We discuss the properties of orbits within the influence sphere of a supermassive black hole (BH), in the case that the surrounding star cluster is nonaxisymmetric. There are four major orbit families; one of these, the pyramid orbits, have the interesting property that they can approach arbitrarily closely to the BH. We derive the orbit-averaged equations of motion and show that in the limit of weak triaxiality, the pyramid orbits are integrable: the motion consists of a two-dimensional libration of the major axis of the orbit about the short axis of the triaxial figure, with eccentricity varying as a function of the two orientation angles, and reaching unity at the corners. Because pyramid orbits occupy the lowest angular momentum regions of phase space, they compete with collisional loss cone repopulation and with resonant relaxation in supplying matter to BHs. General relativistic advance of the periapse dominates the precession for sufficiently eccentric orbits, and we show that relativity imposes an upper limit to the eccentricity: roughly the value at which the relativistic precession time is equal to the time for torques to change the angular momentum. We argue that this upper limit to the eccentricity should apply also to evolution driven by resonant relaxation, with potentially important consequences for the rate of extreme-mass-ratio inspirals in low-luminosity galaxies. In giant galaxies, we show that capture of stars on pyramid orbits can dominate the feeding of BHs, at least until such a time as the pyramid orbits are depleted; however this time can be of order a Hubble time.
Triaxial Orbit-based Dynamical Modeling of Galaxies with Supermassive Black Holes and an Application to Massive Elliptical Galaxy NGC 1453
在轴对称假设下，利用轨道叠加技术得到了大部分邻近超大质量黑洞质量的恒星动力学测定。然而，很少有星系--特别是大质量的早期星系--遵守精确的轴对称。在这里，我们提出了一个修改后的轨道叠加代码和一个新的方法来动态确定的内在形状和质量参数的三轴星系的空间分辨恒星运动学数据的基础上。这里描述的三轴TriOS代码纠正了原始van den Bosch等人（2008年）代码中的一个错误，该错误导致三轴模型中大多数轨道的预测不正确，并可能显著影响参数碰撞参量。修订后的代码还包含轨道采样，质量约束和运行时间的显着改进。此外，我们引入了两个新的参数搜索策略-一组新的三轴形状参数和一种新的无网格采样技术-一起导致在定位最佳拟合模型的效率显着提高。我们将更新的代码和搜索方法应用于NGC 1453，一个快速旋转的大质量椭圆星系。完整的6D参数搜索找到固有轴比的$p=b/a=0.933^{+0.014}_{-0.015}$和$q=c/a=0.779\pm0.012$，以及三轴度参数的$T=0.33\pm0.06$。尽管偏离轴对称，最佳拟合SMBH质量，恒星质光比和NGC 1453的暗物质封闭质量与轴对称结果一致。在得出一般性结论之前，需要对轴对称和三轴模型进行更多的比较。
On the robustness of triaxial Schwarzschild modelling: The effects of correcting the orbit mirroring
在过去的15年里，van den Bosch等人（2008）的三轴Schwarzschild轨道叠加程序被广泛应用于星系动力学的研究。最近，Quenneville等人（2022）报告了该代码的轨道计算中的一个错误，特别是在用于加速计算的镜像过程中。我们已经修复了CNOITE中的错误镜像，CNOITE是van den Bosch等人（2008）公开发布的三轴Schwarzschild代码的继承者。在这项研究中，我们提供了一个彻底的量化这个错误如何影响与此代码进行动态分析的结果。我们比较了原始和修正后的结果，讨论了单轨道相空间分布和全局恒星轨道分布的差异，对高三轴星系PGC 46832中心黑洞质量的估计，以及对50多个星系内禀形状和封闭质量的测量。集中在典型的科学应用史瓦西三轴代码，在我们所有的测试中，我们发现，差异是可以忽略不计的统计和系统的不确定性。我们的结论是，以前的结果与van den Bosch等人（2008）三轴史瓦西代码没有显着影响不正确的镜像。
The SAMI Galaxy Survey: The internal orbital structure and mass distribution of passive galaxies from triaxial orbit-superposition Schwarzschild models
Uncover 3D Dark Matter Distribution of the Milky Way by an Empirical Triaxial Orbit-Superposition Model: Method Validation
Shanghai Astronomical Observatory, Chinese Academy of Sciences（中国科学院上海天文台）；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）；Westlake University（西湖大学）；Hainan University（海南大学）；Beijing Normal University（北京师范大学）
