 When ancient man looked to the heavens for guidance from the gods, he noticed star patterns and began to document their movement across the heavens. The ancients believed that the earth was flat, but around 350 BC, Aristotle proved that the earth was round. Later, about 150 AD, Ptolemy presented the geocentric theory, the belief that the earth is stationary at the center of the universe, to the sun, moon, stars and planets revolving around it in complex orbits. In the 1500s, Nicholas Copernicus of Poland presented the heliocentric theory, the belief that the earth revolves around the sun as it rotates on its axis. This aspect of astronomy evolved into an intricate study of planetary motion, known as orbital mechanics. Today, orbital mechanics is applied to space flight and satellites that orbit the earth or travel beyond our solar system. In the early 1600s, Johann Kepler, a German mathematician, using the data on planetary observations collected by the Danish scientist Tycho Brahe, developed three laws of planetary motion. Kepler's first law states, all planets move in elliptical orbits with the sun at one focus and the other focus empty. Applied to earth satellites, the center of the earth becomes one focus with the other focus empty. For circular orbits, the two foci coincide. Kepler's second law, the law of areas, states, the line joining the planet to the sun sweeps over equal areas in equal time intervals. When a satellite orbits, the line joining it to the earth sweeps over equal areas in equal periods of time. If areas 1, 2 and 3 are equal, times 1, 2 and 3 are also equal. Therefore, the speed of the satellite changes depending on its distance from the center of the earth. Speed is greatest at the point in the orbit closest to the earth, called perigee, and is slowest at the point farthest from the earth, called apogee. It is important to note that the orbit followed by a satellite is not dependent on its mass. A large, heavy satellite could be in the same orbit with a small, light one, each sweeping out equal areas in equal periods of time. Kepler's third law, the law of periods, relates the time required for a planet to make one complete trip around the sun to its mean distance from the sun. For any planet, the square of its period of revolution is directly proportional to the cube of its mean distance from the sun. Applied to earth satellites, Kepler's third law explains that the farther a satellite is from the earth, the longer it will take to complete an orbit, the greater the distance it will travel to complete an orbit, and the slower its average speed will be. Isaac Newton, the father of classical mechanics, laid the groundwork for orbital mechanics. He combined the work of Kepler and others to formulate the law of universal gravitation, and the three Newtonian laws of motion. While Kepler's laws provided a conceptual model of orbital motion, Newton's laws provided the foundation for the mathematical description of orbits. They explained why a satellite stays in orbit. Newton's law of universal gravitation. Any two objects in the universe, such as the earth and the moon, attract each other with a force directly proportional to the product of their masses, and inversely proportional to the square of the distance between them. Stated more simply, the more massive the objects are, or the closer they are, the greater the gravitational pull between them. Newton's first law of motion. A body in motion will keep moving at the same speed and in the same direction, unless acted upon by an external force. A satellite moves in a curved path around the earth because the earth's gravitational pull acts as an external force on it. Newton's second law of motion. If the sum of the forces acting on an object is not zero, the object will have an acceleration proportional to the magnitude and in the direction of the net force. Newton's second law states that force equals mass times acceleration. It is this mathematical equation and the equation for universal gravitation that forms the basis for calculating orbits. Newton's third law of motion explains how a satellite gets into orbit. For every action, there is an equal and opposite reaction. If you blow up a balloon and let it go, the balloon is pushed forward by the action of the air rushing out of it. A rocket's exhaust gases are like the air rushing out of the balloon. The following illustrates how a satellite stays in orbit. If a man stands on a mountain and fires a projectile horizontally, gravity will cause the path of the projectile to curve downward and it will strike the earth. However, if the man fires the projectile fast enough at a specific speed, the curvature of its path due to gravity will match the curvature of the earth under it. The projectile will then fall around the earth becoming an earth-orbiting satellite. A projectile fired even faster will have a flight path away from the earth, but gravity will act to