Protostellar Collapse Research Articles

Protostellar Collapse and Fragmentation: Describing and Testing a Second‐Order–accurate Radiation Hydrodynamic Code

This paper deals with the description and testing of the numerical methods employed in the construction of a new multidimensional, explicit, radiation hydrodynamic code for studying protostellar collapse and fragmentation. The code is written in Eulerian spherical coordinates, and the determining equations are solved using second-order-accurate, finite-difference (FD) methods on a radially moving, nonstaggered grid. The set of hydrodynamic equations is advanced using a multistep solution procedure along with either a first-order or a second-order time-integration scheme. On a fixed grid, the code is effectively second-order accurate as demonstrated through convergence testing. For advection on a moving orthogonal mesh, however, the use of standard FD methods results in systematic errors that progressively affect the accuracy of the solution. In order to minimize these errors, a new FD formulation has been derived for both the temporally first-order and second-order versions of the code that yields solutions for the pressureless collapse test case that are numerically invariant under grid motion. Results are also presented for a number of isothermal and nonisothermal protostellar collapse tests that validate this conclusion. Within this new formulation, the radial dependence of the fluid variables is exactly preserved near the origin of the coordinate system. This aspect results in improved local conservation of angular momentum during axisymmetric collapse and in a significant decrease of the level of nonaxisymmetric noise as compared with a previous version of the code. The reduction of systematic sources of error in the new schemes should then allow for a more precise evaluation of the importance of fragmentation during protostellar collapse.

Collapse and Fragmentation Models of Prolate Molecular Cloud Cores. II. Initial Differential Rotation

The prevalence of companions to pre-main-sequence stars and the emerging observational evidence for binary and multiple protostellar condensations suggest that fragmentation during protostellar collapse is a mechanism that may occur frequently in the star formation process. Here a second-order accurate hydrodynamic code has been used to investigate the gravitational (postmagnetic) collapse and fragmentation of low-mass (~1 M☉), small (~0.05 pc) molecular cloud cores, starting from moderately centrally condensed (Gaussian), prolate (2:1 and 4:1 axial ratios) configurations with varying thermal energies (α) and degrees of differential rotation (ν = 1/3 and 2/3). To facilitate comparisons with previous collapse calculations of uniformly rotating prolate cloud cores (Sigalotti & Klapp), all the models were made to start with a ratio of rotational to gravitational energy of β ≈ 0.036. The results indicate that prolate clouds are highly susceptible to binary fragmentation and that with respect to uniformly rotating initial conditions, differential rotation plays no role in either determining or enhancing fragmentation in initially slowly rotating clouds. In contrast to the fragmentation criteria previously established by Boss and Myhill, the results also indicate that clouds with α = 0.56 and varied prolateness collapse in a similar fashion, producing intermediate central condensations of oblate spheroidal shape before fragmenting into either a binary (2:1 clouds) or multiple protostellar core (4:1 clouds). The models with α ≤ 0.45 all produced binary systems after having formed intermediate central condensations, which might be of prolate ellipsoidal (2:1 clouds) or narrow cylindrical (4:1 clouds) shape. The mass and separation of the binary fragments increase with decreasing α and with an increase of both the degree of differential rotation and the cloud elongation. The results imply that for initial low β, the degree of cloud prolateness has a greater effect on the outcome than does differential rotation.

Diamagnetic effects during the early stages of star formation

The role of magnetic fields during the initial stages of protostellar cloud collapse is investigated, in particular with respect to the scenario where the magnetic force can be an effective compressor due to diamagnetic effects. A multifluid approach involving electrons, ions of different atomic masses and neutrals is adopted, where each species is treated separately. The electron fluid is compressed by the magnetic pressure force, and the other ion species are pulled by the collective electric field developed by the space charge separation. The neutrals are also dragged owing to collisions with the ions. The difference in charge-to-mass ratio ensures that each ion species is accelerated differently, resulting in a distribution following their atomic masses. This model explores the scenario where the electromagnetic forces can achieve a supercritical mass-to-flux ratio in a magnetized cloud before dynamical collapse due to gravity takes over.

Millimeter Interferometry of Class 0 Sources: Rotation and Infall Towards L1448N

In order to study the kinematics of protostellar collapse we present high-spatial resolution millimeter interferometer data of the class 0 source L1448 N. The L1448 N cloud core has fragmented into two rapidly rotating subcores containing three protostars. One subcore is a protobinary system with 2000 AU separation. Strong evidence for infall is seen towards the N(B) component.

