黑洞周围自重物质环：ISCO的轨道频率研究

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"围绕黑洞的轨道上的自重物质环：最内层的稳定圆形轨道" 这篇学术文章探讨了一个独特的天体物理学问题，即自重物质环如何在黑洞周围形成稳定的轨道，特别是关注最内层的稳定圆形轨道（ISCO）。研究的核心是分析一个由质量为M的扰动Kerr黑洞和一个静止、轴对称的粒子环组成的系统。在这个系统中，粒子环自身的质量m对轨道特性产生了显著影响。 Kerr黑洞是一种旋转的黑洞模型，其旋转导致空间时间结构发生扭曲，形成了复杂的轨道动力学。文章指出，由于环的有限质量m，ISCO的轨道频率会发生偏移。ISCO是物体可以绕黑洞稳定公转的最内界线，对于理解黑洞吸积盘的性质至关重要。在薄环（半厚度r远小于M）的情况下，主要的质量修正效应来自环的自我引力势能。这个势能项在环的质量m的二次方程中占主导地位，对ISCO频率的修正起着关键作用。这种主导的修正项的阶数为O(μln(M))，其中μ可能是与环质量和黑洞质量相关的比例系数。文章通过详尽的分析揭示了自我引力如何改变环的动态行为，这对于理解黑洞附近复杂动力学以及可能的观测现象具有重要意义。这项工作还可能对黑洞吸积盘的理论模型产生影响，帮助科学家更准确地预测和解释天文观测数据。该研究深化了我们对黑洞周围复杂物质分布的理解，特别是当这些物质具有一定质量且受到自身引力影响时。这不仅对天体物理学家，也对那些研究黑洞物理、广义相对论以及宇宙结构形成的学者有着深远的影响。通过开放访问的形式发布，这篇文章使得更多科研人员能够访问和利用这些研究成果。

Eur. Phys. J. C (2014) 74:2840

DOI 10.1140/epjc/s10052-014-2840-4

Regular Article - Theoretical Physics

Self-gravitating ring of matter in orbit around a black hole:

the innermost stable circular orbit

Shahar Hod

1,2,a

The Ruppin Academic Center, Emeq Hefer 40250, Israel

The Hadassah Institute, Jerusalem 91010, Israel

Received: 2 February 2014 / Accepted: 27 March 2014 / Published online: 11 April 2014

Abstract We study analytically a black-hole-ring system

which is composed of a stationary axisymmetric ring of par-

ticles in orbit around a perturbed Kerr black hole of mass M.

In particular, we calculate the shift in the orbital frequency

of the innermost stable circular orbit (ISCO) due to the ﬁnite

mass m of the orbiting ring. It is shown that for t hin rings of

half-thickness r  M, the dominant ﬁnite-mass correction

to the characteristic ISCO frequency stems from the self-

gravitational potential energy of the ring (a term in the energy

budget of the system which is quadratic in the mass m of the

ring). This dominant correction to the ISCO frequency is of

order O(μ ln(M/r)), where μ ≡ m/M is the dimensionless

mass of the ring. We show that the ISCO frequency increases

(as compared to the ISCO frequency of an orbiting test-ring)

due to the ﬁnite-mass effects of the self-gravitating ring.

1 Introduction

The geodesic motions of test particles in black-hole space-

times are an important source of information on the structure

of the spacetime geometry [1–11]. Of particular importance

is the innermost stable circular orbit (ISCO). This orbit is

deﬁned by the onset of a dynamical instability for circular

geodesics. In particular, the ISCO separates stable circular

orbits from orbits that plunge into the central black hole [2].

This special geodesic therefore plays a central role in the

two-body dynamics of in-spiralling compact binaries since

it marks the critical point where the character of the motion

sharply changes [5]. In addition, this marginally stable orbit

is usually regarded as the inner edge of accretion disks in

composed black-hole-disk systems [2].

An important physical quantity which characterizes the

ISCO is the orbital angular frequency 

isco

as measured by

asymptotic observers. This characteristic frequency is often

regarded as the end-point of the in-spiral gravitational tem-

e-mail: shaharhod@gmail.com

plates [5]. For a test particle in the Schwarzschild-black-hole

spacetime, this frequency is given by the well-known relation

[1–11]

M

isco

= 6

−3/2

, (1)

where M is the mass of the central black hole.

Realistic astrophysical scenarios often involve a com-

posed two-body system in which the mass m of the orbiting

object is smaller but non-negligible as compared to the mass

M of the central black hole [5]. In these situations the zeroth-

order (test-particle) approximation is no longer valid and one

should take into account the gravitational self-force (GSF)

corrections to the orbit [12–26]. These ﬁrst-order corrections

take into account the ﬁnite mass m of the orbiting object. The

gravitational self-force has two distinct contributions: (1) It

is responsible for dissipative (radiation-reaction) effects that

cause the orbiting particle to emit gravitational waves. The

location of the ISCO may become blurred due to these non-

conservative effects [5,12,13]. (2) The self-force due to the

ﬁnite mass of the particle is also responsible for conserva-

tive effects which preserve the characteristic constants of the

orbital motion. These conservative effects produce a non-

trivial shift in the ISCO frequency, which characterizes the

two-body dynamics.

It should be emphasized that the computation of the con-

servative shift in the characteristic ISCO frequency (due to

the ﬁnite mass of the orbiting object) is a highly non-trivial

task. A notable event in the history of the two-body prob-

lem in general relativity took place three years ago: after two

decades of intensive efforts by many groups of researches to

evaluate the conservative self-force corrections to the orbital

parameters, Barack and Sago [23,24] have succeeded in com-

puting the shift in the ISCO frequency due to the ﬁnite mass

of the orbiting object. Their numerical result for the corrected

ISCO frequency can be expressed in the form [22–24]

M

isco

= 6

−3/2

(1 + c · μ) with c  0.251, (2)

123

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