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In the framework of the elliptic restricted three-body problem, using a semi-analytic approach, we investigate the effects of oblateness, radiation and eccentricity of both primaries on the periodic orbits around the triangular Lagrangian points of oblate and luminous binary systems. The frequencies of the long and short orbits of the periodic motion are affected by the oblateness and radiation of both primaries, so are their eccentricities, semi-major and semi-minor axes.

There exist five co-planar equilibrium points in the restricted three-body problem (R3BP), three collinear with the primaries (collinear points) and two, form equilateral triangles with the line (

It is a well-known fact that when at least one of the primary bodies is a source of radiation, the classical restricted three-body problem fails to adequately discuss the motion of the infinitesimal body. Radzievsky [

The classical restricted three-body problem considers the bodies to be strictly spherical, but in the solar (e.g., Earth, Jupiter and Saturn) and stellar (e.g., Achernar, Alfa Arae, Regulus, VFTS 102, Vega and Altair) systems, some planets and stars are sufficiently oblate to justify the inclusion of oblateness in the study of motion of celestial bodies. Therefore, [_{4} in the planar CR3BP and proved that the positions and stability of the triangular points are affected by this perturbation.

The orbits of most celestial and stellar bodies are elliptic rather than circular; as a result, the study of the elliptic restricted three-body problem (ER3BP) can have significant effects. When the primaries’ orbit is elliptic, a nonuniformly rotating-pulsating coordinate system is commonly used. These new coordinates have the felicitous property that, the positions of the primaries are fixed; however, the Hamiltonian is explicitly time-dependent [

A vast number of researches [

In this communication, we investigate in the elliptic framework the long and short periodic orbits around the triangular points when both primary bodies emit light energy simultaneously and are oblate spheroids as well. The analytic results obtained are applied to the binary systems of mass ratio

The paper is organized as follows: Section 2 provides the equations of motion for the system under investigation; Section 3 computes the long and short periodic orbits and Section 4 describes the eccentricities, semi-ma- jor and semi-minor axes; while Sections 5 & 6 are the numerical analysis and conclusion respectively.

The radiation pressure force

where

_{ }we have

The force due to oblateness of the primaries of masses _{ }is given by [

where

Let

Equating their right hand sides, and then integrating with respect to _{ }, we obtain

Thus, the equations of motion of the test particle are presented here in a dimensionless-rotating-pulsating coordinate system as:

with the force function

The mean motion,

with,

The triangular Lagrangian points

We give these points a small displacement

The superscript 0 indicate that the partial derivatives are to be evaluated at the equilibrium points

where

and

The terms_{ }

The function

Which is a quadratic form in

Using the transformation,

where

But

where

The function around the triangular point is given by Equation (7), but,

The determinant of which is

The characteristic equation of the associated matrix is thus

The roots are

The eccentricities of the ellipses are given by (Szebehely 1967)

where

And therefore,

The semi-major and semi-minor axes of the periodic orbits are given by

(13)

and

Following Singh and Umar [

Mass Ratio (μ) | Oblateness_{ } | Frequencies_{ } | Eccentricities | Semi-major Axes_{ } | Semi-Minor Axes_{ } | |||||
---|---|---|---|---|---|---|---|---|---|---|

A_{1} | A_{2} | S_{1} | S_{2} | e_{1} | e_{2} | a_{1} | a_{2} | b_{1} | b_{2} | |

0.01 | 0 | 0 | 0.280544 | 0.9306418 | 0.983702 | 0.880847 | 7.72103 | 1.65024 | 0.743672 | 0.7861 |

0.001 | 0.002 | 0.281020 | 0.9289990 | 0.983676 | 0.881309 | 7.70196 | 1.65134 | 0.742681 | 0.785095 | |

0.010 | 0.020 | 0.285268 | 0.9141610 | 0.983443 | 0.885464 | 7.53036 | 1.66120 | 0.733698 | 0.776051 | |

0.100 | 0.200 | 0.324708 | 0.7657820 | 0.981114 | 0.927014 | 5.81436 | 1.75984 | 0.636948 | 0.685612 | |

