Celestial and stellar dynamics

Computation and a wealth of new observational techniques have reinvigorated dynamical studies of galaxies and star clusters. These objects are examples of the gravitational n-body problem with n in the range from a few hundred to 1011. Relaxation effects dominate at the low end and are completely negligible at the high end. The gravitational n-body problem is chaotic, and the principal challenge in doing physics where that problem is involved (whether computationally or with analytic theory) is to ensure that chaos has not vitiated the results. Enforcing a Liouville theorem accomplishes this with collision-free large-n problems, but equivalent recipes are not in common use for smaller n. We describe some important insights and discoveries that have come from computation in stellar dynamics, discuss chaos briefly, and indicate the way the physics that comes up in different astronomical contexts is addressed in numerical methods currently in use. Graphics is a vital part of any computational approach. The long range prospects are very promising for continued high scientific productivity in stellar dynamics.

Self-similar gravitational collapse of a globular cluster is studied using the nonlinear kinetic standard model of stellar dynamics consisting of the Fokker-Planck-Vlasov transport equation coupled self-consistently to Poisson’s equation for the Newtonian gravitational potential. It is shown rigorously that any locally integrable self-similar solution to these equations must approach a mass density profile , , in the final 2a r(r, t) / r a p 3 stage of the collapse. The discrepancy between the exact value and previous results in the range a p 32 ! obtained from the orbit-averaged approximation to the kinetic model raises some questions about the a ! 2.5 validity of this popular approximation. Subject headings: celestial mechanics, stellar dynamics — galaxies: star clusters — globular clusters: general

We present a galaxy classification system for 238 (E1-Sdm) CALIFA (Calar Alto Legacy Integral Field Area) galaxies based on the shapes and amplitudes of their circular velocity curves (CVCs). We infer the CVCs from the de-projected surface brightness of the galaxies, after scaling by a constant mass-to-light ratio based on stellar dynamics - solving axisymmetric Jeans equations via fitting the second velocity moment $V_{\mathrm{rms}}=\sqrt{V^2+\sigma^2}$ of the stellar kinematics. We use principal component analysis (PCA) applied to the CVC shapes to find characteristic features and use a $k$-means classifier to separate circular curves into classes. This objective classification method identifies four different classes, which we name slow-rising (SR), flat (FL), round-peaked (RP) and sharp-peaked (SP) circular curves. SR are typical for low-mass, late-type (Sb-Sdm), young, faint, metal-poor and disc-dominated galaxies. SP are typical for high-mass, early-type (E1-E7), old, bright, metal-rich and bulge-dominated galaxies. FL and RP appear presented by galaxies with intermediate mass, age, luminosity, metallicity, bulge-to-disk ratio and morphologies (E4-S0a, Sa-Sbc). The discrepancy mass factor, $f_d=1-M_{*}/M_{dyn}$, have the largest value for SR and SP classes ($\sim$ 74 per cent and $\sim$ 71 per cent, respectively) in contrast to the FL and RP classes (with $\sim$ 59 per cent and $\sim$ 61 per cent, respectively). Circular curve classification presents an alternative to typical morphological classification and appears more tightly linked to galaxy evolution.

Recently, Israel and Kandrup (1984; Kandrup 1984 a,b,c,d) have formulated a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object here is to indicate how that formalism may be used to construct a theory of ‘collisional’ stellar dynamics, valid for a collection of point mass stars in the limit that incoherent radiative effects may be neglected.

We consider a special case of the three dimensional Vlasov-Poisson system where the particles are restricted to a plane, a situation that is used in astrophysics to model extremely flattened galaxies. We prove the existence of steady states of this system. They are obtained as minimizers of an energy-Casimir functional from which fact a certain dynamical stability property is deduced. From a mathematics point of view these steady states provide examples of partially singular solutions of the three dimensional Vlasov-Poisson system.