slow the projectile down, change its flight path, and pull it back toward earth. If the projectile's velocity increased enough, a velocity sufficient to escape the earth's gravitational pull will be reached. This velocity is known as the escape velocity. It is equal to about 7 miles per second at the earth's surface. The preceding description did not consider atmospheric drag and the earth's rotation, both of which will affect the trajectory of the projectile. It illustrated the principles governing a satellite's orbit. There are six numbers called the orbital elements which specify the size, shape, and orientation of an orbit in space, as well as the location of the spacecraft in the orbit. Based on an orbit which is an ellipse, the six orbital elements are length of the semi-major axis, eccentricity, inclination, great ascension of the ascending node, argument of perigee, time of perigee passage. The major axis of an elliptical orbit is the line joining the perigee and apogee. This line is also referred to as the line of absides. The first orbital element is the semi-major axis. It is simply one-half the major axis. Circular orbits have no apogee or perigee. Therefore, the semi-major axis is simply one-half the diameter of the orbit. The semi-major axis is used to define the size of the orbit. From this, the orbital period, or time that it takes for the satellite to complete one orbit, can be calculated. The shape of an orbit is defined by the second orbital element called eccentricity. For all ellipses, the value of eccentricity lies between zero and one. The larger the value, the more elliptical the orbit. A spacecraft in Earth orbit with an eccentricity equal to or greater than one will escape the Earth's gravitational field. When orienting an orbit in space, a three-dimensional coordinate system must be defined. The coordinate system commonly used is the geocentric equatorial coordinate system, which has its origin at the Earth's center. This coordinate system is a non-rotating reference system in which a satellite's orbital plane tends to remain fixed relative to the stars while the Earth turns beneath it. The xy plane is the Earth's equatorial plane. The positive x-axis points to the vernal equinox. This is the point where the sun appears to cross the Earth's equator on its way north on the first day of spring each year. The z-axis is along the Earth's spin axis toward the north pole. Nodes are points in a satellite's orbit which intersect the Earth's equatorial plane. The ascending node is the point at which the spacecraft crosses the equator going from south to north. The descending node is where the spacecraft crosses the equator going from north to south. The line joining the two nodes is called the line of nodes. The orientation of an orbit is determined by three orbital element angles. The right ascension of the ascending node is the angle between the x-axis and the ascending node. It is always measured eastward from the direction away from the vernal equinox in the Earth's equatorial plane. The argument of perigee is the angle between the ascending node and the point of perigee. It is measured in the orbital plane in the direction of spacecraft motion. Inclination is the angle between the equatorial plane and the orbital plane. A satellite which has an eastward velocity component at the ascending node has an orbital inclination lying between 0 and 90 degrees. Such an orbit is called a prograde orbit. A satellite which moves due north at the ascending node is in a polar orbit. Polar orbits have an orbital inclination of exactly 90 degrees. A satellite with a westward velocity component at the ascending node is in a retrograde orbit and has an orbital inclination between 90 and 180 degrees. The five orbital elements explained thus far describe the size, shape, and orientation of the orbit in space. The final element is a time value used to locate the satellite in its orbit. A satellite moves in a very predictable manner. It stays on schedule. Thus, if the time at which a satellite passes a particular point is known, the time when it will pass any other point can be determined. The particular point chosen is perigee and the time of perigee passage is the last of the six orbital elements. The six orbital elements depict a spacecraft's orbit in non-rotating coordinates. To visualize an orbit relative to the rotating Earth, a projection traces the spacecraft's position on the Earth's surface. The projected path is called the ground track. As a satellite orbits the Earth, the ground track shifts westward. There are two causes for this. First, the primary contributor is the Earth's rotation toward the east under the orbital plane. Second, because the Earth is not a uniform sphere and bulges at the equator, its gravity is greatest at the equator. This causes the orbital plane to rotate slowly around the Earth's polar axis in a motion called precession. Precession is