Dynamical Properties of Perturbed Hill’s System

Abstract

In this work, some dynamical properties of Hill’s system are studied under the effect of continued fraction perturbation. The locations and kinds of equilibrium points are identified, and it is demonstrated that these points are saddle points and the general motion in their proximity is unstable. Furthermore, the curves of zero velocity and the regions of possible motion are defined at different Jacobian constant values. It is shown that the regions of forbidden motion increase with increasing Jacobian constant values and there is a noticeable decrease in the permissible regions of motion, leading to the possibility that the body takes a path far away from the primary body and escapes to take an unknown trajectory. Furthermore, the stability of perturbed motion is analyzed from the perspective of a linear sense, and it is observed that the linear motion is also unstable.

1. Introduction

The “Modified restricted three-body problem (RTBP) or N-body problem” is a variation of the classic three-body problem in celestial mechanics. In the standard RTBP, two bodies (often referred to as the primary and secondary bodies) move in circular orbits around their common center of mass, and a third body, which has negligible mass compared to the other two, moves under their gravitational influence without affecting them [1,2,3,4,5].

In the RTBP and N-body problem, the modification could involve several potential changes or additions to the classical problem, such as:

• Non-circular orbits: The primary and secondary bodies may move in elliptical orbits instead of circular ones [6,7,8].
• Perturbation forces: Additional forces might be considered, such as radiation pressure, drag forces, or relativistic effects [9,10,11,12].
• Variable masses: The masses of the primary, secondary, and third bodies could change over time, affecting the dynamics [13,14,15].
• Non-Newtonian gravity: Modifications to the gravitational force law, such as those arising from general relativity and the Yarkovsky effect or other alternative theories of gravity, could be included [16,17,18,19,20,21,22].
• Third-body influence: The influence of a third body with non-negligible mass might be considered, making the problem more complex than the standard restricted case [23,24,25].

However, one of the most important modified models for the three-body problem is Hill’s systems, which is often referred to as the Hill problem, a simplified model used in celestial mechanics. It is particularly useful for understanding the motion of a satellite around a planet while the planet itself orbits around a much larger body, like the Earth–Moon–Sun system [26,27,28].

Hill’s problem simplifies the general three-body problem by making the following assumptions [29,30]:

• Circular restricted three-body problem: The primary and secondary bodies (e.g., the Sun and the Earth) are in circular orbits around their common center of mass.
• Restricted: The mass of the third body (e.g., a satellite or the Moon) is so small that its gravitational influence on the primary and secondary bodies is negligible.

The key aspects of Hill’s problem include:

• Rotating reference frame: The problem is often analyzed in a rotating frame of reference that moves with the secondary body (e.g., the Earth), which simplifies the equations of motion.
• Hill’s equations: These are the differential equations derived from the simplifications mentioned above, describing the motion of the third body in the rotating frame.
• Hill’s region: This is the region within which the third body can move without escaping the gravitational influence of the primary bodies.

Hill’s system can be used in various applications in both astrodynamics and celestial mechanics, such as: analyzing the stability of satellite orbits around planets, studying the motion of moons around planets, understanding the dynamics of planetary rings, and designing space missions, such as Lagrange point missions (e.g., the James Web Space Telescope). By simplifying the complex interactions in a three-body system, Hill’s system provides valuable insights into the motion and stability of celestial objects. Hill’s problem provides a more manageable mathematical framework compared to the full three-body problem, making it a valuable tool in both theoretical and applied celestial mechanics [31,32,33,34,35].

Hill’s problem has been a significant topic in celestial mechanics, leading to numerous publications and advancements in the field. Below is a summary of some key literature and developments related to Hill’s problem. Hill’s seminal paper, “Researches in the Lunar Theory”, laid the groundwork for what is now known as Hill’s problem. This work focused on lunar motion and provided a method to approximate the motion of the Moon by considering a restricted three-body problem [36]. Henri Poincaré expanded on Hill’s work by developing qualitative methods for analyzing the stability and behavior of dynamical systems, which included Hill’s problem [37]. Modern textbooks on celestial mechanics and dynamical systems often include chapters or sections on Hill’s problem, discussing Hill’s problem, its mathematical formulation, solutions, and applications [38]. With the advent of computers, numerical methods have been extensively used to solve Hill’s equations and study the orbits of celestial bodies in details. These methods are crucial for mission planning and satellite deployment [39]. Contemporary research continues to explore various aspects of Hill’s problem, such as perturbations, stability analysis, and applications in exo-planetary systems [40].

There are many methods that can be employed to compute or calculate periodic orbits, for example, the improved grid search approach is used to calculate the periodic orbits and identify their classifications to periodic orbit families [41]. This approach was first constructed in [42] as a methodical technique to calculate entire families of periodic orbits for nonintegrable dynamical systems located in a certain region of space related to the initial conditions. Ever since, it has been used for numerical development of many differential equation systems, like the restricted three-body problem system and its modified versions, such as Hill’s problem and its variants [34,43,44,45,46,47].

