Binding energy of a satellite

Problem

A $500$ kg satellite orbits the Earth at $r = 2R_T$ in a circular orbit. Knowing that for a circular orbit $E_\text{tot} = -GMm/(2r)$, calculate the total energy and the binding energy (the energy needed to take it to infinity).

The gravitational potential energy grows (towards zero) as $r$ increases.

Solution

For a circular orbit the given relation holds (kinetic $+$ potential energy): $E_\text{tot} = -\frac{G M_T\, m}{2r}$ With $r = 2R_T$, using $M_T = 5{,}97\cdot 10^{24}$ kg and $R_T = 6{,}37\cdot 10^6$ m gives $r = 1{,}27\cdot 10^7$ m. Calculating $G M_T\, m = 6{,}67\cdot 10^{-11}\cdot 5{,}97\cdot 10^{24}\cdot 500 = 1{,}99\cdot 10^{17}$. So: $E_\text{tot} = -\frac{1{,}99\cdot 10^{17}}{2\cdot 1{,}27\cdot 10^7} = -\frac{1{,}99\cdot 10^{17}}{2{,}55\cdot 10^7} \approx -7{,}8\cdot 10^9~\text{J} = -7{,}8~\text{GJ}$ The binding energy is the energy that must be supplied to take the satellite to infinity, where $E = 0$. It is the opposite of the total energy: $E_\text{legame} = 0 - E_\text{tot} = +\frac{G M_T\, m}{2r} \approx 7{,}8~\text{GJ}$ $\ev{E_\text{tot} \approx -7{,}8~\text{GJ}, \qquad E_\text{legame} \approx 7{,}8~\text{GJ}}$

Links

Topics: Gravitation Concepts: Universal gravitational potential energy · Kinetic energy · Orbits Skills: Symbolic setup Objects: Satellite