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Recent high-resolution simulations demonstrate that disks around primordialprotostars easily fragment in the accretion phase before the protostars accreteless than a solar mass. To understand why the gravitational instabilitygenerally causes the fragmentation so early, we develop a one-dimensional (1D)non-steady model of the circumstellar disk that takes the mass supply from anaccretion envelope into account. We also compare the model results to athree-dimensional (3D) numerical simulation performed with a code employing theadaptive mesh refinement. Our model shows that the self-gravitating disk,through which the Toomre $Q$ parameter is nearly constant at $Q \sim 1$,gradually spreads as the disk is fed by the gas infalling from the envelope. Wefurther find that the accretion rate onto the star is an order of magnitudesmaller than the mass supply rate onto the disk. This discrepancy makes thedisk more massive than the protostar in an early evolutionary stage. Most ofthe infalling gas is used to extend the outer part of the self-gravitating diskrather than transferred inward toward the star through the disk. We find thatsimilar evolution also occurs in the 3D simulation, where the disk becomesthree 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 ratioobserved in the simulation. We argue that the formation of such a massive diskleads to the early disk fragmentation.

Growth of Massive Disk and Early Disk Fragmentation in the Primordial Star Formation