Evolution of the Solar Nebula. II. Thermal Structure during Nebula Formation

Models of the thermal structure of protoplanetary disks are required for understanding the physics and chemistry of the earliest phases of planet formation. Numerical hydrodynamical models of the protostellar collapse phase have not been evolved far enough in time to be relevant to planet formation, i.e., to a relatively low-mass disk surrounding a protostar. One simplification is to assume a pre-existing solar-mass protostar, and calculate the structure of just the disk as it forms from the highest angular momentum vestiges of the placental cloud core. A spatially second-order accurate, axisymmetric (two-dimensional), radiative hydrodynamics code has been used to construct three sets of protoplanetary disk models under this assumption. Because compressional heating has been included, but not viscous or other heating sources, the model temperatures obtained should be considered lower bounds. The first set started from a spherically symmetric configuration appropriate for freely falling gas: ρ ∝ r^-3/2, υ_r ∝ r^-1/2, but with rotation (Ω ∝ r^-1, where r is the spherical coordinate radius). These first models turned out to be unsatisfactory because in order to achieve an acceptable mass accretion rate onto the protostar (M^ṡ ≤ 10^-5 M_sun yr^-1 for low-mass star formation), the disk mass became much too small (∼ 0.0002 M_sun). The second set improved on the first set by ensuring that the late-arriving, high angular momentum gas did not accrete directly onto the protosun. By starting from a disklike cloud flattened about the equatorial plane and flowing vertically toward the midplane, these models led to M^ṡ → 0, as desired. However, because the initial cloud was not chosen to be close to equilibrium, the disk rapidly contracted vertically, producing an effective disk mass accretion rate M^ṡ_d ∼ 10^-2 M_sun yr^-1, again too high. Hence, the third (and most realistic) set started from an approximate equilibrium state for an adiabatic, self-gravitating "fat" Keplerian disk, with surface density σ ∝ r^-1/2, surrounded by a much lower density "halo" infalling onto the disk. This initial condition produced M^ṡ_s → 0 and M^ṡ_d ∼ 10^-6 to 10^-5 M_sun yr^-1, as desired. The resulting nebula temperature distributions show that midplane temperatures of at least 1000 K inside 2.5 AU, falling to around 100 K outside 5 AU, are to be expected during the formation phase of a minimum mass nebula containing ∼0.02 M_sun within 10 AU. This steady state temperature distribution appears to be consistent with cosmochemical evidence which has been interpreted as implying a phase of relatively high temperatures in the inner nebula. The temperature distribution also implies that the nebula would be cool enough outside 5 AU to allow ices to accumulate into planetesimals even at this relatively early phase of nebula evolution.