Gravitational lensing and stellar dynamics are two independent methods, based solely on gravity, to study the mass distributions of galaxies. Both methods suffer from degeneracies, however, that are difficult to break. In this paper, we present a new framework that self-consistently unifies gravitational lensing and stellar dynamics. This approach breaks some of classical degeneracies that have limited their individual usage, in particular in the study of high-redshift galaxies. The methodology is based on the premise that, for any given galaxy potential, the mapping of both the unknown lensed source brightness distribution and the stellar phase-space distribution function on to the photometric and kinematic observables, can be cast as a single set of coupled linear equations. This set of linear equations is solved, maximizing the likelihood penalty function. The evidence penalty function, as derived from Bayesian statistics, subsequently allows the best potential-model parameters to be found and potential-model families, or other model assumptions (e.g. PSF), to be quantitatively ranked. We have implemented a fast algorithm that solves for the maximum-likelihood pixelized lensed source brightness distribution and the two-integral stellar phase-space distribution function f(E, L_z), assuming axisymmetric potentials. To make the method practical, we have devised a new Monte-Carlo approach to Schwarzschild's orbital superposition method, based on the superposition of two-integral (E and L_z) toroidal components, to find the maximum-likelihood two-integral distribution function in a matter of seconds in any axisymmetric potential. The non-linear parameters of the potential are subsequently found through a hybrid MCMC and Simplex optimization of the evidence. (Abridged)

A review is given, in the modern context of applications, of the major important scientific results obtained by the scientists and graduates of St. Petersburg State University in the field of celestial mechanics and stellar dynamics. The second part of the review covers the following topics: estimates and calculation of the MOID parameter, problems of asteroid–comet hazard, dust complexes in the Solar System, rotational dynamics of planetary satellites, circumbinary dynamics, and methods for the discovery and determination of orbits of exoplanets.

We study the motion of stars in a star cluster which revolves in a circular orbit about its parentgalaxy. The star cluster is modelled as an ellipsoid of uniform spatial density. We exhibit two2-parameter families of self-consistent equilibrium models in which the velocity at each pointis confined to a line in velocity space. We exhibit the link bet ween this problem and that of auniform rotating ellipsoidal galaxy. With minimal adaptation, Freeman’s bar models yield athird family.Key words: stellar dynamics - globular clusters: general - open clusters and associations:general 1 INTRODUCTIONIn a star cluster moving in a circular orbit about the axis of sym-metry of a galaxy, a star is subject to the self-gravity of theclusterand the tidal field of the galaxy. If studied in a rotating, acc eler-ating frame following the orbital motion of the cluster, it is alsosubject to centrifugal and Coriolis forces. If the spatial distributionof stars is uniform within an ellipsoid, and if the usual linear ap-proximation of the tidal field is adopted, the accelerations insidethe ellipsoid are all linear in the spatial coordinates. There are thentwo normal modes of oscillation in planes orthogonal to the axisof rotation. Fellhauer & Heggie (2005) studied this problem in thecase when one of the normal frequencies is imaginary. Taking forconvenience the case of an ellipsoid of revolution, they showed thatit was possible to choose the axial ratio so that it was equal to thatof the orbits in the remaining normal mode. By a very simple orbitsuperposition it was possible to construct a self-consistent model.This problem closely resembles that of constructing a modelof a uniform, rotating ellipsoidal bar (Freeman 1966a,b,c; Hunter1974, 1975). This goal has been achieved in three cases: (i) ellip-tical cylinders, in which the semi-major axis parallel to the axis ofrotation is infinite; (ii) so-called “balanced” systems, in which thecentrifugal and gravitational forces are equal along the major axis;and (iii) disks with the surface density that is obtained by projectinga uniform ellipsoid onto one plane.In this paper we set out the connection between these twoproblems, and give a more systematic treatment of the modelswhich Fellhauer & Heggie stumbled upon. In principle there arethree dimensionless parameters: two axial ratios, and one parame-ter which measures the density of the system. The requirement ofchoosing one axial ratio appropriately reduces the family to a two-parameter family. Fellhauer & Heggie chose to work with axisym-

