and M is the energy associated with the magnetic field:
In writing equation (3.8), we have ignored a number of surface integrals, including one representing the effect of any external pressure. (See equation (D.12) in Appendix D.) Such an approximation would not be justified for HI or diffuse clouds, but is valid for the strongly self-gravitating giant complexes. The issue here is which of the terms in (3.8) can balance the gravitational binding energy W. Note that the integrals U, M, and T are all positive, while W is negative. If none of these three terms can match W in magnitude, then a typical giant molecular cloud would be in a state of gravitational collapse.
Let us briefly explore this latter possibility. We denote by R = L/2 the characteristic "radius" for a cloud of mass M. Then, to within a factor of order unity, W is -GM2/R. Equation (3.8) reduces in this case to
If we further approximate I as MR2, then dimensional analysis of this relation tells us that R in the collapsing cloud shrinks by roughly a factor of two over a characteristic free-fall time tff. This is approximately
where we have inserted representative numbers from Table 3.1. Note further that, since M/R3 « p, the time scale can also be written as (Gp)-1/2. It is conventional to define tff precisely as
This expression actually gives the time for a homogeneous sphere with zero internal pressure to collapse to a point (Chapter 12).
Equation (3.14) indicates that giant molecular clouds have free-fall times comparable to their observed lifetimes. Does this mean that the clouds are indeed collapsing? One issue here is our questionable use of a global, volume-averaged density in equation (3.14). The higher density appropriate for an individual clump would have yielded a tff-value closer to 106 yr. In any case, there is no convincing empirical evidence for large-scale shrinking or flattening over such an interval. The complexes' internal velocities also appear to be random, not systematically directed toward a collapsing center. Apparently, the entities survive until they are destroyed from within, by the massive stars they spawn.
If the complexes are in approximate force balance over their lifetimes, we may actually ignore the left side in equation (3.8) and obtain the form of the virial theorem appropriate for longterm stability:
For a cloud in such "virial equilibrium," the question is again how to balance W. In order to gauge the effectiveness of the internal pressure in supporting the cloud, we first note that U is given, in the spirit of our approximations, by MRT/p, where T is a representative gas temperature. We then form the ratio
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