轴对称系统的双积分分布函数
Two-integral distribution functions for axisymmetric systems
摘要：
对于已知轴对称密度的恒星系统，给出了求两个积分分布函数（DF）的公式，该函数只依赖于能量和角动量关于对称轴的两个经典积分。它们来自于Eddington和Edmke的思想的结合，也是Jiang和Ossipkov为寻找球形星系的各向异性DF所展示的思想的扩展。系统的密度需要表示为势能函数和径向坐标函数的乘积之和。对应于这种密度的解又是能量函数和关于对称轴的角动量的大小函数的乘积之和。密度和径向速度弥散的乘积也可以表示为势函数和径向坐标函数的乘积之和。可以进一步知道，密度乘以其旋转速度色散等于势函数和径向坐标函数的乘积之和减去密度与其平均旋转速度的平方的乘积。这些公式可应用于Binney模型和Lynden-Bell模型。宾尼模型的奇数DF的无穷大也可以在旋转速度定律的假设下找到。
英文摘要：
Some formulae are presented for finding two-integral distribution functions (DFs) which depends only on the two classical integrals of the energy and the magnitude of the angular momentum with respect to the axis of symmetry for stellar systems with known axisymmetric densities. They come from an combination of the ideas of Eddington and Fricke and they are also an extension of those shown by Jiang and Ossipkov for finding anisotropic DFs for spherical galaxies. The density of the system is required to be expressed as a sum of products of functions of the potential and of the radial coordinate. The solution corresponding to this type of density is in turn a sum of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. The product of the density and its radial velocity dispersion can be also expressed as a sum of products of functions of the potential and of the radial coordinate. It can be further known that the density multipied by its rotational velocity dispersion is equal to a sum of products of functions of the potential and of the radial coordinate minus the product of the density and the square of its mean rotational velocity. These formulae can be applied to the Binney and the Lynden-Bell models. An infinity of the odd DFs for the Binney model can be also found under the assumption of the laws of the rotational velocity.
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