## Orbital Elements

Seven different numbers are needed to accurately describe an orbit in space and time. Many different sets of numbers will do including the position and velocity at a given moment of time. There is a standard set called the Keplerian elements that are most used and most useful. These are stored in a few formats.

The Keplerian orbital elements define an orbital ellipse around the Earth, orient it three dimensionally, and place the satellite along the ellipse in time. In Keplerian mechanics, all orbits are ellipses. Reality is more complex and the models used include small corrections called perturbations which add extra parameters.

### Epoch

Orbital elements specify the position of the satellite at a certain time called the epoch. The elements are only accurate for a limited period around the epoch.

### Inclination

The orbit ellipse lies in a plane, and this plane forms an angle with the plane of the equator. This angle is called the inclination. Think of it as the tilt between the orbit and the equator.

Inclinations of near 0 degrees are called equatorial orbits, and those near 90 degrees are called polar orbits. By convention, orbits that go the same way as the Earth rotates (prograde or counter-clockwise from above) have inclinations of 0 to 90 degrees. Satellites that orbit retrograde, opposite to the rotation of the Earth, have inclinations great than 90 degrees. For example, a satellite with an inclination of 180 degrees is in an equatorial orbit going east to west.

### Right Ascension of Ascending Node (RAAN)

This is the second parameter that aligns the orbit ellipse in space. The intersection of the orbit plane and the equatorial plane is called the line of nodes. The point where the satellite's orbit crosses the equator going south to north is called the ascending node. The one on the opposite side of the Earth, where the satellite passes into the Southern Hemisphere is called the descending node.

Since the orbit is fixed relative to the stars and not to the surface of the Earth, the astronomical coordinate system of right ascension and declination is used to measure the position of the ascending node. Right ascension is an angle measured in the equatorial plane from a fixed point in space, called the point of Ares (which is also the point of the vernal equinox, where there Sun crosses the equator in the spring).

### Argument of Perigee (ARGP)

Now that the orbital plane is oriented in space, then the position of the orbital ellipse in the orbital plane must be defined. This parameter is called the argument of perigee and is the angle between the major axis and line of nodes. The perigee is the point on the orbit that satellite is closest to the Earth. On the opposite side of the orbit, the satellite is at its farthest point from the Earth called the apogee. The line through the apogee and perigee is called the major axis; it is the long axis of the ellipse.

The angle between the major axis and the line of nodes is the argument of perigee. This is measured in the plane of the orbit. It ranges from 0 to 360 degrees, and is 0 degrees when the perigee is at the ascending node and 180 degrees when the satellite is farthest from the Earth when it rises up over the equator.

### Eccentricity

The eccentricity determines the shape of the orbital ellipse. An ellipse with an eccentricity of 0 is a circle, and when eccentricty is close to 1, the ellipse is long and skinny. Formally, this is the ratio between distance from the center of the ellipse (which isn't the center of the Earth) to the focus of the ellipse (which is the center of the Earth) and the semi-major axis.

### Mean Motion

Next, we need to determine the size of the orbit, or the distance from the satellite to the Earth. This is usually done by giving the mean motion. This is the average angular rotation rate of the satellite. The speed of the satellite is related by Kepler's Third Law to the semi-major axis of the orbit. Satellites that are farther from the Earth move more slowly.

A satellite in an eccentric orbit will move faster at perigee and slower at apogee. The mean motion is the average rate of the satellite's motion. The mean motion is usually given in revolutions per day for Earth satellites. The period is just the inverse of the mean motion, so a mean motion of 2 revs per day gives a period for one orbit of 12 hours.

Sometimes instead of the mean motion, the semi-major axis is given. This is the distance from the apogee or perigee to the center of the ellipse. It is also half the distance from the perigee to the apogee. The perigee or apogee distances could also be used.

### Mean Anomaly

Finally, we need to specify where in the orbit the satellite was at the specific time. This is done with the eccentric anomaly which the angle between the satellite's position and the perigee in the plane of the orbit. It is better to use the "average" position because, the mean anomaly at a future time is easily calculated as the time since the epoch times the mean motion, plus the original mean anomaly.

A satellite in a circular orbit moves at a constant rate around the orbit. A satellite in a non-circular orbit doesn't move at a constant velocity so the mean anomaly doesn't point directly at the satellite. The mean anomly is the anuglar position that a satellite in a circular orbit with the same period (and mean motion). Kepler's equation relates the eccentric anomaly and the mean anomaly for an eccentric orbit.