General Relativistic Effect on Sitnikov Three-Body Problem: Restricted Case

Abstract

We investigate the effect of general relativity on the Sitnikov problem. The Sitnikov problem is one of the simplest three-body problems, in which the two primary bodies (a binary system) have equal mass m and orbit their barycenter, while the third body is treated as a test particle under Newtonian gravity. The trajectory of the test particle is perpendicular to the orbital plane of the binary (along z-axis) and passes through the barycenter of the two primaries. To study the general relativistic contributions, we first derive the equations of motion for both the binary and the test particle based on the first post-Newtonian Einstein–Infeld–Hoffmann equation, and integrate these equations numerically. We examine the behavior of the test particle (third body) as a function of the orbital eccentricity of the central binary e, the dimensionless gravitational radius $\lambda$, which characterizes the strength of general relativistic effect, and the initial position of the test particle ${\overline{z}}_0$. Our numerical calculations reveal the following; as general relativistic effects $\lambda$ increase and the eccentricity e of the binary orbit grows, the distance $\overline{r}$ between the test particle and the primary star undergoes complicated oscillations over time. Consequently, the gravitational force acting on the test particle also varies in a complex manner. This leads to a resonance state between the position $\overline{z}$ of the test particle and the distance $\overline{r}$, causing the energy E of the test particle to become $E\ge 0$. This triggers the effective ejection of the test particle due to the gravitational slingshot effect. In this paper, we shall refer to this ejection mechanism of test particle as the “Sitnikov mechanism.” As a concrete phenomenon that becomes noticeable, the increase in general relativistic effects and the eccentricity of the binary orbit leads to the following: (a) ejection of test particles from the system in a shorter time, and (b) increasing escape velocity of the test particle from the system. As an astrophysical application, we point out that the high-velocity ejection of test particles induced by the Sitnikov mechanism could contribute to elucidating the formation processes of astrophysical jets and hyper-velocity stars.

1. Introduction

In 1911, MacMillan [1] demonstrated the existence of a periodic solution to the Newtonian restricted three-body problem under the following configuration: the two central primaries (a binary system) have equal mass m and revolve around their barycenter in circular orbits, while the third body is a test particle that moves along a trajectory perpendicular to the orbital plane of the binary (along z-axis) and passes through their common barycenter, under the influence of Newtonian gravitational attraction. In 1960, within the same configuration, Sitnikov [2] showed that a periodic solution also exists when the two primaries move along elliptical orbits, see Figure 1. This type of Newtonian-restricted three-body problem is often referred to as “the Sitnikov problem.”

Although the Sitnikov three-body problem is one of the simplest dynamical models, it is known to exhibit a wide range of chaotic behaviors. As a result, the Sitnikov problem has been extensively investigated both analytically and numerically; see, for example, [3,4,5,6,7,8,9,10,11,12,13,14] and references therein.

In many previous studies, the three-body problem, including the Sitnikov problem, has been investigated under Newtonian gravity. However, in recent years, with the successful detection of gravitational waves and increasing interest in black hole formation, general relativistic effect in the three-body problem have attracted growing attention. For instance, the general relativistic effect on Euler’s collinear solution was investigated in [15], and the papers, e.g., [16,17], studied the contribution of general relativity to Lagrange’s triangle solution, and the stability of the Lagrange points $L_4$ and $L_5$ was examined through linear analysis by [18]. Further, It has been shown that the figure-eight orbit of the three-body system, discovered by Moore [19] under Newtonian gravity (see also [20,21,22,23]) exists under general relativistic gravity, see, e.g., [24,25]. And some papers such as [26,27,28] considered gravitational wave emission in the three-body problem forming Lagrange’s triangular solution.

On the other hand, the discovery of PSR J0337+1715 has drawn increasing attention to test of general relativity in hierarchical triple systems [29]. The effect of general relativity in hierarchical three-body systems has been actively discussed, for example, [30,31,32,33,34,35] and references therein.

The effect of general relativity on the Sitnikov problem has been considered by [36,37]. Kovács et al. [36] explore the Sitnikov three-body problem within the first post-Newtonian approximation to assess how relativistic corrections influence its chaotic phase-space structure. Their numerical analysis shows that increasing the dimensionless gravitational radius $\lambda$ leads to bifurcations in phase-space; altering chaotic regions and escape dynamics, and results in systematic changes to the chaotic saddle, including shifts in the positions of KAM islands and escape corridors. This work demonstrates that even weak relativistic effects can significantly impact chaotic scattering in the classical Sitnikov model.

Bernal et al. [37] investigated a relativistic Sitnikov problem using a first post-Newtonian approximation, exploring how the dimensionless gravitational radius $\lambda$ affects asymptotic behavior and chaotic scattering in the system. They identified (i) a metamorphosis of the KAM islands and associated escape regions as $\lambda$ increases, and (ii) two distinct inflection points in the unpredictability of the final state at $\lambda \simeq 0.02$ and $\lambda \simeq 0.028$. These inflection features are quantified using basin entropy analysis, providing deeper insight into relativistic chaotic scattering in the Sitnikov framework.

In this paper, we numerically investigate the Sitnikov three-body problem within the framework of the first post-Newtonian approximation and present practical and physically concrete results, rather than focusing on chaotic behavior and phase-space perspectives mainly discussed by [36,37].

This paper is organized as follows. In Section 2, we derive the general relativistic equations of motion used in this study from the Einstein–Infeld–Hoffmann equations. In Section 3, we present the results of numerical integration of these equations of motion. In Section 4, we discuss the astrophysical applicability of the Sitnikov mechanism. Section 5 is devoted to conclusions.

2. Equation of Motion

First, we derive the equations of motion used in this paper from the Einstein–Infeld–Hoffmann (EIH) equations, which describe the motion of point masses within the first post-Newtonian approximation. The EIH equations are given by, see, e.g., [38,39], obtained by imposing harmonic coordinates as gauge conditions, ${\nabla }_{\nu }{g}^{\mu \nu }=0$,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathit{v}}_k}{dt}}& =-\sum _{i\ne k}{\displaystyle \frac{G{m}_i}{r_{ki}^3}}{\mathit{r}}_{ki}+{\displaystyle \frac{1}{c^2}}\left\{\sum _{i\ne k}{\displaystyle \frac{G{m}_i}{r_{ki}^3}}{\mathit{r}}_{ki}\left[\sum _{j\ne i}{\displaystyle \frac{G{m}_j}{r_{ij}}}+4\sum _{j\ne k}{\displaystyle \frac{G{m}_j}{r_{kj}}}\right.\right.\hfill \\ \hfill & -\left.{\displaystyle \frac{1}{2}}\sum _{j\ne i}{\displaystyle \frac{G{m}_j}{r_{ij}^3}}({\mathit{r}}_{ki}·{\mathit{r}}_{ij})+{\displaystyle \frac{3}{2}}{\displaystyle \frac{{({\mathit{r}}_{ki}·{\mathit{v}}_i)}^2}{r_{ki}^2}}-2{\mathit{v}}_i·{\mathit{v}}_i-{\mathit{v}}_k·{\mathit{v}}_k+4{\mathit{v}}_k·{\mathit{v}}_i\right]\hfill \\ \hfill & +\left.\sum _{i\ne k}{\displaystyle \frac{G{m}_i}{r_{ki}^3}}{\mathit{v}}_{ki}[(4{\mathit{v}}_k-3{\mathit{v}}_i)·{\mathit{r}}_{ki}]-{\displaystyle \frac{7}{2}}\sum _{i\ne k}{\displaystyle \frac{G{m}_i}{r_{ki}}}\sum _{j\ne i}{\displaystyle \frac{G{m}_j}{r_{ij}^3}}{\mathit{r}}_{ij}\right\}+\mathcal{O}\left(c^{-4}\right),\hfill \end{array}$$

(1)

where G is the Newtonian gravitational constant, c is the speed of light in vacuum, $m_i$ is the mass of body i, ${\mathit{r}}_i$ is the position vector of body i in the barycentric coordinates system, ${\mathit{r}}_{ij}={\mathit{r}}_i-{\mathit{r}}_j$, $r_{ij}=\left|{\mathit{r}}_{ij}\right|$, ${\mathit{v}}_i=d{\mathit{r}}_i/dt$ is the coordinate velocity of body i (t is the coordinate time), and ${\mathit{v}}_{ij}={\mathit{v}}_i-{\mathit{v}}_j$. From this equation, we derive the equations of motion for the central binary and the third body.

