Effects of perturbations on the stability of equilibrium points in the CR3BP with luminous and heterogeneous spheroid primaries

This paper studies the position and stability of equilibrium points in the circular restricted three-body problem under the influence of small perturbations in the Coriolis and centrifugal forces when the primaries are radiating and heterogeneous oblate spheroids. It is seen that there exist five libration points as in the classical restricted three-body problem, three collinear \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{i} ,(i = 1,2,3)$$\end{document}Li,(i=1,2,3) and two triangular \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{i} ,(i = 4,5)$$\end{document}Li,(i=4,5). It is also seen that the triangular points are no longer to form equilateral triangles with the primaries rather they form simple triangles with line joining the primaries. It is further observed that despite all perturbations the collinear points remain unstable while the triangular points are stable for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0 < \mu < \mu _{c} $$\end{document}0<μ<μc and unstable for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mu _{c} \le \mu \le \frac{1}{2} $$\end{document}μc≤μ≤12, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mu _{c} $$\end{document}μc is the critical mass ratio depending upon aforementioned parameters. It is marked that small perturbation in the Coriolis force, radiation and heterogeneous oblateness of the both primaries have destabilizing tendencies. Their numerical examination is also performed.


Equations of motion
We consider the primaries of masses m 1 and m 2 (m 1 > m 2 ) as heterogeneous spheroid with three layers of different densities, source of radiation and we assume that the infinitesimal mass m is moving under the gravitational attraction of the said primaries. Using dimensionless variables, the equations of motion of the infinitesimal body in a synodic coordinate system can be written as 3,7 : where and x , denote the partial derivative of with respect to x and y respectively. M = m 1 + m 2 R = dimensional distance between the primaries. The mean motion n of the primaries is where here r 1 and r 2 are distances of the infinitesimal body from the primaries, ρ i are the densities for the layers, q 1 and q 2 are the radiation factors of the primaries. The resultant force acting on the infinitesimal body is , ρ (ρ (1) i ρ (2) i − ρ (1) i+1 ρ (2) i+1 )a (1) i (a (2) i ) 2 c (2) i (σ (1) i + σ (2) i ) (ρ (2) i − ρ (2) i+1 )(a (2) i ) 2 c (2) i , k 3 << 1, Fg i ,0 < δ i << 1 and q i = 1 − δ i , (i = 1, 2) Fg i and Fp i are gravitational and radiation pressure forces respectively 6 , α = 1 + ǫ , |ǫ| << 1 and β = 1 + ǫ ′ , ǫ ′ << 1 are the parameters for the Coriolis and centrifugal forces respectively for which ǫ and ǫ ′ are small perturbations.
From Eq. (1) we obtain Jacobi integral as where ẋ and ẏ are the velocity components and C is the Jacobian constant.

