Two sets of potential-density basis pairs for the study of radial perturbations in collisionless spherical stellar systems
Abstract
The Kalnajs matrix method is a widely used framework for studying the global linear stability and possible Landau damping of collisionless stellar systems. However, for radial perturbations (l=0) in three-dimensional spherical models with infinite boundaries, standard biorthogonal sets such as the Clutton-Brock basis often converge slowly. This stems from a physical constraint: mass-conserving radial modes force the perturbed potential to decay faster than a point mass at infinity, whereas individual Clutton-Brock elements carry a fictitious net mass and decay only as 𝒪(1/r), producing unphysical asymptotic tails. We construct two new families of potential-density basis pairs that are free of these tails by construction. The first modifies the Clutton-Brock set: a specific linear combination of adjacent elements analytically cancels the leading 𝒪(1/r) term, giving potentials that decay as 𝒪(1/r3) and densities as 𝒪(1/r5). This breaks strict biorthogonality but yields a compact tridiagonal Gram matrix that enters the response equation at negligible additional cost. The second family is built from Jacobi polynomials that embed the required 𝒪(1/r2) potential decay directly into their construction while retaining strict diagonal biorthogonality. Numerical tests demonstrate convergence that is uniform in radius for both expansions: the truncation error decreases exponentially when the asymptotic tail of the expanded potential matches the parity of the basis elements, and algebraically otherwise. Both bases reduce the dimension of the response matrix required for a given accuracy and are suited to studying radial perturbations in open stellar systems.
Related
- Suppression of spiral density waves in collisionless accretion flows
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BibTeX
@misc{polyachenko2026sets,
title = {{Two sets of potential-density basis pairs for the study of radial perturbations in collisionless spherical stellar systems}},
author = {E. V. Polyachenko and I. G. Shukhman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04012},
eprint = {2609.04012},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04012},
}
RIS
TY - UNPB
TI - Two sets of potential-density basis pairs for the study of radial perturbations in collisionless spherical stellar systems
AU - E. V. Polyachenko
AU - I. G. Shukhman
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04012
UR - https://arxiv.org/abs/2609.04012
AB - The Kalnajs matrix method is a widely used framework for studying the global linear stability and possible Landau damping of collisionless stellar systems. However, for radial perturbations (l=0) in three-dimensional spherical models with infinite boundaries, standard biorthogonal sets such as the Clutton-Brock basis often converge slowly. This stems from a physical constraint: mass-conserving radial modes force the perturbed potential to decay faster than a point mass at infinity, whereas individual Clutton-Brock elements carry a fictitious net mass and decay only as 𝒪(1/r), producing unphysical asymptotic tails. We construct two new families of potential-density basis pairs that are free of these tails by construction. The first modifies the Clutton-Brock set: a specific linear combination of adjacent elements analytically cancels the leading 𝒪(1/r) term, giving potentials that decay as 𝒪(1/r^3) and densities as 𝒪(1/r^5). This breaks strict biorthogonality but yields a compact tridiagonal Gram matrix that enters the response equation at negligible additional cost. The second family is built from Jacobi polynomials that embed the required 𝒪(1/r^2) potential decay directly into their construction while retaining strict diagonal biorthogonality. Numerical tests demonstrate convergence that is uniform in radius for both expansions: the truncation error decreases exponentially when the asymptotic tail of the expanded potential matches the parity of the basis elements, and algebraically otherwise. Both bases reduce the dimension of the response matrix required for a given accuracy and are suited to studying radial perturbations in open stellar systems.
ER -