Recent observations and stellar dynamics simulations suggest that thousand\nsolar mass black holes can form in compact massive young star clusters. Any\nsuch clusters in the bulge of their host galaxy will spiral to the center\nwithin a few hundred million years, where their intermediate-mass black holes\nare likely to merge eventually with the galaxy's supermassive black hole. If\nsuch mergers are common, then future space-based gravitational wave detectors\nsuch as the Laser Interferometer Space Antenna will detect them with such a\nhigh signal to noise ratio that towards the end of the inspiral the orbits will\nbe visible in a simple power density spectrum, without the need for matched\nfiltering. We discuss the astrophysics of the inspiral of clusters in the\nnuclear region of a galaxy and the subsequent merger of intermediate-mass with\nsupermassive black holes. We also examine the prospects for understanding the\nspacetime geometry of rotating black holes, based on phase connection of the\nstrong signals visible near the end of these extreme mass ratio inspirals.\n

The dynamical evolution of stellar systems containing a few hundred stars can be studied by numerically integrating the equations of motion of all the stars as an N-body problem. From a methodic point of view, these numerical N-body experiments belong to two branches of dynamical astronomy: celestial mechanics and stellar dynamics. As in celestial mechanics, the basic procedure is the computation of individual orbits of mass points in the exact gravitational field of all the other bodies. However, these individual orbits are not the information at which we aim in stellar dynamics. Hence, we finally deduce from all the individual orbits the desired statistical information about the evolution of the system as a whole (‘macroscopic properties’). Of course, the individual orbits also give valuable insight into the ‘microscopic’ behaviour of a stellar system.

A novel way of looking at the evolution of star clusters is presented. With a dynamicaltemperature, given by the mean kinetic energy of the cluster stars, and a dynamicalluminosity, which is defined as the kinetic energy of the stars leaving the cluster inanalogy to the energy of photons emitted by a star, the dissolution of star clusters isstudied using a new dynamical temperature-luminosity diagram for star clusters. Theinvestigation contains a parameter-space study of open clusters of up to N = 32768single-mass stars with different initial density distributions, half-mass radii, tidal con-ditions and binary fractions. The clusters show a strong correlation between dynamicaltemperature and dynamical luminosity and most of the investigated cluster familiesshare a common sequence in such a dynamical temperature-luminosity diagram. Devi-ations from this sequence are analyzed and discussed. After core collapse, the positionof a cluster within this diagram can be defined by three parameters: the mass, thetidal conditions and the binary fraction. Due to core collapse all initial conditions arelost and the remaining stars adjust to the given tidal conditions. Binaries as internalenergy sourcesinfluence this adjustment. A further finding concerns the Lagrangeradiiof star clusters: Throughout the investigated parameter space nearly all clusters showa constant half-mass radius for the time after core collapse until dissolution. Further-more, the ratio of half-mass radius to tidal radius evolves onto a common sequencewhich only depends on the mass left in the cluster.Keywords: stellar dynamics – galaxies: star clusters – methods: N-body simulations– Hertzsprung–Russel (HR) diagram – open clusters and assoziations: general

