On the perturbed photogravitational restricted five-body problem: the analysis of fractal basins of convergence

TL;DR

This paper investigates the stability and convergence properties of the photogravitational restricted five-body problem, analyzing how perturbations and radiation parameters influence equilibrium points and the basins of convergence.

Contribution

It introduces a numerical analysis of equilibrium stability and basin topology in the perturbed photogravitational five-body problem using a multivariate Newton-Raphson scheme.

Findings

Existence and stability of in-plane equilibrium points analyzed.

Basins of convergence depend on radiation, Coriolis, and centrifugal parameters.

Fractal structures observed in the convergence basins.

Abstract

In the framework of photogravitational version of the restricted five-body problem, the existence and stability of the in-plane equilibrium points, the possible regions for motion are explored and analysed numerically, under the combined effect of small perturbations in the Coriolis and centrifugal forces. Moreover, the multivariate version of the Newton-Raphson iterative scheme is applied in an attempt to unveil the topology of the basins of convergence linked with the libration points as function of radiation parameters, and the parameters corresponding to Coriolis and centrifugal forces.

Equations

$$(x_0,y_0,z_0)$$

$$(x_1,y_1,z_1)$$

$$(x_2,y_2,z_2)$$

$$(x_3,y_3,z_3)$$

$$\ddot{x}-2\dot{y}$$

$$\ddot{y}+2\dot{x}$$

$$\ddot{z}$$

$$U(x,y,z)=\kappa\sum_{i=0}^{3}\frac{m_{i}q_{i}}{r_{i}}+\frac{1}{2}\big{(}x^{2}+y^{2}\big{)}.$$

$$\ddot{x}-2\vartheta \dot{y}$$

$$\ddot{y}+2\vartheta \dot{x}$$

$$\ddot{z}$$

$$\Omega (x,y,z)$$

$$\kappa$$

$$r_i$$

$$\widetilde{x_i}$$

$$J(x,y,z,\dot{x},\dot{y},\dot{z})=2\Omega (x,y,z)-({\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2)=C,$$

$$\dot{x}=$$

$${\Omega }_x(x,y,z)=0,{\Omega }_y(x,y,z)=0,{\Omega }_z(x,y,z)=0,$$

$${\Omega }_x(x,y,z)$$

$${\Omega }_y(x,y,z)$$

$${\Omega }_z(x,y,z)$$

$${\Omega }_{xx}$$

$${\Omega }_{yy}$$

$${\Omega }_{zz}$$

$${\Omega }_{xy}$$

$${\Omega }_{xz}$$

$${\Omega }_{yz}$$

$$\ddot{\varphi }-2\vartheta \dot{\phi }$$

$$\ddot{\phi }+2\vartheta \dot{\varphi }$$

$$({\lambda }^2-{{\Omega }_{xx}}_0){\xi }_1-(2\vartheta \lambda +{{\Omega }_{xy}}_0){\xi }_2$$

$$({\lambda }^2-{{\Omega }_{yy}}_0){\xi }_2+(2\vartheta \lambda -{{\Omega }_{yx}}_0){\xi }_1$$

$${\wp }_1{\Xi }^2+{\wp }_2\Xi +{\wp }_3=0,$$

$${\wp }_1$$

$${\wp }_2$$

$${\wp }_3$$

$${\wp }_2>0,{\wp }_3>0,{\wp }_2^2-4{\wp }_1{\wp }_3>0.$$

$${\mathbf{x}}_{n+1}={\mathbf{x}}_n-{\mathbf{J}}^{-1}f(x_n),$$

Full text

On the perturbed photogravitational restricted five-body problem: the analysis of fractal basins of convergence

Md Sanam Suraj, Rajiv Aggarwal, Amit Mittal, Md Chand Asique, Prachi Sachan

Corresponding author: Md Sanam Suraj

Department of Mathematics, Sri Aurobindo College, University of Delhi, New Delhi-110017, Delhi, India

Email: mdsanamsuraj@gmail.com, Email: mdsanamsuraj@aurobindo.du.ac.in

Abstract:

In the framework of photogravitational version of the restricted five-body problem, the existence and stability of the in-plane equilibrium points, the possible regions for motion are explored and analysed numerically, under the combined effect of small perturbations in the Coriolis and centrifugal forces. Moreover, the multivariate version of the Newton-Raphson iterative scheme is applied in an attempt to unveil the topology of the basins of convergence linked with the libration points as function of radiation parameters, and the parameters corresponding to Coriolis and centrifugal forces.

Keywords: Five-body problem, Radiation forces, The Coriolis and centrifugal forces, Libration points and Zero-velocity curves, Newton-Raphson method.

