Relaxation and stochasticity in a truncated Toda lattice

He, Kai; Mahon, M. Elaine

doi:10.1103/physreve.49.3735

Cited by 48 publications

(87 citation statements)

References 20 publications

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“…In general, because of the existence of semipermeable phase-space barriers and other complications (see, e.g., Wiggins 1992), such assertions are hard to prove in realistic dynamical systems and may be difficult to ascertain numerically (because trajectories may take very long before they fill their allowed phase-space region). Nevertheless, test results by Kandrup & Mahon (1994) suggest its basic plausibility by showing that the local Liapunov indicators mentioned above, averaged over ensembles of initial conditions in a connected phase-space domain, approximate well the long-time Liapunov exponent of a single trajectory of the same region.…”

Section: Liapunov Exponents and Stability Of Trajectoriesmentioning

confidence: 96%

Dark Halo Shapes and the Fate of Stellar Bars

El-Zant¹,

Shlosman²

2002

ApJ

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We investigate the stability properties of trajectories in barred galaxies with mildly triaxial halos by means of Liapunov exponents. This method is perfectly suitable for time-dependent three-dimensional potentials where surfaces of sections and other simple diagnostics are not applicable. We find that when halos are centrally concentrated most trajectories starting near the plane containing the bar become chaotic. The spatial density distribution of these orbits does not match that of the bar, being overextended in and out of the plane compared to the latter. Moreover, the shape of many of the remaining regular trajectories does not match the bar density distribution, being too round. Therefore, time-independent self-consistent solutions are highly unlikely to be found. When the nonrotating nonaxisymmetric perturbation in the potential reaches 10%, almost all trajectories integrated are chaotic and have large Liapunov exponents. No regular trajectories aligned with the bar have been found. Hence, if the evolution of the density figure is directly related to the characteristic timescale of orbital instability, bar dissolution would take place on a timescale of a few dynamical times. The slowly rotating nonaxisymmetric contribution to the potential required for the onset of widespread chaotic behavior is remarkably small. Even a potential axis ratio of 0.99 results in large connected chaotic regions dominating the space of initial conditions. Systems consisting of centrally concentrated axisymmetric halos and stellar bars thus appear to be structurally unstable, and small ($1%) deviations from perfect axisymmetry should result in a bar dissolution on a timescale significantly smaller than the Hubble time. Since halos found in cold dark matter simulations of large-scale structure are both centrally concentrated and triaxial, it is unlikely that stellar bars embedded in such halos would form and survive unless the halos are modified during the formation of the baryonic component.

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Section: Liapunov Exponents and Stability Of Trajectoriesmentioning

confidence: 96%

Dark Halo Shapes and the Fate of Stellar Bars

El-Zant¹,

Shlosman²

2002

ApJ

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“…Gerhard & Binney (1985) first proposed that the box orbits which form the back bone of triaxial elliptical galaxies would become chaotic due to scattering by the divergent central force arising from a central black hole. The presence of a significant fraction of chaotic orbits results in chaotic mixing which can cause secular relaxation of orbits in phase space (Kandrup & Mahon 1994) resulting in a change in the shape of the galaxy from triaxial to axisymmetric on timescales of order ∼30-50 dynamical times (Merritt & Valluri 1996;Merritt & Quinlan 1998). Gerhard & Binney (1985) had also argued, however, that if the triaxial potential had figure rotation, then box orbits (which are crucial to maintaining triaxiality) would be less affected by the central force in the rotating frame due to "envelope doubling" (de Zeeuw & Merritt 1983).…”

Section: Introductionmentioning

confidence: 99%

The Orbital Structure of Triaxial Galaxies With Figure Rotation

Deibel¹,

Valluri²,

Merritt³

2011

ApJ

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We survey the properties of all orbit families in the rotating frame of a family of realistic triaxial potentials with central supermassive black holes (SMBHs). In such galaxies, most regular box orbits (vital for maintaining triaxiality) are associated with resonances which occupy two-dimensional surfaces in configuration space. For slow figure rotation all orbit families are largely stable. At intermediate pattern speeds a significant fraction of the resonant box orbits as well as inner long-axis tubes are destabilized by the "envelope doubling" that arises from the Coriolis forces and are driven into the destabilizing center. Thus, for pattern rotation periods 2×10 8 yr T p 5×10 9 yr, the two orbit families that are most important for maintaining triaxiality are highly chaotic. As pattern speed increases there is also a sharp decrease in the overall fraction of prograde short-axis tubes and a corresponding increase in the retrograde variety. At the highest pattern speeds (close to that of triaxial bars), box-like orbits undergo a sudden transition to a new family of stable retrograde loop-like orbits, which resemble orbits in three-dimensional bars, and circulate about the short axis. Our analysis implies that triaxial systems (with central cusps and SMBHs) could either be very rapidly rotating like fast bars or slowly rotating like triaxial elliptical galaxies or dark matter halos found in N-body simulations. Collisionless triaxial systems that remain in stable dynamical equilibrium are unlikely to have intermediate pattern speeds which produce a high level of stochasticity in both the box and inner long-axis tube orbit families.

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“…Generically, a small ensemble of chaotic orbits will lose coherence much faster than for regular orbits (see, e.g., Kandrup & Mahon 1994;Merritt & Valluri 1996;Kandrup & Siopis 2003), however this depends on the details of the resonant structure around the ensemble and the chaotic evolution of the individual orbits and is thus difficult to predict. For example, ensembles of orbits 'stuck' between resonances may quickly spread to fill the allowed volume, but then the orbits must escape this confinement and diffuse through the Arnold web to reach a fully mixed state (Merritt & Valluri 1996).…”

Section: Mixing Of Orbit Ensemblesmentioning

confidence: 99%

Chaotic dispersal of tidal debris

Price-Whelan

Johnston

Valluri

et al. 2015

Monthly Notices of the Royal Astronomical Society

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Several long, dynamically cold stellar streams have been observed around the Milky Way Galaxy, presumably formed from the tidal disruption of globular clusters. In integrable potentials-where all orbits are regular-tidal debris phase-mixes close to the orbit of the progenitor system. However, the Milky Way's dark matter halo is expected not to be fully integrable; an appreciable fraction of orbits will be chaotic. This paper examines the influence of chaos on the phase-space morphology of cold tidal streams. Streams even in weakly chaotic regions look very different from those in regular regions. We find that streams can be sensitive to chaos on a much shorter time-scale than any standard prediction (from the Lyapunov or frequency-diffusion times). For example, on a weakly chaotic orbit with a chaotic timescale predicted to be >1000 orbital periods (>1000 Gyr), the resulting stellar stream is, after just a few 10's of orbits, substantially more diffuse than any formed on a nearby but regular orbit. We find that the enhanced diffusion of the stream stars can be understood by looking at the variance in orbital frequencies of orbit ensembles centered around the parent (progenitor) orbit. Our results suggest that long, cold streams around our Galaxy must exist only on regular (or very nearly regular) orbits; they potentially provide a map of the regular regions of the Milky Way potential. This suggests a promising new direction for the use of tidal streams to constrain the distribution of dark matter around our Galaxy.

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Relaxation and stochasticity in a truncated Toda lattice

Cited by 48 publications

References 20 publications

Dark Halo Shapes and the Fate of Stellar Bars

Dark Halo Shapes and the Fate of Stellar Bars

The Orbital Structure of Triaxial Galaxies With Figure Rotation

Chaotic dispersal of tidal debris

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