3.2 The Motion of Satellites
Humans have been observing celestial bodies and their motion since ancient times, with some of the earliest records dating back to around 1200 BC.
The term satellite is used in astronomy to describe an astronomical body (like a moon) that orbits a planet. They are also known as natural satellites since the invention of artificial satellites like those used in GNSS. Regardless of whether they’re natural or artificial, all satellite orbits are governed by a set of scientific laws, most commonly known as Kepler’s laws of planetary motion.
Johannes Kepler was a German mathematician and astronomer in the 17th century, and he published a series of works that outlined how the Earth and other planets orbited the Sun. His work built on the theories of Nicolaus Copernicus, who had suggested that planets orbited the Sun in circular orbits. Kepler realised that the orbits were actually elliptical and proposed three laws that explained how these orbits worked.
In GNSS, Kepler’s laws of planetary motion are used to predict the position of satellites, which is a critical component of positioning, and is included in the signals transmitted by GNSS satellites. This information is in the ephemeris, which we will discuss further in the signals section of this module.
We already know that ellipses are important to GNSS, and how to define them, the fact that satellites have an elliptical orbit makes it pretty easy to understand the maths in Kepler’s laws.
However, before we discuss Kepler’s laws, it’s helpful to get our heads around some basic astronomical terms that relate to elliptical orbits. Because we’re going to be talking about satellites orbiting the Earth, we’ll look at the basics from this perspective.
The basics of elliptical orbits
Kepler proved that the orbit, or path that a satellite took around a body, was an ellipse. As we already know, an ellipse is defined by the semi major (a) and semi minor (b) axes, as shown in Figure 3.2(a). The centre of the ellipse is represented by C.
Perigee and apogee
The perigee and the apogee are points on the ends of the major axis, as shown in Figure 3.2(a). The prefix Peri means “near”, so the perigee is the closer of the two points to the Earth, while Ap means “away from”, so the apogee is the furthest point on the major axis from the Earth.
Depending on what the body being orbited is, depends on what these concepts are called. When orbiting the Sun we use the term “–hellion”, giving us aphelion and perihelion, while stars use “-astron” – apastron and periastron. The generic version is “-apsis”, giving apapsis and periapsis.
Foci
An ellipse also has two foci points (focus is the singular), represented by , as shown in Figure 3.2(b). The positions of the foci are such that at any point on the outside of the ellipse, the sum of the distances from that point to each of the foci will always be the same.
d1 + d2 = 2a
Where d1 = distance from one foci to the satellite
d2 = distance from the othe