Equation (10.31) states that, to within a factor of order unity, the asymptotic accretion rate depends only on the ambient temperature. To some extent, this simple result reflects our omission of magnetic support prior to collapse. However, the effect of the additional force is subtle. As we shall discuss in § 10.3, the cloud portion that finally collapses is that which was not magnetically supported, and equation (10.31) remains our best quantitative estimate. This important relation thus sets the basic time scale for the protostar phase. Supplying numerical values, we have
Thus a protostar of 1 Mq accumulates its mass in about 5 x 105 yr. Within the scope of stellar evolution, this period is exceedingly brief, even compared to the 3 x 107 yr pre-main-sequence contraction time for a star of the same mass.
1 The free-fall velocity Vff should not be confused with the similar virial velocity Vv;r, introduced in equation (3.20). The former refers to a test particle falling toward a point mass, while the latter is the characteristic speed within a distributed volume of self-gravitating gas.
Radial Distance log r
Figure 10.7 Rarefaction wave in inside-out collapse (schematic). An interior region of diminished pressure advances from radius n at time 11 to r2 at t2. Within this region, gas falls onto the central protostar of growing mass.
The freely falling portion of the cloud has a structure determined by the strong gravity of the protostar, rather than by conditions prior to collapse. If we focus on some fixed volume relatively close to the star, gas crosses this region in an interval brief compared with the evolutionary time scale, M*/M. Since there is no chance for appreciable mass buildup in any such volume, we may ignore the left-hand side of the continuity equation (10.27) and conclude that r2 pu is a spatial constant throughout the collapsing interior. Setting u = -Vff and utilizing equation (10.30), we solve for the density to find
For comparison, we recall from § 9.1 that all spherical equilibria have p declining as r-2 in their outer regions. Thus, both the density and pressure profiles become flatter in the region of collapse, as illustrated in Figure 10.7.
The numerical calculations we have discussed assume spherical symmetry not only in the initial configuration, but at all subsequent times. It is not difficult to relax the second restriction. That is, one still begins with a spherical, thermally supported cloud, but now follows its collapse with the full, three-dimensional equations of mass continuity (3.7) and momentum conservation
(3.3), together with Poisson's equation (9.3) for the gravitational potential. The result is that the cloud evolution is virtually unchanged. In three dimensions, small, localized density enhancements inevitably arise. Once these enter the collapsing region, however, the straining motion induced by the protostar's gravity (i. e., the increase of \ Vff \ with decreasing r) tends to pull them apart. Inside-out collapse is thus stable against fragmentation. The situation is quite different for clouds that are initially far out of force balance, as we discuss in Chapter 12.2
One reason that pressure is ultimately ineffective in halting collapse is that the gas temperature has been assumed constant. Building up an adverse pressure gradient thus requires a steep inward rise of density. High density, on the other hand, only enhances the effect of self-gravity. It is for this reason, of course, that isothermal equilibria can only tolerate a modest density contrast before they are unstable to collapse.
How realistic, though, is the isothermal assumption? In the case of hydrostatic configurations, we have seen that the temperature responds rather sluggishly to cosmic-ray heating because of efficient cooling by CO and dust grains (recall Figure 8.6). A fluid element within a collapsing cloud has two additional sources of energy input. One is the compressional work performed by the surrounding gas. Here, the power input per unit volume is r comp
where we have assumed steady-state flow in the second form of this relation.
Suppose we now utilize p(r) from equation (10.34) to evaluate rcomp. Then, at radii where this rate is appreciable, we find it is overwhelmed by the second new source of energy, the radiation from the protostar and its surrounding disk. This luminosity stems from the kinetic energy of infall and is generated at the stellar and disk surfaces (Chapter 11). It is the dust grains within the flow that are actually irradiated and they respond, as usual, by emitting their own infrared photons. The temperature of the infalling envelope does not climb steeply until the ambient density is large enough to trap this cooling radiation. As we will see in § 11.1, such trapping occurs at a radius of roughly 1014 cm. The gas at this distance is already traveling at such high speed that the infall process cannot be impeded. Thus, the departures from isothermality, while both interesting physically and critical observationally, do not affect the overall dynamics of inside-out collapse.
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