1 Introduction

The three-body problem is a complex dynamic problem, which refers to the motion of three particles interacting in a particle system first proposed by the French astronomer Laplace in the 18th century when studying the planetary motion in the solar system. Later, French mathematician Poincaré conducted extensive research on this issue. Due to the inability to obtain analytical solutions for the general three-body problems, a new branch of the R3BP (abbreviated form of restricted three-body problem) is obtained by simplifying the problem. The R4BP refers to the motion of an infinitesimal mass under the gravitational force of three large celestial bodies, often called the primaries, and the tiny body does not affect the primaries’ motion. Furthermore, if the mass of an infinitesimal body changes over time, the problem becomes a variable-mass R4BP.

The study of Lagrangian points is of great significance in the study and application of celestial mechanics, including in variable-mass R4BP. The Lagrangian point refers to the points in a celestial system where an object can remain stationary relative to the system reference frame. Italian mathematician Joseph Lagrange first studied it, which has great significance and applications. For example, in a three-body system such as the Sun, Earth, and Moon, five Lagrangian points can be used to place space stations, artificial satellites, astronomical telescopes, and so on [1, 2], which are very useful for aerospace dynamics and deep space exploration. In recent years, scientists have conducted many studies on Lagrangian points and made many of the latest advances [3].

The numerical methods are used to study how the oblate primary and prolate primary parameters affect Lagrangian points’ positions and linear stability [4, 5]. The Lagrangian point dynamics of the R3BP with equally massed prolate radiating bodies were studied in [6]. The linear stability and positions of the co-planar Lagrangian points were determined using numerical methods. The results showed that these two parameters have a great influence on the system’s Lagrangian point dynamics. In [7], the authors mainly studied the linear stability of Lagrangian points in the generalized photo-gravitational Chermnykh-like issue with the power-law. The positions and velocity sensitivities under the effect of radiation and oblate primaries in the R3BP are also studied in [8]. Recently in [9], the authors considered the elliptical R3BP under radiation pressure and the primaries oblateness perturbation, obtained the location of non-collinear Lagrangian points and approximate analytical solutions nearby, and applied them to real astronomical systems. It was found that the stability and locations of the Lagrangian points will be significantly affected by changes in disturbance parameters. While within the framework of quantized Hill’s three-body problem, the stability of equilibrium points and their in-plane and out-of-plane motion was also studied [10].

In the framework of R4BP, the position and existence of the Lagrangian points are studied under the effect of non-spherical shapes of the primaries in [11, 12]. The stability of these points is also investigated with photo-gravitational and Stokes drag perturbations, where three primary bodies have radiation, and the Stokes force is a dissipative force in [13]. Many studies have addressed on this problem under the effect of various perturbations. For example, but not limited to, the existence and position of Lagrangian points in variable mass, as well as the zero-velocity curve, were studied in [14], where the position lines of the three primary bodies form an equilateral triangle and the second and third bodies have the same mass. It was found that there are eight Lagrangian points, and the parameter of variable mass affects the position of the points. While considerable studies are preformed in [15, 16] under the effect of photo-gravitational and Stokes forces to analysis these effects on the positions change of Lagrangian points and the variation of the zero-velocity curves.

The periodic solution of R3BP provides a theoretically rich and complex system, and the study of periodic solutions can be useful for planning space missions. Some artificial satellites and space probes are placed on Lagrangian points or other stable periodic solutions better to observe targets such as the Sun and Earth or to provide a more stable environment for scientific experiments. These points and related periodic solutions are evaluated as the optimal positions to transfer the spacecraft to the nominal periodic solution or related stable manifold. In this context, the heteroclinic connections between quasi-periodic solutions are calculated [17]. Also outstanding study are carried out to analysis the positions of collinear Lagrangian points and their periodic solutions under the effect of triaxial rigid body parameters in the R3BP, and numerical simulation results were provided [18]. But the periodic solutions in the framework of quantized R3BPs are investigated [19]. The LP and differential corrector methods are used to find a family of halo solutions at artificial Sun–Earth \(L_2\) points [20]. Furthermore, periodic solutions and bifurcation analysis of R3BPs under triaxial and radial perturbations are calculated in [21]. Also, the LP method is used to calculate the approximate analytical periodic solutions of R3BP under oblateness, radiation pressure, and variable mass perturbations [22].

The analysis of periodic solutions is not exclusive to R3BP but also a great appearance in four-body problem. In [23], the authors used numerical methods to study the existence of periodic solutions for R4BPs with equilateral triangle configurations. While in [24], the authors used the Fourier series method to determine the periodic solutions around collinear Lagrangian points in the spatial collinear R4BP with non-spherical primaries. Recently in [25], the authors studied perturbed two-body solutions and R3BPs orbits and proved that if the perturbation force is conserved or the corresponding motion has its extended Jacobian integral, the first and second types of orbits in the rotating Kepler problem will always exist. In [26], the authors studied the perturbed relative motion, obtained the interior loops of the periodic solutions using the numerical method, and analyzed its stability. In addition, the Poincaré surface of sections is used to study the stability of periodic solutions under the parameters of the eccentricity of the primaries’ trajectory, solar radiation pressure, and the Jacobian constant perturbations [27]. Some interesting research that enriches readers’ knowledge of space dynamics and celestial mechanics are addressed in [28,29,30,31,32,33,34,35,36,37,38].

When studying Lagrangian points in R4BPs, existing literature has predominantly focused on individual perturbation factors such as photo-gravitational effects, variable mass, and Stokes drag, with limited attention given to their combined influence. While scholars have extensively examined periodic solutions in proximity to collinear Lagrange points, there remains a notable gap in the exploration of periodic solutions near non-collinear Lagrangian points. In light of this, our study aims to analyze Lagrangian points and periodic solutions near non-collinear points, considering the combined perturbations of photo-gravitational forces, variable mass effects, and Stokes drag. We anticipate that this study will contribute helpful insights to this domain.

In this paper, Sect. 1 is introduction. Section 2 presents the motion equations of the variable-mass R4BP under photo-gravitational and Stokes force perturbations. Section 3 calculates the Lagrangian points and their evolution under the variation of variable mass, photogravitational, and Stokes force parameters. Section 4 is about the linear stability of Lagrangian points. Section 5 is the zero-velocity curve, and the Sect. 6 calculates periodic solutions near non-collinear Lagrangian points. The last Sect. 7 is the conclusion.

2 Equations of motion

We will now explore the motion of variable mass within the framework of a R4BP in a Lagrangian configuration. Here, the three primaries involved are radiating and located at the vertices of an equilateral triangle. Let \(m_i\) \((i=1,2,3)\) represent the masses of the three primaries, assuming that \(m_1 \ge m_2 = m_3\), and they are moving along circular orbits around their common center of masses. We denote the mass of a fourth body by m, which is negligible compared to the primaries’ masses. In this context, the fourth body is moving under the gravitational attraction forces of the three primaries without affecting their motion.

