Basins of Convergence in a Multi-Perturbed CR3BP ^†

Abstract

The circular restricted three-body problem (CR3BP) is analyzed to introduce additional factors into the dynamic model, such as radiation forces, flattening of the primary bodies, relativity effects, and the presence of natural satellites. The introduction of these factors increases the accuracy when obtaining the position of the Lagrange points and the basins of convergence of the system. The Newton–Raphson methodis used to implement a searching algorithm. Finally, an application to the Sun–Mars system including the presence of Phobos and Deimos is developed.

1. Introduction

The interest in space missions to Mars has increased significantly in recent years. Since several successful missions have reached the Martian surface [1,2,3], many countries and institutions have designed new missions with exploratory and scientific purposes that could pave the way for future colonization [4]. In the design of these space missions, it is essential to determine the trajectories of the spacecrafts correctly. For this purpose, the inclusion of perturbing accelerations is more and more necessary. On the other hand, optimizing the fuel consumption and resources is also very important. Placing satellites or space stations at points that could be relatively stationary with respect to nearby celestial bodies is a good solution for this optimization. Moreover, locating spacecrafts at these points can provide a strategic advantage in communication and guidance systems [5].

The study of these stationary points, known as Lagrange or libration points, is based on the N-body problem, with $N\ge 3$. As the difficulty of finding a solution to this problem, even for $N=3$, is significant, several simplifying hypotheses are usually incorporated with the aim of reducing the complexity of this problem in the literature. The restricted circular three-body problem (CR3BP) [6] reduces the complexity of the general three-body problem by only considering the masses of the two primary bodies, $m_1$ and $m_2$, respectively, while the mass of the third body, $m_3$, known as the infinitesimal mass, is considered negligible, so it does not alter the dynamics of the primary masses (a restricted problem). Also, the primary bodies are considered to orbit circularly (a circular problem) around their center of masses, and it is assumed that all of the bodies considered orbit on the same plane.

Different modified CR3BP models have been used for analyses of the orbital behavior of the infinitesimal mass which involve additional perturbing accelerations by modifying the potential function [7,8,9,10,11,12,13]. In these studies, the effects of irregularities on the shapes of the primaries, such as the radiation pressure or relativity theory, are considered to obtain more realistic models. For example, in [14], an effective potential application within the framework of multiple planets around a single star, or a binary system, can be found.

However, the dynamic properties of a constrained three-body model are not limited to determining the locations of the equilibrium points but also extend to the respective basins of convergence (BCs) [15,16]. A BC is defined as the set of initial conditions in phase space that numerically converge to a specific Lagrange point under the chosen iterative method. These attractors are determined by the gravitational and centrifugal balance in the vicinity of the equilibrium points. These dynamics tend to drive points with an infinitesimal mass and null initial velocity masses towards the Lagrange points under idealized conditions, but perturbations or transient effects can lead to divergence from these points in the short term. These regions and the libration points are calculated using iterative methods, such as Newton–Raphson’s (NR) method [17] or Broyden’s method [18]. However, these numerical methods’ convergence strongly depends on the initial conditions. Moreover, in regions near the fractal boundaries, some points may fail to converge within a finite number of iterations. These non-convergent points are excluded from the analysis of BCs. This divergence is particularly significant when the model incorporates additional perturbing forces, as these can affect both the convergence speed and the stationary behavior near the equilibrium points. The timescales and stability regions for these BCs depend on the specific parameters of the system and require further analysis in modified CR3BP formulations.

Studies investigating the BCs within the framework of different restricted problems with three bodies can be found in [12,19,20], where the effects of radiation pressure and oblateness have been studied using the iterative NR method. Also, the BCs associated with the equilibrium points in an N-body problem ($N=3, 4, 5$) under different types of perturbing accelerations can be found in [12,21,22]. The basins of the out-of-plane equilibrium points are also studied in [17].

In this paper, a modified CR3BP simultaneously including the effect of radiation forces, the oblateness of the primary bodies, and the effects of relativity is studied. Moreover, the presence of natural satellites is also considered to obtain a more accurate model, mainly when the relation between the primary mass and its satellites is significant. Under these conditions, the BCs are calculated by means of an adaptive algorithm based on the iterative NR method. This paper builds upon and significantly expands our previous work [23], including validation of the results and providing a more comprehensive analysis of our findings.

This paper is structured as follows: In Section 2, a description of the dynamical model is provided, with both classical and modified formulations. In this section, the modifications implemented in the model are presented. In Section 3, the basins of convergence, the Lagrange points, and the chosen iterative method are introduced. The development of the algorithm and its validation are also presented in this section. Next, in Section 4, an application of the model to the Sun–Mars system is obtained considering the different effects and analyzing the speed of convergence of the implemented code. Finally, in Section 5, the conclusions can be found.

