发表机构
Institute of Astronomy, Russian Academy of Sciences; Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, Siberian Branch（俄罗斯科学院天文研究所; 俄罗斯科学院西伯利亚分支太阳-地球物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为球状恒星系统径向扰动构造了一组无质量势密度对基，其势按指定幂次衰减，Gram矩阵为带状，能同时分辨中心与外围，且展开无寄生质量。
AI 中文摘要
在球状恒星系统线性扰动的矩阵方法中，扰动势和密度在一组势密度对上展开。对于径向扰动，质量守恒使得扰动势中$1/r$的系数消失，因此势的衰减快于$1/r$。从$\u2113=0$的Hernquist--Ostriker族出发，我们以闭合形式构造了一组势密度对，其势按$r^{-p}$衰减，其中$p\ge2$为预先指定，尾部包含所有后续整数幂。对于$p=2$，每一对都是单独无质量的，第$n$对的势通过单个Jacobi多项式表示。我们解析地推导了组合权重、主导尾部系数和Gram矩阵；Gram矩阵是半带宽为$p-1$的带状矩阵（在$p=2$时为三对角）。第$n$个势元素恰好有$n-1$个节点，分布在从$r\sim n^{-2}$到$r\sim n^{2}$的范围内，因此该集合能够同时分辨中心和远外围。作为测试，等时模型膨胀模态势的展开指数收敛，并且在任何截断处都不携带寄生质量，这与标准集合形成对比。
英文摘要
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 $\ell=0$ Hernquist--Ostriker family, we construct in closed form a set of potential-density pairs whose potentials decay as $r^{-p}$ with a prescribed $p\ge2$, 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 $r\sim n^{-2}$ to $r\sim n^{2}$, 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.
Comments6 pages, 3 figures. Submitted to Astron.Rep