Furthermore, we have chosen the combined mass of the primaries and the distance between them as our units for mass and length and the time unit is selected to set the gravitational constant to unity. Assuming \(m_2/(m_1+m_2+m_3) = m_3/(m_1+m_2+m_3) = \mu \), \(m_1+m_2+m_3=1\), then \(m_1 = 1 - 2\mu \). In synodic coordinates, we impose that also the coordinates of the infinitesimal body (fourth body) are represented by (X, Y), while those of the primaries, denoted as \((X_i, Y_i)\), where

$$\begin{aligned} \begin{aligned}&(X_1, Y_1)=(\sqrt{3}\, \mu , 0), \\&(X_2, Y_2)=\left( -\dfrac{\sqrt{3}}{2}(1-2\, \mu ), -\dfrac{1}{2}\right) ,\\&(X_3, Y_3)=\left( -\dfrac{\sqrt{3}}{2}(1-2\, \mu ), \dfrac{1}{2}\right) . \end{aligned} \end{aligned}$$
(1)

Thus, the equations of motion of the infinitesimal mass m under the effects of photo-gravitational and Stokes drag in a rotating coordinate can be described in the dimensionless variables as [13, 16]

$$\begin{aligned} \begin{aligned}&\ddot{X} - 2\dot{Y} + \frac{\dot{m}}{m}(\dot{X} - Y) = W_X + S_X,\\&\ddot{Y} + 2\dot{X} + \frac{\dot{m}}{m}(\dot{Y} + X) = W_Y + S_Y, \end{aligned} \end{aligned}$$
(2)

where the function W is defined by

$$\begin{aligned} W(X,Y)= & \dfrac{1}{2}(X^2 + Y^2)+\dfrac{(1 - 2 \mu ) q_1}{d_1}\nonumber \\ & +\dfrac{\mu q_2}{d_2} + \dfrac{\mu q_3}{d_3}, \end{aligned}$$
(3)

here \(q_i\in (0, 1]\ (i=1,2,3)\) being constant implies the neglect of fluctuations in radiation beams, the shadow effect of primaries, and the assumption of purely radial radiation. The radiation pressure, denoted as \(F_{p_i}\), acting on a primary can be expressed in terms of the gravitational attraction force, \(F_{g_i}\), as \(F_{p_i}=F_{g_i}(1-{q_i})\). Here, \(F_{p_i}\) represents the radiation pressure due to the primary, and \(F_{g_i}\) signifies the gravitational force acting on the primary. The parameter \({q_i}=1-{p_i}=1-F_{p_i}/F_{g_i}\), a constant specific to the given primary, serves as a reduction factor determined by the primary’s radius a, density \(\delta \), and radiation-pressure efficiency factor x in the cgs system, where \(q_i=1-5.6 \times 10^{-3}x/(a\delta )\) (See [39] and the references therein for more details).

By using Eq. (1), the separation distances among the infinitesimal mass and primaries \(d_1\), \(d_2\), and \(d_3\) are defined by

$$\begin{aligned} \begin{aligned} d_1^2&=\left( X-\sqrt{3}\mu \right) ^2+Y^2,\\ d_2^2&=\left[ X+\dfrac{\sqrt{3}}{2}(1-2\mu )\right] ^2+\left( Y+\dfrac{1}{2}\right) ^2,\\ d_3^2&=\left[ X+\dfrac{\sqrt{3}}{2}(1-2\mu )\right] ^2+\left( Y-\dfrac{1}{2}\right) ^2. \end{aligned} \end{aligned}$$
(4)

The perturbed acceleration components of the Stokes drag force in the X and Y directions are given by Murray [40]

(5)

where H and d are two functions in variables X and Y and are defined by

$$\begin{aligned} \begin{aligned} H(X,Y)=&\, \left( X^2+Y^2\right) ^{-3/4}, \\ d(X,Y)=&\sqrt{X^2+Y^2}, \end{aligned} \end{aligned}$$
(6)

k is the dissipative constant with rang values Beaugé (1993) \((0<k<1)\) [41], \(\sigma \) is the ratio of the gas and Kepler velocities Murray [40].

The Jeans’ law and Meshcherskii space-time transformations preserve the dimension of space and time. Now, we impose that the mass of the infinitesimal body varies with time t according to Jean’s law: \(d m/dt =-\alpha \, m^n\), where \(\alpha \) is constant and the value \(n \in [0.4, 4.4]\) (for the star of the main sequence). For a rocket, \(n = 1\), and the mass of the rocket varies exponentially as \(m=m_0 e^{-\alpha \, t}\), \(m_0\) is the initial mass. The classic law mainly taken from James Jeans’ monumental work [42]. Despite its empirical basis, a meticulous examination of the monograph reveals a minimal deviation between theoretical calculations and observed data. This precision underscores its status as a classic law that continues to be widely utilized. In recent years, researchers have utilized nonparametric statistical testing methods, rather than empirical approaches, to analyze and interpret natural satellite data, resulting in the derivation of alternative variable mass laws with high credibility. For further details, please refer to [43,44,45,46,47].

Now, the Meshcherskii space-time transformations are used to simplify the equations: \(u=X \gamma ^q\), \(v=Y \gamma ^q\), \( d \tau =\gamma ^{\bar{k}}dt\), \(d_i=\gamma ^{-q} R_i\), \(i=1,2,3\), where \(\gamma \) represents the ratio of the primary’s mass at time t to its initial mass [48].

Let \(n=1, \bar{k}=0, q=1/2\) in keeping with the work by Singh and Ishwar [49], the velocity and acceleration components can be written as

$$\begin{aligned} \begin{aligned} \gamma ^{1/2}\dot{X} =&\, u'+\dfrac{1}{2} \alpha \, u, \\ \gamma ^{1/2}\dot{Y} =&\, v'+\dfrac{1}{2} \alpha \, v, \\ \gamma ^{1/2}\ddot{X} =&\, u''+ \alpha u' + \dfrac{1}{4} \alpha ^2 \,u,\\ \gamma ^{1/2}\ddot{Y} =&\, v''+\alpha v' + \dfrac{1}{4} \alpha ^2 \, v, \end{aligned} \end{aligned}$$
(7)

where the dot (.) and prime \((')\) represent the derivative with respect to t and \(\tau \), respectively, after utilizing Eqs. (2–7), then equations of motion can be rewritten in the following form

$$\begin{aligned} \begin{aligned}&\ddot{u} - 2\dot{v} = \Omega _u+S_u,\\&\ddot{v} + 2\dot{u} =\Omega _v+S_v, \end{aligned} \end{aligned}$$
(8)

where

$$\begin{aligned} \begin{aligned} \Omega (u,v) =&\,\dfrac{1}{2} \left( 1+\dfrac{\alpha ^2}{4} \right) \left( u^2 + v^2\right) \\&+\gamma ^{3/2} \left[ \dfrac{(1-2 \mu ) q_1}{R_1} + \dfrac{\mu q_2}{R_2} + \dfrac{\mu q_3}{R_3}\right] , \end{aligned} \end{aligned}$$
(9)
(10)

and

$$\begin{aligned} \begin{aligned} R_0 =&\, \sqrt{u^2 + v^2}, \\ R_1^2 =&\, \left( u - \sqrt{3}\mu \sqrt{\gamma }\right) ^2 + v^2,\\ R_2^2 =&\, \left[ u+\dfrac{\sqrt{3}}{2}(1-2\mu )\sqrt{\gamma } \right] ^2 + \left( v+\dfrac{\sqrt{\gamma }}{2}\right) ^2,\\ R_3^2 =&\, \left[ u+\dfrac{\sqrt{3}}{2}(1-2\mu )\sqrt{\gamma } \right] ^2 + \left( v - \dfrac{\sqrt{\gamma }}{2}\right) ^2. \end{aligned} \end{aligned}$$
(11)

