What Are Keplerian Elements?
This script from the Houston AMSAT Net was written by AMSAT Area
Coordinator Bruce Paige, KK5DO. Authorization is given for the use
of this information over any ham band. Please give credit for the
script where credit is due.
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...
- Orbital Inclination
- Right Ascension of Ascending Node
- Argument of Perigee
- Mean Motion
- Mean Anomaly, and the optional...
- 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
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
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
- 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
- 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
"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
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
- 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
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.