[en] The 1997 and 1998 studies by Truelove and colleagues introduced the Jeans condition as a necessary condition for avoiding artificial fragmentation during protostellar collapse calculations. They found that when the Jeans condition was properly satisfied with their adaptive mesh refinement (AMR) code, an isothermal cloud with an initial Gaussian density profile collapsed to form a thin filament rather than the binary or quadruple protostar systems found in previous calculations. Using a completely different self-gravitational hydrodynamics code introduced by Boss and Myhill in 1992 (B and M), we present here calculations that reproduce the filamentary solution first obtained by Truelove et al. in 1997. The filamentary solution only emerged with very high spatial resolution with the B and M code, with effectively 12,500 radial grid points (R12500). Reproducing the filamentary collapse solution appears to be an excellent means for testing the reliability of self-gravitational hydrodynamics codes, whether grid-based or particle-based. We then show that in the more physically realistic case of an identical initial cloud with nonisothermal heating (calculated in the Eddington approximation with code B and M), thermal retardation of the collapse permits the Gaussian cloud to fragment into a binary protostar system at the same maximum density where the isothermal collapse yields a thin filament. However, the binary clumps soon thereafter evolve into a central clump surrounded by spiral arms containing two more clumps. A roughly similar evolution is obtained using the AMR code with a barotropic equation of state--formation of a transient binary, followed by decay of the binary to form a central object surrounded by spiral arms, though in this case the spiral arms do not form clumps. When the same barotropic equation of state is used with the B and M code, the agreement with the initial phases of the AMR calculation is quite good, showing that these two codes yield mutually consistent results. However, the B and M barotropic result differs significantly from the B and M Eddington result at the same maximum density, demonstrating the importance of detailed radiative transfer effects. Finally, we confirm that even in the case of isothermal collapse, an initially uniform density sphere can collapse and fragment into a binary system, in agreement with the 1998 results of Truelove et al. Fragmentation of molecular cloud cores thus appears to remain as a likely explanation of the formation of binary stars, but the sensitivity of these calculations to the numerical resolution and to the thermodynamical treatment demonstrates the need for considerable caution in computing and interpreting three-dimensional protostellar collapse calculations. (c) (c) 2000. The American Astronomical Society

[en] Core accretion and disk instability require giant protoplanets to form in the presence of disk gas. Protoplanet migration models generally assume disk masses low enough that the disk's self-gravity can be neglected. However, disk instability requires a disk massive enough to be marginally gravitationally unstable (MGU). Even for core accretion, an FU Orionis outburst may require a brief MGU disk phase. We present a new set of three-dimensional, gravitational radiation hydrodynamics models of MGU disks with multiple protoplanets, which interact gravitationally with the disk and with each other, including disk gas mass accretion. Initial protoplanet masses are 0.01 to 10 M _⊕ for core accretion models, and 0.1 to 3 M _Jup for Nice scenario models, starting on circular orbits with radii of 6, 8, 10, or 12 AU, inside a 0.091 M _☉ disk extending from 4 to 20 AU around a 1 M _☉ protostar. Evolutions are followed for up to ∼4000 yr and involve phases of relative stability (e ∼ 0.1) interspersed with chaotic phases (e ∼ 0.4) of orbital interchanges. The 0.01 to 10 M _⊕ cores can orbit stably for ∼1000 yr: monotonic inward or outward orbital migration of the type seen in low mass disks does not occur. A system with giant planet masses similar to our solar system (1.0, 0.33, 0.1, 0.1 M _Jup) was stable for over 1000 yr, and a Jupiter-Saturn-like system was stable for over 3800 yr, implying that our giant planets might well survive an MGU disk phase.

[en] Observations support the hypothesis that gas disk gravitational instability might explain the formation of massive or wide-orbit gas giant exoplanets. The situation with regard to Jupiter-mass exoplanets orbiting within ∼20 au is more uncertain. Theoretical models yield divergent assessments often attributed to the numerical handling of the gas thermodynamics. Boss used the β cooling approximation to calculate three-dimensional hydrodynamical models of the evolution of disks with initial masses of 0.091 M _⊙ extending from 4 to 20 au around 1 M _⊙ protostars. The models considered a wide range (1–100) of β cooling parameters and started from an initial minimum Toomre stability parameter of Q _i = 2.7 (gravitationally stable). The disks cooled down from initial outer disk temperatures of 180 K to as low as 40 K as a result of the β cooling, leading to fragmentation into dense clumps, which were then replaced by virtual protoplanets (VPs) and evolved for up to ∼500 yr. The present models test the viability of replacing dense clumps with VPs by quadrupling the spatial resolution of the grid once dense clumps form, sidestepping in most cases VP insertion. After at least ∼200 yr of evolution, the new results compare favorably with those of Boss: similar numbers of VPs and dense clumps form by the same time for the two approaches. The results imply that VP insertion can greatly speed disk instability calculations without sacrificing accuracy.

