Advances in available computing facilities, judiciously employed by our colleagues, have undoubtedly enhanced our understanding of the processes by which stars (and planets) form, from very diffuse gas to something you could almost live around. Nevertheless we remain very far from being able to describe (let alone explain) what is going on in many cases. And this is likely to remain true even as Moore's Law growth begins to hit its head against energy and power considerations. A large fraction of long-standing questions appear to have honest answers of the general form, “yes and no”, “all of the above”, or “some of them are and some of them aren't”. This includes many of my favorites, like triggering, formation of binary populations, and the role of magnetic fields. Rather few questions have actually been retired from the universe of discourse in recent years.

Observations show that expanding H ii regions may trigger star formation. We discuss several aspects of this type of star formation, and try to estimate its prevalence. We show how LMC H ii regions may help us to understand what we see in our Galaxy.

A large fraction of brown dwarfs and low-mass stars may form by gravitational fragmentation of relatively massive (a few 0.1 M⊙) and extended (a few hundred AU) discs around Sun-like stars. We present an ensemble of radiative hydrodynamic simulations that examine the conditions for disc fragmentation. We demonstrate that this model can explain the low-mass IMF, the brown dwarf desert, and the binary properties of low-mass stars and brown dwarfs. Observing discs that are undergoing fragmentation is possible but very improbable, as the process of disc fragmentation is short lived (discs fragment within a few thousand years).

The feedback form pre-main sequence and young stars influences their vicinity. The stars are formed in clusters, which implies that the winds of individual stars collide with each other. Inside of a star cluster, winds thermalize a fraction of their kinetic energy, forming a very hot medium able to escape from the cluster in the form of a large-scale wind. Outside of the cluster, the cluster wind forms a shock front as it interacts with the ambient medium which is accreted onto the expanding shell. A variety of instabilities may develop in such shells, and in some cases they fragment, triggering second generation of star formation. However, if the cluster surpasses a certain mass (depending on the radius and other parameters) the hot medium starts to be thermally unstable even inside of the cluster, forming dense warm clumps. The formation of next generations of stars may start if the clumps are big enough to self-shield against stellar radiation creating cold dense cores.

We simulated an isolated quiescent Milky Way-type galaxy with a maximum effective resolution of 7.8 pc. Clouds formed in the interstellar medium through gravitational fragmentation and became the sites for star formation. We tracked the evolution of the clouds through 300 Myr in the presence of star formation, photoelectric heating and feedback from Type II supernovae. The cloud mass distribution agreed well with observational results. Feedback suppressed star formation but did not destroy the surrounding cloud. Collisions between clouds were found to be sufficiently frequent to be a significant factor in determining the star formation rate.

We present the first results from a project to model the prestellar cores in Ophiuchus, using initial conditions constrained as closely as possible by observation. The prestellar cores in Ophiuchus appear to be evolving in isolation — in the sense that the timescale on which an individual prestellar core collapses and fragments is estimated to be much shorter than the timescale on which it is likely to interact dynamically with another core. Therefore it is realistic to simulate individual cores separately, and this in turn makes it feasible (a) to perform multiple realisations of the evolution of each core (to allow for uncertainties in the initial conditions which persist, even for the most comprehensively observed cores), and (b) to do so at high resolution (so that even the smallest protostars are well resolved). The aims of this project are (i) to address how best to convert the observations into initial conditions; (ii) to explore, by means of numerical simulations, how the observed cores are likely to evolve in the future; (iii) to predict the properties of the protostars that they will form (mass function, multiplicity statistics, etc.); and (iv) to compare these properties with the properties of the observed pre-Main Sequence stars in Ophiuchus. We find that if the observed non-thermal velocities in the Ophiuchus prestellar cores are attributed to purely solenoidal turbulence, they do not fragment; they all collapse to form single protostars. If the non-thermal velocities are attributed to a mixture of solenoidal and compressive turbulence, multiple systems form readily. The turbulence first generates a network of filaments, and material then tends to flow along the filaments, at first into a primary protostar, and then onto a compact accretion disc around this protostar; secondary protostars condense out of the material flowing into the disc along the filaments. If the turbulence is purely solenoidal, but part of the non-thermal velocity dispersion is attributed to solid-body rotation, then again multiple systems form readily, but the pattern of fragmentation is quite different. A primary protostar forms near the centre of the core, and then an extended accretion disc forms around the primary protostar, and eventually becomes so massive that it fragments to produce low-mass secondaries; these frequently end up in hierarchical multiple systems.

