on a spacecraft equals that of its central body, which is where D is the mean distance from the satellite to the central body, and mc and ms are the masses of the central body and satellite, respectively. This value is approximately from Earth's Moon. An accurate solution of the trajectory requires treatment as a three-body problem, but a preliminary estimate may be made using a patched conic approximation of orbits around the Earth and Moon, patched at the SOI point and taking into account the fact that the Moon is a revolving frame of reference around the Earth. Translunar injection This must be timed so that the Moon will be in position to capture the vehicle, and might be modeled to a first approximation as a Hohmann transfer. However, the rocket burn duration is usually long enough, and occurs during a sufficient change in flight path angle, that this is not very accurate. It must be modeled as a non-impulsive maneuver, requiring integration by finite element analysis of the accelerations due to propulsive thrust and gravity to obtain velocity and flight path angle: where: F is the engine thrust; α is the angle of attack; m is the vehicle's mass; r is the radial distance to the planet's center; and g is the gravitational acceleration, which varies with the inverse square of the radial distance: Altitude , downrange distance , and radial distance from the center of the Earth are then computed as: Mid-course corrections A simple lunar trajectory stays in one plane, resulting in lunar flyby or orbit within a small range of inclination to the Moon's equator. This also permits a "free return", in which the spacecraft would return to the appropriate position for reentry into the Earth's atmosphere if it were not injected into lunar orbit. Relatively small velocity changes are usually required to correct for trajectory errors. Such a trajectory was used for the Apollo 8, Apollo 10, Apollo 11, and Apollo 12 manned lunar missions. Greater flexibility in lunar orbital or landing site coverage (at greater angles of lunar inclination) can be obtained by performing a plane change maneuver mid-flight; however, this takes away the free-return option, as the new plane would take the spacecraft's emergency return trajectory away from the Earth's atmospheric re-entry point, and leave the spacecraft in a high Earth orbit. This type of trajectory was used for the last five