标题： 具有有限范围的自洽动力学模型——III. 截断幂律球体 显示英文标题

标题： Self-consistent dynamical models with a finite extent -- III. Truncated power-law spheres

摘要： 完全解析的动力学模型通常具有无限的范围，而真实的恒星团簇、星系和暗物质晕则具有有限的范围。 生成有限范围动力学模型的标准方法是采用具有无限范围的模型，并在结合能上施加截断。 然而，这种方法不能用于生成具有预设解析质量密度分布的模型。 我们研究了一族具有广义切向Cuddeford（TC）轨道结构的幂律球体的自洽性和动力学性质。 通过改变密度幂律指数 $\gamma$ 和中心各向异性 $\beta_0$，这些模型覆盖了密度和各向异性剖面的广泛参数空间。 我们明确计算了各种参数组合下的相空间分布函数，并根据束缚轨道的能量分布来解释我们的结果。 我们发现，如果且仅当 $\gamma \geqslant 2\beta_0$ 成立时，截断的幂律球体才能由TC轨道结构支持，这意味着中心密度斜率-各向异性不等式对于这一族模型来说既是充分条件也是必要条件。 我们提供了结构和动力学性质的闭合表达式，例如径向和切向速度弥散剖面，这些可以与更复杂的数值建模结果进行比较。 这项工作极大地丰富了具有有限范围和解析描述的自洽动力学模型的可用集合。 显示英文摘要

摘要： Fully analytical dynamical models usually have an infinite extent, while real star clusters, galaxies, and dark matter haloes have a finite extent. The standard method for generating dynamical models with a finite extent consists of taking a model with an infinite extent and applying a truncation in binding energy. This method, however, cannot be used to generate models with a pre-set analytical mass density profile. We investigate the self-consistency and dynamical properties of a family of power-law spheres with a general tangential Cuddeford (TC) orbital structure. By varying the density power-law slope $\gamma$ and the central anisotropy $\beta_0$, these models cover a wide parameter space in density and anisotropy profiles. We explicitly calculate the phase-space distribution function for various parameter combinations, and interpret our results in terms of the energy distribution of bound orbits. We find that truncated power-law spheres can be supported by a TC orbital structure if and only if $\gamma \geqslant 2\beta_0$, which means that the central density slope-anisotropy inequality is both a sufficient and a necessary condition for this family. We provide closed expressions for structural and dynamical properties such as the radial and tangential velocity dispersion profiles, which can be compared against more complex numerical modelling results. This work significantly adds to the available suite of self-consistent dynamical models with a finite extent and an analytical description.