Boundary Conditions

Solution of the stellar structure equations requires specification of four boundary conditions. Two of these are the statements that both r (Mr) and Lint (Mr) vanish at the center of the configuration:

A third condition is that P (Mr) must equal the appropriate postshock value when Mr = Mt. This value is the ram pressure due to infalling matter, given by pu2. Using equation (10.34) for p just outside the shock and (11.6) for u = —Vff, we find

The fourth boundary condition concerns the surface value of the temperature and its relation to the luminosity. For a main-sequence star, this relation would be the standard photospheric one given by equation (1.5). The total luminosity of a protostar, however, is that released by accretion plus the amount radiated from the interior:

Here, Lpost is the value of Lint obtained by integration of equation (11.19) from Mr =0 to M„ and refers to the inner border of the postshock relaxation region (recall Figure 8.10). Let Tpost denote the corresponding temperature, as found by outward integration of (11.17). Note that this value is much higher than that indicated in Figure 8.10, which concerns the surface layers of molecular clouds rather than optically thick stars. In any case, our task now is to relate Tpost to Lpost and L

acc shock front

Jout

interior hot postshock radiative region gas precursor

Figure 11.5 Contributions to the luminosity of a low-mass protostar. The two regions in front of and behind the accretion shock front (thick vertical line) have been separated for clarity. The net surface luminosity, called Lpost in the text, is the sum of Lout, the outward contribution from the interior, and inward contributions in X-rays and optical photons from the hot, postshock gas and radiative precursor, respectively.

We first remark that Lpost is actually the sum of inward and outward contributions, as illustrated in Figure 11.5. It is only the outward luminosity, stemming from the deeper interior, that is given by the blackbody formula 4nRlaBTpost. This contribution is denoted Lout in the figure. The hot gas immediately behind the shock emits its soft X-rays isotropically. Thus, the inward contribution to Lpost includes Lacc/2 in X-rays. Additionally, there are optical photons stemming from the precursor. Assuming that this layer, which completely absorbs the X-rays entering it, also radiates equally in both directions, the postshock point receives an additional Lacc/4 in optical photons.1 In summary, we find that

which is the desired boundary condition.

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