What Are Keplerian Elements?

Seven numbers are required to define a satellite orbit. This set of seven numbers is called the satellite orbital elements, or sometimes "Keplerian" elements (after Johann Kepler [1571-1630]), or just elements. These numbers define an ellipse, orient it about the earth, and place the satellite on the ellipse at a particular time. In the Keplerian model, satellites orbit in an ellipse of constant shape and orientation.

Orbital elements remain a mystery to most people. This is due to the aversion many people have to thinking in three dimensions, and second to the horrible names the ancient astronomers gave these seven simple numbers and a few related concepts. To make matters worse, sometimes several different names are used to specify the same number. Vocabulary is the hardest part of celestial mechanics!

The basic orbital elements are...

1. Epoch
2. Orbital Inclination
3. Right Ascension of Ascending Node
4. Argument of Perigee
5. Eccentricity
6. Mean Motion
7. Mean Anomaly, and the optional...
8. Drag

1. Epoch [aka Epoch Time or T0]

A set of orbital elements is a snapshot, at a particular time, of the orbit of a satellite. Epoch is simply a number which specifies the time at which the snapshot was taken.

2. Orbital Inclination [aka Inclination or I0]

The orbit ellipse lies in a plane known as the orbital plane. The orbital plane always goes thru the center of the earth, but may be tilted any angle relative to the equator. Inclination is the angle between the orbital plane and the equatorial plane. By convention, inclination is a number between 0 and 180 degrees.

3. Right Ascension of Ascending Node [aka RAAN or RA of Node or O0 and occasionally called Longitude of Ascending Node]

RAAN wins the prize for most horribly named orbital element. Two numbers orient the orbital plane in space. The first number was Inclination. This is the second. After we've specified inclination, there are still an infinite number of orbital planes possible. The "line of nodes" can poke out anywhere along the equator. If we specify where along the equator the line of nodes pokes out, we will have the orbital plane fully specified. The line of nodes pokes out two places, of course. We only need to specify one of them. One is called the ascending node (where the satellite crosses the equator going from south to north. The other is called the descending node (where the satellite crosses the equa- tor going from north to south). By convention, we specify the location of the ascending node.

"Right ascension of ascending node" is an angle, measured at the center of the earth, from the vernal equinox to the ascending node.

I know this is getting complicated. Here's an example. Draw a line from the center of the earth to the point where our satellite crosses the equa- tor (going from south to north). If this line points directly at the vernal equinox, then RAAN = 0 degrees.

By convention, RAAN is a number in the range 0 to 360 degrees.

4. Argument of Perigee [aka ARGP or W0]

Argument is yet another fancy word for angle. Now that we've oriented the orbital plane in space, we need to orient the orbit ellipse in the orbital plane. We do this by specifying a single angle known as argument of perigee.

A few words about elliptical orbits. The point where the satellite is closest to the earth is called perigee, although it's sometimes called periapsis or perifocus. We'll call it perigee. The point where the satellite is farthest from earth is called apogee (aka apoapsis, or apifocus). If we draw a line from perigee to apogee, this line is called the line-of-apsides. (Apsides is, of course, the plural of apsis.) I know, this is getting complicated again. Sometimes the line-of-apsides is called the major-axis of the ellipse. It's just a line drawn through the ellipse the "long way".

The line-of-apsides passes through the center of the earth. We've previously identified another line passing through the center of the earth. That was the line-of-nodes. The angle between these two lines is called the argument of perigee. Where any two lines intersect, they form two complimentary angles, so to be specific, we say that argument of perigee is the angle (measured at the center of the earth) from the ascending node to perigee.

By convention, ARGP is an angle between 0 and 360 degrees.

5. Eccentricity [aka ecce or E0 or e]

This one is simple. In the Keplerian orbit model, the satellite orbit is an ellipse. Eccentricity tells us the "shape" of the ellipse. When e=0, the ellipse is a circle. When e is very near 1., the ellipse is very long and skinny.

6. Mean Motion [aka N0]

So far we've nailed down the orientation of the orbital plane, the orientation of the orbit ellipse in the orbital plane, and the shape of the orbit ellipse. Now we need to know the "size" of the orbit ellipse. In other words, how far away is the satellite?

Kepler's third law of orbital motion gives us a precise relationship between the speed of the satellite and its distance from the earth. Satellites that are close to the earth orbit very quickly. Satellites far away orbit slowly. This means that we could accomplish the same thing by specifying either the speed at which the satellite is moving, or its distance from the earth!

Satellites in circular orbits travel at a constant speed. Simple. We just specify that speed, and we're done. Satellites in non-circular orbits move faster when they are closer to the earth, and slower when they are farther away. The common practice is to average the speed. You could call this number "average speed", but astronomers call it the "Mean Motion". Common units are revolutions per day.

Typically, satellites have Mean Motions in the range of 1 rev/day to about 16 rev/day.

7. Mean Anomaly [aka M0 or MA or Phase]

Now that we have the size, shape, and orientation of the orbit firmly established, the only thing left to do is specify where exactly the satellite is on this orbit ellipse at some particular time. Our very first orbital element (Epoch) specified a particular time, so all we need to do now is specify where, on the ellipse, our satellite was exactly at the Epoch time.

"Anomaly" is yet another astronomer-word for angle. Mean anomaly is simply an angle that marches uniformly in time from 0 to 360 degrees during one revolution. It is defined to be 0 degrees at perigee, hence is 180 degrees at apogee.

If you had a satellite in a circular orbit (therefore moving at constant speed) and you stood in the center of the earth and measured this angle from perigee, you would point directly at the satellite. Satellites in non-circular orbits move at a non-constant speed, so this simple relation doesn't hold. This relation does hold for two important points on the orbit, however, no matter what the eccentricity. Perigee always occurs at MA = 0, and apogee always occurs at MA = 180 degrees.

It has become common practice with radio amateur satellites to use Mean Anomaly to schedule satellite operations. Satellites commonly change modes or turn on or off at specific places in their orbits, specified by Mean Anomaly. They use 256ths because this is a magic number in the computer world.

8. Drag [aka N1]

Drag caused by the earth's atmosphere causes satellites to spiral downward. As they spiral downward, they speed up. The Drag orbital element simply tells us the rate at which Mean Motion is changing due to drag or other related effects. Precisely, Drag is one half the first time derivative of Mean Motion.

Its units are revolutions per day per day. It is typically a *very* small number. Common values for low-earth-orbiting satellites are on the order of 10^-4. Common values for high-orbiting satellites are on the order of 10^-7 or smaller.

Thank you to Franklin Antonio, N6NKF, the author of InstantTrack where I acquired the information for this segment.

Updated 26 March 1995. Article courtesy of Bruce Paige, KK5DO ([email protected]). Feedback to KB5MU.