论文标题
各向同性球形密度簇的自相似轨道平均福克 - 普兰克方程（i）精确的爆发前解决方案
Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense clusters (i) accurate pre-collapse solution
论文作者
论文摘要
这是我们关于自相似轨道平均的Fokker-Planck（OAFP）方程的一系列作品的第一篇论文，用于恒星在致密的各向同性恒星簇中的分布功能。在集群放松演变的后期，标准恒星动力学预测，簇以自相似的方式形成构成核心的构成岩心。但是，相应的数学模型是自相似的OAFP方程，从未在整个能量域$（-1 <e <0）$上解决。基于有限差异方法的现有作品仅在截短的域$ -1 <e <-0.2 $上提供解决方案。为了扩大截断域的范围，目前的工作措施依靠（高度准确，高效）高斯 - chebyshev伪谱法。我们提供了一个光谱解决方案，其在整个域上的重要数字为四个。同样，该溶液可以降低到半分析形式，其多项式的程度只有18个具有三个重要数字的形式。我们还提供新的特征值； $ c_ {1} = 9.0925 \ times10^{ - 4} $，$ c_ {2} = 1.1118 \ times10^{ - 4} $，$ c_ {3} = 7.1975 \ times10^{ - times10^ $ξ= 3.64 \ times10^{ - 3} $，缩放逃脱能量$χ_\ text {ens c} = 13.881 $和powerlaw指数$α= 2.2305 $。由于整个域上的解决方案对Chebyshev多项式的程度不稳定，因此我们还提供了截断域（$ -1 <e <e_ <e_ \ text {max} $的光谱解决方案，其中$ -0.35 <e_ \ e_ \ text {max} {max} <-0.03 $）来解释如何处理强度。通过以几种方式重新重新设计OAFP方程，我们提高了光谱解决方案的准确性，并重现了现有的自相似解决方案，该解决方案充其量只有一个重要的数字，该解决方案充其量只有一个重要的数字。
This is the first paper of a series of our works on the self-similar orbit-averaged Fokker-Planck (OAFP) equation for distribution function of stars in dense isotropic star clusters. At the late stage of relaxation evolution of the clusters, standard stellar dynamics predicts that the clusters evolve in a self-similar fashion forming collapsing cores. However, the corresponding mathematical model, the self-similar OAFP equation, has never been solved on the whole energy domain $(-1< E < 0)$. The existing works based on kinds of finite difference methods provide solutions only on the truncated domain $-1< E<-0.2$. To broaden the range of the truncated domain, the present work resorts to a (highly accurate and efficient) Gauss-Chebyshev pseudo-spectral method. We provide a spectral solution, whose number of significant figures is four, on the whole domain. Also, the solution can reduce to a semi-analytical form whose degree of polynomials is only eighteen holding three significant figures. We also provide the new eigenvalues; $c_{1}=9.0925\times10^{-4}$, $c_{2}=1.1118\times10^{-4}$, $c_{3}=7.1975\times10^{-2}$ and $c_{4}=3.303\times10^{-2}$, corresponding to the core collapse rate $ξ=3.64\times10^{-3}$, scaled escape energy $χ_\text{esc}=13.881$ and power-law exponent $α=2.2305$. Since the solution on the whole domain is unstable against degree of Chebyshev polynomials, we also provide spectral solutions on truncated domains ( $-1< E<E_\text{max}$, where $-0.35<E_\text{max}<-0.03$) to explain how to handle the instability. By reformulating the OAFP equation in several ways, we improve the accuracy of the spectral solution and reproduce an existing self-similar solution, which infers that existing solutions have only one significant figure at best.
