COMBINED EFFECTS OF OBLATENESS AND RADIATION ON THE NONLINEAR STABILITY OF L₄ IN THE RESTRICTED THREE-BODY PROBLEM

The restricted three-body problem describes the motion of an infinitesimal mass moving under the gravitational effect of the two finite masses, called primaries, which move in circular orbits around their center of mass on account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries. It is originally formulated due to the approximately circular motion of the planets around the sun, and the small masses of the asteroids and the satellites of the planets compared to the planets' masses.

This classical restricted three-body problem is not valid when at least one of the interacting bodies is an intense emitter of radiation. In this connection, it is reasonable to modify the model by superposing a light repulsion field whose source coincides with the source of the gravitational field provided by the radiating body on the gravitational field of the main bodies. According to Radzievsky (1950, 1953), the problem in such a statement is called the photogravitational problem. He discussed it for three specific bodies: the sun, a planet, and a dust particle. It was found that an allowance for direct solar radiation pressure results in a change in the positions of the libration points.

In certain stellar dynamics problems it is altogether inadequate to consider solely the gravitational interaction force. For example, when a star acts upon a particle in a cloud of gas and dust, the dominant factor is by no means gravity, but the repulsive force of the radiation pressure. Since a large fraction of all stars belong to binary systems, the particle motion in the field of a double star offers special interest. Probably the simplest dynamical model of such a system is the restricted circular photogravitational three-body problem. Among the possible motions of a particle, the equilibrium states at the libration points in a reference frame sharing the orbital motion of the stars were studied by Schuerman (1980) and Kunitsyn & Tureshbaev (1985).

In stellar systems numerous examples are available where a body is moving under the gravitational field of two radiating bodies. Simmons et al. (1985) obtained a complete solution of the restricted three-body problem. They discussed the existence and linear stability of the equilibrium points for all values of radiation pressures of both luminous bodies and all values of mass ratios. The participating bodies in the classical restricted three-body problem are strictly spherical in shape, but in actual situations we find that several heavenly bodies, such as Saturn and Jupiter, are sufficiently oblate. The minor planets and meteoroids have irregular shapes. In these cases, on account of the small dimensions of the bodies in comparison with their distances from the primaries, they are considered to be point masses, but in many cases the dimensions of the bodies are larger than the distances from their respective primaries. Thus, the above assumption is not justified, and the results obtained are far from a realistic approach. The lack of sphericity, or the oblateness, of the planet causes large perturbations from a two-body orbit. The motions of artificial Earth satellites are examples of this. This enables many researchers to study the restricted problem by taking into account the shapes (oblateness) of the bodies (Subba Rao & Sharma, 1975; Bhatnagar & Chawla, 1979, Elipe & Ferrer, 1985; Elipe 1992; Markellos et al. 1996; Chandra & Kumar, 2004).

An investigation of the positions of libration points, when the more massive primary is a source of radiation and the smaller one is an oblate spheroid, was carried out by, e.g., Sharma (1987). He showed that the triangular points are linearly stable for the mass parameter 0 < μ < μ_crit and the critical mass value μ_crit decreases with the increase in oblateness and radiation force.

The effect of oblateness and radiation pressure forces of the primaries on the location and the linear stability of the triangular points in the restricted three-body problem was analyzed by Singh & Ishwar (1999). They considered both primaries as sources of radiation as well as oblate spheroids, and observed that these points are stable for 0 < μ < μ_co and unstable for μ_co < μ < 1/2, where μ_co is the critical value of the mass parameter and depends on the radiating and oblateness coefficients. The same problem under the influence of small perturbations in the Coriolis and the centrifugal forces was studied by AbdulRaheem & Singh (2006). Dealing with the overall effect they observed that the range of stability of triangular points decreases.

Many mathematicians and astronomers have been interested in the study of the stability of an equilibrium point for all time and all the orders of the terms in the expansion of the Hamiltonian. The nonlinear stability of the trianglar libration points in the classical restricted three-body problem was investigated by Deprit & Deprit-Bartholome (1967). Bhatnagar & Hallan (1983) discussed the effect of perturbations in the Coriolis and centrifugal forces on the nonlinear stability of equilibrium points. Later, analytical studies of the nonlinear stability of L₄ under different aspects were also carried out by Niedzielska (1994), Subba Rao & Sharma (1997), Hallan et al. (2000), Gozdziewski (2003), and Chandra & Kumar (2004).

Our aim is to study the combined effects of oblateness and radiation of the primaries on the nonlinear stability of the libration point L₄ as it has been found that they produce significant changes in the location and stability (Singh & Ishwar 1999) of the triangular points. We also know that the inclusion of the nonlinear terms sometimes changes the entire pattern of the stability and, hence, we intend to study the stability in the nonlinear sense as well. By applying Lyapunov's theorem (Lyapunov 1956) to the linear stability results in Singh & Ishwar (1999) mentioned earlier, we can state that L₄ for μ_co < μ < 1/2 is unstable in the nonlinear sense also. So, we need to study the nonlinear stability of L₄ for 0 < μ < μ_co. For this, we will apply Arnold's theorem (Arnold 1961) and follow the procedure adopted by Bhatnagar & Hallan (1983). Arnold (1961) proved that if

• 1.  
  k₁ω₁ + k₂ω₂ ≠ 0 for all pairs (k₁, k₂) of rational integers where ω₁, ω₂ are the basic frequencies for the linear dynamical system and
• 2.  
  The determinant D ≠ 0, where D = det(b_ij) (i, j = 1, 2, 3), $b_{ij} = \big({\frac{{\partial ^2 H}}{{\partial I_i \partial I_j }}} \big)_{I_i = I_j = 0}$, $(i,j = 1,2),\,\,b_{i3} = b_{3i} = \big({\frac{{\partial H}}{{\partial I_i }}} \big)_{I{}_i = I_i = 0} (i,j = 1,2)$,

b₃₃ = 0 and ${H} = \omega _1 I_1 - \omega _2 I_2 + \frac{1}{2}\big({AI_1^2 + 2BI_1 I_2 + CI_2^2 } \big) + \cdots$ is the normalized Hamiltonian with I₁, I₂ as the action momenta coordinates, and A, B, C as second-order coefficients in the frequencies, then on each energy manifold H = h in the neighborhood of equilibrium, there exists invariant tori of quasi-periodic motions which divide the manifolds, and consequently, the equilibrium is stable. This is valid for a system with two degrees of freedom, which is the case under consideration. Moser (1962) showed that Arnold's theorem is true if the first condition of the theorem is replaced by k₁ω₁ + k₂ω₂ ≠ 0 for all pairs (k₁, k₂) of rational integers such that |k₁| + |k₂| ⩽ 4.

Markellos et al. (1996), Subba Rao & Sharma (1997), and Chandra & Kumar (2004) studied the nonlinear stability of L₄ when the bigger primary is an oblate spheroid. We consider here the case where both primaries are oblate spheroids as well as sources of radiation. We use A_i(i = 1, 2) for the oblateness coefficients of the bigger and smaller primaries, respectively, such that 0 < A_i ≪ 1 (McCuskey 1963). We denote the radiation factors as q_i(i = 1, 2) for the bigger and smaller primaries and these are given by F_p = F_g(1 − q_i) such that 0 < 1 − q_i ≪ 1 (Radzievsky 1950). For simplicity we further substitute q₁ = 1 − ∈, q₂ = 1 − ∈', | ∈ | ≪ 1, | ∈ '| ≪ 1 and restrict ourselves to linear terms in A_i, ∈, and ∈'. Then the coefficients of A₁ throughout the paper are exactly the same as worked out in the paper of Markellos et al. (1996), Subba Rao & Sharma (1997), Chandra & Kumar (2004).

This paper should be read in conjunction with the papers by Bhatnagar & Hallan (1983) and Subba Rao & Sharma (1997) as, to save space, we are not mentioning the values of various variables given in those papers, although they are used in this paper.

