I review what has been learnt so far regarding the origin of stellar properties from numerical simulations of the formation of groups and clusters of stars. In agreement with observations, stellar properties are found to be relatively robust to variations of initial conditions in terms of molecular cloud structure and kinetics, as long as extreme initial conditions (e.g. strong central condensation, weak or no turbulence) and small-scale driving are avoided, but properties may differ between bound and unbound clouds. Radiative feedback appears crucial for setting stellar masses, even for low-mass stars, while magnetic fields can provide low star formation rates.

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 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.

We review the various theories which have been proposed along the years to explain the origin of the stellar initial mass function. We pay particular attention to four models, namely the competitive accretion and the theories based respectively on stopped accretion, MHD shocks and turbulent dispersion. In each case, we derive the main assumptions and calculations that support each theory and stress their respective successes and failures or difficulties.

Firstly, we give a historical overview of attempts to incorporate magnetic fields into the Smoothed Particle Hydrodynamics method by solving the equations of Magnetohydrodynamics (MHD), leading an honest assessment of the current state-of-the-art in terms of the limitations to performing realistic calculations of the star formation process. Secondly, we discuss the results of a recent comparison we have performed on simulations of driven, supersonic turbulence with SPH and Eulerian techniques. Finally we present some new results on the relationship between the density variance and the Mach number in supersonic turbulent flows, finding σ2ln ρ = ln(1 + b22 with b = 0.33 up to Mach 20, consistent with other numerical results at lower Mach number (Lemaster & Stone 2008) but inconsistent with observational constraints on σρ and in Taurus and IC5146.

We provide a brief overview of recent advances and outstanding issues in simulations of interstellar turbulence, including isothermal models for interior structure of molecular clouds and larger-scale multiphase models designed to simulate the formation of molecular clouds. We show how self-organization in highly compressible magnetized turbulence in the multiphase ISM can be exploited in simple numerical models to generate realistic initial conditions for star formation.

I provide a pedagogic review of adaptive mesh refinement (AMR) radiation hydrodynamics (RHD) methods and codes used in simulations of star formation, at a level suitable for researchers who are not computational experts. I begin with a brief overview of the types of RHD processes that are most important to star formation, and then I formally introduce the equations of RHD and the approximations one uses to render them computationally tractable. I discuss strategies for solving these approximate equations on adaptive grids, with particular emphasis on identifying the main advantages and disadvantages of various approximations and numerical approaches. Finally, I conclude by discussing areas ripe for improvement.

In this paper, we review the radiative hydrodynamics methods based upon Smoothed Particle Hydrodynamics(SPH). There are already various implementations so far, which can be categorized into three types: moment equation solvers, Monte Carlo methods, and ray-tracing schemes. These codes have been applied to various astrophysical problems including dynamics of dense proto-stellar cores, photoionization feedback of massive stars on molecular clouds, radiative feedback in the early universe, etc. Among these different methods, we focus on the ray-tracing schemes. We also describe one particular ray-tracing code RSPH in some details.

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.

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.

We present radiation hydrodynamics simulations of the collapse of massive pre-stellar cores. We treat frequency dependent radiative feedback from stellar evolution and accretion luminosity at a numerical resolution down to 1.27 AU. In the 2D approximation of axially symmetric simulations, it is possible for the first time to simulate the whole accretion phase of several 105 yr for the forming massive star and to perform a comprehensive scan of the parameter space. Our simulation series show evidently the necessity to incorporate the dust sublimation front to preserve the high shielding property of massive accretion disks. Our disk accretion models show a persistent high anisotropy of the corresponding thermal radiation field, yielding to the growth of the highest-mass stars ever formed in multi-dimensional radiation hydrodynamics simulations. Non-axially symmetric effects are not necessary to sustain accretion. The radiation pressure launches a stable bipolar outflow, which grows in angle with time as presumed from observations. For an initial mass of the pre-stellar host core of 60, 120, 240, and 480⊙ the masses of the final stars formed in our simulations add up to 28.2, 56.5, 92.6, and at least 137.2⊙ respectively.

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.

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).

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.

Forming stars emit a significant amount of radiation into their natal environment. While the importance of radiation feedback from high-mass stars is widely accepted, radiation has generally been ignored in simulations of low-mass star formation. I use ORION, an adaptive mesh refinement (AMR) three-dimensional gravito-radiation-hydrodynamics code, to model low-mass star formation in a turbulent molecular cloud. I demonstrate that including radiation feedback has a profound effect on fragmentation and protostellar multiplicity. Although heating is mainly confined within the core envelope, it is sufficient to suppress disk fragmentation that would otherwise result in low-mass companions or brown dwarfs. As a consequence, turbulent fragmentation, not disk fragmentation, is likely the origin of low-mass binaries.

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.

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.

Protostellar outflows can inject sufficient mass, momentum, and kinetic energy into their parent star-forming clumps to dramatically alter their structure, generate turbulence, and even to disrupt them. Outflows represent the lowest rung on a ‘feedback ladder’ consisting of increasingly powerful mechanisms which kick-in if star formation escalates towards the production of more massive stars, higher efficiency, and larger clusters. Outflow feedback may dominate turbulence generation and cloud disruption on the scale of cluster-forming clumps having dimensions up to a few parsecs. Outflows inject energy and momentum on a wide-range of length-scales from less than 0.01 pc to over 30 pc. However, they fail by several orders of magnitude to inject sufficient momentum and kinetic energy to drive turbulent motions on the size and mass-scales of GMCs. Injection from higher rungs on the feedback ladder or momentum injected by Galactic-scale processes are needed to power the observed turbulence on the 10 to 100 pc scales of GMCs.

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.

We present simulations of stable isothermal clouds exposed to ionizing radiation from a discrete external source, and identify the conditions that lead to Radiatively Driven Implosion and Star Formation. We use the Smoothed Particle Hydrodynamics code SEREN (Hubber et al. 2010) and the HEALPix-based photoionization algorithm described in Bisbas et al. (2009). We find that the incident ionizing flux is the critical parameter determining the evolution; high fluxes disperse the cloud, whereas low fluxes trigger star formation. We find a clear connection between the intensity of the incident flux and the parameters of star formation.