Location of equilibrium points
The position of equilibrium points can be found by setting ẋ =ẏ =ẍ =ÿ = 0 in the equations of motion.
Putting these values of r 1 , r 2 into system (3) and neglecting the second and highest order terms in ε 1 , ε 2 , we have Substituting the values of r 1 , r 2 from Eqs. (6) and x, y from Eqs. (7) into Eqs. (5) and using α = 1 + ǫ , β = 1 + ǫ ′ , q i = 1 − δ i and restricting ourselves to only linear terms in ǫ, ǫ ′ δ 1 , δ 2 , k 1 , k 2 , k 3 , ε 1 and ε 2 , we obtain Putting these values of Eqs. (8) into Eqs. (7) then we get Thus the coordinates x, ±y denoted by L 4 and L 5 are known as triangular points. One can observe that the locations of the triangular points depend on the small perturbation in the centrifugal force, heterogeneous oblateness and radiation of both primaries. We now compute the position of triangular points for the binary system Upsilon 4 Eridani numerically. The radiation pressure force q i = 1 − δ i i = 1, 2 10 .Taking k = 1 , on the basis of Stefan-Boltzmann's law,q = 1 − AkL rρM , where M and L are the mass and luminosity of a star respectively; r and ρ are the radius and density of moving body; k is the radiation pressure efficiency factor of star; In the CGS system,A = 2.9838 × 10 −5 , Suppose r = 2 × 10 −2 cm and ρ = 1.4 g cm −3 . The relevant numerical data of the system obtain from Wikipedia are given in Tables 1, 2, 3, 4 and 5 presented the effect of radiation pressure of the primaries and a small perturbation in centrifugal force on the position of triangular points with and/or without heterogeneous oblate spheroid of the binary system. Figures 1 and 2 shows the triangular points move closer to the x-axis as observed by the effects of perturbations.
Location of the collinear points. The positions of collinear points are the solutions of the equations.
Scientific Reports | (2022) 12:2819 | https://doi.org/10.1038/s41598-022-06328-y www.nature.com/scientificreports/ It shows that the collinear points lie on the x-axis .Substituting y = 0 in the left hand side of the first equation of system (5) and denoting the resulting expression by f (x) , we have where r 1 = |x − µ| and r 2 = |x − µ + 1|  Table 2. Location of the triangular equilibrium points for µ = 0.4918 , k 1 = k 2 = k 3 = 0.  Table 3. Effect of radition of the bigger primary on the location of triangular equilibrium points for  Table 4. Effect of radiation of the smaller primary on the location of triangular equilibrim points for www.nature.com/scientificreports/ To locate the collinear equilibrium points, we divide the orbital plane into three parts: Let the collinear point L 1 be on the left-hand side of the smaller primary at a distance h 1 from it on the x -axis. Then. Now putting, Since the distance between the primaries is unity, substituting all the above values in Eq. (10), we have This is an algebraic polynomial equation of degree nine in h 1 and there is only one change of sign, therefore, there exists at least one real root in it. The collinear point . L 1 .located by Table 5. Effect of a small perturbation in centrifugal force on the location of triangular equilibrium points for www.nature.com/scientificreports/ Let the collinear point L 2 be on the right-hand side of the smaller primary at a distance h 2 from it on the x-axis.
We put Now, substituting the values of r 1 , r 2 , x in to Eq. (10), we have Let the collinear point L 3 be on the right-hand side of the bigger primary at a distance h 3 from it on the x-axis.
Putting,x = µ + h 3 , x − µ = h 3 = r 1 and substituting the values of r 1 , r 2 and x into Eq. (10),we get Equations (11), (12) and (13) are ninth degree equations and there is only one positive real root in each case. This is physically acceptable root corresponds to one of the three collinear points Li(i = 1, 2, 3) . The Figs. 3, 4 and 5 are the disposition of the collinear equilibrium points in which the Figs. 3 and 5 describe the collinear points L 1 and L 3 moves closer to the primaries and the collinear point L 2 is located between the primaries in Fig. 4.
(i) Absence of radiation pressure of the both primaries, small perturbations in the Coriolis and centrifugal forces and heterogeneous oblate spheroid (classical case) (ii) A small perturbation in the centrifugal force with heterogeneous oblate spheroid only (12)      Figure 5. Position of the collinear equilibrium point L 3 . Table 6. Effect of the cases on the coordinates of collinear libration points of Li (i = 1,2,3).  consequently, there is only one value of µ call µ c in the interval 0, 1 2 for which vanishes. This µ c is called the critical mass ratio parameter. Therefore, we consider the following three regions of possible cases. (iii) When µ c < µ ≤ 1 2 , the discriminant (�) is negative. This shows that the real parts of the two characteristic roots are positive and equal. Hence the triangular point is unstable. Therefore we have stability for the first case and instability for the last case.
Critical mass. The solution of the equation = 0 obtained from (19) for µ gives the critical mass ratio value µ c of the mass parameter. Then we have where here µ c represents the combined effects of small perturbations in the Coriolis and centrifugal forces, heterogeneous oblateness and radiation pressures of both primaries. However, in the absence of perturbations in the Coriolis and centrifugal forces µ c agrees with the critical mass value of 7 when both primaries are radiating and heterogeneous spheroids with three layers of different densities. In this case µ c < µ 0 , this indicates that the range of stability decreases. Further more if the primaries are neither radiating nor heterogeneous oblate spheroid, the critical mass value µ c confirms the results of 2 . Also in the absence of the heterogeneous spheroids (k 1 = k 2 = k 3 = 0) , the critical mass value µ c verifies the result of 6 when neglecting the triaxiality of the primaries (σ 1 = σ 2 = σ ′ 1 = σ ′ 2 = 0) . In this case µ c > µ 0 , which indicate that the range of stability increases. We observe that if the primaries are non luminous and spherical bodies and small perturbations are also absent, the critical mass value µ c reduces to the classical value µ 0 of CR3BP of the 1 . It is obvious that all perturbing parameters except ǫ have the destabilizing effects. Figures 6 and 7 shows the effects of a small perturbation in the Coriolis and centrifugal forces on the sizes of the regions of stability with and/or without the effects of heterogeneous oblateness for an arbitrary system given to the radiation pressure force. Table 7 showcase the various effects in the region of stability with the influence of the effect due heterogeneous oblate spheroids, radiation pressure, Coriolis and centrifugal forces.
Stability of collinear points. Now we consider the stability of the collinear points. At first we consider the point corresponding to L 1 .
For this point, here � 0 xx � 0 yy < 0 , implies that the discriminant of (22) is positive and therefore its four characteristic roots and can be expressed as where s and t are real. Thus the motion around the collinear points is unbounded therefore the collinear points are unstable. It shows that the nature of the stability of collinear points is not affected by the changes in the Coriolis and centrifugal forces, heterogeneous oblateness, or radiation pressure forces of the both primaries and they remain unstable. The Tables 3, 4 and 5 shows the behaviors of the instability of the collinear libration points L 1 , L 2 and L 3 .