Using dimensionless variables and a synodic coordinate system (x, y) as Szebehely (1967) and Singh & Ishwar (1999) did, the equations of motion of the infinitesimal mass can be written as

$$\begin{equation}\ddot x - 2n\dot y = U_{x}, \qquad \ddot y + 2n\dot x = U_y, \end{equation} \tag{ 1 }$$

where

$$\begin{eqnarray} U &=& \frac{{n^2 }}{2}({x^2 + y^2 }) + \frac{{q_1 ({1 - \mu })}}{{r_1 }} + \frac{{\mu q_2 }}{{r_2 }} + \frac{{A_1 q_1 ({1 - \mu })}}{{2r_1^3 }}+ \frac{{A_2 q_2 \mu }}{{2r_2^3 }}\nonumber\\ r_1 ^2 &=& (x - \mu)^2 + y^2,\qquad r_2^2 = (x + 1 - \mu)^2 + y^2, \end{eqnarray} \tag{ 2 }$$

where r₁ and r₂ are the distances of the infinitesimal mass from the primaries and μ is the ratio of the mass of the smaller primary to the total mass of the primaries and 0 < μ ⩽ 1/2. The perturbed mean motion of the primaries, n, is given by

$$\begin{equation} n^2 = 1 + \case{3}{2}(A_1 + A_2). \end{equation} \tag{ 3 }$$

The coordinates of the triangular point L₄, obtained by Singh & Ishwar (1999), are

$$\begin{eqnarray} x_0 &=& {-} \frac{1}{2}\left[ {\gamma + \frac{2}{3}({q_1 - q_2 }) + A_1 q_1 - A_2 q_2 } \right] \ {\rm with\ }\gamma = 1 - 2\mu \nonumber\\ y_0 &=& \left({\frac{{\sqrt 3 }}{2}} \right)\bigg[ 1 - \frac{2}{3}({A_1 + A_2 }) + \frac{1}{3}({A_1 q_1 + A_2 q_2 }) \\ &&- \frac{2}{9}({1 - q_1 }) - \frac{2}{9}({1 - q_2 })\bigg].\nonumber \end{eqnarray} \tag{ 4 }$$

The Lagrangian function of the problem can be written as

$$\begin{eqnarray} L &=& 1/2(\dot x^2 + \dot y^2) + n(x\dot y - \dot xy) + (n^2 /2)(x^2 + y^2) \nonumber\\ &+& (1 - \mu)q_1 /r_1 + \mu q_2 /r_2 + A_1 q_1 (1 - \mu)\big/2r_1^3 + A_2 q_2 \mu \big/2r_2^3. \nonumber \end{eqnarray}$$

Shifting the origin to L₄ and expanding in power series of x and y, we have

$$\begin{eqnarray} L_{2} &=& (\dot x^2 + \dot y^2)/2 +n(x\dot y - \dot xy) + (n^2 /2)\nonumber\\ &&\times (x^2 + y^2) - Ex^2 - Fy^2 - Gxy,\nonumber\\ L_{3} &=& T_1 x^3 /3 + 1/2(T_2 x^2 y) + T_3 xy^2 + 1/3(T_4 y^3)\\ L_{4} &=& N_1 x^4 + N_2 x^3 y + N_3 x^2 y^2 + N_4 xy^3 + N_5 y^4.\nonumber \end{eqnarray} \tag{ 5 }$$

Consequently,

$$\begin{eqnarray} &&\hspace{-1.8pc}H_{2} = \big(P_x ^2 + P_y ^2\big)\big/2 + n(yP_x - xP_y) + Ex^2 + Fy^2 + Gxy, \nonumber\\ &&\hspace{-1.8pc}H_{2} = (\dot x^2 + \dot y^2)/2 - n^2 (x^2 + y^2)/2 + Ex^2 + Fy^2 + Gxy, \\ &&\hspace{-1.8pc}H_3 = - L_3,H_4 = - L_4 \nonumber \end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray} E \!\!\!\!\!&=\!\!\!\!& [ 2 - 3A_1 - 12A_1 \gamma - 3A_2 + 12A_2 \gamma - 2\in \, {+}\, 6 \in \gamma\nonumber\\ && -\, 2\in ^\prime\, {-}\, 6 \in ^\prime \gamma ]/16\nonumber\\ F \!\!\!\!\!&=\!\!\!\!& - [ 10 + 21A_1 + 21A_2 - 2 \in - 2 \in ^\prime +\, 6 \in \gamma\nonumber\\ && -\, 6 \in ^\prime \gamma ]/16,\nonumber\\ G \!\!\!\!\!&=\!\!\!\!& \sqrt 3 [ 6A_1 + 13A_1 \gamma - 6A_2 + 13A_2 \gamma - 2 \in \nonumber\\ && +\, 2 \in ^\prime\, {+}\, 2 \in \gamma /3 + 2 \in ^\prime \gamma /3 + 6\gamma ]/8, \nonumber\\ T_{1} \!\!\!\!\!\!\!&=\!\!\!\!\!\!& [ {-} 24\gamma + 18A_1\, {-} \,75A_1 \gamma + 79A_2 /2 \,{-}\, 35A_2 \gamma /2\nonumber\\ && +\, 21 \in+ 21 \in \gamma /3 - 21 \in ^\prime + 21 \in ^\prime \gamma ]/13, \nonumber\\ T_{2} \!\!\!\!\!&=\!\!\!\!& - \sqrt 3 [ 6 + 43A_1 + 60A_1 \gamma + 43A_2 - 60A_2 \gamma\nonumber\\ &&-\, 3 \in - 3 \in \gamma - 3 \in ^\prime + 3 \in ^\prime \gamma ]/16 \nonumber\\ T_3 \!\!\!\!\!&=\!\!\!\!& [33\gamma /4 + 33A_1 /4 + 195A_1 \gamma /8 - 33A_2 /4\nonumber\\ &&+\, 195A_2 \gamma /8 - 33 \in /8 - 33 \in \gamma /8 \nonumber\\ &&+\, 33 \in ^\prime /8 - 33 \in ^\prime \gamma /8]/4 \nonumber\\ T_{4} \!\!\!\!\!&=\!\!\!\!& - \sqrt 3 [ 18 + 69A_1 - 41A_2 - 9 \in -\ 9 \in ^\prime \nonumber\\ &&+\, \gamma ({ - 110A_2 - 9 \in +\ 9 \in ^\prime }) ]/32. \nonumber\\ N_1 \!\!\!\!\!&=\!\!\!\!& {-} 37/128 - 285A_1 /256 - 25A_1 \gamma /32\nonumber\\ &&-\, 36A_2 /64 + 1545A_2 \gamma /256 - 29 \in /128\nonumber\\ &&+\, 113 \in \gamma /384 - 29 \in ^\prime /128 - 113 \in ^\prime \gamma /384. \nonumber\\ N_2 \!\!\!\!\!&=\!\!\!\!& \sqrt 3 [25\gamma /32 - 45A_1 /32 + 265A_1 \gamma /192\nonumber\\ &&+\, 135A_2 /96 + 265A_2 \gamma /192 \nonumber\\ &&+\, 735 \in /384 + 925 \in \gamma /1152\nonumber\\ && - \,35 \in ^\prime /32 - 5 \in ^\prime \gamma /288]\nonumber\\ N_3 \!\!\!\!\!&=\!\!\!\!& 123\,64 + 255A_1 /32 + 1215A_1 \gamma /128 + 1245A_2 /128\nonumber\\ &&-\, 525A_2 \gamma /64 - 583 \in / 128 \nonumber\\ &&+ \,467 \in \gamma /128 - 303 \in ^\prime /128 + 863 \in ^\prime \gamma /128 \nonumber\\ N_4 \,\, \!\!\!\!\!&=\!\!\!\!& - \sqrt 3 [ 45\gamma /32 + 45A_1 /32 + 355A_1 \gamma /64 + 435A_2 /64\nonumber\\ &&-\, 85A_2 \gamma /32 + 35 \in /32 \nonumber\\ &&+\, 25 \in \gamma /96 + 105 \in ^\prime /64 - 475 \in ^\prime \gamma /192 ] \nonumber\\ N_{5}\!\!\!\!&=&\!\!\!\! 3/128 - 225A_1 /256 - 105A_1 \gamma /64 + 435A_2 /64\nonumber\\ &&-\, 85A_2 \gamma /32 + 35 \in /32 + 25 \in \gamma /96 \nonumber\\ &&+\, 105 \in ^\prime /64 - 475 \in ^\prime \gamma /192 \end{eqnarray} \tag{ 7 }$$

with considerations

$$\begin{eqnarray} &&q_1 = 1 - \in,\quad q_2 = 1 - \in ^\prime,\quad \| {\in} | \ll 1,\quad | { \in ^\prime }| \ll 1.\nonumber \end{eqnarray}$$

All these values are different from the classical case due to the radiation and oblateness of both primaries. When A₂ = ∈ = ∈' =  0, these values agree with those found by Subba Rao & Sharma (1997).