The Evolution of Flows and Protostars

In this paper, I summarize the key properties of Class 0 protostars, discuss the observational evidence for a decline of outflow/inflow power with evolutionary stage, point out a possible connection with the initial conditions of fast protostellar collapse, and suggest a new collapse scenario in regions of induced, multiple star formation. The main thesis put forward here is that Class 0 objects have unusually powerful outflows because they accrete at a substantially higher rate than their Class I descendants.

Star formation and the singular isothermal sphere

We caution against approximating the initial conditions for protostellar collapse with a singular isothermal sphere. First, it is very unlikely that nature can assemble anything like a singular isothermal sphere. Secondly, collapse of a singular isothermal sphere would seem to be severely prejudiced against binary formation - either by fragmentation during collapse, or by disc instability following collapse. It is therefore appropriate to explore star formation paradigms which (a) involve initial conditions which are far less focused than the singular isothermal sphere, and (b) take into account impulsive dynamical and thermodynamic processes - reflecting the rapidly varying environments in which stars are formed. Even in relatively quiescent regions like Taurus, such processes must be reckoned with; they cannot realistically be relegated to the status of small perturbations on an essentially quasistatic theme.

Sheet Models of Protostellar Collapse

view Abstract Citations (77) References (60) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Sheet Models of Protostellar Collapse Hartmann, Lee ; Calvet, Nuria ; Boss, Alan Abstract Recognizing that protostellar clouds are unlikely to be completely spherical, we explore some effects of initial cloud geometry by considering collapse from a sheet initially in hydrostatic equilibrium. A qualitatively different feature of sheet collapse compared with spherical contraction is the development of relatively evacuated cavities in the infalling dusty cloud, which arise because material falls in first along the shortest dimension to the central gravitating mass. Using analytic models of collapse, which reproduce the main features of our previous numerical time-dependent simulations, we perform detailed radiative transfer calculations, which suggest that these collapse cavities can naturally explain the morphological appearance of many reflection nebulae around young stars on small distance scales without requiring initially diverging outflows. Sheet collapse models can simultaneously explain small-scale reflection nebula morphologies and dust envelope emission properties of many young stellar objects more easily than the standard spherical collapse models. The sheet collapse picture suggests that protostars, i.e., young stellar objects still accreting a large fraction of their mass from infalling envelopes, may be optically visible over a substantial range of system inclinations to the line of sight. These results may be especially relevant to cases where fragmentation and collapse has been triggered by an external impulse, such as a shock wave. We show how many properties of the flat-spectrum T Tauri star HL Tau can be interpreted in terms of flattened protostellar cloud collapse and draw some distinctions between the flattened toroids resulting in our calculations and the "pseudodisk" of Galli & Shu. Publication: The Astrophysical Journal Pub Date: June 1996 DOI: 10.1086/177330 Bibcode: 1996ApJ...464..387H Keywords: STARS: CIRCUMSTELLAR MATTER; STARS: FORMATION; STARS: PRE-MAIN-SEQUENCE full text sources ADS | data products SIMBAD (4)

Wave Motions in Molecular Clouds: Results in Two Dimensions

We study the linear evolution of small perturbations in self-gravitating fluid systems in two spatial dimensions; we consider both cylindrical and cartesian (i.e., slab) geometries. The treatment is general, but the application is to molecular clouds. We consider a class of equations of state which heuristically take into account the presence of turbulence; in particular, we consider equations of state which are {\it softer} than isothermal. We take the unperturbed cloud configuration to be in hydrostatic equilibrium. We find a class of wave solutions which propagate along a pressure supported cylinder (or slab) and have finite (trapped) spatial distributions in the direction perpendicular to the direction of propagation. Our results indicate that the dispersion relations for these two dimensional waves have similar forms for the two geometries considered here. Both cases possess a regime of instability and a fastest growing mode. We also find the (perpendicular) form of the perturbations for a wide range of wavelengths. Finally, we discuss the implications of our results for star formation and molecular clouds. The mass scales set by instabilities in both molecular cloud filaments and sheets are generally much larger than the masses of stars. However, these instabilities can determine the length scales for the initial conditions for protostellar collapse.

