Total energy of an orbiting Satellite and its Binding Energy

Definition:

The total mechanical energy associated with an orbiting satellite around the planet is the sum of kinetic energy (i.e., due to orbital motion) and potential energy (i.e., the gravitational potential energy of the satellite).

Derivation of the total mechanical energy of the orbiting satellite around the planet:

Let us consider

The mass of the satellite = $m$

The mass of the planet = $M$

The distance between satellite to a planet 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

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

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

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

The total energy (i.e. mechanical energy) of the satellite is

$E= K+U$

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

$E= \frac{1}{2} \left[ \frac{G M m}{r} \right]+ \left[ -\frac{G M m}{r} \right]$

$E= -\frac{1}{2} \left[ \frac{G M m}{r} \right]$

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

The above equation shows that the total mechanical energy associated with orbiting satellite is negative.

The total mechanical energy associated with 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 mechanical 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 mechanical 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 revolving satellite's total mechanical energy in an orbit of the planet is

$E= -\frac{1}{2} \left[ \frac{G M m}{R+h} \right]$

At infinite distance between a satellite to a planet, the total mechanical energy of the satellite is zero. Therefore, if an orbiting satellite is provided postive energy that is equal to the total mechanical energy, its total mechanical energy becomes zero, and the satellite escapes from the planet's orbit. This total positive mechanical energy is called the gravitational binding energy of the satellite. i.e.

$E= +\frac{1}{2} \left[ \frac{G M m}{R+h} \right]$

The gravitational binding energy associated with 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 the binding energy

$E= +\frac{1}{2} \left[ \frac{G M_{e} m}{R_{e}} \right]$