## Total energy of an orbiting Satellite and its Binding Energy

Definition: The total energy of a satellite revolving around the planet is the sum of kinetic energy (i.e. due to orbital motion) and potential energy (i.e. due to the gravitational potential energy of the satellite).

Derivation of the total energy of the satellite revolving around the planet:

Let us consider

The mass of the satellite = $m$

The mass of the planet = $M$

The satellite revolving around a planet at distance from the center of the planet = $r$

The radius of planet =$R$

The potential energy of the satellite is

$U=-\frac{G \: M \: m}{r} \qquad(1)$

The kinetic energy of the satellite is

$K=\frac{1}{2}m v_{e}^{2}$

$K=\frac{1}{2} m \left( \sqrt{\frac{G \: M}{r}} \right)^{2} \qquad \left( \because v_{e}^{2}=\sqrt{\frac{G \: M}{r}} \right)$

$K=\frac{1}{2} \left( \frac{G \: M \: m}{r} \right) \qquad(2)$

The total energy of the satellite is

$E= K+U$

Now substitute the value of the kinetic energy and potential energy in the above equation from equation $(1)$ and equation $(2)$

$E= \frac{1}{2} \left( \frac{G \: M \: m}{r} \right)+ \left( -\frac{G \: M \: m}{r} \right)$

$E= -\frac{1}{2} \left( \frac{G \: M \: m}{r} \right)$

$E= -\frac{1}{2} \left( \frac{G \: M \: m}{R+h} \right) \left( \because r=R+h \right)$

This is the expression for the total energy of the satellite revolving around the planet. The above expression shows that the total of a revolving satellite is negative.

The total energy of the orbiting satellite around the Earth:

Put $M=M_{e}$ and $R=R_{e}$ then

$E= -\frac{1}{2} \left( \frac{G \: M_{e} \: m}{R_{e}+h} \right) \left( \because r=R_{e}+h \right)$

$E= -\frac{1}{2} \left( \frac{g \: R_{e}^{2} \: m}{R_{e}+h} \right) \left( \because GM_{e}=g R_{e}^{2} \right)$

If the satellite revolves around near the earth (i.e. $h=0$) then the total energy of the satellite

$E= -\frac{1}{2} \left( \frac{g \: R_{e}^{2} \: m}{R_{e}+0} \right)$

$E= -\frac{1}{2} \left( \frac{g \: R_{e}^{2} \: m}{R_{e}} \right)$

$E= -\frac{1}{2} \left( g \: R_{e} \: m \right)$

Binding Energy of the Satellite:

The minimum amount of energy required to free the revolving satellite around the planet from its orbit is called the binding energy of the revolving satellite.

We know that the total energy of the revolving satellite is

$E= -\frac{1}{2} \left( \frac{G \: M \: m}{R+h} \right)$

The total energy of the satellite at infinity is zero. When an equal positive amount of energy of the total energy of the satellite is given to the satellite, the total energy of the satellite is become zero and the planet leaves its orbit. So this total positive energy is called the binding energy of the satellite. i.e.

$E= +\frac{1}{2} \left( \frac{G \: M \: m}{R+h} \right)$

The binding energy of the orbiting satellite around the earth

$E= +\frac{1}{2} \left( \frac{G \: M_{e} \: m}{R_{e}+h} \right)$

If a satellite revolves around near earth ($h=0$) then binding energy

$E= +\frac{1}{2} \left( \frac{G \: M_{e} \: m}{R_{e}} \right)$