## Susceptibility to Collapse

Figure 9.8 Nondimensional cloud mass as a function of density contrast in rotating, isothermal clouds. The two curves correspond to the indicated values of the rotational parameter ¡3.

If we keep 3 fixed at 0.16 and consider models of increasing density contrast, the mass m first rises and then turns over (Figure 9.8). As before, the peak marks the onset of gravitational instability. To see why, we again select two models with the same m-value close to, but on either side of, the maximum. At fixed aT and PD, these clouds have the same dimensional mass. Hence, the spherical reference states from which they contracted had the same radius Ra and, from equation (9.21), the same Qa. It follows from equation (9.30) that their j-distributions are also identical. The two clouds are thus the extreme states in an oscillation of zero frequency. We see from Figure 9.8 that the model with p/pc = 34.0 and m = 2.42 is just on the edge of instability. Hence the one pictured in Figure 9.7a is indeed stable, although not by much. Figure 9.7b shows the unstable model with the same 3 and m, but with the higher density contrast of pc/pc = 100. If we were to use this configuration as the initial state in the full dynamical equations, rather than just those for hydrostatic balance, the cloud would collapse within a few free-fall times.

Figure 9.8 also shows the mass variation for 3 = 0.33. It is apparent that all such curves exhibit a similar stability transition. A useful means of summarizing these results is to plot the peak mass, which we will denote mcrit, as a function of 3 (Figure 9.9). The quantity mcrit, a generalized, nondimensional Jeans mass, starts at the Bonnor-Ebert value mi = 1.18 for 3 = 0 and monotonically rises with 3. This steady increase demonstrates quantitatively how rotation tends to stabilize a cloud against gravitational collapse. The most massive stable cloud, that b 11- 