toward the west for prograde orbits and toward the east for retrograde orbits. For low Earth orbits such as those of the space shuttle at 150 miles altitude, the westward shift of the ground track due to the Earth's rotation is about 22.5 degrees. While the shift due to precession is only about a half degree, the inclination of a satellite orbit determines the north and south latitude limits of its ground track. The minimum orbital inclination is equal to the latitude of the launch site and is achieved by launching due east. For example, if a satellite is launched due east out of the Kennedy Space Center, which is located at 28.5 degrees north latitude, its orbital inclination will be 28.5 degrees and the limits of its ground track will vary between 28.5 degrees north latitude and 28.5 degrees south latitude. If launch azimuth or direction of flight at launch measured eastward from due north is increased from due east, the orbital inclination angle increases as well as the maximum latitude of the north-south ground track. Therefore, the latitude limits of the ground track equal the new launch inclination. Similarly, if launch azimuth is decreased from due east, orbital inclination once again increases as well as the latitude limits of the ground track. The maximum practical inclination from a Kennedy Space Center launch is 57 degrees. This limit is imposed for safety considerations in order to keep the spacecraft and its booster system from flying over land masses during the ascent phase. To obtain an orbit with an inclination greater than 57 degrees, the spacecraft is launched from Vandenberg Air Force Base in California. Vandenberg offers the opportunity for southerly launches with orbit inclinations between approximately 70 degrees prograde through 138 degrees retrograde. A significant advantage of launching from Vandenberg is the capability to economically achieve polar orbits with ground tracks covering all latitudes from the north pole to the south pole. The earth is constantly turning and all points on its surface have an eastward velocity with the greatest velocity occurring at the equator. The farther the launch site is from the equator or as launch azimuth is increased or decreased from due east, less of the earth's rotational velocity will be imparted to the launch vehicle. This requires more fuel to get into orbit or payload weight will have to be decreased. Launches due east from a position on or near the equator such as the Kuru launch site in French Guyana used by the European Space Agency acquire the advantage of a free velocity gain of about 1,500 feet per second. This compares to the approximate 1,300 feet per second gain available at the further north latitude of the Kennedy Space Center. Launching from an equatorial site offers a significant advantage in payload weight capability and minimizes the amount of fuel needed to achieve an equatorial orbit. Since many satellites operate in equatorial orbits these are important considerations. Spacecraft are launched within a specified time interval called the launch window. Some of the factors affecting the launch window are launch and orbit lighting conditions, sun angles, payload orbit requirements, rendezvous phasing if a rendezvous is planned, tracking and communication requirements, and collision avoidance with other orbiting objects to name a few. One of the factors defining the launch window for the Space Shuttle is launch lighting conditions which can be illustrated by plotting time versus day of year. On this plot we see daylight and darkness at the launch site. The longer daylight hours occur in the middle of the year, summertime. If daylight conditions are required for a convenient emergency landing site for the Space Shuttle, the launch window would now look like this. During the winter months, the available launch window for lighting conditions alone can be as little as three hours per day. When combined with the many other launch factors, the launch window becomes even more constrained. The choice of a particular launch vehicle for a mission depends upon the weight and size of the payload and the desired orbit. Expendable rockets used to place spacecraft in orbit usually consist of several stages that may incorporate both solid and liquid propellants for propulsion. When the fuel in each stage is depleted, the spent stage is jettisoned. Staging offers the advantage of discarding weight when it is no longer needed. The Space Shuttle is a two-stage system. At liftoff, the two solid rocket boosters and three Space Shuttle main engines are all producing thrust. After approximately two minutes of flight, at an altitude of 25 miles, the fuel in the solid rocket boosters is depleted and they are jettisoned. The three main engines, fueled by liquid oxygen and liquid hydrogen, carried in the external tank, continue to burn for several minutes until the shuttle reaches its cutoff velocity. At this time, the main