view Abstract Citations (64) References (37) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Mapping Spherical Potentials with Discrete Radial Velocities Merritt, David ; Saha, Prasenjit Abstract We present a technique for inferring the gravitational potential {PHI}(r) of a hot stellar system from line-of-sight velocity data, without any assumptions about the relative distribution of dark and luminous components. We overcome the indeterminacy of mass estimates based on line-of-sight velocity dispersions by fitting the full distribution of radial velocities. The phase-space distribution function f(E,L^2^) of the test sample is recovered as well. Unlike parametric mass estimation methods based on the virial theorem or the core-fitting formula, our technique does not require that the spatial distribution of the test sample bear any relation to that of the matter determining the potential; that is, the mass-to-light ratio may be an arbitrary function of radius. The technique also does not require binning, or the computation of velocity moments. Even for relatively modest samples (N~300), the technique places narrower constraints on the matter distribution than can be inferred from the projected velocity dispersion and number density profiles alone. Strong, model-independent constraints on the form of the potential require somewhat larger samples, of order N = 10^3^ or more. We use our algorithm to infer, in a nearly model-independent way, the distribution of dark matter near the center of the Coma galaxy cluster. We find that the core radius of the dark matter is very unlikely to exceed about 500 kpc (H_0_ = 100) but could be much smaller. The matter distribution at larger radii is poorly constrained. Publication: The Astrophysical Journal Pub Date: May 1993 DOI: 10.1086/172643 Bibcode: 1993ApJ...409...75M Keywords: Celestial Mechanics; Galactic Clusters; Gravitational Fields; Mass To Light Ratios; Radial Velocity; Virial Theorem; Dark Matter; Line Of Sight; Spatial Distribution; Stellar Systems; Astrophysics; CELESTIAL MECHANICS; STELLAR DYNAMICS; GALAXIES: KINEMATICS AND DYNAMICS; TECHNIQUES: RADIAL VELOCITIES full text sources ADS | data products SIMBAD (1)

Context. Dynamical friction is an important phenomenon in stellar dynamics (and plasma physics) resulting in the slowing down of a massive test particle upon many two-body scatters with background particles. Chandrasekhar’s original formulation, developed for idealized infinite and homogeneous systems, has been found to be sufficiently accurate even in models of finite extent and radially dependent density profiles. However, in some cases N-body simulations have evidenced a breakdown of Chandrasekhar’s formalism. In particular, in the case of cored stellar systems, the analytical predictions underestimate the rate of in-fall of the test particle. Moreover, the orbital decay appears to stop at a critical radius where in principle the effect of dynamical friction should still be non-negligible. Aims. Several explanations for such discrepancy have been proposed so far. In spite of this, it remains unclear whether the origin is a finite N effect or an effect arising from the resonance (or near-resonance) of the orbits of the test and field particles, which is independent of N, such as dynamical buoyancy. Here we aim to shed some light on this issue with tailored numerical experiments. Methods. We performed ad hoc simulations of a massive tracer initially placed on a low-eccentricity orbit in spherical equilibrium models with increasing resolution. We used an N-body code for which the self-consistent interaction among the background particles can be substituted with the effect of the static smooth potential of the system’s continuum limit, so that the higher-order contributions to the dynamical friction arising from the formation of a wake can be neglected if needed. Results. We find that, contrary to what has been reported in previous literature, a suppression of dynamical friction happens in both cuspy and cored models. When neglecting the interaction among field particles, we observe in both cases a clear N−1/2 scaling of the radius at which dynamical friction ceases to be effective. This hints at a granularity-induced origin of the so-called core-stalling of the massive tracer in cored models.

We summarize the methods of a new “hybrid” computer code for stellar dynamics. All particles in the inner spatial region are followed exactly via a direct N-body code and all particles in the outer spatial region are treated statistically via a distribution function and Fokker-Planck type methods. An intermediate region, with features of both, allows exchange of particles and energy between the outer and inner regions. We apply our code to the period just before core collapse and just after and summarize the results.

Stellar Dynamics![Figure][1] Some fast-moving stars in our Milky Way Galaxy came from nearby dwarf galaxies. CREDIT: NASA/JPL-CALTECH/R. HURT (SSC/CALTECH) Dynamical interactions and supernovae can accelerate stars to high velocities, sometimes even fast enough that they are no longer gravitationally bound to their host galaxy and escape from it. Marchetti et al. have combined astrometric data with radial velocity measurements to determine the three-dimensional motions of 7 million stars within the Milky Way Galaxy. Within that sample, they identify 20 stars that are not bound to the Galaxy. Only seven of them are moving away from the Milky Way's disc; 13 stars originated elsewhere. The authors postulate that these apparently extragalactic stars may have been ejected or tidally stripped from nearby dwarf galaxies. Mon. Not. R. Astron. Soc. 10.1093/mnras/sty2592 (2018). [1]: pending:yes