We review a range of statistical methods for analysing the structures of star clusters, and derive a new measure Q, which both quantifies and distinguishes between a (relatively smooth) large-scale radial density gradient and multiscale (fractal) subclustering.The distribution of separations p(s) is considered, and the normalized correlation lengths (i.e. the mean separation between stars, divided by the overall radius of the cluster) is shown to be a robust indicator of the extent to which a smooth cluster is centrally concentrated. For spherical clusters having volume-density n ∝ r −α (with α between 0 and 2)s decreases monotonically with α, from ∼0.8 to ∼0.6. Sinces reflects all star positions, it implicitly incorporates edge effects. However, for fractal star clusters (with fractal dimension D between 1.5 and 3)s decreases monotonically with D (from ∼0.8 to ∼0.6). Hences, on its own, can quantify, but cannot distinguish between, a smooth large-scale radial density gradient and multiscale (fractal) subclustering.The minimal spanning tree (MST) is then considered, and it is shown that the normalized mean edge lengthm [i.e. the mean length of the branches of the tree, divided by (N total A) 1/2 /(N total − 1), where A is the area of the cluster and N total is the number of stars] can also quantify, but again cannot on its own distinguish between, a smooth large-scale radial density gradient and multiscale (fractal) subclustering.However, the combination Q =m/s does both quantify and distinguish between a smooth large-scale radial density gradient and multiscale (fractal) subclustering. IC348 has Q = 0.98 and ρ Ophiuchus has Q = 0.85, implying that both are centrally concentrated clusters with, respectively, α 2.2 ± 0.2 and α 1.2 ± 0.3. Chamaeleon and IC2391 have Q = 0.67 and 0.66, respectively, implying mild substructure with a notional fractal dimension D 2.25 ± 0.25. Taurus has even more substructure, with Q = 0.45 implying D 1.55 ± 0.25. If the binaries in Taurus are treated as single systems, Q increases to 0.58 and D increases to 1.9 ± 0.2.

We suggest that a high proportion of brown dwarf (BD) stars are formed by gravitational fragmentation of massive extended discs around Sun-like primary stars. We argue that such discs should arise frequently, but should be observed infrequently, precisely because they fragment rapidly. By performing an ensemble of radiation-hydrodynamic simulations, we show that such discs typically fragment within a few thousand years, and produce mainly BD stars, but also planetary-mass (PM) stars and very low-mass hydrogen-burning (HB) stars. Subsequently most of the lower mass stars (i.e. the PM and BD stars) are ejected by mutual interactions. We analyse the statistical properties of these stars, and compare them with observations.After a few hundred thousand years the Sun-like primary is typically left with a close lowmass HB companion, and two much wider companions: a low-mass HB star and a BD star, or a BD-BD binary. The orbits of these companions are highly eccentric, and not necessarily coplanar, either with one another, or with the original disc. There is a BD desert extending out to at least ∼100 au; this is because BDs tend to be formed further out than low-mass HB stars, and then they tend to be scattered even further out, or even into the field.BDs form with discs of a few Jupiter masses and radii of a few tens of au, and they are more likely to retain these discs if they remain bound to the primary star. Binaries form by pairing of the newly formed stars in the disc, giving a low-mass binary fraction of ∼0.16. These binaries include close and wide BD/BD binaries and BD/PM binaries. Binaries can be ejected into the field and survive, even if they have quite wide separations. BDs that remain as companions to Sun-like stars are more likely to be in BD/BD binaries than are BDs ejected into the field. The presence of close and distant companions around Sun-like stars may inhibit planet formation by core accretion.We conclude that disc fragmentation is a robust mechanism for BD formation. Even if only a small fraction of Sun-like stars host the required massive extended discs, this mechanism can produce all the PM stars observed, most of the BD stars and a significant proportion of the very low-mass HB stars.

