# An introduction to Orbital Mechanics

For this new article, let’s not focus on law but on an introduction to Orbital Mechanics. Orbital mechanics (also called astrodynamics) is the application of the laws of physics to describing the motion of spacecraft. It is one of the fundamental topics in astronautics and is essential to the design, implementation, and operation of a space mission.

As well as defining the sorts of orbits that are possible, orbital mechanics is needed to determine spacecraft trajectories and manoeuvres. In this article on Space Legal Issues, the basic principles of orbital mechanics will be explained, together with the classical orbital elements used to describe orbits.

#### Kepler’s Laws

1st Law: The orbits of the planets are ellipses with the Sun at one focus.

2nd Law: The line joining a planet to the Sun sweeps out equal areas in equal times.

3rd Law: The square of the orbital period (how long it takes the planet to go around the Sun) is directly proportional to the cube of the mean distance between the Sun and the planet.

#### Newton’s Law of Universal Gravitation, an introduction to Orbital Mechanics

Any two bodies attract each other with a force proportional to the product of their masses and inversely proportional to the square distance between them.

#### Restricted Two-body Problem, an introduction to Orbital Mechanics

If we restrict the situation to two spherically-symmetric bodies with fixed mass, which interact only gravitationally and where one body is very large and the other very small, the smaller one will travel around the big one on a trajectory which is one of: circle, ellipse, parabola, and hyperbola.

The first two of these (circle and ellipse) are “closed” and the small body will orbit indefinitely. The second two (parabola and hyperbola) are “open” and the small body has enough energy to escape from Earth and will not return. All orbits are located in the orbit plane, which passes through the centre of mass of the attracting body.

#### Orbits, an introduction to Orbital Mechanics

There are a number of parameters we use to describe an orbit. Semi-major axis (a): half the length of the long (major) axis of the orbit ellipse. Eccentricity (e): the ratio between the short (minor) axis and the (major) long axis of the ellipse. Radius of perigee (Rp): the distance from the centre of the Earth to the point of closest approach. Radius of apogee (Ra): the distance from the centre of the Earth to the point of farthest approach.

To describe the position of the spacecraft in the orbit we use the true anomaly (the angle between the perigee and the spacecraft. There is a direct relation between the energy of an orbit (more correctly, the total mechanical energy per unit mass) and the semi-major axis. This relationship is independent of the shape of the orbit.

#### Classical Orbital Elements, an introduction to Orbital Mechanics

To fully describe the orbit/position of a spacecraft we need six orbital elements. We have already seen three of these; we need three more to describe the orientation of the orbit plane in space; therefore, we need to define a 3D coordinate system.

For spacecraft orbiting the Earth, we use the geocentric, equatorial co-ordinate system. The reference plane is defined by the Earth’s equator with the prime direction defined by the vernal equinox. This is also called the Earth-centered inertial (ECI) system because it is fixed in space (it does not rotate with the Earth). Inclination is the angle between the orbital plane and the reference plane (0° to 180°). Right ascension of the ascending node is the angle between the inclination and the ascending node (the point where a satellite passes through the equatorial plane from the southern hemisphere to the northern one).

### Space Legal Issues and Orbital Mechanics

An orbit is the curved path through which objects in space move around a planet or a star. The 1967 Treaty’s regime and customary law enshrine the principle of non-appropriation and freedom of access to orbital positions. Space Law and International Telecommunication Laws combined to protect this use against any interference.

The majority of space-launched objects are satellites that are launched in Earth’s orbit (a very small part of space objects – scientific objects for space exploration – are launched into outer space beyond terrestrial orbits). It is important to precise that an orbit does not exist: satellites describe orbits by obeying the general laws of universal attraction. Depending on the launching techniques and parameters, the orbital trajectory of a satellite may vary.

Sun-synchronous satellites fly over a given location constantly at the same time in local civil time: they are used for remote sensing, meteorology or the study of the atmosphere. Geostationary satellites are placed in a very high orbit; they give an impression of immobility because they remain permanently at the same vertical point of a terrestrial point (they are mainly used for telecommunications and television broadcasting).

#### List of orbits

A geocentric orbit or Earth orbit involves any object orbiting Planet Earth, such as the Moon or artificial satellites. Geocentric (having the Earth as its centre) orbits are organised as follow:

1) Low Earth orbit (LEO): geocentric orbits with altitudes (the height of an object above the average surface of the Earth’s oceans) from 100 to 2 000 kilometres. Satellites in LEO have a small momentary field of view, only able to observe and communicate with a fraction of the Earth at a time, meaning a network or constellation of satellites is required in order to provide continuous coverage. Satellites in lower regions of LEO also suffer from fast orbital decay (in orbital mechanics, decay is a gradual decrease of the distance between two orbiting bodies at their closest approach, the periapsis, over many orbital periods), requiring either periodic reboosting to maintain a stable orbit, or launching replacement satellites when old ones re-enter.

2) Medium Earth orbit (MEO), also known as an intermediate circular orbit: geocentric orbits ranging in altitude from 2 000 kilometres to just below geosynchronous orbit at 35 786 kilometres. The most common use for satellites in this region is for navigation, communication, and geodetic/space environment science. The most common altitude is approximately 20 000 kilometres which yields an orbital period of twelve hours.

3) Geosynchronous orbit (GSO) and geostationary orbit (GEO) are orbits around Earth at an altitude of 35 786 kilometres matching Earth’s sidereal rotation period. All geosynchronous and geostationary orbits have a semi-major axis of 42 164 kilometres. A geostationary orbit stays exactly above the equator, whereas a geosynchronous orbit may swing north and south to cover more of the Earth’s surface. Communications satellites and weather satellites are often placed in geostationary orbits, so that the satellite antennae (located on Earth) that communicate with them do not have to rotate to track them, but can be pointed permanently at the position in the sky where the satellites are located.

4) High Earth orbit: geocentric orbits above the altitude of 35 786 kilometres.

#### The freedom of use of orbits

Near-Earth space is formed of different orbital layers. Terrestrial orbits are limited common resources and inherently repugnant to any appropriation: they are not property in the sense of law. Orbits and frequencies are res communis (a Latin term derived from Roman law that preceded today’s concepts of the commons and common heritage of mankind; it has relevance in international law and common law). It’s the first-come, first-served principle that applies to orbital positioning, which without any formal acquisition of sovereignty, records a promptness behaviour to which it grants an exclusive grabbing effect of the space concerned.

Geostationary orbit is a limited but permanent resource: this de facto appropriation by the first-comers – the developed countries – of the orbit and the frequencies is protected by Space Law and the International Telecommunications Law. The challenge by developing countries of grabbing these resources is therefore unjustified on the basis of existing law. Denying new entrants geostationary-access or making access more difficult does not constitute appropriation; it simply results from the traditional system of distribution of access rights. The practice of developed States is based on free access and priority given to the first satellites placed in geostationary orbit.

The geostationary orbit is part of outer space and, as such, the customary principle of non-appropriation and the 1967 Space Treaty apply to it. The equatorial countries have claimed sovereignty, then preferential rights over this space. These claims are contrary to the 1967 Treaty and customary law. However, they testify to the concern of the equatorial countries, shared by developing countries, in the face of saturation and seizure of geostationary positions by developed countries.

The regime of res communis of outer space in Space Law (free access and non-appropriation) does not meet the demand of the developing countries that their possibilities of future access to the geostationary orbit and associated radio frequencies are guaranteed. New rules appear necessary and have been envisaged to ensure the access of all States to these positions and frequencies.