There have been a number of theoretical and computational models which state that magnetic fields play an important role in the process of star formation. Competing theories instead postulate that it is turbulence which is dominant and magnetic fields are weak. The recent installation of a polarimetry system at the Submillimeter Array (SMA) has enabled us to conduct observations that could potentially distinguish between the two theories. Some of the nearby low mass star forming regions show hour-glass shaped magnetic field structures that are consistent with theoretical models in which the magnetic field plays a dominant role. However, there are other similar regions where no significant polarization is detected. Future polarimetry observations made by the Submillimeter Array should be able to increase the sample of observed regions. These measurements will allow us to address observationally the important question of the role of magnetic fields and/or turbulence in the process of star formation.

This brief review emphasizes the wide range of environments where interaction induced star formation occurs. In these environments we can study the numerous elaborations of a few basic physical processes, including: gravitational instability, accretion and large-scale shocks.

At present, hydrodynamical simulations in computational star formation are either carried out with Eulerian mesh-based approaches or with the Lagrangian smoothed particle hydrodynamics (SPH) technique. Both methods differ in their strengths and weaknesses, as well as in their error properties. It would be highly desirable to find an intermediate discretization scheme that combines the accuracy advantage of mesh-based methods with the automatic adaptivity and Galilean invariance of SPH. Here we briefly describe the novel AREPO code which achieves these goals based on a moving unstructured mesh defined by the Voronoi tessellation of a set of discrete points. The mesh is used to solve the hyperbolic conservation laws of ideal hydrodynamics with a finite volume approach, based on a second-order unsplit Godunov scheme with an exact Riemann solver. A particularly powerful feature is that the mesh-generating points can in principle be moved arbitrarily. If they are given the velocity of the local flow, an accurate Lagrangian formulation of continuum hydrodynamics is obtained that features a very low numerical diffusivity and is free of mesh distortion problems. If the points are kept fixed, the scheme is equivalent to a Eulerian code on a structured mesh. The new AREPO code appears especially well suited for problems such as gravitational fragmentation or compressible turbulence.

The time evolution of protostellar disks in the embedded phase of star formation (EPSF) is reviewed based on numerical hydrodynamics simulations of the gravitational collapse of two cloud cores with distinct initial masses. Special emphasis is given to disk, stellar, and envelope masses and also mass accretion rates onto the star. It is shown that accretion is highly variable in the EPSF, in agreement with recent theoretical and observational expectations. Protostellar disks quickly accumulate mass upon formation and may reach a sizeable fraction of the envelope mass (~35%) by the end of the Class 0 phase. Systems with disk-to-star mass ratio ξ≈0.5 are common but systems with ξ≥1.0 are rare because the latter quickly evolve into binary or multiple systems. Embedded disks are characterized by radial pulsations, the amplitude of which increases with growing core mass.

The interstellar medium (ISM) in galaxies is multiphase and cloudy, with stars forming in the very dense, cold gas found in Giant Molecular Clouds (GMCs). Simulating the evolution of an entire galaxy, however, is a computational problem which covers many orders of magnitude, so many simulations cannot reach densities high enough or temperatures low enough to resolve this multiphase nature. Therefore, the formation of GMCs is not captured and the resulting gas distribution is smooth, contrary to observations. We investigate how star formation (SF) proceeds in simulated galaxies when we obtain parsec-scale resolution and more successfully capture the multiphase ISM. Both major mergers and the accretion of cold gas via filaments are dominant contributors to a galaxy's total stellar budget and we examine SF at high resolution in both of these contexts.

The idea that stars are formed by gravity goes back more than 300 years to Newton, and the idea that gravitational instability plays a role goes back more than 100 years to Jeans, but the idea that stars are forming at the present time in the interstellar medium is more recent and did not emerge until the energy source of stars had been identified and it was realized that the most luminous stars have short lifetimes and therefore must have formed recently. The first suggestion that stars may be forming now in the interstellar medium was credited by contemporary authors to a paper by Spitzer in 1941 in which he talks about the formation of interstellar condensations by radiation pressure, but then oddly says nothing about star formation. That may be because, as Spitzer later told me, when he first suggested very tentatively in a paper submitted to The Astrophysical Journal that stars might be forming now from interstellar matter, this was considered a radical idea and the referee said it was much too speculative and should be taken out of the paper. So Spitzer removed the speculation about star formation from the published version of his paper.

I'll overview the past, present, and future of the GRAPE project, which started as the effort to design and develop specialized hardware for gravitational N-body problem. The current hardware, GRAPE-DR, has an architecture quite different from previous GRAPEs, in the sense that it is a collection of small, but programmable processors, while previous GRAPEs had hardwired pipelines. I'll discuss pros and cons of these two approaches, comparisons with other accelerators and future directions.