The characteristic equation corresponding to Hamiltonian H₂ is given by

$$\begin{equation}\lambda ^4 + 2({E + F + n^2 })\lambda ^2 + 4EF - G^2 + n^4 - 2n^2 ({E + F}) = 0.\end{equation} \tag{ 8 }$$

The roots of Equation (8) are purely imaginary if its discriminant Δ>0 and thus it is a necessary condition for the stability, in the linear sense, around the L₄ point. The solution of the equation Δ = 0 for μ gives the critical mass value μ_co of the mass parameter.

This implies that (Singh & Ishwar 1999)

$$\begin{eqnarray}\mu _{{\rm co}} &=& 0.03852089 \ldots - 0.28500178 \ldots A_1 - 0.00891747 \ldots \in \nonumber\\ &&- 0.0627795 \ldots A_2 - 0.00891747 \ldots \in ^\prime.\nonumber\end{eqnarray}$$

For A₂ = ∈ = ∈' =  0 the value of μ_co agrees with that given by Subba Rao & Sharma (1975), Markellos et al. (1996), and Chandra & Kumar (2004). The range of stability, in the linear sense, can be written as 0 < μ < μ_co If Equation (8) has four imaginary roots, ±iω_i, ± iω₂ where ω₁, ω₂ are the basic frequencies, then we have

$$\begin{eqnarray} \omega _1^2 + \omega _2^2 &=& ({1/16})[ 16 + \gamma (24A_2 - 24A_1) ], \nonumber\\ &&0 < \omega _2< 1/\sqrt 2 < \omega _1 < 1 \nonumber\\ \omega _1^2 \omega _2^2 &=& ({1/16})27 + 117A_1 + 117A_2 + 6 \in +\, 6 \in ^\prime \nonumber\\ &&- \gamma ^2 (27 + 117A_1 + 117A_2 + 6 \in +\, 6 \in ^\prime), \end{eqnarray} \tag{ 9 }$$

which give

$$\begin{eqnarray} \gamma ^2 &=& \big({16\omega _1^4 - 16\omega _1^2 + 27} \big)\big/27 + \big\{208\omega _1^2 \big({1 - \omega _1^2 } \big)\big/81\nonumber\\ &&+ 8\gamma \omega _1^2 \big/9)\big\} A_1 + \big\{ 208\omega _1^2 \big({1 - \omega _1^2 } \big)\big/81\nonumber\\ && - 8\gamma \omega _1^2 \big/9\big\} A_2 + \big\{ 32\omega _1^2 \big(1 - \omega _1^2\big)\big/243\big\}\in \nonumber\\ &&+ \big\{ 32\omega _1^2 \big(1 - \omega _1^2\big)\big/243 \big\} \in ^\prime. \end{eqnarray} \tag{ 10 }$$

The results given by Equations (9) and (10) in the case A₂ = ∈ = ∈' =  0 agree with those found by Subba Rao & Sharma (1997).

Following the method given in Whittaker (1965), we use a canonical transformation from the phase space x, y, P_x, P_y into the phase space of the angles (ϕ₁,  ϕ₂) and the actions (I₁, I₂) so that H₂ be normalized:

$$\begin{equation}X = JT \end{equation} \tag{ 11 }$$

where

$$\begin{eqnarray}&&X = \left({\begin{array}{@{}c@{}} x \nonumber\\ y \nonumber\\ {P_x } \nonumber\\ {P_y } \nonumber\\ \end{array}} \right),\quad J = (J_{ij})\,1 \le i,j \le 4,\;\;\;\;\;T = \left({\begin{array}{@{}c@{}} {Q_1 } \nonumber\\ {Q_2 } \nonumber\\ {P_1 } \nonumber\\ {P_2 } \nonumber\\ \end{array}} \right)\end{eqnarray}$$

Q_j = (2Ij/ω_j)^1/2sin ϕ_j Pj = (2I_j.ω_j)^1/2 cos ϕ_j(j = 1, 2), and the elements of the dyadic J can be obtained by adopting the method of Breakwell & Pringle (1966). Elements of J necessary for further analysis are given in Appendix A.

The transformation changes the second-order part of the Hamiltonian into the normal form H₂ = ω₁I₁ − ω₂I₂.

For transforming the Hamiltonian H to the Birkhoff's normal forms, we have utilized Henrard's method (Deprit & Deprit-Bartholome 1967). We have expanded the coordinates (x, y) of the infinitesimal body in double d'Alembert series:

$$\begin{equation}x = \sum\limits_{j \ge 1} {B_j^{1.0} } ({\phi _{1,} \phi _{2,} I_1,I_2 }),\quad y = \sum\limits_{j \ge 1} {B_j^{0,1} ({\phi _{1,} \phi _{2,} I_1,I_2 })}\end{equation} \tag{ 12 }$$

where the homogeneous components B^1,0_J and B^0,1_J are of degree j, I₁ and I₂ are to be taken as constants of integration, while ϕ₁ and ϕ₂ are to be determined as linear functions of time such that

$$\begin{eqnarray}&&\dot \phi _1 = \omega _1 + \sum\limits_{j \ge 1} {f_{2j} ({I_{1,} I_2 }),} \quad \quad \dot \phi _2 = - \omega _2 + \sum\limits_{j \ge 1} {f_{2j} } ({I_1,I_2 }).\nonumber\end{eqnarray}$$

The first-order components B^1,0₁ and B^0,1₁ are the values of x and y obtained by Equation (11). Proceeding as in Deprit & Deprit-Bartholome (1967), we observe that the second-order components B^1,0₂ and B^0,1₂ are solutions of the partial differential equations

$$\begin{equation}\Delta _1 \Delta _2 B_2^{1,0} = \Phi _2,\quad \quad \Delta _1 \Delta _2 B_2^{0,1} = - \Psi _2,\end{equation} \tag{ 13 }$$

with

$$\begin{eqnarray} \Delta _i &=& D^2 + \omega _i^2,i = 1,2;\;\;\;D = \omega _1 ({\partial /\partial \phi _1 }) - \omega _2 ({\partial /\partial \phi _2 }),\nonumber\\ \Phi _2 &=& ({D^2 + S_1 })X_2 + ({S_2 D + S_3 })Y_2,\nonumber\\ \Psi _2 \, &=& ({S_2 D - S_3 })X_2 - ({D^2 - S_4 })Y_2,\;\;S_1 = 2F - n^2, \nonumber\\ S_2 &=& 2[1 + (3/4)(A_1 + A_2)],\quad S_3 = - G,\quad S_4 = n^2 - 2E. \nonumber \end{eqnarray}$$

X₂, Y₂ are obtained by substituting respectively in ∂L₃/∂x, ∂L₃/∂y the first-order components for x and y.

Equation (13) can be solved for B^1,0₂ and B^0,1₂ by using the formula

$$\begin{eqnarray}&&\frac{1}{{\Delta _1 \Delta _2 }}\left\{ {\begin{array}{@{}c@{}} {{\rm cos}(m\phi _1 + n\phi _2)}\nonumber \\ {{\rm or}}\nonumber \\ {{\rm sin}(m\phi _1 + n\phi _2)}\nonumber \\ \end{array}} \right\} = \frac{1}{{\Delta _{m,n} }}\left\{ {\begin{array}{@{}c@{}} {{\rm cos}(m\phi _1 + n\phi _2)}\nonumber \\ {{\rm or}}\nonumber \\ {{\rm sin}(m\phi _1 + n\phi _2)}\nonumber \end{array}} \right\}\end{eqnarray}$$

where

$$\begin{eqnarray}&&\Delta _{m,n} = \big[\omega _1^2 - ({m\omega _1 - n\omega _2 })^2 \big]\big[\omega _2^2 - ({m\omega _1 - n\omega _2 })^2 \big]\nonumber\end{eqnarray}$$

provided Δ_m,n ≠ 0.

