The Orbital Velocity of Satellites: A Comprehensive Guide

The orbital velocity of a satellite is a crucial factor in determining its motion around a planet. It is the velocity at which the satellite must travel to maintain a stable orbit and not fall towards the planet or escape its gravitational pull. Understanding the principles and calculations behind orbital velocity is essential for satellite design, launch, and operation.

Understanding the Orbital Velocity Formula

The orbital velocity of a satellite can be calculated using the formula:

v = √(G*M/R)

Where:

• v is the orbital velocity
• G is the gravitational constant (6.674 × 10^-11 N(m/kg)^2)
• M is the mass of the central body (e.g., Earth)
• R is the radius of the orbit

This formula is derived from the balance between the centripetal force (due to the satellite’s circular motion) and the gravitational force acting on the satellite.

Derivation of the Orbital Velocity Formula

The centripetal force acting on the satellite is given by:

F_c = m*v^2/R

Where m is the mass of the satellite.

The gravitational force acting on the satellite is given by:

F_g = G*M*m/R^2

At the stable orbital velocity, these two forces are equal, so we can set them equal to each other:

m*v^2/R = G*M*m/R^2

Rearranging the terms, we get the orbital velocity formula:

v = √(G*M/R)

This formula shows that the orbital velocity is independent of the satellite’s mass, but it is dependent on the mass of the central body and the radius of the orbit.

Calculating Orbital Velocity for Different Orbits

Let’s explore the orbital velocity calculations for different types of orbits.

Low Earth Orbit (LEO)

For a satellite in a circular orbit 6,770 km above the Earth’s surface (a typical altitude for a low Earth orbit), the orbital velocity can be calculated as follows:

v = √(G*M/R)
v = √((6.674 × 10^-11 N(m/kg)^2) * (5.972 × 10^24 kg) / (6,770,000 m + 6,371,000 m))
v = 7,676 m/s

This means that the satellite must travel at a speed of 7,676 m/s, or approximately 27,602 km/h, to maintain its orbit.

Geostationary Orbit (GEO)

In a geostationary orbit, the satellite’s orbital period is equal to the Earth’s rotation period, allowing the satellite to remain stationary relative to a point on the Earth’s surface. The orbital radius for a geostationary orbit is approximately 42,164 km (the distance from the Earth’s center to the satellite).

Calculating the orbital velocity for a geostationary orbit:

v = √(G*M/R)
v = √((6.674 × 10^-11 N(m/kg)^2) * (5.972 × 10^24 kg) / (42,164,000 m))
v = 3,075 m/s

The satellite in a geostationary orbit must travel at a speed of 3,075 m/s, or approximately 11,070 km/h, to maintain its position.

Highly Elliptical Orbit (HEO)

Highly elliptical orbits, such as Molniya orbits, have a high apogee (the farthest point from the Earth) and a low perigee (the closest point to the Earth). The orbital velocity varies significantly between these two points.

For example, in a Molniya orbit with an apogee of 40,000 km and a perigee of 500 km, the orbital velocities would be:

At apogee:

v = √(G*M/R)
v = √((6.674 × 10^-11 N(m/kg)^2) * (5.972 × 10^24 kg) / (40,000,000 m))
v = 3,875 m/s

At perigee:

v = √(G*M/R)
v = √((6.674 × 10^-11 N(m/kg)^2) * (5.972 × 10^24 kg) / (500,000 m))
v = 10,072 m/s

The significant difference in orbital velocity between the apogee and perigee is a key characteristic of highly elliptical orbits.

Factors Affecting Orbital Velocity

Several factors can influence the orbital velocity of a satellite:

1. Altitude: As the altitude of the orbit increases, the orbital velocity decreases. This is because the gravitational force acting on the satellite decreases with the square of the distance from the Earth’s center.

2. Central Body Mass: The mass of the central body (e.g., Earth) directly affects the orbital velocity. A more massive central body will result in a higher orbital velocity for a given orbit.

3. Eccentricity: For elliptical orbits, the orbital velocity varies throughout the orbit. The velocity is highest at the perigee (closest point to the central body) and lowest at the apogee (farthest point from the central body).

4. Satellite Mass: As mentioned earlier, the mass of the satellite does not affect its orbital velocity. The orbital velocity is determined solely by the mass of the central body and the radius of the orbit.

Practical Applications and Considerations

The understanding of orbital velocity is crucial for various applications in satellite technology and space exploration:

1. Satellite Launching: Accurate calculation of the required orbital velocity is essential for the successful launch and insertion of a satellite into its desired orbit.

2. Satellite Maneuvers: Adjusting the satellite’s velocity is necessary for orbital maintenance, station-keeping, and changing the satellite’s orbit, such as raising or lowering the altitude.

3. Satellite Deorbiting: When a satellite reaches the end of its operational life, its orbital velocity must be reduced to allow it to re-enter the Earth’s atmosphere and safely burn up or land.

4. Interplanetary Missions: The orbital velocity of a spacecraft is a critical factor in achieving the necessary escape velocity to leave the Earth’s gravitational influence and travel to other planets or beyond.

5. Satellite Constellations: Coordinating the orbital velocities of multiple satellites in a constellation is essential for maintaining the desired coverage and communication patterns.

Understanding the principles and calculations of orbital velocity is a fundamental aspect of satellite technology and space engineering. By mastering this knowledge, engineers and scientists can design, launch, and operate satellites more effectively, contributing to the advancement of space exploration and communication technologies.

References

1. Gravitation (5 of 17) Calculating Orbital Velocity of a Satellite, YouTube: https://www.youtube.com/watch?v=V5-xqAjVaAM
2. orbital velocity of satellite unit 5 new.pptx, SlideShare: https://www.slideshare.net/SagarSharma1992/orbital-velocity-of-satellite-unit-5-new
3. Mathematics of Satellite Motion – The Physics Classroom: https://www.physicsclassroom.com/class/circles/Lesson-4/Mathematics-of-Satellite-Motion
4. Orbital Velocity – an overview, ScienceDirect Topics: https://www.sciencedirect.com/topics/engineering/orbital-velocity
5. HSC Physics: Orbital Velocity Explained, Science Ready: https://scienceready.com.au/hsc-physics-orbital-velocity-explained/