Collinear Libration Points in the Photogravitational CR3BP with Zonal Harmonics and Potential from a Belt

We have studied a reformed type of the classic restricted three-body problem where the bigger primary is radiating and the smaller primary is oblate; and they are encompassed by a homogeneous circular cluster of material points centered at the mass center of the system (belt). In this dynamical model, we have derived the equations that govern the motion of the infinitesimal mass under the effects of oblateness up to the zonal harmonics J4 of the smaller primary, radiation of the bigger primary and the gravitational potential generated by the belt. Numerically, we have found that, in addition to the three collinear libration points Li (i = 1, 2, 3) in the classic restricted three-body problem, there appear four more collinear points Lni (i = 1, 2, 3, 4). Ln1 and Ln2 result due to the potential from the belt, while Ln3 and Ln4 are consequences of the oblateness up to the zonal harmonics J4 of the smaller primary. Owing to the mutual effect of all the perturbations, L1 and L3 come nearer to the primaries while Ln3 advances away from the primaries; and L2 and Ln1 tend towards the smaller primary whereas Ln2 and Ln4 draw closer to the bigger primary. The collinear libration points Li (i = 1, 2, 3) and Ln2 are linearly unstable whereas the Ln1, Ln3 and Ln4 are linearly stable. A practical application of this model could be the study of motion of a dust particle near a radiating star and an oblate body surrounded by a belt.


Introduction
In celestial mechanics, one amidst various inspiring subject is the restricted three-body problem (R3BP).The problem entails three bodies: two primary bodies having finite masses moving under their mutual gravitational attraction and the third with a negligible-mass (infinitesimal) body, whose motion is influenced by the primaries.If the primaries move on circular orbits about their common centre of mass, it is termed as the circular R3BP (CR3BP).Then, the objective of this CR3BP is to determine the motion of the infinitesimal mass.[1] and [2] gave a detailed description of the solution of the CR3BP.They showed that if the primary bodies were fixed in a rotating coordinate system, five libration points existed.That is the points where the infinitesimal mass can remain permanent, if placed there with zero velocity.Three of the points 1 2 3 , , L L L are on the line linking the primaries, whereas the other two 4 5 , L L are in equilateral triangular alignment with the primaries.The collinear points 1 2 3 , , L L L are linearly unstable, while the triangular points 4 5 , L L are linearly stable for the mass ratio of the primaries less than 0.03852.
Researches on the sites and stability of the libration points of the CR3BP with perturbations have achieved ample attention in recent times.[3] indicated that small particles were equally influenced by the gravitation and light radiation force as they moved toward luminous celestial bodies.[4] [5] established that the presence of direct solar radiation pressure caused a variation in the sites of the libration points of the CR3BP.He called the CR3BP, photogravitational when one or both of the masses of the primaries were discharges of radiation.Researchers [6]- [10] have examined the existence of libration points and their linear stability in the photogravitational CR3BP.
[11] [12] studied a modified CR3BP by considering the influence from a belt (circular cluster of material points) for planetary systems and found that the likelihood to get libration points around the inner part of the belt was greater than the one nigh the outer part.The impact of the belt makes the configuration of the dynamical system altered such that new libration points emerge under certain condition [13]- [16].
The primaries in CR3BP are generally considered to be spherical in shape, whereas in real situations, numerous celestial bodies are non-spherical (e.g. the Earth, Jupiter, Saturn, Regulus stars are oblate).The oblateness of the planets causes large deviations from a two-body orbit.The most salient instance of disturbance due to oblateness in the solar system is the orbit of the fifth satellite of Jupiter, Amalthea.This planet is extremely oblate and the satellite's orbit is exceptionally small that its line of apsides progresses approximately 900˚ in one year [17].This vindicates the incorporation of oblateness of the primaries in the study of CR3BP [18]- [25].
The orbital effects of the oblateness up to the quadrupole, i.e.J 2 , and the octupole, i.e.J 4 , on the orbital motion of a particle in the field of a non-spherical body have been worked out in the general case of an arbitrarily oriented spin axis [26].[22] certified that the sites of the triangular libration points and their linear stability were influenced by the oblateness up to J 4 of the bigger primary in the CR3BP.[27] examined the effects of photogravitational force and oblateness in the perturbed restricted three-body problem.[15] analyzed analytically and numerically the effects of oblateness up to J 2 of the smaller primary and gravitational potential from the belt on the linear stability of libration points in the photogravitational CR3BP.[16] explored the combined effect of radiation and oblateness up to J 2 of both primaries, together with additional gravitational potential from the circumbinary belt on the motion of an infinitesimal body in the binary stellar systems within the frame work of CR3BP.[9] studied the effects of oblateness up to J 4 of the smaller primary and gravitational potential from a belt, on the linear stability of triangular libration points in the photogravitational CR3BP.[24] looked at the effects of oblateness of both primaries up to zonal harmonic J 4 and gravitational potential from the belt on the linear stability of the triangular libration points in the CR3BP.
Here, our intention is to look into the resultant effect of radiation of the bigger primary, oblateness up to the zonal harmonic J 4 of the smaller primary and gravitational potential from the belt on the sites and stability of collinear libration points in the CR3BP .
The manuscript is structured in five units.Unit 2 deals with the mathematical formulation of the problem, while Unit 3 is dedicated to the determination of the sites of the collinear libration points.The linear stability of collinear points and the conclusion are presented in Units 4 and 5 respectively.

