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Lifetime of the embedded phase of low-mass star formation and the envelope depletion rates

Motivated by a considerable scatter in the observationally inferred lifetimes of the embedded phase of star formation, we study the duration of the Class 0 and Class I phases in upper-mass brown dwarfs and low-mass stars using numerical hydrodynamics simulations of the gravitational collapse of a large sample of cloud cores. We resolve the formation of a star/disk/envelope system and extend our numerical simulations to the late accretion phase when the envelope is nearly totally depleted of matter. We adopted a classification scheme of Andre et al. and calculate the lifetimes of the Class 0 and Class I phases (τ_C0 and τ_CI, respectively) based on the mass remaining in the envelope. When cloud cores with various rotation rates, masses, and sizes (but identical otherwise) are considered, our modeling reveals a sub-linear correlation between the Class 0 lifetimes and stellar masses in the Class 0 phase with the least-squares fit exponent m=0.8 \pm 0.05. The corresponding correlation between the Class I lifetimes and stellar masses in the Class I is super-linear with m=1.2 \pm 0.05. If a wider sample of cloud cores is considered, which includes possible variations in the initial gas temperature, cloud core truncation radii, density enhancement amplitudes, initial gas density and angular velocity profiles, and magnetic fields, then the corresponding exponents may decrease by as much as 0.3. The duration of the Class I phase is found to be longer than that of the Class~0 phase in most models, with a mean ratio τ_CI / τ_C0 \approx 1.5--2. A notable exception are YSOs that form from cloud cores with large initial density enhancements, in which case τ_C0 may be greater than τ_CI. Moreover, the upper-mass (>= 1.0 Msun) cloud cores with frozen-in magnetic fields and high cloud core rotation rates may have the τ_CI / τ_C0 ratios as large as 3.0--4.0. (Abdridged).