2.1. Motion of Primary Bodies

In this paper, we assume that the third body has an infinitesimally small mass, setting $m_3=0$. Then, the motion of the central binary, consisting of two massive primary bodies (1 and 2), can be described by the relativistic two-body problem. Therefore, it is convenient to express in relative coordinates as

$$\begin{array}{c}\hfill \mathit{R}={\mathit{r}}_1-{\mathit{r}}_2,\mathit{V}={\displaystyle \frac{d\mathit{R}}{dt}}={\mathit{v}}_1-{\mathit{v}}_2,\end{array}$$

(2)

and from Equation (1), the equation of motion of the binary becomes,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\mathit{V}}{dt}}& =-{\displaystyle \frac{GM}{R^3}}\mathit{R}\hfill \\ \hfill & +{\displaystyle \frac{GM}{c^2{R}^3}}\{\left[{\displaystyle \frac{GM}{R}}(4+2\nu )+{\displaystyle \frac{3}{2}}\nu {\displaystyle \frac{{(\mathit{R}·\mathit{V})}^2}{R^2}}-(1+3\nu )(\mathit{V}·\mathit{V})\right]\mathit{R}\hfill \\ \hfill & +(4-2\nu )(\mathit{R}·\mathit{V})\mathit{V}\}\hfill \end{array}$$

(3)

in which $R=\left|\mathit{R}\right|$, $M=m_1+m_2$, and $\nu =m_1{m}_2/M^2$. In the Sitnikov problem, the central binary stars have equal masses, then we set $m_1=m_2=m, M=2m$ and $\nu =1/4$. Equation (3) can be rewritten as

$$\begin{array}{c}\hfill {\displaystyle \frac{d\mathit{V}}{dt}}=-{\displaystyle \frac{2Gm}{R^3}}\mathit{R}+{\displaystyle \frac{2Gm}{c^2{R}^3}}\left\{\left[9{\displaystyle \frac{Gm}{R}}+{\displaystyle \frac{3}{8}}{\displaystyle \frac{{(\mathit{R}·\mathit{V})}^2}{R^2}}-{\displaystyle \frac{7}{4}}(\mathit{V}·\mathit{V})\right]\mathit{R}+{\displaystyle \frac{7}{2}}(\mathit{R}·\mathit{V})\mathit{V}\right\}.\end{array}$$

(4)

Here, we transform the coordinate time t, $\mathit{R}$ and $\mathit{V}$ into a dimensionless variables T, $\overline{\mathit{R}}$ and $\overline{\mathit{V}}$ as follows:

$$\begin{array}{c}\hfill t={\displaystyle \frac{1}{n}}T,\mathit{R}=a\overline{\mathit{R}},R=a\overline{R},\mathit{V}=na\overline{\mathit{V}},\overline{\mathit{V}}={\displaystyle \frac{d\overline{\mathit{R}}}{dT}},n=\sqrt{{\displaystyle \frac{2Gm}{a^3}}},\end{array}$$

(5)

where a is the semi-major axis and n is the mean motion of central binary.

Further let us introduce dimensionless gravitational radius $\lambda$ as follows:

$$\begin{array}{c}\hfill \lambda \equiv {\displaystyle \frac{2Gm}{a{c}^2}}.\end{array}$$

(6)

Then, the case, $\lambda =0$ corresponds to the Newtonian gravity. The value of $\lambda$ increases as the mass m of the central bodies increases or as the two primary bodies come closer together, and represents the strength of relativistic effect.

Using Equations (5) and (6), Equation (4) is rewritten as

$$\begin{array}{c}\hfill {\displaystyle \frac{d\overline{\mathit{V}}}{dT}}=-{\displaystyle \frac{\overline{\mathit{R}}}{{\overline{R}}^3}}+{\displaystyle \frac{\lambda }{{\overline{R}}^3}}\left\{\left[{\displaystyle \frac{9}{2}}{\displaystyle \frac{1}{\overline{R}}}+{\displaystyle \frac{3}{8}}{\displaystyle \frac{{(\overline{\mathit{R}}·\overline{\mathit{V}})}^2}{{\overline{R}}^2}}-{\displaystyle \frac{7}{4}}(\overline{\mathit{V}}·\overline{\mathit{V}})\right]\overline{\mathit{R}}+{\displaystyle \frac{7}{2}}(\overline{\mathit{R}}·\overline{\mathit{V}})\overline{\mathit{V}}\right\}.\end{array}$$

(7)

Equation (7) is more suitable for numerical integration than Equation (4) since the variables $\overline{\mathit{R}},\overline{\mathit{V}}$ are of order $\mathcal{O}\left(1\right)$.

2.2. Motion of Third Body

Next, let us write down the equation of motion of the third body (test particle). In the Sitnikov problem, the test particle moves along a trajectory perpendicular to the orbital plane of the central binary (along z-axis) and passes through the center of mass of binary. We therefore assume that the motion of the primary bodies (binary) is confined to the $\overline{X}\overline{Y}$-plane. And because $m_1=m_2=m$, we set the positions and velocities of the primaries as follows:

$$\begin{array}{c}\hfill {\mathit{r}}_1=-{\mathit{r}}_2={\displaystyle \frac{1}{2}}\mathit{R},\mathit{R}=(X,Y,0),{\mathit{v}}_1=-{\mathit{v}}_2={\displaystyle \frac{1}{2}}\mathit{V},\mathit{V}=(V_X,V_Y,0).\end{array}$$

(8)

Since, in this paper, we assume the third body is a test particle, $m_3=0$, we set the position and velocity of the third body, respectively, as follows:

$$\begin{array}{c}\hfill {\mathit{r}}_3=(0,0,z),{\mathit{v}}_3=(0,0,v_z),\end{array}$$

(9)

From Equation (1), the equation of motion of third body becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_z}{dt}}& =-{\displaystyle \frac{2Gm}{r^3}}z\hfill \\ & +{\displaystyle \frac{Gm}{c^2{r}^3}}\left\{\left[{\displaystyle \frac{5}{2}}{\displaystyle \frac{Gm}{R}}+16{\displaystyle \frac{Gm}{r}}+{\displaystyle \frac{3}{16}}{\displaystyle \frac{{(\mathit{R}·\mathit{V})}^2}{r^2}}-\mathit{V}·\mathit{V}+6v_z^2\right]z+{\displaystyle \frac{3}{2}}(\mathit{R}·\mathit{V})v_z\right\},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_x}{dt}}& ={\displaystyle \frac{d{v}_y}{dt}}=0,\hfill \end{array}$$

(11)

where

$$\begin{array}{c}\hfill r = |{\mathit{r}}_{31}| = |{\mathit{r}}_{32}| =\sqrt{z^2+{\left({\displaystyle \frac{R}{2}}\right)}^2}.\end{array}$$

(12)

