具有可分密度的轴对称恒星系统的双积分分布函数
Two-integral distribution functions for axisymmetric stellar systems with separable densities
摘要：
对于已知轴对称密度的恒星系统，我们给出了仅依赖于能量和角动量相对于对称轴的两个经典积分的分布函数（DF）的不同表达式。要求系统的密度是在势和径向坐标中可分离的函数的乘积，其中径向坐标的函数是径向坐标的平方和及其单位尺度的幂。对应于这类密度的二积分DF的偶数部分又是能量函数和关于对称轴的角动量的大小函数的乘积的和或无穷级数。在转动定律的假设下，也可以得到其奇部的类似表达式。进一步证明了这些表达式实际上与用Hunter和Qian的围道积分公式得到的表达式是等价的。这种方法在计算上一般优于围道积分法。给出了两个例子，求出了它们的两个整数DF的奇、偶部分。一个是长Jaffe模型，另一个是长Plummer模型。还可以发现，轴对称系统的二次积分偶DF的Hunter-Qian围道积分公式，可以利用由Dejonghe提出的Laplace-Mellin积分变换恢复。
英文摘要：
We show different expressions of distribution functions (DFs) which depend 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. The density of the system is required to be a product of functions separable in the potential and the radial coordinate, where the functions of the radial coordinate are powers of a sum of a square of the radial coordinate and its unit scale. The even part of the two-integral DF corresponding to this type of density is in turn a sum or an infinite series of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. A similar expression of its odd part can be also obtained under the assumption of the rotation laws. It can be further shown that these expressions are in fact equivalent to those obtained by using Hunter and Qian's contour integral formulae for the system. This method is generally computationally preferable to the contour integral method. Two examples are given to obtain the even and odd parts of their two-integral DFs. One is for the prolate Jaffe model and the other for the prolate Plummer model. It can be also found that the Hunter-Qian contour integral formulae of the two-integral even DF for axisymmetric systems can be recovered by use of the Laplace-Mellin integral transformation originally developed by Dejonghe.
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