How do satellites move around the Earth? Let’s start at the beginning! Klepero’s famous three laws were formulated by Johannes von Kepler between 1609 and 1619, based on astronomical observations of the planets by Tycho Brahe. **The first law states that the planets describe closed trajectories around the Sun**: these are ellipses and the Sun occupies one of the two foci. The second law concerns the velocity of the planets along their orbit.

In particular, the segment joining the Sun to one of the planets sweeps equal areas in equal times. This implies that **the speed of the bodies is not constant along their orbit**, but increases when close to the Sun and decreases otherwise. Finally, Kepler’s third law relates the size of the orbit (through the cube of the semi-major axis of the ellipse) proportionally to the square of the period of revolution. In other words, the larger the orbit, the more naturally the period of revolution increases.

It was later to be Isaac Newton who motivated Kepler’s laws mathematically through the introduction of the gravitational force and the study of the solutions of the second equation of motion (force equals mass times acceleration). Kepler formulated the three laws for the motion of the planets around the Sun. However, these apply to a pair of bodies: the Earth with respect to the Sun, the Moon with respect to the Earth, or artificial satellites with respect to the Earth. **By exploiting Kepler’s laws and Newton’s dynamics it is possible to construct orbits to observe the Earth or deep space**, to place satellites in space and to conduct scientific experiments.

## Orbits around the Earth

Earth orbits can be classified according to their distance from the planet’s surface. The orbits in which satellites move around the Earth can be:

- LEO (Low Earth Orbit)
- HEO (High Earth Orbit)
- GEO (Geostationary Equatorial Orbit)

**An orbit is called a Low Earth Orbit (LEO) if it is confined between 200 and 600 km altitude**. The lower limit depends mainly on the residual atmospheric drag, which influences the satellite’s motion, causing it to gradually lose altitude. For example, the International Space Station (ISS) is at an altitude of about 404 km.

**The upper limit of 600 km depends mainly on Van Allen radiation belts, which can damage solar cells and on-board electronics**. As a consequence, low orbits tend to be circular, as the eccentricity is confined below 0.03. LEO orbits therefore ensure safe space flight (with respect to the radiation hazard) and are often used as parking orbits for interplanetary probes before they are launched on their journey beyond Earth’s gravity. In addition, LEOs are used for Earth observation, as they allow high resolutions but pay for a limited visual range.

An orbit is called a high orbit (or HEO, from High Earth Orbit) if it extends beyond an altitude of 10,000 km. At these altitudes, atmospheric resistance is absent, radiation is low and much of the Earth is visible. **These, however, are not used for terrestrial observation, since at high altitudes resolution suffers, but for telecommunications**. The high visibility allows antennas on the ground (even very far apart) to be in constant communication with the satellite, which remains in the same portion of the sky for a long time, thus facilitating pointing.

Between LEOs and HEOs lie the so-called Medium Earth Orbits (or MEOs), a sort of compromise that allows for adequate resolution and limited power required for ground transmissions, while maintaining wide visibility.

## How satellites move around the Earth: ‘special’ orbits

Among the most important cases of ‘special’ orbits on which satellites move around the Earth, we would like to mention two examples:

- Sun-synchronous orbits (SSO, from Sun-Synchronous Orbit)
- Geostationary Equatorial Orbits (GEO)

SSOs are LEO orbits that take advantage of a particular orbital perturbation: the polar flattening of the Earth. The Earth is not a perfect sphere, and its mass is not uniformly distributed. Therefore, the gravitational field it generates is not symmetrical. **The orbits, therefore, will not be exactly as described by the Keplerian conics, but will be slightly perturbed**. For LEOs, one of the main gravitational perturbations is polar flattening. It can be shown that this deformation causes the plane of the orbit to rotate.

This effect may seem annoying, as the satellite, after a full period, will not be at the same starting point, but slightly displaced. However, SSO orbits use this mechanism to make the orbital plane rotate with the same period as the Earth rotates with respect to the Sun. In this way, **one can construct orbits whereby the satellite has at least one face always to the Sun, and then orient the solar panels accordingly**. In addition, SSOs guarantee overflights of the same point on the ground at the same local time. To ensure this, SSOs are highly inclined (94 to 98 degrees) and have an altitude of about 500 km. They are therefore very useful for earth observation.

The GEOs, on the other hand, are located at an altitude of about 36000 km, and have a period of revolution equal to one Earth day. Thus, **also being circular and equatorial, their position in the sky remains fixed, allowing continuous transmissions**. From this position, the satellite covers about half the planet and three satellites, offset by about 120 degrees from each other, manage to cover the whole Earth, except for a small region at high latitudes, and remain in constant communication with each other. Telecommunication satellites generally occupy these orbits.