As Equation (5), we use following the transformations:

$$\begin{array}{c}\hfill z=a\overline{z},r=a\overline{r},\overline{r}=\sqrt{{\left({\displaystyle \frac{\overline{R}}{2}}\right)}^2+{\overline{z}}^2},v_z=na{\overline{v}}_z,{\overline{v}}_z={\displaystyle \frac{d\overline{z}}{dT}},\end{array}$$

(13)

and using Equation (6), Equation (10) is replaced by

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\overline{v}}_z}{dT}}& =-{\displaystyle \frac{\overline{z}}{{\overline{r}}^3}}\hfill \\ & +{\displaystyle \frac{\lambda }{{\overline{r}}^3}}\left\{\left[{\displaystyle \frac{5}{8}}{\displaystyle \frac{1}{\overline{R}}}+{\displaystyle \frac{4}{\overline{r}}}+{\displaystyle \frac{3}{32}}{\displaystyle \frac{{(\overline{\mathit{R}}·\overline{\mathit{V}})}^2}{{\overline{r}}^2}}-{\displaystyle \frac{1}{2}}(\overline{\mathit{V}}·\overline{\mathit{V}})+3{\overline{v}}_z^2\right]\overline{z}+{\displaystyle \frac{3}{4}}(\overline{\mathit{R}}·\overline{\mathit{V}}){\overline{v}}_z\right\},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\overline{v}}_x}{dT}}& ={\displaystyle \frac{d{\overline{v}}_y}{dT}}=0.\hfill \end{array}$$

(15)

The configuration of the dynamical system considered in this paper is shown in Figure 1.

Under Newtonian gravity ($\lambda =0$), the Sitnikov problem can be analytically treated such that its equation of motion is written in the form of harmonic oscillator under the approximation, $1\ll \left|z\right|$,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d^2\overline{z}}{d{T}^2}}& =-{\displaystyle \frac{\overline{z}}{{\sqrt{{\displaystyle {\overline{z}}^2+\frac{1}{4}}}}^3}}\hfill \\ & =-8\overline{z}+\mathcal{O}\left({\overline{z}}^3\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \overline{z}& ={\overline{z}}_{0}cos2\sqrt{2}T,\hfill \end{array}$$

(17)

see Equation (14), and we assumed here that the orbit of the central binary star is circular with $\overline{R}=1$. Considering general relativistic effects on Equation (17), the equation of motion for $\overline{z}$ takes the form of a perturbed harmonic oscillator, then a perturbative treatment is possible If $\left|z\right|\ll 1$. However, when $\left|z\right|>1$, it becomes difficult to analytically treat the motion of the third body using the perturbed harmonic oscillator model. Therefore, we examine the motion of the third body by integrating the equations of motion for the central binary, Equation (7) and the third body, (14) numerically.

3. Numerical Experiments

In this section, we present the results of numerical integration for the Sitnikov problem when general relativistic effects are present.

3.1. Overview of Numerical Integration

The setup of our numerical integration is summarized in

Table 1. Setup of numerical integration.

Range of initial value of third body ${\overline{z}}_0$	$0\le {\overline{z}}_0\le 6,\Delta {\overline{z}}_0=0.05$
Initial velocity of third body	${\dot{\overline{z}}}_0=0$ (fixed)
Range of dimensionless gravitational radius $\lambda$	$0\le \lambda \le 0.035,\Delta \lambda =0.0005$
Range of Eccentricity of binary	$0\le e\le 0.9,\Delta e=0.1$

.

The initial condition of the binary is chosen as follows:

$$\begin{array}{c}\hfill \overline{X}=1-e,\overline{Y}=0,\dot{\overline{X}}=0,\dot{\overline{Y}}={\displaystyle \frac{\sqrt{1-e^2}}{(1-e)}},\end{array}$$

(18)

where e is the orbital eccentricity of the binary, with $0\le e\le 0.9$, varied in intervals of $\Delta e=0.1$. The initial values of the third body, ${\overline{z}}_0$ and ${\dot{\overline{z}}}_0$, are taken as $0\le {\overline{z}}_0\le 6$ with intervals of $\Delta {\overline{z}}_0=0.05$, and ${\dot{\overline{z}}}_0=0$. The range of the dimensionless gravitational radius $\lambda$ is $0\le \lambda \le 0.035$.

Numerical integration is performed using Gragg’s extrapolation method based on the Aitken–Neville algorithm (see, e.g., [40,41,42]) and the calculation time per orbit is set to a maximum of 1000 Keplerian periods for the binary, $T_K=2\pi /n$.

Before presenting the numerical integration results, we describe the verification of the validity of the numerical results we obtained. First, we compared the numerical calculation accuracy at the step size $\Delta t=h=0.02$ and $\Delta t=h/2=0.01$, and confirmed that the results were identical for both. Then, the numerical calculation results presented in this paper were computed using step size $\Delta t=0.02$. The results of numerical integration showed no difference even when the error tolerance $\delta$ for the extrapolation method was changed from $\delta ={10}^{-11}$ to $\delta ={10}^{-12}$. We also confirmed that Jacobi constant C is conserved within the numerical integration error range thus verifying the validity of the computational code. The Jacobi constant C is a conserved quantity in the circular restricted three-body problem. For the Sitnikov problem, it is represented by the element $(e,\lambda )=(0.0,0.0)$ as follows:

$$\begin{array}{c}\hfill C\equiv {\displaystyle \frac{2}{\sqrt{{\displaystyle {\overline{z}}^2+\frac{1}{4}}}}}-{\dot{\overline{z}}}^2,\overline{R}=1.\end{array}$$

(19)

3.2. Criteria for Ejection Judgment and Underlying Mechanisms

In our numerical integration, the test particle is considered to have escaped from the system when the energy of the test particle, E, becomes positive

$$\begin{array}{c}\hfill E={\displaystyle \frac{1}{2}}{\overline{v}}_z^2-{\displaystyle \frac{1}{\sqrt{{\displaystyle {\overline{z}}^2+{\left(\frac{\overline{R}}{2}\right)}^2}}}}\ge 0,for{\overline{v}}_z\overline{z}>0.\end{array}$$

(20)

If this condition holds when a test particle is ejected from the system, we can determine that the test particle was ejected due to the gravitational slingshot effect. The gravitational slingshot effect, also known as a gravity assist, is a process in which an celestial object, such as a spacecraft or a star, gains energy and velocity by passing close to a massive moving body (i.e., a planet or a star). In this encounter, the smaller object (test particle in our case) is accelerated through the exchange of energy and momentum with the gravitational field of massive body. Thus, the kinetic energy of the smaller celestial body increases. In this paper, we refer to such a test particle ejection mechanism as “the Sitnikov mechanism”. As we will see below, the Sitnikov mechanism arises from resonance between the position $\overline{z}$ of the test particle and the distance $\overline{r}$ between the test particle and the primary star.