I briefly review recent observations of regions forming low mass stars. The discussion is cast in the form of seven questions that have been partially answered, or at least illuminated, by new data. These are the following: where do stars form in molecular clouds; what determines the IMF; how long do the steps of the process take; how efficient is star formation; do any theories explain the data; how are the star and disk built over time; and what chemical changes accompany star and planet formation. I close with a summary and list of open questions.

Magnetic diffusion plays a vital role in star formation. We trace its influence from interstellar cloud scales down to star-disk scales. On both scales, we find that magnetic diffusion can be significantly enhanced by the buildup of strong gradients in magnetic field structure. Large scale nonlinear flows can create compressed cloud layers within which ambipolar diffusion occurs rapidly. However, in the flux-freezing limit that may be applicable to photoionized molecular cloud envelopes, supersonic motions can persist for long times if driven by an externally generated magnetic field that corresponds to a subcritical mass-to-flux ratio. In the case of protostellar accretion, rapid magnetic diffusion (through Ohmic dissipation with additional support from ambipolar diffusion) near the protostar causes dramatic magnetic flux loss. By doing so, it also allows the formation of a centrifugal disk, thereby avoiding the magnetic braking catastrophe.

Major progress has been made over the last few years in understanding hydrodynamical processes on cosmological scales, in particular how galaxies get their baryons. There is increasing recognition that a large part of the baryons accrete smoothly onto galaxies, and that internal evolution processes play a major role in shaping galaxies – mergers are not necessarily the dominant process. However, predictions from the various assembly mechanisms are still in large disagreement with the observed properties of galaxies in the nearby Universe. Small-scale processes have a major impact on the global evolution of galaxies over a Hubble time and the usual sub-grid models account for them in a far too uncertain way. Understanding when, where and at which rate galaxies formed their stars becomes crucial to understand the formation of galaxy populations. I discuss recent improvements and current limitations in “resolved” modeling of star formation, aiming at explicitly capturing star-forming instabilities, in cosmological and galaxy-sized simulations. Such models need to develop three-dimensional turbulence in the ISM, which requires parsec-scale resolution at redshift zero.

Stars form predominantly in clusters inside dense clumps of molecular clouds that are both turbulent and magnetized. The typical size and mass of the cluster-forming clumps are ~1 pc and ~102 – 103 M⊙, respectively. Here, we discuss some recent progress on numerical simulations of clustered star formation in such parsec-scale dense clumps with emphasis on the role of magnetic fields. The simulations have shown that magnetic fields tend to slow down global gravitational collapse and thus star formation, especially in the presence of protostellar outflow feedback. Even a relatively weak magnetic field can retard star formation significantly, because the field is amplified by supersonic turbulence to an equipartition strength. However, in such a case, the distorted field component dominates the uniform one. In contrast, if the field is moderately-strong, the uniform component remains dominant. Such a difference in the magnetic structure is observed in simulated polarization maps of dust thermal emission. Recent polarization measurements show that the field lines in nearby cluster-forming clumps are spatially well-ordered, indicative of a rather strong field. In such strongly-magnetized clumps, star formation should proceed relatively slowly; it continues for at least several global free-fall times of the parent dense clump (tff ~ a few × 105 yr).

We review computational approaches to understanding the origin of the Initial Mass Function (IMF) during the formation of star clusters. We examine the role of turbulence, gravity and accretion, equations of state, and magnetic fields in producing the distribution of core masses - the Core Mass Function (CMF). Observations show that the CMF is similar in form to the IMF. We focus on feedback processes such as stellar dynamics, radiation, and outflows can reduce the accreted mass to give rise to the IMF. Numerical work suggests that filamentary accretion may play a key role in the origin of the IMF.

Radiative Transfer (RT) is considered to be one of the four Grand Challenges in Computational Astrophysics aside of Astrophysical Fluid Dynamics, N-Body Problems in Astrophysics, and Relativistic Astrophysics. The high dimensionality (7D instead of 4D for MHD) and the underlying integro-differential transport equation have forced coders to implement approximative RT methods in order to fit spectra and images or to treat RT in their HD and MHD codes.

The central role of RT in star formation (SF) is based on several facts: a) The dense dusty gas in SF regions alters the radiation substantially making SF one of the most complex applications of RT. b) Radiation transports energy within the object and is therefore an essential part of any dynamical SF model. c) RT calculations tell us which of the processes/structures are visible at what wavelength by which telescope/instrument. Hence, RT is the central tool to analyze simulation results or to explore the scientific capabilities of planned instruments. d) With inverse RT, we can obtain the 1D-3D density and temperature structure from observations, completely decoupled from any (M)HD modeling (and the approximations made within).

In this review, we summarize the main difficulties and the currently used computational techniques to calculate the RT in SF regions. Recent applications of 3D continuum RT in molecular clouds and disks around young massive stars are discussed to illustrate the capabilities and limits of current RT modeling.