Turbulence is a remarkable subject in physics. The underlying equations, which are in their simplest formulation the Euler equations, were published 250 years ago (Euler 1757). Yet a theoretical grasp of the phenomenology emerging from these equations had not been achieved before the mid-twentieth century, when Heisenberg (1923) and Kolmogorov (1941) obtained their first analytical results. Eventually, it took the capabilities of modern supercomputers to obtain a full appreciation of the complexity that is inherent to the Euler equations. Astrophysics is now at the very frontier of numerical turbulence modelling. Among the additional ingredients for making turbulence in astrophysics even more complex are supersonic flow, self-gravity, magnetic fields and radiation transport. In contrast, terrestrial turbulence is mostly incompressible or only weakly compressible. External gravity is, of course, an issue in the computation of atmospheric processes on Earth. Self-gravity, however, is only encountered on large, astrophysical scales. The dynamics of turbulent plasma has met vivid attention in research related to nuclear fusion reactors but, otherwise, is not encountered under terrestrial conditions.

In this chapter, I give an overview of the various approaches towards the numerical modelling of turbulence, particularly, in the interstellar medium (ISM). The discussion is placed in a physical context, i.e. computational problems are motivated from basic physical considerations. Presenting selected examples for solutions to these problems, I introduce the basic ideas of the most commonly used numerical methods. For detailed methodological accounts, the reader is invited to follow the references.

Introduction

Although not precisely in its infancy, the question of building planets and planetary systems still faces many challenges: how are planetesimals assembled from micrometre-sized grains? What does radial transport do to growing dust aggregates? Do solids concentrate close to the star or do (metre size) objects vanish rapidly into the central star? How are planets formed from planetesimals? And how do giant planets form that have to acquire hydrogen and helium before the gas is accreted onto the star or is swept away by stellar winds and photoevaporation? Are many generations of planets formed and then lost? Can we explain the compositions of planets in the structure of our present solar system?

These are but a few of the basic questions which are currently the focus of a highly active research field. Presenting a complete overview of the problem is beyond the scope of this short chapter and would not be a long-lasting one as the field is rapidly evolving. We present here what we believe are some important pieces of the puzzle, in the domain of planetesimal formation and giant planet composition.

Planetesimals

There are several important steps of structure formation after the solar nebula formed and before full-size planets came into existence. One of them is the formation of planetesimals. There is some ambiguity in the term planetesimal as it is used throughout the literature.

We have heard a lot about probability functions p(M) at this meeting for mass M of planets or stars or clouds or clusters under various conditions. Since we have covered such an enormous range of masses, it is not surprising that power-law distributions close to the scale-invariant power have recurred so often. A power law differing from this distribution in the direction of favouring either low or high masses must of course have a turnover (or termination) towards this end to avoid a divergence. The physical reason for such a turnover is of interest, as is the question of continuity between the various types of objects. Bingelli and Hascher (PASP 119, 592, 2007) have followed this power-law continuity over 36 orders of magnitude in mass from asteroids to galaxy superclusters. It is instructive to look at similar probability distribution functions in quite different fields. I will give only the examples of two different kinds of human aggregates. One example, which has been discussed for more than a century or so, is the probability distribution for the size (i.e. the number of inhabitants) of a village, town or city. Near the end of the nineteenth century, the deviation from scale invariance was a slight increase towards the bottom end, i.e. overall slightly more people lived in a village of population 100-200 than in a city of 250 000 to 500 000.

Introduction

The formation of molecular clouds (MCs) from the diffuse interstellar gas is a necessary step for star formation, as young stars invariably occur within them. However, the mechanisms controlling the formation of MCs remain controversial. In this contribution, we focus on their formation in compressive flows driven by interstellar turbulence and large-scale gravitational instability.

Turbulent compression driven by supernovae appears insufficient to explain the bulk of cloud and star formation. Rather, gravity must be important at all scales, driving the compressive flows that form both clouds and cores. Cooling and thermal instability allow the formation of dense gas out of moderate, transonic compressions in the warm diffuse gas and drive turbulence into the dense clouds. MCs may be produced by an overshoot beyond the thermal-pressure equilibrium between the cold and the warm phases of atomic gas, caused by some combination of the ram pressure of compression and the self-gravity of the compressed gas.

In this case, properties of the clouds such as their mass, mass-to-magnetic flux ratio, and total kinetic and gravitational energies are in general time-variable quantities. MCs may never enter a quasi-equilibrium or virial equilibrium state but rather continuously collapse to stars. Gravitationally collapsing clouds exhibit a pseudo-virial energy balance |Egrav|~2Ekin, which, however, is representative of contraction rather than of virial equilibrium in this case. However, compression-driven cloud and core formation still involves significant delays as additional material accretes, leading to lifetimes longer than the free-fall time.

