This paper investigates the triangular libration points in the photogravitational restricted three-body problem of variable mass, in which both the attracting bodies are radiating as well and the infinitesimal body vary its mass with time according to Jeans' law. Firstly, applying the space-time transformation of Meshcherskii in the special case when q=1/2, k=0, n=1, the differential equations of motion of the problem are given. Secondly, in analogy to corresponding problem with constant mass, the positions of analogous triangular libration points are obtained, and the fact that these triangular libration points cease to be classical ones when alpha not equal 0, but turn to classical L (4) and L (5) naturally when alpha=0 is pointed out. Lastly, introducing the space-time inverse transformation of Meshcherskii, the linear stability of triangular libration points is tested when alpha > 0. It is seen that the motion around the triangular libration points become unstable in general when the problem with constant mass evolves into the problem with decreasing mass.

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On the triangular libration points in photogravitational restricted three-body problem with variable mass

In the three-dimensional restricted three-body problem, by considering the more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion as well as source of radiation, it is found that the collinear point L
 1 comes nearer to the primaries with the increase in oblateness and radiation pressure, while L
 2 and L
 3 move away from the more massive primary with the increase in oblateness and come nearer to it with the increase in radiation pressure It is noted that the angular frequency s
 1 at L
 1 increases with oblateness as well as with radiation pressure s
 2 increases with oblateness and decreases with radiation pressure and s
 3 decreases with oblateness and increases with radiation pressure A study on the norms of the characteristic roots λ and s at L
 1, L
 2 and L
 3 is carried out It is established that for certain oblateness and radiation pressure parameters there is a one-to-one commensurability at the collinear points L
 2, L
 3 between the planar angular frequencies (s
 2,3) and the corresponding angular frequency (s
 z
 ) in the z-direction, and that at L
 1 no such commensurability exists At L
 2 and L
 3, the value of oblateness parameter providing the commensurability decreases with the increase in the radiation pressure However, the commensurable angular frequencies and eccentricity of the periodic orbits decrease at L
 2 and increase at L
 3, with the increase in the radiation pressure

Effect of oblateness and radiation pressure on angular frequencies at collinear points

The case of the planar circular photogravitational restricted three-body problem where the more massive primary is an emitter of radiation is numerically investigated. A thorough numerical analysis takes place in the configuration $(x,y)$
 and the $(x,C)$
 space in which we classify initial conditions of orbits into three main categories: (i) bounded, (ii) escaping and (iii) collisional. Our results reveal that the radiation pressure factor has a huge impact on the character of orbits. Interpreting the collisional motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape as well as the collisional basins and we managed to correlate them with the corresponding escape and collision times. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting version of the restricted three-body problem.

Unveiling the influence of the radiation pressure in nature of orbits in the photogravitational restricted three-body problem

The case of the planar circular photogravitational restricted three-body problem where the more massive primary is an emitter of radiation is numerically investigated. A thorough numerical analysis takes place in the configuration $(x,y)$ and the $(x,C)$ space in which we classify initial conditions of orbits into three main categories: (i) bounded, (ii) escaping and (iii) collisional. Our results reveal that the radiation pressure factor has a huge impact on the character of orbits. Interpreting the collisional motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape as well as the collisional basins and we managed to correlate them with the corresponding escape and collision times. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting version of the restricted three-body problem.

The low-energy lunar landing trajectory design using the invariant manifolds of restricted three-body problem is studied. Considering angle between the ecliptic plane and lunar orbit plane, the four-body problem of sun-earth-moon-spacecraft is divided into two three-body problems, the sun-earth-spacecraft in the ecliptic plane and the earth-moon-spacecraft in the lunar orbit plane. Using the orbit maneuver at the place where the two planes and the invariant manifolds intersect, a general method to design low energy lunar landing trajectory is given. It is found that this method can save the energy about 20% compared to the traditional Hohmann transfer trajectory. The mechanism that the method can save energy is investigated in the point of view of energy and the expression of the amount of energy saved is given. In addition, some rules of selecting parameters with respect to orbit design are provided. The method of energy analysis in the paper can be extended to energy analysis in deep space orbit design.

Lunar landing trajectory design based on invariant manifold

The photogravitational restricted three-body problem in which the mass reduction factors of two primaries q1, q2 e −∞,1] are studied and an analytic method to estimate the number of libration points and to calculate their location is given in this paper. The results show that in photogravitational restricted three-body problem, the number of libration points is from one to seven for different q1 and q2. As application, the motion of dust grain like comet tail in the solar system is also discussed.

Yu Li-zhong

Papers

The Libration Points in Photogravitational Restricted Three-Body Problem