistent with the helium-detonation scenario on small CO cores. Above a certain density (larger CO cores) the detonation leaves mainly Ni-56 and unburnt helium, and the predicted spectrum will unlikely fit the unique features of this class of SNe. Finally, none of our studied models reproduces the bright, fast-evolving light curves of another type of peculiar SNe suggested to originate in helium detonations (SNe 1885A, 1939B, and 2002bj).
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