Introduction

This conference has brought together researchers studying structure formation in astrophysics, at scales ranging from planet and star formation through galaxy formation to cosmic structure formation. Aside from gravity, these fields require knowledge about many further physical processes and phenomena, such as turbulent gas dynamics, magnetic fields, non-equilibrium chemistry and the interaction of radiation with matter. The different communities also all rely on numerical simulations and the same modern, general-purpose, ground-based and space-borne telescopes.

In this proceedings contribution, we attempt to identify some of the major challenges for the future. We furthermore debate whether the physical processes relevant for each field exhibit sufficient overlap to warrant concerted cross-disciplinary efforts or whether the features that define and distinguish these fields prevail and make successful cross-fertilization less likely.

Planet formation

With the first discovery of a planet around another star in 1995, we have begun to place our solar system in the context of other planetary systems. More than 250 extrasolar planets have been identified, most with characteristics vastly different from our own solar system. Planets around stars such as our Sun may be the rule, rather than the exception, but the observed properties exhibit an enormous spread (see Udry & Santos 2007).

In the core instability model, planet formation begins with the coagulation of dust in protoplanetary disks, forming larger aggregates of solid material through a sequence of collisions and agglomeration.

Introduction

The collapse of a molecular cloud core leads to the creation of a newborn star and attendant circumstellar accretion disk. The action of disk accretion during and after collapse likely controls the initial mass and angular momentum of these young stellar objects (YSOs). It also has a hand in shaping the conditions under which planetary systems are born. Determining the observational properties of YSOs, particularly their variations with environment, mass and time, can therefore place stringent constraints on the physical processes at play during both stellar and planetary formation. In this chapter, we give an overview of the current knowledge of YSO properties, with an emphasis on their implications for both processes.

Circumstellar disks are a ubiquitous outcome of the star formation process, making them a powerful probe of YSO evolution. In addition, disks are the birthsite of planetary systems and can therefore be used to constrain the overall planet formation process (via statistical analyses) and some of the key physical mechanisms, such as grain growth, vertical settling or radial migration (via detailed studies of individual objects). We focus our discussion on the analysis of young, optically thick, gas-rich protoplanetary disks. In particular, we focus on their dust component which, although it amounts to only a tiny fraction (on the order of 1%) of the total mass, represents the building blocks of planetesimals and planets. We also address some observations of more evolved debris disks.

Introduction

The dust and gas disks surrounding many pre-main-sequence stars are thought to be the birthplaces of planets. They are therefore the locations of structure formation in the universe, albeit on small scales. In recent years, the topic of protoplanetary disks has gained an increasing amount of attention in astrophysics, which is in large part due to the enormous increase of high-quality observational data that have been published recently, from the Spitzer Space Telescope, from the Very Large Telescope, from millimetre interferometers and so forth. Moreover, studies of the properties of extrasolar planetary systems have put new constraints on the formation and migration of newly born planets in such disks. The topic of the structure and evolution of protoplanetary disks is wide open, and significant new developments, both from theory and from observations, can be expected in the near future.

This chapter gives an overview of some aspects of theoretical modelling of protoplanetary disks. The review consists of four parts. First, we discuss the overall formation, evolution and dispersal of protoplanetary disks. We show that much of what we know of the long timescale disk evolution is still based on relatively simple models. We discuss their vertical structure, as derived from radiative transfer models and their comparison to observations.We then show that magneto-rotational and gravitational instabilities (GIs) can introduce complex dynamics and non-linear structure to these disks, and we discuss an on-going debate about the role of GIs in planet formation.

Turbulence in fluids is a topic of great interest. First and foremost, most flows in nature are turbulent and this is particularly true in the astrophysical context (Kritsuk & Norman 2004). Also, turbulence leads to very peculiar mechanics that still escapes to a great extent from our understanding. Since the pioneering works conducted by Osborne Reynolds at the end of the nineteenth century (around 1895), turbulence in fluids has become a rich and challenging research subject in which scientists from engineering, theoretical and experimental physics have been involved with many different perspectives. There is no doubt that bridging ideas from one field to another, and therefore stimulating new interdisciplinary approaches, should provide a fruitful means of gaining understanding on turbulence in the future.