Protostellar collapse with RHD in spherical symmetry

Protostellar collapse with RHD in spherical symmetry

The formation and evolution of low mass protostars

A review is presented of the earliest stages of protostellar evolution. Observations of prestellar cores, which are believed to represent the initial conditions for protostellar collapse, depart significantly from the scale-free density distribution which is usually taken as the starting point for the formation of a low-mass protostar. Pre-stellar cores are observed to have radial density profiles which have flat inner regions, steepening towards their edges. This is seen to qualitatively match the predictions of the Bonnor-Ebert stability criterion for pressure-bounded self-gravitating gas clouds. From these initial conditions, theoretical modelling of cores threaded by magnetic fields predicts that quasi-static evolution by the process of ambipolar diffusion will lead to a significantly different starting point for collapse than the static singular isothermal sphere.

Collapse and Fragmentation of Molecular Cloud Cores. III. Initial Differential Rotation

view Abstract Citations (37) References (40) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Collapse and Fragmentation of Molecular Cloud Cores. III. Initial Differential Rotation Boss, Alan P. ; Myhill, Elizabeth A. Abstract Fragmentation during protostellar collapse is the leading mechanism for explaining the formation of binary and multiple stars. Most calculations of protostellar fragmentation have assumed the initial molecular cloud cores to be in solid body rotation, whereas differential rotation might characterize some molecular cloud cores. Here we use a second-order accurate, radiative hydrodynamics code to calculate the self-gravitational collapse of three-dimensional protostellar clouds, starting from centrally condensed (Gaussian), prolate (1.5:1 and 2:1 ratios) configurations, with varying degrees of differential rotation. The initial differential rotation is characterized by an exponent γι specifying the dependence of the initial angular velocity profile on the initial density (Ωι ∝ ργιι); γι = 0 produces initially solid body rotation, while γι = ⅔ results in the maximum degree of differential rotation consistent with spherical contraction at conserved mass and angular momentum. Depending on the value of γι, and the initial ratios of thermal (αι) and rotational (βι) to gravitational energy, three types of outcomes result: slow contraction, vigorous collapse without fragmentation, or collapse leading to fragmentation. Fragmentation occurs for γι = ⅔ clouds with initial 1.5:1 axis ratios when αι < 0.45 and βι ≍ 0.1, whereas γι = 0 clouds only fragment for αι < 0.3. For initial 2:1 axis ratios, fragmentation requires αι < 0.5 and βι ≍ 0.1 for γι = ⅔, compared to αι < 0.4 for clouds with γι = 0. These criteria quantify the extent to which initial differential rotation enhances rotationally driven fragmentation by concentrating the angular momentum per unit mass in the densest regions of the cloud. Gaussian clouds in rapid differential rotation can fragment even if they start collapse from a configuration close to virial equilibrium (αι + βι = ½). Publication: The Astrophysical Journal Pub Date: September 1995 DOI: 10.1086/176213 Bibcode: 1995ApJ...451..218B Keywords: HYDRODYNAMICS; ISM: CLOUDS; ISM: KINEMATICS AND DYNAMICS; ISM: MOLECULES; STARS: BINARIES: GENERAL; STARS: FORMATION full text sources ADS | Related Materials (9) Part 1: 1993ApJ...410..157B Part 2: 1995ApJ...439..224B Part 4: 1996ApJ...468..231B Part 5: 1997ApJ...483..309B Part 6: 1999ApJ...520..744B Part 7: 2002ApJ...568..743B Part 8: 2005ApJ...622..393B Part 9: 2007ApJ...658.1136B Part 10: 2009ApJ...697.1940B

Use of multiply nested grids for the solution of flux-limited radiation diffusion and hydrodynamics

Local grid refinement techniques allow improved spatial resolution for a large class of astrophysical problems involving radiation hydrodynamics. We describe here the implementation of multiply nested grids in a 2D radiation hydrodynamic code. The basic hydrodynamic code is explicit (except for the equation of energy conservation), spatially second-order, invokes artificial viscosity for the treatment of shocks, and through the use of operator-splitting is greater than first-order accurate in time. Radiation transfer is treated in the flux-limited diffusion approximation and solved implicitly in a separate substep. Tests and the following astrophysical applications are presented and briefly discussed: (a) expansion of a blast wave through several nested grids; (b) protostellar collapse with the formation of a circumstellar disk around a central source; (c) the interaction of circumstellar disk with a hot, hydrogen-ionizing star with a stellar wind.

