由可分离增广密度确定的恒星动力学模型的相空间一致性
Phase-space consistency of stellar dynamical models determined by separable augmented densities
摘要：
假设增广密度是可分的，则对于任意给定的密度和各向异性，都可以构造球形总体的分布函数。我们考虑在什么条件下这样构造的分布在可及相空间中处处实际上是非负的。我们首先推广已知的必要条件，增广密度使用分数阶微积分。半径部分R（r^2）（其对数导数是各向异性参数）的条件等价于R（1/w）/w的完全单调性。另一方面，关于势部分的条件由其导数给出，直到不大于（3/2-β）的任何阶次都是非负的，其中β是中心各向异性参数。我们还从可分离增广密度的分布推导出一个专门的反演公式，这导致了可分离增广密度的分布的非负性的充分条件。最后一个条件将前面得到的广义Cuddeford系统的相似条件推广到任意可分系统。
英文摘要：
Assuming the separable augmented density, it is always possible to construct a distribution function of a spherical population with any given density and anisotropy. We consider under what conditions the distribution constructed as such is in fact non-negative everywhere in the accessible phase-space. We first generalize known necessary conditions on the augmented density using fractional calculus. The condition on the radius part R(r^2) (whose logarithmic derivative is the anisotropy parameter) is equivalent to the complete monotonicity of R(1/w)/w. The condition on the potential part on the other hand is given by its derivative up to any order not greater than (3/2-beta) being non-negative where beta is the central anisotropy parameter. We also derive a specialized inversion formula for the distribution from the separable augmented density, which leads to sufficient conditions on separable augmented densities for the non-negativity of the distribution. The last generalizes the similar condition derived earlier for the generalized Cuddeford system to arbitrary separable systems.
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