This study is implemented as follows: The introduction to Hill’s problem and equations of motion of the dynamical system are stated in Section 1 and Section 2. The locations of equilibrium points are calculated in Section 3. Section 4 is devoted to identifying equilibrium points and the nature of motion around these points. Curves of zero velocity and regions of permissible and forbidden motion are explored in Section 5. The linear stability of perturbed motion is investigated in Section 6, Finally, the proposed work is summarized in the Conclusions of Section 7.

2. Equations of Motion

The Hill problem can be considered a specialized version of the restricted three-body problem that describes the motion of a small body under the gravitational influence of two larger bodies typically used to analyze the motion of a satellite around a planet while the planet itself orbits a much larger body. Thus, the key points of the standard Hill problem are as follows:

• The restricted three-body problem, in which the third body (the test particle) has a small mass and the two bigger bodies (the primary and secondary) have significant masses, is the source of the Hill problem.
• By assuming that the main and secondary bodies are in circular orbits around their common center of mass, it simplifies the equations and concentrates on a tiny area surrounding the smaller of the two big entities.
• The differential equations (also known as Hill’s equations) that explain the relative motion of the third body in this frame are obtained by deriving the equations in a rotating reference frame that rotates together with the secondary body.

Thus, the equations of motion of the perturbed Hill’s system using the proposed nomenclature in [48], in the framework of the continued fraction potential (CFP) are

$$\begin{array}{cc}\hfill \ddot{x}-2(1-{\displaystyle \frac{3}{2}}\delta )\dot{y}=& V_x,\hfill \\ \hfill \ddot{y}+2(1-{\displaystyle \frac{3}{2}}\delta )\dot{x}=& V_y,\hfill \\ \hfill \ddot{z}=& V_z,\hfill \end{array}$$

(1)

where V is the effective potential in Hill’s system, which combines the gravitational potentials of the two larger bodies and the centrifugal potential due to the rotating reference frame.

$$\begin{array}{c}\hfill V={\displaystyle \frac{1}{2}}\left[3(1-5\delta )x^2-(1-3\delta )z^2\right]+{\displaystyle \frac{1}{\rho }},\end{array}$$

(2)

and

$${\rho }^2=x^2+y^2+z^2.$$

(3)

The parameter $\delta$ in Equations (1) and (2) represents the perturbation of continued fraction [49], where the perturbed Hill’s system can be reduced to the unperturbed one when $\delta =0$. Furthermore, the first integral of perturbed motion, or what is called the Jacobi constant, can be obtained from integrating the set of equations in System (1) by utilizing Equation (2) in the following form

$$v^2=2V-\mathcal{C},$$

(4)

where $\mathcal{C}$ is the Jacobian constant, and the velocity v is identified by

$$v^2={\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2.$$

(5)

The Jacobian constant defines regions in space where the small body can move. These regions are called Hill’s regions, and they are bounded by zero-velocity surfaces. Inside these regions, the small body is gravitationally bound to the primary bodies, while outside these regions, it may escape their influence. It is important to note that the Jacobi constant is a key parameter in celestial mechanics, making it possible to analyze the stability of planetary orbits and define authorized or prohibited regions of motion for celestial bodies. It is possible to accurately estimate and predict the Jacobi constant, thus offering crucial information on the dynamics of celestial systems.

3. Locations of Equilibrium Points

Hill’s system includes equilibrium points, known as libration points (Lagrange points), where the gravitational and centrifugal forces balance. These points are crucial for understanding the stability of orbits and for planning space missions (e.g., placing satellites in stable positions). To identify these locations we utilize Equations (1)–(3) to rewrite the equations of motion as

$$\begin{array}{cc}\hfill \ddot{x}-2(1-{\displaystyle \frac{3}{2}}\delta )\dot{y}=& \left[3(1-5\delta )-{\displaystyle \frac{1}{{\rho }^3}}\right]x,\hfill \\ \hfill \ddot{y}+2(1-{\displaystyle \frac{3}{2}}\delta )\dot{x}=& -\left[{\displaystyle \frac{1}{{\rho }^3}}\right]y,\hfill \\ \hfill \ddot{z}=& -\left[\left(1-3\delta \right)+{\displaystyle \frac{z}{{\rho }^3}}\right]z.\hfill \end{array}$$

(6)

If the components of the velocities and accelerations in the left hand side of System (6) are equal to zero, then each expression on the right side of the same system will also equal zero, hence we obtain

$$\begin{array}{cc}\hfill 3(1-5\epsilon )x-{\displaystyle \frac{x}{{\rho }^3}}=& 0,\hfill \\ \hfill {\displaystyle \frac{y}{{\rho }^3}}=& 0,\hfill \\ \hfill \left(1-3\epsilon \right)z-{\displaystyle \frac{z}{{\rho }^3}}=& 0.\hfill \end{array}$$

(7)

The solutions of Equation (7) give the locations of equilibrium points $(x_e,y_e,z_e)$, where

$$\begin{array}{cc}\hfill x_e=& \pm {\displaystyle \frac{1}{\sqrt[3]{3(1-5\delta )}}},\hfill \\ \hfill y_e=& 0,\hfill \\ \hfill z_e=& 0.\hfill \end{array}$$

(8)

It is clear that from Equation (8), the locations of the equilibrium points will be affected by the perturbation parameter and further far away from the origin of coordinates. Although this parameter is very small, it is positive and cannot be equal to 1/5 because, in this case, the equilibrium points are undefined and they will move to take a place at infinity, see Figure 1.