System (8) represents a generalized dynamical system for the motion of variable mass in the framework of R4BP under the perturbation effects of Stokes drag and photo-gravitational forces. Its mass variation is also a source for motion’s perturbation. This system reduced to many sub-models or special cases as follows:

  • When \(q_1=q_2=q_3=1\), the equations are convenient with the obtained system in [16].

  • When \(k=0\), the equations are convenient with the obtained system in [15].

  • When \(\alpha =0,\ {\gamma =1}\), the equations are convenient with the obtained system in [13].

  • When \(k=0,\ q_1=q_2=q_3=1\), the equations are convenient with the obtained system in [14].

By multiplying the first and second equations of (8) by \(2 \dot{u}\) and \(2 \dot{v}\), respectively, and adding them together, after integration, we obtain a quasi-integral for motion which is similar to Jacobian integral.

$$\begin{aligned} \dot{u}^2+\dot{v}^2=2 (S+\Omega )-C -2\int _0^t\dfrac{\partial }{\partial t}(S+\Omega )dt, \end{aligned}$$
(12)

where C denotes the Jacobian integral constant when the last term in Eq. (12) is either equals zero or can be calculated.

3 Lagrangian points

Investigating the Lagrangian points and their stability is very meaningful because many qualitative analyses of dynamic systems are carried around them, including the computation of periodic solutions in their vicinity. A Lagrangian point refers to a location where both velocity and acceleration are zero, i.e. \(\dot{x}=\dot{y}=\ddot{x}=\ddot{y}\). Lagrangian points lie on the u-axis are called the collinear Lagrangian points, otherwise called the non-collinear ones. In this section, we will investigate the Lagrangian points of the infinitesimal mass body with photo-gravitational, Stokes drag, and variable mass perturbations

Substituting Eqs. (9–11) into righthand sides of Eq. (8), we get

$$\begin{aligned} \begin{aligned} \Omega _u+S_u =&\, T_{u1} + T_{u2} +T_{u3} + T_{u4} +T_{u5},\\ \Omega _v+S_v =&\, \, T_{v1} + T_{v2} +T_{v3} + T_{v4} +T_{v5}, \end{aligned} \end{aligned}$$
(13)
Fig. 1
figure 1

The positions of Lagrangian points with \(\gamma =0.15,\alpha =2.2,\sigma =0.018,k=0.000005,q_1=0.98,q_2=0.975,q_3=0.975\), and different mass ratio parameters \(\mu \)

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where

$$\begin{aligned} \begin{aligned} T_{u1} =&\, \left( \dfrac{\alpha ^2}{4}+1\right) u, \\ T_{u2} =&\, \dfrac{ \gamma ^{3/2} q_1 (2 \mu -1)\left( u-\sqrt{3\,\gamma } \mu \right) }{\left[ \left( u- \sqrt{3 \,\gamma } \mu \right) ^2+v^2\right] ^{3/2}}, \\ T_{u3} =&\,\dfrac{\gamma ^{3/2} q_2\, \mu \left[ \dfrac{ \sqrt{3\,\gamma }}{2} (2 \mu -1)-u\right] }{\left[ \left( \dfrac{\sqrt{3\,\gamma }}{2} (1-2 \mu )+u\right) ^2+\left( v + \dfrac{\sqrt{\gamma }}{2}\right) ^2\right] ^{3/2}}, \\ T_{u4} =&\, \dfrac{\gamma ^{3/2} q_3 \,\mu \left[ \dfrac{\sqrt{3\,\gamma }}{2}(2 \mu -1 )-u\right] }{\left[ \left( \dfrac{\sqrt{3\,\gamma }}{2} (1-2 \mu )+u\right) ^2+\left( v-\dfrac{\sqrt{\gamma }}{2}\right) ^2\right] ^{3/2}}, \\ T_{u5} =&\,k \left[ \dfrac{\alpha \,u}{2} - v \left( 1+ \dfrac{3 \gamma ^{7/4} \sigma }{2 \left( u^2+v^2\right) ^{7/4}}\right) \right] , \end{aligned} \end{aligned}$$
(14)

and

$$\begin{aligned} \begin{aligned} T_{v1} =&\, \left( \dfrac{\alpha ^2}{4}+1\right) v, \\ T_{v2}=&\, \dfrac{\gamma ^{3/2} q_1 (2 \mu -1 ) v}{\left[ \left( u- \sqrt{3\,\gamma } \mu \right) ^2+v^2\right] ^{3/2}},\\ T_{v3} =&\,\dfrac{- \, \gamma ^{3/2} q_2 \, \mu \left( v +\dfrac{\sqrt{\gamma }}{2}\right) }{\left[ \left( \dfrac{\sqrt{3\,\gamma }}{2} (1-2 \mu ) +u\right) ^2+\left( v+ \dfrac{\sqrt{\gamma }}{2}\right) ^2\right] ^{3/2}},\\ T_{v4} =&\, \dfrac{-\, \gamma ^{3/2} q_3 \, \mu \left( v-\dfrac{\sqrt{\gamma }}{2}\right) }{\left[ \left( \dfrac{\sqrt{3\,\gamma }}{2}(1-2 \mu ) +u\right) ^2+\left( v-\dfrac{\sqrt{\gamma }}{2}\right) ^2\right] ^{3/2}},\\ T_{v5} =&-k \left[ \dfrac{\alpha \, v}{2}+u \left( 1+ \dfrac{3 \gamma ^{7/4} \sigma }{2 \left( u^2+v^2\right) ^{7/4}}\right) \right] . \end{aligned} \end{aligned}$$
(15)

Now, substitute Eqs. (14, 15) into Eq. (13) and let \(\Omega _u+S_u=0\) and \(\Omega _v+S_v =0\). Then, the obtained roots are the Lagrangian points.

The parameters’s values are within the range \(0<\mu \le 1/3\), \(0<\gamma <1\), \(0<\alpha \le 2.2\), \(0<\sigma <1\), \(0<q_1,q_2,q_3\le 1\). Constraining \(\alpha \) within this specified range aims to effectively capture the intricate dynamic characteristics and evolutionary processes within the celestial system, thereby ensuring the rationality and accuracy of the model employed. For further details, please refer to [14, 50, 51] and the references therein.