2. Description of the Dynamic Model

As stated in the introduction, the restricted circular three-body problem (CR3BP) only considers the masses of the two primary bodies, $m_1$ and $m_2$, orbiting circularly around their centers of mass and on the same plane, together with the third infinitesimal body. These equations are usually expressed in a rotating coordinate system, whose origin is located at the centers of mass of the system of primary bodies. It has an angular velocity equal to the mean motion of the orbits of the primaries, which can be calculated as follows:

$$n_D=\sqrt{{\displaystyle \frac{G(m_1+m_2)}{R^3}}}$$

(1)

where R is the distance between the primary bodies, and G is the gravitational constant. Equally, the equations of the CR3BP are expressed in dimensionless terms, taking the appropriate unit measuring system.

The dynamics of the infinitesimal mass can be obtained from classical mechanics and Newton’s law of universal gravitation, taking into account the initial hypotheses (see Appendix A for more details). Then, the resulting equations are

$$\ddot{x}-2\dot{y}={\displaystyle \frac{\partial \Omega }{\partial x}}$$

(2)

$$\ddot{y}+2\dot{x}={\displaystyle \frac{\partial \Omega }{\partial y}}$$

(3)

with x and y being the coordinates of the infinitesimal mass and the dot (˙) the derivative with respect to dimensionless time, while $\Omega$ is the potential function, defined as

$$\Omega \left(x,y\right)={\displaystyle \frac{1}{2}}\left(x^2+y^2\right)+{\displaystyle \frac{1-\mu }{r_1}}+{\displaystyle \frac{\mu }{r_2}}$$

(4)

where $r_1$ and $r_2$ are the distances between the infinitesimal body and each one of the primary masses, and $\mu =m_2/(m_1+m_2)$ is the dimensionless mass parameter.

It should be taken into account that when the coordinates are dimensionless, the positions of the primary bodies become

$$\begin{array}{cc}{x}_1=-\mu & x_2=1-\mu \\ y_1=0& y_2=0\end{array}$$

(5)

It is worth noting that any additional phenomena capable of modifying the dynamics of the system have not yet been considered in Equations (2) and (3). In recent decades, studies on the CR3BP have been carried out by modifying the effective potential to obtain more complex and accurate models [10,11,19,21,24,25,26,27].

Next, some extra phenomena are formulated separately. The associated terms that modify the effective potential are added cumulatively, so that the final potential (${\Omega }^{*}$) incorporates all of these effects. The physical and geometrical properties of the primary masses also affect the mean motion of the orbit, so a modified mean motion $n_D^{*}$ must be used. Defining $n=n_D^{*}/n_D$ as the dimensionless corrected mean motion, the dynamic model is modified according to the following expressions:

$$\ddot{x}-2n\dot{y}={\displaystyle \frac{\partial {\Omega }^{*}}{\partial x}}$$

(6)

$$\ddot{y}+2n\dot{x}={\displaystyle \frac{\partial {\Omega }^{*}}{\partial y}}$$

(7)

2.1. Radiation Forces

Various authors have studied the effect of radiation forces on the dynamics of celestial bodies. The first studies carried out in this field, and specifically applied to the CR3BP, gave rise to what is known as the restricted three-body photogravitational problem [7,20]. These models consider one or both of the primary masses to act as a source of radiation, in addition to generating the corresponding gravitational potential. This phenomenon is introduced into the dynamic system by means of a radiation factor $q_i$, in which the ratio between the gravitational effects $F_{g_i}$ and the effects of the forces associated with radiation $F_{r_i}$ emitted by $m_i$ is established:

$$q_i=1-{\displaystyle \frac{F_{r_i}}{F_{g_i}}}$$

(8)

According to Chernikov [7], it is assumed that this radiation factor remains constant, so fluctuations in the pressure forces or variations in the emitted radiation are not considered. The modified potential function, considering these radiative effects, includes the radiation factor as a multiplicative factor in the terms associated with the primary masses:

$${\Omega }^{*}(x,y)={\displaystyle \frac{n^2}{2}}(x^2+y^2)+{\displaystyle \frac{q_1(1-\mu )}{r_1}}+{\displaystyle \frac{q_2\mu }{r_2}}$$

(9)

Considering the radiation emitted by the primary bodies as isotropic, the associated forces do not imply a change in the mean motion of their orbits, so $n=1$.

2.2. Oblateness of the Primary Masses

The original CR3BP problem established the gravitational potential generated by two primary masses when they were considered as spherical shapes. However, not all celestial bodies have a perfectly spherical form but rather have certain flattening or oblateness, so several authors have modified the effective potential in order to consider these new geometries [11,21,24,26,28]. To define these parameters, three semi-axes are considered for each primary mass, denoted as $a_j$, $b_j$, and $c_j$ and measured on a right-hand trihedral axis, as are the relationships between them. In this work, it is assumed that the x-axis of the reference frame aligns with the line joining the primary masses. This alignment is consistent with tidal effects, which naturally orient the primary masses along their principal axes due to mutual gravitational interactions. Thus, the triaxiality parameters ${\sigma }_{i_j}$ are defined as follows:

$${\sigma }_{1_j}={\displaystyle \frac{a_j^2-c_j^2}{5R^2}}{\sigma }_{2_j}={\displaystyle \frac{b_j^2-c_j^2}{5R^2}}$$

(10)

where $a_j$ and $b_j$ represent the semi-axes in the x and y directions, respectively, and $c_j$ is perpendicular to them.