engines are shut down and the external tank is jettisoned. Two additional burns, using the orbiter's maneuvering system referred to as ohms, are required to place the orbiter in its final orbit. The ohms one burn occurs about two minutes after main engine shutdown and establishes the orbital apogee point. The ohms two burn takes place approximately 30 minutes later and circularizes the orbit. Once satellites are launched and put into orbit, it is often necessary to change the orbit with an on-orbit burn. The common term used in describing on-orbit burns or engine firings is delta V. Delta V is the incremental change in spacecraft velocity measured in feet per second, resulting from the burn. The amount of fuel used during a burn depends on the desired delta V change and the mass of the spacecraft. Because the amount of fuel carried is limited, fuel consumption is one of the primary considerations in spacecraft mission planning and is critical to orbit lifetime. On orbit, a spacecraft can thrust in any direction. Burns along the flight path forward and backward are the most common. A unique feature of any orbital burn is that if no other burns occur, the spacecraft will later always pass again through the point of burn. Forward burns increase the spacecraft's velocity and are known as posigrade burns. With posigrade burns, the flight path of the vehicle will be raised at all points except the burn point. Burns opposite the direction of flight which slow the spacecraft down are called retrograde burns. For retrograde burns, the orbit will be lowered at all points except the burn point. The greater the delta V, the greater the difference between the preburn and postburn orbits. Burns can be combined into maneuver sequences to change orbit size, shape or orientation. One of the most common maneuver sequences is made up of two burns and is used to accomplish an orbit transfer between two circular orbits in the same orbital plane. The most energy efficient transfer between two orbits of this type is the Hohmann transfer. The Hohmann transfer is actually one half of an elliptical orbit with its perigee in one of the orbits and its apogee in the other. The burns occur at the perigee and apogee of the transfer orbit. The use of the Hohmann transfer minimizes the delta V required, thus having the advantage of using minimum fuel. The disadvantage of the Hohmann transfer is that it takes longer than most other transfers. The type of the transfer sequence depends on the mission and the amount of fuel available. For example, a space rescue where time is critical might use a fast transfer while a routine satellite deployment where fuel saved for later use is important would most likely use a Hohmann transfer. The burns discussed so far have all been maneuvers in the original orbital plane and do not affect orbit inclination or node position. There are situations which require an orbital plane change such as setting up a rendezvous or placing a satellite in an equatorial orbit. To change the inclination the thrust vector must be directed at an angle to the orbital plane. A thrust with a component that is perpendicular to the orbital plane at either the ascending or descending node will rotate the orbital plane about the line of nodes. A northerly out-of-plane thrust at the ascending node will increase the inclination of a prograde orbit while a southerly thrust will decrease it. Out-of-plane thrusts require considerable amounts of fuel and are performed only when absolutely required. The space shuttle, for example, using all of its onboard propellant is capable of an on-orbit plane change of less than three degrees. Satellite orbital planes and altitudes are determined by their design mission which very often includes a field of view requirement for optical or communications purposes. The field of view of a satellite is defined as the area of the Earth's surface that is in view from the satellite at any given time. Satellites in high orbits have greater fields of view than those in lower orbits. For example, a satellite at an altitude of 800 nautical miles has a circular field of view with a diameter of about 4100 nautical miles. A satellite at 200 nautical miles has a circular field of view with a diameter of about 2,000 nautical miles. Low-orbit satellites are often used for photography and other types of Earth observation. A satellite placed in a low-inclination circular orbit at an altitude of about 19,300 nautical miles will have an angular velocity exactly equal to that of the Earth's. The satellite would seem to remain stationary in longitude as viewed from the ground. Such orbits are called geosynchronous and are used to provide a continuous communications capability among any system of ground stations within their field of view. The geosynchronous orbit field of view is constant and is limited to a latitude zone of about 70 degrees north and south of the equator. Effective satellite