Since Δ_1,0 = Δ_0,1 = 0, the terms cos ϕ₁, sin ϕ₁, cos ϕ₂, and sin ϕ₂ are the critical terms. Φ₂ and Ψ₂ are free from such terms. By condition (1) of Moser's theorem (Moser 1962), none of the divisors Δ_2,0,  Δ_0,2,  Δ_1,1 Δ_1,−1, is zero. The second-order components B^1,0₂ and B^0,1₂ are as follows:

$$\begin{eqnarray} B_2^{1,0} &=& r_1 I_1 + r_2 I_2 + r_3 I_1 \cos 2\phi _1 + r_4 I_2 \cos 2\phi _2 + r_5 I_1 \sin 2\phi _1\nonumber\\ &&+ r_6 I_2 \sin 2\phi _2 + r_7 I_1 ^{1/2} I_2^{1/2} \cos ({\phi _1 + \phi _2 })\nonumber\\ &&+ r_8 I_1 ^{1/2} I_2^{\frac{1}{2}} \cos ({\phi _1 - \phi _2 }) + r_9 I_1^{1/2} I_2^{1/2} \sin ({\phi _1 + \phi _2 }) \nonumber\\ &&+ r_{10} I_1^{1/2} I_2^{1/2} \sin ({\phi _1 - \phi _2 }) \nonumber\\ B_2^{0,1} &=& s_1 I_1 + s_2 I_2 + s_3 I_1 \cos 2\phi _1 + s_4 I_2 \cos 2\phi _2\nonumber\\ &&+ s_5 I_1 \sin 2\phi _1 + s_6 I_2 \sin 2\phi _2 \nonumber\\ &&+ s_7 I_1 ^{1/2} I_2^{1/2} \cos ({\phi _1 + \phi _2 }) + s_8 I_1^{1/2} I_2^{\frac{1}{2}} \cos ({\phi _1 - \phi _2 })\nonumber\\ &&+ s_9 I_1^{1/2} I_2^{1/2} \sin ({\phi _1 + \phi _2 })\nonumber\\ &&+ s_{10} I_1^{1/2} I_2^{1/2} \sin ({\phi _1 - \phi _2 }) \nonumber \end{eqnarray}$$

The coefficients r_k and s_k for k = 1, 2, ..., 10 are given in Appendix B.

Following Henrard's method, we find that the third-order components B^1,0₃ and B^0,1₃ in the coordinates x, y and the second-order polynomials f₂ and g₂ in the frequencies $\dot \phi _1$ and $\dot \phi _2$ satisfy the partial differential equations:

$$\begin{equation} \begin{array}{c} \Delta _1 \Delta _2 B_3^{1,0} = \Phi _3 - 2f_2 P - 2g_2 Q,\\ \ms \Delta _1 \Delta _2 B_3^{0,1} = \psi _3 - 2f_2 U - 2g_2 V,\end{array}\end{equation} \tag{ 14 }$$

with

$$\begin{eqnarray} \Phi _3 &=& [D^2 - \{ 1 + 3({A_1 + A_2 })/2 - 2F\} ]X_3\nonumber\\ &&+ [ {2\{ 1 + 3({A_1 + A_2 })/4\} D - G} ]Y_3, \nonumber\\ \psi _3 &=& [D^2 - \{ 1 + 3({A_1 + A_2 })/2 - 2E\} ]Y_3\nonumber\\ &&- [ {2\{ 1 + 3({A_1 + A_2 })/4\} D + G} ]X_3 \nonumber\\ P &=& ({\partial /\partial \phi _1 })\big[\big\{ \omega _1^2 \partial ^2 \big/\partial \phi _1^2 - 1 - 3({A_1 + A_2 })/2\nonumber\\ &&+ 2F\big\} \big\{ \omega _1 \partial B_1^{1,0} \big/\partial \phi _1 - ({1 + 3A_1 /4 + 3A_2 /4})B_1^{0,1} \big\} \nonumber\\ &&+ \{ 2({1 + 3A_1 /4 + 3A_2 /4})\omega _1 \partial /\partial \phi _1 - G\}\nonumber\\ &&\times \big\{ \omega _1 \partial B_1 ^{0,1} \big/\partial \phi _1+ ({1 + 3A_1 /2 + 3A_2 /2})B_1^{0,1} \big\} \big] \nonumber\\ Q &=& ({\partial /\partial \phi _2 })\big[\big\{ \omega _2^2 \partial ^2 \big/\partial \phi _2^2 - 1 - 3({A_1 + A_2 })/2 + 2F\big\} \nonumber\\ &&\times \big\{ {-} \omega _2 \partial B_1^{1,0} \big/\partial \phi _2 - ({1 + 3A_1 /4 + 3A_2 /4})B_1^{0,1} \big\} \nonumber\\ &&+ \{ - 2({1 + 3A_1 /4 + 3A_2 /4})\omega _2 \partial /\partial \phi _2 - G\}\nonumber\\ &&\times \big\{ {-} \omega _2 \partial B_1 ^{0,1} \big/\partial \phi _2+ ({1 + 3A_1 /4 + 3A_2 /4})B_1^{0,1} \big\} \big] \nonumber\\ U &=& ({\partial /\partial \phi _1 })\big[ \big\{ \omega _1^2 \partial ^2 \big/\partial \phi _1^2 - 1 - 3A_1 /2 - 3A_2 /2+ 2E\big\} \nonumber\\ &&\times\big\{ \omega _1 \partial B_1^{1,0} \big/\partial \phi _1 + ({1 + 3A_1 /4 + 3A_2 /4})B_1^{0,1} \big\} \nonumber\\ &&- \{ 2({1 + 3A_1 /4 + 3A_2 /4})\omega _1 \partial /\partial \phi _1 + G\} \big\{ \omega _1 \partial B_1 ^{0,1} \big/\partial \phi _1\nonumber\\ &&- ({1 + 3A_1 /4 + 3A_2 /4})B_1^{0,1} \big\} \big] \nonumber\\ V &=& ({\partial /\partial \phi _2 })\big[\big\{ \omega _2^2 \partial ^2 \big/\partial \phi _2^2 - 1 - 3A_1 /2 - 3A_2 /2 + 2E\big\}\nonumber\\ &&\times\big\{ {-} \omega _2 \partial B_1^{1,0} \big/\partial \phi _2 + ({1 + 3A_1 /4 + 3A_2 /4})B_1^{0,1} \big\} \nonumber\\ &&+ \{ - 2({1 + 3A_1 /4 + 3A_2 /4})\omega _2 \partial /\partial \phi _2 + G\}\nonumber\\ &&\times\big\{ {-} \omega _2 \partial B_1^{0,1} \big/\partial \phi _2 + ({1 + 3A_1 /4 + 3A_2 /4})B_1^{0,1} \big\} \big], \nonumber \end{eqnarray}$$

X₃, Y₃ are the homogeneous components of order 3 obtained by substituting, respectively,

$$\begin{eqnarray}x &=& B_1^{1,0} \, + B_2^{1,0},\, \quad y = B_1^{0,1} + B_2^{0,1},\nonumber\\ &&{\rm into}\, \frac{\partial }{{\partial x}}(L_3 + L_4),\quad \frac{\partial }{{\partial y}}(L_3 + L_4).\nonumber\end{eqnarray}$$

We do not require to find out the components B^1,0₃ and B^0,1₃. We find the coefficients of cos ϕ₁ , sin ϕ₁, cos ϕ₂, and sin ϕ₂ on the right-hand sides of Equation (14). They are the critical terms. We eliminate these terms by properly choosing the coefficients in the polynomials

$$\begin{eqnarray}&&f_2 = f_{2,0} I_1 + f_{0,2} I_2,\quad {\rm and}\quad g_2 = g_{2,0} I_1 + g_{0,2} I_2.\nonumber\end{eqnarray}$$

Further, we find that

$$\begin{eqnarray} f_{2,0} &=& \frac{1}{2}\frac{{({\rm coefficient\ of}\cos \phi _1\ {\rm in }\ \Phi _3)}}{{({\rm coefficient\ of}\cos \phi _1\ {\rm in\ }P)}} = A,\nonumber \\ f_{2,0} &=& \,g_{2,0} = \frac{1}{2}\frac{{({\rm coefficient\ of\ }\cos \phi _2 {\rm in\ }\Phi _3)}}{{({\rm coefficient\ of\ }\cos \phi _2\ {\rm in\ }Q)}} = B,\nonumber \\ \,g_{2,0} &=& \frac{1}{2}\frac{{({\rm coefficient\ of\ }\cos \phi _2\ {\rm in\ }\Psi _3)}}{{({\rm coefficient\ of\ }\cos \phi _2\ {\rm in\ }Q)}} = C.\nonumber \end{eqnarray}$$

with

$$\begin{eqnarray} A &=& A_{_1,_1 } + (A_{1,2} + A_{1,3} \gamma)A_1 + (A_{1,4} + A_{1,5}\gamma)A_2\nonumber\\ &&+ (A_{1,6} + A_{1,7}\gamma)\in \,{+ }\,(A_{1,8} + A_{1,9 }\gamma) \in ^\prime, \nonumber\\ B &=& B_{1,1} + (B_{1,2} + B_{1,3} \gamma)A_1 + (B_{1,4} + B_{1,5} \gamma)A_2 \nonumber\\ &&+ (B_{1,6} + B_{1,7} \gamma)\in +\, (B_{1,8} + B_{1,9} \gamma) \in ^\prime, \nonumber\\ C &=& C_{1,1} + (C_{1,2} + C_{1,3} \gamma)A_1 + (C_{1,4} + C_{1,5} \gamma)A_2 \nonumber\\ &&+ (C_{1,6} + C_{1,7} \gamma) \in +\, (C_{1,8} + C_{1,9} \gamma) \in ^\prime, \nonumber \end{eqnarray}$$

where the coefficients A, B, C are given in Appendix C.

