Eventos Anais de eventos
COBEM 2017
24th ABCM International Congress of Mechanical Engineering
Earth-Moon Trajectories based on Variational Equation for Jacobi Integral
Submission Author:
Luiz Arthur Gagg Filho , SP
Co-Authors:
Sandro da Silva Fernandes, Luiz Arthur Gagg Filho
Presenter: Luiz Arthur Gagg Filho
doi://10.26678/ABCM.COBEM2017.COB17-0377
Abstract
This work proposes a new two-point boundary value problem to determine Earth-Moon bi-impulsive trajectories in the planar bi-circular restricted four-body model, in which the gravitational attraction of Earth, Moon and Sun are considered. For mission analysis, it is assumed that, initially, the space vehicle is inserted at a prescribed altitude of a low Earth orbit (LEO). After an application of the first impulsive velocity increment, the space vehicle is inserted into a transfer trajectory. The second velocity increment is applied to decelerate and circularize the movement of the space vehicle at a prescribed altitude of a low Moon orbit (LMO). The velocity increments are applied tangentially to the terminal orbits. The fuel consumption used in the mission is closely related to the arithmetic sum of the velocity increments. So, LEO-LMO transfers with smaller total velocity increment are of interest in mission analysis. On the other hand, the Jacobi integral is a first integral of motion in the three-body problem, and, therefore, it remains constant during the motion of the space vehicle. However, in the four-body problem, the Jacobi integral is not a first integral anymore; hence, its value changes during the space vehicle motion. Therefore, a variational equation for the Jacobi integral in the four-body model is deduced in this work, and, the Jacobi integral is taken as an additional state variable in the description of the dynamics of the space vehicle. Using the initial and the final conditions of the differential equation of motion, analytical expression for the velocity increments can be deduced from the Jacobi integral expression. In this way, an estimation of the fuel consumption, which is represented by the total velocity increment, is provided before the resolution of the boundary value problem. To solve the Earth-Moon transfer problem a new two-point boundary value problem with four constraints and four unknowns is then proposed. The first constraint is related to the altitude of the LMO; the second constraint is related to the circular velocity that the vehicle must have after the application of the second velocity increment; the third constraint is related to the orthogonality condition between the position and the velocity vectors of the space vehicle at the final time providing the direction of motion in the LMO; and, the fourth constraint specifies the value of the Jacobi integral at the final time. The four unknowns are: the initial phase angle of the space vehicle related to Earth; the second velocity increment applied at the arrival at the LMO; the total time of flight; and, the initial value of the Jacobi integral. The state equations of the space vehicle is described by the differential equations for the position vector, the velocity vector and the variational equation for the Jacobi integral, expressed in a reference frame centered at the barycenter of the Earth-Moon system. The first velocity increment is calculated using the analytical expressions deduced from Jacobi integral at the initial time. So, the boundary value problem involves the initial value of the Jacobi integral as an additional unknown and the final value of the Jacobi integral as a new constraint. Moreover, the initial phase angle of the Sun must be given as a parameter in order to solve the transfer problem. The analysis of the analytical expressions for the velocity increments provides that, if larger values of Jacobi integral at the final time are set, smaller values of the second velocity increment are achieved independently from the values of the Jacobi integral at other time instant and considering fixed the remaining parameters. This fact motivates a procedure to determine sets of solutions of the TPBVP, in which trajectories are obtained with close values of Jacobi integral at the initial time and increasingly values of Jacobi integral at the final time. Thus, trajectories with smaller fuel consumption are determined. The TPBVP is solved by means of the Newton-Raphson algorithm, and, in order to keep the initial value of the Jacobi integral inside a close interval, a heuristic over is established during the iterations of the Newton-Raphson algorithm. Moreover, an analysis of the Kepler’s energy at the arrival of the LMO with the Jacobi integral and the Hill’s regions can be done using the development of this work. This kind of analysis permits to establish some limits values of the Jacobi integral about possibilities of ballistic capture by the Moon.
Keywords
Earth-Moon trajectories, Jacobi Integral, four-body problem, ballistic capture
