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Keplerian Elements

Things You Need to Know in Order to Calculate a Satellite's Orbit

What are the Keplerian Elements

The Keplerian Elements are a set of numbers which allow satellite tracking programs to calculate a satellite's position in space. The keplerian elements come in two formats - either the NASA 2-line elements (TLE) or the AMSAT verbose format elements.

Keps, as they are sometimes called, give us specific information about a satellite's orbit at a specific moment. Once these elements are known for a specific time, the satellite's position in space can be predicted using complex mathematical calculations. There is, however, one problem. Keps are given for a specific time. The accuracy of the position prediction degrades as time goes by. The predicted postion of a satellite using 7 days old keps is more accurate than a prediction using 3 months old keps

There are 8 elements that you need to define an orbit. These elements are also called Keplerian Elements after the German astronomer Johannes Kepler (1571-1630). Kepler discovered that planets moved in elliptical orbits rather than circular orbits. The following are Keplerian Elements:

1. Epoch Time2. Orbital Inclination 3. Right Ascension of Ascending Node 4. Eccentricity 5. Argument of Perigee 6. Semi Major Axis 7. Mean Anomaly 8. Drag

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Keplerian element describe an orbit’s physical attributes. Satellites orbit the Earth in elliptical path’s with the center of the Earth at one of the ellipse’s (mathematical) focus points. The position in the orbit closest to the Earth is called perigee; the position in the orbit furthest from the Earth is called apogee.

Apogee = In the orbit, The more distant point between satellite and the Earth

Perigee = In the orbit, The nearer point between satellite and the Earth

1) Epoch Time

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Τ0 = 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

The first thing you need to define an orbit is the time at which the Keplerian Elements were defined. You need a snapshot of where and how fast the satellite was going.

2) Orbital Inclination (degrees)

Ι0 = The angle between the equator and the orbit planet

This element tells you what the angle is between the equator and the orbit when looking from the center of the Earth. If the orbit went exactly around the equator from left to right, then the inclination would be 0. The inclination ranges from 0 to 180 degrees.

3) Right Ascension of Ascending Node (degrees)

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Ω0 = The angle between vernal equinox and the point where the orbit crosses the equatorial plane (going north)

This is probably one of the most difficult of the elements to describe. The ascending node is the place where the satellite crosses the equator while going from the Southern Hemisphere to the Northern Hemisphere. Now since the Earth rotates, you need to specify a fixed object in space. We use Aries (this is also the same location as the vernal equinox). The angle, from the center of the Earth, between Aries and the ascending node is called the right ascension of ascending node.

4) Eccentricity

Ε0 = A constant defining the shape of the orbit

Eccentricity is shape of the ellipse.  Eccentricity is computed as the linear eccentricity (the distance from the center of the ellipse to the center of the Earth) divided by the semi major axis. A zero eccentricity describes a circular orbit; an eccentricity approaching one describes a highly elliptical orbit.

The eccentricity tells you how flat the orbit is. If the orbit is a perfect circle, then the eccentricity is 0. When the eccentricity is close to 1, then the orbit is very flat.

5) Argument of Perigee (degrees)

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ω0 = the angle between the ascending node and the orbit's point of closest approach to the earth (perigee)

Since an orbit usually has an elliptical shape, the satellite will be closer to the Earth at one point than at another. The point where the satellite is the closest to the Earth is called the perigee. The point where the satellite is the furthest from the Earth is called the apogee.

The argument of perigee is the angle formed between the perigee and the ascending node. If the perigee would occur at the ascending node, the argument of perigee would be 0.

6) Semi Major Axis

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The semi major axis describes the size of the ellipse. Semi-major axis is one half the longest distance across the ellipse (or one half the distance between apogee and perigee).  By Kepler's third law of planetary motion, you can compute the orbital period (the time it takes to make one complete orbit) from the semi major axis.  The mean motion (or orbital frequency) is the reciprocal of the period. The commonly available Keplerian elements use mean motion

The mean motion tells you how fast the satellite is going. According to Kepler's Law:

as the satellite gets closer to the Earth, its velocity increases. If we know how fast the satellite is going, we also know the altitude of the satellite.

7) Mean Anomaly

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θ0 = True Anomaly (degrees) The angle between perigee and the vehicle (in the orbit plane)

True anomaly is the angle measured in the direction of motion from perigee to the satellite's position at some defined epoch time.  Mean anomaly describes what the satellite's true anomaly would be if it were in a circular orbit.  You can compute mean anomaly from the orbit's true anomaly and eccentricity. The commonly available Keplerian elements use mean anomaly.

The mean anomaly tells you where the satellite is in its orbital path. The mean anomaly ranges from 0 to 360 degrees. The mean anomaly is referenced to the perigee. If the satellite were at the perigee, the mean anomaly would be 0.

8) Drag (optional)

Several factors can affect the velocity of a satellite. If the satellite were in a low orbit, then the atmosphere would produce drag. This would cause the satellite to come closer to the Earth therefore speeding up (Kepler's Law). Another factor that can affect satellite orbits is gravitational pull from stellar bodies such as the sun or the moon. These bodies could pull the satellite away from the Earth causing it to slow down.