The perturbed Hill’s problem can be reduced to the unperturbed one in the case of ignoring the perturbation effect of the continued fraction perturbation ($\delta =0$). Consequently, the relations of dynamical properties such as the Jacobian constant, velocity, and equilibrium points will also be reduced to the unperturbed one. Thus, from Equation (8), the locations of equilibrium points $L_1$ and $L_2$ are reduced to the locations of unperturbed system ${\overline{L}}_1=(-{\rho }_e,0,0)$ and ${\overline{L}}_2=({\rho }_e,0,0)$ where ${\rho }_e=1/\sqrt[3]{3}$, hence $x_e=\pm 1/\sqrt[3]{3}$, $y_e=z_e=0$.

4. Analysis of Motion around the Proximity of Equilibrium Points

Hill’s problem is a simplified version of the restricted three-body problem; however, it possesses two collinear points instead of three points as in the three-body problem. Meanwhile, the third collinear point in the restricted three-body problem lies at the right side of the larger primary body. However, this point dose not appear in Hill’s problem since this body moved to infinity. However, the locations of equilibrium points in Hill’s problem are affected by the perturbation of the continued fraction, the configuration of these points are not changed and are still colinear, taking place all the time on the X-axis.

4.1. Analysis of Motion around the Proximity of Equilibrium Points in Two Dimensions

To check the kind of equilibrium points and the stability of motion around these points in two dimensions, we have to find the Hessian (${\mathcal{H}}_V$) of the potential function (V), which is identified in the plane motion by

$${\mathcal{H}}_V=\left(\begin{array}{cc}{V}_{xx}& V_{xy}\\ V_{xx}& V_{yy}\end{array}\right),$$

(9)

where

$$\begin{array}{cc}\hfill V_{xx}=& {\displaystyle \frac{3x^2}{{\rho }^5}}-{\displaystyle \frac{1}{{\rho }^3}}+3(1-5\delta ),\hfill \\ \hfill V_{xy}=& V_{yx}={\displaystyle \frac{3xy}{{\rho }^5}},\hfill \\ \hfill V_{yy}=& {\displaystyle \frac{3y^2}{{\rho }^5}}-{\displaystyle \frac{1}{{\rho }^3}}.\hfill \end{array}$$

(10)

Utilizing Equations (9) and (10), the Hessian determinant ($D_{\mathcal{H}}$) is given by

$$\begin{array}{c}\hfill D_{\mathcal{H}}={\displaystyle \frac{1}{{\rho }^6}}\left[3(2y^2-x^2)(1-5\delta )\rho -2\right].\end{array}$$

(11)

In the case of unperturbed motion ($\delta =0$) using Equations (8), (10), and (11), then $D_{\mathcal{H}}=-27<0$ at both equilibrium points ${\overline{L}}_1$ and ${\overline{L}}_2$ where $V_{xx}=9$, $V_{xy}=0$ and $V_{yy}=-3$; therefore, both points are saddle points and the motion is unstable around these points. In the case of perturbed motion, the value of the determinant is $D_{\mathcal{H}}=-27{(1-5\delta )}^2<0$, which is negative for all $\delta$ values, while $V_{xx}=(9-45\delta$), $V_{xy}=0$ and $V_{yy}=-(3-15\delta )$. Hence, both points are still saddle points and the motion is still unstable around the perturbed equilibrium points.

4.2. Analysis of Motion around the Proximity of Equilibrium Points in Three Dimensions

In three-dimensional motion, the Hessian (${\mathcal{H}}_V$) of the potential function (V) is given by

$${\mathcal{H}}_V=\left(\begin{array}{ccc}{V}_{xx}& V_{xy}& V_{xz}\\ V_{yx}& V_{yy}& V_{yz}\\ V_{zx}& V_{zy}& V_{zz}\end{array}\right),$$

(12)

where

$$\begin{array}{cc}\hfill V_{xx}=& {\displaystyle \frac{1}{{\rho }^5}}\left[3x^2-{\rho }^2+3(1-5\delta ){\rho }^5\right],\hfill \\ \hfill V_{xy}=& V_{yx}={\displaystyle \frac{3xy}{{\rho }^5}},\hfill \\ \hfill V_{xz}=& V_{zx}={\displaystyle \frac{3xz}{{\rho }^5}},\hfill \\ \hfill V_{yy}=& {\displaystyle \frac{1}{{\rho }^5}}\left[3y^2-{\rho }^2\right],\hfill \\ \hfill V_{yz}=& V_{zy}={\displaystyle \frac{3yz}{{\rho }^5}},\hfill \\ \hfill V_{zz}=& {\displaystyle \frac{1}{{\rho }^5}}\left[3z^2-{\rho }^2-(1-3\delta ){\rho }^5\right].\hfill \end{array}$$