Fig. 2
figure 2

The positions of Lagrangian points for \(\mu =0.019\), \(\gamma = 0.4\), \(\alpha = 0.6\), \(\sigma = 0.05\), \(k = 0.00015\), \(q_ 2 =q_ 3= 0.9985\), for different \(q_1\) values, \(q_ 1 = 0.6 \) (blue, green), \(q_ 1 = 0.7 \) (blue, gray), \(q_ 1 = 0.8 \) (blue, orange), \(q_ 1 = 0.9 \) (blue, purple), \(q_ 1 = 1 \) (blue, pink). (Color figure online)

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We employed a numerical method to determine the positions of Lagrangian points within the proposed system. Typically, there exist eight or ten Lagrangian points, all of which are non-collinear. The number of Lagrangian points obtained varies with different parameter values, as illustrated in Figs. 1, 2, 3, 4 and 5. In these figures, the blue dots represent the primaries, while the red dots indicate the Lagrangian points.

Fig. 3
figure 3

The positions of Lagrangian points for \(\mu =0.01\), \(\gamma = 0.6\), \(\alpha = 0.4\), \(\sigma = 0.05\), \(k = 0.00001\), \(q_ 2 = 0.8\), \(q_ 3 = 0.6\), \(q_ 1 = 0.6 \) (blue, green), \(q_ 1 = 0.7 \) (blue, gray), \(q_ 1 = 0.8 \) (blue, orange), \(q_ 1 = 0.9 \) (blue, purple), \(q_ 1 = 1 \) (blue, pink). (Color figure online)

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Fig. 4
figure 4

The positions of Lagrangian points for \(\mu =0.019\), \(\gamma =0.1,\sigma =0.05\), \(q_1=0.95\), \(q_2=0.9\), \(q_3=0.9\), \(k=0.00015\), \(\alpha = 0.2\) (blue, green), \(\alpha = 0.6\) (blue, gray), \(\alpha = 1 \) (blue, orange), \(\alpha = 1.5 \) (blue, purple), \(\alpha = 2.2 \) (blue, pink). (Color figure online)

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In Fig. 1, set \(\gamma =0.15,\alpha =2.2,\sigma =0.018,k=0.000005,\) \(q_1=0.98,q_2=0.975,q_3=0.975\). When \(\mu =0.015\), there are eight non-collinear Lagrangian points in Fig. 1a and no collinear Lagrangian points. When \(\mu \) increased to 0.2 and 0.32, there were still only eight non-collinear Lagrangian points, as shown in Fig. 1b, c. However, when \(\mu =0.33\), ten non-collinear Lagrangian points appeared, adding two more non-collinear Lagrangian points, \(L_1\) and \(L_9\), as shown in Fig. 1d. Figure 1a–d shows the process of the number of Lagrangian points increasing from eight to ten with increasing \(\mu \) value.

In Fig. 2, we present the positional evolution of the Lagrangian points as the parameter \(q_1\), given \(\mu =0.019,\ \sigma =0.05, \gamma =0.4\), \(\alpha =0.6\), \(k=0.00015\), \(q_2=q_3=0.9985\). In this case, there are eight non-collinear points, and all Lagrangian points are not symmetric about the u-axis. When \(q_1\) increases at (0.6,1), the Lagrangian points \(L_1\), \(L_2\), \(L_7\), and \(L_8\) move away from the bigger primary \(m_1\) in Fig. 2a. \(L_3\) moves towards \(m_3\) and \(L_5\) moves away from \(m_3\) in Fig. 2b. \(L_4\) moves towards \(m_2\) and \(L_6\) moves away from \(m_2\) in Fig. 2c. In addition, we also investigate the case of primaries \(m_ 2\) and \(m_ 3\) having different radiation effects (\(q_2\ne q_3\)), and the Lagrangian points under the conditions are shown in Fig. 3. When \(q_1\) increases at (0,1) and fixed parameters of \(\mu =0.01\), \(\gamma = 0.6\), \(\alpha = 0.4\), \(\sigma = 0.05\), \(k = 0.00001\), \(q_ 2 = 0.8\), \(q_ 3 = 0.6\), the Lagrangian points undergo a similar change in Fig. 3a–d.

The evolution of Lagrangian points is studied when \(\mu =0.019,\ \sigma =0.05\), \(\gamma =0.1\), \(q_1=0.95\), \(q_2=0.9\), \(q_3=0.9\), \(k=0.00015\), and \(\alpha \) increases at (0.2,2.2) in Fig. 4. It can be observed that eight non-collinear points appear at this parameter value. As the \(\alpha \) increases, all Lagrangian points move towards the origin, and the point \(L_3\) moves toward \(m_3\) and \(L_5\) moves away from \(m_3\) in Fig. 4b. \(L_4\) moves towards \(m_2\) and \(L_6\) moves away from \(m_2\), as shown in Fig. 4c. The Lagrangian points \(L_1\) and \(L_2\) move toward the primary \(m_1\) on both sides of the origin near the u-axis in Fig. 4d–e, respectively.

In Fig. 5, the Lagrangian points are identified under the effect of k variation, with fixed values for \(\mu =0.019,\ \sigma =0.05\), \(\gamma =0.12\), \(\alpha =0.19\), \(q_1=0.99\), \(q_2=0.98\), \(q_3=0.98\) and \(k \in (0.05,0.4)\), six non-collinear points were found, and as the value of k increases, \(L_1\) moves away from the primary \(m_3\) and upwards to the left, \(L_3\) moves away from primary \(m_3\) and upwards to the right. \(L_5\) moves towards primary \(m_3\) and downwards to the left. These three Lagrangian points tend to merge. \(L_2\) moves downwards to the primary \(m_2\), while \(L_4\) and \(L_6\) move away from primary \(m_2\) and upwards to the right and left, respectively. These three Lagrangian points also tend to merge.

