Nonlinear Stability of the Triangular Libration Points for Radiating and Oblate Primaries in CR3BP in Nonresonance Condition
This paper investigates the existence of resonance and nonlinear stability of the triangular equilibrium points when both oblate primaries are luminous. The study is carried out near the resonance frequency, satisfying the conditions , and in circular cases by the application of Kolmogorov-Arnold-Moser (KAM) theory. The study is carried out for the various values of radiation pressure and oblateness parameters in general. It is noticed that the system experiences resonance at for different values of radiation pressures and oblateness parameter. The case corresponds to the boundary region of the stability for the system. It is found that, except for some values of the radiation pressure, and oblateness parameters and for , the triangular equilibrium points are stable.
The three-body problem, which describes three masses interacting through Newtonian gravity, has attracted many scientists for more than 300 years. The three-body problem is one of the most challenging problems in the history of science. In celestial mechanics the general three-body problem deals with gravitationally interacting astronomical bodies and intends to predict their motions. The restricted three-body problem (R3BP) is a particular case of the general three-body problem. The problem is restricted in the sense that the infinitesimal mass does not influence the motion of two gravitating primaries but is rather influenced by them. In R3BP, there are two possibilities; namely, the two bodies with dominant masses move around their common centre of mass along either circular or elliptic orbits, which lead to the respective circular or elliptic restricted three-body problems (ER3BP). The ER3BP describes the three-dimensional motion of small particle called the infinitesimal mass under the gravitational force of two finite bodies called primaries around their common centre of mass. The ER3BP generalizes the original circular restricted three-body problems, while some useful properties of circular model still can be satisfied in the elliptical case. The ER3BP describes the dynamical system more accurately as the primaries move along the elliptical orbit.
Modern applications of the three-body problem and restricted three-body problem have been extended to include the Earth, the Moon, and the artificial satellites as well as recently discovered exoplanets. The study of the stability of an elliptical or circular restricted three- body problem of the Hamiltonian system is generally performed by using Kolmogorov-Arnold-Moser (KAM) theorem. The KAM theory is very useful for studying the global stability in three-body problem. The KAM theorem is suitable for the motion undergoing small perturbations which preserves the features of the unperturbed motion.
The nonlinear stability for resonance as well as for the nonresonance cases of the triangular libration points, taking one of the bodies as radiating, was studied by Manju and Choudhry . Kumar and Choudhry  investigated the stability of the triangular libration points for nonresonance as well as resonance case, taking both the bodies as radiating in circular restricted three-body problem in the presence of the third and fourth order resonance. Bhatnagar et al.  discussed the nonlinear stability of the triangular equilibrium points in circular restricted three bodies, considering bigger primary as a source of radiation. The nonlinear stability of the triangular Lagrangian points, considering the bigger primary as oblate spheroid in circular case, was examined by Markellos et al. . Recently Narayan and Singh  studied the nonlinear stability of higher order for both radiating primaries and found that binary systems are stable.
The detailed description and the behavior of equilibrium points in ER3BP are touched upon by Danby , Bennett , Szebebely , Markeev , Selaru and Cucu-Dumitrescu , and Hallan and Rana . The influence of the eccentricity of the orbits of the primaries with or without radiation pressure on the existence and stability of the equilibrium points was studied by Györgyey , Grebenikov , Kumar and Choudhry [2, 14], Markellos [4, 15], Sahoo and Ishwar , Roberts , Zimovshchikov and Tkhai , Ammar , Érdi et al. , Kumar and Ishwar , Singh and Umar [22, 23], Usha et al. , and Narayan and Singh [5, 25, 26].
It may be noted that the case when the frequencies are equal to zero or are equal to each other usually corresponds to the boundary of stability of the linear system, since, in the absence of oblateness parameters and radiation pressures the critical value of mass ratio denoted by is . Also, if it is considered that , then it is found that the inequality , provided ( are integers), is invalid for , which gives rise to the resonance cases.
The present paper investigates the existence of the resonance and the stability of the infinitesimal mass about the triangular equilibrium points by taking both primaries as radiating and oblate in nonresonance condition. The existence of the resonance and the stability of infinitesimal near the resonance frequency satisfying the conditions , , are studied in the circular cases. The study is carried out at various values of radiation pressures and oblateness parameters.
This paper has been organized in various sections.
Section 1 gives introduction, Section 2 describes the equations of motion of the problem, and Section 3 deals with the characteristics roots and first order stability of the triangular equilibrium points. The existence of resonance is discussed in Section 4, while Section 5 deals with normalization and higher order stability of the libration points. Finally Section 6 summarizes the discussion and conclusion of the paper.
2. Equations of Motion
The differential equations of the motion of the infinitesimal mass in elliptical restricted three-body problem under radiating primaries in pulsating system as given by Narayan and Shrivastava  arewhere the force function is defined aswhere is the radiation pressure, is a true anomaly of the primaries, , are the oblateness parameter, is the eccentricity of the orbits, and is the mass ratio defined asThe coordinates of the triangular equilibrium points and as given by Narayan and Shrivastava  are
3. Characteristics Roots and First Order Stability of the Triangular Equilibrium Points
The stability of the elliptical restricted three-body problem is restricted to planar case only. Since the nature of stability about and is similar it is sufficient to study the stability only about . The Hamiltonian as described by Narayan and Shrivastava  is given bywhere , , , are the variations in the coordinates .