Therefore, each primary mass j has two triaxiality functions $f_{i_j},i=1,2$, given by

$$f_{1_j}=2{\sigma }_{1_j}-{\sigma }_{2_j}{f}_{2_j}={\sigma }_{2_j}-{\sigma }_{1_j}$$

(11)

The terms of the potential function associated with each of the primary masses are modified by means of these triaxiality functions:

$${\Omega }^{*}(x,y)={\displaystyle \frac{n^2}{2}}(x^2+y^2)+{\displaystyle \frac{1-\mu }{r_1}}\left(q_1+{\displaystyle \frac{f_{1_1}}{2r_1^2}}+{\displaystyle \frac{3y^2{f}_{2_1}}{2r_1^4}}\right)+{\displaystyle \frac{\mu }{r_2}}\left(q_2+{\displaystyle \frac{f_{1_2}}{2r_2^2}}+{\displaystyle \frac{3y^2{f}_{2_2}}{2r_2^4}}\right)$$

(12)

where the corrected dimensionless mean movement is

$$n=\sqrt{1+{\displaystyle \frac{3}{2}}(2{\sigma }_{1_1}-{\sigma }_{2_1})+{\displaystyle \frac{3}{2}}(2{\sigma }_{1_2}-{\sigma }_{2_2})}.$$

(13)

2.3. Non-Newtonian Potential—Strong Gravity (Relativistic Effects)

In this section, certain relativistic effects are introduced into the effective gravitational potential. The modified potential, known as the effective Schwarzschild potential, introduces an additional term of the form $1/r^3$ which assumes that the gravitational field generated by a mass is stronger than that predicted by Newtonian theory of gravity.

According to the development shown in [12], the parameter $ϵ\in [0,1]$ is used to incorporate these relativistic effects into the term of the second primary mass’s potential function, while the first primary mass always generates a classical Newtonian gravitation field. When $ϵ=0$, the effective potential will not include relativistic terms (classical Newtonian potential), while when $ϵ>0$, the term associated with strong gravity will acquire greater importance. Then, the expression of the modified effective potential is given as follows:

$$\begin{array}{cc}\hfill {\Omega }^{*}(x,y)=& {\displaystyle \frac{n^2}{2}(x^2+y^2)+\frac{1-\mu }{r_1}\left(q_1+\frac{f_{1_1}}{2r_1^2}+\frac{3y^2{f}_{2_1}}{2r_1^4}\right)+}\hfill \\ \hfill +& {\displaystyle \frac{\mu }{r_2}\left(q_2+\frac{f_{1_2}}{2r_2^2}+\frac{3y^2{f}_{2_2}}{2r_2^4}+\frac{ϵ}{r_2^2}\right)}\hfill \end{array}$$

(14)

The inclusion of relativistic terms also modifies the mean motion. The correction of the average movement is indicated in red:

$$n=\sqrt{\left[1+{\displaystyle \frac{3}{2}}(2{\sigma }_{1_1}-{\sigma }_{2_1})+{\displaystyle \frac{3}{2}}(2{\sigma }_{1_2}-{\sigma }_{2_2})\right]\left(1+3ϵ\right)}$$

(15)

2.4. Effects of Natural Satellites

In this section, the effect of the presence of natural satellites in the CR3BP model is studied in a similar way to that shown in [29]. The modification of the potential is based on the original equations of classical mechanics (Equations (A2) and (A3)). It is assumed that the satellites are perfectly spherical bodies, that they are not a source of radiation, and that their mass is significantly lower than the mass of the primary bodies so that they do not affect the dynamics of the primaries. For each satellite $i=[1,n_s]$, its mass is denoted by ${m_s}_i$, and it is considered to orbit circularly with a radius ${a_s}_i$ around one of the primary bodies, with a mean motion of ${n_{Ds}}_i$. Employing this notation, the position of each satellite at any time is given as follows:

$$\begin{array}{c}{x}_i=x_j+{\displaystyle \frac{{a_s}_i}{R}}cos\left(({n_{Ds}}_i-n_D^{*})t\right)\\ y_i=y_j+{\displaystyle \frac{{a_s}_i}{R}}sin\left(({n_{Ds}}_i-n_D^{*})t\right)\end{array}$$

(16)

where subscript j refers to the primary mass around which satellite i orbits. The modifications to the potential are deduced from the equations of classical mechanics and the mathematical development set forth in Section 2 and Appendix A. The effective potential considering the effect of natural satellites is then

$$\begin{array}{ccc}{\Omega }^{*}(x,y,\tau )\hfill & =\hfill & {\displaystyle \frac{n^2}{2}(x^2+y^2)+\frac{1-\mu }{r_1}\left(q_1+\frac{f_{1_1}}{2r_1^2}+\frac{3y^2{f}_{2_1}}{2r_1^4}\right)+}\hfill \\ \hfill & +\hfill & {\displaystyle \frac{\mu }{r_2}\left(q_2+\frac{f_{1_2}}{2r_2^2}+\frac{3y^2{f}_{2_2}}{2r_2^4}+\frac{ϵ}{r_2^2}\right)+\sum _{i=1}^{n_s}\frac{{\alpha }_i}{r_{s_i}}}\hfill \end{array}$$

(17)

where ${\alpha }_i=m_{si}/(m_1+m_2)$, and $r_{s_i}$ is the distance between the infinitesimal body and satellite i.

