Recent high-resolution simulations demonstrate that disks around primordial protostars easily fragment in the accretion phase before protostars accrete less than a solar mass. To understand why gravitational instability generally causes fragmentation so early, we develop a one-dimensional (1D) nonsteady model of the circumstellar disk that takes the mass supply from an accretion envelope into account. We also compare the model results to a three-dimensional (3D) numerical simulation performed with a code employing adaptive mesh refinement. Our model shows that the self-gravitating disk, through which the Toomre Q parameter is nearly constant at Q ∼ 1, gradually spreads as the disk is fed by the gas infalling from the envelope. We further find that the accretion rate onto the star is an order of magnitude smaller than the mass supply rate onto the disk. This discrepancy makes the disk more massive than the protostar in an early evolutionary stage. Most of the infalling gas is used to extend the outer part of the self-gravitating disk rather than transferred inward toward the star through the disk. We find that similar evolution also occurs in 3D simulations, where the disk becomes three times more massive than the star before the first fragmentation occurs. Our 1D disk model well explains the evolution of the disk-to-star mass ratio observed in the simulation. We argue that the formation of such a massive disk leads to early disk fragmentation.
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