The final ingredient needed to construct protostar models is the mass accretion rate M. This quantity enters the boundary conditions of equations (11.23) and (11.25), and tells us, of course, how much to increase M* from one time step to the next. Ideally, one should take this rate directly from collapse calculations, such as the ones depicted in Figure 10.6. Thus far, however, the most detailed studies of protostar interiors have considered only constant rates, effectively treating M as a free parameter. The range of this parameter follows by inserting plausible cloud temperatures (or, equivalently, sound speeds), into equation (10.31). For cloud temperatures from 10 to 20 K, one finds that M should span an order of magnitude, from about 10-6 to 10-5 Mq yr-1.
1 In low-mass protostars, the precursor is opaque to X-rays generated behind the shock, but transparent to the optical photons emitted locally. Thus, equation (11.8) provides only a rough approximation to the gas temperature in this region. The precursors in higher-mass protostars are opaque to both the X-rays and to their own cooling radiation.
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