(13)

Utilizing Equations (12) and (13), the Hessian determinant ($D_{\mathcal{H}}$), in this case, is identified by

$$D_{\mathcal{H}}={\displaystyle \frac{1}{{\rho }^9}}\left[\begin{array}{cc}\hfill & 2+\left(9x^2-3z^2-4{\rho }^2\right)\rho \hfill \\ \hfill & -\left(15x^2-3z^2-8{\rho }^2\right)\rho \delta \hfill \\ \hfill & +3(1-3\delta )(1-5\delta )(y^2+{\rho }^2){\rho }^2\hfill \end{array}\right].$$

(14)

Substituting the values of the equilibrium point into Equations (13) and (14), we get

$$\begin{array}{cc}\hfill V_{xx}=& 9-45\delta ,\hfill \\ \hfill V_{xy}=& V_{yx}=0\hfill \\ \hfill V_{xz}=& V_{xz}=0\hfill \\ \hfill V_{yy}=& -3+15\delta ,\hfill \\ \hfill V_{yz}=& V_{zy}=0,\hfill \\ \hfill V_{zz}=& -4+14\delta .\hfill \end{array}$$

(15)

and

$$D_{\mathcal{H}}=54{(1-5\delta )}^2(2-9\delta )$$

(16)

In three-dimensional motion, the values of Hessian’s elements of the potential function (V) and its determinant under the effect of the perturbation parameter ($\delta$) can be evaluated as functions in this parameter from Equations (15) and (16) at the equilibrium points $(x_e,y_e,z_e)$, and the numerical values can be also computed using the equations for all $\delta$ values.

In the unperturbed three-dimensional motion ($\delta =0$), the Hessian Matrix for both equilibrium points is identified by

$${\left.{\mathcal{H}}_V\right|}_{(1/\sqrt[3]{3},0,0)}=\left(\begin{array}{ccc}9& 0& 0\\ 0& -3& 0\\ 0& 0& -4\end{array}\right),$$

(17)

the determinant value at the same two points is $D_{\mathcal{H}}=108>0$, $V_{xx}=9>0$, and the determinant of the first $2\times 2$-principal sub-matrix ($S_D=-27<0$) is negative while the value of the second derivative $V_{xx}=9>0$. Hence, we conclude from Equation (17) that $V_{xx}>0$, $S_D<0$ and $D_{\mathcal{H}}=108>0$, which yields that the system manifold or the potential function has three eigenvalues with different signs (i.e., there is at least one negative and one positive). So both equilibrium points are saddle points and we can deduce that the unperturbed three-dimensional motion in the proximity of both equilibrium points is unstable.

To analyze the kind of equilibrium points in the case of the perturbed three-dimensional motion under the effect of continued fraction parameter ($\delta$), we have to analyze the possible values for the elements of the Hessian matrix ($\mathcal{H}$), its determinant ($D_{\mathcal{H}}$), and the determinant of the first $2\times 2$ principal sub-matrix ($S_D$), as well as the second partial derivative $V_{xx}$ as a function in the perturbation parameter ($\delta$). In this context, the aforementioned quantities are captured in Figure 2. Whereas the equilibrium points are not defined when $\delta =1/5$, and the recommended value is very small, so the diagram is sketched in the domain of $\delta \in [0,0.5]$. We observe that $V_{xx}$ and ($D_{\mathcal{H}}$) are positive for all $\delta \in [0,1/5)$ and their values are negative for all $\delta \in (1/5,0.5]$. $S_D$ has negative values for all $\delta \in [0,1/5)\cup (1/5,0.5]$. Thus, both of the two equilibrium points are also saddle points and we can conclude that the perturbed three-dimensional motion in the proximity of both equilibrium points is still unstable for all perturbation values.

5. Regions of Permissible Motion

The “region of permissible motion” refers to the area within which a moving object can travel over a given period, considering some factors like its initial position, speed, direction, and any constraints on its motion. This concept is often used in fields like robotics, physics, and navigation to predict the possible locations an object can reach. Constraints might include physical barriers, maximum speed, and other environmental or system-specific limitations.

In celestial mechanics, the “region of permissible motion” refers to the area or volume of space that a celestial body, such as a planet, moon, or satellite, can occupy over a given period, considering its initial position, velocity, and the gravitational influences of other bodies. This concept is crucial for predicting the future positions and trajectories of celestial bodies and for understanding their past and current dynamics. Key factors influencing the region of possible motion in celestial mechanics include:

• Gravitational forces: The primary forces acting on celestial bodies, especially from massive bodies like stars and planets.
• Initial conditions: The initial position and velocity of the body determine its trajectory according to Kepler’s laws and Newton’s laws of motion.
• Orbital parameters: Elements such as semi-major axis, eccentricity, inclination, argument of periapsis, longitude of the ascending node, and true anomaly.
• Perturbations: Effects from additional forces or influences, such as gravitational interactions with other bodies (e.g., other planets, moons), relativistic effects, or non-gravitational forces (e.g., solar radiation pressure).

