一个均匀球形山羊（问题）：同态坍缩随时间径向演化的显式解
A Uniform Spherical Goat (Problem): Explicit Solution for Homologous Collapse's Radial Evolution in Time
摘要：
均匀密度球在自身引力作用下的同系坍缩是从恒星到星系等天体形成动力学的著名模型。同样众所周知的是，由于微分方程解的超越性质，半径随时间的演化不能明确获得。相反，半径和时间都是根据展开角$θ$参数化地编写的。在这里，我们提出了一个明确的积分解半径作为时间的函数，利用复分析的方法，最近应用到几何相似的“几何山羊问题”。我们的解决方案可以使用快速傅立叶变换进行有效评估，并允许在时间上进行任意采样，使用简单的Python实现$\sim$ $100\times$比使用数值求根实现任意采样更快。我们的显式解决方案是有利的，相对于通常的方法，首先生成一个均匀的网格在$θ$，因为后者的结果在一个非均匀的径向或时间采样，不太有用的应用程序，如生成的子网格物理模型。
英文摘要：
The homologous collapse from rest of a uniform density sphere under its self gravity is a well-known toy model for the formation dynamics of astronomical objects ranging from stars to galaxies. Equally well-known is that the evolution of the radius with time cannot be explicitly obtained because of the transcendental nature of the differential equation solution. Rather, both radius and time are written parametrically in terms of the development angle $θ$. We here present an explicit integral solution for radius as a function of time, exploiting methods from complex analysis recently applied to the mathematically-similar 'geometric goat problem'. Our solution can be efficiently evaluated using a Fast Fourier Transform and allows for arbitrary sampling in time, with a simple Python implementation that is $\sim$$100\times$ faster than using numerical root-finding to achieve arbitrary sampling. Our explicit solution is advantageous relative to the usual approach of first generating a uniform grid in $θ$, since this latter results in a non-uniform radial or time sampling, less useful for applications such as generation of sub-grid physics models.
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