Let 1
m and 2 m be the masses of the primaries with 1 2 m m > , and let m be the mass of the infinitesimal body moving in the plane of motion of the primaries.The positions of the primaries are defined with respect to a rotating coordinate frame oxyz whose x-axis overlaps with the line connecting them and whose origin coincides with the center of mass of 1 m and 2 m .The y-axis is perpendicular to the x-axis and the z-axis is normal to the orbital plane of the primaries.Let r 1 be the distance between m and m 1 , r 2 the distance between m and m 2 ; and R the distance between 1 m and 2 m .The coordinates of m 1 , m 2 and m are (x 1 , 0), (x 2 , 0) and (x, y) correspondingly.Our aim is to find the equations of motion of m under the influence of radiation of 1 m , oblateness up to J 4 of the smaller primary, and a circumbinary belt centred at the origin of the coordinate system oxyz (see Figure 1).

The Kinetic Energy
The kinetic energy (K.E) of the infinitesimal body in the barycentric coordinate system oxyz rotating about z-axis with uniform angular velocity n Figure 1, is given as where over dot represents differentiation with respect to time t.

Force Due to Radiation Pressure
Now, since the radiation pressure force p F varies with distance by the same law as the gravitational attraction force g F and works opposite to it, it is likely that this force will lead to a decrease of the effective mass of the bigger primary.Furthermore this decrease relies on the properties of the particle; it is therefore tolerable to talk about a reduced mass.Hence, the consequential force on the particle is [4] 1 ; where 1 , a constant for a particular particle, is the mass reduction factor.We represent the radiation factor for the bigger primary as 1

Potential Due to an Oblate Body
In free space the gravitational potential exterior to an oblate body with its mass distributed symmetrically about its equator, can be expanded in terms of Legendre polynomials in the form Figure 1.The planar configuration of the problem.[28].Equation ( 3) is expressed in standard spherical coordinates, with f the longitude and q representing the angle between the body's symmetry axis and the vector to a particle r o (i.e., the colatitudes).R o is the mean radius of the oblate body.The terms ( ) θ are the Legendre polynomials, given by J 2n are dimensionless coefficients that characterize the size of non spherical components of the potential, called the zonal harmonic coefficients.Since the present study is concerned with planar problem, assuming the equatorial plane of the smaller primary coincides with the plane of motion, then with 90 We denote the oblateness coefficient for the smaller primary as B i ,

Potential Due to the Belt
The gravitational potential from belt (circular cluster of material points) centered at the origin of a coordinates system oxyz, Figure 1 as specified by [29] is where b M is the total mass of the belt, 2

The Potential Energy of the Infinitesimal Body
The potential energy of the infinitesimal body, under the influence of the oblateness up to J 4 of smaller primary, radiation of the bigger primary and the circumbinary belt, now takes the form ( ) with ( ) G is the gravitational constant.

