Total Energy of a Satellite

Satellites, in simple words, are artificial objects which are launched into space using rockets. These satellites are placed in the orbit of the Earth, the moon, or any other planet to gather information. The world’s first rocket to launch was Sputnik 1 by the Soviet Union. Presently, there are over three thousand active satellites in space. 

Once the satellite is successfully fixed in the orbit of the Earth, it begins to revolve under the gravitational attraction of the Earth. Due to its revolving motion, it gains kinetic energy. In addition, because it is revolving in the Earth’s gravitational field, it also gains in potential energy. We will now discuss these two energies contained by the satellite broadly. In this study note on the total energy, we will be assuming the reference planet to be Earth.

Potential and Kinetic energy of a Satellite

According to the work-energy theorem, the final total mechanical energy is equal to the sum of the initial total mechanical energy of the system and the work done by any external force. So this can be explained by the equation – 

KEi + PEi + Wext = KEf + PEf

Here, the expression Wext  is zero because the gravitational force is the only source of external force in the case of satellites. The force of gravity is considered to be a conservative force.

Therefore, this simplifies the equation further, and the total energy of a satellite can be described as the sum of its potential and kinetic energy. The equation can be written as –

KEi + PEi  = KEf + PEf

The tangential velocity of a satellite revolving around the Earth is expressed as 

V = GM/(R+h)

Here, 

M = mass of the Earth,

R = radius of the Earth, and

h = height of the object from the surface of the Earth.

So for a satellite of mass ‘m’ moving at speed ‘v’, the kinetic energy can be shown by the equation – 

KE = 12 mv2 = GmM2( R+h)

Now, since the potential gravitational energy at infinity is considered to be 0, the potential energy can be written as 

PE = – GmMR+h

Now, earlier in this study material, it has been mentioned that the total energy of a satellite is the sum of its potential energy and kinetic energy. Therefore, the total energy of a satellite (E) can be expressed as – 

E = kinetic energy + potential energy = GmM2( R+h) + –GmMR+h

We can see that the value of the total energy of the satellite is negative. This indicates that the satellite will not escape the gravitational pull of the Earth.

Type of Orbits and its Effects on the Total Energy of a Satellite

There are two orbits in the discussion here – circular orbit and elliptical orbit. 

Effect of a circular orbit

Since the orbit is circular, there is no alteration in the satellite’s distance from the Earth in its radial movement. In such cases, both the potential and the kinetic energy remain constant. The velocity remains constant as it is the function of the radius of the orbit, and thus it makes the kinetic energy constant. On the other hand, the potential energy is made constant as it depends upon the object’s height, which is constant because of the circular orbit.

Effect of the elliptical orbit

Variation occurs due to the elliptical shape of the orbit. The square root of the radius is indirectly related to the tangential velocity of a body orbiting around the Earth. This means the kinetic energy decreases with the increase in the radius and increases with the decrease in the radius. Similarly, the height of the object increases with the increase in height, as a result of which the potential energy increases and vice versa.

Conclusion

Satellites are important to us. They are a source of valuable information that keep us abreast with the weather change forecast. They also help in long-distance communication and for radio and television broadcasts. The total energy of the satellite is negative, indicating that it will not be able to escape the gravitational pull of the planet. The total energy of a satellite is an important concept to be learned.