The positions of the satellites, unlike the positions of the primary masses, are not fixed in the rotating reference system in which the equations are defined. This means that the potential function depends not only on the coordinates of the infinitesimal mass but also on time. However, their presence does not affect the average movement of the primaries, so this modification does not imply any extra term within n, given by (15).

3. Lagrange Points and Basins of Convergence

This section describes the Lagrange points of the system and the calculation methodology implemented to obtain them, which involves the use of the NR method. As shown in Section 2, the CR3BP is based on defining a potential function whose value depends on the coordinates of the infinitesimal mass. Given the definition of this effective potential, it is possible to find points where this function presents maximums and/or minimums. These points are known as Lagrange points or libration points and determine the places in space where an infinitesimal mass, affected only by the gravitational field exerted by the two primary masses, can reach a stationary position with respect to the latter. At these points, the position of the third mass can remain stationary since the gravitational field exerted by the primary bodies is compensated for by the centrifugal acceleration of the infinitesimal mass.

In general, there are a total of five libration points, denoted as $L_1$, $L_2$, $L_3$, $L_4$, and $L_5$. The first three are known as collinear points and are located on axis x ($y=0$), that is, on the line joining the two primary masses. On the other hand, points $L_4$ and $L_5$, known as triangular points, are points that lie outside the mentioned axis, so their coordinate y is not zero.

Analogously, when considering natural satellites and other effects, a modified CR3BP results in a new effective potential function (see Section 2). The introduction of new terms into the dynamic model does not affect the definition of the Lagrange points, but now, all of the added phenomena are considered in addition to the gravitational field. Thus, more than five libration points can appear. Some authors have shown how additional libration points appear when modifying the dynamic model of the CR3BP with the effect of the oblateness of the primary masses [21].

As is known, the libration points act as attractors within the gravitational field exerted by the two primary masses (including any modifications deemed appropriate). That is why an infinitesimal mass located at $(x_0,y_0)$ will naturally tend to approach one of the aforementioned points. The basins of convergence determine which Lagrange point an infinitesimal mass evolves towards given its initial conditions $(x_0,y_0)$ and assuming that its initial velocity is zero $({\dot{x}}_0={\dot{y}}_0=0)$. While these regions mathematically define the dynamic attractors for specific initial conditions, real objects may not remain stable at the Lagrange points due to perturbations or orbital eccentricities. Instead, they may exhibit periodic or quasi-periodic motion in their vicinity.

Taking into account the fact that the condition that must be met for a point in space to be a libration point is for the potential to reach the maximum or the minimum, the equations that define them are the following:

$$\begin{array}{c}{{\Omega }^{*}}_x=0\\ {{\Omega }^{*}}_y=0\end{array}$$

(18)

In order to solve these equations and determine the convergence basins, a certain portion of the plane is taken and meshed, dividing it into $n_x\times n_y$ uniformly distributed elements. The centroid of each of these cells represents position $(x_0,y_0)$. Then, two different iterative methods are used to find the libration point towards which an infinitesimal mass without velocity and located at $(x_0,y_0)$ will tend. So, Equation (18) is solved by means of a searching algorithm, taking as the centroid of each cell as the initial solution.

3.1. Developing the Searching Algorithm

The NR method is one of the most common methods used to find zeros of functions. Thus, many studies have employed it to solve Equation (18) [12,17,20,30]. This method is based on using a first-order linear approximation of the function [20]. Then, considering vector $\mathbf{x}={\left\{x,y\right\}}^T$ and function $\mathbf{F}\left(\mathbf{x}\right)={\left\{{{\Omega }^{*}}_x,{{\Omega }^{*}}_y\right\}}^T$, the NR method is formulated as follows:

$$\mathbf{F}\left({\mathbf{x}}_{n+1}\right)=\mathbf{F}\left({\mathbf{x}}_n\right)+{\mathbf{F}}^{\prime }\left({\mathbf{x}}_n\right)({\mathbf{x}}_{n+1}-{\mathbf{x}}_n)$$

(19)

where ${\mathbf{F}}^{\prime }\left({\mathbf{x}}_n\right)$ represents the Jacobian matrix of function $\mathbf{F}\left(\mathbf{x}\right)$ evaluated at point ${\mathbf{x}}_n$. Then, the zeros of function ($\mathbf{F}\left({\mathbf{x}}_{n+1}\right)$) are iteratively obtained as follows:

$${\mathbf{x}}_{n+1}={\mathbf{x}}_n-{\left({\mathbf{F}}^{\prime }\left({\mathbf{x}}_n\right)\right)}^{-1}\mathbf{F}\left({\mathbf{x}}_n\right).$$

(20)

The first-order NR method does not always converge towards a specific solution, at least not with an acceptable level of precision, when the complexity of the model is increased.

