Satellites: Orbits and Energy — Question 1

Question 1

A bound circular orbit has positive kinetic energy and negative total energy

A satellite of mass $m$ moves in a circular orbit of radius $r$ about a fixed spherical source. Write $\mu=GM$ and choose potential zero at infinity.

1. Derive its speed, kinetic energy, potential energy and total energy.

2. Find the additional orbital energy required to escape with zero speed at infinity, and compare it with the kinetic energy already present.

Question 1 - Solution

A bound circular orbit has positive kinetic energy and negative total energy

(a) Derivation and calculation

Circular force balance gives $$\frac{mv_c^2}{r}=\frac{\mu m}{r^2},\qquad \boxed{v_c=\sqrt{\mu/r}}.$$ With zero potential at infinity, $$K=\frac12mv_c^2=\frac{\mu m}{2r},\quad U=-\frac{\mu m}r,$$ $$\boxed{E=K+U=-\frac{\mu m}{2r}},\qquad U=-2K.$$

(b) Result and interpretation

Marginal escape requires $E_f=0$: $$\boxed{\Delta E=E_f-E_i=\frac{\mu m}{2r}=K_i}.$$ At fixed radius, this raises the speed to $$v_{\mathrm{esc}}^2=2\mu/r=2v_c^2.$$ Initial kinetic energy is positive, but the larger negative potential keeps the original circular orbit bound.

Physical check

The final escape energy is zero with zero speed at infinity, consistent with the chosen potential reference.