The Equations of Motion
We start from Lagrangian (L) of the problem which is the kinetic energy minus the potential energy of the infinitesimal body.That is where ( ) Subsequently, we obtain the equations of motion of the infinitesimal body as To covert the variables to non dimensional, we choose unit for the mass as the sum of the masses of the primaries, the unit of length as the distance between the primaries and unit of time is such that the gravitational constant is unit.Consequently, 1 1 is the mass ratio.Thus, in the dimensionless synodic coordinate system, the equations of motion (10) reduce to 2 , 2 , x y x ny y nx with and n is the mean motion, given by [24] as ( ) c r is the radial distance of the infinitesimal body in the classical restricted three-body problem.

Locations of Collinear Libration Points
We now search for possible collinear libration points of the infinitesimal mass in the rotating reference frame.The libration points are positions of gravitational balance between the primaries.At these points the two finite masses would exert zero net force on the infinitesimal mass, in effect, allowing the infinitesimal mass to have zero velocity in the rotating frame of reference.That is the libration points satisfy . It thus follows, from Equation (11), that the libration points are the solutions of Now, an evident solution of Equation ( 15) is y = 0, corresponding to the collinear libration points (the libration points which lie on the x-axis).This deciphers to Equation ( 16) reduces to those of [1], in the absence of the perturbations.That is when 1 with three collinear points 1 2 , L L and 3 .
L Only the collinear point 2 L is located between the primaries 2).If we consider the effects of the potential from the belt only (i.e.

(
)( ) ( ) ( ) [16] showed that whenever 2 T µ < and ( ) in the interval ( ) , 0 , µ − Equation (18) will have five collinear points (Figure 3).Now, using Equation ( 16) and with the help of the MATLAB (R2007b) software package, we obtain the coordinates of the collinear libration points for different cases as classified in the following order which are portrayed in Table 1: 1) Absence of radiation, oblateness and potential from the belt (classical case).
2) Radiation of the bigger primary only.
3) Potential from the belt only.4) Oblateness of the smaller primary up to J 2 only.5) Oblateness of the smaller primary up to J 4 only.6) Radiation of the bigger primary, oblateness of the smaller primary up to J 4 and potential from the belt.The combined effect of these perturbations on the collinear points is given in Table 2.
In the absence of the perturbations (i.e. 1 1 2

1, 0
) Table 1 Case 1, it is observed that there are three collinear libration points (L i , i = 1, 2, 3) which correspond to the classical case of [1].Owing to the effect of the radiation of the bigger primary only (i.e. ) Case 3, there surface five collinear libration points (L n1 , L n2 and L i , i = 1, 2, 3), this confirms those of [14]- [16].The collinear points L 1 and L 3 shifted nearer to the primaries while L 2 moved away from the bigger primary, due to the potential from the belt.In the presence of the oblateness of the smaller primary up to J 2 only (i.e. 1 ) Case 4, the collinear point L 1 sifted away from the primaries while L 2 and L 3 stepped closer to the bigger primary.In Case 5, due oblateness of the smaller primary up to J 4 only (i.e. ), L n1 moved away from the bigger primary while L n2 stepped towards it.Similarly, owing to the oblateness of the smaller primary up to J 2 with Table 1.Positions of the collinear points when µ = 0.35, q 1 = 0.98, B 1 = 0.01, B 2 = 0.005 and M b = T = 0.01, r c = 0.8789.Table 2. Combined effects of the perturbations on the collinear points when m = 0.35, T = 0.01, r c = 0.8789.
) Case 5, collinear points L 1 and L 3 moved nigh to the primaries while L 2 stepped away from the bigger primary; and there emerge additional two new collinear points L n3 , L n4 .In the presence of all these perturbations (i.e.  2, the collinear points L 1 , L 3 draw closer to the primaries while L n3 moves away from the them; L 2 , L n1 move away from the bigger primary while L n2 , L n4 tend towards it.