However, it has a significantly lower computational cost than that of other methods and performs the calculations faster. These considerations are taken into account when developing the libration point searching methodology.

Firstly, as mentioned before, a rectangular domain is discretized, and the NR method is applied, with the centroids of each of the cells as the starting points. The solution is considered to converge when the distance between two consecutive iterations is below a precision threshold. Then, the method is used to get as close as possible to the solution. If the method is not precise enough or its convergence velocity is too slow, then the algorithm stops.

After that, the solutions obtained are analyzed. Two solutions are considered equal if the distance between them is less than a given threshold. Once all possible solutions have been determined, each of them is assigned to a corresponding identifier (called $L_i$), which corresponds to a Lagrange point. Subsequently, the solution reached in each cell is compared with each one of the Lagrange points so that each convergent cell is assigned to the Lagrange point to which it converges. Representing these identifiers graphically, the BCs of the system are obtained. In regions near the boundaries of the basins of convergence (BCs), where the dynamics are highly sensitive, some points may fail to converge within a fixed number of iterations. These points are assigned as "non-convergent" and are excluded from the analysis of the BCs.

3.2. Validating the Algorithm: The Sun–Earth System

In this subsection, an analysis of the results is carried out to validate the implemented code. Given the modular construction of the dynamic model and assuming the independence hypothesis discussed in Section 2, the validations are presented separately. For this purpose, meshes with a resolution of $500\times 500$ are used for the validation, without losing significant precision in the solution.

All of the effects presented in the previous section can be validated against the literature except for perturbation due to natural satellites. This last effect was studied theoretically in [29] using a different formulation for a single satellite, but, to our knowledge, there are no data provided to compare. Furthermore, we note that our algorithm allows us to consider more than one satellite.

3.2.1. The Effect of Radiation Forces and Mean Motion

Firstly, the algorithm is tested against the effective potential given in (9). In this case, we try to reproduce the results obtained by R. Aggarwal et al. [20], computing our algorithm under the same conditions. So, the BCs without radiation forces, that is, taking $q_1=q_2=1[-]$ (the symbol $[-]$ is used to indicate that the quantity is dimensionless) but $\mu =0.05[-]$ and $n=0.5[-]$, are shown in Figure 1. In addition, the BCs obtained for $\mu =0.5[-]$, $q_1=0.15[-]$, $q_2=0.25[-]$, and $n=0.25[-]$ are shown in Figure 2, while Figure 3 shows the BCs with the same mass and radiation parameters but with the mean motion changed to $n=0.95[-]$. These cases would be equivalent to studying a classic CR3BP in which the masses orbit more slowly since $n<1$. In

Table 1. Libration point coordinates for the Sun–Earth system without radiation forces, that is, for $q_1=q_2=1[-]$, $\mu =0.05[-]$, and $n=0.5[-]$.

Libration Points	Coordinates $({\mathrm{x}}_{\mathrm{L}},{\mathrm{y}}_{\mathrm{L}})[-]$
$L_1$	(0.75280529, 0)
$L_2$	(1.67007870, 0)
$L_3$	(−1.60407046, 0)
$L_4$	(0.44999999, 1.50659952)
$L_5$	(0.44999999, −1.50659952)

,

Table 2. Libration point coordinates for the Sun–Earth system obtained for $q_1=0.15[-]$, $q_2=0.25[-]$, $\mu =0.5[-]$, and $n=0.25[-]$.

Libration Points	Coordinates $({\mathrm{x}}_{\mathrm{L}},{\mathrm{y}}_{\mathrm{L}})[-]$
$L_1$	(−0.06229089, 0)
$L_2$	(1.68242540, 0)
$L_3$	(−1.5540698, 0)
$L_4$	(−0.36364010, 1.33190385)
$L_5$	(−0.36364010, −1.33190385)

and

Table 3. Libration point coordinates for the Sun–Earth system obtained for $q_1=0.15[-]$, $q_2=0.25[-]$, $\mu =0.5[-]$, and $n=0.95[-]$.

Libration Points	Coordinates $({\mathrm{x}}_{\mathrm{L}},{\mathrm{y}}_{\mathrm{L}})[-]$
$L_1$	(−0.04943902, 0)
$L_2$	(0.90150065, 0)
$L_3$	(−0.83200116, 0)
$L_4$	(−0.06132351, 0.33144656)
$L_5$	(−0.06132351, −0.33144656)

, the coordinates of the Lagrange points obtained in each of the cases analyzed are given.

As can be observed, the Lagrange points change depending on the values of parameters $\mu$, $q_1$, $q_2$, and n. These results agree with those obtained by R. Aggarwal et al. [20] under the same conditions.