3.3. Dimensionless Gravitational Radius

Here, we note that the value of the dimensionless gravitational radius, $0.0\le \lambda \le 0.035$, adopted in this paper is applicable to phenomena ranging from weak to strong gravitational fields. In this paper, general relativistic effects are characterized by the dimensionless gravitational radius $\lambda$. Since $\lambda$ couples the mass m of primary star and the semi-major axis a of binary, see Equation (6), for astrophysical applications, we relate it to observational quantities using a black hole binary with

$$\begin{array}{c}\hfill m_{\mathrm{bh}}=30M_{\odot },a_{\mathrm{bh}}=0.01\mathrm{AU},\end{array}$$

(21)

and the value of ${\lambda }_{\mathrm{bh}}$ in this case is

$$\begin{array}{c}\hfill {\lambda }_{\mathrm{bh}}\approx 6.0\times {10}^{-5}.\end{array}$$

(22)

While the supermassive black hole binary in the galaxy center with

$$\begin{array}{c}\hfill m_{\mathrm{gc}}={10}^9{M}_{\odot },a_{\mathrm{gc}}=0.005\mathrm{pc},\end{array}$$

(23)

then ${\lambda }_{\mathrm{gc}}$ becomes

$$\begin{array}{c}\hfill {\lambda }_{\mathrm{gc}}\approx 0.013.\end{array}$$

(24)

As we will see below, it is worth noting that the black hole mass $m_{\mathrm{bh}}=30M_{\odot }$ adopted here is comparable to the masses of the two black holes estimated from observations of the gravitational wave event GW150914. If the Sitnikov mechanism occurs in a binary black hole system with $m_{rmbh}=30M_{\odot }$, the effect would be sufficiently observable.

3.4. Results of Numerical Integrations

The following figures show the results of numerical integrations of Equations (7) and (14). As mentioned in Section 1, numerical studies in previous works [36,37], primarily focused on revealing the chaotic aspects inherent in the Sitnikov problem. Our interest is to reveal practical and physically concrete results for the Sitnikov problem. As far as we know, we are the first to perform such a comprehensive calculation.

Figure 2 plots the time evolution of the position $\overline{z}$ of the test particle (red), the orbital radius $\overline{R}$ of the binary (blue), and the distance $\overline{r}=\sqrt{{\overline{z}}^2+{(\overline{R}?2)}^2}$ between the test particle and the primary star of binary (dark-green). In Figure 2, the left vertical axis represents $\overline{z}$, and the right vertical axis presents $\overline{R}$ and $\overline{r}$. The left column represents the Newtonian case, $\lambda =0.0$, while the right column shows the result considering general relativistic effects, $\lambda =0.015$. Except for the difference in the value of $\lambda$, the same initial values are used for ${\overline{z}}_0,{\overline{\dot{z}}}_0=0$ as well as the orbital eccentricity e of the binary. The calculation time is up to $1000T_K$ where $T_K$ is the Keplerian orbital period of the binary, and for stable orbits (where the test particle remains in the system), we displayed results up to $20T_K$.

The periodic variation in the position $\overline{z}$ of the test particle, combined with the change in the orbital radius $\overline{R}$ of the binary, results in the distance $\overline{r}$ between the test particle and the primary star of binary exhibiting complex variations even in the Newtonian case. Consequently, the gravitational force acting on the test particle undergoes complex changes.

Figure 3 illustrates a phase-space plot of the motion of test particle $(\overline{z},{\overline{v}}_z)$ with fixed eccentricity $e=0.2$ of binary orbit, comparing the dimensionless gravitational radius $\lambda$ in the Newtonian case ($\lambda =0.0000$) and cases where relativistic effects are acting ($\lambda =0.0100,0.0200,0.0300$). Figure 3 plots $(\overline{z},{\overline{v}}_z)$ when the binary star is at periastron. Figure 3 corresponds to Figure 2, Figure 3, Figure 4 and Figure 5 in [36].

In Figure 5 of [36], and our Figure 3, we find that the structure of phase-space changes as the dimensionless gravitational radius $\lambda$ increases. In particular, the size of the stable region at the center tends to decrease as $\lambda$ increases, and this result agrees with Figure 5 presented in [36].

As shown in Figure 2 above, in the Sitnikov problem, test particles can be ejected from the system depending on the initial conditions used such as $e,\lambda$ and ${\overline{z}}_0$ as well as the orbital eccentricity e of binary. Therefore, we will first investigate at which parameter values the test particle escapes from the system. Figure 4 shows the parameters under which test particles remain in the system (white area), particles are ejected from the system (blue area), and binary star collisions (red area), with a maximum computational time of $1000T_k$.

Through numerical experiments, we observed that the collision of the binary is caused by large values of $\lambda$ as well as by large orbital eccentricity e of binary. And we found that collisions between the primaries occur when the periastron distance of the binary becomes smaller than several tens of times the Schwarzschild radius:

$$\begin{array}{c}\hfill a(1-e)\lesssim \delta {\displaystyle \frac{2Gm}{c^2}}=\delta a\lambda ,\delta =\mathcal{O}\left(10\right).\end{array}$$

(25)

Here, $\delta$ represents a multiple of the Schwarzschild radius $r_g$. This condition can be considered in agreement with the prediction that the collision of binary systems accompanying gravitational wave emission occurs at tens of Schwarzschild radii, $\delta r_g$.

Next, we will clarify the kinematic and positional characteristics between the test particle and the binary when the test particle is ejected, in order to gain insight into the mechanism behind the test particle ejection.

Figure 5 shows the distance $\overline{r}$ between the test particle and the primary star, and the kinetic condition of test particles when the test particle is ejected, in four different cases:

• Case 1 (Red): When the test particle is away from the binary orbital plane, and the distance $\overline{r}$ between the test particle and the primary star is close.
• Case 2 (Yellow): When the test particle approaches the binary orbital plane, and the distance $\overline{r}$ between the test particle and the primary star is close.
• Case 3 (Cyan): When the test particle is away from the binary orbital plane, and the distance $\overline{r}$ between the test particle and the primary star is distant.
• Case 4 (Blue): When the test particle approaches the binary orbital plane, and the distance $\overline{r}$ between the test particle and the primary star is distant.

When the orbital eccentricity of a binary is $e=0.0$, the binary orbit becomes circular, $\overline{R}=1$. Therefore, the gravitational force experienced by the test particle can be determined solely by $\overline{z}$. As a result, the ejection of the test particle occurs randomly, largely independent of the distance $\overline{r}$ between the test particle and the primary star. See Equation (19).

However, as the orbital eccentricity e of the binary increases, the ejection of the test particle tends to occur more prominently near the point where the distance $\overline{r}$ between the binary and the test particle is near the minimum value or the test particle and the primary star are in a close proximity (red and yellow regions). From Figure 5, the ejection of test particles is found to occur more prominently when the particles are moving away from the binary orbital plane (red) than when they are approaching it (yellow). Moreover, Figure 5 shows that in Case 1 (red) and Case 2 (yellow), the test particle and the primary star are in close positional proximity. Furthermore, Figure 2 reveals that immediately before ejection of the test particle, the phases appear to be synchronized in $\overline{z}$ and $\overline{r}$. Our numerical calculations indicate that the distance $\overline{r}$, rather than the orbital radius $\overline{R}$ of binary, is the key factor when test particles are emitted from the system. For example, see the blue lines ($\overline{r}$) in Figure 2D,F–H. And Figure 6 is a close-up of Figure 2G,H.

Here, let us calculate the Lyapunov exponent ${\lambda }_L$ for case (F) of Figure 2 (unstable case). Figure 7 shows the Lyapunov exponent obtained from two orbits: one with initial conditions $e=0.4$, $\lambda =0.0015$, $z_1=2.20$, and another with initial conditions $z_2=z_1+0.000001$.

