The nature of very low luminosity objects with the internal luminosity L_obj <= 0.1 Lsun is investigated by means of numerical modeling coupling the core collapse simulations with the stellar evolution calculations. The gravitational collapse of a large sample of model cores in the mass range 0.1-2.0 Msun is investigated. Numerical simulations were started at the pre-stellar phase and terminated at the end of the embedded phase when 90% of the initial core mass had been accreted onto the forming protostar plus disk system. The disk formation and evolution was studied using numerical hydrodynamics simulations, while the formation and evolution of the central star was calculated using a stellar evolution code. Three scenarios for mass accretion from the disk onto the star were considered: hybrid accretion in which a fraction of accreted energy absorbed by the protostar depends on the accretion rate, hot accretion wherein a fraction of accreted energy is constant, and cold accretion wherein all accretion energy is radiated away. Our conclusions on the nature of VeLLOs depend crucially on the character of protostellar accretion. In the hybrid accretion scenario, most VeLLOs (90.6%) are expected to be the first hydrostatic cores (FHSCs) and only a small fraction (9.4%) are true protostars. In the hot accretion scenario, all VeLLOs are FHSCs due to overly high photospheric luminosity of protostars. In the cold accretion scenario, on the contrary, the majority of VeLLOs belong to the Class I phase of stellar evolution. The reason is that the stellar photospheric luminosity, which sets the floor for the total internal luminosity of a young star, is lower in cold accretion, thus enabling more VeLLOs in the protostellar stage. VeLLOs are relatively rare objects occupying 7%-11% of the total duration of the embedded phase and their masses do not exceed 0.3 Msun. (abridged).
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