Galaxy models comprising several components (including dark matter) that are bound by the self-consistently generated gravitational field are readily constructed from distribution functions (DFs) that are analytic functions of the action integrals J. We explain why such models have unphysical velocity distributions unless the DFs of hot components satisfy certain conditions asJ(phi) -> 0. We show how DFsfor both isotropic and radially biased sphericalsystems can be constructed with specified f (J). We show how to construct DFsfor flattened systems with significant velocity anisotropy. Construction of self-consistent models rather than populations that are confined by an external potential leads to the conclusion that radially-biased sphericalsystems are generically unstable to quadrupolar perturbations. Chaosislikely key to maintenance of these constraints during adiabatic disc growth. If the DFs of dark haloes are radially biased, as simulations of cosmic clustering suggest, then models presented here suggest that dark haloes should be significantly oblate

Code is presented that computes and exploits orbital tori for any axisymmetric gravitational potential. The code is a development of the agama software package for action-based galaxy modelling and can be downloaded as the agamab code library. Although coded in c++ , most of its functions can be accessed from python. We add to the package functions that facilitate confronting models with data, which involve sky coordinates, lines of sight, distances, extinction, etc. The new torus generator can produce tori for both highly eccentric and nearly circular orbits that lie beyond the range of the earlier torus-mapping code. Tori can be created by interpolation between tori at very low cost. Tori are fundamentally devices for computing ordinary phase-space coordinates from angle-action coordinates, but agamab includes an action finder that returns angle-action coordinates from any given phase-space location. This action finder yields the torus through the given point, so it includes the functionality of an orbit integrator. The action finder is more accurate and reliable but computationally more costly than the widely used St & auml;ckel Fudge. We show how agamab can be used to generate sophisticated but cheap models of tidal streams and use it to analyse data for the GD1 stream. With the most recently published distances to the stream, energy and angular momentum imply that the end that must be leading is trailing, but extremely small changes to the distances rectify the problem.

The challenge presented by computing actions for eccentric orbits in axisymmetric potentials is discussed. In the limit of vanishing angular momentum about the potential's symmetry axis, there is a clean distinction between box and loop orbits. We show that this distinction persists into the regime of non-zero angular momentum. In the case of a Stackel potential, there is a critical value I-3crit(E) of the third integral I-3 below which I(3 )does not contribute to the centrifugal barrier. An orbit is of box or loop type according as its value of I-3 is smaller or greater than I-3crit. We give algorithms for determining I-3crit(E) and the critical action J(zcrit) below which orbits in any given potential are boxes. It is hard to compute the actions and especially the frequencies of orbits that have J(z) similar or equal to J(zcrit )using the St & auml;ckel Fudge. A modification of the Fudge that alleviates the problem is described.

The challenge presented by computing actions for eccentric orbits in axisymmetric potentials is discussed. In the limit of vanishing angular momentum about the potential's symmetry axis, there is a clean distinction between box and loop orbits. We show that this distinction persists into the regime of non-zero angular momentum. In the case of a Staeckel potential, there is a critical value I_3crit(E) of the third integral I_3 below which I_3 does not contribute to the centrifugal barrier. An orbit is of box or loop type according as its value of I_3 is smaller or greater than I_3crit. We give algorithms for determining I_3crit(E) and the critical action Jzcrit below which orbits in any given potential are boxes. It is hard to compute the actions and especially the frequencies of orbits that have Jz Jzcrit using the Staeckel Fudge. A modification of the Fudge that alleviates the problem is described.

We outline the rationale for holding a Discussion Meeting thus titled at the Royal Society, London during 15 and 16 April 2024, and summarize what we learnt there.This article is part of the discussion meeting issue 'Challenging the standard cosmological model'.

Entropy is as important a concept as energy, but less well known. The two concepts were first distinguished in the middle of the 19th century. By 1900 entropy was understood to be a measure of disorder and been widely applied in physics and chemistry, but in 1947 it was discovered to be the correct measure of ignorance. In an increasingly data-driven world the ‘principle of maximum entropy’ is playing a key role in data analysis and decision making. Quantum mechanics is a scheme for computing probability distributions, which are quantified by entropy. ‘entanglement entropy’, which plays a key role in cutting-edge quantum technologies, is revealing aspects of entropy that are not evident in the classical world. The thermodynamics of gravitating systems such as star clusters profoundly differs from that of the every-day systems from which entropy emerged. From the thermodynamics of black holes, which involves relativity and quantum mechanics in addition to gravity, an idea, the ‘holographic principle’, emerges that calls into question our current model of material reality. This book traces the emergence of entropy and energy as key concepts and illustrates their practical importance by discussing aspects of the energy transition such as heat pumps, solar cells, carbon capture, and liquified natural gas. Then it explains how entropy quantifies ignorance and illustrates the resulting principle of maximum entropy by reference to image reconstruction. Chapters on the role of entropy in quantum mechanics and gravity follow.