By substituting we haveNow expand the Hamiltonian function aswhere is the sum of the terms of the th degree which is homogeneous in the variables , , , . is constant and . Consider where Restricting to alone, the characteristics equation can be given in the formwhere , , and are given by ((9), (10), and (11)). After further calculations the characteristics equation reduces to the formIf and are the frequencies then putting in (15), roots can be written asThe correlation between and and is shown in Figures 1–5. It is found that for fixed values of radiation pressure increases with an increase in whereas decreases.
4. Existence of Resonance in Circular Cases
In order to discuss the existence of resonance, we consider the following three cases.
Case 1. The first case is when ; that is, Solving the above equation we obtain that is,Solving for we obtain Since , the positive sign is inadmissible. Hence the region of stability in the first approximation can be written asThus, the value of responsible for stable equilibrium points is given by It is clear from (22) that in the absence of radiation pressures and oblateness parameters the critical value of mass ratio is when , which usually corresponds to a boundary of the region of stability of the system.
Case 2. It is when ; that is,that is,Solving the above equation for the resonance value is obtained as
Case 3. It is when ; that is,that is,Solving the above equation for the resonance value is obtained as Table 1 shows the values of corresponding to , , and . Figures 6–10 show correlation between and by varying , , and . Correlation between and is depicted in Figure 11 by varying and taking and . Figure 12 shows combined figures between and by taking , , and , 0.02, 0.03.
5. Normalization and Higher Order Stability of the Libration Points
In order to investigate the stability the Hamiltonian is normalized by Birkhoff’s method to the following form: where If is Hamiltonian of 2nd order, defined by (8), (9), (10), and (11) and is of a positive definite form, then the equilibrium position is stable by virtue of Lyapunov theorem . Otherwise the problem of stability can be solved by KAM theorem as given by Arnold [29, 30]. To apply KAM theorem, linear canonical transformation of variations as given by Manju and Choudhry  is used which is given as where whereThe transformation (31) reduces the Hamiltonian to the following form:The Hamiltonian can be expanded in the following form:Similarly can be expanded asThe coefficient of third and fourth order terms of and given by Kumar and Choudhry  can be listed as Again the Hamiltonian is reduced to a more convenient form which is suitable for further investigation by using the following canonical transformation:Thus, the Hamiltonian (34) may be written asAs explained by Kumar and Choudhry , consider the following.
If then The other ten coefficients of third order terms of (39) are obtained by the formulaIf we denote and consider and if it is found that ≠ an integer, then it implies that resonance of the third order is absent (Kumar and Choudhry ).
Using Birkhoff’s transformation , all the third order terms from the Hamiltonian (39) are nullified provided third order resonance does not occur. This transformation is introduced by means of the generating function which is given as follows:whereFrom (39) and (43), expanding and equating the terms of the same degree on the two sides, we obtainwhere is the term other than the homogeneous ones in and . In (44) the new variables and can be replaced by and on both sides of (44) by implicit function theorem.
Since our system is autonomous .
If we put using (44), we getWith the help of third equation of (44), the new Hamiltonian, inclusive of the fourth order terms, is given aswherenow, KAM theorem is applied which is best suited for the problem which is being defined and is stated as follows.
If the Hamiltonian of the perturbed motion satisfy the following given conditions mentioned below as:
The characteristics equation of the system with has pure imaginary roots and the frequencies , satisfy the inequality
Consider , as given by Kumar and Choudhry .
If all the above mentioned conditions are satisfied, then the equilibrium points are stable.
The value of is calculated with the help of the following formula:where , , and are defined in (48).
Table 2 shows the values for different values of radiation pressures and oblateness parameters.
6. Discussion and Conclusion
The stability of the triangular equilibrium points in CR3BP is investigated when both primaries are radiating and oblate, under the nonresonance case using KAM theory. Recently Narayan and Singh  studied nonlinear stability considering both primaries as radiating in ER3BP and found that the binary systems are stable. It has been observed that in general the stability character remains the same even if oblateness factor is considered apart from radiation factor in circular cases. The following observation has been made assuming both primaries as radiating and oblateness:(i)It is found that corresponds to the boundary region of the stability for the system, whereas the other two cases and correspond to the resonant cases. It has been observed that resonance of the third and the fourth order exists for all values of , , , , and taken.(ii)It is noticed that for fixed values of radiation pressures and by varying oblateness parameters increases by increasing whereas decreases and becomes equal to the critical value (i.e., ).(iii)From Table 2 it is clear that for , , , , and , , , , there is change in the sign of value of .This change in sign suggests that for fixed and by slightly changing radiation pressure and oblateness parameters vanishes, where KAM theory is not applicable. It is also concluded that except at this point the equilibrium points are stable. Consider the following:(i)It is clear from Table 2 that for , , , , and , , , , again there is change in sign of . This suggests that for slight variation in radiation pressure and oblateness parameters and for , the value of must be zero.(ii)It is observed that for , , , , and , , , , changes sign. This implies that when , , and and for slight variation in oblateness parameters, vanishes.(iii)Also, it is found that when , , , , and , , , , , there is change in the sign of .From the above discussion the following conclusion can be drawn.
In all the above four cases considered, it is noticed that KAM theory fails. From the above discussion it is also clear that by varying one of the parameters and taking another parameter as fixed, there may be many cases where KAM theory fails and their motion can be found numerically which needs further investigation. It is observed that, except for some values of radiation pressure and oblateness parameters and for