Binding Energy

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Table Of Content

Definition


The energy required to remove a satellite from its orbit around the earth to infinity is called Binding energy of a satellite.

Introduction


We know that the satellite revolves around the earth in an orbit near the surface of earth. The satellite gets the necessary centripetal force from the gravitational force of earth. Assume a satellite, revolving around the earth of mass ‘Me’ and radius ‘Re’. It is height ‘h’ above the surface of earth. If ‘v’ is the orbital velocity of the satellite and ‘m’ is the mass of satellite then

binding-energy

Where ‘g’ is acceleration due to gravity

binding-energy-1

The numerical value of orbital velocity for a satellite launched near the surface of earth is 7.2 km/s. A satellite revolving around the earth has potential energy as well as kinetic energy. It has potential energy due to its position above the earth’s surface & it remains within the gravitational field of the earth. It has kinetic energy because it is moving around the earth. The total mechanical energy of a satellite is the sum of its potential energy (U) and kinetic energy (K).

Expression for total energy


If a satellite is revolving around the earth in a circular orbit close to the surface of earth, the radius of its orbit can be taken as Re. Let ‘m’ be the mass of satellite and ‘v’ be its orbital velocity then its gravitational potential energy is given by:

binding-energy-2

Where Me is mass of earth and ‘G’ is universal gravitational constant. The Kinetic energy of a satellite is given by

binding-energy-3

As, satellite gets necessary centripetal force from the earth’s gravitational force, therefore

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Putting eq. (3) in eq. (2), we get

binding-energy-5

Therefore total energy of a satellite is

binding-energy-6

Thus, the total energy of a satellite is negative. At infinity (that is, Re = ∞), both potential energy and Kinetic energy  becomes zero. Hence at infinity, the total energy becomes zero. The Kinetic energy can never be negative. Thus a negative total energy means, in order to send a satellite to infinity, we have to give energy to the satellite. Unless a revolving satellite gets extra energy, it would not leave its orbit.

Expression for Binding Energy


The energy required for a satellite to leave its orbit around the earth and escape to infinity is called the ‘binding energy‘ for that satellite. The total energy of a satellite revolving close to earth is

binding-energy-8

Therefore in order to escape, the satellite would require an amount of energy

binding-energy-9

So that the total energy ‘E’ becomes zero. Thus Binding Energy of satellite is

binding-energy-10

References


  1. Pradeep’s Fundamental (XI)
  2. Physics (Part I)