Geostationary Orbit: Calculation and Explanations

Image description: Geostationary satellites are telecommunications, broadcasting, observation, and meteorological satellites. They are located at an altitude of 35,796 km and have a fixed position relative to the Earth's surface. Their altitude allows them to "see" more than a third of the Earth; three satellites are enough to ensure almost total coverage of the Earth's surface.

What is a Geostationary Orbit?

The geostationary orbit is a circular orbit at a specific altitude above the Earth's equator, where a satellite remains fixed relative to a point on the Earth's surface. This orbit allows for a constant position in the sky, which is particularly useful for communication and meteorological satellites.

N.B.: The geostationary orbit is a circular orbit that allows a satellite to complete one orbit around its planet while the planet completes one rotation around itself. Since its inclination relative to the Earth's equatorial plane is 0, the satellite appears "stationary," suspended in the sky always in the same position above the equator. Geostationary satellites are used for continuous observation of a specific area of the Earth.

Fundamental Physical Principles

To understand and calculate a geostationary orbit, it is essential to master certain concepts of orbital mechanics:

• Orbital Period (T): the time required for a satellite to complete one orbit around the Earth.
• Standard Gravitational Parameter (µ): the product of the gravitational constant (G) and the mass of the Earth (M).
• Altitude (h): the distance between the satellite and the Earth's surface.

Calculation of the Altitude of the Geostationary Orbit

The orbital period (T) for a geostationary orbit must match the Earth's rotation period, which is 24 hours or 86,400 seconds. Using Kepler's third law, the formula for the orbit radius (⃒a⃓) is:

\[ T = 2\pi \sqrt{\frac{a^3}{\mu}} \]

• T: orbital period (in seconds)
• a: orbit radius (distance between the Earth's center and the satellite, in meters)
• µ: standard gravitational parameter (µ = GM, approximately 3.986 × 10ⁱ⁴ m³/s² for Earth)

By isolating a, we get:

\[ a = \left( \frac{\mu T^2}{4\pi^2} \right)^{1/3} \]

For Earth, with T = 86,400 s, the calculation gives an orbit radius of 42,164 km. The satellite's altitude is obtained by subtracting the Earth's radius (6,378 km):

\[ h \approx 42,164 - 6,378 = 35,786 \, \text{km} \]

Main Applications

Geostationary orbits are used in many fields:

• Communication Satellites: for international telecommunications.
• Meteorological Satellites: real-time observation of weather conditions.
• Navigation Systems: fixed points for location.
• Security and Defense: secure military communications, surveillance, and strategic intelligence.
• Environmental Monitoring: forest fires, volcanic eruptions, atmospheric or marine pollution.
• Emergency Alert Broadcasting: broadcasting alerts in case of natural disasters, such as tsunamis, earthquakes, or hurricanes.

Differences Between Synchronous Orbit and Geostationary Orbit

All geostationary satellites are synchronous, but not all synchronous satellites are geostationary.

Synchronous Orbit

• A synchronous orbit is one where the satellite completes one revolution around its planet in exactly the same time the planet takes to complete one rotation on its axis.
• This means the satellite returns to the same relative position above a given point on the planet after each revolution.
• A synchronous orbit can be inclined (the satellite oscillates between the northern and southern hemispheres), elliptical (the satellite appears to oscillate from east to west), or non-equatorial (the satellite is not aligned with the equator).

Geostationary Orbit

A geostationary orbit is a special case of synchronous orbit that meets very specific criteria:

• The satellite must be located above the equator (inclination = 0°).
• The orbit must be circular (eccentricity = 0).
• The satellite appears totally fixed in the sky when observed from the planet's surface.

This configuration is only possible for satellites around Earth, at an altitude of ~35,786 km, which corresponds to a semi-major axis of approximately 42,164 km.