4 Linear stability analysis of the Lagrangian points

Only the Lagrangian points around which an infinitesimal body can maintain motion are stable. Otherwise, it is unstable. Here, we have referred to the literature in [14]. If the coordinates of the Lagrangian points are \((u_0,v_0)\), and (x, y) is the small displacement relative to the Lagrangian point, then this small displacement can be written as \(x=u-u_0,y=v-v_0\). We expand the right-hand side of Eq. (8) to first-order by Taylor series, then Eq. (8) can be linearized to the following system

$$\begin{aligned} \begin{aligned}&\ddot{x}-2 \dot{y}= \left( S_{uu}+\Omega _{uu}\right) _0 x+ \left( S_{uv}+\Omega _{uv}\right) _0 y -k \dot{x},\\&\ddot{y}+2 \dot{x}= \left( S_{vu}+\Omega _{vu}\right) _0 x+ \left( S_{vv}+\Omega _{vv}\right) _0 y -k \dot{y}, \end{aligned}\nonumber \\ \end{aligned}$$
(16)

where, subscript (0) represents the partial derivative at the Lagrangian point, let \(\left( S_{uu}+\Omega _{uu}\right) _0=N_1\), \(\left( S_{uv}+\Omega _{uv}\right) _0=N_2\), \(\left( S_{vu}+\Omega _{vu}\right) _0=F_1\), \(\left( S_{vv}+\Omega _{vv}\right) _0=F_2\) in “Appendix I”. For the problem of variable mass, the location of the primary will change over time t, and their distances to the Lagrangian point \((u_0,v_0)\) decrease with time t. Therefore, conventional methods cannot determine linear stability.That is why we have used Meshcherskii space time transformations \(u=X \gamma ^{\frac{1}{2}},v=Y \gamma ^{\frac{1}{2}}\). This fixes the positions of the primaries and the distance to the Lagrangian point. Let \(\dot{x}=x_1,\dot{y}=y_1\), Eq. (16) in phase-space as

$$\begin{aligned} \begin{aligned} \dot{x_1}-2 y_1&= \left( S_{uu}+\Omega _{uu}\right) _0 x+ \left( S_{uv}+\Omega _{uv}\right) _0 y -k x_1,\\ \dot{y_1}+2 x_1&= \left( S_{vu}+\Omega _{vu}\right) _0 x+ \left( S_{vv}+\Omega _{vv}\right) _0 y -k y_1. \end{aligned}\nonumber \\ \end{aligned}$$
(17)

Consider the Meshcherskii inverse transform, and take \(X '=\gamma ^{-1/2} x,Y '=\gamma ^{-1/2} y,x'=\gamma ^{-1/2} x_1,y'=\gamma ^{-1/2} y_1\). The Eq. (17) can be rewritten as the matrix form

$$\begin{aligned} \begin{bmatrix} \dfrac{dX'}{dt} \\ \dfrac{dY'}{dt} \\ \dfrac{dx'}{dt} \\ \dfrac{dy'}{dt} \end{bmatrix} =\begin{bmatrix} \dfrac{\alpha }{2} & 0 & 1 & 0 \\ 0 & \dfrac{\alpha }{2} & 0 & 1 \\ N_1 & N_2 & \dfrac{\alpha }{2}-k & 2 \\ F_1 & F_2 & -2 & \dfrac{\alpha }{2}-k \end{bmatrix} \begin{bmatrix} X' \\ Y' \\ x' \\ y' \end{bmatrix}. \end{aligned}$$
(18)
Fig. 5
figure 5

The positions of Lagrangian points for \(\mu =0.019\), \(\gamma =0.12\), \(\alpha =0.19\), \(\sigma =0.05\), \(q_1=0.99\), \(q_2=0.98\), \(q_3=0.98\) when \(0.05\le k\le \ 0.4.\)

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The linear stability of Eqs. (8) and (18) is consistent with each other. We determine the linear stability of the Lagrangian points by solving the eigenvalues of the coefficient matrix of Eq. (18) numerically. The characteristic equation of the linearized Eq. (18) corresponding to the equilibrium point is

$$\begin{aligned}&\lambda ^4+ 2(k-\alpha )\lambda ^3+ \left[ f_1+\frac{1}{2} (3 \alpha ) (\alpha -2 k)\right] \lambda ^2 \nonumber \\&- \left[ \alpha f_1+f_2-\dfrac{1}{2} \left( 3 \alpha ^2\right) (k-\alpha )\right] \lambda \nonumber \\&+\dfrac{\alpha ^4}{16}-\dfrac{1}{4} \left( \alpha ^3 k\right) \nonumber \\&+\dfrac{\alpha ^2 f_1}{4}+\dfrac{\alpha f_2}{2}+f_3=0, \end{aligned}$$
(19)

where

$$\begin{aligned} \begin{aligned}&f_1=-F_2+k^2-N_1+4,\\&f_2=k \left( F_2+N_1\right) +2 F_1-2 N_2,\\&f_3=F_2 N_1-N_2F_1. \end{aligned} \end{aligned}$$
(20)

If four complex roots of the characteristic Eq. (19) all have negative real parts or are pure imaginary roots, then the corresponding Lagrangian point is asymptotically stable or stable. If there are roots with positive real parts, it is unstable. Under the influence of photo-gravitational, variable mass, and Stokes drag, all Lagrangian points are unstable.

5 Zero-velocity curves

This section studies the zero-velocity curve of the variable mass R4BP under the perturbation of photo-gravitational and Stokes drag. The expression for the possible motion region of the fourth body is

$$\begin{aligned} \begin{aligned} (\Omega +S )(\gamma )&\,\ge C(\gamma )\quad \text {for} \quad \dfrac{\partial }{\partial \gamma }(\Omega +S) \geqslant 0,\\ (\Omega +S )(\gamma _0)&\, \ge C(\gamma _0)\quad \text {for} \quad \dfrac{\partial }{\partial \gamma }(\Omega +S )\leqslant 0, \end{aligned} \end{aligned}$$
(21)

where

$$\begin{aligned} C(\gamma )= -\dfrac{V_0^2}{2}+S+\Omega \left( \gamma ,u_0,v_0\right) ,V_0^2=u_0^2+v_0^2\nonumber \\ \end{aligned}$$
(22)

Using the obtained relation in Eq. (12), these curves can be identified by

$$\begin{aligned} 2 (S+\Omega )-C -2\int _0^t\dfrac{\partial }{\partial t}(S+\Omega )dt>0. \end{aligned}$$
(23)

In Fig. 6, we plot the zero-velocity curve under the Jacobian constant C value variation for fixed parameters \(\mu =0.019,\ \alpha =0.4,\ \gamma =0.3,\ k=0.00015,\ q_1=0.9,\ q_2=q_3=0.8\). The blue and red dots denote the primaries and Lagrangian points, respectively. The yellow and white regions represent the Hill and forbidden regions, respectively. The Hill region refers to the region where infinitesimal body may move, and the forbidden region is the region where the infinitesimal body cannot reach. In Fig. 6, there are eight non-collinear Lagrangian points, all of which are not symmetric about the u-axis.