Analyzing the region of permissible motion involves solving the equations of motion that describe the body’s trajectory under the influence of these forces. This analysis can be complex, requiring numerical methods and simulations, especially when considering multiple interacting bodies and perturbations.

5.1. Curves of Zero Velocity

Curves of zero velocity, also known as zero-velocity curves or surfaces, are important concepts in the N-body problem in celestial mechanics. They represent the boundaries within which a smaller body (like a spacecraft or an asteroid) can move under the gravitational influence of two or more larger primary bodies (such as the Earth and the Moon). In general, the Jacobi integral is a conserved quantity in the restricted 3-body problem, and its reduction models as Hill’s body problem. This integral can be used to identify both the possible regions of motion and the curves of zero velocity. Utilizing Equations (4) and (5), the Jacobi integral ($\mathcal{C}$) is given by: $[\mathcal{C}=2V-v^2]$ where (V) is the effective potential energy and (v) is the velocity of the smaller body. We would like to refer to the effective potential as the potential function in the rotating frame of reference. It is a combination of the gravitational potentials of the two primary bodies and the centrifugal potential due to the rotating frame. The zero-velocity curves are defined by setting the velocity (v) to zero in the Jacobi integral where $[\mathcal{C}=2V]$. These curves show where the Jacobi constant is equal to double the effective potential.

The curves of zero velocity in the $XY$-plane are sketched in Figure 3, Figure 4 and Figure 5 when the perturbed parameter $\delta =0$, $0.1$, $0.3$, respectively. The curves of unperturbed motion are represented by the value of $\delta =0$, while the curves under the effect of perturbation are shown when $\delta =0.1$ and $0.3$. The curves in these figures identify the separation boundaries between the regions of possible and forbidden motion for a smaller body like a spacecraft or an asteroid. The significant and main idea for each moving body is its own Jacobi constant, the associated zero-velocity curves, and the specified region of possible motion. This means that the motion is possible only inside the boundary of zero-velocity curves. Otherwise, if the body takes a place on a zero velocity curve, it may travel along the normal to the curve, which leads to the motion in three dimensions or in the $XYZ$-space, as in Figure 6. This is the key motion in the plane and out of the plane of primary bodies, see Figure 6a,b. For clarity of the perturbation effect on curves of zero velocity, Figure 3c, Figure 3d, Figure 4c, Figure 4d, Figure 5c, and Figure 5d are zoomed in of Figure 3a, Figure 3b, Figure 4c, Figure 4d, Figure 5a, and Figure 5b, respectively. It is clear from the zoomed-in figures, that there are increases in the number of curves of zero velocity with increasing values of the continuation fraction parameter, which also leads to an increase in the number of permissible regions of motion.

Understanding zero-velocity curves is crucial for mission planning in space exploration, such as determining stable orbits, transfer trajectories, and regions where spacecraft can operate with minimal fuel consumption. By analyzing zero-velocity curves, scientists and engineers can predict the motion of celestial bodies and design efficient space missions that take advantage of natural gravitational forces.

5.2. Permissible Regions of Motion

The zero-velocity curves divide the space into allowed and forbidden regions. The smaller body can only move within the allowed regions where its velocity would be real and positive. The forbidden regions are areas where the smaller body cannot reach because it would require a negative velocity squared, which is not physically possible. The points where the zero-velocity curves intersect with the line connecting the two primary bodies are known as Lagrange points.

There are five such points, which are positions of equilibrium in the restricted three-body problem and two in Hill’s problem. Near these points, the zero-velocity curves form distinct shapes that indicate regions where the smaller body can be trapped or move in complex trajectories. Thus, regions of possible motion are explored in Figure 7 at different Jacobian values when the parameter of perturbation $\delta =0.1$. The colored blue areas in the Sub-Figures represent permissible regions of motion, while the colored red areas represent the forbidden regions of motion, but the separation orange curve between both regions identifies the curves of zero velocity.

In the first Figure 7a, the regions of motion are sketched when the Jacobian constant $\mathcal{C}=2.5$. This case shows that there are two forbidden regions, and one large permissible region for motion where a spacecraft (infinitesimal body) or an asteroid can freely move inside the boundaries of this area, and in the vicinity of both equilibrium points $L_1$ and $L_2$, it can easily exchange its motion around the two points. In the second Figure 7b, where the value of the constant is increased and $\mathcal{C}=5$, the situation is inverted where the infinitesimal body has one large forbidden region for motion compared with the first case, and three unconnected small regions for motion. One of these regions forms a circle inside the forbidden region where its center coincides with the origin of coordinates, while the other two permissible regions take the left and right sides of the forbidden region of motion. It is clear that the body cannot move in the neighborhood of any two equilibrium points, and it will be trapped inside one permissible region for motion with the inability to travel from one region to another.

