Instability of the Gravitational<i>N</i>‐Body Problem in the Large‐<i>N</i>Limit

Hemsendorf, Marc; Merritt, David

doi:10.1086/343027

Cited by 41 publications

(52 citation statements)

References 11 publications

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“…Although all orbits in an N-body system are mildly chaotic (e.g. , Miller 1964;Hemsendorf & Merritt 2002),for the purpose of testing our automated classification scheme, we also attempt to visually classify chaotic orbits, although this is not possible to do in a robust manner. An orbit was visually classified as chaotic only if it could not be easily identified with a major orbit family (box, short-axis tube, long-axis tube), or if it showed signs of changing from one family to another during the integration time (indicating that it is sticky chaotic).…”

Section: Orbit Classificationmentioning

confidence: 99%

A Unified Framework for the Orbital Structure of Bars and Triaxial Ellipsoids

Valluri

Shen

Abbott

et al. 2016

ApJ

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We examine a large random sample of orbits in two self-consistent simulations of N-body bars. Orbits in these bars are classified both visually and with a new automated orbit classification method based on frequency analysis. The well-known prograde x1 orbit family originates from the same parent orbit as the box orbits in stationary and rotating triaxial ellipsoids. However, only a small fraction of bar orbits (∼4%) have predominately prograde motion like their periodic parent orbit. Most bar orbits arising from the x1 orbit have little net angular momentum in the bar frame, making them equivalent to box orbits in rotating triaxial potentials. In these simulations a small fraction of bar orbits (∼7%) are long-axis tubes that behave exactly like those in triaxial ellipsoids: they are tipped about the intermediateaxis owingto the Coriolis force, with the sense of tipping determined by the sign of their angular momentum about the long axis. No orbits parented by prograde periodic x2 orbits are found in the pure bar model, but a tiny population (∼2%) of short-axis tube orbits parented by retrograde x4 orbits are found. When a central point mass representing a supermassive black hole (SMBH) is grown adiabatically at the center of the bar, those orbits that lie in the immediate vicinity of the SMBH are transformed into precessing Keplerian orbits thatbelong to the same major families (short-axis tubes, long-axis tubes and boxes) occupying the bar at larger radii. During the growth of an SMBH,the inflow of mass and outward transport of angular momentum transformsome x1 and long-axis tube orbits into prograde short-axis tubes. This study has important implications for future attempts to constrain the masses of SMBHs in barred galaxies using orbit-based methods like the Schwarzschild orbit superposition scheme and for understanding the observed features in barred galaxies.

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Section: Orbit Classificationmentioning

confidence: 99%

A Unified Framework for the Orbital Structure of Bars and Triaxial Ellipsoids

Valluri

Shen

Abbott

et al. 2016

ApJ

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“…This point is particularly important for a system with long-range interactions, which can relax even if it is collisionless 4 . In fact, taking the presence of chaotic motions as a diagnostic of relaxation, it is interesting to remember that N -body self-gravitating systems can exhibit large (and increasing as ln N ) rates of growth of small perturbations, even for large N -see Hemsendorf & Merritt (2002). In other words: these systems seem to be more chaotic for larger N , i.e.…”

Section: Final Remarksmentioning

confidence: 99%

“…In addition, it has been already shown (see Valluri & Merritt 2000;Kandrup & Sideris 2001;Hemsendorf & Merritt 2002) that the gravitational N -body problem does not converge to the continuous limit for N → ∞, at least when using Lyapunov exponents as a diagnostic. Note that being a discrete sample with N bodies (stars or dark matter particles), as opposed to the continuous limit, is not a feature of N -body simulations only.…”

Section: Introductionmentioning

confidence: 99%

“…What does, however, seem apparent from the preceding is that, even for very large N, true N -body trajectories could differ significantly from CBE characteristics". Hemsendorf & Merritt (2002) claim that "if the rate of growth of small perturbations remains substantial even for large N, there would be an important sense in which the CBE does not correctly describe the behavior of N -body systems". Finally, Bindoni & Secco (2008) say that: "any further relaxation of the system should be therefore considered in terms of the coarse-grained phase-space density which, as we have seen, would yield results different from the predictions based on the initial fine-grained phase-space density.…”