In this chapter, the background physics of turbulence will be discussed spontaneously at a (very) basic level, i.e. without getting into details or precise formulation. The discussion will be limited to incompressible hydrodynamics governed by the Navier-Stokes (NS) equations. Firstly, general comments on turbulence (as a statistical-mechanical problem) will be made. Then, I shall attempt to provide some hints (rather than definite answers) to a series of questions: What is generally the source of turbulence? What are the main statistical features of turbulence? How to deal with turbulence? Much more elaborated developments and references may be sought in the following books (among many others) dealing with turbulence:

• a reference book on the physics of turbulence: A first course in turbulence by H. Tennekes and J. L. Lumley, MIT Press, Cambridge, USA (1972)

• […]

Introduction

A significant fraction of all efforts in astronomy is expended on studying the properties of galaxies and their spatial distributions over a very large fraction of all of cosmic time. Much of this is motivated by the mysteries of understanding dark matter and dark energy. The accuracy at which we can use galaxy clustering and other properties to determine the more fundamental parameters describing the universe is determined by the sophistication of our understanding of galaxies themselves.

As is by now well known, hierarchical cosmological models like ΛCDM predict that the first objects to collapse out of the expanding homogeneous universe are also the smallest. Since the early universe was also generally a simpler place (fewer heavy elements, etc.), this produces the anti-intuitive result that it is often easier to model the early universe. For the present-day universe, the significant complexities of the star formation process and the enormous range of relevant spatial and temporal scales forces us to employ phenomenological sub-grid models. In this contribution, we explore some aspects of our current understanding of galaxy formation with a distinctly numerical bias. We cannot hope to comprehensively review this topic, so instead we begin with a general overview of some of the important physical processes which are operating and then delve into a few current issues in more detail.

Abstract

Cosmology faces three distinct challenges in the next decade. (i) The dark sector, both dark matter and dark energy, dominates the universe. Key questions include determining the nature of the dark matter and whether dark energy can be identified with, or if dynamical, replace, the cosmological constant. Nor, given the heated level of current debates about the nature of gravity and string theory, can one yet unreservedly accept that dark matter or the cosmological constant/dark energy actually exists. Improved observational probes are crucial in this regard. (ii) Galaxy formation was initiated at around the epoch of reionization: we need to understand how and when the universe was reionized, as well as to develop probes of what happened at earlier epochs. (iii) Our simple dark matter-driven picture of galaxy assembly is seemingly at odds with several observational results, including the presence of ultraluminous infrared galaxies (ULIRGS) at high redshift, the ‘downsizing’ signature whereby massive objects terminate their star formation prior to those of lower masses, chemical signatures of α-element ratios in early-type galaxies and suggestions that merging may not be important in defining the Hubble sequence. Any conclusions, however, are premature, given current uncertainties about possible hierarchy-inverting processes involved with feedback. Understanding the physical implications of these observational results in terms of a model of star formation in galaxies is a major challenge for theorists and refining the observational uncertainties is a major goal for observers.

Introduction: dense cores and the origin of the IMF

Stars form from the gravitational collapse of dense cloud cores in the molecular interstellar medium of galaxies. Studying and characterizing the properties of dense cores is thus of great interest to gain insight into the initial conditions and initial stages of the star formation process.

Our observational understanding of low-mass dense cores has made significant progress in recent years, and three broad categories of cores can now be distinguished within nearby molecular clouds, which possibly represent an evolutionary sequence: starless cores, prestellar cores and ‘Class 0’ protostellar cores. Starless cores are possibly transient concentrations of molecular gas and dust without embedded young stellar objects (YSOs), typically observed in tracers such as C18O (Onishi et al. 1998), NH3 (Jijina et al. 1999) or dust extinction (Alves et al. 2007), and which do not show evidence of infall. Prestellar cores are also starless (M* = 0) but represent a somewhat denser and more centrally concentrated population of cores which are self-gravitating, hence unlikely to be transient. They are typically detected in (sub)millimetre dust continuum emission and dense molecular gas tracers such as NH3 or N2H+ (Benson & Myers 1989; Ward-Thompson et al. 1994; Caselli et al. 2002), are often seen in absorption at mid- to far-infrared wavelengths (Bacmann et al. 2000; Alves et al. 2001) and frequently exhibit evidence of infall motions (Gregersen & Evans 2000). Conceptually, all prestellar cores are starless but only a subset of the starless cores evolve into prestellar cores; the rest are presumably ‘failed’ cores that eventually disperse and never form stars.