3.2.2. Effects of Oblateness

Next, the code is checked for the effective potential given in (12) and the mean movement (13) when the effects of oblateness are considered through a comparison with the results given in [21]. The results obtained for a mass parameter $\mu =0.1[-]$ and the triaxiality parameters ${\sigma }_{1_1}=0.7[-]$, ${\sigma }_{2_1}=0.5[-]$, and ${\sigma }_{1_2}={\sigma }_{2_2}=0$ are shown in Figure 4 and

Table 4. Libration point coordinates for the Sun–Earth system considering the effects of oblateness with $\mu =0.1[-]$, ${\sigma }_{1_1}=0.7[-]$, ${\sigma }_{2_1}=0.5[-]$, and ${\sigma }_{1_2}={\sigma }_{2_2}=0[-]$.

Libration Points	Coordinates $({\mathrm{x}}_{\mathrm{L}},{\mathrm{y}}_{\mathrm{L}})[-]$
$L_1$	(0.70490520, 0)
$L_2$	(1.14837969, 0)
$L_3$	(−1.0544736, 0)
$L_4$	(0.06123130, 0.85300535)
$L_5$	(0.06123130, −0.85300535)

. Figure 5 and

Table 5. Libration point coordinates for the Sun–Earth system considering the effects of oblateness with $\mu =0.1[-]$, ${\sigma }_{1_1}=0.5[-]$, ${\sigma }_{2_1}=0.7[-]$, and ${\sigma }_{1_2}={\sigma }_{2_2}=0[-]$.

Libration Points	Coordinates $({\mathrm{x}}_{\mathrm{L}},{\mathrm{y}}_{\mathrm{L}})[-]$
$L_1$	(0.66297092, 0)
$L_2$	(1.20457368, 0)
$L_3$	(−1.0496936, 0)
$L_4$	(0.79265246, 0.57347612)
$L_5$	(0.79265246, −0.57347612)
$L_6$	(−0.4440806, 1.02755166)
$L_7$	(−0.4440806, −1.02755166)

show a case with seven Lagrange points and the corresponding BCs.

Both the BCs and the coordinates of the Lagrange points obtained in this study are similar to those shown in [21]. A slight deviation in the coordinates of the points is observed, which may be due to the precision and the software employed in the calculations since in [21], the high-precision algorithms in Fortran 77 were used, while the software used in this work was Matlab R2024a.

It is worth noting that new attractors appear when considering the effects of oblateness. These “perturbed” equilibrium points could be interpreted as quasi-stable gravitational regions. Under this interpretation, one could redefine the concept of Lagrange points as “Lagrangian attractors”, acknowledging the broader influence of perturbations in shaping the dynamics of these zones. This perspective aligns with the notion that real-world systems are rarely idealized and often exhibit dynamical behavior, where the stability is relative within a defined perturbation neighborhood.

3.2.3. Relativistic Effects

In this section, relativistic effects are applied when calculating the BCs, considering Equations (14) and (15). Two cases are presented with different values for the strong gravity parameter $ϵ$, while the rest of the parameters remain the same. That is, Figure 6 and Figure 7 show the results for $\mu =0.5[-]$, with a unitary dimensionless mean motion ($n=1[-]$) and a strong gravity parameter equal to 0 and 1, respectively. The coordinates of the libration points can be seen in

Table 6. Libration point coordinates for the Sun–Earth system considering relativistic effects with $\mu =0.5[-]$, $n=1[-]$, and $ϵ=0[-]$.

Libration Points	Coordinates $({\mathrm{x}}_{\mathrm{L}},{\mathrm{y}}_{\mathrm{L}})[-]$
$L_1$	(0, 0)
$L_2$	(1.19840614, 0)
$L_3$	(−1.19840614, 0)
$L_4$	(0, 0.86602540)
$L_5$	(0, −0.86602540)

and

Table 7. Libration point coordinates for the Sun–Earth system considering relativistic effects with $\mu =0.5[-]$, $n=1[-]$, and $ϵ=1[-]$.

Libration Points	Coordinates $({\mathrm{x}}_{\mathrm{L}},{\mathrm{y}}_{\mathrm{L}})[-]$
$L_1$	(−0.21134724, 0)
$L_2$	(1.27677761, 0)
$L_3$	(−0.90827430, 0)
$L_4$	(−0.30157487, 0.59789441)
$L_5$	(−0.30157487, −0.59789441)

. It can be verified that these results agree with those obtained in [12] for the same conditions.

4. Application to the Sun–Mars System: Adding the Effects of Phobos and Deimos

In this section, an application of the algorithm to the Sun–Mars system is shown and compared to the classical CR3BP solution. The effects of the triaxiality of the primary masses, the radiation forces, and the presence of the natural satellites of Mars, Phobos and Deimos, are included.

Therefore, a system that considers the Sun and Mars as the primary masses is studied, with the initial data shown in

Table 8. Sun–Mars problem data [31,32].

Parameter	Value
Mass of Sun [kg]	$1.98850\times {10}^{30}$
Mass of Mars [kg]	$6.41710\times {10}^{23}$
Sun–Mars distance [km]	$2.27923\times {10}^8$

. The distance between the primary masses is considered equal to the semi-major axis of the elliptical orbit that Mars follows around the Sun.

