Numerical investigation of collision orbits of lunar satellites

Abstract

In the present study an investigation of the collision orbits of natural satellites of the Moon (considered to be of finite dimensions) is developed, and the tendency of natural satellites of the Moon to collide on the visible or the far side of the Moon is studied. The collision course of the satellite is studied up to its impact on the lunar surface for perturbations of its initial orbit arbitrarily induced, for example, by the explosion of a meteorite. Several initial conditions regarding the position of the satellite to collide with the Moon on its near (visible) or far (invisible) side is examined in connection to the initial conditions and the direction of the motion of the satellite. The distribution of the lunar craters-originating impact of lunar satellites or celestial bodies which followed a course around the Moon and lost their stability -is examined. First, we consider the planar motion of the natural satellite and its collision on the Moon's surface without the presence of the Earth and Sun. The initial velocities of the satellite are determined in such a way so its impact on the lunar surface takes place on the visible side of the Moon. Then, we continue imparting these velocities to the satellite, but now in the presence of the Earth and Sun; and study the forementioned impacts of the satellites but now in the Earth-Moon-Satellite system influenced also by the Sun. The initial distances of the satellite are taken as the distances which have been used to compute periodic orbits in the planar restricted three-body problem (cf. Gousidou-Koutita, 1980) and its direction takes different angles with the x-axis (Earth-Moon axis). Finally, we summarise the tendency of the satellite's impact on the visible or invisible side of the Moon.

A study of collision orbits with the Moon by the method of surface of section

“…The Moon's radius in The initial conditions will be x0 = (1 -CL) + az, yo = 0, xo = V cos cp, io = V sin cp, with a2 the initial distance (at t = 0) of the satellite from the centre of the Moon, I/ the velocity value for which collision takes place after a large number of revolutions before impact (cf. Gousidou-Koutita, 1985b), and cp is the angle of ejection of the satellite at time t = 0 (the angle between the velocity's vector V and the x-axis).…”

“…In previous work (Gousidou-Koutita, 1985b) the collision orbits of the satellite were studied up to their impact on the lunar surface for perturbations of their initial orbits arbitrarily induced. Here we study the case of the explosion of a meteorite in the neighbourhood of the Moon which was executing periodic orbits around the Moon but by the chance of the explosion the pieces of the meteorite followed different unstable courses with different angles of explosion.…”

“…Several astronomical applications of the restricted three-body problem in the plane were dealt with in previous works (see Gousidou-Koutita, 1980, 1985bKopal and Lyttleton, 1963;Szebehely, 1967). Periodic orbits were investigated for the circular and elliptic cases by the author (Gousidou-Koutita, 1980, 1985 and other researchers (Broucke, 1969;Hadjidemetriou, 1975;Moulton, 1920;Szebehely, 1967).…”

“…Periodic orbits were investigated for the circular and elliptic cases by the author (Gousidou-Koutita, 1980, 1985 and other researchers (Broucke, 1969;Hadjidemetriou, 1975;Moulton, 1920;Szebehely, 1967). The influence on those periodic orbits of a fourth body was subsequently investigated by the author with special application on the Earth-Moon-Satellite system induced by the Sun (Gousidou-Koutita, 1980, 1985b. According to the 'external theory' for the origin of Moon's craters (see Kopal, 1966), it is postulated that those craters originated from the collision of bodies orbiting around the Moon.…”

“…Such orbits have been studied by the author in the past (cf. Gousidou-Koutita, 1985b), especially those involving a limited number of revolutions before impact.…”