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 n1 and L n2 result due to the potential from the belt, while L n3 and L n4 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 n3 advances away from the primaries; and L 2 and L n1 tend towards the smaller primary whereas L n2 and L n4 draw closer to the bigger primary. The collinear libration points L i ( i = 1, 2, 3) and L n2 are linearly unstable whereas the L n1, L n3 and L n4 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.

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.  and  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.  indicated that small particles were equally influenced by the gravitation and light radiation force as they moved toward luminous celestial bodies.   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  - have examined the existence of libration points and their linear stability in the photogravitational CR3BP.

  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  -  .

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  . This vindicates the incorporation of oblateness of the primaries in the study of CR3BP  -  .

The orbital effects of the oblateness up to the quadrupole, i.e. J2, and the octupole, i.e. J4, 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  .  certified that the sites of the triangular libration points and their linear stability were influenced by the oblateness up to J4 of the bigger primary in the CR3BP.  examined the effects of photogravitational force and oblateness in the perturbed restricted three-body problem.  analyzed analytically and numerically the effects of oblateness up to J2 of the smaller primary and gravitational potential from the belt on the linear stability of libration points in the photogravitational CR3BP.  explored the combined effect of radiation and oblateness up to J2 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.  studied the effects of oblateness up to J4 of the smaller primary and gravitational potential from a belt, on the linear stability of triangular libration points in the photogravitational CR3BP.  looked at the effects of oblateness of both primaries up to zonal harmonic J4 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 J4 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 Model2.1. 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 r1 be the distance between m and m1, r2 the distance between m and m2; and R the distance between and. The coordinates of m1, m2 and m are (x1, 0), (x2, 0) and (x, y) correspondingly. Our aim is to find the equations of motion of under the influence of radiation of, oblateness up to J4 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

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 

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

The planar configuration of the problem

 . 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 ro (i.e., the colatitudes). Ro is the mean radius of the oblate body. The terms are the Legendre polynomials, given by

J2n 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

We denote the oblateness coefficient for the smaller primary as Bi,.

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  is

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

2.6. The Potential Energy of the Infinitesimal Body

The potential energy of the infinitesimal body, under the influence of the oblateness up to J4 of smaller primary, radiation of the bigger primary and the circumbinary belt, now takes the form

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

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

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

with

and n is the mean motion, given by  as

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

and

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  , in the absence of the perturbations. That is when), we have

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

 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 J2 only.

5) Oblateness of the smaller primary up to J4 only.

6) Radiation of the bigger primary, oblateness of the smaller primary up to J4 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 (Li, i = 1, 2, 3) which correspond to the classical case of  . Owing to the effect of the radiation of the bigger primary only (i.e.) Case 2, L1 and L3 stepped closer to the primaries while L2 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 (Ln1, Ln2 and Li, i = 1, 2, 3), this confirms those of  -  . The collinear points L1 and L3 shifted nearer to the primaries while L2 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 J2 only (i.e.) Case 4, the collinear point L1 sifted away from the primaries while L2 and L3 stepped closer to the bigger primary. In Case 5, due oblateness of the smaller primary up to J4 only (i.e.), Ln1 moved away from the bigger primary while Ln2 stepped towards it. Similarly, owing to the oblateness of the smaller primary up to J2 with

Disposition of the collinear points in the classical case Disposition of the collinear points under the effects of the belt Positions of the collinear points when µ = 0.35, q<sub>1</sub> = 0.98, B<sub>1</sub> = 0.01, B<sub>2</sub> = 0.005 and M<sub>b</sub> = T = 0.01, r<sub>c</sub> = 0.8789
CaseL1L2L3Ln1Ln2Ln3Ln4
11.2448130.213295−1.142867
21.2437140.210813−1.137286
31.2393620.224700−1.137090−0.000451−0.038855
41.2495640.205046−1.138453
51.2355820.245494−1.1412670.9619310.319350
61.2284440.259431−1.129916−0.000441−0.0392470.9625370.314837
Combined effects of the perturbations on the collinear points when m = 0.35, T = 0.01, r<sub>c</sub> = 0.8789
q1B1B2MbL1L2L3Ln1Ln2Ln3Ln4
10001.244810.21329−1.14287
0.990.0010.00050.011.237950.22579−1.13420−0.000445−0.039050.824210.47525
0.980.0020.00060.021.232400.23420−1.12557−0.000219−0.053650.830680.46863
0.970.0030.00070.031.227070.24145−1.11719−0.000144−0.063910.836340.46280
0.960.0040.00080.041.221940.24787−1.10906−0.000107−0.072070.841380.45758

potential from the belt only (i.e.) Case 5, collinear points L1 and L3 moved nigh to the primaries while L2 stepped away from the bigger primary; and there emerge additional two new collinear points Ln3, Ln4. In the presence of all these perturbations (i.e.) Case 6, there appeared seven collinear points: L1, L2, L3, Ln1, Ln2, Ln3, Ln4 as shown in Figure 4. With increase in these perturbations Table 2, the collinear points L1, L3 draw closer to the primaries while Ln3 moves away from the them; L2, Ln1 move away from the bigger primary while Ln2, Ln4 tend towards it.

