A radial basis of massless potential-density pairs
Abstract
In the matrix method for linear perturbations of spherical stellar systems, the perturbed potential and density are expanded over a set of potential-density pairs. For radial perturbations, mass conservation makes the coefficient of 1/r in the perturbed potential vanish, so that the potential decays faster than 1/r. Starting from the ℓ=0 Hernquist--Ostriker family, we construct in closed form a set of potential-density pairs whose potentials decay as r−p with a prescribed p≥2, the tail containing all subsequent integer powers. For p=2 each pair is individually massless, and the potential of the n-th pair is expressed through a single Jacobi polynomial. We derive the combination weights, the leading tail coefficients and the Gram matrix analytically; the Gram matrix is banded with half-width p−1 (tridiagonal at p=2). The n-th potential element has exactly n−1 nodes, spread from r∼ n−2 to r∼ n2, so that the set resolves both the centre and the far periphery. As a test, the expansion of the dilation-mode potential of the isochrone model converges exponentially and carries no parasitic mass at any truncation, in contrast to the standard set.
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BibTeX
@misc{polyachenko2026radial,
title = {{A radial basis of massless potential-density pairs}},
author = {E. V. Polyachenko and I. G. Shukhman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25308},
eprint = {2609.25308},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25308},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A radial basis of massless potential-density pairs
AU - E. V. Polyachenko
AU - I. G. Shukhman
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25308
UR - https://arxiv.org/abs/2609.25308
AB - In the matrix method for linear perturbations of spherical stellar systems, the perturbed potential and density are expanded over a set of potential-density pairs. For radial perturbations, mass conservation makes the coefficient of 1/r in the perturbed potential vanish, so that the potential decays faster than 1/r. Starting from the ℓ=0 Hernquist--Ostriker family, we construct in closed form a set of potential-density pairs whose potentials decay as r^−p with a prescribed p≥2, the tail containing all subsequent integer powers. For p=2 each pair is individually massless, and the potential of the n-th pair is expressed through a single Jacobi polynomial. We derive the combination weights, the leading tail coefficients and the Gram matrix analytically; the Gram matrix is banded with half-width p−1 (tridiagonal at p=2). The n-th potential element has exactly n−1 nodes, spread from r∼ n^−2 to r∼ n^2, so that the set resolves both the centre and the far periphery. As a test, the expansion of the dilation-mode potential of the isochrone model converges exponentially and carries no parasitic mass at any truncation, in contrast to the standard set.
ER -