1 Introduction

In the recent decades, the few-body problem and particularly the circular restricted five-body problem is one of the most fascinated as well as the challenging problem in the fields of dynamical astronomy and Celestial mechanics. The restricted problem of five bodies describes the dynamics of the test particle moving under the gravitational influence of the four primaries. For describing the problem in very realistic manner, the dynamics of the test particles has been studied by adding several perturbing terms in the effective potential of the circular restricted five-body problem. The study of the five-body problem with several modification is motivated by the various previous study of the restricted three or four-body problem where the same modifications are proposed and significant results are obtained (e.g., [4], [20], [7], [17], [1], [3], [10], [2]).

A particular restricted five-body problem is analyzed by [18] where he explored the dynamics of the infinitesimal body ( i.e., the fifth body) of negligible mass, in comparison to remaining four bodies, and showed the existence and stability of libration points. Further, [16] have extended the study of same problem by including the effect of perturbation due to the radiation of the some or even all of the primaries. The Newton-Raphson iterative scheme is applied to analyze the domain of the basins of convergence associated with the libration points in restricted five-body problem by the Ref. [32]. This problem is one of the special case where $N=4$ of the ring problem ( [12]) which defines the motion of a test particle, moving in the gravitational attraction of $N$ much bigger bodies. In which $N-1$ bodies are located in equal distances on a fictitious circle. This ring configuration was first formulated by [13] to study the rings of Saturn.

One of the important study is the analysis of the dynamics of test particle under the effect of small perturbations in the Coriolis and centrifugal forces in various dynamical system where the primaries are also source of radiation. Various literature is available where the effect of these forces are discussed in the restricted three as well as four-body problems.

The restricted three-body problem with the effect of small perturbation in the Coriolis force has been studied by [31] where he has stated that the Coriolis force is a stabilizing force. It was observed that all the collinear libration points $L_{i},i=1,2,3$ are unstable whereas the non-collinear libration points $L_{4,5}$ are stable when the relation $\mu_{c}=\mu_{0}+\frac{16\epsilon}{2\sqrt{69}}$ is satisfied where $\mu_{0}=0.03852...$, $\mu_{c}$ is the critical mass ratio and the change in the Coriolis force is controlled by $\epsilon$. Moreover, in the paper, only the effect of small perturbation in the Coriolis force has been considered to analyze the stability of the libration points whereas the centrifugal force remains constant. In an extension of this study, [8], [9] have discussed the stability of the libration points in the linear as well in non-linear sense by taking the effect of small perturbations in the Coriolis as well as in centrifugal forces. They further stated that the Coriolis force is not always a stabilizing force. The collinear libration points are always unstable whereas the non-collinear libration points are stable for all mass ratios in the range of linear stability except the three mass ratios. Moreover, many authors have also extended their study to unveil the effect of small perturbations in the Coriolis and centrifugal forces in the frame of the restricted four-body problem as well as in their photogravitational version, where they have discussed the existence and stability of libration points, regions of possible motions, and the basins of convergence connected to the libration points of the system (e.g., [5], [14], [15], [21], [26] ).

Recently, the existence as well as stability of the libration points are discussed by various authors in the context of the five-body problem, i.e., the axisymmetric five-body problem, see Ref. [11], the basins of the convergence associated with the equilibrium points, which acts as attractors, in the axisymmetric five-body problem: the convex case, see Ref. [22], in the concave case, see Ref. [23], the five-body problem when the mass of the test particle is variable, see Ref. [24] and the effect of Coriolis and centrifugal force in the axisymmetric five-body problem, see Ref. [25].

In the past few years, the Newton-Raphson basins of convergence have been studied by many authors in various dynamical system, i.e., the restricted three-body problem (e.g., [29], [33], [34]), the restricted four-body problem (e.g., [26, 27, 28, 30]), the axisymmetric restricted five-body problem (e.g., [22, 23]), restricted five-body problem (e.g., [24], [32]). The basins of convergence, linked with the libration points of the dynamical system, provide some of the most intrinsic properties of these systems. The Newton-Raphson iterative scheme is applied to scan the set of initial conditions in an attempt to unveil the final states. The basins of convergence is collections of all those initial conditions which converge to one of the particular attractor.

Therefore, we believe that it would be interesting to analyze the effect of small perturbations in the Coriolis and centrifugal forces on the existence as well as on the positions of the libration points in the photogravitational version of the restricted problem of five bodies. Moreover, the effect of the radiation parameter on the basins of convergence associated with the libration points are illustrated in a systematic manner. Therefore, we believe that the present study and obtained results are novel and this is exactly the contribution of our work. The paper is constructed in following pattern: the Sec. 2 provides the idea about of the mathematical model as well as the equations of motion of the test particle. In the following section, the positions of the libration points as function of involved parameters and their stability are discussed, whereas, the regions of possible motion are explored in Sec. 4. In addition, a systematic study of the topology of the domain of basins of convergence connected with the libration points are explored in Sec. 5. The paper ends with Sec. 6 where the discussion and conclusions regarding the main results are presented.