Since condition (1) of Moser's theorem (Moser 1962) is applicable, Birkloff's normalization up to third order can be obtained. Condition (1) of Moser's theorem is satisfied in the interval 0 < μ < μ_co if the mass ratio does not take the critical values:

$$\begin{eqnarray} \mu _{{\rm c}_{1} } &=& 0.0242938 \ldots - 0.1790727 \ldots A_1 - 0.0368505 \ldots A_2\nonumber\\ &&- 0.0055364 \ldots \in - 0.0055364 \ldots \, \in ^\prime \nonumber\\ \mu _{{\rm c}_{2} } &=& 0.013516 \ldots - 0.099383 \ldots A_1 - 0.019383 \ldots A_2 \nonumber\\ &&- 0.0030452 \ldots \in - 0.0030452 \ldots \, \in ^\prime. \nonumber \end{eqnarray}$$

These results, in the case A₂ = ∈ = ∈' =  0, agree with those found by Markellos et al. (1996), Subba Rao & Sharma (1997), and Chandra & Kumar (2004).

Normalized Hamiltonian up to fourth order is

$$\begin{eqnarray}&&H = \omega _1 I_1 - \omega _2 I_2 + \case{1}{2}\big({AI_1^2 + 2BI_1 I_2 + CI_2^2 } \big) + \cdots,\nonumber \end{eqnarray}$$

where the coefficients A, B, C are given in Appendix C.

Calculating the determinant D occurring in condition (2) of the Moser (1962) theorem, we have

$$\begin{eqnarray}&&D = - \big({A\omega _2^2 + 2B\omega _1 \omega _2 + C\omega _1^2 } \big).\nonumber\end{eqnarray}$$

Putting the values of A, B, and C, we have

$$\begin{eqnarray} D &=& [(644u^4 - 541u^2 + 36)/8(4u^2 - 1)(25u^2 - 4)] \nonumber\\ &&+ (D_2 + D_3 \gamma)A_1 + (D_4 + D_5 \gamma)A_2 \nonumber\\ &&+ (D_6 + D_7 \gamma) \in + (D_8 + D_9 \gamma) \in ^\prime \nonumber \end{eqnarray}$$

with

$$\begin{eqnarray} D_{2} &=& [(39176u^4 - 14359u^2 + 492)]/[48(4u^2 - 1)(25u^2 - 4)], \nonumber\\ D_{3} &=& [3(1593600u^{10} + 2122096u^8 - 13052000u^6 + 5408175u^4\nonumber\\ &&- 840076u^2 + 23616)]/[16(4u^2 - 1)^2\nonumber\\ &&\times(25u^2 - 4)^2 (16u^2 + 117)],\nonumber\\ D_{4} &=& [59944u^6 + 14269689u^4 - 12215187u^2\nonumber\\ &&+ 166284]/[1962(4u^2 - 1)2(25u^2 - 4)^2 ], \nonumber\\ D_{5} &=& [55977008u^{10} + 16915980u^8 - 694540170u^6\nonumber\\ &&+ 171574880u^4 - 31023250u^2 + 732927]/\nonumber\\ &&[208(4u^2 - 1)^2 (25u^2 - 4)^2 (16u^2 + 117)], \nonumber\\ D_{6} &=& [195560u^6 - 549558u^4 + 24575u^2\nonumber\\ &&+ 48]/[432(4u^2 - 1)^2 (25u^2 - 1)]\nonumber\\ D_7 &=& [48947712u^{12} + 617901664u^{10} - 454674128u^8\nonumber\\ &&+ 245406416u^6 - 73692732u^4 + 8118909u^2 - 71472]/\nonumber\\ &&[96(4u^2 - 1)^3 (25u^2 - 4)^2 (16u^2 + 117] \nonumber\\ D_{8} &=& [19440u^8 - 578504u^6 + 272091u^4 - 33970u^2\nonumber\\ &&+ 216]/[654(4u^2 - 1)^2 (25u^2 - 4)], \nonumber\\ D_{9} &=& [252817792u^{12} + 2444733120u^{10} - 1258751060u^8\nonumber\\ &&- 45728424u^6 + 32733556u^4 - 5336499u^2 + 631382]/\nonumber\\ &&[48(4u^2 - 1)^3(25u^2 - 4)^2 (16u^2 + 117)]. \nonumber \end{eqnarray}$$

Moser's second condition is violated for the unperturbed problem, i.e., for A₁ = A₂ = ∈ = ∈' = 0 when μ₀ = 0.0109136.... When A₁, A₂ ∈, ∈ ' ≠ 0, we take μ = μ₀ + αA₁ + α'A₂ + α'' ∈ +α'^'' ∈ ' + such thatD = 0. It is seen that the second condition of Moser's theorem is satisfied, that is D ≠ 0 if in the interval 0 < μ < μ_co, the mass ratio does not take the value

$$\begin{eqnarray}&&\mu _{c_3 } = \mu _0 + \alpha A_1 + \alpha ^\prime A_2 + \alpha ^{\prime \prime} \in + \alpha ^{\prime \prime \prime} \in ^\prime,\nonumber \end{eqnarray}$$

where

$$\begin{eqnarray} \alpha &=& {-} 0.1036878 \ldots,\quad \alpha ^\prime = 0.5456468\ldots, \nonumber\\ \alpha ^{\prime \prime} &=& {-} 0.4748319 \ldots,\quad \alpha ^{\prime \prime \prime} = - 0.1432604 \ldots. \nonumber \end{eqnarray}$$

By taking both the primaries as oblate spheroids as well as sources of radiation, it has been seen that, in the nonlinear sense, the triangular point L₄ is stable for all mass ratios in the range of linear stability except for three mass ratios,

$$\begin{eqnarray} \mu _{{\rm c}_{1} } &=& 0.0242 \ldots - 0.1790 \ldots A_1 - 0.0368 \ldots A_2\nonumber\\ &&- 0.0055 \ldots \in - 0.0055 \ldots \in ^\prime, \nonumber\\ \mu _{{\rm c}_{2} } &=& 0.0135 \ldots - 0.0993 \ldots A_1 - 0.093 \ldots A_2\nonumber\\ &&- 0.0030 \ldots \in - 0.0030 \ldots \in ^\prime \,, \nonumber\\ \mu _{{\rm c}_3 } &=& 0.0109 \ldots - 0.1036 \ldots A_1 + 0.5456 \ldots A_2\nonumber\\ &&- 0.4748 \ldots \in - 0.1432 \ldots \in ^\prime, \nonumber \end{eqnarray}$$

at which the Moser (1962) theorem fails.

When A₁A₂ = ∈ = ∈' = 0, the values of μ_ci(i = 1, 2, 3) agree with those found by Deprit & Deprit-Bartholome (1967).

When A₂ = ∈ = ∈' = 0, the values of μ_ci are the same as those found by Markellos et al. (1996), Subba Rao & Sharma (1997), and Chandra & Kumar (2004).