The science of thermodynamics started with the recognition that a body has a well-defined energy content, which can be increased either by applying heat or doing work. After this breakthrough, the state of a fluid could be characterised by four related variables: volume, pressure, temperature and energy. Subtle arguments then showed that a fifth variable, ‘entropy’ could be assigned. Whereas Carnot considered that caloric flowed through a steam engine from boiler to condenser, in the revised picture it was entropy that flowed. From the five based ‘functions of state’ other functions were derived that opened up quantitative descriptions of melting, boiling and chemical reactions. The new functions of state also deliver remarkable relations between things that can be measured, such as a fluid’s compressibility and its heat capacity.

Maxwell develops the kinetic theory of gases without mentioning entropy. Decades later Gibbs presents a theory of any thermal system in which entropy plays a central role. Entropy proves to be a measure of how widely the probability of the system’s location in phase space is spread. Boltzmann’s earlier, more specialized definition of entropy proves to be an approximation to Gibbs’s definition. The distributions of Maxwell and Gibbs imply that there’s an absolute zero of temperature. Nernst shows that this temperature isn’t reachable. Einstein uses entropy to understand fluctuations—fluctuations in the density of air give rise to the blueness of a clear sky. The chaotic nature of the equations of motion of large systems explains why we perceive an ‘arrow of time’ even though the equations are time-symmetric.

We revisit the dynamics of razor-thin, stone-cold, self-gravitating discs. By recasting the equations into standard cylindrical coordinates, the linearised vertical dynamics of an exponential disc can be followed for several gigayears on a laptop in a few minutes. An initially warped disc rapidly evolves into a flat inner region and an outward-propagating spiral corrugation wave that rapidly winds up and would quickly thicken a disc with non-zero radial velocity dispersion. The Sgr dwarf galaxy generates a similar warp in the Galactic disc as it passes through pericentre, and the warp generated by the dwarf's last pericentre ~ 35 Myr ago is remarkably similar to the warp traced by the Galaxy's HI disc. The resemblance to the observed warp is fleeting but its timing is perfect. For the adopted parameters the amplitude of the model warp is a factor 3 too small, but there are several reasons for this being so. The marked flaring of our Galaxy's low-alpha disc just outside the solar circle can be explained as a legacy of earlier pericentres.

A chemodynamical model of our Galaxy is fitted to data from DR17 of the APOGEE survey supplemented with data from the StarHorse catalogue and Gaia DR3. Dynamically, the model is defined by action-based distribution functions for dark matter and six stellar components plus a gas disc. The gravitational potential jointly generated by the model's components is used to examine the Galaxy's chemical composition within action space. The observational data probably cover all parts of action space that are populated by stars. The overwhelming majority of stars have angular momentum J(phi) > 0 implying that they were born in the Galactic disc. High-alpha stars dominate in a region that is sharply bounded by J(phi) less than or similar to J(phi)(solar). Chemically the model is defined by giving each stellar component a Gaussian distribution in ([Fe/H],[Mg/Fe]) space about a mean that is a linear function of the actions. The model's 47 dynamical parameters are chosen to maximize the likelihood of the data given the model in 72 three-dimensional velocity spaces while its 70 chemical parameters are similarly chosen in five-dimensional chemodynamical space. The circular speed falls steadily from 237 km s(-1) at R = 4 kpc to 218 km s(-1) at R = 20 kpc. Dark matter contributes half the radial force on the Sun and has local density 0.011 M-circle dot pc(-3), there being 24.5 M-circle dot pc(-2) in dark matter and 26.5 M-circle dot pc(-2) in stars within 1.1 kpc of the plane.

