The Universe in the Classroom

Up, Up, and Away

Hooking Up

Many human space flights involve a meeting, or rendezvous, with another spacecraft, such as the docking of the space shuttle with the Mir space station (see sidebar). During the rendezvous maneuvers, the astronauts must change their orbit to match the orbit of the spacecraft they'd like to visit. And changing orbits can be counterintuitive.

Firing the rocket engines ultimately makes an orbiting spaceship go slower. The reason is that the engines move the ship into a higher orbit -- which is a slower orbit, because gravity is weaker. Satellites in low orbit experience a stronger pull of gravity and therefore move faster. At an altitude of 200 kilometers (120 miles), a satellite zips by at 28,000 kilometers per hour, completing one orbit every 90 minutes.

At an altitude of 35,800 kilometers (22,200 miles), a satellite travels at a more modest 11,100 kilometers per hour, completing one orbit every 24 hours. It keeps pace with the Earth's rotation. Such an orbit, termed a geostationary or a geosynchronous orbit, is used for weather and communications satellites that need to hang over one point on the Earth's surface. At an altitude of 390,000 kilometers (242,000 miles), our natural satellite -- the Moon -- orbits at 3,600 kilometers per hour and completes one orbit every 28 days.

For each altitude, there is a unique velocity that a satellite must have in order to keep to a circular orbit. If the rocket engines force the satellite to go faster or slower than this velocity, the satellite goes into an elliptical orbit. A satellite in an elliptical orbit keeps changing its distance from Earth (see diagram). Its closest point to Earth is called the perigee; its most distant point, the apogee. As the satellite's altitude changes, so does its speed.
Swingin' at the Savoy
Swingin' at the Savoy. Unless you're a daredevil, you swing back and forth in an arc -- exchanging height for momentum, and then vice versa. At the top of the arc ('A'), your swing comes to a halt and reverses direction. As you swing downward, gravity causes your speed to build up, reaching a maximum at the bottom of the arc ('B'). As you swing back upwards, you slow down and eventually reach the top of the arc ('C'). A satellite in an elliptical orbit moves the same way for the same reason. At its farthest point from Earth (apogee), the satellite stops moving away from Earth and starts moving downward. As the satellite descends, it accelerates, reaching a maximum speed at its closest point to Earth (perigee). As the satellite ascends, it slows down until it reaches apogee again.

The satellite moves fastest at perigee and slowest at apogee. The speed varies just as the speed of a child on a swing (see diagram). Near the top of the swing arc, the swing is going slow; at the bottom, it goes very fast. At the bottom, the swing's inertia carries it upward; as the swing climbs, its speed decreases; at the top, gravity brings the swing back down. Likewise, a satellite goes faster when near Earth (perigee) and slower when farther away (apogee).

Almost all satellites start off in elliptical orbits because it is difficult to get the launch speed just right for a circular orbit. To shift the satellite into a circular orbit, the mission controllers fire the rocket engines once the satellite is above the atmosphere. The space shuttle regularly does this with an "OMS burn'' 46 minutes into the flight. The OMS burn ensures that the shuttle moves at exactly the right speed for a circular orbit at the desired altitude.

To get from one circular orbit to another, a satellite must first go into an elliptical orbit. There are many elliptical orbits that will do the trick, but the one that takes the least effort is called a Hohmann transfer orbit. Suppose you want to move your communications satellite from a low orbit into a geosynchronous orbit at 35,800 kilometers (see figure). The first step is to increase the velocity by firing the rockets. This boosts the satellite into an elliptical orbit with an apogee of 35,800 kilometers. The second step is to fire the rockets again, just as the satellite reaches apogee. This second burn changes the elliptical orbit into a circular one.
give me a transfer
Give me a transfer. Boosting a communications satellite from low Earth orbit into geosynchronous orbit is a two-step procedure. First, fire the rockets to increase the satellite's speed from 28,000 to 36,900 kilometers per hour ('A'). This puts the satellite on an elliptical orbit, known as a Hohmann transfer. The perigee of the new elliptical orbit is the altitude of the low orbit; the apogee is the altitude of the geosynchronous orbit. The satellite will move outward on its new orbit, slowing down as it goes. When the satellite reaches its apogee, it is moving at 5,800 kilometers per hour. At this point, fire the rockets again to increase the speed to 11,100 kilometers per hour ('B'). This puts the satellite into a circular orbit at that altitude.

When astronauts use Hohmann transfers to rendezvous with another spacecraft, they must time the transfer precisely so that they will arrive in their new orbit at the place where the other spacecraft will be.

Inclined to Agree

The altitude and circularity of the orbit are not the only things that must match before two spacecraft can meet. The craft must have orbits with the same inclination.

When the first astronauts went into orbit, most people wondered why the ground path displayed on a world map made a strange up-and-down looping pattern (see map). We all had assumed the orbit would be directly above the equator. In actuality, nearly all orbits are inclined to the equator. Usually the angle of inclination equals the latitude of the launch site: 28.5 degrees for the Kennedy Space Center in Florida (see activity). This means that the space shuttle generally goes 28.5 degrees to the north and 28.5 degrees to the south. NASA sometimes uses orbits with greater than 28.5 degrees of inclination, but those orbits are less efficient.
Ground traces of Mir and Hubble. This map show the points on Earth immediately below these two satellites. The tracks change over the course of a day; this map shows them on April 20, 1996 from 9 to 10:45 p.m. Pacific time. Mir completes one orbit every 92 minutes 26 seconds and reaches latitudes between 51.65 degrees north and 51.65 degrees south. Hubble orbits once every 96 minutes 35 seconds and reaches latitudes lower than 28.47 degrees. Map generated by "OrbiTrack'' version 2.1.4 with NASA orbital elements set 795.
Click here for a larger version of this image.

Currently, the United States, Russia, Canada, Europe, and Japan are building a space station (see image). In an orbit inclined 51.6 degrees to the equator, the station will be easy for the Russians to reach, since their launch site -- the Baikonur Cosmodrome in Kazakhstan -- is at a high latitude. But the American shuttles will have a harder time. Because the shuttle will require extra fuel to reach the station, it will be unable carry as much payload as it could to a less inclined orbit.
The International Space Station
The International Space Station. It doesn't quite live up to "2001: A Space Odyssey,'' but it's a start. Like a high-tech Habitrail, the station will consist of interlocking modules from the United States, Canada, Europe, and Russia. If all goes as planned, six people will live on board -- conducting all sorts of scientific and engineering experiments and, space buffs hope, demonstrating that people can live for long times in space. Assembly is due to start in November 1997 and finish in June 2002. Artwork courtesy of NASA Johnson Space Center.

On the other hand, low-latitude launches have an important advantage. When the shuttle blasts off due east from Kennedy, it receives a free boost of 1,465 kilometers per hour (908 miles per hour) because Earth is rotating. People use the same principle when they run forward while throwing a ball. The ball gets the speed that you're running at, in addition to the velocity that your arm imparts.

A launch pad nearer the equator, such as the European Space Agency's Kourou launch site in French Guiana, is even better. A point on the equator is already moving east at 1,670 kilometers per hour (1,035 miles per hour). Remember that Earth turns on its axis once in 24 hours. Since Earth is 40,080 kilometers (24,850 miles) around, a point on the equator must go 40,080 kilometers in 24 hours. Launch sites at higher latitudes go slower because the distance around is not as great as it is at the equator.

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