With increasing values of the Jacobian constant, as in the third and fourth Figure 7c,d, where $\mathcal{C}=7.5$ and 10, respectively, we observe that the area of forbidden motion is also increased and there is a noticeable decrease in the permissible regions of motion. The analysis of permissible regions of motion in the second and third sub-figures is the same, but the only difference is that the forbidden region in the third is larger than the region of the second. However, there are two permissible regions of motion and one forbidden region in the fourth sub-figure. The areas of two regions are marginal with respect to the forbidden area and there is a possibility that the body takes a path away from the primary body and is freed from having its motion subject to the gravitational influence of the primary body and escapes away to take an unknown trajectory.

6. Linear Stability of Equilibrium Points

The stability of motion refers to the ability of a system or object in motion to maintain or return to a desired state or path when subjected to disturbances. This concept is important in various fields, including physics, engineering, and biomechanics. Stability can be classified as static and dynamic, within each of which there is stable, neutral, and unstable:

• Static stability: The tendency of a stationary object to return to its original position after being slightly displaced.
• Dynamic stability: The ability of a moving object to return to its desired path after a disturbance.
• Neutral stability: When an object, after being disturbed, neither returns to its original position nor continues to move away but stays in its new position.
• Unstable motion: When any disturbance causes the object to move further away from its original position.

For example, in the context of an aircraft, dynamic stability is crucial. If the aircraft encounters turbulence (a disturbance), dynamic stability would allow it to return to its intended flight path without constant input from the pilot. Similarly, in robotics, the stability of motion is critical to ensure robots can navigate environments without falling or deviating from their intended course. The stability of spacecraft is a critical aspect of their design and operation, ensuring they can maintain their intended orientation and trajectory during missions. Stability in spacecraft can be categorized into several types, one of the most important kinds is orbital stability, where a spacecraft has the ability to maintain its intended orbit. This involves countering perturbations caused by gravitational forces from celestial bodies, atmospheric drag (for Low Earth Orbit), and other factors. Maintaining stability in all of these aspects is crucial for the successful operation of spacecraft, whether they are satellites, space probes, or crewed vehicles.

Equilibrium points are the points where the system does not change, i.e., where the derivatives of the system variables are zero. The stability of equilibrium points in a dynamical system refers to the behavior of the system when it is slightly perturbed from those points. In this section, we intend to study the linear stability of Hill’s system, where it can often be approximated by a linear system for small perturbations like the one created by the continued fraction potential.

Researchers, especially celestial mechanics scientists have been interested in developing mathematical models that describe the restricted three-body problem and its own modified versions for the purpose of obtaining an accurate description of the dynamic behavior of the third body’s motion. It is certain that these models are strongly influenced by the initial conditions and the diversity of parameters that affect the measurement processes or the type of dynamic motion. In fact, there is a major difficulty in finding solutions to these models directly for any of the variables because they are nonlinear in their structures in most cases. Therefore, we always resort to finding their corresponding linear models in order to find simplified systems to describe nonlinear motion.

Studying the motion in the proximity of equilibrium points often involves linearizing the system and analyzing the resulting linear system. Near these equilibrium points, the set of differential equations that describes the dynamical system of motion can often be approximated by a linear system using a Taylor expansion. In this context, we imposed that the relation between the variation and initial state vectors are $\mathit{\rho }={\mathit{\rho }}_0$ + Δ$\mathit{\rho }$, where $\mathit{\rho }=(x,y,z)$ and ${\mathit{\rho }}_0=(x_e,y_e,z_e)$ defined the locations of equilibrium points, while Δ$\mathit{\rho }=(\overline{\xi },\eta ,\zeta )$ is the variation vector, which represents the new location of the body with respect to the state vector, while $\overline{\xi }$, $\eta$ and $\zeta$ are very small quantities of displacements along the frame axes. Inserting the relation of the state vector into Equation (1) with applying Taylor expansion, we obtain

$$\begin{array}{cc}\hfill \ddot{\overline{\xi }}-2\left(1-{\displaystyle \frac{3}{2}}\delta \right)\dot{\eta }=& V_x(\overline{\xi },\eta ,\zeta ),\hfill \\ \hfill \ddot{\eta }+2\left(1-{\displaystyle \frac{3}{2}}\delta \right)\dot{\overline{\xi }}=& V_y(\overline{\xi },\eta ,\zeta ),\hfill \\ \hfill \ddot{\zeta }=& V_z(\overline{\xi },\eta ,\zeta ),\hfill \end{array}$$

