Kerr度规中低angular momentum轴对称吸积中激波形成的滞后效应和诊断
Hysteresis effects and diagnostics of the shock formation in low angular momentum axisymmetric accretion in the Kerr metric
摘要：
研究了旋转黑洞周围纯广义相对论性低角动量吸积流的长期演化，发现吸积流具有滞后效应。这证实了在克尔度规中，定常激波是这种吸积盘的一个组成部分。描述吸积物质动力学流速空间梯度的方程等价于一阶自治动力系统。固定点分析保证了对于某些天体物理学相关的初始边界条件，这种流动必须是多跨音速的。与文献中已有的共识相反，证明了临界点和声学点一般不是同构的。具有多个临界点的流的同宿轨道选择了具有较高熵吸积率的临界点，证实了熵吸积率是系统中简并的消除剂。然而，在某些特殊情况下，也观察到异宿轨道，其中两个鞍型临界点的流配置具有相同的熵吸积率。拓扑异宿轨道是唯一允许的非可移动的退化解的吸积流与多个临界点，并被证明是结构不稳定。根据适当的初始边界条件，同宿轨道可以通过能量守恒的Rankine-Hugoniot型定常激波与标准的非同宿轨道相结合。本文提出了一个有效的李雅普诺夫指数，用以分析确定为什么某些跨音速流即使产生多个临界点也不能容纳激波解。（节录）
英文摘要：
The secular evolution of the purely general relativistic low angular momentum accretion flow around a spinning black hole is shown to exhibit hysteresis effects. This confirms that a stationary shock is an integral part of such an accretion disc in the Kerr metric. The equations describing the space gradient of the dynamical flow velocity of the accreting matter have been shown to be equivalent to a first order autonomous dynamical systems. Fixed point analysis ensures that such flow must be multi-transonic for certain astrophysically relevant initial boundary conditions. Contrary to the existing consensus in the literature, the critical points and the sonic points are proved not to be isomorphic in general. Homoclinic orbits for the flow flow possessing multiple critical points select the critical point with the higher entropy accretion rate, confirming that the entropy accretion rate is the degeneracy removing agent in the system. However, heteroclinic orbits are also observed for some special situation, where both the saddle type critical points of the flow configuration possesses identical entropy accretion rate. Topologies with heteroclinic orbits are thus the only allowed non removable degenerate solutions for accretion flow with multiple critical points, and are shown to be structurally unstable. Depending on suitable initial boundary conditions, a homoclinic trajectory can be combined with a standard non homoclinic orbit through an energy preserving Rankine-Hugoniot type of stationary shock. An effective Lyapunov index has been proposed to analytically confirm why certain class of transonic flow can not accommodate shock solutions even if it produces multiple critical points. (Abridged)
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