communications from geosynchronous orbit is not possible at either pole. However, because of their altitude, their field of view covers nearly half the globe. A special type of geosynchronous orbit with an inclination of zero degrees is called a geostationary orbit. It appears to hover over a fixed point on the Earth's surface at the equator. Most U.S. communication satellites are in geosynchronous orbits providing near-worldwide communications coverage. For effective communications at high latitudes, the molnia orbit is used. Molnia is the Russian word for lightning and is an orbit used extensively by the Soviet Union for its communication satellites. The molnia orbit is highly eccentric with an apogee that is near the geosynchronous altitude and an inclination of about 63 degrees. The satellite slows down at apogee in the northern hemisphere and whips through perigee in the southern hemisphere. This provides communications in the northern hemisphere for up to 75% of its orbital period. Several satellites properly spaced in molnia orbits can provide constant communications at the northern latitudes. Navigation satellites such as the U.S. Navy's transit system and the Joint Service NavStar GPS Global Positioning System use lower orbits so that a user can receive signals from more than one satellite at any time. Another frequently used orbit is known as a sun-synchronous orbit. These take advantage of the precession of the orbital plane caused by the Earth not being a perfect sphere. All sun-synchronous orbits are highly inclined retrograde orbits which precess eastward around the Earth's polar axis at the rate of one revolution per year. Since the Earth's sun-line also revolves eastward at the rate of one revolution per year, the orbital plane will maintain a constant orientation relative to the Earth's sun-line. If the satellite's period is then synchronized with the rotation of the Earth, it will pass over the same point on the Earth's surface at the same local time at a regular interval. A sun-synchronous satellite ensures that a constant sun angle and uniform lighting exist for the same field of view from pass to pass. Satellites such as those in the Defense Meteorological Satellite Program and Landsat are sun-synchronous, imaging the entire Earth on a regular schedule. The gravitational attraction of the Earth on a spacecraft causes it to move in its orbit around the Earth. There are other much smaller forces which will cause a spacecraft to deviate from its desired orbit. These forces cause what are known as orbital perturbations. Orbital precession, which is used to obtain sun-synchronous orbits, results from the perturbing effects of the Earth's non-spherical shape. Other perturbing forces are the gravitational pull of the sun, the moon, and planets, and solar winds, which are charged streams of protons and electrons that heat the Earth's atmosphere and increase atmospheric drag. In most cases, perturbing forces can be compensated for in the spacecraft and orbit design and present no major problems. If the forces disturb the orbit too much, thrusters can be fired to re-establish its desired orbital orientation or altitude. This is particularly true for spacecraft orbiting at very low altitudes where the effects of atmospheric drag are greater and, if not compensated for, will eventually cause the spacecraft to de-orbit. A spacecraft's operational lifetime is frequently limited only by the amount of fuel available to maintain its desired orbit. When its useful life is complete, a satellite is left in orbit or is de-orbited, burning up when re-entering the Earth's atmosphere. When the space shuttle completes its orbital mission, it executes a precise retrograde burn to initiate its controlled return to Earth. This burn occurs nearly halfway around the Earth of the landing site. The new orbit established by the retrograde burn causes the orbiter to enter the Earth's atmosphere about 4,000 miles from the landing site. During the period the orbiter descends from its orbital altitude to atmospheric re-entry, its attitude is maintained by the use of reaction control jets located in the nose and tail of the orbiter. Once the orbiter enters the Earth's atmosphere, its wing and tail aero surfaces begin to become effective and gradually replace the jets for attitude control. As the orbiter nears the landing field, it maneuvers to a long straight-in approach at an angle of 17 to 19 degrees. Nearing the runway, it executes a flare maneuver to reduce its sink rate and glides to a touchdown at approximately 230 miles per hour. As the orbiter rolls to a stop, our journey into the world of orbital mechanics comes to an end for now. This is only the basics of orbital mechanics, an intricate study of planetary and satellite motion. The next time you see a launch, you will see it from a different, somewhat knowledgeable perspective. You will understand the fundamentals of spaceflight.