(18)

where

$$\begin{array}{cc}\hfill V_x(\overline{\xi },\eta ,\zeta )=& V_x^0+{\displaystyle \frac{1}{1!}}\left[\overline{\xi }{\displaystyle \frac{\partial }{\partial x}}+\eta {\displaystyle \frac{\partial }{\partial y}}+\zeta {\displaystyle \frac{\partial }{\partial z}}\right]V_x^0+{\displaystyle \frac{1}{1!}}\left[\overline{\xi }{\displaystyle \frac{\partial }{\partial x}}+\eta {\displaystyle \frac{\partial }{\partial y}}+\zeta {\displaystyle \frac{\partial }{\partial z}}\right]V_x^0+\mathcal{O}\left(3\right),\hfill \\ \hfill V_y(\overline{\xi },\eta ,\zeta )=& V_y^0+{\displaystyle \frac{1}{1!}}\left[\overline{\xi }{\displaystyle \frac{\partial }{\partial x}}+\eta {\displaystyle \frac{\partial }{\partial y}}+\zeta {\displaystyle \frac{\partial }{\partial z}}\right]V_y^0+{\displaystyle \frac{1}{1!}}\left[\overline{\xi }{\displaystyle \frac{\partial }{\partial x}}+\eta {\displaystyle \frac{\partial }{\partial y}}+\zeta {\displaystyle \frac{\partial }{\partial z}}\right]V_y^0+\mathcal{O}\left(3\right),\hfill \\ \hfill V_z(\overline{\xi },\eta ,\zeta )=& V_z^0+{\displaystyle \frac{1}{1!}}\left[\overline{\xi }{\displaystyle \frac{\partial }{\partial x}}+\eta {\displaystyle \frac{\partial }{\partial y}}+\zeta {\displaystyle \frac{\partial }{\partial z}}\right]V_z^0+{\displaystyle \frac{1}{1!}}\left[\overline{\xi }{\displaystyle \frac{\partial }{\partial x}}+\eta {\displaystyle \frac{\partial }{\partial y}}+\zeta {\displaystyle \frac{\partial }{\partial z}}\right]V_z^0+\mathcal{O}\left(3\right),\hfill \end{array}$$

(19)

here, the superscript (0) means the derivatives are evaluated at equilibrium points, while $\mathcal{O}\left(3\right)$ refers to third and higher order terms in $\overline{\xi }$, $\eta$, and $\zeta$.

Utilizing Equations (1), (18), and (19) while ignoring the higher order terms of $\overline{\xi }$, $\eta$, and $\zeta$, the linear of Hill’s motion can be written as

$$\begin{array}{cc}\hfill \ddot{\overline{\xi }}-2\left(1-{\displaystyle \frac{3}{2}}\delta \right)\dot{\eta }-9\left(1-{\displaystyle \frac{5}{3}}\delta \right)\overline{\xi }+5\sqrt[3]{9}\delta =& 0,\hfill \\ \hfill \ddot{\eta }+2\left(1-{\displaystyle \frac{3}{2}}\delta \right)\dot{\overline{\xi }}+3\eta =& 0,\hfill \\ \hfill \ddot{\zeta }+4\left(1-{\displaystyle \frac{3}{4}}\delta \right)\zeta =& 0.\hfill \end{array}$$

(20)

To write System (20) in the standard linear form, we assume that

$$\xi =\overline{\xi }-{\displaystyle \frac{5\delta }{3\sqrt[3]{3}\left(1-{\displaystyle \frac{5}{3}}\delta \right)}}$$

(21)

Utilizing System (20) and Equation (21), we obtain

$$\begin{array}{cc}\hfill \ddot{\xi }-2\left(1-{\displaystyle \frac{3}{2}}\delta \right)\dot{\eta }-9\left(1-{\displaystyle \frac{5}{3}}\delta \right)\xi =& 0,\hfill \\ \hfill \ddot{\eta }+2\left(1-{\displaystyle \frac{3}{2}}\delta \right)\dot{\xi }+3\eta =& 0,\hfill \\ \hfill \ddot{\zeta }+4\left(1-{\displaystyle \frac{3}{4}}\delta \right)\zeta =& 0.\hfill \end{array}$$

(22)

Equation (22) represents the linear dynamical system of Hill’s motion under the effect of continued fraction perturbation. It is clear that the third sub-equation is uncoupled with $XY$-plane motion; furthermore, it describes a simple harmonic motion with angular speed ${\omega }_z=\sqrt{4\left(1-3\delta /4\right)}$. Therefore, the motion in the Z-axis direction is completely periodic and stable. Thus, it is enough to analyze the stability in the $XY$-plane. In this case, the Jacobian matrix (${\mathcal{J}}_{\mathcal{M}}$) of plane motion is

$${\mathcal{J}}_{\mathcal{M}}=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ \left(9-15\delta \right)& 0& 0& \left(2-3\delta \right)\\ 0& -3& \left(-2+3\delta \right)& 0\end{array}\right)$$

(23)

The eigenvalues of ${\mathcal{J}}_{\mathcal{M}}$ (23) is

$$\begin{array}{cc}\hfill {\lambda }_{1,2}=& \pm i\sqrt{{\displaystyle \frac{1}{2}}\left[\alpha \left(\delta \right)+\beta \left(\delta \right)\right]},\hfill \\ \hfill {\lambda }_{3,4}=& \pm \sqrt{{\displaystyle \frac{1}{2}}\left[\alpha \left(\delta \right)-\beta \left(\delta \right)\right]},\hfill \end{array}$$

