Orbital Mechanics 101

In 1687, Isaac Newton published “A Treatise of the System of the World”, which formed the foundation of classical mechanics. In a chapter of this work Newton visualises a cannon on top of a very high mountain. If there were no forces of gravitation or air resistance, the cannonball should follow a straight line away from Earth, in the direction that it was fired. If a gravitational force acts on the cannonball, it will follow a different path depending on its initial velocity. If the speed was the orbital speed at that altitude, it would go on circling around the Earth along a fixed circular orbit, just like the Moon. This visualisation is key for understanding orbital mechanics.

Figure 1. Image from page 6 from Newton’s Philosophiæ Naturalis Principia Mathematica Volume 3.

There are dozens of different kinds of orbits that are used for multiple purposes, telecommunications, science, technology demonstrations, remote sensing, but in this article, we will explain the most common application orbits.

Figure 2. Representation of multiple orbits.

However, before getting to learn about different orbits we shall understand first some basic parameters and artificial constructions developed to visualise and comprehend the space bodies motion.

The first thing to understand is that the orbits are not circular, they are elliptical, an ellipse is a closed plane curve surrounding two focal points, and for all points on the curve, the sum of the two distances to the focal points is a constant.

Figure 3. Condition for all points of the ellipse.

The elongation of an ellipse is measured by its eccentricity e, a number ranging from e = 0, when it is a circle and the two foci are at the same point, to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola, not an open orbit). To calculate the eccentricity as the ratio between the distance from the centre to the focus and the distance between the co-vertex to the focus, the elements can be found in the following image to visualise it better. The value of the eccentricity of an orbit could be bigger than one but it would no longer be a closed orbit, it would be a hyperbola, but we are going to focus in this article in elliptical orbits.

Figure 4. Ellipse notations.

Equation 1. Mathematical definition of eccentricity.

In the following image we can see several orbits around Earth with different eccentricities. As it is appreciated the smaller the eccentricity the most similar the orbit is to a circle. As a matter of fact, some of the space bodies of the Solar System have very small eccentricities and their orbits are almost circular, this data can be shown in table 1.

Figure 5. Orbits with different eccentricities.

Table 1. Eccentricity of space bodies in the Solar System.

The central body of the system is always in one of the foci of the ellipse, the Sun for the Solar System and Earth for satellite orbits. The periapsis is how it is called the point in the orbit where the distance between the bodies is minimal. And the apoapsis is the point in the orbit where the distance between the bodies is maximum. When talking about Earth these points are called perigee and apogee.

Figure 6. Periapsis and apoapsis locations.

The mean value of the periapsis and apoapsis, also the semisum, results in the value of the semi-major axis, a, which is half the longitude of the distance of the biggest axis of the ellipse.

Figure 7. Semimajor axis.

When defining an orbit, apart from distances there are also angles involved. We will go through some of the most relevant and helpful to understand. But firstly, we shall define the references for defining angles, for such thing we create artificial geometrical constructions to visualise them.

The first of these constructions is the Equatorial plane, which, as the name suggests, contains the Equator of Earth. The second plane is the orbit plane, which, as the name suggests, contains the orbit. The angle between those two planes is the inclination angle, it varies between 0º and 180º. When the inclination angle is less than 90º the orbit is called prograde or direct, when the inclination angle is greater than 90º the orbit is called retrograde or indirect, and when the inclination angle is exactly 90º and the orbit goes over the poles the orbit is called polar orbit.

Figure 8. Inclination angle.

The next angle is the Right Ascension of the Ascending Node (RAAN), Ω, which is the equivalent of terrestrial longitude in space. RAAN is measured from the Sun at the March equinox, which is the place on the celestial sphere where the Sun crosses the celestial Equator from South to North at the Spring equinox and there are the same amount of light hours and darkness at the Equator.

Figure 9. Seasonal configuration of Earth and Sun.

RAAN is measured continuously in a full circle from that alignment of Earth and Sun in space, that equinox, the measurement increasing towards the East.

Figure 10. Concept of Right Ascension of the Ascending Node (RAAN).

Another relevant angle is the argument of the perigee, ω, is the angle from the body’s ascending node to its periapsis, measured in the direction of motion. An argument of perigee of 0° means that the orbiting body will be at its closest approach to the central body at the same moment that it crosses the plane of reference from South to North.

Figure 11. Concept of Argument of the Perigee.

After studying the most relevant angles we will see the most relevant orbits, the ones that are most used. The first three that we are going to see are Low Earth Orbit (LEO), Medium Earth Orbit (MEO) and Geosynchronous Earth Orbit (GEO), being the parameter that differentiates them their distance from the surface of Earth, their altitude. But that will be explained in the next article.

Figure 12. Most used orbits.