Matthew R. Bate

Modelling accretion in protobinary systems

A method for following fragmentation simulations further in time using smoothed particle hydrodynamics (SPH) is presented. In a normal SPH simulation of the collapse and fragmentation of a molecular cloud, high-density regions of gas that form protostars are represented by many particles with small separations. These high-density regions require small time steps, limiting the time for which the simulation can be followed. Thus, the end result of the fragmentation can never be definitively ascertained, and comparisons between cloud fragmentation calculations and the observed characteristics of stellar systems cannot be made. In this paper, each high-density region is replaced by a single, non-gaseous particle, with appropriate boundary conditions, which contains all the mass in the region and accretes any infalling mass. This enables the evolution of the cloud and the resulting protostars to be followed for many orbits or until most of the original cloud mass has been accreted.

The formation of a star cluster: predicting the properties of stars and brown dwarfs

We present results from the largest numerical simulation of star formation to resolve the fragmentation process down to the opacity limit. The simulation follows the collapse and fragmentation of a large-scale turbulent molecular cloud to form a stellar cluster and, simultaneously, the formation of circumstellar discs and binary stars. This large range of scales enables us to predict a wide variety of stellar properties for comparison with observations. The calculation clearly demonstrates that star formation is a highly-dynamic and chaotic process. Star formation occurs in localized bursts within the cloud via the fragmentation both of dense molecular cloud cores and of massive circumstellar discs. Star-disc encounters form binaries and truncate discs. Stellar encounters disrupt bound multiple systems. We find that the observed statistical properties of stars are a natural consequence of star formation in such a dynamical environment. The cloud produces roughly equal numbers of stars and brown dwarfs, with masses down to the opacity limit for fragmentation (5 Jupiter masses). The initial mass function is consistent with a Salpeter slope (Γ = -1.35) above 0.5 M O ., a roughly flat distribution (r = 0) in the range 0.006-0.5 M O ., and a sharp cut-off below 0.005 M O .. This is consistent with recent observational surveys. The brown dwarfs form by the dynamical ejection of low-mass fragments from dynamically unstable multiple systems before the fragments have been able to accrete to stellar masses. Close binary systems (with separations ≤10 au) are not formed by fragmentation in situ. Rather, they are produced by hardening of initially wider multiple systems through a combination of dynamical encounters, gas accretion, and/or the interaction with circumbinary and circumtriple discs. Finally, we find that the majority of circumstellar discs have radii less than 20 au due to truncation in dynamical encounters. This is consistent with observations of the Orion Trapezium cluster and implies that most stars and brown dwarfs do not form large planetary systems.

On the formation of massive stars

A B STR A CT We present a model for the formation of massive (Mz10 M>) stars through accretion-induced collisions in the cores of embedded dense stellar clusters. This model circumvents the problem of accreting on to a star whose luminosity is sufficient to reverse the infall of gas. Instead, the central core of the cluster accretes from the surrounding gas, thereby decreasing its radius until collisions between individual components become sufficient. These components are, in general, intermediate-mass stars that have formed through accretion on to low-mass protostars. Once a sufficiently massive star has formed to expel the remaining gas, the cluster expands in accordance with this loss of mass, halting further collisions. This process implies a critical stellar density for the formation of massive stars, and a high rate of binaries formed by tidal capture.

Competitive accretion in embedded stellar clusters

We investigate the physics of gas accretion in young stellar clusters. Accretion in clusters is a dynamic phenomenon as both the stars and the gas respond to the same gravitational potential. Accretion rates are highly non-uniform with stars nearer the centre of the cluster, where gas densities are higher, accreting more than others. This competitive accretion naturally results in both initial mass segregation and a spectrum of stellar masses. Accretion in gas-dominated clusters is well modelled using a tidal-lobe radius instead of the commonly used Bondi–Hoyle accretion radius. This works as both the stellar and gas velocities are under the influence of the same gravitational potential and are thus comparable. The low relative velocity which results means that Rtidal<RBH in these systems. In contrast, when the stars dominate the potential and are virialized, RBH<Rtidal and Bondi–Hoyle accretion is a better fit to the accretion rates.

