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Register a new userLinear response theory and damped modes of stellar clusters
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Added by
Jean-Baptiste Fouvry
Publication date
2021
fields
Physics
and research's language is
English
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Because all stars contribute to its gravitational potential, stellar clusters amplify perturbations collectively. In the limit of small fluctuations, this is described through linear response theory, via the so-called response matrix. While the evaluation of this matrix is somewhat straightforward for unstable modes (i.e. with a positive growth rate), it requires a careful analytic continuation for damped modes (i.e. with a negative growth rate). We present a generic method to perform such a calculation in spherically symmetric stellar clusters. When applied to an isotropic isochrone cluster, we recover the presence of a low-frequency weakly damped $ell = 1$ mode. We finally use a set of direct $N$-body simulations to test explicitly this prediction through the statistics of the correlated random walk undergone by a clusters density centre.
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This research was stimulated by the recent studies of damping solutions in dynamically stable spherical stellar systems. Using the simplest model of the homogeneous stellar medium, we discuss nontrivial features of stellar systems. Taking them into account will make it possible to correctly interpret the results obtained earlier and will help to set up decisive numerical experiments in the future. In particular, we compare the initial value problem versus the eigenvalue problem. It turns out that in the unstable regime, the Landau-damped waves can be represented as a superposition of van Kampen modes {it plus} a discrete damped mode, usually ignored in the stability study. This mode is a solution complex conjugate to the unstable Jeans mode. In contrast, the Landau-damped waves are not genuine modes: in modes, eigenfunctions depend on time as $exp (-{rm i} omega t)$, while the waves do not have eigenfunctions on the real $v$-axis at all. However, `eigenfunctions on the complex $v$-contours do exist. Deviations from the Landau damping are common and can be due to singularities or cut-off of the initial perturbation above some fixed value in the velocity space.
Rotation is ubiquitous in the Universe, and recent kinematic surveys have shown that early type galaxies and globular clusters are no exception. Yet the linear response of spheroidal rotating stellar systems has seldom been studied. This paper takes a step in this direction by considering the behaviour of spherically symmetric systems with differential rotation. Specifically, the stability of several sequences of Plummer spheres is investigated, in which the total angular momentum, as well as the degree and flavour of anisotropy in the velocity space are varied. To that end, the response matrix method is customised to spherical rotating equilibria. The shapes, pattern speeds and growth rates of the systems unstable modes are computed. Detailed comparisons to appropriate N-body measurements are also presented. The marginal stability boundary is charted in the parameter space of velocity anisotropy and rotation rate. When rotation is introduced, two sequences of growing modes are identified corresponding to radially and tangentially-biased anisotropic spheres respectively. For radially anisotropic spheres, growing modes occur on two intersecting surfaces (in the parameter space of anisotropy and rotation), which correspond to fast and slow modes, depending on the net rotation rate. Generalised, approximate stability criteria are finally presented.
Linear theory provides a reasonable description of the velocity correlations of biased tracers both perpendicular and parallel to the line of separation, provided one accounts for the fact that the measurement is almost always made using pair-weighted statistics. This introduces an additional term which, for sufficiently biased tracers, may be large. Previous work suggesting that linear theory was grossly in error for the components parallel to the line of separation ignored this term.
We present a study of the optical response of compact and hollow icosahedral clusters containing up to 868 silver atoms by means of time-dependent density functional theory. We have studied the dependence on size and morphology of both the sharp plasmonic resonance at 3-4 eV (originated mainly from $sp$-electrons), and the less studied broader feature appearing in the 6-7 eV range (interband transitions). An analysis of the effect of structural relaxations, as well as the choice of exchange correlation functional (local density versus generalized gradient approximations) both in the ground state and optical response calculations is also presented. We have further analysed the role of the different atom layers (surface versus inner layers) and the different orbital symmetries on the absorption cross-section for energies up to 8 eV. We have also studied the dependence on the number of atom layers in hollow structures. Shells formed by a single layer of atoms show a pronounced red shift of the main plasmon resonances that, however, rapidly converge to those of the compact structures as the number of layers is increased. The methods used to obtain these results are also carefully discussed. Our methodology is based on the use of localized basis (atomic orbitals, and atom-centered- and dominant- product functions), which bring several computational advantages related to their relatively small size and the sparsity of the resulting matrices. Furthermore, the use of basis sets of atomic orbitals also brings the possibility to extend some of the standard population analysis tools (e.g., Mulliken population analysis) to the realm of optical excitations. Some examples of these analyses are described in the present work.
We investigate the triggering of star formation and the formation of stellar clusters in molecular clouds that form as the ISM passes through spiral shocks. The spiral shock compresses gas into $sim$100 pc long main star formation ridge, where clusters forming every 5-10 pc along the merger ridge. We use a gravitational potential based cluster finding algorithm, which extracts individual clusters, calculates their physical properties and traces cluster evolution over multiple time steps. Final cluster masses at the end of simulation range between 1000 and 30000 M$_{odot}$ with their characteristic half-mass radii between 0.1 pc and 2 pc. These clusters form by gathering material from 10-20 pc size scales. Clusters also show a mass - specific angular momentum relation, where more massive clusters have larger specific angular momentum due to the larger size scales, and hence angular momentum from which they gather their mass. The evolution shows that more massive clusters experiences hierarchical merging process, which increases stellar age spreads up to 2-3 Myr. Less massive clusters appear to grow by gathering nearby recently formed sinks, while more massive clusters with their large global gravitational potentials are increasing their mass growth from gas accretion.
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