0.02 | 0 | 0 | 0.394740 | 0.8920900 | 0.968893 | 0.887374 | 4.42934 | 1.67072 | 0.750408 | 0.782824 |

0.001 | 0.002 | 0.395410 | 0.8902950 | 0.968849 | 0.892307 | 4.42139 | 1.67192 | 0.749439 | 0.781820 | |

0.010 | 0.020 | 0.401388 | 0.8741440 | 0.968455 | 0.892307 | 4.34988 | 1.68312 | 0.740661 | 0.772779 | |

0.100 | 0.200 | 0.456881 | 0.7126300 | 0.964513 | 0.936709 | 3.63474 | 1.79468 | 0.646364 | 0.682375 | |

0.03 | 0 | 0 | 0.480983 | 0.8543270 | 0.955502 | 0.894235 | 3.32419 | 1.69392 | 0.757301 | 0.779489 |

0.001 | 0.002 | 0.481799 | 0.8523900 | 0.955447 | 0.894763 | 3.31991 | 1.69532 | 0.756351 | 0.778485 | |

0.010 | 0.020 | 0.489083 | 0.8349490 | 0.954960 | 0.95496 | 3.28141 | 1.70792 | 0.747743 | 0.769448 | |

0.100 | 0.200 | 0.556700 | 0.6605420 | 0.950082 | 0.947021 | 2.89637 | 1.83394 | 0.655473 | 0.679074 |

Eccentricity e | Frequencies_{ } | Eccentricities | Semi-Major Axes_{ } | Semi-Minor Axes_{ } | Roots | |||||
---|---|---|---|---|---|---|---|---|---|---|

S_{1} | S_{2} | e_{1} | e_{2} | a_{1} | a_{2} | b_{1} | b_{2} | λ_{1} | λ_{2} | |

0 | 0.390246 | 0.908554 | 0.969008 | 0.882729 | 4.74520 | 1.67227 | 0.766838 | 0.797088 | 2.9905 | 0.0050079 |

0.1 | 0.393061 | 0.899951 | 0.96887 | 0.885123 | 4.64637 | 1.67498 | 0.760378 | 0.791011 | 3.0204 | 0.0053074 |

0.2 | 0.401388 | 0.874144 | 0.968455 | 0.892307 | 4.34988 | 1.68312 | 0.740661 | 0.772779 | 3.1102 | 0.0053074 |

0.3 | 0.414894 | 0.831131 | 0.967763 | 0.904281 | 3.85573 | 1.69667 | 0.706578 | 0.742394 | 3.2598 | 0.0056816 |

0.4 | 0.433096 | 0.770914 | 0.966704 | 0.921044 | 1.71564 | 1.71564 | 0.655893 | 0.699854 | 3.4693 | 0.0062056 |

0.5 | 0.455431 | 0.693491 | 0.965549 | 0.942596 | 2.27445 | 1.74004 | 0.584301 | 0.64516 | 3.7387 | 0.0068792 |

0.6 | 0.481323 | 0.598864 | 0.964027 | 0.968938 | 1.18733 | 1.76985 | 0.482586 | 0.578312 | 4.0678 | 0.0077026 |

Semi-Major Axis a | Frequencies_{ } | Eccentricities | Semi-Major Axes_{ } | Semi-Minor Axes_{ } | ||||
---|---|---|---|---|---|---|---|---|

S_{1} | S_{2} | e_{1} | e_{2} | a_{1} | a_{2} | b_{1} | b_{2} | |

0.8 | 0.414672 | 0.851848 | 0.966990 | 0.901721 | 3.37718 | 1.61406 | 0.678415 | 0.722047 |

1.0 | 0.400241 | 0.857728 | 0.969296 | 0.893670 | 4.87783 | 1.76437 | 0.761752 | 0.796165 |

1.2 | 0.385270 | 0.863608 | 0.971692 | 0.885618 | 6.37846 | 1.91469 | 0.851345 | 0.870284 |

1.4 | 0.369693 | 0.869488 | 0.973908 | 0.877567 | 7.87913 | 2.06501 | 0.925775 | 0.944402 |