4.1. The Sun–Mars Problem: The Classical Formulation

Firstly, the classical CR3BP is studied. The Lagrange points in this case are obtained assuming that Mars orbits circularly around the Sun. Taking into account the initial data given in Table 8, it can be estimated the value of the mass parameter $\mu =3.22710\times {10}^{-7}[-]$. The BCs of the problem configured with these data are shown in Figure 8, Figure 9 and Figure 10, where different domain sizes have been considered. The coordinates of the Lagrange points estimated in this case are those shown in

Table 9. Coordinates of the Sun–Mars libration points in the classical formulation.

Libration Points	Obtained Coordinates $({\mathrm{x}}_{\mathrm{L}},{\mathrm{y}}_{\mathrm{L}})[-]$
$L_1$	(0.99525, 0)
$L_2$	(1.00476, 0)
$L_3$	(−1.00000, 0)
$L_4$	(0.50000, 0.86603)
$L_5$	(0.50000, −0.86603)

.

As can be seen, the region covered by the BCs is significantly larger than that obtained in the validation process. This is fundamentally due to the fact that the dimensionless parameter of the masses is remarkably smaller. However, it can also be seen that the libration points are located in a region very close to the primary masses. Moreover, it is possible to observe that all of the points outside of the regions resembling fractals converge towards Lagrange point $L_1$.

Finally, it is also worth noting the proximity of libration points $L_1$ and $L_2$ to Mars (see Figure 10). Considering Equations (2) and (3), it can be deduced that this fact is also related to the very low mass parameter. As the mass parameter decreases, the collinear points $L_1$ and $L_2$ tend to approach the smallest mass ($m_2$), while point $L_3$ moves away from it. The Lagrange points $L_4$ and $L_5$ remain at the midpoint between the primary masses, as can be seen in Table 9.

4.2. Triaxiality Effects

Triaxiality effects are now introduced. It is assumed that the semi-axes, contained in the $\{x,y\}$ plane, are equal, while the z semi-axis means that the primary masses are not perfectly spherical. The semi-axes of Mars were extracted from NASA’s data sheets [32]. However, this information is not available for the Sun, so an estimation of the corresponding semi-axes was made using its mean volumetric radius and its ellipticity [31]. This information is reflected in

Table 10. Data on the primary masses used for the generation of the BCs. They allow us to calculate the triaxiality parameters using Equation (10).

Parameter	Sun [31]	Mars [32]
Ellipticity [-]	0.00005	0.00589
Average volumetric radius [km]	$6.95700\times {10}^5$	$3.38950\times {10}^3$
Equatorial semi-axis [km]	$6.95712\times {10}^5$	$3.39620\times {10}^3$
Polar semi-axis [km]	$6.95677\times {10}^5$	$3.37620\times {10}^3$

.

Using these data, it is possible to calculate the oblateness parameters, which modify the equations according to the model shown in Section 2.2. Note that the terms of the potential associated with the Sun hardly change when introducing the effects of triaxiality due to its very low ellipticity. However, the oblateness of Mars is significantly greater than that of the Sun, which implies a substantial variation in the terms associated with Mars.

In Figure 11, Figure 12 and Figure 13, the BCs of the Sun–Mars problem are shown, taking into account their oblateness.

Table 11. Coordinates of the Sun–Mars libration points considering triaxiality effects.

Libration Points	Obtained Coordinates $({\mathrm{x}}_{\mathrm{L}},{\mathrm{y}}_{\mathrm{L}})[-]$
$L_1$	(0.99525, 0)
$L_2$	(1.00476, 0)
$L_3$	(d1.00000, 0)
$L_4$	(0.50000, 0.86603)
$L_5$	(0.50000, −0.86603)

shows the coordinates of the libration points in this case.

As can be seen in Table 11, the location of the Lagrange points has not changed due to the inclusion of the effects of oblateness. Nor have the BCs in the environment around Mars (Figure 12 and Figure 13). This can be explained because the terms associated with the Sun have greater weight than those associated with this planet, and the star is practically spherical. However, an appreciable change in the shape of the BCs can be observed in far-field solutions (see Figure 11) since the effects of the oblateness of Mars begin to become more noticeable.

4.3. Effects of Phobos and Deimos

As discussed before, the BCs are also analyzed when considering the presence of the natural satellites of Mars, Phobos and Deimos. They are assumed to be perfectly spherical masses that follow circular orbits whose radii are equal to the semi-major axes of their real orbits. Equally, the approximate mean motions are obtained as the inverse of their orbital periods. The corresponding data are shown in

Table 12. Mars satellite data: Phobos and Deimos.

Parameter	Phobos [33]	Deimos [34]
Mass [kg]	$1.06590\times {10}^{16}$	$1.47620\times {10}^{15}$
Major axis of the orbit [km]	$9.37600\times {10}^3$	$2.34580\times {10}^4$
Orbital period [H]	7.65	30.00
Average movement (approximate) [1/s]	$3.63108\times {10}^{-5}$	$9.25925\times {10}^{-6}$

.

