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院太阳-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
该研究针对无碰撞球恒星系统径向扰动，构造了两组无渐近尾的势-密度基对，解决了标准基收敛慢的问题，提升了响应矩阵计算效率。
AI 中文摘要
Kalnajs矩阵方法是研究无碰撞恒星系统全局线性稳定性和可能的朗道阻尼的广泛使用的框架。然而，对于具有无限边界的三维球形模型中的径向扰动（l=0），Clutton-Brock基等标准双正交集的收敛速度较慢。这源于一个物理约束：守恒质量的径向模式迫使扰动势比无穷远处的点质量衰减得更快，而单个Clutton-Brock基元带有虚构的净质量，仅以O(1/r)衰减，产生非物理的渐近尾。我们构造了两组新的势-密度基对，其构造本身就没有这些尾。第一组对Clutton-Brock集进行了修改：相邻基元的特定线性组合解析地抵消了主导的O(1/r)项，得到的势以O(1/r³)衰减，密度以O(1/r⁵)衰减。这打破了严格的双正交性，但产生了紧凑的三对角Gram矩阵，其进入响应方程的额外成本可忽略不计。第二组由Jacobi多项式构建，这些多项式在构造时直接嵌入了所需的O(1/r²)势衰减，同时保持严格的对角双正交性。数值测试表明，两种展开的收敛性在半径上是均匀的：当展开势的渐近尾与基元的宇称匹配时，截断误差呈指数下降，否则呈代数下降。两种基都降低了给定精度下响应矩阵所需的维度，适用于研究开放恒星系统中的径向扰动。
英文摘要
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 $\mathcal{O}(1/r)$, 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 $\mathcal{O}(1/r)$ term, giving potentials that decay as $\mathcal{O}(1/r^3)$ and densities as $\mathcal{O}(1/r^5)$. 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 $\mathcal{O}(1/r^2)$ 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.
Comments6 pages, 3 figures. Accepted for publication in MNRAS
