Richard I. Klein

The formation of massive star systems by accretion.

Massive stars produce so much light that the radiation pressure they exert on the gas and dust around them is stronger than their gravitational attraction, a condition that has long been expected to prevent them from growing by accretion. We present three-dimensional radiation-hydrodynamic simulations of the collapse of a massive prestellar core and find that radiation pressure does not halt accretion. Instead, gravitational and Rayleigh-Taylor instabilities channel gas onto the star system through nonaxisymmetric disks and filaments that self-shield against radiation while allowing radiation to escape through optically thin bubbles. Gravitational instabilities cause the disk to fragment and form a massive companion to the primary star. Radiation pressure does not limit stellar masses, but the instabilities that allow accretion to continue lead to small multiple systems.

Radiation-hydrodynamic simulations of collapse and fragmentation in massive protostellar cores

We simulate the early stages of the evolution of turbulent, virialized, high-mass protostellar cores, with primary attention to how cores fragment and whether they form a small or large number of protostars. Our simulations use the Orion adaptive mesh refinement code to follow the collapse from ~0.1 pc scales to ~10 AU scales, for durations that cover the main fragmentation phase, using three-dimensional gravito-radiation hydrodynamics. We find that for a wide range of initial conditions radiation feedback from accreting protostars inhibits the formation of fragments, so that the vast majority of the collapsed mass accretes onto one or a few objects. Most of the fragmentation that does occur takes place in massive, self-shielding disks. These are driven to gravitational instability by rapid accretion, producing rapid mass and angular momentum transport that allows most of the gas to accrete onto the central star rather than forming fragments. In contrast, a control run using the same initial conditions but an isothermal equation of state produces much more fragmentation, both in and out of the disk. We conclude that massive cores with observed properties are not likely to fragment into many stars, so that, at least at high masses, the core mass function probably determines the stellar initial mass function. Our results also demonstrate that simulations of massive star-forming regions that do not include radiative transfer, and instead rely on a barotropic equation of state or optically thin heating and cooling curves, are likely to produce misleading results.

Self-gravitational Hydrodynamics with Three-dimensional Adaptive Mesh Refinement: Methodology and Applications to Molecular Cloud Collapse and Fragmentation

We describe a new code for numerical solution of three-dimensional self-gravitational hydrodynamics problems. This code utilizes the technique of local adaptive mesh refinement (AMR), employing multiple grids at multiple levels of resolution and automatically and dynamically adding and removing these grids as necessary to maintain adequate resolution. This technology allows solution of problems that would be prohibitively expensive with a code using fixed resolution, and it is more versatile and efficient than competing methods of achieving variable resolution. In particular, we apply this technique to simulate the collapse and fragmentation of a molecular cloud, a key step in star formation. The simulation involves many orders of magnitude of variation in length scale as fragments form at positions that are not a priori discernible from general initial conditions. In this paper, we describe the methodology behind this new code and present several illustrative applications. The criterion that guides the degree of adaptive mesh refinement is critical to the success of the scheme, and, for the isothermal problems considered here, we employ the Jeans condition for this purpose. By maintaining resolution finer than the local Jeans length, we set new benchmarks of accuracy by which to measure other codes on each problem we consider, including the uniform collapse of a finite pressured cloud. We find that the uniformly rotating, spherical clouds treated here first collapse to disks in the equatorial plane and then, in the presence of applied perturbations, form filamentary singularities that do not fragment while isothermal. Our results provide numerical confirmation of recent work by Inutsuka & Miyama on this scenario of isothermal filament formation.

THE EFFECTS OF RADIATIVE TRANSFER ON LOW-MASS STAR FORMATION

Forming stars emit a substantial amount of radiation into their natal environment. We use ORION, an adaptive mesh refinement (AMR) three-dimensional gravito-radiation-hydrodyanics code, to simulate low-mass star formation in a turbulent molecular cloud. We compare the distributions of stellar masses, accretion rates, and temperatures in the cases with and without radiative transfer, and we demonstrate that radiative feedback has a profound effect on accretion, multiplicity, and mass by reducing the number of stars formed and the total rate at which gas turns into stars. We also show that once the star formation reaches a steady state, protostellar radiation is by far the dominant source of energy in the simulation, exceeding viscous dissipation and compressional heating by at least an order of magnitude. Calculations that omit radiative feedback from protstars significantly underestimate the gas temperature and the strength of this effect. Although heating from protostars is mainly confined to the protostellar cores, we find that it is sufficient to suppress disk fragmentation that would otherwise result in very low-mass companions or brown dwarfs. We demonstrate that the mean protostellar accretion rate increases with the final stellar mass so that the star formation time is only a weak function of mass.

Embedding Lagrangian Sink Particles in Eulerian Grids

We introduce a new computational method for embedding Lagrangian sink particles into a Eulerian calculation. Simulations of gravitational collapse or accretion generally produce regions whose density greatly exceeds the mean density in the simulation. These dense regions require extremely small time steps to maintain numerical stability. Smoothed particle hydrodynamics (SPH) codes approach this problem by introducing nongaseous, accreting sink particles, and Eulerian codes may introduce fixed sink cells. However, until now there has been no approach that allows Eulerian codes to follow accretion onto multiple, moving objects. We have removed that limitation by extending the sink particle capability to Eulerian hydrodynamics codes. We have tested this new method and found that it produces excellent agreement with analytic solutions. In analyzing our sink particle method, we present a method for evaluating the disk viscosity parameter α due to the numerical viscosity of a hydrodynamics code and use it to compute α for our Cartesian adaptive mesh refinement (AMR) code. We also present a simple application of this new method: studying the transition from Bondi to Bondi-Hoyle accretion that occurs when a shock hits a particle undergoing Bondi accretion.