[en] A long-standing problem in the collisional accretion of terrestrial planets is the possible loss of m-size bodies through their inward migration onto the protostar as a result of gas drag forces. Such inward migration can be halted, and indeed even reversed, in a protoplanetary disk with local pressure maxima, such as marginally gravitationally unstable (MGU) phases of evolution, e.g., FU Orionis events. Results are presented for a suite of three-dimensional models of MGU disks extending from 1 to 10 AU and containing solid particles with sizes of 1 cm, 10 cm, 1 m, or 10 m, subject to disk gas drag and gravitational forces. These hydrodynamical models show that over disk evolution time scales of $\sim <!--\; ???\; -->6\times <!--\; ??\; -->{10}^3$ years or longer, during which over half the gaseous disk mass is accreted by the protostar, very few 1 and 10 m bodies are lost through inward migration: most bodies survive and orbit stably in the outer disk. A greater fraction of 1 and 10 cm particles are lost to the central protostar during these time periods, as such particles are more closely tied to the disk gas accreting onto the protostar, but even in these cases, a significant fraction survive and undergo transport from the hot inner disk to the cold outer disk, perhaps explaining the presence of small refractory particles in Comet Wild 2. Evidently MGU disk phases offer a means to overcome the m-sized migration barrier to collisional accumulation.

[en] Analyses of primitive meteorites and cometary samples have shown that the solar nebula must have experienced a phase of large-scale outward transport of small refractory grains as well as homogenization of initially spatially heterogeneous short-lived isotopes. The stable oxygen isotopes, however, were able to remain spatially heterogeneous at the ∼6% level. One promising mechanism for achieving these disparate goals is the mixing and transport associated with a marginally gravitationally unstable (MGU) disk, a likely cause of FU Orionis events in young low-mass stars. Several new sets of MGU models are presented that explore mixing and transport in disks with varied masses (0.016 to 0.13 M_☉) around stars with varied masses (0.1 to 1 M_☉) and varied initial Q stability minima (1.8 to 3.1). The results show that MGU disks are able to rapidly (within ∼10⁴ yr) achieve large-scale transport and homogenization of initially spatially heterogeneous distributions of disk grains or gas. In addition, the models show that while single-shot injection heterogeneity is reduced to a relatively low level (∼1%), as required for early solar system chronometry, continuous injection of the sort associated with the generation of stable oxygen isotope fractionations by UV photolysis leads to a sustained, relatively high level (∼10%) of heterogeneity, in agreement with the oxygen isotope data. These models support the suggestion that the protosun may have experienced at least one FU Orionis-like outburst, which produced several of the signatures left behind in primitive chondrites and comets

[en] Forming giant planets by disk instability requires a gaseous disk that is massive enough to become gravitationally unstable and able to cool fast enough for self-gravitating clumps to form and survive. Models with simplified disk cooling have shown the critical importance of the ratio of the cooling to the orbital timescales. Uncertainties about the proper value of this ratio can be sidestepped by including radiative transfer. Three-dimensional radiative hydrodynamics models of a disk with a mass of 0.043 M_sun from 4 to 20 AU in orbit around a 1 M_sun protostar show that disk instabilities are considerably less successful in producing self-gravitating clumps than in a disk with twice this mass. The results are sensitive to the assumed initial outer disk (T_o ) temperatures. Models with T_o = 20 K are able to form a single self-gravitating clump, whereas models with T_o = 25 K form clumps that are not quite self-gravitating. These models imply that disk instability requires a disk with a mass of at least ∼0.043 M_sun inside 20 AU in order to form giant planets around solar-mass protostars with realistic disk cooling rates and outer-disk temperatures. Lower mass disks around solar-mass protostars must rely upon core accretion to form inner giant planets.

[en] Recent meteoritical analyses support an initial abundance of the short-lived radioisotope (SLRI) ⁶⁰Fe that may be high enough to require nucleosynthesis in a core-collapse supernova, followed by rapid incorporation into primitive meteoritical components, rather than a scenario where such isotopes were inherited from a well-mixed region of a giant molecular cloud polluted by a variety of supernovae remnants and massive star winds. This paper continues to explore the former scenario, by calculating three-dimensional, adaptive mesh refinement, hydrodynamical code (FLASH 2.5) models of the self-gravitational, dynamical collapse of a molecular cloud core that has been struck by a thin shock front with a speed of 40 km s⁻¹, leading to the injection of shock front matter into the collapsing cloud through the formation of Rayleigh–Taylor fingers at the shock–cloud intersection. These models extend the previous work into the nonisothermal collapse regime using a polytropic approximation to represent compressional heating in the optically thick protostar. The models show that the injection efficiencies of shock front materials are enhanced compared to previous models, which were not carried into the nonisothermal regime, and so did not reach such high densities. The new models, combined with the recent estimates of initial ⁶⁰Fe abundances, imply that the supernova triggering and injection scenario remains a plausible explanation for the origin of the SLRIs involved in the formation of our solar system.

