The motion of planets or spacecraft around the sun is governed primarily by the gravitational attraction between the central body (the sun or planet) and the secondary body (the planet or spacecraft). The same is true for a satellite traveling around the Earth or other planet, where the Earth is now the central body and the satellite is the secondary body. The trajectory, or path, of a planet or spacecraft is referred to as an orbit. In the early 1600s, Johannes Kepler described the orbits of planets in his famous "Three Laws":

1. The orbit of each planet is an ellipse with the sun at the focus.
2. The line joining the planet to the sun sweeps our equal areas in equal times.
3. The square of the period of a planet is proportional to the cube of its mean distance to the sun.

Kepler's laws were based on empirical evidence: observations of the planet Mars made by Tycho Brahe. At the time, Keple hoped that he would be able to describe planetary motion by circles, but the experimental evidence proved otherwise, and he was forced to resort to a little-known geometrical shape — the ellipse. It is interesting to note that Kepler did not understand the physics of why planets followed these paths; but rather just described mathematically what he found in the data. Roughly half a century later, Sir Isaac Newton filled in the gap, explaining the physics of the motion with his famous laws of motion and gravitation. Orbits of planets about the sun and satellites about the Earth are typically described by six Keplerian elements (Figure 1).

They can be divided into parameters that describe the size and shape of the orbit:

a - The semimajor axis (a) is the longest axis of the ellipse going through the two foci.

e - The eccentricity (e) describes the elongation of the ellipse. The distance between the two foci can be computed as 2ae.

And the orientation of the orbit plane:

Ω - The right ascension of the ascending node (Ω) describes where the orbit crosses the equatorial plane relative to a reference direction known as the vernal equinox.

i - The inclination of the orbit (i) describes how the orbit plane is tipped relative to the equatorial plane.

ω - The argument of perigee (ω) describes where the point of closest approach to the planet lies within the orbit plane. It is an angle measured in the plane of the orbit from the ascending node to the point of closest approach.

θ - The last parameter describes where the planet or spacecraft is within its orbit, measured from the perigee and given the unusual name of "true anomaly" (θ).

When an orbit is not circular, it has only one point that is closest to the central body and one point that is farthest from the central body. The closest point is known as the periapse and the farthest point is known as the apoapse. If the central body is the Earth, these points are also referred to as the perigee and apogee, respectively. Similarly, if the central body is the sun, the names change to perihelion and aphelion.

Very clear and accessible information on orbits and definitions can be found at: http://www.jpl.nasa.gov/basics/

Planets in the solar system all travel in orbits with the sun at the focus. Most planets are in almost circular orbits with both foci at the sun. The orbits of Mercury and Pluto are both somewhat elliptical with eccentricities of 0.2 and 0.5, respectively. The eccentricity of Pluto's orbit causes it to sometimes travel inside the nearly circular adjacent orbit of Neptune.

According to Kepler's laws, if you know the semimajor axis of the orbit, you can find the orbit period (i.e., how long it takes to travel around the central body). The period of an orbit is given by:

P = 2 π SQRT( a³ / μ)

Where a is the semimajor axis of the orbit and μ is the gravitational constant for the central body. For Earth it is: μ_E=398600 km³/s². From this, you can see that the larger the orbit, the longer the period. For a circular orbit you can also compute the speed at which the object travels as follows:

s = SQRT( μ/ a )

This shows that the smaller the semimajor axis (i.e., the smaller the orbit), the faster the speed at which the satellite must travel. For example, the International Space Station in orbit around the Earth has a semimajor axis of about 6756 km. So, using the equation for the period given above, we find that its period is about 5,526 seconds or 92 .1 minutes. Using the speed equation, we can calculate its speed as 7.68 km/s. In comparison, a TV broadcasting satellite, in a geosynchronous orbit around the Earth, has an orbital period of 24 hours!

To get from one orbit to another, a spacecraft must transfer to an elliptical path that connects its current orbit to the desired orbit. A simple way to do this is the Hohmann transfer. To execute a Hohmann transfer, a spacecraft fires thrusters tangentially along the direction of travel for a short burst. This initial "delta-v" changes the energy of the orbit to put it on an elliptical path. The periapse of the new orbit is at the starting orbit, and the apoapse of the orbit is at the target orbit. After half an orbital period, the satellite is at the right altitude for its new orbit, but it must again increase its speed to stay in the new higher orbit. A second delta-v is performed to circularize the orbit at the higher altitude. The Hohmann transfer is efficient for fairly small orbital changes. More complex maneuvers using gravity-assist fly-bys are also used to provide more efficient transfers to distant planets.

