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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.

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. [

Researches on the sites and stability of the libration points of the CR3BP with perturbations have achieved ample attention in recent times. [

[

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 [

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 [_{4} of the bigger primary in the CR3BP. [_{2} of the smaller primary and gravitational potential from the belt on the linear stability of libration points in the photogravitational CR3BP. [_{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. [_{4} of the smaller primary and gravitational potential from a belt, on the linear stability of triangular libration points in the photogravitational CR3BP. [_{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} be the distance between m and m_{1}, r_{2} the distance between m and m_{2}; and R the distance between _{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 _{4} of the smaller primary, and a circumbinary belt centred at the origin of the coordinate system oxyz (see

The kinetic energy (K.E) of the infinitesimal body in the barycentric coordinate system oxyz rotating about z-axis with uniform angular velocity

where over dot represents differentiation with respect to time t.

Now, since the radiation pressure force

where

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

[_{o} (i.e., the colatitudes). R_{o} is the mean radius of the oblate body. The terms

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

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

The gravitational potential from belt (circular cluster of material points) centered at the origin of a coordinates system oxyz,

where

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

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,

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

with

and n is the mean motion, given by [

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

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 [

with three collinear points

If we consider the effects of the potential from the belt only (i.e.

[

(18) will have five collinear points (

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

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

In the absence of the perturbations (i.e._{i}, i = 1, 2, 3) which correspond to the classical case of [_{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._{n}_{1}, L_{n}_{2} and L_{i}, i = 1, 2, 3), this confirms those of [_{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} 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._{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

Case | L_{1} | L_{2} | L_{3} | L_{n}_{1} | L_{n}_{2} | L_{n}_{3} | L_{n}_{4} |
---|---|---|---|---|---|---|---|

1 | 1.244813 | 0.213295 | −1.142867 | ||||

2 | 1.243714 | 0.210813 | −1.137286 | ||||

3 | 1.239362 | 0.224700 | −1.137090 | −0.000451 | −0.038855 | ||

4 | 1.249564 | 0.205046 | −1.138453 | ||||

5 | 1.235582 | 0.245494 | −1.141267 | 0.961931 | 0.319350 | ||

6 | 1.228444 | 0.259431 | −1.129916 | −0.000441 | −0.039247 | 0.962537 | 0.314837 |

q_{1} | B_{1} | B_{2} | M_{b} | L_{1} | L_{2} | L_{3} | L_{n}_{1} | L_{n}_{2} | L_{n}_{3} | L_{n}_{4} |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | 0 | 0 | 1.24481 | 0.21329 | −1.14287 | − | − | − | − |

0.99 | 0.001 | 0.0005 | 0.01 | 1.23795 | 0.22579 | −1.13420 | −0.000445 | −0.03905 | 0.82421 | 0.47525 |

0.98 | 0.002 | 0.0006 | 0.02 | 1.23240 | 0.23420 | −1.12557 | −0.000219 | −0.05365 | 0.83068 | 0.46863 |

0.97 | 0.003 | 0.0007 | 0.03 | 1.22707 | 0.24145 | −1.11719 | −0.000144 | −0.06391 | 0.83634 | 0.46280 |

0.96 | 0.004 | 0.0008 | 0.04 | 1.22194 | 0.24787 | −1.10906 | −0.000107 | −0.07207 | 0.84138 | 0.45758 |

potential from the belt only (i.e._{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._{1}, L_{2}, L_{3}, L_{n}_{1}, L_{n}_{2}, L_{n}_{3}, L_{n}_{4} as shown in _{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.

To study the stability of a libration point (x_{0}, y_{0}), we employ small displacement _{0}, y_{0}). So, the variations

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

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), 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 L_{i} (i = 1, 2, 3), L_{nj} (j = 1, 2, 3, 4) of

Studying Tables 3-9, we find that all the collinear libration points L_{i} (i = 1, 2, 3) and L_{n}_{2} are unstable (_{n}_{1,} L_{n}_{3} and L_{n}_{4} are stable (

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

Case | L_{1} | Remark | ||||
---|---|---|---|---|---|---|

1 | 1.244813 | 4.6468 | −0.8234 | ±1.3674 | ±1.4305i | Unstable |

2 | 1.243714 | 4.6595 | −0.8297 | ±1.3722 | ±1.4329i | Unstable |

3 | 1.239362 | 4.7796 | −0.8510 | ±1.3897 | ±1.4512i | Unstable |

4 | 1.249564 | 4.8515 | −0.8355 | ±1.4112 | ±1.4267i | Unstable |

5 | 1.235582 | 4.2890 | −0.8377 | ±1.2772 | ±1.4841i | Unstable |

6 | 1.228444 | 4.3987 | −0.8739 | ±1.2973 | ±1.5113i | Unstable |

Case | L_{2} | Remark | ||||
---|---|---|---|---|---|---|

1 | 0.213295 | 16.6783 | −6.8391 | ±3.7405 | ±2.8552i | Unstable |

2 | 0.210813 | 16.4862 | −6.7431 | ±3.7147 | ±2.8383i | Unstable |

3 | 0.224700 | 18.7266 | −7.8271 | ±3.9966 | ±3.0293i | Unstable |

4 | 0.205046 | 17.7676 | −7.0603 | ±3.8738 | ±2.8912i | Unstable |

5 | 0.245494 | 8.5669 | −5.9936 | ±2.5451 | ±2.8155i | Unstable |

6 | 0.259431 | 7.6595 | −6.4396 | ±2.3914 | ±2.9368i | Unstable |

Case | L_{3} | Remark | ||||
---|---|---|---|---|---|---|

1 | −1.142867 | 3.7297 | −0.3648 | ±0.9441 | ±1.2355i | Unstable |

2 | −1.137286 | 3.7334 | −0.3667 | ±0.9463 | ±1.2364i | Unstable |

3 | −1.137090 | 3.8282 | −0.3753 | ±0.9574 | ±1.2519i | Unstable |

4 | −1.138453 | 3.7908 | −0.3726 | ±0.9540 | ±1.2458i | Unstable |

5 | −1.141267 | 3.7523 | −0.3675 | ±0.9476 | ±1.2392i | Unstable |

6 | −1.129916 | 3.8558 | −0.3805 | ±0.9638 | ±1.2568i | Unstable |

Case | L_{n}_{1} | Remark | ||||
---|---|---|---|---|---|---|

3 | −0.000451 | −9874.8 | −9985.0 | ±98.6059i | ±100.7019i | Stable |

6 | −0.000441 | −9879.7 | −9986.0 | ±98.6019i | ±100.7356i | Stable |

Case | L_{n}_{2} | Remark | ||||
---|---|---|---|---|---|---|

3 | −0.038855 | 327.1441 | −176.4617 | ±18.0135 | ±13.3382i | Unstable |

6 | −0.039247 | 319.0101 | −171.7775 | ±17.7859 | ±13.1617i | Unstable |

Case | L_{n}_{3} | Remark | ||||
---|---|---|---|---|---|---|

5 | 0.961931 | −36.7586 | −1.1726 | ±1.0266i | ±6.3953i | Stable |

6 | 0.962537 | −36.0006 | −1.2218 | ±1.0453i | ±6.3447i | Stable |

Case | L_{n}_{2} | Remark | ||||
---|---|---|---|---|---|---|

5 | 0.319350 | −15.5449 | −4.5783 | ±1.8538i | ±4.5507i | Stable |

6 | 0.314837 | −11.8748 | −5.0872 | ±1.8490i | ±4.2035i | Stable |

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.

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