Dynamics of Stellar Systems with Collisions: Eigenvalues and Eigenfunctions in Nearly Collisionless Limit

We examine the decay of perturbations in an infinite homogeneous self-gravitating model with a Maxwellian distribution function (DF) when weak collisions are present. In collisionless systems within the stable parameter range, the eigenvalue spectrum consists of a continuous set of real frequencies...

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Vydáno v:The Astronomical journal Ročník 169; číslo 4; s. 224 - 231
Hlavní autoři: Polyachenko, Evgeny V., Shukhman, Ilia G.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Madison The American Astronomical Society 01.04.2025
IOP Publishing
Témata:
Collisions
Decay
Distribution functions
Eigenvalues
Eigenvectors
Gravitation
Gravitational instability
Landau damping
Maxwellian distribution
Perturbation
Resonant frequencies
Stellar dynamics
Stellar systems
ISSN:0004-6256, 1538-3881
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Shrnutí:We examine the decay of perturbations in an infinite homogeneous self-gravitating model with a Maxwellian distribution function (DF) when weak collisions are present. In collisionless systems within the stable parameter range, the eigenvalue spectrum consists of a continuous set of real frequencies associated with van Kampen (vK) modes, which are singular eigenfunctions of the stellar DF. An initial perturbation in the stellar density and gravitational potential decays exponentially through a superposition of these modes, a phenomenon known as Landau damping. However, the perturbation in the stellar DF does not decay self similarly; it becomes increasingly oscillatory in velocity space over time, indicating the absence of eigenfunctions corresponding to the Landau damping eigenfrequencies. Consequently, we refer to perturbations undergoing Landau damping as quasi-modes rather than true eigenmodes. Even rare collisions suppress the formation of steep DF gradients in velocity space. C. S. Ng & A. Bhattacharjee demonstrated that introducing collisions eliminates vK modes and transforms Landau quasi-modes into true eigenmodes forming a complete set. As the collision frequency approaches zero, their eigenfrequencies converge to those of the collisionless Landau quasi-modes. In this study, we investigate the behavior of the eigenfunction of the least-damped aperiodic mode as the collision frequency approaches zero. We derive analytic expressions for the eigenfunction in the resonance region and for the damping rate as a function of collision frequency. Additionally, we employ the standard matrix eigenvalue problem approach to numerically verify our analytical results.
Bibliografie:Galaxies and Cosmology
AAS61766
ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:0004-6256
1538-3881
DOI:10.3847/1538-3881/adb8da