Fig. 6
figure 6

The zero-velocity curve in the R4BP with the photo-gravitational, variable mass, and Stokes drag perturbations for \(\mu =0.019,\ \alpha =0.4,\ k=0.00015,\ \gamma =0.3,\ q_1=0.9,\ q_2=q_3=0.8\), for different C values

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In Fig. 6a, the Jacobian integral C=0.456375 and the Lagrangian points \(L_1\), \(L_2\), \(L_7\), and \(L_8\) are all located in the forbidden region. The infinitesimal body cannot reach these Lagrangian points, while \(L_3\), \(L_4\), \(L_5,\) and \(L_6\) are located in the Hill region. Especially, \(L_3\) and \(L_4\) are trapped in the Hill region with the primaries \(m_3\) and \(m_2\), respectively. This means the infinitesimal body cannot travel from one primary to another. In Fig. 6b, the Jacobian integral C decreases to 0.454533. It can be observed that there is a gap between \(L_3\) and \(L_5\), \(L_4\) and \(L_6\), and the infinitesimal body can reach either \(L_5\) or \(L_6\) from \(m_3\) and \(m_2\). In Fig. 6c, the Jacobian integral constant C decreases to 0.442488, and two branches appear near the primaries \(m_3\) and \(m_2\). That infinitesimal body can freely move between the primaries, but \(L_1\), \(L_2\), \(L_7\), and \(L_8\) are still trapped in the forbidden zone. In Fig. 6d, C decreases to 0.426496, and all Lagrangian points are located in the Hill region, but forbidden regions block \(L_1\), \(L_7\), and \(L_8\). In Fig. 6e, f, Jacobian integral C decreases to 0.422425 and 0.420288, respectively. The forbidden regions near \(L_1\) disappear first, and then the forbidden regions around \(L_7\) and \(L_8\) disappear. All Lagrangian points are located in the Hill region, meaning the infinitesimal body can shuttle between all primary bodies and all Lagrangian points.

6 Periodic solutions near the non-collinear points

In the framework of R4BP with variable mass, the study of motion near the Lagrangian points is significant. Periodic solutions near unstable non-collinear Lagrangian points can allow the fourth body to maintain longer periods here. The LP method used to remove secular terms and provides periodic solutions for regular motion. A brief introduction of this method can be found in “Appendix II”.

Move the \(-k \dot{u}\) and \(-k \dot{v}\) term in the stokes force on the right side of Eq. (8) to the left side then this equation can be rewritten as

$$\begin{aligned} \begin{aligned}&\ddot{u} - 2\dot{v}+k\dot{u} = \Omega _u+S_u^*,\\&\ddot{v} + 2\dot{u}+k\dot{v}=\Omega _v+S_v^*, \end{aligned} \end{aligned}$$
(24)

where

$$\begin{aligned} \begin{aligned}&S_u^*=-k \left[ \dfrac{\alpha u}{2}-v \left( 1+\dfrac{3 \gamma ^{7/4} \sigma }{2 \left( u^2+v^2\right) ^{7/4}}\right) \right] ,\\&S_v^*=-k \left[ \dfrac{\alpha v}{2}+u \left( 1+\dfrac{3 \gamma ^{7/4} \sigma }{2 \left( u^2+v^2\right) ^{7/4}}\right) \right] . \end{aligned} \end{aligned}$$
(25)

Introducing transformations \(u=u_0+x,v=v_0+y\), then Eq. (24) become the following form

$$\begin{aligned} \begin{aligned}&\ddot{x} - 2\dot{y}+k\dot{x} = \Omega _x+S_x^*, \\&\ddot{y} + 2\dot{x}+k\dot{y} = \Omega _y+S_y^*. \end{aligned} \end{aligned}$$
(26)

Utilizing Eq. (26) and applying the Taylor series up to third-order term, we get

$$\begin{aligned} \ddot{x} - 2\dot{y}+k\dot{x}&= U_x^0 + U_x^0\left( x\dfrac{\partial }{\partial x}+y\dfrac{\partial }{\partial y}\right) \nonumber \\&+\dfrac{1}{2!}U_x^0\left( x\dfrac{\partial }{\partial x}+y\dfrac{\partial }{\partial y}\right) ^2\nonumber \\&+U_x^0\dfrac{1}{3!}\left( x\dfrac{\partial }{\partial x}+y\dfrac{\partial }{\partial y}\right) ^3+O(4), \nonumber \\ \ddot{y} + 2\dot{x}+k\dot{y}&= U_y^0 + U_y^0\left( x\dfrac{\partial }{\partial x}+y\dfrac{\partial }{\partial y}\right) \nonumber \\&+\dfrac{1}{2!}U_y^0\left( x\dfrac{\partial }{\partial x}+y\dfrac{\partial }{\partial y}\right) ^2\nonumber \\&\quad +U_y^0\dfrac{1}{3!}\left( x\dfrac{\partial }{\partial x}+y\dfrac{\partial }{\partial y}\right) ^3+O(4), \end{aligned}$$
(27)

where \( U_x=\Omega _x+S_x^*\), the superscript \((^0)\) represents the value evaluated at the Lagrangian point, O(4) represents fourth-order and higher-order terms, which we ignore here. At the Lagrangian points \(L_i,i=1,2,\cdots ,10,\ U_x^0(L_i)= U_y^0(L_i)=0\). Therefore, the equation can be further expanded and simplified as

$$\begin{aligned} \ddot{x} - 2\dot{y}+k\dot{x} =&\, U_{xx}^0x+ U_{xy}^0y +\dfrac{1}{2} U_{xxx}^0x^2 \nonumber \\&+ U_{xxy}^0xy +\dfrac{1}{2} U_{xyy}^0y^2\nonumber \\&+\dfrac{1}{6} U_{xxxx}^0x^3 +\dfrac{1}{2} U_{xxxy}^0x^2 y \nonumber \\&+\dfrac{1}{2} U_{xxyy}^0xy^2+\dfrac{1}{6} U_{xyyy}^0y^3,\nonumber \\ \ddot{y} + 2\dot{x} +k\dot{y} =&\, U_{yx}^0x+ U_{yy}^0y+\dfrac{1}{2} U_{yxx}^0x^2 \nonumber \\&+ U_{yxy}^0xy +\dfrac{1}{2} U_{yyy}^0y^2\nonumber \\&+\dfrac{1}{6} U_{yxxx}^0x^3+\dfrac{1}{2} U_{yxxy}^0x^2 y\nonumber \\&+\dfrac{1}{2} U_{yxyy}^0xy^2+\dfrac{1}{6} U_{yyyy}^0y^3, \end{aligned}$$
(28)

where \(U_{xx}^0=N_1\), \(U_{xy}^0=N_2\), \(U_{xxx}^0 =2N_3\), \(U_{xxy}^0=N_4\), \(U_{xyy}^0 =2N_5\), \(U_{xxxx}^0 =6N_6\), \(U_{xxxy}^0 =2N_7\), \(U_{xxyy}^0 =2N_8\), \(U_{xyyy}^0=6N_9\), \(U_{yx}^0=F_1\), \(U_{yy}^0=F_2\), \(U_{yxx}^0 =2F_3\), \(U_{yxy}^0=F_4\), \(U_{yyy}^0=2F_5\), \(U_{yxxx}^0=6F_6\), \(U_{yxxy}^0=2F_7\), \(U_{yxyy}^0=2F_8\), \(U_{yyyy}^0=6F_9\), and \(N_i,i=1,2,\ldots ,9\), \(F_i,i=1,2,\ldots ,9\) in “Appendix I”.

