Satellites: Orbits and Energy — Question 5

Question 5

A pure orbital-plane change needs an impulse even when speed is unchanged

At a node, a satellite's velocity is redirected through angle $i$ while keeping speed $v_c$ and radius fixed. Both the old and new orbits are circular about the same spherical source.

1. Derive the magnitude of the velocity increment.

2. For $i=60^\circ$, compare $\Delta v$ with $v_c$ and determine the change in orbital energy.

Question 5 - Solution

A pure orbital-plane change needs an impulse even when speed is unchanged

(a) Derivation and calculation

Let the two velocity vectors have equal magnitude $v_c$ and angle $i$. Then $$|\Delta\vec v|^2=|\vec v_f-\vec v_i|^2 =2v_c^2(1-\cos i).$$ Using $1-\cos i=2\sin^2(i/2)$, $$\boxed{\Delta v=2v_c\sin(i/2)}.$$

(b) Result and interpretation

At $i=60^\circ$, $$\boxed{\Delta v=2v_c\sin30^\circ=v_c}.$$ The orbital energy change is nevertheless $$\Delta E=\tfrac12m(v_f^2-v_i^2)+[U(r)-U(r)]=\boxed{0}.$$ An impulse changes momentum direction even at constant speed; zero net orbital-energy change does not imply zero propulsion cost.

Physical check

The result tends to zero for $i\to0$ and to $2v_c$ for a complete velocity reversal.