Aims. We introduce and test a new and highly efficient method for treating the thermal and radiative effects influencing the energy equation in SPH simulations of star formation. Methods. The method uses the density, temperature and gravitational potential of each particle to estimate a mean optical depth, which then regulates the particle's heating and cooling. The method captures -at minimal computational cost -the effects of (i) the rotational and vibrational degrees of freedom of H 2 ; (ii) H 2 dissociation and H o ionisation; (iii) opacity changes due to ice mantle melting, sublimation of dust, molecular lines, H − , bound-free and free-free processes and electron scattering; (iv) external irradiation; and (v) thermal inertia. Results. We use the new method to simulate the collapse of a 1 M cloud of initially uniform density and temperature. At first, the collapse proceeds almost isothermally (T ∝ ρ 0.08 ; cf. Larson 2005, MNRAS, 359, 211). The cloud starts heating fast when the optical depth to the cloud centre reaches unity (ρ C ∼ 7 × 10 −13 g cm −3 ). The first core forms at ρ C ∼ 4 × 10 −9 g cm −3 and steadily increases in mass. When the temperature at the centre reaches T C ∼ 2000 K, molecular hydrogen starts to dissociate and the second collapse begins, leading to the formation of the second (protostellar) core. The results mimic closely the detailed calculations of Masunaga & Inutsuka (2000, ApJ, 531, 350). We also simulate (i) the collapse of a 1.2 M cloud, which initially has uniform density and temperature, (ii) the collapse of a 1.2 M rotating cloud, with an m = 2 density perturbation and uniform initial temperature, and (iii) the smoothing of temperature fluctuations in a static, uniform density sphere. In all these tests the new algorithm reproduces the results of previous authors and/or known analytic solutions. The computational cost is comparable to a standard SPH simulation with a simple barotropic equation of state. The method is easy to implement, can be applied to both particle-and grid-based codes, and handles optical depths 0 < τ < ∼ 1011 .

Thousands of exoplanets have now been discovered with a huge range of masses, sizes and orbits: from rocky Earth-like planets to large gas giants grazing the surface of their host star. However, the essential nature of these exoplanets remains largely mysterious: there is no known, discernible pattern linking the presence, size, or orbital parameters of a planet to the nature of its parent star. We have little idea whether the chemistry of a planet is linked to its formation environment, or whether the type of host star drives the physics and chemistry of the planet's birth, and evolution. ARIEL was conceived to observe a large number (~1000) of transiting planets for statistical understanding, including gas giants, Neptunes, super-Earths and Earth-size planets around a range of host star types using transit spectroscopy in the 1.25-7.8 μm spectral range and multiple narrow-band photometry in the optical. ARIEL will focus on warm and hot planets to take advantage of their well-mixed atmospheres which should show minimal condensation and sequestration of high-Z materials compared to their colder Solar System siblings. Said warm and hot atmospheres are expected to be more representative of the planetary bulk composition. Observations of these warm/hot exoplanets, and in particular of their elemental composition (especially C, O, N, S, Si), will allow the understanding of the early stages of planetary and atmospheric formation during the nebular phase and the following few million years. ARIEL will thus provide a representative picture of the chemical nature of the exoplanets and relate this directly to the type and chemical environment of the host star. ARIEL is designed as a dedicated survey mission for combined-light spectroscopy, capable of observing a large and welldefined planet sample within its 4-year mission lifetime. Transit, eclipse and phasecurve spectroscopy methods, whereby the signal from the star and planet are differentiated using knowledge of the planetary ephemerides, allow us to measure atmospheric signals from the planet at levels of 10-100 part per million (ppm) relative to the star and, given the bright nature of targets, also allows more sophisticated techniques, such as eclipse mapping, to give a deeper insight into the nature of the atmosphere. These types of observations require a stable payload and satellite platform with broad, instantaneous wavelength coverage to detect many molecular species, probe the thermal structure, identify clouds and monitor the stellar activity. The wavelength range proposed covers all the expected major atmospheric gases from e.g. H 2 O, CO 2 , CH 4 NH 3 , HCN, H 2 S through to the more exotic metallic compounds, such as TiO, VO, and condensed species. Simulations of ARIEL performance in conducting exoplanet surveys have been performedusing conservative estimates of mission performance and a

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