Modeling Line Profiles of Protostellar Collapse in B335 with the Monte Carlo Method

view Abstract Citations (141) References (24) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Modeling Line Profiles of Protostellar Collapse in B335 with the Monte Carlo Method Choi, Minho ; Evans, Neal J., II ; Gregersen, Erik M. ; Wang, Yangsheng Abstract A collapsing dark cloud core, B335, is modeled as an inside-out collapse. The radiative transfer code uses the Monte Carlo method and new collision rates for CS. Line profiles for several observed transitions of CS and H2CO are computed. We confirm that a double-peaked spectrum with a stronger blue peak is a good qualitative signature of infall. Compared to the models using the large velocity gradient method by Zhou et al., our best-fit model using the Monte Carlo method has 20% smaller infall radius, 70% larger CS abundance, and 30% larger H2CO abundance. Also, infall radii found from CS and H2CO agree better than they did in the modeling by Zhou et al. Our model also produces off-center spectra close to the observations. Publication: The Astrophysical Journal Pub Date: August 1995 DOI: 10.1086/176002 Bibcode: 1995ApJ...448..742C Keywords: HYDRODYNAMICS; ISM: GLOBULES; ISM: INDIVIDUAL ALPHANUMERIC: B335; ISM: MOLECULES; METHODS: NUMERICAL; STARS: FORMATION full text sources ADS | data products SIMBAD (1)

Line formation in collapsing cloud cores with rotation and applications to B335 and IRAS 16293-2422

view Abstract Citations (92) References (19) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Line Formation in Collapsing Cloud Cores with Rotation and Applications to B335 and IRAS 16293-2422 Zhou, Shudong Abstract We present radiative transfer models of collapsing clouds with rotation and apply them to B335 and IRAS 16293-2422, the two best candidates of protostellar collapse known to date. By including rotation in the manner of Tereby, Shu, & Cassen (1984), we can reproduce the profiles of several molecular lines not only toward the center position, but also toward a grid of positions near the center. We find that the model of B335 by Zhou et al. (1993) is not affected significantly by the presence of rotation. By including rotation, we can reproduce the observations of IRAS 16293-2422 by Menden et al. (1987); hence, we support the infall interpretations of spectral lines from IRAS 16293-2422 proposed by Walker et al. (1986). We also observed the large-scale rotation of IRAS 16293-2422 in the C(18)O J = 2 approaches 1 line. The observed rotation rate is a factor of 6 smaller than that required to explain the small-scale CS data of Menden et al. (1987). This probably means the precollapse cloud has differential rotation, possibly due to more efficient magnetic braking on large scales. Publication: The Astrophysical Journal Pub Date: April 1995 DOI: 10.1086/175474 Bibcode: 1995ApJ...442..685Z Keywords: Line Spectra; Protostars; Star Formation; Stellar Mass Accretion; Stellar Models; Stellar Rotation; Numerical Analysis; Radiative Transfer; Spectral Signatures; Velocity Distribution; Astrophysics; ISM: INDIVIDUAL ALPHANUMERIC: B335; ISM: INDIVIDUAL ALPHANUMERIC: IRAS 16293-2422; LINE: FORMATION; LINE: PROFILES; RADIATIVE TRANSFER full text sources ADS | data products SIMBAD (2)

Spectral energy of first protostellar cores: Detecting 'class -I' protostars with ISO and SIRTF

view Abstract Citations (72) References (23) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Spectral Energy of First Protostellar Cores: Detecting ``Class -I'' Protostars with ISO and SIRTF Boss, Alan P. ; Yorke, Harold W. Abstract Radiative hydrodynamical models of protostellar collapse are used to calculate the spectral energy distributions of single and binary protostars at the phase of formation of the first (outer) protostellar core. In accordance with the established nomenclature, where classes 0, I, II, and III form a sequence in time, we term these pre-class 0 objects to be class-I ('class minus one') objects. These class -I objects are characterized by central core temperatures of approximately 200 K, envelope temperatures of approximately 10 K, and substantial far-infrared and submillimeter-wave fluxes. While undetectable by IRAS, these objects should be detectable by Infrared Space Observatory (ISO) and Space Infrared Telescope Facility (SITF) at 60 micrometer and longer wavelengths. First protostellar cores exist for times on the order of a few percent of the total collapse time, implying that a small fraction of all protostellar objects should be class -I objects. Publication: The Astrophysical Journal Pub Date: February 1995 DOI: 10.1086/187743 Bibcode: 1995ApJ...439L..55B Keywords: Far Infrared Radiation; Hydrodynamic Equations; Mathematical Models; Protostars; Spectral Energy Distribution; Star Formation; Stellar Cores; Computerized Simulation; Infrared Astronomy; Infrared Astronomy Satellite; Infrared Space Observatory (Iso); Space Infrared Telescope Facility; Astrophysics; INFRARED: STARS; STARS: FORMATION; STARS: PRE--MAIN-SEQUENCE full text sources ADS | data products SIMBAD (1)