1.6 | 0.353431 | 0.875368 | 0.976214 | 0.869516 | 9.37978 | 2.21533 | 0.994651 | 1.018520 |

1.8 | 0.336383 | 0.875368 | 0.97852 | 0.861464 | 10.8804 | 2.36565 | 1.025906 | 1.09264 |

2.0 | 0.318423 | 0.887128 | 0.980826 | 0.853413 | 12.3811 | 2.51597 | 1.11976 | 1.16676 |

Radiation Factor q_{1} | Frequencies_{ } | Eccentricities | Semi-Major Axes_{ } | Semi-Minor Axes_{ } | ||||
---|---|---|---|---|---|---|---|---|

S_{1} | S_{2} | e_{1} | e_{2} | a_{1} | a_{2} | b_{1} | b_{2} | |

0.999 | 0.48548 | 0.7748 | 0.9592 | 0.90233 | 4.5397 | 2.0088 | 0.8435 | 0.8532 |

0.900 | 0.48992 | 0.7728 | 0.95844 | 0.90259 | 4.2773 | 1.9628 | 0.8224 | 0.8306 |

0.850 | 0.49214 | 0.7716 | 0.95801 | 0.90274 | 4.1449 | 1.9396 | 0.81156 | 0.8719 |

0.800 | 0.49435 | 0.7705 | 0.95761 | 0.90289 | 4.0124 | 1.9164 | 0.8006 | 0.8077 |

0.750 | 0.49656 | 0.7694 | 0.95721 | 0.90303 | 3.8799 | 1.8932 | 0.7894 | 0.7963 |

0.700 | 0.49875 | 0.7683 | 0.95681 | 0.90318 | 3.7474 | 1.8700 | 0.7781 | 0.7847 |

0.650 | 0.50093 | 0.7672 | 0.95041 | 0.90333 | 3.6145 | 1.8469 | 0.7666 | 0.7733 |

Oblateness_{ } | Frequencies_{ } | Eccentricities | Semi-Major Axes_{ } | Semi-Minor Axes_{ } | Roots | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

A_{1} | A_{2} | S_{1} | S_{2} | e_{1} | e_{2} | a_{1} | a_{2} | b_{1} | b_{2} | λ_{1} | λ_{2} |

0 | 0 | 1.00213 | 0.497863 | 0.876433 | 0.962865 | 1.54499 | 2.30177 | 0.96389 | 0.84098 | 2.3331 | 0.334757 |

0 | 0 | 1.03476 | 0.434633 | 0.883600 | 0.983990 | 1.52343 | 2.37897 | 0.94898 | 0.81668 | 2.43099 | 0.35691 |

0.001 | 0.002 | 1.03669 | 0.430870 | 0.88402 | 0.985244 | 1.53553 | 2.38407 | 0.94818 | 0.81571 | 2.43118 | 0.35824 |

0.010 | 0.020 | 1.05384 | 0.397007 | 0.88781 | 0.996534 | 1.54535 | 2.43000 | 0.9410 | 0.80698 | 2.4329 | 0.370196 |

0.150 | 0.010 | 1.05069 | 0.396878 | 0.88899 | 0.996905 | 1.54593 | 2.43450 | 0.94157 | 0.80855 | 2.43196 | 0.36798 |

0.018 | 0.01 | 1.05258 | 0.391195 | 0.88998 | 0.998897 | 1.54774 | 2.44350 | 0.9405 | 0.80757 | 2.43196 | 0.36931 |

shown in Figures 1-9. Figures 1-8 show the effects of the semi-major axis of the elliptic orbits (

Equation (12) gives the eccentricities of the long and short periods, the eccentricity of the long period increases with oblateness, while that of the short period decreases (

axis, eccentricity and radiation pressure while their sizes reduce with increase in oblateness parameters of both primaries.

The expressions for the frequencies of the long and short periods around the triangular points with their orientations, eccentricities, semi-major and semi-minor axes has been obtained. They have been found to be influ-

enced by the eccentricities of the orbits of the primaries, radiation pressures, semi-major axis and oblateness. In the absence of these perturbations, the results are in accordance with [