The heating and vaporization of dust grains in the protosolar environment is modeled in order to assess the survivability of interstellar solids during the formation of the solar system. A multidimensional, discrete ordinate radiative transfer code is used to compute thermal transport in the collapsing protosolar cloud. The results are combined with estimates of heating at the shock where infalling material arrives at the surface of the solar nebula/accretion disk, and in the interior of the disk, to determine the distances at which various solid phases are vaporized. The thermal coupling between the envelope and the accretion disk (backheating) is treated self-consistently, so its effect on the disk's radial temperature profile is included. This treatment also permits evaluation of the effect of backheating on the observational inference of disk properties. Calculations are performed for various values of cloud collapse rate, rotation rate, and disk accretion rate. The latter factor is the main determinant of the total luminosity, and we consider both "low-luminosity" cases, in which disk accretion is inefficient, and high-luminosity" cases, in which disk accretion keeps pace with cloud collapse. We also examine situations in which a polar, optically thin cavity is swept clear by a protosolar wind. We conclude that refractory grains, such as silicates, can generally survive the envelope and accretion shock, and enter the nebula at or within 1 AU. Inside the nebula, their vaporization distances are controlled by the disk accretion rate and optical depth. In contrast, the vaporization distances of volatiles such as water ice are sensitive to envelope conditions, which control the thermal state of the outer, optically thin regions of the disk. The ice vaporization distance lies between about 2 and 30 AU, depending on the total source luminosity and characteristics of the collapsing cloud. Moderately volatile organics (methanol, formaldehyde, and polymerized formaldehyde) may survive as solids in the terrestrial planet region; they generally are not vaporized outside of several AU, which supports the idea that comets inherit this material from the parent molecular cloud.
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