On the Role of Disks in the Formation of Stellar Systems: A Numerical Parameter Study of Rapid Accretion

We study rapidly accreting, gravitationally unstable disks with a series of idealized global, numerical experiments using the code ORION. Our numerical parameter study focuses on protostellar disks, showing that one can predict disk behavior and the multiplicity of the accreting star system as a function of two dimensionless parameters which compare the infall rate to the disk sound speed and orbital period. Although gravitational instabilities become strong, we find that fragmentation into binary or multiple systems occurs only when material falls in several times more rapidly than the canonical isothermal limit. The disk-to-star accretion rate is proportional to the infall rate and governed by gravitational torques generated by low-m spiral modes. We also confirm the existence of a maximum stable disk mass: disks that exceed ~50% of the total system mass are subject to fragmentation and the subsequent formation of binary companions.

The Jeans Condition and Collapsing Molecular Cloud Cores: Filaments or Binaries?

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 & Myhill in 1992 (B&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&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&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&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&M barotropic result differs significantly from the B&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.

The formation of stars by gravitational collapse rather than competitive accretion

There are two dominant models of how stars form. Under gravitational collapse, star-forming molecular clumps, of typically hundreds to thousands of solar masses (M[circdot]), fragment into gaseous cores that subsequently collapse to make individual stars or small multiple systems. In contrast, competitive accretion theory suggests that at birth all stars are much smaller than the typical stellar mass (∼0.5M[circdot]), and that final stellar masses are determined by the subsequent accretion of unbound gas from the clump. Competitive accretion models interpret brown dwarfs and free-floating planets as protostars ejected from star-forming clumps before they have accreted much mass; key predictions of this model are that such objects should lack disks, have high velocity dispersions, form more frequently in denser clumps, and that the mean stellar mass should vary within the Galaxy. Here we derive the rate of competitive accretion as a function of the star-forming environment, based partly on simulation, and determine in what types of environments competitive accretion can occur. We show that no observed star-forming region can undergo significant competitive accretion, and that the simulations that show competitive accretion do so because the assumed properties differ from those determined by observation. Our result shows that stars form by gravitational collapse, and explains why observations have failed to confirm predictions of the competitive accretion model.

Radiation Feedback, Fragmentation, and the Environmental Dependence of the Initial Mass Function

The fragmentation of star-forming interstellar clouds, and the resulting stellar initial mass function (IMF), is strongly affected by the temperature structure of the collapsing gas. Since radiation feedback from embedded stars can modify this as collapse proceeds, feedback plays an important role in determining the IMF. However, the effects and importance of radiative heating are likely to depend strongly on the surface density of the collapsing clouds, which determines both their effectiveness at trapping radiation and the accretion luminosities of the stars forming within them. In this paper, we report a suite of adaptive mesh refinement radiation-hydrodynamic simulations using the ORION code in which we isolate the effect of column density on fragmentation by following the collapse of clouds of varying column density while holding the mass, initial density and velocity structure, and initial virial ratio fixed. We find that radiation does not significantly modify the overall star formation rate or efficiency, but that it suppresses fragmentation more and more as cloud surface densities increase from those typical of low-mass star-forming regions like Taurus, through the typical surface density of massive star-forming clouds in the Galaxy, up to conditions found only in super-star clusters. In regions of low surface density, fragmentation during collapse leads to the formation of small clusters rather than individual massive star systems, greatly reducing the fraction of the stellar population with masses 10 M ☉. Our simulations have important implications for the formation of massive stars and the universality of the IMF.

Radiation-Hydrodynamic Simulations of Massive Star Formation with Protostellar Outflows

We report the results of a series of adaptive mesh refinement radiation-hydrodynamic simulations of the collapse of massive star-forming clouds using the ORION code. These simulations are the first to include the feedback effects protostellar outflows, as well as protostellar radiative heating and radiation pressure exerted on the infalling, dusty gas. We find that outflows evacuate polar cavities of reduced optical depth through the ambient core. These enhance the radiative flux in the poleward direction so that it is 1.7-15 times larger than that in the midplane. As a result the radiative heating and outward radiation force exerted on the protostellar disk and infalling cloud gas in the equatorial direction are greatly diminished. This simultaneously reduces the Eddington radiation pressure barrier to high-mass star formation and increases the minimum threshold surface density for radiative heating to suppress fragmentation compared to models that do not include outflows. The strength of both these effects depends on the initial core surface density. Lower surface density cores have longer free-fall times and thus massive stars formed within them undergo more Kelvin contraction as the core collapses, leading to more powerful outflows. Furthermore, in lower surface density clouds the ratio of the time required for the outflow to break out of the core to the core free-fall time is smaller, so that these clouds are consequently influenced by outflows at earlier stages of the collapse. As a result, outflow effects are strongest in low surface density cores and weakest in high surface density ones. We also find that radiation focusing in the direction of outflow cavities is sufficient to prevent the formation of radiation pressure-supported circumstellar gas bubbles, in contrast to models which neglect protostellar outflow feedback.