A new class of models of stellar discs is introduced and used to build a self-consistent model of our Galaxy. The model is defined by the parameters that specify the action-based distribution functions (DFs) f(J) of four stellar discs (three thin-disc age cohorts and a thick disc), spheroidal bulge and spheroidal stellar and dark haloes. From these DFs plus a specified distribution of gas, we solve for the densities of stars and dark matter and the potential they generate. The principal observational constraints are the kinematics of stars with Gaia RVS data and the density of stars in the column above the Sun. The model predicts the density and kinematics of stars and dark matter throughout the Galaxy. We determine the structure of the dark halo prior to the infall of baryons. A simple extension of the DFs of stellar components to include chemistry allows the model to reproduce the way the Galaxy's chemistry is observed to vary in the (R,z) plane. Surprisingly, the data indicate that high-alpha stars are confined to orbits with J_z >~ 50 kpc km/s. The code used to create the model is available on Github.

In the last 15 years different ground-based spectroscopic surveys have been started (and completed) with the general aim of delivering stellar parameters and elemental abundances for large samples of Galactic stars, complementing Gaia astrometry. Among those surveys, the Gaia-ESO Public Spectroscopic Survey (GES), the only one performed on a 8m class telescope, was designed to target 100,000 stars using FLAMES on the ESO VLT (both Giraffe and UVES spectrographs), covering all the Milky Way populations, with a special focus on open star clusters. This article provides an overview of the survey implementation (observations, data quality, analysis and its success, data products, and releases), of the open cluster survey, of the science results and potential, and of the survey legacy. A companion article (Gilmore et al.) reviews the overall survey motivation, strategy, Giraffe pipeline data reduction, organisation, and workflow. The GES has determined homogeneous good-quality radial velocities and stellar parameters for a large fraction of its more than 110,000 unique target stars. Elemental abundances were derived for up to 31 elements for targets observed with UVES. Lithium abundances are delivered for about 1/3 of the sample. The analysis and homogenisation strategies have proven to be successful; several science topics have been addressed by the Gaia-ESO consortium and the community, with many highlight results achieved. The final catalogue has been released through the ESO archive at the end of May 2022, including the complete set of advanced data products. In addition to these results, the Gaia-ESO Survey will leave a very important legacy, for several aspects and for many years to come.

The Gaia-ESO Public Spectroscopic Survey is an ambitious project designed to obtain astrophysical parameters and elemental abundances for 100,000 stars, including large representative samples of the stellar populations in the Galaxy, and a well-defined sample of 60 (plus 20 archive) open clusters. We provide internally consistent results calibrated on benchmark stars and star clusters, extending across a very wide range of abundances and ages. This provides a legacy data set of intrinsic value, and equally a large wide-ranging dataset that is of value for homogenisation of other and future stellar surveys and Gaia's astrophysical parameters. This article provides an overview of the survey methodology, the scientific aims, and the implementation, including a description of the data processing for the GIRAFFE spectra. A companion paper (arXiv:2206.02901) introduces the survey results. Gaia-ESO aspires to quantify both random and systematic contributions to measurement uncertainties. Thus all available spectroscopic analysis techniques are utilised, each spectrum being analysed by up to several different analysis pipelines, with considerable effort being made to homogenise and calibrate the resulting parameters. We describe here the sequence of activities up to delivery of processed data products to the ESO Science Archive Facility for open use. The Gaia-ESO Survey obtained 202,000 spectra of 115,000 stars using 340 allocated VLT nights between December 2011 and January 2018 from GIRAFFE and UVES. The full consistently reduced final data set of spectra was released through the ESO Science Archive Facility in late 2020, with the full astrophysical parameters sets following in 2022.

A seven-parameter distribution function (DF) is fitted to 20 000 RR-Lyrae stars for which only astrometric data are available. The observational data are predicted by the DF in conjunction with the gravitational potential of a self-consistent model Galaxy defined by DFs for the dark halo, the bulge and a four-component disc. Tests of the technique developed to deal with missing line-of-sight velocities show that adding such velocities tightens constraints on the DF only slightly. The recovered model of the RR-Lyrae population confirms that the population is flattened and has a strongly radially biased velocity distribution. At large radii its density profile tends to ρ ∼ r−4.5 but no power law provides a good fit inside the solar sphere. The model is shown to provide an excellent fit to the data for stars brighter than r = 16.5 but at certain longitudes it predicts too few faint stars at Galactocentric radii ∼20 kpc, possibly signalling that the halo is not axisymmetric. The DF is used to predict the velocity distribution of BHB stars for which space velocities are available. The z components are predicted successfully but too much anisotropy in the vRvφ plane is expected.