(24)

where the functions $\alpha \left(\delta \right)$ and $\beta \left(\delta \right)$ are give by

$$\begin{array}{cc}\hfill \alpha \left(\delta \right)=& \sqrt{112-192\delta -27{\delta }^2+54{\delta }^3+81{\delta }^4},\hfill \\ \hfill \beta \left(\delta \right)=& (3\delta -1)(3\delta +2).\hfill \end{array}$$

(25)

To analyze the nature of ${\lambda }_{1,2}$ and ${\lambda }_{3,4}$, we define the following two functions $F\left(\delta \right)$ and $G\left(\delta \right)$ where

$$\begin{array}{cc}\hfill F\left(\delta \right)=& {\displaystyle \frac{1}{2}}\left[\alpha \left(\delta \right)+\beta \left(\delta \right)\right],\hfill \\ \hfill G\left(\delta \right)=& {\displaystyle \frac{1}{2}}\left[\alpha \left(\delta \right)-\beta \left(\delta \right)\right].\hfill \end{array}$$

(26)

Utilizing Equations (25) and (26), we can see that the values of $F\left(\delta \right)$ and $G\left(\delta \right)$ depend on perturbation parameter $\left(\delta \right)$, which is very small and does not equal 0.2. The value of $F\left(\delta \right)$ is positive for all $\delta$ values, while the value $G\left(\delta \right)$ is positive when $0<\delta <0.6$ and has negative values for all $\delta >0.6$, see Figure 8. Therefore, Equation (24) shows that the eigenvalues ${\lambda }_1$ and ${\lambda }_2$ have pure imaginary values where ${\lambda }_1={\lambda }_2$, while ${\lambda }_3$ and ${\lambda }_4$ are real and ${\lambda }_3=-{\lambda }_4$. In this case, we can conclude that the motion is generally unstable, while conditional stability is expected from a footstep perspective linear motion when $\delta >0.6$, but this contradicts the assumption that $\delta$ is very small.

7. Conclusions

The importance of Hill’s system motivated us to present a comprehensive study on the dynamical behavior of the perturbed motion of such a system in the framework of the continued fraction potential. We have found that the perturbed system has two locations for the equilibrium points, as in the unperturbed one, but we showed that these points are affected by the perturbation parameter. The nature of these points is also analyzed in the cases of plane and space motion, and it proved that the points in both cases are saddle points, and we can deduce that the the motion in the proximity of such points is still unstable for all perturbation values. The allowed and prohibited areas of motion based on the Jacobi constant, which depends on the idea of Hill’s regions and zero-velocity curves. Hence, this constant is used to identify both the curves of zero velocity and the possible regions of motion. We have explored these curves and the related regions of permissible and forbidden motion at different Jacobian constants. Through the analysis of the obtained regions of motion, we have observed that the area of forbidden motion is increased, and there is a noticeable decrease in the permissible regions of motion with increasing Jacobian constant values.

The linear stability of the perturbed motion is studied. However, we have proved that the motion along the perpendicular direction on the plane of motion is periodic; the motion in the plane is unstable but it may be conditionally stable in the framework of linear stability. In general, we can state that the motion is unstable, while conditional stability is expected from a perspective of linear motion, but this contradicts the assumption that $\delta$ is very small.

We would like to indicate that the perturbation of the continued fraction potential is considered in a few number papers [48,49,50,51,52,53,54]. In the first paper [50], the authors used the CFP to reformulate the equations of motion for the two-body problem. In the second paper [49], the authors proved that the parameter of CFP may embody the lack of sphericity of the celestial object such as an oblate body. Further, the periodic solution of two-body motion using the KB averaging technique was evaluated. In the third paper [51], a new version was introduced to study the dynamics of the restricted three-body problem, where the large primary body is considered a source of radiation pressure (say Sun), while the smaller primary (say Earth) creates a potential similar to that generated by CFP. In the fourth paper [52], the locations of equilibrium points and their linear stability, curves of zero velocity, as well as the regions of possible motion, are studied with application to the Earth–Moon system, where the Earth creates a potential, such as CFP, while the Moon is considered a point mass. In the fifth one [53], the dynamics of the two-body problem were revisited in the framework of CFP perturbation. In the sixth one [54], a new formulation for relative motion satellites is established by finding their periodic solutions. In the seventh one [48], the proposed model in the fourth paper is used to construct a new version of Hill’s system with shed light on the properties of the Lagrangian and Hamiltonian of Hill’s system. In the current paper, we employed a new version of Hill’s system [50] to explore analytically the dynamical behavior of this system under the influence of continued fraction perturbation. The locations of equilibrium points, their type, and the nature of motion around these points are studied. Furthermore, the stability of linear motion is also investigated, as well as the curves of zero velocity and regions of permissible and forbidden motion, are analyzed.