The main aim of this section is to check the influence of these natural satellites on the libration points and the distribution of the BCs. Although the program is capable of calculating temporal evolutions, in this section, only certain positions of the satellites are analyzed in order to verify whether their presence is able to modify the distribution of the BCs or the libration points.

Firstly, a case in which both satellites are on the x axis and both are eclipsed by Mars, i.e., $x_{si}>1-\mu$, is analyzed. This study leads to very similar results to those obtained when triaxiality effects were considered, both in terms of the coordinates of the Lagrange points and the shape of the BCs. However, they do differ in position of the tenth decimal.

Next, we analyze a case in which both satellites are aligned in the direction perpendicular to the abscissa axis, that is, when the offsets of both satellites with respect to the x axis are $\pm {90}^{\circ }$. Phobos would be on the positive side of the y axis (a $+{90}^{\circ }$ offset), while Deimos would be on the negative side (a $-{90}^{\circ }$ offset). In this case, notably similar results to those in the case of the satellites being aligned were obtained, so they are not shown.

Then, we find that the presence of the satellites modifies neither the distribution of the BCs nor the position of the Lagrange points. This can be explained by their mass ratio: the terms of the potential associated with the satellites are several orders of magnitude lower than those associated with Mars, whose weight in the potential function was already significantly less than that of the Sun. So, in view of Equation (17), it seems that if the distribution of the total mass between the primary masses was more balanced and the masses of the satellites were larger, the satellites’ effects may modify the results.

4.4. Radiation Effects

Now, the effects of the Sun’s radiation forces are studied. In order to simplify the calculations, the satellites are not included in the simulation since their effect has been proven to be negligible. Since the radiation factor depends not only on the emitting body but also on the characteristics of the infinitesimal mass, we chose to use an arbitrary value for the radiation factor $q_1=0.4[-]$ of the Sun and a null radiation factor for Mars ($q_2=0[-]$).

The results are shown in Figure 14, Figure 15 and Figure 16 and

Table 13. Coordinates of the Sun–Mars libration points considering triaxiality and radiation effects.

Libration Points	Obtained Coordinates $({\mathrm{x}}_{\mathrm{L}},{\mathrm{y}}_{\mathrm{L}})[-]$
$L_1$	(0.73680, 0)
$L_2$	(−0.73681, 0)
$L_3$	(0.27144, 0.68498)
$L_4$	(0.27144, −0.68498)

. Clearly, it can be seen that radiation does have an important effect on the BCs and the libration points.

In this particular case, one of the collinear Lagrange points (that corresponding to $x>1-\mu$) does not appear. So, the system has only four libration points, which are renumbered. That is why the numbering of the points in this case differs from that used before now. In addition, the region converging to $L_1$, the Lagrange point located between the Sun and Mars, enlarges with respect to the areas converging to the other libration points, especially in the broad domain. In the reduced domain, more cells of space converge towards the other Lagrange points. The regions of convergence towards the triangular points remain symmetric with respect to the abscissa axis. Finally, it is also convenient to highlight that the triangular points have moved closer to the Sun, so they are no longer at the midpoint between the Sun and Mars, and the collinear points are located almost symmetrically with respect to the Sun.

4.5. Speed of Convergence

Finally, it is interesting to see the number of iterations that are necessary to reach convergence. For the case of the CR3BP being modified by the effects of oblateness without taking into account any other effects, the results are shown in Figure 17, Figure 18 and Figure 19. These figures show that the speed of convergence towards the collinear points is much faster than that of convergence to the triangular points. It can be seen that cells converging towards collinear points generally require a relatively low number of iterations.

The rest of the cases analyzed in this work lead to the same conclusions about the trends in the speed of convergence towards both collinear and triangular points.

5. Conclusions

In this paper, different modifications to the classical CR3BP were made in order to increase the precision of the model. These modifications, which were added in a cumulative way, included the following: the effects of radiation forces, the effects of the oblateness of the primary masses, relativistic effects, and the contribution of satellites to the effective potential.

The iterative, numerical NR method was implemented to solve the equations of the modified complex problem. The algorithm designed allows us to calculate the basins of convergence and the Lagrange points in a considerably versatile way. It can be configured according to the problem conditions and the desired accuracy: the analyzed domain, the precision of the calculations, the primary bodies considered, etc. It has been seen that when reducing the number of effects included in our code, we obtained results similar to the results obtained by other authors.

Once the feasibility of the algorithm was proven, it was applied to the Sun–Mars system. In this case, it has been observed that the convergence basins and the libration points are mainly affected by the effects of radiation. Oblateness and the presence of natural satellites barely affect the results since the mass of the Sun is much bigger than the mass of Mars, so the terms of the effective potential associated with the star are more important than those for Mars. Moreover, this application shows that the number of iterations required to achieve convergence towards collinear points is significantly lower than that required for convergence towards triangular ones.