APPLICATIONS TO SPACEFLIGHTSpacecraft can follow four different types of flight paths in space. One of these, the elliptical
orbit, corresponds to natural planetary motion. The other three are the circular orbit, parabolic trajectory, and hyperbolic trajectory. All four are known as conic sections because they are curves generated by passing a plane through a cone at varying angles. Circular and elliptical orbits are closed and periodic flight paths. All orbits of Earth satellites fall into one of these two categories. The parabolic trajectory is the escape path, and the associated velocity along this route is the escape velocity. The velocity along a hyperbolic trajectory is always greater than the escape velocity at the same point. In contrast to the parabolic trajectory, a hyperbolic path leaves the
spacecraft with some extra velocity after escaping the planet. Planetary probes follow these trajectories when leaving Earth's gravitational influence. LaunchTwo launch sites are currently used by the United States, one at the Kennedy Space Center in Florida and the other at Vandenberg Air Force Base in California. East Coast launches are generally for spacecraft going into low
inclination orbits. (Inclination is simply the angular deviation of the orbit of the spacecraft from the plane of the Earth's equator.) Launches from Kennedy Space Center can place payloads directly into orbits with inclinations between 28.5¡ and 57¡. Vandenberg is used for higher inclination orbits. Launch is achieved by the use of a single-stage or multistage rocket. The rocket must develop sufficient thrust for a sufficient length of time to lift the payload above the atmosphere and place it in orbit or send it into deep space. Orbital ManeuversTwo categories of orbital maneuvers are of interest. The more common one is raising the altitude of a circular orbit. The other is changing the orbital inclination. To raise the radius of a circular orbit, at least two separate thrusting intervals are required. For most such maneuvers, two such thrusts are optimum if applied properly. This maneuver, called a Hohmann transfer, consists of simply increasing the orbital velocity at one point, waiting exactly one-half the transfer orbit period, and then increasing the orbital velocity again. The magnitude of the first increase inserts the spacecraft into a transfer orbit whose apogee just reaches the final desired altitude. When that point is reached, the second increase inserts the vehicle into the desired circular orbit.Most communications satellites are placed in geostationary orbits, or orbits in the equatorial plane with a period precisely equal to that of the Earth about the poles. Thus, a satellite in such an orbit would appear stationary over one point on the equator. Geostationary orbits have a radius of 35,880 km (22,300 mi). Communications satellites are sometimes released into space from a Space Shuttle. In such cases the latitude of the Kennedy Space Center launch site corresponds to the minimum inclination directly reachable by the Shuttle boosters. The satellites are therefore first placed in low parking orbits at an inclination of 28.5¡. A combination Hohmann transfer and inclination change is then required. Rendezvous and DockingMany missions involve the rendezvous and docking of two spacecraft. Such maneuvers were demonstrated in the Apollo program. Specifically, the return of the Lunar Excursion Module to lunar orbit and its rendezvous and docking with the command and service modules were essential to the success of the Apollo missions to the Moon. Docking involves a sequence of maneuvering phases. Initially, the chase vehicle is launched into an orbit quite close to that of the target. Once in the vicinity, special rendezvous maneuvers are carried out, using carefully applied, small thrusts over a period of several minutes. ReentryAs a mission in orbit ends, a spacecraft may simply be shut off, or it may be brought back to Earth. All mannflights, obviously, must end with a return through the atmosphere. This is initiated by slowing the vehicle velocity slightly through the use of retrofiring. A shallow descent is then begun into the upper atmospheric limits. Passage through the upper atmosphere is spread along several thousand kilometers to minimize heat buildup in the spacecraft's thermal protection system. Once the vehicle has slowed, landing can be achieved by parachute deployment or, in the case of vehicles such as the Shuttle, by an unpowered gliding descent. Midcourse CorrectionMany space missions involve planetary probes that leave a low parking orbit on a hyperbolic trajectory away from Earth. As the spacecraft speeds away, the gravitational attraction diminishes. The pull of the Sun, however, quickly becomes a dominant force in the flight. Thus, the probe enters an orbit about the Sun in which it travels until it approaches the target planet. Gravitational forces of the planet on the probe get stronger as the vehicle gets closer. Finally, they become dominant over solar pull. The probe will either crash on or fly by the planet, depending on small adjustments in its flight path. Such minute changes are called midcourse corrections.