Section: Introductionmentioning

confidence: 99%

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The Arrow of Time in the Collapse of Collisionless Self-gravitating Systems: Non-validity of the Vlasov–Poisson Equation during Violent Relaxation

Silva

Pedra

Sodré

et al. 2017

ApJ

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The collapse of a collisionless self-gravitating system, with the fast achievement of a quasi-stationary state, is driven by violent relaxation, with a typical particle interacting with the time-changing collective potential. It is traditionally assumed that this evolution is governed by the Vlasov-Poisson equation, in which case entropy must be conserved. We run N-body simulations of isolated self-gravitating systems, using three simulation codes: NBODY-6 (direct summation without softening), NBODY-2 (direct summation with softening) and GADGET-2 (tree code with softening), for different numbers of particles and initial conditions. At each snapshot, we estimate the Shannon entropy of the distribution function with three different techniques: Kernel, Nearest Neighbor and EnBiD. For all simulation codes and estimators, the entropy evolution converges to the same limit as N increases. During violent relaxation, the entropy has a fast increase followed by damping oscillations, indicating that violent relaxation must be described by a kinetic equation other than the Vlasov-Poisson, even for N as large as that of astronomical structures. This indicates that violent relaxation cannot be described by a time-reversible equation, shedding some light on the so-called "fundamental paradox of stellar dynamics". The long-term evolution is well described by the orbit-averaged Fokker-Planck model, with Coulomb logarithm values in the expected range 10 − 12. By means of NBODY-2, we also study the dependence of the 2-body relaxation time-scale on the softening length. The approach presented in the current work can potentially provide a general method for testing any kinetic equation intended to describe the macroscopic evolution of N-body systems.

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“…This is where N -body modelling become essential, as it supplements other approaches to stellar dynamics through more general three-dimensional coding of the equations of motion. The caveat, of course, is that to date the largest-N calculations performed are limited to N ∼ 10 5 (e.g., Hemsendorf & Merritt 2002;Baumgardt et al , 2008Fujii et al 2008) which is still below the average of 4.3 × 10 5 for Galactic clusters (assuming a standard stellar mass function; Meylan & Heggie 1997) and further still from the large-N regime sought. It is reassuring to find that the simple problem of core collapse for equalmass systems over a time t 15t r is well reproduced by completely different methods (e.g., Takahashi 1995;Makino 1996;Spurzem et al 2005).…”

Section: Basicsmentioning

confidence: 99%

The internal dynamics of stellar clusters

Boily

2009

Proc. IAU

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Abstract. In this brief review I summarise recent progress in the area of stellar dynamics, focusing on the dynamics of bound, self-gravitating stellar associations in isolation and (approximate) equilibrium. The basics of stellar dynamics are first outlined and the importance of stellar evolution is stressed. Subsequently, I argue that the evolution of anisotropic clusters of stars still holds solutions to current outstanding problems, such as the dynamics of galactic nuclei. I take a more personal standpoint when discussing the role of stellar evolution in the dynamics on relaxation timescales and draw from several recent models to underscore that a major step forward has been made in coupling stellar evolution and dynamics. I then briefly visit the issue of multiple stars and highlight some as yet unsolved problems.

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Instability of the GravitationalN‐Body Problem in the Large‐NLimit

Cited by 41 publications

References 11 publications

A Unified Framework for the Orbital Structure of Bars and Triaxial Ellipsoids

A Unified Framework for the Orbital Structure of Bars and Triaxial Ellipsoids

The Arrow of Time in the Collapse of Collisionless Self-gravitating Systems: Non-validity of the Vlasov–Poisson Equation during Violent Relaxation

The internal dynamics of stellar clusters

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