ABSTRACT We investigate the structure of our Galaxy’s young stellar disc by fitting the distribution functions (DFs) of a new family to 5D Gaia data for a sample of $47\, 000$ OB stars. Tests of the fitting procedure show that the young disc’s DF would be strongly constrained by Gaia data if the distribution of Galactic dust were accurately known. The DF that best fits the real data accurately predicts the kinematics of stars at their observed locations, but it predicts the spatial distribution of stars poorly, almost certainly on account of errors in the best-available dust map. We argue that dust models could be greatly improved by modifying the dust model until the spatial distribution of stars predicted by a DF agreed with the data. The surface density of OB stars is predicted to peak at $R\simeq 5.5\, \mathrm{kpc}$, slightly outside the reported peak in the surface density of molecular gas; we suggest that the latter radius may have been underestimated through the use of poor kinematic distances. The velocity distributions predicted by the best-fitting DF for stars with measured line-of-sight velocities v∥ reveal that the outer disc is disturbed at the level of $10\, \mathrm{km\, s}^{-1}$ in agreement with earlier studies, and that the measured values of v∥ have significant contributions from the orbital velocities of binaries. Hence the outer disc is colder than it is sometimes reported to be.

Observed clusters should be modelled by considering the distribution function (DF) to be a random variable that quantifies the degree of excitation of the system’s normal modes. A system of canonical coordinates for the space of DFs are identified so DFs can be weighted in a consistent way.

The excursions of star clusters and galaxies around statistical equilibria are studied. For an ergodic model with monotone decreasing DF Antonov's Hermitian operator on six-dimensional phase space has the normal modes as its eigenfunctions. The excitation energy of the system is just the sum of the (positive) energies associated with each normal mode. The positivity of modal energies opens the way to modelling the thermal properties of clusters in close analogy with those of crystals. Formulae are given for the DFs of modes, which are of the type first described by van Kampen rather than Landau. Each mode comprises the response of non-resonant stars to driving by the gravitational field of stars on a group of resonant tori. The structure of each mode is sensitive to the degree of self gravity. The emergence of global distortions in N-body models when particles are started from an analytical equilibrium is explained in terms of the interplay of normal modes.

Cosmology requires at least half of the baryons in the Univer se to be in the intergalactic medium, much of which is believed to form hot c ronae around galaxies. Star-forming galaxies must be accreting from their coronae . Hi observations of external galaxies show that they have H i halos associated with star formation. These halos are naturally modelled as ensembles of clouds driven up by super nova bubbles. These models can fit the data successfully only if clouds exchange mass and momentum with the corona. As a cloud orbits, it is ablated and forms a turbulent wake whe re cold high-metallicity gas mixes with hot coronal gas causing the prompt cooling of the l atter. As a consequence the total mass of Hi increases. This model has recently been used to model the Lei d nArgentina-Bonn survey of Galactic H i. The values of the model’s parameters that are required to model NGC 891, NGC 2403 and our Galaxy show a remar kable degree of consistency, despite the very di fferent natures of the two external galaxies and the dramatic difference in the nature of the data for our Galaxy and the externa l galaxies. The parameter values are also consistent with hydrodynamical simulation s f the ablation of individual clouds. The model predicts that a galaxy that loses its coolgas disc for instance through a major merger cannot reform it from its corona; it can return t o steady star formation only if it can capture a large body of cool gas, for example by accre ting a gas-rich dwarf. Thus the model explains how major mergers can make galaxies “red a nd dead.”

The excursions of star clusters and galaxies around statistical equilibria are studied. For a stable ergodic model Antonov’s Hermitian operator on six-dimensional phase space has the normal modes as its eigenfunctions. The excitation energy of the system is just the sum of the (positive) energies associated with each normal mode. Formulae are given for the DFs of modes, which are of the type first described by van Kampen rather than Landau, and Landau ‘modes’ can be expressed as sums of van Kampen modes. Each van Kampen mode comprises the response of non-resonant stars to driving by the gravitational field of stars on a group of resonant tori, so its structure is sensitive to the degree of self gravity. The emergence of global distortions in N-body models when particles are started from an analytical equilibrium is explained in terms of the interplay of normal modes. The positivity of modal energies opens the way to modelling the thermal properties of clusters in close analogy with those of crystals.