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.

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.

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.

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.

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.

It is now well established that the majority of young stars are found in multiple systems, so that any theory of stellar formation must account for their existence and properties. Studying the properties of multiple star systems therefore represents a very powerful approach to place observational constraints on star formation theories. Additionally, multiple systems offer other advantages. They provide the most accurate and unambiguous way to measure masses, using orbital fitting and Kepler's laws, and even the stellar radius in the special case of eclipsing binaries. They also allow to compare the properties of 2 coeval objects with different masses, providing important tests for the evolutionary models.

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.

Extinction maps at 8μm from the Spitzer Space Telescope show that many Class 0 protostars exhibit complex, irregular, and non-axisymmetric structure within the densest regions of their dusty envelopes. Many of the systems have highly irregular and non-axisymmetric morphologies on scales ~1000 AU, with a quarter of the sample exhibiting filamentary or flattened dense structures. Complex envelope structure is observed in regions spatially distinct from outflow cavities, and the densest structures often show no systematic alignment perpendicular to the cavities. We suggest that the observed envelope complexity is the result of collapse from protostellar cores with initially non-equilibrium structures. The striking non-axisymmetry in many envelopes could provide favorable conditions for the formation of binary systems. We then show that the kinematics around L1165 as probed with N2H+ are indicative of asymmetric infall; the velocity gradient is not perpendicular to the outflow.

A systematic, high angular-resolution study of IR-quiet Massive Dense Cores (MDCs) of Cygnus-X in continuum and high-density molecular tracers is presented. The results are compared with the quasi-static and the dynamical evolutionary scenario. We find that the fragmentation properties are not compatible with the quasi-static, monolithic collapse scenario, nor are they entirely compatible with the formation of a cluster of mostly low-mass stars. The kinematics of MDCs shows individual velocity components appearing as coherent flows, which indicate important dynamical processes at the scale of the mass reservoir around high-mass protostars.

There has been considerable progress in our understanding of how massive stars form but still much confusion as to why they form. Recent work from several sources has shown that the formation of massive stars through disc accretion, possibly aided by gravitational and Rayleigh-Taylor instabilities is a viable mechanism. Stellar mergers, on the other hand, are unlikely to occur in any but the most massive clusters and hence should not be a primary avenue for massive star formation. In contrast to this success, we are still uncertain as to how the mass that forms a massive star is accumulated. there are two possible mechanisms including the collapse of massive prestellar cores and competitive accretion in clusters. At present, there are theoretical and observational question marks as to the existence of high-mass prestellar cores. theoretically, such objects should fragment before they can attain a relaxed, centrally condensed and high-mass state necessary to form massive stars. Numerical simulations including cluster formation, feedback and magnetic fields have not found such objects but instead point to the continued accretion in a cluster potential as the primary mechanism to form high-mass stars. Feedback and magnetic fields act to slow the star formation process and will reduce the efficiencies from a purely dynamical collapse but otherwise appear to not significantly alter the process.

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.

Low-mass protostars are less luminous than expected. This luminosity problem is important because the observations appear to be inconsistent with some of the basic premises of star formation theory. Two possible solutions are that stars form slowly, which is supported by recent data, and/or that protostellar accretion is episodic; current data suggest that the latter accounts for less than half the missing luminosity. The solution to the luminosity problem bears directly on the fundamental problem of the time required to form a low-mass star. The protostellar mass and luminosity functions provide powerful tools both for addressing the luminosity problem and for testing theories of star formation. Results are presented for the collapse of singular isothermal spheres, for the collapse of turbulent cores, and for competitive accretion.

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.

Stars form from molecular clouds, mostly in clusters with tens to tens of thousands of members, and the mass distribution within these clusters, or the Initial Mass Function, seems to be invariable against many parameters and over a wide range of masses. However, masses are a very difficult quantity to assess, and the precision of our determinations of the IMF is systematically lower than usually quoted. I will discuss the process of determining masses from observations and the type of uncertainties associated with this process.

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.

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.

Massive stars influence the surrounding universe far out of proportion to their numbers through ionizing radiation, supernova explosions, and heavy element production. Their formation requires the collapse of massive interstellar gas clouds with very high accretion rates. We discuss results from the first three-dimensional simulations of the gravitational collapse of a massive, rotating molecular cloud core that include heating by both non-ionizing and ionizing radiation. Local gravitational instabilities in the accretion flow lead to the build-up of a small cluster of stars. These lower-mass companions subsequently compete with the high-mass star for the same common gas reservoir and limit its overall mass growth. This process is called fragmentation-induced starvation, and explains why massive stars are usually found as members of high-order stellar systems. These simulations also show that the H ii regions forming around massive stars are initially trapped by the infalling gas, but soon begin to fluctuate rapidly. Over time, the same ultracompact H ii region can expand anisotropically, contract again, and take on any of the observed morphological classes. The total lifetime of H ii regions is given by the global accretion timescale, rather than their short internal sound-crossing time. This solves the so-called lifetime problem of ultracompact H ii region. We conclude that the the most significant differences between the formation of low-mass and high-mass stars are all explained as the result of rapid accretion within a dense, gravitationally unstable flow.

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).

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.

We use images derived from collapsing, turbulent molecular cloud simulations without sinks to explore the effects of finite image angular resolution and noise on the derived clump mass function. These effects randomly perturb the clump masses, producing a lognormal clump mass function with a Salpeter-like high mass end. We show that the characteristic break mass of the simulated clump mass functions changes with the angular resolution of the images in a way that is entirely consistent with the observations. We also present some cautionary tales regarding sink particles and highlight the need to ensure that sinks actually correspond to distinct collapsing objects. We test several popular numerical sink criteria in the literature and compare to converged, non-sink results.