Stellar, Brown Dwarf, and Multiple Star Properties from Hydrodynamical Simulations of Star Cluster Formation

We report the statistical properties of stars, brown dwarfs and multiple systems obtained from the largest hydrodynamical simulation of star cluster formation to date that resolves masses down to the opacity limit for fragmentation (a few Jupiter masses). The simulation is essentially identical to that of Bate, Bonnell & Bromm except that the initial molecular cloud is larger and more massive. It produces more than 1250 stars and brown dwarfs, providing unprecedented statistical information that can be compared with observational surveys. The calculation uses sink particles to model the stars and brown dwarfs. Part of the calculation is rerun with smaller sink particle accretion radii and gravitational softening to investigate the effect of these approximations on the results. We find that hydrodynamical/sink particle simulations can reproduce many of the observed stellar properties very well. Multiplicity as a function of the primary mass, the frequency of very low mass (VLM) binaries, general trends for the separation and mass ratio distributions of binaries and the relative orbital orientations of triples systems are all in reasonable agreement with observations. We also examine the radial variations of binarity, velocity dispersion and mass function in the resulting stellar cluster and the distributions of disc truncation radii due to dynamical interactions. For VLM binaries, because their separations are typically close, we find that their frequency is sensitive to the sink particle accretion radii and gravitational softening used in the calculations. Using small accretion radii and gravitational softening results in a frequency of VLM binaries similar to that expected from observational surveys (≈20 per cent). We also find that VLM binaries evolve from wide, unequal-mass systems towards close equal-mass systems as they form. The two main deficiencies of the calculations are that they overproduce brown dwarfs relative to stars and that there are too few unequalmass binaries with K- and G-dwarf primaries. The former of these is likely due to the absence of radiative feedback and/or magnetic fields.

Stellar, brown dwarf and multiple star properties from a radiation hydrodynamical simulation of star cluster formation

We report the statistical properties of stars, brown dwarfs and multiple systems obtained from the largest radiation hydrodynamical simulation of star cluster formation to date that resolves masses down to the opacity limit for fragmentation (a few Jupiter masses). The initial conditions are identical to those of previous barotropic calculations published by Bate, but this time the calculation is performed using a realistic equation of state and radiation hydrodynamics. The calculation uses sink particles to model 183 stars and brown dwarfs, including 28 binaries and 12 higher-order multiple systems, the properties of which are compared to the results from observational surveys. We find that the radiation hydrodynamical/sink particle simulation reproduces many observed stellar properties very well. In particular, whereas using a barotropic equation of state produces more brown dwarfs than stars, the inclusion of radiative feedback results in a stellar mass function and a ratio of brown dwarfs to stars in good agreement with observations of Galactic star-forming regions. In addition, many of the other statistical properties of the stars and brown dwarfs are in reasonable agreement with observations, including multiplicity as a function of primary mass, the frequency of very low mass binaries, and general trends for the mass ratio and separation distributions of binaries. We also examine the velocity dispersion of the stars, the distributions of disc truncation radii due to dynamical interactions, and coplanarity of orbits and sink particle spins in multiple systems. Overall, the calculation produces a cluster of stars whose statistical properties are difficult to distinguish from observed systems, implying that gravity, hydrodynamics and radiative feedback are the primary ingredients for determining the origin of the statistical properties of low-mass stars.

The hierarchical formation of a stellar cluster

Recent surveys of star-forming regions have shown that most stars, and probably all massive stars, are born in dense stellar clusters. The mechanism by which a molecular cloud fragments to form several hundred to thousands of individual stars has remained elusive. Here, we use a numerical simulation to follow the fragmentation of a turbulent molecular cloud, and the subsequent formation and early evolution of a stellar cluster containing more than 400 stars. We show that the stellar cluster forms through the hierarchical fragmentation of a turbulent molecular cloud. This leads to the formation of many small subclusters, which interact and merge to form the final stellar cluster. The hierarchical nature of the cluster formation has serious implications in terms of the properties of the new-born stars. The higher number-density of stars in subclusters, compared to a more uniform distribution arising from a monolithic formation, results in closer and more frequent dynamical interactions. Such close interactions can truncate circumstellar discs, harden existing binaries and potentially liberate a population of planets. We estimate that at least one-third of all stars, and most massive stars, suffer such disruptive interactions.

