Question

A problem about the motion of planets?

a 1000 kg satelite is in a circular orbit of radius =3Re about the earth. how much energy is required to transfer the satellite to an orbit of radius =6Re?

Answer 1

For a satellite of mass `m` to be circular orbit, centripetal force must be balanced by the force of gravitation between the earth and satellite. We therefore have

`(mv^2)/r=(GM_em)/r^2`
`mv^2=(GM_em)/r` .....(1)

Total energy of the satellite in an orbit is

`TE=PE+KE`

For the first case

`TE_1=-(GM_em)/(3R_e)+1/2(GM_em)/(3R_e)`
`=>TE_1=-1/2(GM_em)/(3R_e)`

Similarly for the second orbit we have

`TE_2=-1/2(GM_em)/(6R_e)`

Difference in total energy in two orbits is the additional energy required to transfer the satellite to the new orbit

`Delta TE=TE_2-TE_1`
`=>Delta TE=-1/2(GM_em)/(6R_e)-(-1/2(GM_em)/(3R_e))`
`=>Delta TE=(GM_em)/(12R_e)`

Now acceleration due to gravity at surface of the earth
`g-=(GM_e)/(R_e^2)and =9.81ms^-2`, also `R_e=6.371xx10^6\ m`. Hence, inserting values in above equation we get

`Delta TE=1/12xx9.81xx6.371xx10^6xx1000`
`=>Delta TE=5.21xx10^9\ J`, rounded to two decimal places.