2 Mathematical model and equations of motion

The circular restricted five-body problem consists of three primaries where $P_{i}=1,2,3$ move in a plane under the mutual gravitational attraction in circular orbits around their common center of mass whereas the primary $P_{0}$ is present at the center. The fifth body, which act as an infinitesimal test particle has a significantly smaller mass in comparison of the masses of the primaries, does not disturb the motion of the primaries.

We choose the rotating co-ordinate system in which the origin coincides with the center of mass of the system of primaries. The positions of the center of the primaries are:

[TABLE]

while the dimensionless masses of the primaries are $m_{0}=\beta m$, $m_{1}=m_{2}=m_{3}=m=1$. Moreover, the three primaries $P_{1,2,3}$ with mass $m$ are placed at the vertices of an equilateral triangle with unity side, whereas the primary $P_{0}$ with mass $\beta m$, is present at the center of the equilateral triangle (see Fig. 1). The line joining the centre of mass of the primaries $P_{0}$ and $P_{1}$ are taken as the $x-$axis while the line passing through origin and perpendicular to $x-$axis is taken as $y-$axis and the $z-$axis is the line passing through the origin and perpendicular to the plane of motion of the primaries.

We have further assumed that some or all the primary bodies are sources of radiation in context of the restricted five-body problem.

We, further, apply the transformation to scale the physical quantities, from the inertial to the synodic coordinates system. Therefore, in the rotating frame of reference, the motion of infinitesimal test particle is governed by the following equations:

[TABLE]

where

[TABLE]

In addition, we have introduce the small perturbations $\vartheta$ and $\upsilon$ in the Coriolis and centrifugal forces respectively, using the parameters $\epsilon$ and $\epsilon^{\prime}$ , respectively by $\vartheta=1+\epsilon,|\epsilon|\ll 1$ and $\upsilon=1+\epsilon^{\prime},|\epsilon^{\prime}|\ll 1$. Therefore, the Eqs. (1a-1c) become:

[TABLE]

where

[TABLE]

and

[TABLE]

are the distances of the infinitesimal test particle from the respective primaries.

The effect of radiation pressure of a radiating source on a particle is expressed by the radiation factor $q_{i}=1-b_{i}$ where $b_{i}$, always known as radiation coefficient (see [19]), describe the ratio of the radiation force $F_{ri}$ to the gravitational force $F_{gi}$, i.e., $b_{i}=\frac{F_{ri}}{F_{gi}}$, thus there are four reduction factors $q_{i}$, $i=0,1,...,3$.

The above mentioned problem is a particular case of the restricted five body problem $(\beta\neq 0)$ reduces to Ref. [16] when $\vartheta=1$ and $\upsilon=1$, further reduces to Ref. [18] when $q_{i}=1,i=0,1,2,3$ while it reduces to the circular equilateral restricted four-body problem when in addition $\beta=0$. The Eqs. (2a-2c), i.e., the equations of motion of the test particle correspond to an integral of motion, i.e., the Jacobi integral which reads as:

[TABLE]

where $\dot{x},\dot{y}$, and $\dot{z}$ represent the velocities, while $C$ is linked to the conserved numerical value of Jacobian constant.

To compare, we can define the mass parameter $\mu=1/(1+\beta)$ and corresponding to the classical mass parameter of the restricted three-body problem, we have $\mu\in(0,1]$ for $\beta\in[0,\infty)$.

3 The libration points of the system

We will try to illustrate how the effect of small perturbations in the Coriolis and centrifugal forces influence all the dynamical properties of the libration points. The necessary and sufficient conditions, which must satisfy for the existence of libration points, are:

[TABLE]

The linked coordinates $(x,y,z)$ of the equilibrium points can be evaluated by solving the system of following equations, numerically:

[TABLE]

where

[TABLE]

In the same vein, the second order partial derivatives of the effective potential are as follows:

[TABLE]

More precisely, when we restrict this study only to the coplanar libration points, i.e., the points which are located on the configuration $(x,y)$ plane with $z=0$.

In addition for simplicity, we consider only the central primary is radiating, i.e., $q_{i}=1,i=1,2,3$. Therefore, the intersections of the nonlinear equations $\Omega_{x}(x,y,0)=0$ and $\Omega_{y}(x,y,0)=0$ define the locations of the coplanar libration points. In Ref. [16], where the effect of the Coriolis and centrifugal forces has been neglected, it is shown that the critical value of the mass parameter $\mu$ is function of $q_{0}$, i.e., the radiation factor when the central primary $P_{0}$ is source of radiation. When $q_{0}=1$, then $\mu^{*}=0.98617276$, and when $q_{0}=0.5$, then $\mu^{*}=0.97189778$, and when $q_{0}=0.1$, then $\mu^{*}=0.82955666$, therefore it is concluded that when the radiation factor $q_{0}$ increases, the value of $\mu^{*}$ increases. Where the $\mu^{*}$ is the critical value where the number of collinear libration points changes. The problem admits three collinear libration points for $\mu\in(0,\mu^{*})$ on the $x-$axis in which $L_{x1}$ lies out side the primary $m_{1}$ whereas the libration point $L_{x2}$ lies between $m_{0}$ and $m_{1}$ and the libration point $L_{x3}$ lies on negative $x-$axis. For $\mu\in[\mu^{*},1)$ the problem admits two more collinear libration points $L_{x4,5}$ when $\epsilon^{\prime}=0$, i.e., the effect of centrifugal force is neglected and therefore there exists fifteen libration points in total.

