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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Published in:	The Astronomical journal Vol. 169; no. 4; pp. 224 - 231
Main Authors:	Polyachenko, Evgeny V., Shukhman, Ilia G.
Format:	Journal Article
Language:	English
Published:	Madison The American Astronomical Society 01.04.2025 IOP Publishing
Subjects:	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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Abstract	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.
AbstractList	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.
Author	Polyachenko, Evgeny V. Shukhman, Ilia G.
Author_xml	– sequence: 1 givenname: Evgeny V. orcidid: 0000-0002-4596-1222 surname: Polyachenko fullname: Polyachenko, Evgeny V. organization: Institute of Astronomy , Russian Academy of Sciences, 48 Pyatnitskaya st, Moscow 119017, Russia – sequence: 2 givenname: Ilia G. orcidid: 0000-0002-6060-694X surname: Shukhman fullname: Shukhman, Ilia G. organization: Institute of Solar-Terrestrial Physics , Russian Academy of Sciences, Siberian Branch, P.O. Box 291, Irkutsk 664033, shukhman@iszf.irk.ru Russia
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Snippet	We examine the decay of perturbations in an infinite homogeneous self-gravitating model with a Maxwellian distribution function (DF) when weak collisions are...
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SubjectTerms	Collisions Decay Distribution functions Eigenvalues Eigenvectors Gravitation Gravitational instability Landau damping Maxwellian distribution Perturbation Resonant frequencies Stellar dynamics Stellar systems
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Title	Dynamics of Stellar Systems with Collisions: Eigenvalues and Eigenfunctions in Nearly Collisionless Limit
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