Figure 1. The six Keplerian elements of planetary orbits. click for copyright

Unmanned satellites and spacecraft are generally tracked by ground stations to determine their orbits. NORAD (the North American Aerospace Defense Command) currently tracks all objects larger than 30 cm in orbit around the Earth. NASA's Deep Space Network (DSN) is used to track interplanetary craft. Commercial satellites are tracked by private ground stations, and some satellites (like the GPS satellites) have their own dedicated tracking networks. A tracking station receives transmissions from the satellite and may also send out a ranging signal that is reflected or echoed by the satellite.

Three basic types of ground-based measurements are used to determine an orbit: range, Doppler, and angles.

Angle measurements are the most straightforward and generally the least accurate type of measurement. A ground station antenna connected to a receiver is scanned to find the highest signal power from the satellite. The antenna sweeps horizontally across in azimuth and vertically in elevation. When it locks on to the signal, the azimuth and elevation angles of the antenna indicate the direction to the satellite. By making a sequence of angle measurements over time, the orbit of the satellite can be estimated.

Range measurements can be made by timing the arrival of marked pulses or symbols in the communications link. The ground station transmits a signal to the satellite or spacecraft and the spacecraft receives and retransmits this signal. This is known as a transponder. The ground station measures the roundtrip time of the signal, divides by 2, and multiplies by the speed of light to find the range to the satellite. More precise measurements can also be made by using more than one ground station on the Earth. Differences between ranges measured at the different stations allow delays in the transponder to be determined and permit much more precise ranging measurements.

Doppler measurements represent the rate of change of range to the satellite. They are a measure of frequency shift of a transmission. Assume that a transmitter at rest emits a signal or tone of a certain frequency that we will call the base frequency. When a transmitter approaches a receiver, the receiver hears or senses a higher frequency than the base frequency. At the "point of closest approach," that is, when the transmitter and receiver are as close as possible, the received frequency is equal to the base frequency. As the transmitter moves away from the receiver, the received frequency is lower than the base frequency. The figure below illustrates the Doppler shift as a function of time as a satellite in a low Earth orbit moves overhead a ground station. The closer to directly overhead the satellite moves, the more dramatic the Doppler shift will be. Scientists can use a sequence of Doppler measurements to determine the orbit of a satellite or spacecraft.

Today, orbit determination programs use all available measurement types to form the best estimate of a satellite's trajectory. The accuracy of the estimates depends on the type of orbit and the quality and quantity of observations from the ground.

Many near-Earth satellites can be seen using a backyard telescope. A few are also visible to the naked eye. A good example is the International Space Station (ISS). The ISS is a very large structure, 3,300 sq. ft. in area, in an almost circular low-Earth orbit, about 250km above the surface of the Earth. The inclination of the orbit is 51.5 deg. Because of its large size and reflective surfaces, and under good viewing conditions (on a clear night when the sun's rays hit it just right), you can see the station pass across the sky with your naked eye (no telescope).

To observe a satellite (or planet or comet), you will need to know what time and where it will rise and set in the sky for your location on Earth. You will also need to know the quality of the "viewing conditions;" that is, how well it will be illuminated by the sun. A great website maintained by NASA provides observing information for a variety of satellites including the ISS. Instructions on using the website are given in Activity 2.

Question/Answer: Ask the students and discuss as a class:

• Ask students if anyone knows the name of the first artificial satellite launched into orbit around the Earth. (Answer On October 4, 1957, the Soviet Union launched Sputnik I.)
• How many satellites would you guess are currently orbiting the Earth? (Answer: about 3,000)

Voting: Ask a true/false question and have students vote by holding thumbs up for true and thumbs down for false. Count the votes, and write the totals on the board. Give the right answer.

• True or False: Unmanned satellites and spacecraft are generally tracked by ground stations to determine their orbits. (Answer: True)

Bingo: Provide each student with a sheet of paper containing a list of the lesson vocabulary terms. Have each student walk around the room and find a student who can define one vocabulary term. Students must find a different student for each word. When a student has all terms completed s/he shouts "Bingo!" Continue until two or three students have bingo. Ask the students who shouted "Bingo!" to give definitions of the vocabulary terms.