In Fig. 2, we have depicted how the positions of the collinear libration points $L_{x3,4,5}$ change when the radiation parameter as well as the parameter $\epsilon^{\prime}$ which controlled the centrifugal force change. It is observed that as the value of $q_{0}$ increases, the position of libration point $L_{3}$ move far from the origin whereas the position of this libration point moves towards the origin when $\epsilon^{\prime}$ increases. When we have taken the effect of the centrifugal force into the consideration, it can be easily observed that for $\mu^{**}<\mu^{*}$ in all the considered three cases which means that the interval $\mu^{\in}(0,\mu^{**}]$ there exist only three collinear libration points whereas for $\mu^{\in}(\mu^{**},1)$, two more libration points occur. Therefore, the interval shrinks where the nine libration points exist and consequently the interval increases in which the fifteen libration points exist when the effect of the centrifugal force is taken into consideration.

It is observed that circular restricted five body problem has an axis of symmetry, i.e., x-axis. Indeed, if the $x-$axis is rotated successively through an angle of $2\pi/3$, two additional lines of symmetry $y=\pm\sqrt{3}$ are obtained. We have observed that two more symmetry lines of the problem exist where three or five libration points exist on each of them with respective positions as the ones on the $x-$axis. Therefore, we have a total of nine or fifteen libration points of the problem when $q_{i}=1,i=1,2,3$.

In Fig. 3, the movement of the positions of the libration of points are presented, where Fig. 3a, is illustrated for varying value of radiation parameter $q_{0}$, and Fig. 3b, is illustrated for varying value of parameter $\epsilon^{\prime}$, i.e., the effect of perturbation in the coriolis force parameter. It is further observed that as the radiation parameter $q_{0}$ increases all the nine libration points $L_{xi},i=1,2,3$ and $L_{xyi},i=1,2,...,6$ move far from the central primary whereas all these libration points move towards the central primary when the parameter $\epsilon^{\prime}$ increases.

In the case when the radiation pressure due to other primaries are taken into consideration i.e., $q_{i}\neq 1,i=1,2,3$, there exist no collinear libration points when $q_{2}\neq q_{3}$. It is observed that when $q_{2}\neq q_{3}$ there exist only five or seven non collinear libration points depending on the combination of the parameters $\epsilon^{\prime},q_{i},i=0,1,2,3$.

From Eq. 3, it is observed that the first order partial derivatives of $\Omega$ are free from the parameter $\epsilon$ controlling the Coriolis force. Therefore, the positions of the coplanar libration points remains unaffected by this perturbation parameter. However, this parameter effect significantly the stability of these libration points.

The study of the stability of libration points in any dynamical system plays a crucial role to unveil the various properties of that system. In an attempt to study the linear stability of the coplanar libration points $(x_{0},y_{0})$, a displacement is given to the positions of infinitesimal mass as $(x_{0}+\phi,y_{0}+\varphi)$ where $\phi$, and $\varphi$ denote the perturbations along the $ox$ and $oy$ axes respectively.

Therefore, expanding the equations of motion $(\ref{Eq:2a}$ and $\ref{Eq:2b})$ into first-order terms, with respect to $\phi$ and $\varphi$, we have

[TABLE]

where ${{\Omega_{xx}}_{0},\Omega_{yy}}_{0}$, and ${\Omega_{xy}}_{0}$ are the second order derivatives of $\Omega$, with respect to $x$ and $y$, evaluated at the libration points.

Substituting the solutions of the variational equations, i.e., $\phi=\xi_{1}e^{\lambda t}$ and $\varphi=\xi_{2}e^{\lambda t}$ in Eqs. (8a, 8b), we obtain:

[TABLE]

The Eqs. (9a, 9b) have nontrivial solution if the determinant of the coefficients matrix of the system vanishes. Therefore, the characteristic equation associated with the system of linear equations (9a, 9b) is quadratic in $\Xi=\lambda^{2}$ and given as follows:

[TABLE]

where

[TABLE]

The associated libration points are said to be stable if all the four roots of the characteristic equation are pure imaginary. This happens only when the conditions, illustrated below, satisfy simultaneously:

[TABLE]

Indeed, if $\lambda^{2}=\Xi$ then the characteristic equation has two real negative roots which consequently give four pure imaginary roots in $\lambda$.

