Using three-dimensional magnetohydrodynamics simulations, the driving of protostellar jets is investigated in different star-forming cores with the parameters of magnetic field strength and mass accretion rate. Powerful high-velocity jets appear in strongly magnetized clouds when the mass accretion rate onto the protostellar system is lower than $dot{M} lesssim 10^{-3},{rm M}_odot$ yr$^{-1}$. On the other hand, even at this mass accretion rate range, no jets appear for magnetic fields of prestellar clouds as weak as $mu_0 gtrsim 5$--$10$, where $mu_0$ is the mass-to-flux ratio normalized by the critical value $(2pi G^{1/2})^{-1}$. For $dot{M}gtrsim 10^{-3},{rm M}_odot$ yr$^{-1}$, although jets usually appear just after protostar formation independent of the magnetic field strength, they soon weaken and finally disappear. Thus, they cannot help drive the low-velocity outflow when there is no low-velocity flow just before protostar formation. As a result, no significant mass ejection occurs during the early mass accretion phase either when the prestellar cloud is weaky magnetized or when the mass accretion rate is very high. Thus, protostars formed in such environments would trace different evolutionary paths from the normal star formation process.
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