Although the main accretion phase involves dynamical motion, the protostar itself is always in a state of hydrostatic equilibrium. Its central region is also hot enough to power nuclear fusion, at least for part of the time. What distinguishes this entity from more mature objects is the rapid, continual addition of fresh material from the surrounding cloud. With the end of infall, this critical difference vanishes. The further evolution of the star, now of fixed mass, is driven almost entirely by radiation from its surface layers. Let us now examine more closely the course of events.

Radiation drains away internal energy. Since the object is gravitationally bound, its total energy, Etot, is negative and becomes more so with the passage of time. The virial theorem tells us, however, that the thermal contribution, U, is -Etot. Thus, U actually increases, as does the internal temperature. A pre-main-sequence star is therefore an object with negative heat capacity, one whose temperature rises as a result of heat loss. This behavior, odd by terrestrial standards, occurs because of the increased gravitational binding.

The energy loss from radiation causes the star to shrink with the characteristic time scale tKH . For the evolution to be quasi-static, the stellar interior must continually readjust to maintain force balance. This readjustment is achieved through pressure disturbances and takes place over the sound travel time, ts = R*/as, where as is an appropriately averaged sound speed. The justification for treating the star at each instant as a hydrostatic object is that tKH > ts throughout the pre-main-sequence phase.

It is worthwhile to check this inequality through a numerical example. A 1 Mq star near the birthline has, according to Table 16.1, a radius of 4.92 Rq. From equation (11.2a), the volume-averaged sound speed is

IjJZT

hGM*

which is 150 km s 1 for y = 5/3. Thus, ts is only 6.5 hours .Since L* = 7.1 Lq, theKelvin-Helmholtz time is 8.7 x 105 years. Moreover, the ratio tKH/ts varies as R-5/2 and so increases even more as contraction proceeds.

Numerical calculation of pre-main-sequence evolution utilizes the four stellar structure equations, along with the equation of state for an ideal gas, equation (11.16). As developed a s in Chapter 11, the first three structure equations are dr 1

dMr dP ~dM~r dLint 8Mr

4nr2 p GMr

4 nr4

where we have suppressed all subscripts in the partial derivatives. In radiatively stable regions of the star, the fourth equation is

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