Energy of a satellite

Describe the energy of a satellite?

1. A satellite revolving around the earth has potential energy as well as kinetic energy. It has potential energy because of attractive force acting on it due to earth, and kinetic energy because of its motion.

2. We know that gravitational potential energy of a body of mass m placed on the surface of the earth is –[GM_em/R_e²]. Hence, if a satellite is revolving around the earth in a circular orbit close to the surface of the earth, then the radius of its orbit can be taken as R_e. Therefore, if m be the mass of the satellite then its gravitational potential energy will be given by:
U = -[GM_em/R_e²] .............(1)

Energy of a satellite

3.If the orbital speed of the satellite be v₀, then its kinetic energy is

K = (1/2)mv₀² .............(2)

As the satellite gets the necessary centripetal force the gravitational force
mv₀²/R_e=GM_em/R_e² or mv₀² = GM_em/R_e

From eq.(2), K=1/2.GM_em/R_e …….(3)
Thus, total energy of the satellite is
E=U+K= - (GM_em/R_e)+(GM_em/2R_e)
Or E = - GM_em/2R_e

Note:a) The total energy of the satellite is negative: At infinity the potential energy and kinetic energy are both equal to zero i.e., at infinity, the total energy is zero. The kinetic energy can never be negative. Thus a negative total energy implies that in order to send the satellite to infinity (to make the total energy zero), we have to provide an extra energy to satellite i.e., if satellite is not given extra energy, it will go on revolving in close orbit. This implies that the satellite is bound to earth.

b) The energy required for a satellite to leave its orbit around the earth and escape to infinity is called the binding energy. Because the total energy of a satellite revolving close the earth is - GM_em/2R_e hence, in order to escape, the satellite would require an amount of energy GM_em/2R_e so that the total energy becomes zero. Thus binding energy of the = + GM_em/2R_e