The coefficients are represented by \(N_i,F_i,i=1,2,\ldots ,9\), and the Eq. (28) can be rewritten as

$$\begin{aligned} \begin{aligned} \ddot{x}-2 \dot{y}+k \dot{x}=&\,N_1 x+N_2 y+N_3 x^2+N_4 xy+N_5 y^2\\&+N_6 x^3+N_7 x^2 y+N_8 xy^2+N_9 y^3,\\ \ddot{y}+2 \dot{x}+k \dot{y}=&\, F_1 x+F_2 y+F_3 x^2+F_4 xy+F_5 y^2\\&+F_6 x^3+F_7 x^2 y+F_8 xy^2+F_9 y^3. \end{aligned}\nonumber \\ \end{aligned}$$
(29)

Supposing that the form of the solution to Eq. (29) is

$$\begin{aligned} \begin{aligned}&x=x_1 \epsilon +x_2 \epsilon ^2+x_3 \epsilon ^3,\\&y=y_1 \epsilon +y_2 \epsilon ^2+y_3 \epsilon ^3, \end{aligned} \end{aligned}$$
(30)

where \(\epsilon \) is a small parameter and value range is \(| \epsilon | \ll 1\).

Substituting Eqs. (30) into (29) and comparing the powers of \(\epsilon \) on both sides. Thus, we obtain the following three linear systems concerning the coefficients of \(\epsilon \), \(\epsilon ^2\), and \(\epsilon ^3\) in the following forms

$$\begin{aligned} \begin{aligned}&\ddot{x}_1-2 \dot{y}_1+k \dot{x}_1=N_1 x+N_2 y_1,\\&\ddot{y}_1+2 \dot{x_1}+k \dot{y}_1=F_1 x_1+F_2 y_1, \end{aligned} \end{aligned}$$
(31)
$$\begin{aligned} \begin{aligned} \ddot{x}_2-2 \dot{y_2}+k \dot{x}_2=&\, N_1 x_2+N_2 y_2 +N_3 x_1^2\\&+N_4 x_1 y_1+N_5 y_1^2,\\ \ddot{y}_2+2 \dot{x_2}+k \dot{y}_2=&\, F_1 x_2+F_2 y_2 +F_3 x_1^2\\&+F_4 x_1 y_1+F_5 y_1^2. \end{aligned} \end{aligned}$$
(32)
$$\begin{aligned} \begin{aligned} \ddot{x}_3 - 2\dot{y}_3 + k\dot{x}_3 =&\, N_1 x_3 + N_2 y_3 + 2N_3 x_1 x_2 \\&+ N_4 x_1 y_2+ N_4 x_2 y_1 + 2N_5 y_1 y_2 \\&+ N_6 x_1^3 + N_7 x_1^2 y_1\\&+ N_8 x_1 y_1^2 + N_9 y_1^3, \\ \ddot{y}_3 + 2\dot{x}_3 + k\dot{y}_3 =&\, F_1 x_3 + F_2 y_3 + 2F_3 x_1 x_2 \\&+ F_4 x_1 y_2 + F_4 x_2 y_1 + 2F_5 y_1 y_2 \\&+ F_6 x_1^3 + F_7 x_1^2 y_1 \\ &+ F_8 x_1 y_1^2 + F_9 y_1^3. \end{aligned} \end{aligned}$$
(33)

For the Eq. (31), supposing that the form of the solutions are

$$\begin{aligned} \begin{aligned} x_1&= G_1 \cos (\omega t) + H_1 \sin (\omega t), \\ y_1&= G_2 \cos (\omega t) + H_2 \sin (\omega t), \end{aligned} \end{aligned}$$
(34)

where period \(T=2 \pi /\omega \).

Substituting Eq. (34) into (31) yields

$$\begin{aligned} \begin{aligned}&A_{11}\,\sin \,(\omega t ) + B_{11}\,\cos \,(\omega t)=0, \\&A_{21}\, \sin \, (\omega t) + B_{21}\,\cos \, ( \omega t) =0, \end{aligned} \end{aligned}$$
(35)

where

$$\begin{aligned} \begin{aligned}&A_{11}= \left[ 2 G_1 \omega +G_2 k \omega +F_1 H_1+\left( F_2 +\omega ^2\right) H_2\right] , \\&B_{11}= \left[ F_1 G_1+\left( F_2 + \omega ^2\right) G_2 -2 H_1 \omega -H_2 k \omega \right] ,\\&A_{21}= \left[ G_1 k \omega -2 G_2 \omega +H_1\left( N_1+\omega ^2\right) +H_2 N_2 \right] ,\\&B_{21}= \left[ G_1 \left( N_1+\omega ^2\right) +G_2 N_2-H_1 k \omega +2 H_2 \omega \right] . \end{aligned}\nonumber \\ \end{aligned}$$
(36)

In order to make the homogeneous system of Eq. (35) have non-zero solutions, the determinant is zero, as follows

$$\begin{aligned} \left| \begin{array}{cccc} N_1+\omega ^2 & N_2 & -k \, \omega & 2\, \omega \\ k \, \omega & -2 \,\omega & N_1+\omega ^2 & N_2 \\ F_1 & F_2+\omega ^2 & -2\, \omega & -k \,\omega \\ 2 \, \omega & k \, \omega & F_1 & F_2+\omega ^2 \end{array} \right| =0. \end{aligned}$$
(37)

Set \(H_2=1\), obtain \(G_1,G_2,H_1\) in “Appendix III”. So the solutions of Eq. (31) are

$$\begin{aligned} \begin{aligned}&x_1=G_1 \cos (\omega t)+H_1 \sin (\omega t ),\\&y_1=G_2 \cos (\omega t )+\sin (\omega t ). \end{aligned} \end{aligned}$$
(38)

Similarly, to solve Eq. (32), let the solutions take the form of

$$\begin{aligned} \begin{aligned} x_2=&\, G_3 \cos (\omega t )+G_4 \cos (2 \omega t )\\&+H_3 \sin ( \omega t )+H_4 \sin (2 \omega t ),\\ y_2=&\, G_5 \cos (\omega t )+G_6 \cos (2 \omega t )\\&+H_5 \sin ( \omega t )+H_6 \sin (2 \omega t). \end{aligned}\nonumber \\ \end{aligned}$$
(39)

By using the same method as solving Eq. (31) and substituting the already obtained \((x_1,y_1)\) in Eq. (38) into (32), we obtain \(G_3=0\), \(H_3=0\), \(G_5=0\), \(H_5=0\), and \(G_4\), \(G_6\), \(H_4\), \(H_6\) in “Appendix III”. Therefore, the solutions of Eq. (32) are given by

$$\begin{aligned} \begin{aligned}&x_2=G_4 \cos (2 \omega t )+H_4 \sin (2\omega t ),\\&y_2=G_6 \cos (2 \omega t )+H_6 \sin (2 \omega t). \end{aligned} \end{aligned}$$
(40)