Abstract

We describe numerical methods for solving the equations of radiation magnetohydrodynamics (MHD) for astrophysical fluid flow. Such methods are essential for the investigation of the time-dependent and multidimensional dynamics of a variety of astrophysical systems, although our particular interest is motivated by problems in star formation. Over the past few years, the authors have been members of two parallel code development efforts, and this review reflects that organization. In particular, we discuss numerical methods for MHD as implemented in the Athena code, and numerical methods for radiation hydrodynamics as implemented in the Orion code. We discuss the challenges introduced by the use of adaptive mesh refinement (AMR) in both codes, as well as the most promising directions for future developments.

Introduction

The dynamics of astrophysical systems described by the equations of radiation magnetohydrodynamics (MHD) span a tremendous range of scales and parameter regimes, from the interiors of stars (Kippenhahn & Weigert 1994), to accretion disks around compact objects (Turner et al. 2003), to dusty accretion flows around massive protostars (Krumholz et al. 2005, 2007a), to galactic-scale flows onto AGN (Thompson et al. 2005). All of these systems have in common that matter, radiation and magnetic fields are strongly interacting and that the energy and momentum carried by the radiation field is significant in comparison to that carried by the gas. Thus, an accurate treatment of the problem must include analysis of both the matter and the radiation, as well as the magnetic fields, and their mutual interaction.

Understanding the formation of gravitationally bound structures at all scales in the universe is one of the most fascinating challenges of modern astronomy. It is now realized that the initial building blocks of galaxies were small collapsing dark matter halos, produced by the primordial fluctuations. These blocks then merged and were assembled into progressively larger galaxies, a scheme generally described as the hierarchical model of galaxy formation. The modern understanding of star formation involves large-scale turbulent motions producing local overdensities which eventually collapse and form prestellar cores under the action of gravity. The most likely scenario for planet formation is the collapse of a vast gaseous envelope onto a central dense core formed from the aggregation of millimetre-size grains in the original protoplanetary nebula, although disk fragmentation could remain an alternative scenario in some situations. The detailed processes responsible for the formation of these structures, however, remain poorly understood. Many important issues remain unsettled, so the robustness of these general paradigms is still ill determined. All these scenarios for the formation of galaxies, stars and planets, although involving vastly different scales, share many underlying physical mechanisms.They all involved hydrodynamical processes, generally leading to turbulent motions, but the very nature of these motions and their real role in structure formation remains unclear. The role of magnetic fields, in the collapse itself and in the generation of winds and jets, remains one of the major unknowns in the formation of structures.

Abstract

Astrophysical jets are associated with the formation of young stars of all masses, stellar and massive black holes, and perhaps even with the formation of massive planets. Their role in the formation of planets, stars and galaxies is increasingly appreciated and probably reflects a deep connection between the accretion flows – by which stars and black holes may be formed – and the efficiency by which magnetic torques can remove angular momentum from such flows. We compare the properties and physics of jets in both non-relativistic and relativistic systems and trace, by means of theoretical argument and numerical simulations, the physical connections between these different phenomena. We discuss the properties of jets from young stars and black holes, give some basic theoretical results that underpin the origin of jets in these systems, and then show results of recent simulations on jet production in collapsing star-forming cores as well as from jets around rotating Kerr black holes.

Introduction

The goal of this book, to explore structure formation in the cosmos and the physical linkage of astrophysical phenomena on different physical scales, is both timely and important. The emergence of multi-wavelength astronomy in the late twentieth century with its unprecedented ground- and space-based observatories, as well as the arrival of powerful new computational capabilities and numerical codes, has opened up unanticipated new vistas in understanding how planets, stars and galaxies form.

Introduction

This chapter is devoted to planet formation and to the early stages of evolution of low-mass objects, including low-mass stars, brown dwarfs and exoplanets. We first summarize the general properties of current exoplanet observations (Section 15.2) and describe the two main planet formation models based on disk instability and on the core-accretion scenario, respectively (Section 15.3). Recent progress of the latter formation model allows sophisticated population synthesis analyses which provide fully quantitative predictions that can be compared to the observed statistical properties of exoplanets (Section 15.3.5). The last part of this chapter is devoted to the distinction between brown dwarfs and planets, in terms of structure and evolutionary properties. The existence of a mass overlap between these two distinct populations of low-mass objects is highlighted by the increasing discoveries of very massive exoplanets (M ≳ 5MJ) and by the identification of planetary mass brown dwarfs in young clusters (M ≲ 10MJ) These discoveries stress the importance to define signatures which could allow to disentangle a brown dwarf from a planet. We first analyse the effect of accretion on the evolution of young brown dwarfs and the resulting uncertainties of evolutionary models at ages of a few million years. We also analyse different specific signatures of brown dwarfs and planets such as their luminosity at young ages, their radii and their atmospheric properties.