In Fig. 4, the stability diagrams of the collinear libration points are illustrated. In Fig. 4a, the stability diagram is presented for the case when none of the primary is radiating. The pink colour region shows the stability region for the classical case of restricted five-body problem. It can be noticed that the only most negative libration point on the $x-$axis i.e., $L_{x3}$ is stable for $\beta^{*}=43.1810594751$, where $\beta^{*}$ is the critical value of $\beta$ such that the libration point $L_{x3}$ is stable for $\beta\geq\beta^{*}$. The green colour region shows the stability region for the case when the effect of small perturbations in the Coriolis and centrifugal forces have been taken into account. It is observed that the stable regions increase in comparison of the classical case (see [18]) and also the $\beta^{**}=9.3205312844<\beta^{*}$ i.e., the critical value of $\beta$ also decreases significantly in comparison of the $\beta^{*}$. However, for this set of values only the libration point $L_{x3}$ is stable. In Fig. 4b, we have illustrated the stability region when only the central primary is source of radiation. It is observed that as the effect due to small perturbation in the Coriolis force increases the libration points $L_{x3,4}$ become stable. The critical value of $\epsilon$ are denoted by $\epsilon^{*}$ and $\epsilon^{**}$ such that the libration point $L_{x3}$ is stable for $\epsilon\geq\epsilon^{**}$ while $L_{x4}$ is stable for $\epsilon\geq\epsilon^{*}$. Therefore, it can be concluded that the libration point $L_{x4}$ is stable for slightly higher value of $\epsilon$. In Fig. 4c, the stability regions for the collinear libration points are discussed for varying value of the parameter $\epsilon^{\prime}$ and fixed value of the other parameters. It is observed that the collinear libration point $L_{x3}$ is stable for the two different intervals.

In Fig. 4d, the stability region is depicted for varying value of the radiation parameter $q_{0}$ where the other primaries are also source of radiation. It is observed that the libration point $L_{x3}$ is stable in two intervals, i.e., $q_{0}\in(0,q_{0}^{**}]$ and $q_{0}\in[q_{0}^{*},1]$.

In Fig. 4e, the stability region is illustrated only when the central primary is radiating. It is observed that the collinear libration point $L_{x3}$ is stable for the two different intervals, i.e., $q_{0}\in[q_{0}^{***},B]$ and $q_{0}\in[A,1]$.

The movement of the positions of the libration points has been illustrated in Fig. 4f, where the radiation parameter of each primary is different, i.e., $q_{0}=0.15$, $q_{2}=0.35$, $q_{3}=0.4$, and the radiation parameter $q_{1}\in(0,1]$. It can be observed that there exist only non-collinear libration points and the number of libration points varies from five to seven as the value of the radiation parameter $q_{1}$ due to the primary $P_{1}$ increases. The stability analysis shows that the libration points $L_{xy2,3,6,7}$ are stable in some interval of radiation parameter $q_{1}$. The stable libration points are shown in green colour.

For $q_{i}=1,i=1,2,3$; $q_{0}=0.1$; $\epsilon^{\prime}=0.25$; and $\mu=0.628699732$ there exist fifteen libration points in which five are collinear i.e., $L_{xi},i=1,2,...,5$ and ten are non-collinear i.e., $L_{xyi}$, $i=1,2,...10$. When we have analyzed the effect of the parameter $\epsilon$ on the stability of the these libration points, it is unveiled that the libration points $L_{x3}$ and $L_{xy1,2}$ are stable for $\epsilon\in[\epsilon^{**}=0.370814,1]$ while $L_{x4}$ and $L_{xy7,8}$ are stable for $\epsilon\in[\epsilon^{*}=0.65071,1]$.

For $q_{i}=1,i=1,2,3$; $q_{0}=0.1$; $\epsilon=0.35$; $\mu=0.628699732$ and varying value of $\epsilon^{\prime}$, it is observed that in the interval $\epsilon^{\prime}\in(0,D^{*}]$ and $(C^{*},1]$ there exist nine libration points whereas for $\epsilon^{\prime}\in(D^{*},C^{*}]$ there exist fifteen libration points. The numerical investigations for the stability of these libration points unveil that the libration points $L_{x3},L_{xy1,2}$ are stable in $\epsilon^{\prime}\in(0,A^{*}]$, $L_{x5},L_{xy1,2}$ are stable in $\epsilon^{\prime}\in(B^{*},C^{*}]$ and $L_{x5},L_{xy7,8}$ are stable in $\epsilon^{\prime}\in(D^{*},E^{*}]$.