So, the second-order periodic solution is

$$\begin{aligned} \begin{aligned} x=&\, \epsilon \left[ G_1 \cos (\omega t )+H_1 \sin (\omega t )\right] \\&+\epsilon ^2 \left[ G_4 \cos (2 \omega t )+H_4 \sin (2 \omega t )\right] , \\ y=&\,\epsilon \left[ G_2 \cos (\omega t )+\sin (\omega t )\right] \\&+\epsilon ^2 \left[ G_6 \cos (2 \omega t )+H_6 \sin (2 \omega t )\right] . \end{aligned} \end{aligned}$$
(41)
Fig. 7
figure 7

Second- and third-order periodic solutions, for fixed parameter \(\mu =0.25\), \(\sigma =0.05\), \(k=0.00005\), \(q_1=0.99\), \(q_2=0.9985\), \(q_3=0.9985\), \(\alpha =2.2\) at different values for mass variation parameter. The second-order periodic solution is represented by the red dotted line, while the third-order periodic solution is depicted by the thick blue line. (Color figure online)

Full size image

Similarly, to solve Eq. (33), let the solutions take the below forms

$$\begin{aligned} \begin{aligned} x_3=&\, G_7 \cos ( \omega t)+G_8 \cos (2 \omega t )+G_9 \cos (3 \omega t)\\&+H_7 \sin (\omega t )+H_8 \sin (2 \omega t )+H_9 \sin (3 \omega t ),\\ y_3=&\, G_{10} \cos ( \omega t )+G_{11} \cos (2 \omega t )+G_{12} \cos (3 \omega t )\\&+H_{10} \sin (\omega t )+H_{11} \sin (2 \omega t )+H_{12} \sin (3 \omega t). \end{aligned} \end{aligned}$$
(42)

By using the same method as above and substituting the already obtained \((x_1, y_1)\) and \((x_2,y_2)\) in Eqs. (38, 40) into Eq. (33), we obtain \(G_8=0\), \(G_{11}=0\), \(H_7=0\), \(H_8=0\), \(H_{11}=0\), \(H_{12}=1\), and \(G_7\), \(G_9\), \(G_{10}\), \(G_{12}\), \(H_9\), \(H_{12}\) in “Appendix III”. Therefore, the solutions to Eq. (33) are

$$\begin{aligned} \begin{aligned} x_3=&\, G_7 \cos (\omega t)+G_9 \cos (3 \omega t)+H_9 \sin (3 \omega t),\\ y_3=&\, G_{10} \cos (\omega t)+G_{12} \cos (3 \omega t)\\&+H_{10} \sin (\omega t)+\sin (3 \omega t). \end{aligned} \end{aligned}$$
(43)

So, the third-order periodic solution is

$$\begin{aligned} \begin{aligned} x =&\, \epsilon \left[ G_1 \cos (\omega t ) + H_1 \sin (\omega t )\right] \\&+ \epsilon ^2\left[ G_4 \cos (2 \omega t ) + H_4 \sin (2 \omega t )\right] \\&+ \epsilon ^3 \left[ G_7 \cos (\omega t ) + G_9 \cos (3 \omega t ) + H_9 \sin (3 \omega t )\right] ,\\ \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} y =&\, \epsilon \left[ G_2 \cos (\omega t ) + \sin (\omega t )\right] \\&+ \epsilon ^2 \left[ G_6 \cos (2 \omega t ) + H_6 \sin (2 \omega t )\right] \\&+ \epsilon ^3 \left[ G_{10} \cos (\omega t ) + G_{12} \cos (3 \omega t ) \right. \\&\left. \qquad + H_{10} \sin (\omega t ) + \sin (3 \omega t )\right] . \end{aligned} \end{aligned}$$
(44)

Fixed the parameter \(\mu =0.25\), \(\sigma =0.05\), \(k=0.00005\), \(q_1=0.99\), \(q_2=0.9985\), \(q_3=0.9985\), \(\alpha =2.2\), when \(\gamma =0.95\), 0.7, 0.5, 0.3, the coordinates of Lagrangian points \(L_6=(0.132165,-0.567669)\), \((0.113444,-0.487284)\), (0.0958778, \(-0.41183)\) and (0.0742666, \(-0.319002)\) are obtained respectively. The corresponding second- and third-order periodic solutions are shown in Fig. 7a–d, respectively. When \(\gamma \) has a larger value of 0.95, it can be observed that the second- and third-order periodic solutions of the non-collinear Lagrangian point \(L_6\) are very close, and in some areas they are close to overlapping. When \(\gamma \) decreases to 0.7, the difference between the second- and third-order periodic solutions at point \(L_6\) increases. When \(\gamma \) decreases to 0.5, the difference between the second- and third-order periodic solutions increases, and both become somewhat flatter. When \(\gamma \) decreases to 0.3, in addition to the above changes, the third-order periodic solution no longer resembles a smooth ellipse and undergoes significant deformation. Furthermore, as \(\gamma \) decreases, the coordinates of Lagrangian point \(L_6\) approach the origin and the periodic solution shrinks constantly.

7 Conclusions

The equations of motion were obtained for the R4BP with variable mass under the perturbations of photo-gravitational and Stokes forces. The Lagrangian points were calculated with parameters \(0<\mu \le 1/3\), \(0<\alpha \le 2.2\), \(\sigma ,k,\gamma \in (0,1)\), \(0<q_i\le 1\ (i=1,2,3)\) and found that under the joint perturbation of photo-gravitational, Stokes force, and variable mass, there are six, eight, or ten non-collinear Lagrangian points, and there are no collinear Lagrangian points.

As the radiation pressure \(q_1\) of the primary body \(m_1\) increases in (0.6,1), the Lagrangian points are all far from the origin, while for the primary bodies, the Lagrangian points are either closer or farther away from it, regardless of whether the radiation of the second- and third primary bodies are equal. When the dissipative force constant \(k\in (0.05,0.4)\), there are six Lagrangian points, and with the increase of k value, these points are divided into \(L_1\), \(L_3\), \(L_5\), and \(L_2\), \(L_4\), \(L_6\), two groups, and the points in each group are closer and closer. As the variable mass parameter varies in (0.2,2.2), the coordinates of the Lagrangian points all move toward the origin. For different primary bodies, the Lagrangian points near them tend to move away or close to them. Through linear stability calculations, it was found that all Lagrangian points under the parameters considered in this paper are unstable.

Through the zero-velocity curves plotted for the given parameters, it is found that when the Jacobian integral constant changes within the range of \((0.456375,0.420288)\), the zero-velocity surface also changes, the Hill region gradually increases, and it is possible to transition between the primaries. With the help of the LP method, we also present the second- and third-order analytical periodic solutions around the non-collinear Lagrangian point. The numerical simulation results near the point \(L_6\) show that a substantial increase in the mass change parameter \(\gamma \) results in a larger area surrounding the periodic solution of \(L_6\), exhibiting a visually regular elliptical shape. Conversely, a decrease in \(\gamma \) leads to a reduction in the region of periodic solutions, accompanied by notable alterations in shape, particularly concerning third-order periodic solutions.