Discussion
The equations governing the motion of the infinitesimal body in the circular restricted three body problem with the effect of perturbations in the Coriolis and centrifugal forces when the both primaries are heterogeneous oblate spheroids with three layers of different density and the source of radiation are described in (1-3).The Eqs. (1)(2)(3) are same as in the classical case when there are no perturbations in the Coriolis and centrifugal forces and both primaries are neither heterogeneous oblate spheroid nor source of radiation pressure. The system (9) gives the positions of triangular points L 4,5 . This shows that they depend on the effects of a small perturbation in the centrifugal force and heterogeneous oblateness and radiation pressures of the both primaries. They no longer form equilateral triangles with the primaries as in the classical case. Rather they form simple triangles with the primaries. The Eq. (14) differs from 7 due to presence of small perturbations in the Coriolis and centrifugal forces. If the primaries are neither heterogeneous spheroid nor radiating and there are no perturbations in the Coriolis and centrifugal forces, the Eq. (14) represents that of the classical restricted problem. The critical value of the mass parameter µ c given by Eq. (20) shows the combined effect of perturbations in the Coriolis and centrifugal forces, heterogeneous oblateness and radiation pressures of the both primaries. In the absence of all parameters ( k 1 = k 2 = k 3 = δ 1 = δ 2 = ǫǫ ′ ) µ c reduces to µ 0 ∼ = 0.0385208…, which corresponds to 1 . But in the absence of the effect of heterogeneous spheroids ( k 1 = k 2 = k 3 = 0 ) the critical mass value µ c affirms the result of 6 when there are no effects of triaxility ( σ 1 = σ 2 = σ ′ 1 = σ ′ 2 = 0 ). In Table 7 the first result coincides with the 7 and similarly with the 1 for some certain values of the mass parameter while the other results emanate the region of stabilizing form with variation of a small perturbation in Coriolis and centrifugal forces and radiation pressure. It is observed that with the values of the parameters increase, the stability region is slightly reduced. We presented the stability of collinear equilibrium points in Tables 8, 9 and 10 according to the different cases, it is seen that no any case meet the requirement for stability of collinear points, therefore the result of the characteristics of Eq. (22) shows that the two roots are positive and negative real numbers while the other roots are positive and negative imaginary numbers. Hence it is unstable. Figures 6 and 7show that with the effect of perturbation in Coriolis and centrifugal forces the sizes of the region of stability seen to be increased. The graphs are showing that as the perturbation in centrifugal force is reducing µ C is increasing, so also as the perturbation in Coriolis force increasing the µ C increases.  Table 7. Effect of the parameters ( δ 1 , δ 2 ,ǫ, ǫ ′ ) in the stability region of equilibrium points for k 1 = 1.58302 × 10 −7 , k 2 = 9.83933 × 10 −18 , k 3 = 3.13153 × 10 −8 .

Conclusion
We have investigated the location and stability of equilibrium points under the influence of small perturbations in the Coriolis and centrifugal forces when both the primaries are heterogeneous spheroids with three layers and radiating. It is found that the stability behavior of the collinear points remains unchanged despite all the perturbations involved and they are unstable. The triangular points are stable for 0 < µ < µ c and unstable for µ c ≤ µ ≤ 1 2 , where µ c is the critical mass parameter affected by small perturbations in the Coriolis and centrifugal forces, heterogeneous oblateness and radiation pressure of both primaries. Our observation shows that all perturbations except that for Coriolis force have destabilizing tendencies and possess the decreasing size of the region of stability.