In figure 5, the parametric evolution of the movement of position of libration points are illustrated when the radiation parameters for the primaries $m_{2,3}$ are same and for varying value of radiation parameter $q_{0}\in(0.08,1]$. It can be observed that the positions as well as the movement of the non-collinear libration points are symmetrical about the $x-$axis. Also, all the libration points move far from the central primary as the value of the parameter $q_{0}$ increases. The thick green lines show the stable region for the libration points $L_{x3}$ and $L_{xy1,2}$. The non-collinear libration points $L_{xy1,2}$ are stable for $q_{0}\in(0.28494,1]$ with the same set of values for the other parameters while the co-linear libration point $L_{x3}$ is stable for $q_{0}\in(q_{0}^{*},1)$.

4 The regions of possible motion

The zero velocity surface (ZVS) is a three-dimensional surface defined by the relation $2\Omega(x,y,z)=C$. This ZVS provided the freedom to illustrate the energetically possible orbits of the fifth body whereas the Hill’s regions are defined as the projections of these surfaces on the configuration $(x,y)$ plane. Moreover, the zero velocity curves (ZVCs) are the locus where the kinetic energy vanishes, describe the boundaries of the Hill’s regions. Thus, we revealed that how the topology of the ZVSs, and of course the linked ZVCs change as the function of the parameters $q_{0},\epsilon^{\prime}$ and Jacobian constant $C$.

In Fig.6, we have illustrated the zero velocity curves for varying value of the parameter $\epsilon^{\prime}$ and for fixed value of mass parameter $\mu$ where none of the primary is radiating. We have fixed the value of the mass parameter $\mu$ in the interval for which there exist fifteen libration points and also the value of the Jacobian constant $C$ is taken constant. It is observed that when $\epsilon^{\prime}=0$ (see Fig. 6a), i.e., the classical case of restricted five-body problem, the primaries are surrounded by the oval shaped region, inside which the fifth particle is free to move, and consequently, the fifth body (test particle) can not communicate from one primary to either of the primary also the test particle is restricted to move from inner regions to outer region and vice-versa, i.e., there is no communication channels. In Fig. 6b, when $\epsilon^{\prime}=0.15$, the channel at the libration points $L_{x2},L_{xy3}$ and $L_{xy4}$ open and consequently the test particle can orbit to communicate between the primaries $P_{i},i=0,1,2,3$ inside the interior region but it cannot move to exterior region from it. Further increase in the value of the parameter $\epsilon^{\prime}$ to 0.153, the three more channels around the libration points $L_{x1,5,6}$ appear so that the infinitesimal mass can communicate from interior region to the exterior region and vice-versa. In the following panel, it is observed that as the parameter $\epsilon^{\prime}$ increases to 0.52, the forbidden region shrinks significantly and constitutes three branches, each containing $L_{x3,4,5}$, $L_{xy1,7,9}$, and $L_{xy2,8,10}$, respectively. Moreover, for $\epsilon^{\prime}=0.59$, three limiting situations at the libration points $L_{x5},L_{xy9}$ and $L_{xy10}$ occur while these forbidden regions further shrink and constitute six branches each containing one libration point for $\epsilon^{\prime}=0.9$ and these forbidden regions do not disappear completely even for higher value of the parameter $\epsilon^{\prime}$. In Fig. 6g and h, the regions of possible motions are illustrated for the specific value of the mass parameter $\mu$ for which nine libration points exist and observed that as the value of $\varepsilon^{\prime}$ increases, the regions of possible motion also increase but even for higher value of the $\varepsilon^{\prime}=0.8$, the forbidden regions do not disappear completely.

It can be concluded that as the value of the parameter $\epsilon^{\prime}$, which occur due to effect of small perturbation in the centrifugal force, increases the forbidden region decreases significantly and consequently the test particle can move freely on the entire configuration $(x,y)$ plane except the six small islands shaped forbidden region.

In Fig. 7, the zero velocity curves are drawn for the increasing value of the Jacobian constant. It is observed that as the value of the Jacobian constant increases, the regions of possible motion also decrease.

5 The basins of convergence

In this section, we describe how the topology of the basins of convergence linked with the coplanar libration points (i.e., the libation points lie on $(x,y)$ plane) are affected by the effect of small perturbations in the Coriolis and centrifugal forces and by the radiation parameters $q_{i},i=0,1,...,3$. We have adopted the same philosophy and procedure discussed in Ref. [32], and [22]. The basin of convergence associated with the libration points are collections of the set of points which converge to specific attractor after successive iterations. We use the multivariate version of Newton-Raphson iterative method, which is defined by the map

[TABLE]

where $\mathbf{x}=(x,y),f(\mathbf{x}_{n})$ define the system of equations 6a-6b, and $\mathbf{J}^{-1}$ is the Jacobian matrix.

In this section, we will illustrate how the radiation parameter $q_{i},i=0,1,2,3$ and the parameter $\epsilon^{\prime}$ which occurs due to effect of small perturbation in the centrifugal force, affect the topology of the Newton-Raphson basins of convergence linked with the libration points (which act as attractors) in the circular photogravitational restricted five-body problem. We have used the color-coded diagrams to classify the nodes on the configuration $(x,y)$ plane. The different colour is assigned to each pixel according to the final state of the associated initial condition.

