We revisit the role that gravitational scattering off stars plays in establishing the steady-state distribution of collisionless dark matter (DM) around a massive black hole (BH). This is a physically interesting problem that has potentially observable signatures, such as $\gamma$ rays from DM annihilation in a density spike. The system serves as a laboratory for comparing two different dynamical approaches, both of which have been widely used: a Fokker-Planck treatment and a two-component conduction fluid treatment. In our Fokker-Planck analysis we extend a previous analytic model to account for a nonzero flux of DM particles into the BH, as well as a cutoff in the distribution function near the BH due to relativistic effects or, further out, possible DM annihilation. In our two-fluid analysis, following an approximate analytic treatment, we recast the equations as a “heated Bondi accretion” problem and solve the equations numerically without approximation. While both the Fokker-Planck and two-fluid methods yield basically the same DM density and velocity dispersion profiles away from the boundaries in the spike interior, there are other differences, especially the determination of the DM accretion rate. We discuss limitations of the two treatments, including the assumption of an isotropic velocity dispersion.

中文翻译：

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恒星对黑洞周围暗物质尖峰的影响：两种治疗方法的故事

我们重新审视恒星的引力散射在建立大质量黑洞 (BH) 周围无碰撞暗物质 (DM) 的稳态分布中所起的作用。这是一个物理上有趣的问题，具有潜在的可观察特征，例如$\gamma$来自 DM 湮灭的射线在密度尖峰中。该系统用作比较两种不同的动力学方法的实验室，这两种方法都已被广泛使用：Fokker-Planck 处理和双组分传导流体处理。在我们的 Fokker-Planck 分析中，我们扩展了之前的分析模型，以解释进入 BH 的 DM 粒子的非零通量，以及由于相对论效应或可能的 DM 湮灭导致的 BH 附近分布函数的截止。在我们的二流体分析中，经过近似解析处理，我们将方程重铸为“热邦迪吸积”问题，并在没有近似的情况下对方程进行数值求解。虽然 Fokker-Planck 和双流体方法在远离尖峰内部边界的地方产生了基本相同的 DM 密度和速度分散剖面，还有其他差异，特别是 DM 吸积率的确定。我们讨论了两种处理方法的局限性，包括各向同性速度分散的假设。