In this calculation, even when test particle was ejected from the system ($E>0$), the calculation was continued up to $1000T_K$. The Lyapunov exponent ${\lambda }_L$ is given by the following formula:

$$\begin{array}{c}\hfill {\lambda }_L\equiv \underset{t\to \infty }{lim}{\displaystyle \frac{1}{t}}ln{\displaystyle \frac{\delta z\left(t\right)}{\delta z\left(0\right)}}.\end{array}$$

(26)

In practice, we use the following equation to evaluate ${\lambda }_L$:

$$\begin{array}{c}\hfill {\lambda }_L\approx {\displaystyle \frac{1}{t_n-t_0}}ln\left|{\displaystyle \frac{z_2\left(t_n\right)-z_1\left(t_n\right)}{z_2\left(t_0\right)-z_1\left(t_0\right)}}\right|.\end{array}$$

(27)

In Figure 7, we display the value of $\lambda$ averaged over each Keplerian period $T_k$ of the binary orbit. As is evident from this Figure 7, in this numerical calculation example, the Lyapunov exponent is $\lambda >0$, indicating that chaotic behavior is hidden within the Sitnikov problem. As already mentioned in Section 1, many previous studies have addressed the chaotic properties inherent in the Sitnikov problem. Since this paper focuses primarily on investigating the physical characteristics of the Sitnikov problem through comprehensive numerical integrations, we will not cover its chaotic properties in detail below.

Here, we discuss the mechanism by which the test particle is ejected. As is evident from Figure 2 and Figure 6, it can be seen that before the test particle is ejected, the trajectory $\overline{z}$ of the test particle and the distance $\overline{r}$ between the test particle and the primary star are in a resonant state. This resonance phenomenon is thought to occur through the following process. As the eccentricity e of the binary orbit increases, the orbital radius $\overline{R}$ of binary oscillates widely. Then, the combination of $\overline{R}$ and $\overline{z}$ causes complex oscillations in the distance $\overline{r}$ between the test particle and the primary star. Due to the complex variation in the distance $\overline{r}$, the gravitational potential experienced by the test particle also fluctuates in a complex manner. When the oscillation period of the test particle resonates with the oscillation period of the distance $\overline{r}$, energy is periodically transferred into the test particle, acting like a gravitational pump, the test particle gradually increases its energy E. When the energy E of the test particle becomes $E\ge 0$, the test particle is no longer confined in the system and escapes at high velocity. As a results, the ejection process of the test particle is caused by the gravitational slingshot effect and we refer to this emission mechanism as “the Sitnikov mechanism”, as described in Section 3.2. One would expect that the resonance between $\overline{z}$ and $\overline{r}$ causes more effective ejection of the test particle from the system as the eccentricity e of the binary orbit and the dimensionless gravitational radius $\lambda$ increase.

The numerical experimental results below focus on illustrating the physical picture produced by the ejection of test particles due to the Sitnikov mechanism. Throughout Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12, we present the results excluding the red-colored parameter region shown in Figure 4.

Figure 8 illustrates the escape time $T_{\mathrm{esc}}$ of the test particle from the system. In the case of $e=0.0$ (the orbit of the primaries is circular), many particles remain in the system during the calculation period (white region), and the ejection of test particles occurs only in regions where ${\overline{z}}_0$ and $\lambda$ are large. As the eccentricity e of the binary orbit as well as the dimensionless gravitational radius $\lambda$ increase, the tendency for the test particle to escape from the system becomes more pronounced.

As a general trend, when the orbital eccentricity e of the binary is small, it takes longer for the test particle to escape from the system; the region $200\lesssim T_{\mathrm{esc}}\lesssim 400$ (yellow to light orange area) is dominant. However, as the eccentricity e increases, the particle escapes in a shorter time; in fact, as e increases, the gray area becomes increasingly dominant. These trend can be attributed to the effective excitation of the Sitnikov mechanism thereby enhancing the rapid escape of test particles. Furthermore, the region where test particles are ejected in a short time (gray region) in Figure 8 tends to correspond to the region shown in yellow in Figure 5, namely the region where test particles are approaching the binary orbital plane.

We mention here that Figure 7 in [36]. shows similar numerical results to our Figure 8 with an orbital eccentricity $e=0.2$, and the two plots exhibit good agreement.

Using Equations (21) and (23) to express the escape time in years, noting that $T_{\mathrm{esc}}=N{T}_K(0\le N\le 1000)$, we find

$$\begin{array}{c}\hfill T_{\mathrm{esc}}\left(\mathrm{bh}\right)\simeq 0.01N\mathrm{yr},T_{\mathrm{esc}}\left(\mathrm{gc}\right)\simeq 2.7\times {10}^{6}N\mathrm{yr},0\le N\le 1000.\end{array}$$

(28)

In which Keplarian period of binary $T_K$ is $T_K=2\pi /n=2\pi \sqrt{a^3/2Gm}$.

Figure 9 shows the escape velocity ${\overline{v}}_{\mathrm{esc}}$ of the test particle from the system. From Figure 9, it can be observed that when the eccentricity e of the binary orbit is small, e.g., $e=0.0$ and $e=0.1$, the escape velocity ${\overline{v}}_{\mathrm{esc}}$ of the test particle tends to be relatively low, $|{\overline{v}}_z| <1$ (the light-colored, cyan to light-yellow, regions). However, as the orbital eccentricity e increases, the escape velocity ${\overline{v}}_{\mathrm{esc}}$ tends to become larger $|{\overline{v}}_z| >1$; it can be seen that the darker-colored region expands. We find that the parameter regions with high escape velocities in Figure 9 shows good agreement with the regions in Figure 8 where escape times $T_{\mathrm{esc}}$ tend to be short (the gray areas). Therefore, we can reasonably conclude that a high escape velocity ${\overline{v}}_{\mathrm{esc}}$ leads to the test particle being released from the system in a short time.

In addition, comparing with Figure 5, the region of high escape velocities shows good agreement with the region where the test particle approaches the binary orbital plane in Figure 5. At this point, due to the conservation of energy, the test particle already possesses high velocity. Then, it is reasonable to assume that the test particle is then accelerated by the Sitnikov mechanism and ejected from the system at high velocity.

From Equations (21) and (23), expressing the escape velocity ${\overline{v}}_{\mathrm{esc}}$ in units of m/s, we obtain the following:

$$\begin{array}{c}\hfill v_{\mathrm{esc}}\left(\mathrm{bh}\right)\simeq 2.3\times {10}^6{\overline{v}}_{\mathrm{esc}}\mathrm{m}/\mathrm{s},v_{\mathrm{esc}}\left(\mathrm{gc}\right)\simeq 4.1\times {10}^7{\overline{v}}_{\mathrm{esc}}\mathrm{m}/\mathrm{s}.\end{array}$$

(29)

Figure 10 and Figure 11 display the ejection position ${\overline{z}}_{\mathrm{esc}}$ of test particle and the distance ${\overline{r}}_{\mathrm{esc}}$ between the primary star and the test particle at which the test particle is ejected from the system, respectively.

In Figure 10, the ejection position ${\overline{z}}_{\mathrm{esc}}$ of the test particles tends to occur at relatively small values, specifically $|{\overline{z}}_{\mathrm{esc}}| \ll 1$ (the gray region). As the orbital eccentricity of binary e increases, regions with $2\lesssim |{\overline{z}}_{\mathrm{esc}}| \lesssim 3$ become observable, but the observed behavior of ${\overline{z}}_{\mathrm{esc}}$ does not appear to be significantly influenced by the values of e and $\lambda$. Moreover, while a correlation was observed among Figure 5, Figure 8 and Figure 9, no significant correlation was found between Figure 5 and Figure 10.

Figure 11 shows that the distance ${\overline{r}}_{\mathrm{esc}}$ between the test particle and the primary at the time of test particle ejection is observed in the range $2\lesssim {\overline{r}}_{\mathrm{esc}}\lesssim 3$, independent of the orbital eccentricity e of the binary as well as the dimensionless gravitational radius $\lambda$. Then, We cannot identify any correlation between Figure 5 and Figure 11.