Collapse and fragmentation of molecular cloud cores. 2: Collapse induced by stellar shock waves

The standard scenario for low-mass star formation involves 'inside-out' collapse of a dense molecular cloud core following loss of magnetic field support through ambipolar diffusion. However, isotopic anomalies in presolar grains and meteoritical inclusions imply that the collapse of the presolar cloud may have been triggered by a stellar shock wave. This paper explores 'outside-in' collapse, that is, protostellar collapse initiated directly by the compression of quiescent dense cloud cores impacted by relatively slow stellar shock waves. A second-order accurate, gravitational hydrodynamics code has been used to study both the spherically symmetrical and three-dimensional evolution of initially centrally condensed, isothermal, self-gravitating, solar-mass cloud cores that are struck by stellar shock waves with velocities up to 25 km/s and postshock temperatures of 10 to 10,000 K. The models show that such mild shock waves do not completely shred and destroy the cloud, and that the dynamical ram pressure can compress the cloud to the verge of self-gravitational collapse. However, compression caused by a high postshock temperature is a considerably more effective means of inducing collapse. Shock-induced collapse produces high initial mass accretion rates (greater than 10(exp -4) solar mass/yr in a solar-mass cloud) that decline rapidly to much lower values, depending on the presence (approximately 10(exp -6) solar mass/yr) or absence (approximately 10(exp -8) to 10(exp -7) solar mass/yr) of an infinite reservoir of mass. Stellar mass accretion rates approximately 10(exp -7) solar mass/yr have been previously inferred from the luminosities of T Tauri stars; balanced mass accretion (stellar rate = envelope rate) at approximately 10(exp -7) solar mass/yr could then be possible if accretion occurs from a finite mass reservoir. Fluid tracers are used to determine what fraction of the stellar shock material is incorporated into the resulting protostellar object and disk; roughly half the impinging material is injected into the collapsing cloud core when there is a high postshock temperature. The models are consistent with a scenario where an AGB star wind triggered the collapse of the presolar cloud while injecting about 0.01 solar mass of matter derived from the AGB star envelope, as has been separately inferred on the basis of nucleosynthesis calculations.

Utilitarian Models of the Solar Nebula

Models of the primitive solar nebula based on a combination of theory, observations of T Tauri stars, and global conservation laws are presented. The models describe the motions of nebular gas, mixing of interstellar material during the formation of the nebula, and evolution of thermal structure in terms of several characteristic parameters. The parameters describe key aspects of the protosolar cloud (its rotation rate and collapse rate) and the nebula (its mass relative to the Sun, decay time, and density distribution). For most applications, the models are heuristic rather than predictive. Their purpose is to provide a realistic context for the interpretation of solar system data, and to distinguish those nebula characteristics that can be specified with confidence, independently of the assumptions of particular models, from those that are poorly constrained. It is demonstrated that nebular gas typically experienced large radial excursions during the evolution of the nebula and that both inward and outward mean radial velocities on the order of meters per second occurred in the terrestrial planet region, with inward velocities predominant for most of the evolution. However, the time history of disk size, surface density, and radial velocities are sensitive to the total angular momentum of the protosolar cloud, which cannot be constrained by purely theoretical considerations. It is shown that a certain amount of "formational" mixing of interstellar material was an inevitable consequence of nebular mass and angular momentum transport during protostellar collapse, regardless of the specific transport mechanisms involved. Even if the protosolar cloud was initially homogeneous, this mixing was important because it had the effect of mingling presolar material that had experienced different degrees of thermal processing during collapse and passage through the accretion shock. Nebular thermal structure is less sensitive to poorly constrained parameters than is dynamical history. A simple criterion is derived for the condition that silicate grains are evaporated at midplane, and it is argued that this condition was probably fulfilled early in nebular history. Cooling of a hot nebula due to coagulation of dust and consequent local reduction of optical depth is examined, and it is shown how such a process leads naturally to an enrichment of rock-forming elements in the gas phase.

