Orbits of Cosmic Rockets Toward the Moon

T REALIZATION and employment of space flights are based on a theoretical analysis and on numerical calculations of the equations of motion for vehicles in cosmic space. The basic requirements for the powering and guidance of the launching rocket, as well as the optimal and the acceptable conditions for the takeoff, are determined by computation. As the initial coordinate system for describing the travel of cosmic vehicles, we may take a Cartesian coordinate system with the origin at the Earth's center, which is in translational motion relative to the stars. In many practical problems spherical coordinate systems have to be used, rigidly fastened with the Earth, with the origin at the center of the Earth and at various points on the Earth's surface. The movement of bodies relative to the moon or other planets is also to be considered. Into the differential equations of motion of celestial ballistics describing free flight in cosmic space, we must introduce only the integration forces which are determined by Newton's universal gravitation law. In solving problems of hitting the moon and flight around the moon, we must consider the body's movement in a certain gravitational field produced by the sun, the moon and the Earth, taking into account the oblateness of the Earth. In groping for the optimal conditions of a rocket's takeoff for the moon, approximation methods may be used, permitting us to consider the body's Keplerian motion relative to Earth when the distance from the moon is more than 66,000 km, and as Keplerian motion relative to the moon when this distance is less than 66,000 km. For given points in the northern hemisphere, the optimal conditions for a takeoff in a first approximation when only the Earth's gravitational forces are considered may be obtained in the following way. First, let us consider the collision case. Let A be the position of the point of cutoff of the rocket, B the position of the moon's center at the moment of collision, and O the Earth's center (see Fig. 1). Any orbit corresponding to a flight from point A to point B lies in a plane determined by the three points AOB. For a full determination of the orbit and the magnitude of the initial velocity V, it is sufficient to specify the inclination angle # to the horizon of the initial velocity V. From the solution of the two-body problem, we easily find the functional relation