Linear Stability of the Collinear Points
To study the stability of a libration point (x 0 , y 0 ), we employ small displacement , η ξ to the coordinates (x 0 , y 0 ).
So, the variations η and ξ can take the form: The superscript "0" indicates that the partial derivatives have been evaluated at the libration point under consideration (x 0 , y 0 ).
Let solutions of the equations of ( 19) be ( ) where ,  and  are constants.Then, Equation (19) will have a non -trivial solution for A and B when On expanding the determinant we obtain the characteristic equation equivalent to the variational equations of (19) as . Now, we obtain the second partial derivatives as: ( The partial derivatives computed at any collinear libration points (x 0 , 0), Substituting these values in Equation ( 21), the characteristic equation reduces to xx yy c =Ω Ω The libration point is stable if all the roots of the characteristic equation ( 26) are either negative real numbers or distinct pure imaginary numbers or real parts of the complex numbers are negative.

Conclusion
The collinear libration points are investigated in a modified CR3BP when the bigger primary is a source of radiation, the smaller primary is an oblate spheroid; and the bodies are surrounded by a belt (circular cluster of material points).We have established the equations that govern the motion of the infinitesimal body under the  influence of radiation of the bigger primary, oblateness up to the zonal harmonics J 4 of the smaller primary and gravitational potential from the belt.The equations are affected perturbations.Numerically, we have determined the positions of the collinear libration points and investigated the resultant effect of the aforesaid perturbations on them.It is found that in count to the three libration points L 1 , L 2 , L 3 in the classical problem, there emerge four new collinear points which we call L n1 , L n2 , L n3 and L n4 .L n1 and L n2 arise from the effect of the potential from the belt, whereas L n3 and L n4 stem from the influence of the oblateness up to the zonal harmonics J 4 of the smaller primary.Due to the pooled impact of the aforesaid perturbations, the collinear points L 1 and L 3 advance toward the primaries while L n3 moves away from the primaries; and L 2 and L n1 tend towards the smaller primary as L n2 and L n4 come closer to the bigger primary.Despite the influence of radiation of the bigger primary, oblateness up to the zonal harmonics J 4 of the smaller primary and gravitational potential from the belt, the collinear libration points L i (i = 1, 2, 3) as in the classical case, remain unstable.However, all the additional new collinear points are stable except L n2 .The existence of stable new collinear points can be utilized as stations for artificial satellites.

+
, a and b are parameters which determine the density profile of the belt.The parameter a controls the flatness of the profile and is known as the flatness parameter.The parameter b controls the size of the core of the density profile and is called the core parameter.When a = b = 0, the potential reduces to the one by a point mass.Restricting ourselves to the xy -plane, Equation (6) be- comes

2 ,
L 1 and L 3 stepped closer to the primaries while L 2 moved towards the bigger primary.Nevertheless, on taking into account the effect of the potential from the belt only (i.e.

Figure 2 .
Figure 2. Disposition of the collinear points in the classical case.

Figure 3 .
Figure 3. Disposition of the collinear points under the effects of the belt.

4 .
there appeared seven collinear points:L 1 , L 2 , L 3 , L n1 , L n2 , L n3 , L n4as shown in Figure With increase in these perturbations Table

Figure 4 .
Figure 4. Disposition of the collinear points under the combined effects of the perturbations. )