A special class of orbits known to exist around a Kerr black hole are spherical orbits—orbits with constant coordinate radii that are not necessarily confined to the equatorial plane. Spherical time-like orbits were first studied by Wilkins almost 50 years ago. In the present paper, we perform a systematic and thorough study of these orbits, encompassing and extending previous works on them. We first present simplified forms for the parameters of these orbits. The parameter space of these orbits is then analysed in detail; in particular, we delineate the boundaries between stable and unstable orbits, bound and unbound orbits, and prograde and retrograde orbits. Finally, we provide analytic solutions of the geodesic equations, and illustrate a few representative examples of these orbits.

中文翻译：

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围绕克尔黑洞的球形轨道

已知存在于克尔黑洞周围的一类特殊轨道是球面轨道——具有恒定坐标半径的轨道，不一定限于赤道平面。近 50 年前，威尔金斯首次研究了类球体轨道。在本文中，我们对这些轨道进行了系统而彻底的研究，包括并扩展了以前的工作。我们首先给出这些轨道参数的简化形式。然后详细分析这些轨道的参数空间；特别是，我们划定了稳定和不稳定轨道、束缚和未束缚轨道以及顺行和逆行轨道之间的界限。最后，我们提供了测地线方程的解析解，并说明了这些轨道的一些代表性示例。