Massive star formation: nurture, not nature

We investigate the physical processes that lead to the formation of massive stars. Using a numerical simulation of the formation of a stellar cluster from a turbulent molecular cloud, we evaluate the relevant contributions of fragmentation and competitive accretion in determining the masses of the more massive stars. We find no correlation between the final mass of a massive star, and the mass of the clump from which it forms. Instead, we find that the bulk of the mass of massive stars comes from subsequent competitive accretion in a clustered environment. In fact, the majority of this mass infalls on to a pre-existing stellar cluster. Furthermore, the mass of the most massive star in a system increases as the system grows in numbers of stars and in total mass. This arises as the infalling gas is accompanied by newly formed stars, resulting in a larger cluster around a more massive star. High-mass stars gain mass as they gain companions, implying a direct causal relationship between the cluster formation process and the formation of higher-mass stars therein.

Three-dimensional calculations of high- and low-mass planets embedded in protoplanetary discs

We analyse the non-linear, three-dimensional response of a gaseous, viscous protoplanetary disc to the presence of a planet of mass ranging from 1 Earth mass (1 M⊕) to 1 Jupiter mass (1 MJ) by using the ZEUS hydrodynamics code. We determine the gas f ow pattern, and the accretion and migration rates of the planet. The planet is assumed to be in a f xed circular orbit about the central star. It is also assumed to be able to accrete gas without expansion on the scale of its Roche radius. Only planets with masses Mp 0.1 MJ produce significan perturbations in the surface density of the disc. The fl w within the Roche lobe of the planet is fullythree-dimensional.GasstreamsgenerallyentertheRochelobeclosetothediscmid-plane, but produce much weaker shocks than the streams in two-dimensional models. The streams supply material to a circumplanetary disc that rotates in the same sense as the orbit of the planet. Much of the mass supply to the circumplanetary disc comes from non-coplanar fl w. The accretion rate peaks with a planet mass of approximately 0.1 MJ and is highly efficient occurring at the local viscous rate. The migration time-scales for planets of mass less than 0.1 MJ, based on torques from disc material outside the Roche lobes of the planets, are in excellent agreement with the linear theory of type I (non-gap) migration for three-dimensional discs.ThetransitionfromtypeItotypeII(gap)migrationissmooth,withchangesinmigration times of about a factor of 2. Starting with a core which can undergo runaway growth, a planet can gain up to a few MJ with little migration. Planets with fina masses of the order of 10 MJ would undergo large migration, which makes formation and survival difficult

The importance of radiative feedback for the stellar initial mass function

We investigate the effect of radiative feedback on the star formation process using radiation hydrodynamical simulations. We repeat the previous hydrodynamical star cluster formation simulations of Bate et al. and Bate & Bonnell, but we use a realistic gas equation of state and radiative transfer in the flux-limited diffusion approximation rather than the original barotropic equation of state. Whereas star formation in the barotropic simulations continued unabated until the simulations stopped, we find that radiative feedback, even from low-mass stars, were essentially terminates the production of new objects within low-mass dense molecular cloud cores after roughly one local dynamical time. Radiative feedback also dramatically decreases the propensity of massive circumstellar discs to fragment and inhibits fragmentation of other dense gas (e.g. filaments) close to existing protostellar objects. These two effects decrease the numbers of protostars formed by a factor of ≈4 compared with the original hydrodynamical simulations using the barotropic equation of state. In particular, whereas the original simulations produced more brown dwarfs than stars, the radiative feedback results in a ratio of stars to brown dwarfs of approximately 5:1, in much better agreement with observations. Most importantly, we find that although the characteristic stellar mass in the original calculations scaled linearly with the initial mean Jeans mass in the clouds, when radiative feedback is included the characteristic stellar mass is indistinguishable for the two calculations, regardless of the initial Jeans mass of the clouds. We thus propose that the reason the observed initial mass function appears to be universal in the local Universe is due to self-regulation of the star formation process by radiative feedback. We present an analytic argument showing how a characteristic mass may be derived that is relatively independent of initial conditions such as the clouds density.