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It is shown that the set of all multi-homography matrices describing I-element families of interdependent homographies between two views has dimension 4I + 7.
Citations
... , (15) where (16) In Eq. (16), & are defined as (17) The position of and from the lesser primary can be obtained from the root of the polynomial (18) and (19) ...
... Similarly, by integrating the unstable vector (65) forwards in time over [0, ], we generate the unstable manifold's trajectories. By varying and , one can get a better approximation (Mireless, 2006). ...
At collision, the circular restricted 3 body Equations of motion possess singularities which play a key role under computational, theoretical, and physical aspects. In this paper, an attempt is made for finding periodic orbit of regularized circular restricted three body system near the L-points L1 and L2 by applying Kustaanheimo-Stiefel transformation when the infinitesimal body moves closely to smaller primary. The fourth-order approximation is chosen as the starting initial guess for the Newton's method for the computation of halo orbits numerically. The linear stability of the halo orbits is discussed by finding the Eigen values of the monodromy matrix. Using the differential continuation method, it is found that there exists a stable range of periodic orbit near L-point L2 of Sun-Earth, Earth-Moon, and Sun-Mars systems. The invariant manifolds associated to the halo orbit are evaluated by exploiting the Eigen vectors of the monodromy matrix.
... The magnitude of should be small enough to be within the validity of the linear estimate. The invariant manifolds are parameterized by two coordinates, namely the angular coordinate parameterized by the halo orbit itself, and the linear coordinate parameterized by time along the stable (unstable) orbit (Mireless 2006). For the angular coordinate, the halo orbit is divided ...
In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth L1 and L2 -halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at possible intersection between the Earth’s parking orbit of the spacecraft and the manifold.
... In this investigation, we have utilized Newton's method of differential corrections [25] [26] to compute numerically the halo orbits at the Lagrangian points L1 and L2 of the Sun-Mars system for different values of solar radiation pressure. Comparison of the size, shape, location and time period of L1 and L2 halo orbits of Sun-Mars system with respect to radiation pressure is made and some interesting results are drawn. ...
... The method of differential correction is a powerful application of Newton's method that employs the state transition matrix (STM) to solve various boundary value problems. Differential correction method is used to determine the initial conditions of the halo orbits from the initial guess [25] [28]. Taking advantage of the fact that halo orbits are symmetric about xz -plane, the initial state vector takes the form ...
Photogravitational Restricted Three-Body Problem (PRTBP) with smaller primary being an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries is
considered and halo orbits in the vicinity of Sun-Mars Lagrangian points L1 and L2 are computed numerically. The effects of perturbations on size, shape, location and time period of the halo orbits are studied. It is found that the increase in solar radiation pressure at constant oblateness elongates the halo orbits at L1 and the orbits move towards the radiating body. At L2, the halo orbits shrink and move towards the smaller primary with increase in solar radiation pressure at constant oblateness. For constant radiation pressure, increase in oblateness causes the location of L1 and L2 halo orbits to move away from the smaller primary. The time period of L1 halo orbits increases with increase in radiation pressure for constant oblateness and decreases with increase in oblateness for constant radiation pressure. However, the effect of solar radiation pressure and oblateness for L2 halo orbits is reversed.
... Halo orbits in the vicinity of Sun-Earth (SE) L 2 and Sun-Mars (SM) L 1 are designed by identifying the initial conditions of Lyapunov orbits and giving a z-amplitude. Newton's method of differential correction (McInnes 2009;Mireles 2006) are then used to integrate the initial conditions of the modified Lyapunov orbit to obtain the initial conditions for the halo orbit. The initial conditions for SE-L 2 halo are obtained as ...
With the increase in complexities of interplanetary missions, the main focus has shifted to reducing the total delta-V for the entire mission and hence increasing the payload capacity of the spacecraft. This paper develops a trajectory to Mars using the Lagrangian points of the Sun-Earth system and the Sun-Mars system. The whole trajectory can be broadly divided into three stages: (1) Trajectory from a near-Earth circular parking orbit to a halo orbit around Sun-Earth Lagrangian point L2. (2) Trajectory from Sun-Earth L2 halo orbit to Sun-Mars L1 halo orbit. (3) Sun-Mars L1 halo orbit to a circular orbit around Mars. The stable and unstable manifolds of the halo orbits are used for halo orbit insertion. The intermediate transfer arcs are designed using two-body Lambert’s problem. The total delta-V for the whole trajectory is computed and found to be lesser than that for the conventional trajectories. For a 480 km Earth parking orbit, the total delta-V is found to be 4.6203 km/s. Another advantage in the present approach is that delta-V does not depend upon the synodic period of Earth with respect to Mars.
... The photogravitational restricted three-body problem arises from the classical problem if at least one of the bodies is an intense emitter of radiation. Radzievskii (1950) formulated the photogravitational restricted three-body problem and discussed it for three specific bodies: the Sun, a planet and a dust particle. The radiation repulsive force exerted on the particle can be represented in terms of gravitational attraction F g (Radzievskii 1950) as: ...
... As the size of the particles increase, their density decreases. This paper uses Newton's method of differential corrections (Mireles 2006) to compute periodic orbits at the L 1 point of the Sun-Mars system for different values of ε. A comparison of the halo orbits time period and size of the orbits with respect to q is made. ...
Photogravitational Restricted Three-Body Problem (PGRTBP) is considered and halo orbits are generated in the vicinity of the Sun-Mars L1 Lagrangian point. Deviation of properties such as time period, size and velocity variation in the halo orbits with Sun as a source of radiation are discussed. With increase in solar radiation pressure, the halo orbits are found to elongate and move towards the Sun and the time period of the halo orbits is found to increase. The variation in the behaviour of invariant manifolds with change in radiation pressure is also computed and it is found that as the radiation pressure increases, the transition from Mars-centric path to heliocentric path is delayed. Certain implications of the velocity profile of the invariant manifolds are also discussed.