Protostellar collapse in a self-gravitating sheet

We present preliminary calculations of protostellar cloud collapse starting from an isothermal, self-gravitating gaseous layer in hydrostatic equilibrium. This gravitationally unstable layer collapses into a flattened or toroidal density distribution, even in the absence of rotation or magnetic fields. We suggest that the flat infalling envelope recently observed in HL Tau by Hayashi et al.is the result of collapse from an initially nonspherical layer. We also speculate that the later evolution of such a flattened, collapsing envelope can produce a structure similar to the 'flared disk' invoked by Kenyon and Hartmann to explain the infrared excesses of many T Tauri stars.

Protostellar hydrodynamics: Constructing and testing a spacially and temporally second-order accurate method. 2: Cartesian coordinates

view Abstract Citations (32) References (26) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Protostellar Hydrodynamics: Constructing and Testing a Spatially and Temporally Second-Order Accurate Method. II. Cartesian Coordinates Myhill, Elizabeth A. ; Boss, Alan P. Abstract In Boss & Myhill (1992) we described the derivation and testing of a spherical coordinate-based scheme for solving the hydrodynamic equations governing the gravitational collapse of nonisothermal, nonmagnetic, inviscid, radiative, three-dimensional protostellar clouds. Here we discuss a Cartesian coordinate-based scheme based on the same set of hydrodynamic equations. As with the spherical coorrdinate-based code, the Cartesian coordinate-based scheme employs explicit Eulerian methods which are both spatially and temporally second-order accurate. We begin by describing the hydrodynamic equations in Cartesian coordinates and the numerical methods used in this particular code. Following Finn & Hawley (1989), we pay special attention to the proper implementations of high-order accuracy, finite difference methods. We evaluate the ability of the Cartesian scheme to handle shock propagation problems, and through convergence testing, we show that the code is indeed second-order accurate. To compare the Cartesian scheme discussed here with the spherical coordinate-based scheme discussed in Boss & Myhill (1992), the two codes are used to calculate the standard isothermal collapse test case described by Bodenheimer & Boss (1981). We find that with the improved codes, the intermediate bar-configuration found previously disappears, and the cloud fragments directly into a binary protostellar system. Finally, we present the results from both codes of a new test for nonisothermal protostellar collapse. Publication: The Astrophysical Journal Supplement Series Pub Date: December 1993 DOI: 10.1086/191852 Bibcode: 1993ApJS...89..345M Keywords: Cartesian Coordinates; Computerized Simulation; Gravitational Collapse; Hydrodynamics; Numerical Analysis; Protostars; Binary Stars; Fourier Transformation; Interpolation; Poisson Density Functions; Poisson Equation; Radiative Transfer; Thermal Shock; Astrophysics; HYDRODYNAMICS; STARS: FORMATION; METHODS: NUMERICAL full text sources ADS | Related Materials (1) Part 1: 1992ApJS...83..311B

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

view Abstract Citations (63) References (52) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Evolution of the Solar Nebula. II. Thermal Structure during Nebula Formation Boss, Alan P. Abstract 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, υr ∝ 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 (Mṡ ≤ 10-5 Msun yr-1 for low-mass star formation), the disk mass became much too small (∼ 0.0002 Msun). 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 Mṡ → 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 Mṡd ∼ 10-2 Msun 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 Mṡs → 0 and Mṡd ∼ 10-6 to 10-5 Msun 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 Msun 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. Publication: The Astrophysical Journal Pub Date: November 1993 DOI: 10.1086/173318 Bibcode: 1993ApJ...417..351B Keywords: ACCRETION; ACCRETION DISKS; HYDRODYNAMICS; SOLAR SYSTEM: FORMATION full text sources ADS | Related Materials (8) Part 1: 1989ApJ...345..554B Part 3: 1996ApJ...469..906B Part 4: 1998ApJ...503..923B Part 5: 2002ApJ...576..462B Part 6: 2004ApJ...616.1265B Part 7: 2005ApJ...629..535B Part 8: 2007ApJ...660.1707B Part 9: 2011ApJ...739...61B