The author is extremely thankful to Professor B. Ishwar, Department of Mathematics, B. R. A. Bihar University, Muzaffarpur, India for his valuable suggestions.

$$\begin{eqnarray} J_{1,3} &=& l_1 \big[1 + 33A_1 \big/4 l_1^2 - 3A_1 \gamma \big/4k_1^2 + 33A_2 \big/4l_1^2\nonumber\\ && + 3A_2 \gamma \big/4k_1^2- \big({3\omega _1^2 - 7} \big) \in \big/2l_1^2 k_1^2 \nonumber\\ &&- \big({3\omega _1^2 - 7} \big) \in \gamma \big/2 l_1^2 k_1^2 - \big({3\omega _1^2 - 7} \big) \in ^\prime \big/2 l_1^2 k_1^2\nonumber\\ &&+ \big({7\omega _1^2 + 2} \big) \in ^\prime \gamma \big/2 l_1^2 k_1^2 \big]\big/2\omega _1 k_1 \nonumber\\ J_{2,1} \,\, &=& {-} 4\omega _1 \big[l + 6k_1^2 A_1 \big/4 l_1^2 - 3A_1 \gamma \big/4k_1^2 + 6k_1^2 A_2 \big/4 l_1^2\nonumber\\ &&+ 3A_2 \gamma \big/4k_1^2 + \big({7\omega _1^2 + 2} \big) \in \big/2 l_1^2 k_1^2 \nonumber\\ &&+ \big({7\omega _1^2 + 2} \big) \in \gamma \big/2 l_1^2 k_1^2 + \big({7\omega _1^2 + 2} \big) \in ^\prime \big/2 l_1^2 k_1^2\nonumber\\ &&- \big(3\omega _1^2 - 7\big) \in ^\prime \gamma \big/2l_1^2 k_1^2 \big]\big/l_1 k_1 \nonumber\\ J_{2,3} &=& \sqrt 3 \big[6\gamma + \big({ - 16\omega _1^4 + 88\omega _1^2 - 63} \big)A_1 \big/6k_1^2\nonumber\\ &&+ \big(104\omega _1^2 + 135 \big)A_1 \gamma \big/2 l_1^2 \nonumber\\ &&- \big({-} 16\omega _1^4 + 88\omega _1^2 - 63 \big)A_2 \big/6k_1^2 \nonumber\\ &&+ \big(104\omega _1^2 + 135 \big)A_2 \gamma /2l_1^2 + \big(112\omega _1^6 - 136\omega _1^4\nonumber\\ &&- 22l\omega _1^2 - 189 \big) \in \big/9k_1^2 l_1^2 \nonumber\\ &&- \big({24\omega _1^4 + 2l\omega _1^2 - 33} \big) \in \gamma \big/l_1^2 k_1^2 - \big(112\omega _1^6 + 136\omega _1^4\nonumber\\ &&- 221\omega _1^2 - 189 \big) \in ^\prime \big/9k_1^2 l_1^2 \nonumber\\ &&- \big({24\omega _1^4 + 2l\omega _1^2 - 33} \big) \in ^\prime \gamma \big/l_1^2 k_1^2 \big], \nonumber \end{eqnarray}$$

with

$$\begin{eqnarray} &&l_J^2 = \omega _1^2 + 9(J = 1,2)\quad k_1^2 = 2\omega _1^2 - 1,\quad k_2^2 = 1 - 2\omega _2^2.\nonumber\end{eqnarray}$$

The values of J_1,4J_2,2J_2,4 can be obtained respectively from J_1,3, − J_2,1, J_2,3 by replacing ω₁ by ω₂, I₁ by I₂, k₁ by k₂, k²₁ by −k²₂ wherever they occur keeping A₁, A₂, ∈, ∈ ' unchanged.

$$\begin{eqnarray} {\bf r}_{\rm i} &=& r_{_{\rm i},_1 } \gamma + r_{i,2} A_1 + r_{i,3} A_1 \gamma + r_{i,4} A_2 + r_{i,5} A_2 \gamma + r_{i,6} \in \nonumber\\ &&+ r_{i,7} \in \gamma + r_{i,8} \in ^\prime + r_{i,9} \in ^\prime \gamma \nonumber\\ {\bf r}_{\rm j} &=& r_{_{j,1} } + r_{j,2} A_1 + r_{j,3} A_1 \gamma + r_{j,4} A_2 + r_{j,5} A_2 \gamma + r_{j,6} \in \nonumber\\ &&+ r_{j,7} \in \gamma + r_{j,8} \in ^\prime + r_{j,9} \in ^\prime \gamma, \nonumber\\ {\bf s}_i &=& s_{_{i,1} } + s_{i,2} A_1 + s_{i,3} A_1 \gamma + s_{i,4} A_2 + s_{i,5} A_2 \gamma + s_{i,6} \in \nonumber\\ &&+ s_{i,7} \in \gamma + s_{i,8} \in ^\prime + s_{i,9} \in ^\prime \gamma, \nonumber\\ {\bf s}_j &=& s_{_{j,1} } \gamma + s_{j,2} A_1 + s_{j,3} A_1 \gamma + s_{j,4} A_2 + s_{j,5} A_2 \gamma + s_{j,6} \in \nonumber\\ &&+ s_{j,7} \in \gamma + s_{j,8} \in ^\prime + s_{j,9} \in ^\prime \gamma, \nonumber \end{eqnarray}$$

for i = 1, 2, 3, 4, 7, 8 and j = 5, 6, 9, 10.

The values of all r_i,1, r_j,1, r_i,2, r_j,2, r_i,3, r_j,3, s_i,1, s_j,1, s_i,2, s_j,2, s_i,3, s_j,3 are the same as those in the paper by Subba Rao & Sharma (1997):

$$\begin{eqnarray} {\bf r}_{1,4} &=& \big[ {-} 958464\omega _1^{10} - 1709568\omega _1^8 + 14119744\omega _1^6 \nonumber\\ &&+ 15690052\omega _1^4 - 9561825\omega _1^2 \nonumber\\ &&+ 3082941\big]\big/27648\omega _1^3 I_1^2 k_1^4 \big({1 - \omega _1^4 } \big) \nonumber\\ {\bf r}_{1,5} &=& \big[ {-} 612352\omega _1^8 - 780840\omega _1^6 - 25430134\omega _1^4 \nonumber\\ &&+ 2831380\omega _1^2 + 1700271 \big]\big/1024\omega _1^3 I_1^4 k_1^2 \big({1 - \omega _1^2 } \big) \nonumber\\ {\bf r}_{1,6} &=& \big[200704\omega _1^{12} - 2838528\omega _1^{10} - 9627904\omega _1^8 \nonumber\\ &&+ 4520136\omega _1^6 - 5069434\omega _1^4 - 2258551\omega ^2 \nonumber\\ &&+ 9372528\big]\big/13824\omega _1^3 l_1 {}^4k_1^4 (1 - \omega ^2), \nonumber\\ {\bf r}_{1,7} &=& - \big[12672\omega _1^{10} + 29520\omega _1^8 + 32568\omega _1^6 + 95792\omega _1^4 \nonumber\\ &&+ 168210\omega _1^2 - 130491\big]\big/64 \omega _1^3l_1^4k_1^4\big(1 - \omega _1^2\big) \nonumber\\ {\bf r}_{1,8} &=& \big[1024\omega _1^{12} + 1964312\omega _1^{10} - 391680\omega _1^8 \nonumber\\ &&- 5436512\omega _1^6 + 2663658\omega _1^4 - 3674889\omega _1^2 \nonumber\\ &&+ 256962\big]\big/6912\omega _1^3 l_1 {}^4k_1^4 \big(1 - \omega _1^2\big) \nonumber\\ {\bf r}_{1,9} &=& \big[8448\omega _1^{10} - 79008\omega _1^8 - 65455\omega _1^6 - 186064\omega _1^4 \nonumber\\ &&- 89964\omega _1^2 + 88938\big]\big/128\omega _1^3 l_1 {}^4k_1^4 \big(1 - \omega _1^2\big), \nonumber\\ {\bf r}_{3,4} &=& \big[20353\omega _1^{10} - 45342\omega _1^8 + 52839\omega _1^6 - 73059\omega _1^4\nonumber\\ && + 25394\omega _1^2 - 1145\big]\big/\big(288\omega _1 l_1^2 k_1 {}^4z_1^2\big), \nonumber\\ {\rm r}_{3,5} &=& \big[1755\omega _1^{6} - 5809\omega _1^4 - 98766\omega _1^2 - 2544\big]\big/\nonumber\\ &&\big(16\omega _1 l_1 {}^4k_1 {}^2z_1\big), \nonumber\\ {\bf r}_{3,6} &=& \big[16535\omega _1^{12} - 203943\omega _1^{10} + 17528\omega _1^8 + 23197\omega _1^6 \nonumber\\ &&- 889120\omega _1^4 - 336119_1^2 + 786267\big]\big/\big(144\omega _1 {}^4k_1 {}^4z_1 ^2\big), \nonumber\\ {\bf r}_{3,8} &=& r_{3,6}, {\bf r}_{3,9} = - r_{3,7} \nonumber\\ {\bf r}_{5,4} &=& \surd 3\big[2890240\omega _1^{12} + 589184\omega _1^{10} + 1462400\omega _1^8 \nonumber\\ &&- 5521120\omega _1^6 + 2246062\omega _1^4 - 680835\omega _1^2 + 105705\big]\big/\nonumber\\ &&\big(864\omega _1^2 l_1 {}^4k_1 {}^4z_1^2\big), \nonumber\\ {\bf r}_{5,5} &=& \surd 3\big[1800\omega _1^8 - 4496\omega _1^6 - 2731\omega _1^4 \nonumber\\ &&+ 1890\omega _1^2 - 243\big]\big/\big(6\omega _1^2 l_1 {}^2k_1 {}^4z_1 ^2\big), \nonumber\\ {\bf r}_{5,6} &=& {-} \surd 3\big[ {-} 56320\omega _1^{14} - 576128\omega _1^{12} + 620624\omega _1^{10} \nonumber\\ &&- 17304\omega _1^8 - 1638348\omega _1^6 + 1669932\omega _1^4 - 902583\omega _1^2 \nonumber\\ &&+ 124659\big]\big/\big(288\omega _1^4 l_1 {}^4k_1^4 z_1^2\big), \nonumber\\ {\bf r}_{5,7} &=& {-} \surd 3\big[ {-} 160896\omega _1^{12} + 8192\omega _1^{10} - 271752\omega _1^8\nonumber\\ && - 531644\omega _1^6 + 7594492\omega _1^4 - 664848\omega _1^2 + 104976\big]\big/\nonumber\\ &&\big(288\omega _1^4 l_1 {}^4k_1^4 z_1^2\big), \nonumber \end{eqnarray}$$