In Fig. 8, the crooked path line is illustrated by the successive approximation points. The crooked path leads to a desired position of the equilibrium point when the iterative scheme converges for the particular initial condition. The collections of all initial conditions, which lead to same specific libration point, compose a basins of convergence or attracting domain.

The colour code for various attractors are as follows: $L_{x1}\rightarrow$ red, $L_{x2}\rightarrow$ cyan, $L_{x3}\rightarrow$ teal, $L_{x4}\rightarrow$ gray, $L_{x5}\rightarrow$ crimson, $L_{xy1}\rightarrow$ amber, $L_{xy2}\rightarrow$ black, $L_{xy3}\rightarrow$ pink, $L_{xy4}\rightarrow$ indigo, $L_{xy5}\rightarrow$ olive, $L_{xy6}\rightarrow$ yellow, $L_{xy7}\rightarrow$ lime green, $L_{xy8}\rightarrow$ purple, $L_{xy9}\rightarrow$ light blue, $L_{xy10}\rightarrow$ orange, and the non-converging points are shown in white colour.

In Fig. 9, we have depicted the basins of convergence associated with the co-planar libration points by using the multivariate version of the Newton-Raphson iterative scheme for six increasing values of the radiation parameter $q_{0}$ i.e., when the central primary is radiating and also $q_{i}=1,i=1,2,3$ i.e., the other three primaries are non-radiating bodies. In an attempt to observe the effect of the radiation parameter $q_{0}$, the mass parameter $\mu$ is fixed in the interval for which there is only nine libration points. It is observed that in all the panels the extent of the basins of convergence associated with each of the libration point is infinite however the configuration $(x,y)$ plane is fully covered by well-formed domain of the basins of convergence. Additionally, the vicinity of the basins boundaries are composed of the mixture of the initial conditions which have fractal like geometry. It is observed that the slight change in the radiation parameter $q_{0}$ leads to the significant change in the structure of the basins of convergence. In Fig. 9a, the basins of convergence is illustrated for parameter $q_{0}=0.2$, i.e., the central primary is source of radiation and the area enclosed by schematic yellow circle will be named as interior region-I, and area enclosed by schematic green circle will be named as interior region-II, from now. It is observed that, outside the interior region-I, three wings shaped area is very noisy. The domain of the basins of convergence associated with the libration points $L_{x1}$, and $L_{xy5,6}$ look like three exotic bugs, with many legs and antenna whereas the domain of the basins of convergence $L_{x3}$ and $L_{xy1,2}$ look like butterfly wings, which lie inside the region-I, are regular but their boundaries are separated by thin strips which are composed of initial conditions. The region-II is also noisy except the three leaves shaped domain of the basins of convergence associated with the libration points $L_{x2}$ and $L_{xy3,4}$.

Moreover, as the value of radiation parameter $q_{0}$ increases, the three wings shaped regions, which were noisy in previous panel and looks like chaotic sea of initial conditions, now started to convert into some small regular islands. In Fig. 9c-f, we can observe that a large area of these wings shaped regions converted to well shaped basins of convergence as the value of radiation parameter $q_{0}$ increases. Additionally, the three leaves shaped domain of the basins of convergence inside the region-II increase and most of the region inside it, is now covered by well formed basins of convergence.

In Fig. 10, the basins of convergence are illustrated as the function of parameter $\epsilon^{\prime}$ when nine libration points exist. It can be observed that the entire configuration plane is covered by well formed basins of attraction, however the domain of each of the basins of convergence has infinite extent. As we increase $\epsilon^{\prime}$, the interior region-I, shrinks whereas the number of leaves in the wings shaped region increases. Moreover, we observe the vicinity of the basins boundaries are composed of the mixture of initial conditions. This fact unveils that the basins boundaries are highly chaotic and hence, the initial conditions, inside these areas are extremely sensitive to its final state. Indeed, even a very slight change in the value of initial conditions will lead to a entirely unexpected attractor. The domain of the basins of convergence associated with the libration points $L_{x1},L_{xy5,6}$ look like exotic bugs with many legs and antenna, shrink as the parameter $\epsilon^{\prime}$ increases.

In Fig. 11, the basins of convergence are depicted for varying values of the radiation parameter $q_{0}$, when fifteen libration points exist. The configuration plane is covered by well-formed basins of attraction whereas the basins boundaries are composed of the mixtures of the initial conditions. When we compare the Figs. 11a and b, it can be observed that the three heart shaped regions, which occur at the joining point of wing shaped area, separate into two parts and consequently hinges which connect the interior region-I and the wings shaped region shrinks. Moreover, the area inside the interior region-II becomes more regular in comparison of panel-b. Furthermore, the increase in the parameter $q_{0}=0.95$ the basins boundaries which connect the leaves of the wings are now regular and the chaotic basin boundaries disappear.