The range of ${\overline{r}}_{\mathrm{esc}}$ corresponds to the vicinity of the minimum distance ${\overline{r}}_{\min }$. This trend is consistent with the results of orbital evolution where test particle ejection appears to occur at small values of ${\overline{r}}_{\mathrm{esc}}$, see Figure 2 and Figure 6, Figure 11 confirms the conclusion that the ejection of the test particle occurs when the distance ${\overline{r}}_{\mathrm{esc}}$ is small and the gravitational force acting on the test particle becomes strong.

Referring to Figure 11, there are some parameter regions where test particle ejection is triggered for $5\lesssim {\overline{r}}_{\mathrm{esc}}\lesssim 8$, but this does not imply that it occurs in regions where the distance ${\overline{r}}_{\mathrm{esc}}$ is large. We confirmed that immediately before the ejection of the test particle, the distance $\overline{r}$ rapidly increased from $\overline{r}=\mathcal{O}\left(1\right)$ to values in the range $\mathcal{O}\left(10\right)\lesssim r\lesssim \mathcal{O}\left(100\right)$, refer to the results in (D) and (F) of Figure 2 and Figure 6. Then, the distance $\overline{r}$ decreases sharply just before ejection, and as a result, the gravitational force experienced by the test particle also increases sharply, enhancing the ejection of the test particle.

Using Equations (21) and (23), ${\overline{z}}_{\mathrm{esc}}$ and ${\overline{r}}_{\mathrm{esc}}$ are converted to the following quantities, respectively:

$$\begin{array}{cc}\hfill z_{\mathrm{esc}}\left(\mathrm{bh}\right)& =0.01{\overline{z}}_{\mathrm{esc}}\mathrm{AU}=1.5\times {10}^9{\overline{z}}_{\mathrm{esc}}\mathrm{m},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill z_{\mathrm{esc}}\left(\mathrm{gc}\right)& =0.005{\overline{z}}_{\mathrm{esc}}\mathrm{pc}=1.5\times {10}^{14}{\overline{z}}_{\mathrm{esc}}\mathrm{m},\hfill \end{array}$$

(31)

and

$$\begin{array}{cc}\hfill r_{\mathrm{esc}}\left(\mathrm{bh}\right)& =0.01{\overline{r}}_{\mathrm{esc}}\mathrm{AU}=1.5\times {10}^9{\overline{r}}_{\mathrm{esc}}\mathrm{m},\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill r_{\mathrm{esc}}\left(\mathrm{gc}\right)& =0.005{\overline{r}}_{\mathrm{esc}}\mathrm{pc}=1.5\times {10}^{14}{\overline{r}}_{\mathrm{esc}}\mathrm{m}.\hfill \end{array}$$

(33)

However, from Figure 9 showing the escape velocity ${\overline{v}}_{\mathrm{esc}}$, it can be seen that even in parameter regions where the acceleration acting on the test particle during ejection is small, the escape velocity can sometimes take on large values. This might be considered as follows: as mentioned above, before the ejection of the test particle from the system, the distance $\overline{r}$ rapidly increased from $\overline{r}=\mathcal{O}\left(1\right)$ to values in the range $\mathcal{O}\left(10\right)<r<\mathcal{O}\left(100\right)$, see (D) and (F) in Figure 2 and also Figure 6. At this stage, the test particle has already reached a state of $E\sim 1$ and high velocity, due to the gravitational slingshot effect caused by the resonance between $\overline{z}$ and $\overline{r}$, and it may be that even a slight acceleration would cause it to easily escape the system.

Figure 12 shows the acceleration ${\overline{a}}_{\mathrm{esc}}$ acting on the test particle at the moment it escapes from the system. It can be observed that, as the eccentricity e of the binary increases, regions with higher acceleration (yellow or cyan) appear; this parameter region corresponds to the region where test particles are ejected from the system in a short time (Figure 8) and have a high escape velocity (Figure 9). Also, referring to Figure 5, this corresponds to the parameter region where the test particle is approaching the plane of the binary orbit. On the other hand, the region where the acceleration acting on the test particle is small (the gray region), ${\overline{a}}_{\mathrm{esc}}\ll 1$, generally coincides with the region where the test particle is moving away from the binary orbital plane, the red-colored region in Figure 5.

Converting ${\overline{a}}_{\mathrm{esc}}$ to units of $m/s^2$ yields the following: Using Equations (21) and (23),

$$\begin{array}{c}\hfill a_{\mathrm{esc}}\left(\mathrm{bh}\right)=3.6\times {10}^3{\overline{a}}_{\mathrm{esc}}\mathrm{m}/{\mathrm{s}}^2,a_{\mathrm{asc}}\left(\mathrm{gc}\right)=11.1{\overline{a}}_{\mathrm{esc}}\mathrm{m}/{\mathrm{s}}^2.\end{array}$$

(34)

We mention that our numerical results reproduce the distribution of stable and unstable regions similar to that shown in [36] for central binary orbits with eccentricity $e=0.2$; refer to Figure 7 in [36] and our Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 for the results at $e=0.2$. This similarity also serves as check to ensure the validity of our computational findings.

Finally, Figure 13 displays the survival rate of test particles in the system up to 1000 orbital periods of the binary. Given the eccentricity e of the binary orbit and the dimensionless gravitational radius $\lambda$, the number of initial values of ${\overline{z}}_0$ is $N_p=121$, since ${\overline{z}}_0$ varies from zero to six in increments of 0.05. Figure 13 represents the ratio $n_p\left(T\right)/N_p$ as a function of the time, T where $n_p\left(T\right)$ means the number of particles remaining in the system at time T. Figure 13 is an extension of Figure 8 and Figure 10 in [36] and Figure 5 in [37].

For the range $0.0\le e\le 0.4$, the number of surviving test particles decreases with increasing values of both the dimensionless gravitational radius $\lambda$ and the orbital eccentricity of the binary e. However, $e\ge 0.5$, there exists a region where the number of surviving particles increases despite the increase in $\lambda$; for example, see the dark-colored area around $\lambda =0.03$ in the upper left of Figure 13, where $e=0.5$.

Looking at Figure 4, the region where the number of surviving particles increases with increasing $\lambda$ is the region where there are many test particles remaining in the system. This region exists near the lower limit of the region where the central binary collision occurs.

Throughout the numerical experiments above, it can be concluded that as general relativistic gravity becomes stronger as well as the increasing eccentricity of the binary orbit, the ejection of test particles from the system is stimulated. The underlying mechanism is that as orbital eccentricity e and general relativistic effects $\lambda$ increase, the distance $\overline{r}$ between the test particle and the primary star undergoes complex oscillations over time. This makes the position $\overline{z}$ of the test particle and the distance $\overline{r}$ more easily lead to a resonant state. As a result, the energy of the test particle increases, and the gravitational slingshot effect causes the test particle to be ejected from the system.

Lastly, we briefly summarize a simple comparison between present study and previous research in

Table 2. Brief comparison between present study and previous studies.

	Kovács et al. (2011) [36]	Bernal et al. (2020) [37]	Arakida (2025) (This Paper)
Main Objective	Phase-space structure and bifurcation analysis	Predictability of chaotic scattering	Astrophysical applications (Jets and HVSs)
Novelty	Discovery of bifurcations due to variation in $\lambda$	Introduction of basin entropy analysis	Proposal of physical applications of the relativistic Sitnikov effect
Range of $\lambda$	$0\le \lambda \le 0.035$	$0\le \lambda \le 0.035$	$0\le \lambda \le 0.035$
Main Output	Poincaré maps	Escape basin structures	Escape positions and escape velocity distributions

.

4. Applications of Sitnikov Mechanism

We will comment on the astrophysical applications of the Sitnikov mechanism in this section. In particular, we address two problems: the formation of astrophysical jets and the formation of hyper-velocity stars, focusing on the Sitnikov mechanism that allows test particles to be ejected at high escape velocities.

