Understanding Keplerian Elements

To completely and uniquely specify an orbit in space we need a set of 6 parameters, that may vary depending on the approach we use. If we refer to the classic two-body problem (i.e. assuming there are only two bodies - the Earth and the satellite - in space and the Earth is a perfect sphere), the most common one is the Keplerian element set. Each element has its own effect on orbit, and I'd like to discuss these effects separately, discussing differences in orbits that have only one parameter modified at time.

Semimajor Axis

Also called SMA, it defines the size of the orbit. Every body in space describe an ellipse around its central body (the Earth, in our case), and the SMA is one half of the ellipse's major axis. By increasing it, we actually increase the ellipse area and, accordingly with the third Kepler's law (the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit), we get an increase of the orbital period when we increase the SMA.

LEO satellites have a SMA of about 7000 km and, accordingly with the orbit equations, the relevant period is about 1 hr 40 min. GEO satellites have a SMA of about 42000 km, with an orbit period of 24 hr. Between those two orbit classed we can found the MEO satellites (e.g. GPS), that have a SMA of about 26000 km and a period of 12 hr.

In the video above the blue satellite has a SMA of 7000 km, while the purple one as a SMA of 10000 km. The satellite velocity is also affected by the size of the orbit: when we increase it, the satellite speed decreases accordingly (the lower satellite here is going 60% faster than the other). This is really important, and explains why the geostationary satellites appear to be stationary respect to the Earth: their SMA is designed in such a way the orbital period is the same as the Earth, so they rotate with the same angular speed and the relative geometry don't change over time.

Eccentricity

The eccentricity defines the shape of the orbit. Its value can vary between 0 (circular orbit) and 1 (parabolic orbit).

The two satellites in the video above have the same SMA (so they have the same orbital period), but they have a different eccentricity. The blue satellite (eccentricity = 0) flies along a circular path around the Earth center. The purple satellite (eccentricity = 0.3) flies along an elliptical path, with the Earth center located at on of the ellipse foci. In this case there is a point where the satellite is closest to the Earth more then in any other point. This point is called perigee, and it is 180 deg apart of the apogee, that is the farthest point from the Earth.

While the satellite speed is constant for circular orbits, it continuously varies for elliptical orbits. At perigee the speed is higher than in other point, and at apogee is lower than in any other point. HEO (Highly Elliptical Orbit - eccentricity > 0.7) orbits are widely used for communications: when the satellite is close to the apogee, its speed is very low and it appears to be stationary respect to the Earth for a small amount of time.

Inclination

Inclination is one of the two parameters (the other is the RAAN) that define the orientation of the orbital plane in inertial space. It varies from 0 degrees (equatorial orbit) to 90 degrees (polar orbit), even if sometimes, when satellites need to fly backwards respect to the Earth's rotation direction, it falls between 90 and 180 degrees.

The video above shows two satellites: the blue one is on an equatorial orbit (inclination = 0 degrees), while the purple one has 45 degrees inclination. The inclination has also an impact on the satellite ground track: its value defines the maximum latitude that the satellite can overflies on Earth (in this case, the purple satellite's ground track will reach 45 degrees as maximum value).

RAAN

It stands for Right Ascension of Ascending Node and is the other parameter that defines the orientation of the orbit in the inertial space. Its value defines the angle between the reference direction (the X inertial axis) and the satellite's ascending node, that is the point where the satellite crosses the equatorial plane from the south towards the north.

The video below shows two satellites, both with 45 degrees of inclination: the blue one has a RAAN of 0 degrees ( the direction of X inertial axes and the ascending node is the same), while the purple one has a RAAN of 45 degrees.

For real orbits the RAAN value changes over time (RAAN precession): this is due to the perturbing forces other Earth gravitational field that "pull" the satellite orbit and let it rotate about the reference Z axes. The RAAN precession rate is function of the satellite altitude and inclination, and sometimes it is used to design special orbits named sun-synchronous, where the orbit plane remains stationary respect to the incoming Sun rays.

Argument of Perigee

It defines the angle between the line of nodes and the perigee location. In other words, how much the orbit is tilted around its normal axis respect to the nominal position (along the semimajor axis direction):

The video above shows two orbits having different argument of perigee (0 and 90 degrees respectively). The semimajor axis of the two orbits define an angle of 90 degrees.

True Anomaly

Last but not least, the true anomaly defines the initial position of the satellite along its orbit, starting from the perigee. The video below shows a true anomaly difference of 90 degrees (in this case, the angular separation is not constant due to the orbit eccentricity):