It is possible to extract, from the observations, distribution functions of the birth dynamical properties of a stellar population, and to also infer that these are quite invariant to the physical conditions of star formation. The most famous example is the stellar IMF, and the initial binary population (IBP) seems to follow suit. A compact mathematical formulation of the IBP can be derived from the data. It has three broad parts: the IBP of the dominant stellar population (0.08–2M⊙), the IBP of the more-massive stars and the IBP of brown dwarfs. These three mass regimes correspond to different physical regimes of star formation but not to structure in the IMF. With this formulation of the IBP it becomes possible to synthesize the stellar-population of whole galaxies.

We briefly review ground-based (sub)millimeter dust continuum observations of the prestellar core mass function (CMF) and its connection to the stellar initial mass function (IMF). We also summarize the first results obtained on this topic from the Herschel Gould Belt survey, one of the largest key projects with the Herschel Space Observatory. Our early findings with Herschel confirm the existence of a close relationship between the CMF and the IMF. Furthermore, they suggest a scenario according to which the formation of prestellar cores occurs in two main steps: 1) complex networks of long, thin filaments form first, probably as a result of interstellar MHD turbulence; 2) the densest filaments then fragment and develop prestellar cores via gravitational instability.

Graphics Processing Units (GPUs) offer a new way to accelerate numerical calculations by means of on-board massive parallelisation. We discuss two examples of GPU implementation relevant for cosmological simulations, an N-Body Particle-mesh solver and a radiative transfer code. The latter has also been ported on multi-GPU clusters. The range of acceleration (x30-x80) achieved here offer bright perspective for large scale simulations driven by GPUs.

Modern hydrodynamic simulations of galaxy formation are able to predict accurately the rates and locations of the assembly of giant molecular clouds in early galaxies. These clouds could host star clusters with the masses and sizes of real globular clusters. I describe current state-of-the-art simulations aimed at understanding the origin of the cluster mass function and metallicity distribution. Metallicity bimodality of globular cluster systems appears to be a natural outcome of hierarchical formation and gradually declining fraction of cold gas in galaxies. Globular cluster formation was most prominent at redshifts z > 3, when massive star clusters may have contributed as much as 20% of all galactic star formation.

Stars form in the densest regions of clouds of cold molecular hydrogen. Measuring structure in these clouds is far from trivial as 99% of the mass of a molecular cloud is inaccessible to direct observation. Over the last decade we have been developing an alternative, more robust density tracer technique based on dust extinction measurements towards background starlight. The new technique does not suffer from the complications plaguing the more conventional molecular line and dust emission techniques, and when used with these can provide unique views on cloud chemistry and dust grain properties in molecular clouds. In this brief communication we summarize the main results achieved so far using this technique.

We use numerical simulations to investigate how the expansion of an HII region is affected by an ambient magnetic field. First we consider the test problem of expansion in a uniform medium with a unidirectional magnetic field. We then describe the expansion of an HII region in a turbulent medium, taking as our initial conditions the results of and MHD turbulence simulation. We find that although in the uniform medium case the magnetic field does produce interesting effects over long length and timescales, in the turbulent medium case the main effect of the magnetic field is to reduce the efficiency of fragmentation of the molecular gas.

This contribution contains the introductory remarks that I presented at IAU Symposium 270 on “Computational Star Formation” held in Barcelona, Spain, May 31–June 4, 2010. I discuss the historical development of numerical MHD methods in astrophysics from a personal perspective. The recent advent of robust, higher-order accurate MHD algorithms and adaptive mesh refinement numerical simulations promises to greatly improve our understanding of the role of magnetic fields in star formation.

The study of the internal structure of star clusters provides important clues concerning their formation mechanism and dynamical evolution. There are both observational and numerical evidences indicating that open clusters evolve from an initial clumpy structure, presumably a direct consequence of the formation in a fractal medium, toward a centrally condensed state. This simple picture has, however, several drawbacks. There can be very young clusters exhibiting radial patterns maybe reflecting the early effect of gravity on primordial gas. There can be also very evolved clusters showing fractal patterns that either have survived through time or have been generated subsequently by some (unknown) mechanism. Additionally, the fractal structure of some open clusters is much clumpier than the average structure of the interstellar medium in the Milky Way, although in principle a very similar structure should be expected. Here we summarize and discuss observational and numerical results concerning this subject.

Radiative feedback and magnetic field are understood to have a strong impact on the protostellar collapse. We present high resolution numerical calculations of the collapse of a 1 M⊙ dense core in solid body rotation, including both radiative transfer and magnetic field. Using typical parameters for low-mass cores, we study thoroughly the effect of radiative transfer and magnetic field on the first core formation and fragmentation. We show that including the two aforementioned physical processes does not correspond to the simple picture of adding them separately. The interplay between the two is extremely strong, via the magnetic braking and the radiation from the accretion shock.

Galaxies cover a wide range of masses and star formation histories. In this review, I summarize some of the evolutionary key features of common galaxy types. At the high-mass end, very rapid, efficient early star formation is observed, accompanied by strong enrichment and later quiescence, well-described by downsizing scenarios. In the intermediate-mass regime, early-type galaxies may still show activity in low-mass environments or when being rejuvenated by wet mergers. In late-type galaxies, we find continuous, though variable star formation over a Hubble time. In the dwarf regime, a wide range of properties from bursty activity to quiescence is observed. Generally, stochasticity dominates here, and star formation rates and efficiencies tend to be low. Morphological types and their star formation properties correlate with environment.

Solving radiative transfer problems with ray casting methods is compared with the commonly used ‘Flux Limited Diffusion’ approximation. Whereas ray casting produces solutions that converge to the exact one as the number of rays is increased, flux-limited-diffusion is fundamentally a ‘look-alike’ method, which produces solutions that are reminiscent of the correct solution but which cannot be made to converge to it.

