**International Journal of Astronomy and Astrophysics**

Vol.05 No.03(2015), Article ID:59484,10 pages

10.4236/ijaa.2015.53020

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

Jagadish Singh^{1}, Joel John Taura^{2}

^{1}Department of Mathematics, Faculty of Science, Ahmadu Bello University, Zaria, Nigeria

^{2}Department of Mathematics and Computer Science, Federal University, Kashere, Nigeria

Email: jgds2004@yahoo.com, taurajj@yahoo.com

Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 29 April 2015; accepted 6 September 2015; published 9 September 2015

ABSTRACT

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 J_{4} 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 L_{i} (i = 1, 2, 3) in the classic restricted three-body problem, there appear four more collinear points L_{ni} (i = 1, 2, 3, 4). L_{n}_{1} and L_{n}_{2} result due to the potential from the belt, while L_{n}_{3} and L_{n}_{4} are consequences of the oblateness up to the zonal harmonics J_{4} of the smaller primary. Owing to the mutual effect of all the perturbations, L_{1} and L_{3} come nearer to the primaries while L_{n}_{3} advances away from the primaries; and L_{2} and L_{n}_{1} tend towards the smaller primary whereas L_{n}_{2} and L_{n}_{4} draw closer to the bigger primary. The collinear libration points L_{i} (i = 1, 2, 3) and L_{n}_{2} are linearly unstable whereas the L_{n}_{1}, L_{n}_{3} and L_{n}_{4} 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.

**Keywords:**

Circular Restricted Three-Body Problem, Photogravitational, Zonal Harmonic Effect, Potential from the Belt

1. 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 are on the line linking the primaries, whereas the other two are in equilateral triangular alignment with the primaries. The collinear points are linearly unstable, while the triangular points 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.

2. Mathematical Formulation of Model

2.1. The Problem

Let
and
be the masses of the primaries with, and let
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
and. 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
and. 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
under the influence of radiation of, 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).

2.2. 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 Figure 1, is given as

(1)

where over dot represents differentiation with respect to time t.

2.3. Force Due to Radiation Pressure

Now, since the radiation pressure force varies with distance by the same law as the gravitational attraction force 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]

(2)

where, a constant for a particular particle, is the mass reduction factor. We represent the radiation factor for the bigger primary as,

2.4. 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

(3)

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
are the Legendre polynomials, given by

(4)

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, Equation (3) becomes

(5)

We denote the oblateness coefficient for the smaller primary as B_{i},.

2.5. 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

(6)

where is the total mass of the belt, , and 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 -plane, Equation (6) becomes

(7)

2.6. 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

(8)

with G is the gravitational constant.

2.7. 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

or

(9)

where

Subsequently, we obtain the equations of motion of the infinitesimal body as

(10)

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, , where is the mass ratio. Thus, in the

dimensionless synodic coordinate system, the equations of motion (10) reduce to

(11)

with

(12)

and n is the mean motion, given by [24] as

(13)

is the radial distance of the infinitesimal body in the classical restricted three-body problem.

3. 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

(14)

and

(15)

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

(16)

Equation (16) reduces to those of [1] , in the absence of the perturbations. That is when), we have

(17)

with three collinear points and Only the collinear point is located between the primaries (Figure 2).

If we consider the effects of the potential from the belt only (i.e.), the Equation (17) reduces to

(18)

[16] showed that whenever and

in the interval 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.) 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 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.) Case 3, there surface five collinear libration points (L_{n}_{1}, L_{n}_{2} 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.) 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_{n}_{1} moved away from the bigger primary while L_{n}_{2} stepped towards it. Similarly, owing to the oblateness of the smaller primary up to J_{2} with

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

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

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.

potential from the belt only (i.e.) 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_{n}_{3}, L_{n}_{4}. In the presence of all these perturbations (i.e.) Case 6, there appeared seven collinear points: L_{1}, L_{2}, L_{3}, L_{n}_{1}, L_{n}_{2}, L_{n}_{3}, L_{n}_{4} as shown in Figure 4. With increase in these perturbations Table 2, the collinear points L_{1}, L_{3} draw closer to the primaries while L_{n}_{3} moves away from the them; L_{2}, L_{n}_{1} move away from the bigger primary while L_{n}_{2}, L_{n}_{4} tend towards it.

4. Linear Stability of the Collinear Points

To study the stability of a libration point (x_{0}, y_{0}), we employ small displacement
to the coordinates (x_{0}, y_{0}). So, the variations
and
can take the form:
and
and the equations of the motion (5) become

(19)

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 A,B and λ are constants. Then, Equation (19) will have a non ?trivial solution for A and B when

(20)

On expanding the determinant we obtain the characteristic equation equivalent to the variational equations of (19) as

(21).

Now, we obtain the second partial derivatives as:

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

(22)

The partial derivatives computed at any collinear libration points (x_{0}, 0), are

(23)

(24)

(25)

Substituting these values in Equation (21), the characteristic equation reduces to

(26)

where,

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.

The roots of the characteristic equation (26) for the libration points L_{i} (i = 1, 2, 3), L_{nj} (j = 1, 2, 3, 4) of Table 1 are presented in Tables 3-9 correspondingly.

Studying Tables 3-9, we find that all the collinear libration points L_{i} (i = 1, 2, 3) and L_{n}_{2} are unstable (Table 3, Table 4, Table 5, Table 7), whereas the additional new collinear points L_{n}_{1,} L_{n}_{3} and L_{n}_{4} are stable (Table 6, Table 8, Table 9).

5. 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

Table 3. Stability of L_{1}.

Table 4. Stability of L_{2}.

Table 5. Stability of L_{3}.

Table 6. Stability of L_{n}_{1}.

Table 7. Stability of L_{n}_{2}.

Table 8. Stability of L_{n}_{3}.

Table 9. Stability of L_{n}_{4}.

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 by the aforementioned 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_{n}_{1}, L_{n}_{2}, L_{n}_{3} and L_{n}_{4}. L_{n}_{1} and L_{n}_{2} arise from the effect of the potential from the belt, whereas L_{n}_{3} and L_{n}_{4} 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_{n}_{3} moves away from the primaries; and L_{2} and L_{n}_{1} tend towards the smaller primary as L_{n}_{2 }and L_{n}_{4} 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_{n}_{2}. The existence of stable new collinear points can be utilized as stations for artificial satellites.

Cite this paper

JagadishSingh,Joel JohnTaura, (2015) Collinear Libration Points in the Photogravitational CR3BP with Zonal Harmonics and Potential from a Belt. *International Journal of Astronomy and Astrophysics*,**05**,155-165. doi: 10.4236/ijaa.2015.53020

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