Centrifugal Radius

These considerations indicate that magnetic braking is both rapid and efficient. On the other hand, the braking must fail within the deeper interior of a more realistic, collapsing cloud. As the density climbs, the matter and field decouple because of the drop in ionization fraction. In addition, much of the remaining field is left behind once it is severely pinched and reconnects. Thus, the matter inside some volume of the equatorial region indeed conserves angular momentum and spins up as it approaches the central protostar. Since the centrifugal force rises faster than gravity, each fluid element inevitably veers away from the geometrical center of the cloud. Whether or not this element lands on the protostar or misses it and joins the equatorial disk depends in part on its specific angular momentum j.

In fact, a range of specific angular momenta is present in the infalling matter at any time. Those elements with the highest j-values depart from radial trajectories earliest and ultimately land farthest from the center. The maximum impact distance in the equatorial plane is known as the centrifugal radius, here denoted rocen. As we will see in ยง 11.3, rocen also sets the scale

Figure 10.15 Parabolic orbit in rotating infall. A fluid element, with instantaneous polar coordinates (r, 0) in the orbital plane, falls into the radial distance req, where it impacts the disk. If the disk were absent, the element would have reached the smaller distance rmin before swinging back out.

of the circumstellar disk. The inside-out nature of collapse implies that fluid elements arriving at some later epoch begin falling from more distant locations within the cloud. These regions include higher j-values (recall Figure 9.6), so that some new elements cross the plane even farther from the center than before. In other words, rocen increases with time.

Exploring this idea more quantitatively requires that we determine the actual fluid trajectories. Suppose the magnetic field has effectively decoupled and the velocity is supersonic, so that thermal pressure is no longer significant. Then only gravity and rotation affect the infall. The former is supplied mostly by the protostar and its disk, which we may lump together as a point mass for our purposes. Under these conditions, the trajectory must be an ellipse, i. e., the conic section corresponding to a bound orbit of negative energy. In practice, both the gravitational potential and kinetic energies at the start of infall are tiny compared to their magnitudes when the element nears the star or equatorial plane. The trajectory is thus closely approximated as a zero-energy conic section, i. e., a parabola.6

Figure 10.15 shows a typical parabolic orbit, with the protostar located at the focus. Here, we have specified the instantaneous location of the fluid element through the radius r and angle 0. The angle starts at n and decreases to n/2 by the time the element reaches the plane, when it is a distance req from the protostar. At this point, the fluid velocity normal to the plane abruptly goes to zero, as the element either collides with a pre-existing disk or else with a streamline approaching from the opposite direction. We will consider both these possibilities further in Chapter 11. In any case, the dashed portion of the parabola, corresponding to 0 < n/2, is never actually traversed.

The functional relation between r and 0 is

Notice that we are neglecting any increase in the protostar or disk mass during the relatively brief time a fluid element crosses the distance of interest. The same qualification applies to equation (10.34), which pertains to spherical collapse. In both cases, we are making a steady-state approximation for the interior region of the flow. Mathematically, we ignore the explicit time derivative d/dt in the fluid equations and solve for the spatial variation of all variables.

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