4. Linear Stability of the Collinear Points

To study the stability of a libration point (x0, y0), we employ small displacement to the coordinates (x0, y0). So, the variations and can take the form: and and the equations of the motion (5) become

The superscript “0” indicates that the partial derivatives have been evaluated at the libration point under consideration (x0, y0).

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

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:

Disposition of the collinear points under the combined effects of the perturbations

The partial derivatives computed at any collinear libration points (x0, 0), are

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

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 Li (i = 1, 2, 3), Lnj (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 Li (i = 1, 2, 3) and Ln2 are unstable (Table 3, Table 4, Table 5, Table 7), whereas the additional new collinear points Ln1, Ln3 and Ln4 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

Stability of L<sub>1</sub>
CaseL1Remark
11.2448134.6468−0.8234±1.3674±1.4305iUnstable
21.2437144.6595−0.8297±1.3722±1.4329iUnstable
31.2393624.7796−0.8510±1.3897±1.4512iUnstable
41.2495644.8515−0.8355±1.4112±1.4267iUnstable
51.2355824.2890−0.8377±1.2772±1.4841iUnstable
61.2284444.3987−0.8739±1.2973±1.5113iUnstable
Stability of L<sub>2</sub>
CaseL2Remark
10.21329516.6783−6.8391±3.7405±2.8552iUnstable
20.21081316.4862−6.7431±3.7147±2.8383iUnstable
30.22470018.7266−7.8271±3.9966±3.0293iUnstable
40.20504617.7676−7.0603±3.8738±2.8912iUnstable
50.2454948.5669−5.9936±2.5451±2.8155iUnstable
60.2594317.6595−6.4396±2.3914±2.9368iUnstable
Stability of L<sub>3</sub>
CaseL3Remark
1−1.1428673.7297−0.3648±0.9441±1.2355iUnstable
2−1.1372863.7334−0.3667±0.9463±1.2364iUnstable
3−1.1370903.8282−0.3753±0.9574±1.2519iUnstable
4−1.1384533.7908−0.3726±0.9540±1.2458iUnstable
5−1.1412673.7523−0.3675±0.9476±1.2392iUnstable
6−1.1299163.8558−0.3805±0.9638±1.2568iUnstable
Stability of L<sub>n</sub><sub>1</sub>
CaseLn1Remark
3−0.000451−9874.8−9985.0±98.6059i±100.7019iStable
6−0.000441−9879.7−9986.0±98.6019i±100.7356iStable
Stability of L<sub>n</sub><sub>2</sub>
CaseLn2Remark
3−0.038855327.1441−176.4617±18.0135±13.3382iUnstable
6−0.039247319.0101−171.7775±17.7859±13.1617iUnstable
Stability of L<sub>n</sub><sub>3</sub>
CaseLn3Remark
50.961931−36.7586−1.1726±1.0266i±6.3953iStable
60.962537−36.0006−1.2218±1.0453i±6.3447iStable
Stability of L<sub>n</sub><sub>4</sub>
CaseLn2Remark
50.319350−15.5449−4.5783±1.8538i±4.5507iStable
60.314837−11.8748−5.0872±1.8490i±4.2035iStable

influence of radiation of the bigger primary, oblateness up to the zonal harmonics J4 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 L1, L2, L3 in the classical problem, there emerge four new collinear points which we call Ln1, Ln2, Ln3 and Ln4. Ln1 and Ln2 arise from the effect of the potential from the belt, whereas Ln3 and Ln4 stem from the influence of the oblateness up to the zonal harmonics J4 of the smaller primary. Due to the pooled impact of the aforesaid perturbations, the collinear points L1 and L3 advance toward the primaries while Ln3 moves away from the primaries; and L2 and Ln1 tend towards the smaller primary as Ln2 and Ln4 come closer to the bigger primary. Despite the influence of radiation of the bigger primary, oblateness up to the zonal harmonics J4 of the smaller primary and gravitational potential from the belt, the collinear libration points Li (i = 1, 2, 3) as in the classical case, remain unstable. However, all the additional new collinear points are stable except Ln2. 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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