Abstract This is the first paper of a series of our works on the self-similar orbit-averaged Fokker-Planck (OAFP) equation to model distribution function for stars in isotropic dense spherical star clusters. At the late stage of the 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 ) . 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 accurate to four significant figures on the whole domain. 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 × 10 − 4 , c 2 = 1.1118 × 10 − 4 , c 3 = 7.1975 × 10 − 2 , and c 4 = 3.303 × 10 − 2 , corresponding to the core-collapse rate ξ = 3.64 × 10 − 3 , scaled escape energy , χ esc = 13.881 and power-law exponent α = 2.2305 . Since the solution is unstable against the degree of Chebyshev polynomials on the whole domain, we also provide spectral solutions on truncated domains ( − 1 E E max , where − 0.35 ≤ E max ≤ − 0.03 ) to explain how to handle the instability. By reformulating the self-similar OAFP equation in several ways, we improve the accuracies of the spectral solutions and reproduce an existing self-similar solution, which infers that the existing solutions are accurate up to only one significant figure.

中文翻译：

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各向同性球面致密星团的自相似轨道平均福克-普朗克方程（i）精确预坍缩解

摘要 这是我们关于自相似轨道平均福克-普朗克 (OAFP) 方程对各向同性致密球形星团中恒星的分布函数进行建模的一系列工作的第一篇论文。在星团弛豫演化的后期，标准恒星动力学预测星团以自相似的方式演化，形成坍缩的核心。然而，相应的数学模型，自相似 OAFP 方程，从未在整个能量域 (- 1 E 0 ) 上求解。基于各种有限差分方法的现有工作仅在截断域 - 1 E ⪅ - 0.2 上提供解决方案。为了扩大截断域的范围，目前的工作采用（高度准确和高效）高斯-切比雪夫伪谱方法。我们提供了在整个域上精确到四位有效数字的频谱解决方案。该解决方案可以简化为半解析形式，其多项式的次数仅为 18 次，保留三位有效数字。我们还提供了新的特征值 c 1 = 9.0925 × 10 − 4 、c 2 = 1.1118 × 10 − 4 、c 3 = 7.1975 × 10 − 2 和 c 4 = 3.303 × 10 − 2 ，对应于核心坍塌率ξ = 3.64 × 10 − 3 ，标度逃逸能 ，χ esc = 13.881 和幂律指数 α = 2.2305 。由于该解相对于整个域上的 Chebyshev 多项式的次数是不稳定的，我们还提供了截断域上的谱解（ − 1 EE max ，其中 − 0.35 ≤ E max ≤ − 0.03 ）来解释如何处理不稳定性。通过以多种方式重新表述自相似的 OAFP 方程，