$$\begin{eqnarray} {\bf r}_{5,8} &=& r_{5,6}\quad {\bf r}_{5,9} = - r_{5,7} \nonumber\\ {\bf r}_{7,4} &=& {-} [156u^5 - 536325u^4 - 81902u^3 \nonumber\\ &&+ 15809u^2 - 34119u - 16140]/8uz_3 (5u - 2)^2 \nonumber\\ {\bf r}_{7,5} &=& 3[6156u^4 + 42688u^3 + 24721u^2 \nonumber\\ &&+ 145069u + 27438]/8z_3 (5u - 2)(16u^2 + 117), \nonumber\\ {\bf r}_{7,6} &=& [184u^5 + 36964u^4 + 83154u^3 - 20520u^2 \nonumber\\ &&+ 48117u + 16560]/96uz_3 (5u - 2)^2, \nonumber\\ {\bf r}_{7,7} &=& 3[1296u^4 + 52996u^3 + 72849u^2 + 168072u \nonumber\\ &&+ 53848]/64z_3 (5u - 2)(16u^2 + 117), \nonumber\\ {\bf r}_{7,8} &=& r_{7,6}, {\bf r}_{7,9} = - r_{7,7} \nonumber\\ {\bf r}_{9,4} &=& \surd 3[32774u^4 - 3568u^3 - 24u^2 + 36882u \nonumber\\ &&+ 85368]\surd (1 - 2u)/12z_3 (5u - 2)(16u^2 + 117) \nonumber\\ {\bf r}_{9,5} &=& 3\surd 3[1284u^4 + 4055u^3 + 1650u^2 \nonumber\\ &&- 1924u + 27]/16uz_3 (5u - 2)2\surd (1 - 2u), \nonumber\\ {\bf r}_{9,6} &=& \surd 3[1012u^4 - 5468u^3 - 84u^2 + 53972u \nonumber\\ &&+ 48956]\surd (1 - 2u)/8z_3(5u - 2)(16u^2 + 117), \nonumber\\ {\bf r}_{9,7} &=& 3\surd 3[5462u^4 + 6412u^3 + 2436u^2 \nonumber\\ &&- 184u + 81]/8uz_3 (5u - 2)2\surd (1 - 2u) \nonumber \end{eqnarray}$$

$$\begin{eqnarray} {\bf r}_{9,8} &=& r_{9,6}, {\bf r}_{9,9} = - r_{9,7} \nonumber\\ {\bf s}_{1,4} &=& \surd 3\big[853234\omega _1^{10} + 313266\omega _1^8 + 11536480\omega _1^{6}\nonumber\\ && + 20235825\omega _1^4 - 9284569\omega _1^2 + 1032645\big]\big/\nonumber\\ &&6912\omega _1^3 l_1 {}^2k_1^4 \big(1 - \omega _1^2\big). \nonumber\\ {\bf s}_{1,5} &=& \surd 3\big[324860\omega _1^8 - 12158552\omega _1^6 + 13235261\omega _1^4 \nonumber\\ &&- 2413835\omega _1^2 - 2300032\big]\big/\big(2048\omega _1^3 l_1 ^4 k_1^2 \big(1 - \omega _1^2\big), \nonumber\\ {\bf s}_{1,6} &=& \surd 3\big[100802\omega _1^{12} + 1535328\omega _1^{10} - 322955458\omega _1^8 \nonumber\\ &&+ 19231146\omega _1^6 - 8256628\omega _1^4 + 3381658\omega _1^2 \nonumber\\ &&- 8727857\big]\big/6912\omega _1^3 l_1 {}^4k_1^4 \big(1 - \omega _1^2\big), \nonumber\\ {\bf s}_{1,7} &=& \surd 3\big[24686\omega _1^{10} - 92130\omega _1^8 + 65323\omega _1^6 - 59944\omega _1^4 \nonumber\\ &&- 452820\omega _1^2 + 222465\big]\big/32\omega _1^3 l_1 {}^4k_1^4 \big(1 - \omega _1^2\big), \nonumber\\ {\bf s}_{1,8} &=& \surd 3\big[1256\omega _1^{12} + 8553678\omega _1^{10} - 134730\omega _1^8 + 3162025\omega _1^6 \nonumber\\ &&- 865856\omega _1^4 + 7489056\omega _1^2 - 852967\big]\big/\nonumber\\ &&3456\omega _1^3 l_1 {}^4k_1^4 \big(1 - \omega _1^2\big), \nonumber\\ {\bf s}_{1,9} &=& \surd 3\big[5616\omega _1^{10} + 82809\omega _1^8 - 73525\omega _1^6 + 825292\omega _1^4 \nonumber\\ &&+ 16568 - 63598\big] \big/256\omega _1^3 l_1 {}^4k_1^4 \big(1 - \omega _1^2\big), \nonumber\\ {\bf s}_{3,4} &=& \surd 3\big[16881\omega _1^{10} - 64001\omega _1^8 + 632153\omega _1^6 \nonumber\\ &&+ 16593\omega _1^4 - 11657\omega _1^2 + 6352\big]\big/\big(288\omega _1 l_1 {}^2k_1 {}^4z_1^2\big), \nonumber\\ {\bf s}_{3,5} &=& \surd 3\big[1656\omega _1^6 + 5701\omega _1^4 - 23459\omega _1^2 + 2862\big]\big/\nonumber\\ &&\big(32\omega _1 l_1 {}^4k_1^2 z_1\big) \nonumber\\ {\bf s}_{3,6} &=& \surd 3\big[24954\omega _1^{12} - 669953\omega _1^{10} + 63527\omega _1^8 + 32286\omega _1^6 \nonumber\\ &&- 465130\omega _1^4 + 118316\omega _1^2 - 956210\big]\big/\big(288\omega _1^4 l_1 {}^4k_1 {}^4z_1^2\big), \nonumber\\ {\bf s}_{3,7} &=& \surd 3\big[ {-} 502810\omega _1^{10} + 115652\omega _1^8 - 15418 \omega _1^6 + 468880\omega _1^4 \nonumber\\ &&- 365710\omega _1^2 + 325795\big]\big/\big(144\omega _1^4 l_1 {}^4k_1^4 z_1^2\big), \nonumber \end{eqnarray}$$