The basins of the convergence associated with the libration points, for the case when all the primaries are radiating and the values of these radiation parameters are different, are illustrated in Fig. 12. As we have observed that in this case the symmetry of the libration points about the $x-$axis is destroyed and only five non-collinear libration points exist, the basins of convergence also look strange. We observe that a majority of the $(x,y)$ plane is covered by the highly fractal111The term fractal simply unveils the fact that the specific area has a fractal-like geometry, without conducting extra calculations, as in [6]. mixture of initial conditions. The only regular island shown in* green* and red colour corresponds to those libration points which exist near the origin, i.e., $L_{xy2,3}$ (see Fig. 4). However, the basins of convergence associated with the libration points $L_{xy1,4,5}$ look like exotic bugs without legs and antennas. The Fig. 12b and c look like mirror images about the $x-$axis when the green and red colors are interchanged. This happens only because the value of the radiation parameter $q_{2}$ and $q_{3}$ are interchanged.

6 Discussions and conclusions

In the present manuscript, we have successfully illustrated the influence of the effect of small perturbations in the Coriolis and centrifugal forces on the positions of the libration points, their stability and the regions of possible motion in context of photogravitational version of restricted five-body problem. We have further, investigated numerically the basins of convergence, linked to the libration points, in the problem. We have used the multivariate version of the Newton-Raphson iterative scheme to unveil the associated basins of attraction on the configuration $(x,y)$ plane. The role of the attracting domains is very crucial as it can explain how each initial point of the $(x,y)$ plane is attracted by the attractors of the dynamical system, which are, indeed, a libration point of the system. We have successfully managed to detect how the geometry of the Newton-Raphson basins of attraction changes as a function of the radiation parameters and the $\epsilon^{\prime}$.

In the problem, we have taken either the central or all of the primaries as source of radiations. The main conclusions can be summarized as follows:

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The number as well as the positions of the libration points are highly sensitive with the change in the value of the parameter $\epsilon^{\prime}$. The value of $\mu^{*}$, ( i.e., the value of mass parameter is critical value since it delimits the point where the number of the equilibrium points changes) decreases as $\epsilon^{\prime}$ increases.

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If only the central primary radiates, i.e., $q_{i}=1,i=1,2,3,$ and other perturbations are neglected, all the libration points move away along a straight line from the central primary when $q_{0}\in(0,1]$ whereas, all the libration points move towards it along a straight line if the effect in the centrifugal force increases where none of the primary is radiating.

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The non collinear libration points $L_{xyi},i=1,2,3,4$ move away from the central primary along a curve whereas the other libration points move along the straight line when the radiation parameter $q_{0}$ increases where the other primaries are also the source of radiation such that $q_{1}\neq q_{2}=q_{3}$.

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The stability analysis of the libration points have unveiled that in most of the cases only the collinear libration point $L_{x3}$ is stable. Moreover, for various combination of parameters, the libration points $L_{x3,4,5}$ and $L_{xy7,8,9,10}$ are also stable sometime or another. It is interesting to note that only those non-collinear libration points are stable, which lie in the first or fourth quadrant.

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If we compare our result from the classical five-body problem (see [18]), it is observed that the critical value of $\beta$ decreases significantly when the effect of the Coriolis and centrifugal forces is taken into consideration.

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If we increase the parameter $\epsilon^{\prime}$, the possible regions of motion increase whereas they decrease when the Jacobian constant $C$ increases. Further, these regions are unaffected by the change in value of $\epsilon$, i.e., parameter occur due to presence of the Coriolis force as the effective potential is independent from this parameter.

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The basins of convergence associated with all the libration points have infinite extent. The the interior region-I, interior region-II, and the wing shaped region become more regular as the radiation parameter due to the central primary increases. Moreover, the entire basins of convergence shrinks in whole as the parameter $\epsilon^{\prime}$ increases while the basins of convergence remains unaffected with the change in the value of $\epsilon$ which control the Coriolis force.

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The majority of the area of basins of convergence look like chaotic sea, composed of different type of initial conditions when the value of the radiation parameters $q_{i},i=0,1,..,3$ are different. Therefore, in these cases it is almost impossible to predict that which initial conditions will converge to which of the libration points.

The latest version 11.3 of Mathematica® is used for the graphical illustration of the paper. It is interesting to use other types of iterative schemes in the present problem to discuss the similarities and differences in the context of the basins of convergence linked with the libration points of the dynamical system. We believe that the above-mentioned results and ideas will be useful in the active field of attracting domains of libration points in various dynamical system.

Compliance with Ethical Standards

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Funding: The authors state that they have not received any research grants.

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Conflict of interest: The authors declare that they have no conflict of interest.