Figure 11.1 Evolution of central temperature in the first core. The temperature is plotted as a function of central density.

infrared, cooling radiation. Further compression then causes its internal temperature to rise steadily. The enhanced pressure decelerates material drifting inward, which settles gently onto the hydrostatic structure. The settling gas can still radiate rather freely in the infrared, at least before it is smothered by successive layers of incoming matter. This energy loss from the outer skin then further enhances compression. The calculations show, in fact, that the core eventually stops expanding and begins to shrink, even as fresh material continues to arrive. The total compressed mass is still small at this stage, about 5 x 10-2 Me, but the radius is large by stellar standards, roughly 5 AU (8 x 1013 cm).

The interior of the central object, like its surroundings, consists mostly of molecular hydrogen. This fact alone seals the fate of the first core and ensures its early collapse. To see why, let us first estimate the mean internal temperature, utilizing the virial theorem in the version of equation (3.16). The object builds up from that portion of the parent cloud that was least supported rotationally and magnetically. We therefore tentatively ignore both the bulk kinetic energy T and the magnetic term M in the virial theorem. We further approximate the gravitational potential energy W as -GM2/R, for a core of mass M and radius R. The internal energy becomes where T and p are the volume-averaged temperature and molecular weight, respectively. Ap-

plying equation (3.16) and solving for the temperature, we find

Here we have set p equal to 2.4, the value appropriate for molecular gas.

The internal temperature, while very low compared to true stars, is higher than in quiescent molecular clouds, as is the average mass density, which is now of order 10-10 g cm-3. With the addition of mass and shrinking of the radius, T soon surpasses 2000 K, and collisional dissociation of H2 begins. At this point, the temperature starts to level off. The effect is evident in Figure 11.1, which tracks the temperature as a function of density at the center. Viewing the situation energetically, we note that the number of H2 molecules in the core is XM/2mH, where X = 0.70 is the interstellar hydrogen mass fraction. From equation (11.1), the thermal energy per molecule is therefore 3kBT/X, or 0.74 eV when T = 2000 K. This figure is small compared to the 4.48 eV required to dissociate a single molecule. During the transition epoch, therefore, even a modest rise in the fraction of dissociated hydrogen absorbs most of the com-pressional work of gravity, without a large increase in temperature.

As the density of the first core keeps climbing, the region containing atomic hydrogen spreads outward from the center. We recall from ยง 9.1 that purely isothermal configurations can tolerate only a modest density contrast before they become gravitationally unstable. The reason is that the compression arising from any perturbations can no longer be effectively opposed by a rise in the internal pressure, once the temperature is held fixed. The interior temperature of the first core is not a fixed constant, but its rise is severely damped by the dissociation process. Hence, the partially atomic region can only spread and increase its mass by a limited amount before the entire configuration becomes unstable and collapses. This event marks the end of the first core.

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