Question #b5921

1 Answer
Apr 9, 2017

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#