We are aware of Earth’s gravitational pull as we have seen objects fall towards the surface of the Earth innumerable times. However, Earth’s gravitational pull can work in myriad more ways, allowing satellites to orbit around itself. Just like Earth and other planets of the Solar System orbit the Sun, these satellites too can orbit planets in a circular motion.
Binding energy refers to the energy required for a satellite to remain bound to a planet. Satellites revolve around planets in a circular orbit with the help of centripetal force. This force allows the satellite to stay in a stable orbit around the planet due to its gravitational attraction.
Thus, the binding energy is the minimum amount of energy required to free a satellite from the gravitational pull of a planet. Once the satellite is free from the planet’s gravitational pull, it will be able to leave the orbit to a point at infinity.
Objectives and Functions
The main objective of finding the binding energy of a satellite is to understand the amount of energy it takes to maintain its position in the planet’s orbit. For example, consider a satellite orbiting our planet Earth. The gravitational pull of Earth provides the required centripetal force to the satellite, enabling a circular orbit.
The centripetal force is created with the help of two mechanical energies within the satellite. The circular motion of the satellite is due to the existence of kinetic energy. On the other hand, its position in orbit within Earth’s gravitational field is due to Earth’s potential energy.
Derivation for the binding energy of a satellite in orbit
Let’s consider the mass of Earth to be M, the mass of the satellite as m, the radius of Earth as R, the height of the satellite from Earth as h, the radius of the circular orbit of the satellite as r or (R+h), and the critical velocity of the satellite as v.
The centripetal force would thus be equal to the gravitational force. This is because it is the gravitational force that provides the required centripetal force to allow the circular motion of the satellite.
mv2/r = GMm/r2
mv2 = GMm/r
Here G is the Universal gravitational constant.
Thus, the kinetic energy can be solved as:
K.E. = GMm/2r
Furthermore, the potential energy between the satellite and Earth can be solved as:
P.E = -GMm/r
Here, the negative sign symbolised the force between the satellite and Earth as attractive. However, the kinetic energy of the satellite will be half of the potential energy. Thus, the total energy of a satellite will be the sum of kinetic energy and potential energy.
E = K.E + P.E
= GMm/2r + (-GMm/r)
= GMm/2r – GMm/r
Thus, this expression will help us find the binding energy of a satellite revolving around the Earth in a stable and circular orbit.
Deriving an equation for a stationary satellite
Similarly, we can also arrive at an equation to explain the binding energy of a satellite that is stationary on Earth’s surface.
Here, let’s consider the mass of the stationary satellite to be m. The kinetic energy of the satellite is zero.
Thus, the potential energy can be written as:
P.E = -GMm/r
The total energy will be the sum of kinetic energy and potential energy. Hence, the total energy will be:
E = K.E + P.E
E = 0 + (-GMm/r)
E = 0 – GMm/r
E = – GMm/r
Here, the negative sign refers to the attractive nature of the force between Earth and the satellite.
To explain the binding energy of a satellite, it is the energy that enables the satellite to stay in a stable and circular orbit around a planet. We know that the total energy of satellite importance is a sum of its kinetic and potential energy.
Thus, if we know the value of the satellite and planet’s masses, the radius of the planet and the height at which the satellite is located from the planet’s surface, the velocity at which the satellite is orbiting, and the value of the gravitational constant- we shall be able to know the binding energy of the satellite.