Note that the assumption of equal-mass binaries in Sitniko problem is highly unusual, and such an ideal configuration may be difficult to form. It is possible, however, that a configuration similar to the Sitniko problem could be formed through the following process; three-body exchange in dense clusters, e.g., [43,44], the formation and evolution of triple supermassive black hole systems, e.g., [45,46], the circumbinary disk interactions, e.g., [47], the binary disruption, e.g., [48,49], and xxial ejection via gravitational wave radiation and asymmetric nerger [50,51] which may be the most likely formation scenario, namely, immediately before the merger of a binary black holes, the center of mass of binary receivesa “kick” due to the asymmetry of gravitational wave radiation, accelerating surrounding celestial bodies along the center-of-mass axis (z-axis). This instantaneous dynamical configuration may induce axial scattering.

As mentioned above, it is not easy to present a useful scenario here that would produce the structure assumed by the Sitnikov problem. However, assuming such a configuration exists, we consider the following two applications.

4.1. Astrophysical Jets

Astrophysical jets are highly collimated, relativistic outflows observed in various systems, including active galactic nuclei (AGN), microquasars, gamma-ray bursts (GRBs), and young stellar objects (YSOs). Although their scales and environments differ, the underlying physics appears to involve accretion disks, magnetic fields, and rotating compact objects (black holes or neutron stars). To explain the observed jets in the universe, the following scenarios have been proposed; Blandford–Znajek (BZ) mechanism (black hole spin extraction), e.g., [52,53], Blandford–Payne (BP) mechanism (magneto-centrifugal disk winds), e.g., [54,55], magnetically arrested disk model, e.g., [56,57], radiation-driven/thermally driven jets, e.g., [58].

Despite intensive studies mentioned above, several challenges remain to be addressed; the origin of magnetic field, the mass loading and composition, jet stability and collimation, energy conversion efficiency, and jet–accretion coupling. Among these theoretical challenges, the Sitnikov mechanism may contribute to resolving the jet stability and collimation problem. Typical astrophysical jets are thought to be accelerated to velocities around,

$$\begin{array}{c}\hfill v_{\mathrm{jet}}\approx 0.9c,\end{array}$$

(35)

where c is the speed of light in vacuum.

As seen in Equation (29), if a supermassive black hole, $m_{\mathrm{gc}}={10}^9{M}_{\odot },a_{\mathrm{gc}}=0.005\mathrm{pc}$, binary exists at the galactic center, the escape velocity of particles becomes nearly equal to the jet velocity,

$$\begin{array}{c}\hfill v_{\mathrm{esc}}\left(\mathrm{gc}\right)=\mathcal{O}\left({10}^8\right)\mathrm{m}/\mathrm{s},\end{array}$$

(36)

due to the Sitnikov mechanism.

While it may be hard to explain astrophysical jets solely by the Sitnikov mechanism, this mechanism causes celestial bodies to be ejected along the z-direction (perpendicular to the binary orbital plane). Then, this mechanism may contribute to the formation of extremely narrow, collimated astrophysical jets. However, to explain astrophysical jets using the Sitnikov mechanism, it is necessary to incorporate electro-magnetic fields and radiation effects through a general relativistic radiative hydrodynamic dynamics, as the jet velocity is slightly lower than the observed velocity. However, if we consider it as an extended model incorporating MHD effects and disk–jet coupling, it may still hold the potential to lead to an understanding as a gravitational mechanism for the initial and periodic triggers of jet formation. For example, if we formulate the Sitnikov problem as a vertical 1D model of a thin disk (a vertical fluid element), the equations of the Sitnikov problem may be expressed as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}z}{d{t}^2}}+{\omega }_z^{2}z+{\Gamma }_{\mathrm{mag}}(z,\dot{z})=F_{\mathrm{grav}}\left(t\right),\end{array}$$

(37)

where ${\omega }_z$ is the disk’s vertical restoration frequency (locally derived from the vertical gravity and pressure gradient), ${\Gamma }_{\mathrm{mag}}$ is a modeling term for magnetic field-induced recovery (magnetic pressure/tension) and decay (e.g., friction/viscosity), and $F_{\mathrm{grav}}$ is the time-varying gravitational force of a binary system. Further linearizing and simplifying magnetic recovery and decay, introducing linear viscosity $\gamma$ and magnetic field-dependent spring constant $k_{\mathrm{mag}}$, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}z}{d{t}^2}}+2\gamma {\displaystyle \frac{dz}{dt}}+{\omega }_0^{2}z=F\left(t\right),\end{array}$$

(38)

in which ${\omega }_0^2\equiv {\omega }_z^2+k_{\mathrm{mag}}/\rho$ ($\rho$ is a density), and $F\left(t\right)$ is the periodic gravitational term due to the binary motion and should take the following form using a Fourier expansion:

$$\begin{array}{c}\hfill F\left(t\right)=\sum _n{A}_{n}cos\left(n{\Omega }_{\mathrm{bin}}t+{\varphi }_n\right),\end{array}$$

(39)

here, ${\Omega }_{\mathrm{bin}}$ is the frequency characterizing the periodic motion of a binary star and $A_n$ is the amplitude (determined by the eccentricity and mass ratio of the binary star).

Furthermore, in the above 1D model, the presence of a magnetic field induces the significant effects such as collimation by magnetic fields, particle acceleration by magnetic fields, and stabilization/destabilization by magnetic tension. Adding magnetic field pressure and tension, and averaging, the equation governing the test particle may be written as follows:

$$\begin{array}{c}\hfill \rho {\displaystyle \frac{d^{2}z}{d{t}^2}}=-\rho {\Omega }_K^{2}z-{\displaystyle \frac{\partial p}{\partial z}}+{\displaystyle \frac{1}{{\mu }_0}}\left[B_z{\displaystyle \frac{\partial B_z}{\partial z}}+B_r{\displaystyle \frac{\partial B_r}{\partial z}}-{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{B^2}{2}}\right)\right]+\rho F_{\mathrm{bin}}\left(t\right),\end{array}$$

(40)

where $B_z$ and $B_r$ are the magnetic fields in the z-direction and r-direction, respectively, and ${\Omega }_K$ is the local Keplerian frequency.

Based on these equations, it may be possible that the periodic gravitational forces of the binary resonantly excite local vertical modes in the disk, inducing localized ejections of material under certain conditions. Furthermore, if a sufficiently strong magnetic field is present, the ejected material could be collimated along the magnetic field lines, potentially creating jet-like structures. However, since the model described above is a simple one-dimensional model, it is necessary to consider mechanisms for sustained energy supply, such as black hole spin, to maintain the elongated jet structure stably over long distances. Furthermore, to account for kinks and rotational symmetry breaking, it will be essential to perform three-dimensional MHD simulations.

It is also difficult to directly correlate the observed jet radiation properties (synchrotron radiation, polarization, time-varying structure) with the ejection of test particles via the Sitnikov mechanism. Therefore, it would be necessary to perform simulations that couple particle motion due to gravity with radiative transport in order to compare with actual jet observations.

Finally, since the chaotic scattering in the Sitnikov mechanism exhibits a broad distribution, we need to introduce an external magnetic field or disk plasma to suppress chaotic diffusion in order to explain the extremely high collimation observed.

4.2. Hyper-Velocity Stars

Hyper-velocity stars are stars moving at extremely high velocities, fast enough to be unbound by the gravity of galaxy, and were first observed in 2005 [59]. To date, several dozen of hyper-velocity stars have been observed through, e.g., precise measurements by the GAIA.

