^{1}

^{2}

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^{1}

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^{3}

We have studied periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are triaxial rigid bodies and source of radiation pressure. We have determined periodic orbits for different values of ( h is energy constant; μ is mass ratio of the two primaries; are parameters of triaxial rigid bodies and are radiation parameters). These orbits have been determined by giving displacements along the tangent and normal at the mobile co-ordinates as defined in our papers (Mittal et al. [1]-[3]). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of triaxial bodies and source of radiation pressure on the periodic orbits by taking fixed value of μ.

This paper is the extension of our papers, Mittal et al. [_{2} and J_{4} for the more massive body. They showed that the triangular points in the restricted three-body problem have long or short periodic orbits in the range 0 ≤ µ < µ_{c}. Perdios et al. [

The celestial bodies are in general axis-symmetric bodies, so its shape should be taken into account as well. The replacement of mass point by rigid-body is quite important because of its wide applications. The re-entry of artificial satellite has shown the importance of periodic orbits.

That is why, we have thought of studying, in this paper, the periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are triaxial rigid bodies and source of radiation pressure. We determine the periodic orbits by giving displacements at the mobile co-ordinates along the tangent and normal. We have also determined family of periodic orbits by fixing μ (mass ratio of the two primaries) and changing the values of _{ }and the radiation pressure on the energy constant (h).

Most of the authors have not taken into account the effect of the solar radiation pressure in the motion of the third body whereas we have taken both the primaries as radiating triaxial rigid bodies. Besides taking both the primaries as triaxial rigid bodies and the source of radiation, we have used mobile-coordinates and given the displacement along the normal and the tangent to the orbit which has wider applications in space dynamics. We have drawn the periodic orbits by using the predictor-corrector method which is given in detail in our papers [

Following the procedure of our papers [

Equations of motion with Lagrangian function L are given by

where

a, b, c = length of the semi axes of the triaxial body of mass

R = dimensional distance between the primaries,

U = constant to be so chosen such that h (energy constant) will vanish at

The coordinates of

where

and

Equations of motion can also be written as

where

The Jacobi integral is

We consider the system of generalized coordinates

(2) with Jacobi integral given by (3). We consider the solutions of Equations (2) for which C is zero. If we consider the solutions of Equation (2) given by (4) for some fixed parameters value p then there may exist

another solution given by (5) with another parameter value

and

Solution (5) will reduce to Solution (4) as

Now we give the displacements

i.e.,

and

(6a)

We consider

with the integral constructed from Equation (3), retaining the first order terms only, we get

The modulus of momentary velocity on the orbit is defined as

is not corresponding to the equilibrium state, i.e.,

In the new coordinate system, we consider the transition matrix S as follows:

Consider the first column of S as

So, we have

It can be easily verified that

It may be noted that,

Now, we may further define

^{−}^{1},

^{−1}.

We write

where N is displacement along the normal to the orbit and M is displacement along the tangent to the orbit.

Then, the new coordinates are given by

Substituting these values into the integral (7a), we have

Equation (10) can be solved for

Equations of motion (2) for the new coordinates are

where

Since

and

Thus, we have derived the equation in

For determining the periodic orbits, the required equations of motion and the variational equations are given as:

where

So, for finding the new periodic motion it is necessary to integrate the system (14) of the differential equations from t = 0 and t = T. In the formulae (i) to (vi) of (14), it may be noted that I_{2J} = (e_{1}… e_{2J}), Z = (Z_{1}…Z_{2J}), μ = (μ_{1}, …, μ_{2J}) and the initial conditions x(0), y(0),

After solving the above equations of motion (i) and (ii), the variational Equations (iii)-(vi) of (14) and applying the predictor-corrector method, we have determined the periodic orbits.

We have drawn the periodic orbits for the following:

1) for fixed μ = 0.001, A_{1} = 0.0, A_{2} = 0.0,

2) for fixed μ = 0.001, A_{1} = 0.001, A_{2} = 0.0,

3) for fixed μ = 0.001, A_{1} = 0.001, A_{2} = 0.001,

4) for fixed μ = 0.001, A_{1} = 0.001, A_{2} = 0.001,

5) for fixed μ = 0.001, A_{1} = 0.002, A_{2} = 0.003,

In each figure, we have drawn 5 periodic orbits corresponding to different values of h. These orbits have been numbered 1, 2, 3, 4 and 5 corresponding to different values of h.

The above analysis is summed up in

Karimov and Sokolsky [

μ = 0.001 | Oblate body | Triaxial body | Triaxial body | Triaxial body | Triaxial body |
---|---|---|---|---|---|

Values of Energy Constant h | |||||

0.05 | 0.15 | 0.25 | 0.28 | 0.30 | |

0.10 | 0.20 | 0.20 | 0.23 | 0.22 | |

0.15 | 0.23 | 0.15 | 0.18 | 0.16 | |

0.20 | 0.28 | 0.10 | 0.12 | 0.15 | |

0.3258 | 0.31205 | 0.0895 | 0.09225 | 0.10215 |

five periodic orbits in a family for fixed value of the mass parameter μ, the triaxial parameters

We have observed the following effects on the periodic orbits and on the energy constant h due to triaxial rigid bodies and radiation pressure if we compare it with the results of Karimov and Sokolsky [

1) The energy constant h increases in a family (for

2) As we increase the radiation parameters P and

3) The periodic orbits go away from the libration point

We have investigated the family up to the member which touches the point

We are thankful to the Centre for Fundamental Research in Space Dynamics and Celestial Mechanics (CFRSC), Delhi and the Deanship of Scientific Research, College of Science in Zulfi, Majmaah University, KSA for providing all the research facilities in the completion of this research work.

Preeti Jain,Rajiv Aggarwal,Amit Mittal, Abdullah, (2016) Periodic Orbits in the Photogravitational Restricted Problem When the Primaries Are Triaxial Rigid Bodies. International Journal of Astronomy and Astrophysics,06,111-121. doi: 10.4236/ijaa.2016.61009