南方基地知识库

St?ckel引力势是一类最普遍形式的可分离势。具有St?ckel形式的星系是完全可积系统,即其中的恒星轨道都是规则的,其运动积分可以解析求出。运动积分——特别是作用量（J,一种特殊的运动积分）,能简化恒星运动的描述;研究星系中恒星的运动学和动力学的重要途径是作用量空间,比如使用以作用量作为参量的分布函数f（JR, Lz, Jz）。通过将一般星系的引力势局部地近似成St?ckel势,人们可以估算一般星系中恒星运动的作用量。介绍天文学家在拓展St?ckel引力势的研究上取得的重大进展:在基于St?ckel势理论估算一般星系中的作用量（或运动积分）方面,开发出了若干快速数值算法（例如St?ckel作弊法[St?ckel Fudge]、St?ckel拟合法）,基于环面映射的收敛性高精度数值算法（例如轨道积分拟合生成函数法、迭代环面构造法）,以及提出作为近似的运动积的解析表达式;在分布函数的构造方面,基于上述估算的作用量或直接利用St?ckel势的运动积分公式（主要是I3表达式）,提出了一些分布函数模型,f（JR, Lz, Jz）或f（E, Lz, I3）。这些进展使得人们可以对星系开展基于分布函数的建模。此外,还介绍近几年把这些方法应用到银河系观测数据的若干代表性工作。

St¨ackel gravitational potentials are the most general class of separable potentials. Galaxies with a St¨ackel form belong to complete integrable systems, in which the stellar orbits are all regular and their integrals of motion (I) can be expressed analytically. Integrals of motion, particularly actions (J) as a special kind of integrals, can simplify the description of stellar motions. An important approach to study the kinematics and dynamics of stars in galaxies is by means of action space; e. g. , using actions as the arguments of distribution functions (DFs), f(JR, Lz, Jz). One can estimate actions for the stellar motions in general galaxies by locally approximating the gravitational potentials with St¨ackel models. Recent years, there have been important progresses in extending the use of St¨ackel potential theory in general galaxies: in the respect of computing actions (or integrals of motion) for general galaxies, researchers have developed several fast algorithms (e. g. , St¨ackel Fudge and St¨ackel Fitting), and several convergent high-precision algorithms based on torus mapping (e. g. , generating function from orbit integration method and iterative torus construction method), as well as formalisms of analytically expressing “approximate integrals of motion” of general galaxies (e. g. , the quasi-I3); in the respect of constructing DFs for galaxies and galactic components, with the actions estimated by the above methods or directly using integrals formulae of St¨ackel potentials (particularly the I3 expression), some DF models are proposed in the form of f(JR, Lz, Jz) or f(E, Lz, I3). These theoretical and methodological developments make possible DF modelling for galaxies. This article reviews the above developments, and introduces several recent representative studies that use action- or integral-based DF models to fit the MW data, as well as the recent developments in extending the use of the (action, angle) methodology to non-integrable phenomena (such as galactic resonances and bars) and out-of-equilibrium processes (such as the formation of tidal streams), with a hope of being helpful for Chinese astronomical community to welcome this “Gaia + LAMOST” era.