The measured velocities of the hyper-velocity stars are in the range,

$$\begin{array}{c}\hfill 6.0\times {10}^5\lesssim v_{\mathrm{hvs}}\lesssim 2.0\times {10}^6\mathrm{m}/\mathrm{s}.\end{array}$$

(41)

The mechanism for hyper-velocity star formation is based on the Hills mechanism, e.g., [48,49], recoil escape from Supernova explosions, e.g., [60], binary black hole slingshot mechanism, e.g., [49,61,62,63], and acceleration due to AGN activity, e.g., [64].

Although the Sitnikov mechanism is a simple model, it can reproduce the velocities of observed hyper-velocity stars; based on Equation (29) and assuming black hole binary with $m_{\mathrm{bh}}=30M_{\odot }$ and $a_{\mathrm{bh}}=0.01$, the escape velocity is estimated as

$$\begin{array}{c}\hfill |v_{\mathrm{hvs}}| \lesssim 7.2\times {10}^6\sim 1.2\times {10}^7\mathrm{m}/\mathrm{s},\end{array}$$

(42)

and this value can adequately explain the observed velocities of hyper-velocity stars, Equation (41). Therefore, in a binary black hole system with a mass of about 30 solar masses, if general relativistic effects are significant and the binary orbit has a high eccentricity, it may be possible to effectively reproduce hyper-velocity stars through the Sitnikov mechanism, as demonstrated in this paper.

In this paper, the third body was treated as a test particle; however, observed hyper-velocity stars possess masses of approximately $1\sim 10N_{\odot }$. Therefore, since the binary orbit is also influenced by the third body’s gravitational interaction, it is necessary to perform realistic calculations with $m_3\ne 0$. It is also necessary to consider whether the Sitnikov mechanism can be driven even when the masses of the two bodies in a binary are unequal, as the masses of the two bodies are not necessarily equal. Furthermore, we must consider the possibility that the strong tidal forces of the black hole could cause stellar disruption before the Sitnikov mechanism operates when the third body passes near the black hole binary.

4.3. Supplementary Notes on Higher-Order Relativistic Effects and Gravitational Radiation Reaction

When discussing the Sitnikov mechanism in strong gravitational fields (for example, the 30 solar mass black hole binaries and ${10}^9$ solar mass supermassive black hole binaries exemplified in this paper), it is important to consider the effects of higher-order post-Newtonian (2 pN) terms and gravitational wave radiation (2.5 pN) terms. The equations of motion for the three-body problem in the 2 pN approximation are obtained from the ADM Hamiltonian described in, e.g., [65], see also [25]. The gravitational wave radiation (2.5 pN) causes the binary orbit to shrink and circularize, e.g., [66],

$$\begin{array}{cc}\hfill 〈{\displaystyle \frac{da}{dt}}〉& =-{\displaystyle \frac{64}{5}}{\displaystyle \frac{G^3{m}_1{m}_2(m_1+m_2)}{c^5{a}^3{(1-e^2)}^{7/2}}}\left(1+{\displaystyle \frac{73}{24}}{e}^2+{\displaystyle \frac{37}{96}}{e}^4\right),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill 〈{\displaystyle \frac{de}{dt}}〉& =-{\displaystyle \frac{304}{15}}e{\displaystyle \frac{G^3{m}_1{m}_2(m_1+m_2)}{c^5{a}^4{(1-e^2)}^{5/2}}}\left(1+{\displaystyle \frac{121}{304}}{e}^2\right).\hfill \end{array}$$

(44)

As the orbital radius of the binary star decreases (Equation (43)), the stars come closer together, thereby increasing the gravitational force experienced by the test particle. As a result, the test particle may become more tightly bound to the system, potentially suppressing its ejection. On the other hand, it is also possible that the ejection of test particle could be enhanced by the effect of the “kick” caused by orbital shrinkage of the binary, or by the complex interplay between gravitation amplified by shrinkage and effect of the eccentrnicity of the binary orbit.

Since the binary orbital period $T_{\mathrm{bin}}$ relates to the semi-major axis a of the binary system via Equation $T_{\mathrm{bin}}=a^{3/2}$, $T_{\mathrm{bin}}$ decreases as a also decreases. Therefore, the frequency of the periodic gravitational potential experienced by the test particle gradually increases, causing the positions of the resonant islands to slowly shift. The variation in the orbital period of this binary star system complicates the behavior of the gravitational force acting on the test particle. Therefore, it is necessary to verify whether this effect enhances or, conversely, suppresses the ejection of the test particle.

Moreover, the decrease in the semi-major axis a of the binary orbit reduces the total energy of the system, resulting in the following damping term acting on the test particle as a reaction effect, e.g., [67,68],

$$\begin{array}{c}\hfill {\mathbf{a}}_{\mathrm{RR}}=-{\displaystyle \frac{2G}{5c^5}}{\displaystyle \frac{d^5{Q}_{ij}}{d{t}^5}}{x}^j.\end{array}$$

(45)

Here, $Q_{ij}$ is the (traceless) quadrupole moment of the binary star system. Although this term is very small, over the long term it will alter the orbital energy of test particles, amplifying or suppressing chaotic scattering processes.

On the other hand, because the binary orbit becomes circular, namely the orbital eccentricity e becomes smaller (Equation (44)), the fluctuation in the binary orbital radius (the difference between the periastron and apastron) becomes smaller. This effect may act to suppress the ejection of test particles, refer to the case where the orbital eccentricity e is small, e.g., in Figure 4.

Therefore, if gravitational waves are emitted from a binary orbit, the orbital parameters a and e decrease. As a result, although the effect of radiation reaction themselves is extremely small, the aforementioned effects interact in complex ways and influence the motion of the test particle. To clarify the effects of these effects on the motion of the test particle, it is necessary to perform comprehensive numerical integration.

5. Conclusions

In this paper, we investigated the effect of general relativity on the Sitnikov problem. To study the general relativistic contributions, we first derived the equations of motion for both the binary and the test particle based on the first post-Newtonian Einstein–Infeld–Hoffmann equation, and integrated these equations numerically. We examine the behavior of the test particle (third body) as a function of the orbital eccentricity of the central binary e, the dimensionless gravitational radius $\lambda$, which characterizes the strength of general relativistic effect, and the initial position of the test particle $z_0$.

Our numerical calculations revealed the following: as general relativistic effects $\lambda$ increase and the eccentricity e of the binary orbit grows, the distance $\overline{r}$ between the test particle and the primary star undergoes complicated oscillations over time. Consequently, the gravitational force acting on the test particle also varies in a complex manner. This leads to a resonance state between the position $\overline{z}$ of the test particle and $\overline{r}$, causing the energy E of the test particle to become $E\ge 0$. This triggers the effective ejection of the test particle due to the gravitational slingshot effect. In this paper, we shall refer to this mechanism of test particle ejection as the “Sitnikov mechanism.” As a concrete phenomenon that becomes noticeable, the increase in general relativistic effects and the eccentricity of the binary orbit leads to the following: (a) ejection of test particles from the system in a shorter time, and (b) increasing escape velocity of the test particle from the system.

As an astrophysical application, we pointed out that the high-velocity ejection of test particles due to the Sitnikov mechanism could contribute to elucidating the formation processes of astrophysical jets and hyper-velocity stars. In fact, the escape velocity of test particles arising from the Sitnikov mechanism is approximately consistent with observed astrophysical jets and hyper-velocity stars, see Equations (36) and (42).

In this paper, we have considered the third body as a test particle, but if we regard the third body as a massive particle, the central binary will be affected by the gravity of this third body. Then, the behavior of the system will become more complex. Gravitational waves from such three-body system have not been studied so far, therefore it would be worthwhile to investigate them in the future.