Question

Question #b5921

Answer 1

Binding energy of a satellite is the energy which must be added to planet-satellite system to free the duo from their gravitational attraction.
If a satellite of mass `m` orbits a planet having mass `M` at a radius `R_O` with orbital velocity `v`, the total energy of the system is given by
`E_"Total"=GPE+KE`
`=>E_"Total"=-(GMm)/R_O+1/2mv^2` .....(1)
where `G` is Universal Gravitational constant.

We know that for circular motion
`F_"Centripetal" = (m v^2) / R_O`
and force of gravity is
`F_"grav" = ( G Mm ) / R_O^2`

As the system is in equilibrium, the centripetal force must be balanced by the gravitational attraction force between the two. Therefore we get
`(m v^2) / R_O= ( G Mm ) / R_O^2`
`=>v^2 = ( G M ) / R_O` .....(2)

Inserting this value in (1) we get
`E_"Total"=-(GMm)/R_O+1/2m( ( G M ) / R_O)`
`E_"Total"=-1/2(GMm)/R_O` ....(3)

When the planet-satellite duo are free from each others gravitational pull, (implies that `R_O=oo`), we have

`E_"Total"+BE=0`
`=>BE=-(E_"Total")`

Using (3) we have
`BE=1/2(GMm)/R_O`