[en] Doppler surveys have shown that more massive stars have significantly higher frequencies of giant planets inside ∼3 AU than lower mass stars, consistent with giant planet formation by core accretion. Direct imaging searches have begun to discover significant numbers of giant planet candidates around stars with masses of ∼1 M_sun to ∼2 M_sun at orbital distances of ∼20 AU to ∼120 AU. Given the inability of core accretion to form giant planets at such large distances, gravitational instabilities of the gas disk leading to clump formation have been suggested as the more likely formation mechanism. Here, we present five new models of the evolution of disks with inner radii of 20 AU and outer radii of 60 AU, for central protostars with masses of 0.1, 0.5, 1.0, 1.5, and 2.0 M_sun, in order to assess the likelihood of planet formation on wide orbits around stars with varied masses. The disk masses range from 0.028 M_sun to 0.21 M_sun, with initial Toomre Q stability values ranging from 1.1 in the inner disks to ∼1.6 in the outer disks. These five models show that disk instability is capable of forming clumps on timescales of ∼10³ yr that, if they survive for longer times, could form giant planets initially on orbits with semimajor axes of ∼30 AU to ∼70 AU and eccentricities of ∼0 to ∼0.35, with initial masses of ∼1 M_Jup to ∼5 M_Jup, around solar-type stars, with more protoplanets forming as the mass of the protostar (and protoplanetary disk) is increased. In particular, disk instability appears to be a likely formation mechanism for the HR 8799 gas giant planetary system.

[en] Observational evidence exists for the formation of gas giant planets on wide orbits around young stars by disk gravitational instability, but the roles of disk instability and core accretion for forming gas giants on shorter period orbits are less clear. The controversy extends to population synthesis models of exoplanet demographics and to hydrodynamical models of the fragmentation process. The latter refers largely to the handling of radiative transfer in three-dimensional (3D) hydrodynamical models, which controls heating and cooling processes in gravitationally unstable disks, and hence dense clump formation. A suite of models using the β cooling approximation is presented here. The initial disks have masses of 0.091 M _⊙ and extend from 4 to 20 au around a 1 M _⊙ protostar. The initial minimum Toomre Qi values range from 1.3 to 2.7, while β ranges from 1 to 100. We show that the choice of Q _i is equal in importance to the β value assumed: high Q_i disks can be stable for small β , when the initial disk temperature is taken as a lower bound, while low Q_i disks can fragment for high β . These results imply that the evolution of disks toward low Q_i must be taken into account in assessing disk fragmentation possibilities, at least in the inner disk, i.e., inside about 20 au. The models suggest that if low Q_i disks can form, there should be an as yet largely undetected population of gas giants orbiting G dwarfs between about 6 au and 16 au.

[en] Short-lived radioisotopes (SLRIs) such as ⁶⁰Fe and ²⁶Al were likely injected into the solar nebula in a spatially and temporally heterogeneous manner. Marginally gravitationally unstable (MGU) disks, of the type required to form gas giant planets, are capable of rapid homogenization of isotopic heterogeneity as well as of rapid radial transport of dust grains and gases throughout a protoplanetary disk. Two different types of new models of an MGU disk in orbit around a solar-mass protostar are presented. The first set has variations in the number of terms in the spherical harmonic solution for the gravitational potential, effectively studying the effect of varying the spatial resolution of the gravitational torques responsible for MGU disk evolution. The second set explores the effects of varying the initial minimum value of the Toomre Q stability parameter, from values of 1.4 to 2.5, i.e., toward increasingly less unstable disks. The new models show that the basic results are largely independent of both sets of variations. MGU disk models robustly result in rapid mixing of initially highly heterogeneous distributions of SLRIs to levels of ∼10% in both the inner (<5 AU) and outer (>10 AU) disk regions, and to even lower levels (∼2%) in intermediate regions, where gravitational torques are most effective at mixing. These gradients should have cosmochemical implications for the distribution of SLRIs and stable oxygen isotopes contained in planetesimals (e.g., comets) formed in the giant planet region (∼5 to ∼10 AU) compared to those formed elsewhere.