$$\begin{eqnarray} {\bf s}_{3,8} &=& s_{3,6}, {\bf s}_{3,9} = - s_{3,7} \nonumber\\ {\bf s}_{5,4} &=& \big(1658\omega _1^8 + 2853\omega _1^6 + 5258\omega _1^4 \nonumber\\ &- &1628\omega _1^2 + 1768\big)\big/\big(54l_1 {}^2k_1 {}^4z_1^2\big), \nonumber\\ {\bf s}_{5,5} &=& \big(2816\omega _1^6 + 3639\omega _1^4 - 1159\omega _1^2 + 5308\big)\big/8l_1 {}^4k_1 {}^2z_1\big), \nonumber\\ {\bf s}_{5,6} &=& \big(8752\omega _1^8 + 1153\omega _1^6 - 8816\omega _1^4 \nonumber\\ &&+ 5353\omega _1^2 + 1844\big)\big/\big(12l_1 {}^2k_1^4 z_1^2\big), \nonumber\\ {\bf s}_{5,7} &=& \big(6751\omega _1^6 - 8559\omega _1^4 + 6353\omega _1^2 - 1008\big)\big/\big(8l_1 {}^4k_1^2 z_1\big), \nonumber\\ {\bf s}_{5,8} &=& s_{5,6}, {\bf s}_{5,9} = - s_{5,7} \nonumber\\ {\bf s}_{7,4} &=& - \surd 3(1180u^5 - 45081u^4 + 15932u^3 + 283362u^2 \nonumber\\ &&+ 148157u + 638264)/72(5u - 2)(16u^2 + 117)z_3 \nonumber\\ {\bf s}_{7,5} &=& - \surd 3(84u^4 + 8212u^3 - 1654u^2 \nonumber\\ &&+ 1553u - 3372)/16u(54 - 2)^2 z_3 \nonumber\\ {\bf s}_{7,6}& =& - \surd 3(114u^5 - 39368u^4 + 80456u^3 - 10820u^2 \nonumber\\ &&+ 34168u - 33582)/36(5u - 2)(16u^2 + 117)z_3 \nonumber\\ {\bf s}_{7,7} &=& - \surd 3(3496u^4 - 82756u^3 - 92346u^2 \nonumber\\ &&+ 148052u + 91637)/64(5u - 2)^2 z_3 \nonumber\\ {\bf s}_{9,4} &=& (20278u^5 + 16643u^4 - 30128u^3 + 14286u^2 \nonumber\\ &&+ 4802u + 4563)/48u(5u - 2)^2 z_3 \nonumber\\ {\bf s}_{9,5} &=& 3(3641u^3 + 1814u^2 + 5263u + 7209)/\nonumber\\ &&8(5u - 2)(16u^2 + 117)z_3, \nonumber\\ {\bf s}_{9,6} &=& (72082u^5 + 68809u^4 - 23756u^3 - 28270u^2 \nonumber\\ &&+ 32883u - 3609)/36u(5u - 2)^2 z_3 \surd (1 - 2u), \nonumber\\ {\bf s}_{9,7} &=& 3(3786u^3 + 3606u^2 + 3949u - 8008)\surd (1 - 2u)/\nonumber\\ &&8(5u - 2)(16u2 + 117)z_3, \nonumber\\ {\bf s}_{9,8} &=& s_{9,6}, {\bf s}_{9,9} = - s_{9,7} \nonumber\\ z_1 &=& 1 - 5\omega _1^2, z_2 = 1 - 5\omega _2^2 \nonumber\\ z_3 &=& l_1 l_2 k_1 k_2 \surd u, u = \omega _1 \omega _2. \nonumber \end{eqnarray}$$

The values of r_k, s_k, for k = 2, 4, 6 can be obtained respectively from those for k = 1, 3, 5 by replacing ω₁ by −ω₂, l₁ by l₂k²₁ by −k²₂z₁ by z₂ wherever they occur and the values of r_k, s_k for k = 8, 10 can be obtained respectively from those for k = 7, 9 by replacing u by −u, keeping z₃ unchanged, wherever they occur.

The values of all A_1,1, A_1,2, A_1,3, B_1,1, B_1,2, B_1,3, C_1,1, C_1,2, and C_1,3 are the same as those in the paper by Subba Rao & Sharma (1997):

$$\begin{eqnarray} A_{1,4} &=& \big(1319 - 12639\omega _1^2 + 18275\omega _1^4 - 1598\omega _1^6\big)\big/\nonumber\\ &&436\big(1 - 2\omega _1^2\big)^2 \big(1 - 5\omega _1^2\big), \nonumber\\ {\rm A}_{1,5} &=& \big(57 + 525\omega _1^2 - 950\omega _1^4 + 3118\omega _1^6 + 1132\omega _1^8\big)\big/\nonumber\\ &&52\big(1 - 2\omega _1^2\big)^3 \big(1 - 5\omega _1^2\big) \nonumber\\ {\rm A}_{1,6} &=& \big(24 - 624\omega _1^2 + 416\omega _1^4 - 2004\omega _1^6 + 856\omega _1^8 \big)\big/\nonumber\\ &&864\big(1 - 2\omega _1^2\big)^2 \big(1 - 5\omega _1^2\big) \nonumber\\ {\rm A}_{{1,7}} &=& \big(32 - 842\omega _1^2 - 608\omega _1^4 - 1628\omega _1^6 + 1856\omega _1^8\big)/\nonumber\\ &&64\big(1 - 2\omega _1^2\big)^3 \big(1 - 5\omega _1^2\big)^2 \nonumber\\ {\rm A}_{1,8} &=& \big(72 - 884\omega _1^2 - 204\omega _1^4 + 1600\omega _1^6 + 648\omega _1^8\big)\big/\nonumber\\ &&872\big(1 - 2\omega _1^2\big)^2 \big(1 - 5\omega _{1}^{2}\big) \nonumber\\ {\rm A}_{1,9} &=& \big(36 - 432\omega _1^2 + 448\omega _1^4 + 1208\omega _1^6 + 896\omega _1^8\big)\big/\nonumber\\ &&64\big(1 - 2\omega _1^2\big)^3 \big(1 - 5\omega _1^2\big)^2, \nonumber\\ {\rm B}_{1,4} &=& \omega _1 \omega _2 \big(2223 - 6817\omega _1^2 \omega _2^2\big)\big/36\big(1 - 2\omega _1^2\big)\nonumber\\ &&\times\big(1 - 2\omega _2^2\big)\big(1 - 5\omega _1^2\big)\big(1 - 5\omega _2^2\big) \nonumber \end{eqnarray}$$

$$\begin{eqnarray} {\rm B}_{1,5} &=& \big(89211 - 2042998\omega _1^2 \omega _2^2 + 10285776\omega _1^4 \omega _2^4 \nonumber\\ &&- 16052098\omega _1^6 \omega _2^6 - 1215804\omega _1^8 \omega _2^8\big)\big/32\omega _1 \omega _2 \big(4\omega _1^2 + 9\big)\nonumber\\ &&\times\big(4\omega _2^2 + 9\big)\big(1 - 2\omega _1^2\big)\big(1 - 2\omega _2^2\big)\big(1 - 5\omega _1^2\big)^2 \big(1 - 5\omega _2^2\big)^2 \nonumber\\ {\rm B}_{1,6} &=& \omega _1 \omega _2 \big(1854 - 3256\omega _1^2 \omega _2^2\big)\big/48\big(1 - 2\omega _1^2\big)\nonumber\\ &&\times\big(1 - 2\omega _2^2\big)\big(1 - 5\omega _1^2\big)\big(1 - 5\omega _2^2\big) \nonumber\\ {\rm B}_{1,7} &=& \big(40332 - 2456280\omega _1^2 \omega _2^2 + 1603200\omega _1^4 \omega _1^4 \nonumber\\ &&- 12140864\omega _1^6 \omega _2^6 - 1043208\omega _1^8 \omega _2^8\big)\big/48\omega _1 \omega _2 \big(4\omega _1^2 + 9\big)\nonumber\\ &&\times\big(4\omega _2^2 + 9\big)\big(1 - 2\omega _1^2\big) \big(1 - 2\omega _2^2\big)\big(1 - 5\omega _1^2\big)^2 \big(1 - 5\omega _2^2\big)^2 \nonumber\\ {\rm B}_{1,8} &=& \omega _1 \omega _2 \big(1264 - 4868\omega _1^2 \omega _2^2\big)\big/48\big(1 - 2\omega _1^2\big)\nonumber\\ &&\times\big(1 - 2\omega _2^2\big)\big(1 - 5\omega _1^2\big)\big(1 - 5\omega _2^2\big) \nonumber\\ {\rm B}_{1,9} &=& \big(16004 + 1284430\omega _1^2 \omega _2^2 - 1216028\omega _1^4 \omega _2^4 \nonumber\\ &&- 81442042\omega _1^6 \omega _2^6 - 8034956\omega _1^8 \omega _2^8\big)\big/48\omega _1 \omega _2 \big(4\omega _1^2 + 9\big)\nonumber\\ &&\times\big(4\omega _2^2 + 9\big)\big(1 - 2\omega _1^2\big)\big(1 - 2\omega _2^2\big) \big(1 - 5\omega _1^2\big)^2 \big(1 - 5\omega _2^2\big)^2. \nonumber \end{eqnarray}$$

C_1,i(i = 1, 2, 3, ..., 9) can be obtained respectively from A_1i by replacing ω₁ by ω₂.