Gravitational torques

In short, there is still much to learn about the specific processes that drive disk evolution. Currently, the two best candidates, the MRI and gravitational torques, likely produce values of a that vary with time and location in a protoplanetary disk. In order to account properly for these variations more detailed models than those discussed in this chapter are required, and due to their complexity and computational rigor, the amount of model time that could be investigated is limited to 103-104 yr. While adopting a constant or effective value of a overlooks the details of these variations, it greatly reduces the numerical complexity of disk models, allowing the evolution of disks for times > 106 yr to be calculated. This provides a way for the timescales that are needed to study meteoritic materials or dust around other stars to be modeled. 

Accretion. Viscous stresses and gravitational torques within the disk transport angular momentum to the outer regions allowing disk matter to flow inward and accrete onto the star. Because the source of viscosity is still not well understood (see also Chapter 4), it is common to describe the viscosity via a dimensionless parameter a (Shakura Syunyaev 1973). Using this simplification, the viscous dispersal timescale, i.e. the time for the disk to disperse via accretion, becomes inversely proportional to a and increases linearly with the radial distance from the star (Hartmann et al. 1998). While the inner-disk material accretes onto the star, material further out moves in and replenishes the inner disk. Thus, the disk-dispersal timescale from accretion alone is set by the timescale to disperse the mass at the outer disk. 

In the case of nonmagnetic collapse of a spherical cloud (Yorke and Bodenheimer, 1999), the protostar that forms is orbited by a protostellar disk with a similar mass. When angular momentum is transported outward by assumed gravitational torques, and therefore mass is transported inward onto the protostar, the amount of mass remaining in the disk is still so large that most of this matter must eventually be accreted by the protostar through other processes. Hence, the disk at this phase must still be considered a protostellar disk, not a relatively late phase, protoplanetary disk where any objects which form have some hope of... 

The dimensionless group y is the ratio of the applied torque to the maximum gravitational torque. If / > 1 then the applied torque can never be balanced by the gravitational torque and the pendulum will overturn continually. The rotation rate is nonuniform, since gravity helps the applied torque on one side and opposes it on the other (Figure 4.4.2). 

The gravitational torque is like that of an inverted pendulum, since water is pumped in at the top of wheel (Figure 9.1.6). 

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