Numerical method of optimal dates getting moon libration point L1, with the influence of noncentral earth's gravitational field and perturbing acceleration of the Sun

Space technologies


Okishev Y. A.

Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia



The aim of this work is to find the optimal date of getting a spacecraft (SC) to the libration point L1 of the «Earth — Moon» system with the influence of the second zonal harmonic of the Earth and the perturbing acceleration of the Sun. A ballistic flight happens with two inclusions of the chemical rocket engine. We assume that the criterion of the optimization is the minimum value of the total pulse of the velocity, which is necessary for the flight with a low near-earth orbit to the lunar libration point L1. Ephemerides of the Moon and the Sun are taken from the planetarium DE-403 of the production JPL.
A numerical solution is found for an every date under consideration. We examine a SC movement in the geocentric equatorial coordinate system. A SC movement is described by the differential equation system, which is a special case of the limited three or more body problem. We have got the numerical values by the numerical integration of Runge-Kutta method during the date of leaving the base orbit and the date of getting to the libration point. For the first approach the vector of the initial values ​​(coordinates of the starting point and the velocity components of the spacecraft in the geocentric equatorial coordinate system) is determined by the Homanov’s condition of the flight. The solution of the boundary value and the optimization problem is described in the article. This solution gives us the direct hit of the spacecraft to the libration point. The optimization problem presents the optimal time of the flight for each date of getting into the libration point.
We analyzed the Sun’s position effect on the power of the flight. The choice the regarding period is based on an analysis of the Sun and the Moon.
Results of this work are
· the selection of the optimal date of getting into the libration point L1 of the «Earth — Moon» system;
· the values ​​of the optimal time of the flight;
· the values of the longitude of ascending node of the orbit base;
· the values of the first, the second and the total impulse of the velocity;
· the coordinates of the starting point in the geocentric equatorial coordinate system.
These results can be used for the space project’s development for using the lunar libration point L1.
The comparative analysis of the numerical values ​​of the total impulse of the velocity with and without the influence of the Sun lets us show the importance of the selection of the date. The choice of the optimal date allows us to reduce the total impulse of the velocity for about 4 m/s.


mathematical modeling ballistics analysis, Runge-Kutta method, libration point, L1, Earth – Moon


  1. Konstantinov M.S., Kamenkov E.F., Perelygin B.P., Bezverbyj V.K., Mishin V.P. Mehanika kosmicheskogo poleta (Space flight mechanics), Moscow, Mashinostroenie, 1989, 407 p.
  2. Markeev A.P. Tochki libracii v nebesnoj mehaniki i kosmodinamike (Libration points in celestial mechanics and cosmodynamics), Moscow, Nauka, 1978, 312 p.
  3. Okishev Yu.A., Klinaev Yu.V. Vestnik Saratovskogo gosudarstvennogo tekhnicheskogo universiteta, 2013, no. 1(69), pp. 56-61.

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