Understanding and calculating the synchronous orbit

What is a Synchronous Orbit?

The synchronous orbit is the orbit that allows a satellite to complete one orbit around its planet while the planet completes one rotation around itself. This means that if this orbit has an inclination and eccentricity equal to 0, the satellite will appear, from the planet's surface, to be "stationary," suspended in the sky always in the same position, above the equator.

When the inclination of the satellite's orbital plane is not equatorial (inclination ≠ 0), the satellite appears to oscillate from north to south, above the planet's equator.

When the satellite's orbit is elliptical (eccentricity ≠ 0), the satellite appears to oscillate from East to West.

When both the inclination and eccentricity of the satellite's orbit are different from 0, the satellite moves in the sky producing a figure-eight shape, called an analemma.

For Earth, the most well-known synchronous orbit is the geostationary orbit, where the satellite remains fixed relative to a point on the Earth's equator. This orbit is particularly useful for communication and meteorological satellites.

Principles of Orbital Mechanics

To calculate the synchronous orbit, it is essential to understand some basic principles of orbital mechanics:

• The orbital period (T) is the time required for a satellite to complete one orbit around the planet.
• The rotation period of the planet (T_planet) is the time required for the planet to complete one rotation on its axis.
• For a synchronous orbit, T must be equal to T_planet.

Orbital Period Formula

The orbital period of a satellite can be calculated using Kepler's third law:

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

• \( T \) is the orbital period,
• \( a \) is the semi-major axis of the orbit (average distance between the satellite and the center of the planet),
• \( \mu \) is the standard gravitational parameter of the planet (\( \mu = GM \), where \( G \) is the gravitational constant and \( M \) is the mass of the planet).

Calculating the Synchronous Orbit

For a synchronous orbit, we must equate the orbital period \( T \) to the planet's rotation period \( T_planet \).

\[ T = T_planet \]

Using the orbital period formula, we can solve for \( a \):

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

For Earth, \( T_planet \) is approximately 86,400 seconds (24 hours), and \( \mu \) is approximately \( 3.986 \times 10^{14} \, \text{m}^3/\text{s}^2 \).

Substituting these values, we get:

\[ a \approx 42,164 \, \text{km} \]

This means that for a geostationary orbit around Earth, the satellite must be at an altitude of approximately 35,786 km above the Earth's surface (since the Earth's radius is approximately 6,378 km).

Applications of Synchronous Orbit

Synchronous orbits are widely used for various applications:

• Communication satellites: Satellites in geostationary orbit provide continuous coverage of a large portion of the Earth's surface, which is ideal for telecommunications.
• Meteorological satellites: These satellites provide continuous images of a specific region, enabling real-time weather monitoring.
• Navigation: Synchronous orbit satellites can aid in navigation by providing fixed reference points.

N.B.:
Do not confuse the synchronous orbit of a satellite with the synchronous rotation of a satellite. The synchronous orbit is the orbit that allows a satellite to complete one orbit around its planet while the planet completes one rotation around itself. In synchronous rotation, the satellite takes the same time to rotate around its axis as it does to orbit around its planet. In the case of synchronous orbit, the satellite is always in the same place in the sky, and in the case of synchronous rotation, the satellite always presents the same face to its planet. This is the case with the Moon, which has a rotation period equal to its orbital period (27.3217 days), always presenting the same face to Earth but moving in the sky because it is not in a synchronous orbit.

Table of Synchronous Orbits of Planets.