Satellites: Orbits and Energy — Question 2

Question 2

Raising a circular orbit adds total energy while reducing kinetic energy

A satellite moves between circular orbits of radii $r$ and $3r$ around the same fixed source. Compare the final and initial states; details of the transfer are not specified. Use $E=-GMm/(2r)$ and $K=GMm/(2r)$.

1. Find the changes in $E$, $K$ and $U$.

2. Explain how positive supplied energy is consistent with a lower final speed.

Question 2 - Solution

Raising a circular orbit adds total energy while reducing kinetic energy

(a) Derivation and calculation

Let $C=GMm/r$. The endpoint energies are $$E_i=-C/2,\quad K_i=C/2,\quad U_i=-C,$$ $$E_f=-C/6,\quad K_f=C/6,\quad U_f=-C/3.$$ Subtracting, $$\boxed{\Delta E=C/3,\qquad\Delta K=-C/3,\qquad\Delta U=2C/3}.$$

(b) Result and interpretation

The speed ratio and energy balance are $$\boxed{v_f/v_i=\sqrt{r/(3r)}=1/\sqrt3},$$ $$\Delta K+\Delta U=-C/3+2C/3=C/3=\Delta E.$$ The potential-energy increase exceeds the supplied total energy because kinetic energy decreases. Endpoint energies alone do not specify a burn sequence.

Physical check

The higher orbit is less tightly bound: its negative total energy has smaller magnitude.