In star forming regions, we can observe different evolutionary stages of various objects and phenomena such as molecular clouds, protostellar jets and outflows, circumstellar disks, and protostars. However, it is difficult to directly observe the star formation process itself, because it is veiled by the dense infalling envelope. Numerical simulations can unveil the star formation process in the collapsing gas cloud. Recently, some studies showed protostar formation from the prestellar core stage, in which both molecular clouds and protostars are resolved with sufficient spatial resolution. These simulations showed fragmentation and binary formation, outflow and jet driving, and circumstellar disk formation in the collapsing gas clouds. In addition, the angular momentum transfer and dissipation process of the magnetic field in the star formation process were investigated. In this paper, I review recent developments in numerical simulations of low-mass star formation.

Using the AMR code ENZO we are simulating the formation of massive star clusters within turbulent Giant Molecular Clouds (GMCs). Here we discuss the simulations from the first stages of building realistic turbulent GMCs, to accurate star formation, and ultimately comprehensive feedback. These simulations aim to build a better understanding of how stars affect GMCs, helping to answer the questions of how long GMCs live and why only a small fraction of the GMC gas becomes stars.

We describe an algorithm for constructing fractal molecular clouds that obeys prescribed mass and velocity scaling relations.The algorithm involves a random seed, so that many different realisations corresponding to the same fractal dimension and the same scaling relations can be generated. It first generates all the details of the density field, and then position the SPH particles, so that the same simulation can be repeated with different numbers of particles to explore convergence. It can also be used to initialise finite-difference simulations. We then present preliminary numerical simulations of Hii regions expanding into such clouds, and explore the resulting patterns of star formation. If the cloud has low fractal dimension, it already contains many small self-gravitating condensations, and the principal mechanism of star formation is radiatively driven implosion. This results in star formation occurring quite early, throughout the cloud. The stars resulting from the collapse and fragmentation of a single condensation are often distributed in a filament pointing radially away from the source of ionising radiation; as the remainder of the condensation is dispersed, these stars tend to get left behind in the Hii region. If the cloud has high fractal dimension, the cloud does not initially contain dense condensations, and star formation is therefore delayed until the expanding Hii region has swept up a sufficiently massive shell. The shell then becomes gravitationally unstable and breaks up into protostars. In this collect-and-collapse mode, the protostars are distributed in tangential arcs, they tend to be somewhat more massive, and as the expansion of the shell stalls they move ahead of the ionisation front.

Supernovae are the most energetic stellar events and influence the interstellar medium by their gasdynamics and energetics. By this, both also affect the star formation positively and negatively. In this paper, we review the development of the complexity of investigations aiming at understanding the interchange between supernovae and their released hot gas with the star-forming molecular clouds. Commencing from analytical studies the paper advances to numerical models of supernova feedback from superbubble scales to galaxy structure. We also discuss parametrizations of star-formation and supernova-energy transfer efficiencies. Since evolutionary models from the interstellar medium to galaxies are numerous and apply multiple recipes of these parameters, only a representative selection of studies can be discussed here.

We developed a fast numerical scheme for solving ambipolar diffusion MHD equations with the strong coupling approximation, which can be written as the ideal MHD equations with an additional ambipolar diffusion term in the induction equation. The mass, momentum, magnetic fluxes due to the ideal MHD equations can be easily calculated by any Godunov-type schemes. Additional magnetic fluxes due to the ambipolar diffusion term are added in the magnetic fluxes, because of two same spatial gradients operated on the advection fluxes and the ambipolar diffusion term. In this way, we easily kept divergence-free magnetic fields using the constraint transport scheme. In order to overcome a small time step imposed by ambipolar diffusion, we used the super time stepping method. The resultant scheme is fast and robust enough to do the long term evolution of star formation simulations. We also proposed that the decay of alfen by ambipolar diffusion be a good test problem for our codes.

The formation of high-mass stars represents a challenge from both a theoretical and an observational point of view. Here, we present an overview of the current status of the observational research on this field, outlining the progress achieved in recent years on our knowledge of the initial phases of massive star formation. The fragmentation of cold, infrared-dark clouds, and the evidence for star formation activity on some of them will be discussed, together with the kinematics of the gas in hot molecular cores, which can give us insights on the mechanism leading to the birth of an OB star.

Gas materials in the inner Galactic disk continuously migrate toward the Galactic center (GC) due to interactions with the bar potential, magnetic fields, stars, and other gaseous materials. Those in forms of molecules appear to accumulate around 200 pc from the center (the central molecular zone, CMZ) to form stars there and further inside. The bar potential in the GC is thought to be responsible for such accumulation of molecules and subsequent star formation, which is believed to have been continuous throughout the lifetime of the Galaxy. We present 3-D hydrodynamic simulations of the CMZ that consider self-gravity, radiative cooling, and supernova feedback, and discuss the efficiency and role of the star formation in that region. We find that the gas accumulated in the CMZ by a bar potential of the inner bulge effectively turns into stars, supporting the idea that the stellar cusp inside the central 200 pc is a result of the sustained star formation in the CMZ. The obtained star formation rate in the CMZ, 0.03–0.1 M⊙, is consistent with the recent estimate based on the mid-infrared observations by Yusef-Zadeh et al. (2009).

In this contribution, I briefly review our empirical knowledge of disks around ≲2M⊙ pre-main sequence (T Tauri) stars, focusing first on the dichotomic question of their frequency before moving on to some more detailed disk properties (overall orientation, total mass, outer radius). Finally, I conclude with a brief discussion of disks around embedded protostars, which will play in the next few years a major role in testing star formation theory and simulations.