Orbital Motion
The Earth and some other planets revolve around the sun in nearly circular paths. The artificial satellites launched by men also adopt nearly circular courses around the Earth. This type of motion is called orbital motion.
Mathematical Derivation
The mass of the satellite is ms and v is its orbital speed. The mass of the Earth is M and r represents the radius of the orbit. A centripetal force is required to hold a satellite in orbit. This force is provided by the gravitational force of attraction between Earth and satellite.
When equating the gravitational force to the required centripetal force, we get
This equation shows that the mass of the satellite is unimportant in describing the satellite’s orbit. Thus, any satellite orbiting at distance r from Earth’s center must have the orbital speed as in the above equation.
The speed less than this will bring the satellite tumbling back to Earth.
Geostationary orbits
An interesting case study is the study of geo-stationary satellite. This type of satellite is the one whose orbital motion is synchronized with the rotation of Earth. In this way, the synchronous satellite remains always over the same point on the equator as the Earth spins on its axis.
Such a satellite is very useful for worldwide communication, weather observations, navigation, GPS and other military uses.
In this case, the speed necessary for the circular orbit is given as
Where T is the period of revolution of satellite, that is equal to one day. This means the satellite must move in one complete orbit in a time of exactly one day.
As the Earth rotates in one day and the satellite will revolve around the Earth in one day.
So, by equating the equations we get
On calculating the values, the value of r = 4.23 x 10 4 km
A satellite at this height will always stay directly above a particular point on the surface of the Earth. the height above the equator comes to be 36000 km.
FAQs Related to Orbital Velocity and Geostationary Orbits:
1. What is orbital motion?
- Orbital motion refers to the circular paths followed by celestial bodies, such as planets orbiting the sun or artificial satellites orbiting the Earth.
2. How is the centripetal force responsible for maintaining a satellite in orbit derived mathematically?
- The centripetal force required to hold a satellite in orbit is derived by equating the gravitational force between the Earth and the satellite to the centripetal force needed for circular motion.
3. Why is the mass of the satellite considered unimportant in describing its orbit?
- The mass of the satellite does not affect its orbit because in the gravitational force equation, the mass of the satellite cancels out, indicating that the orbit only depends on the distance from Earth’s center and the orbital speed.
4. What is a geostationary orbit?
- A geostationary orbit is one in which a satellite’s orbital motion is synchronized with the rotation of the Earth, causing the satellite to remain stationary relative to a fixed point on Earth’s surface.
5. What are the applications of geostationary satellites?
- Geostationary satellites are used for various purposes such as worldwide communication, weather observations, navigation systems like GPS, and military applications due to their ability to remain stationary over a specific point on Earth’s surface.
6. How is the orbital speed for a geostationary orbit calculated?
- The orbital speed for a geostationary orbit is calculated by equating the period of revolution of the satellite (which is one day) with the period of Earth’s rotation, allowing the satellite to complete one orbit in exactly one day.
7. What is the significance of a satellite’s height above the equator being 36000 km for a geostationary orbit?
- A satellite at this height will remain directly above a specific point on the Earth’s equator, making it appear stationary relative to an observer on the ground.
8. How does the altitude of a satellite affect its orbital period?
- The higher the altitude of a satellite, the longer its orbital period, as it takes more time to complete one revolution around the Earth at greater distances from the Earth’s center.
9. Why is a geostationary orbit preferred for communication satellites?
- Geostationary orbits are preferred for communication satellites because they remain fixed relative to a point on Earth’s surface, allowing for continuous communication without the need for satellite tracking.
10. How does the speed of a satellite in a geostationary orbit compare to the rotational speed of the Earth?
- The speed of a satellite in a geostationary orbit matches the rotational speed of the Earth, ensuring that it remains stationary relative to a point on the Earth’s surface.
Summary:
In summary, this tutorial delves into the concepts of orbital velocity and geostationary orbits, both fundamental aspects of celestial mechanics and satellite technology.
- Orbital Motion: The tutorial begins by explaining orbital motion, describing how celestial bodies like the Earth and artificial satellites travel in nearly circular paths around larger bodies like the sun or the Earth itself.
- Mathematical Derivation: It then explores the mathematical derivation of orbital velocity, highlighting the equilibrium between the gravitational force acting on a satellite and the centripetal force required to keep it in orbit. This derivation emphasizes the insignificance of the satellite’s mass in determining its orbit, focusing instead on the distance from the Earth’s center and the orbital speed.
- Geostationary Orbits: The tutorial proceeds to discuss geostationary orbits, a special type of orbit where a satellite’s motion synchronizes with the rotation of the Earth. This synchronization allows the satellite to remain fixed relative to a specific point on the Earth’s equator, making it ideal for applications such as global communication, weather observation, and navigation systems.
The tutorial concludes by presenting the mathematical equation for calculating the orbital speed required for a geostationary orbit, which ensures that the satellite completes one orbit in exactly one day, matching the Earth’s rotational period. A specific height above the equator, approximately 36000 km, is identified for satellites to maintain geostationary orbits.
Overall, this tutorial provides insights into the mechanics behind orbital motion, the mathematical principles governing satellite orbits, and the practical applications of geostationary satellites in modern technology and communication systems.