Satellites: Orbits and Energy — Question 9

Question 9

Hyperbolic arrival requires energy removal before circular capture

A probe approaches a spherical source with speed at infinity $v_\infty$. At periapsis $r_p$ it makes a short retrograde burn into a circular orbit there. Let $v_c=\sqrt{\mu/r_p}$ and take $v_\infty=v_c$. Ignore mass loss in calculating specific orbital energy.

1. Find the arrival speed and required velocity decrement.

2. Find the specific orbital energy removed and compare with the energy needed merely to reach the zero-energy escape threshold.

Question 9 - Solution

Hyperbolic arrival requires energy removal before circular capture

(a) Derivation and calculation

Before the burn, specific energy is $\epsilon_i=v_\infty^2/2$. At periapsis, $$\frac{v_p^2}{2}-\frac\mu{r_p}=\frac{v_\infty^2}{2}.$$ With $v_\infty=v_c$ and $v_c^2=\mu/r_p$, $$\boxed{v_p=\sqrt3v_c},\qquad \boxed{\Delta v=v_p-v_c=(\sqrt3-1)v_c}.$$

(b) Result and interpretation

The initial and final specific energies are $$\epsilon_i=+\frac12v_c^2,\qquad \epsilon_f=-\frac12v_c^2.$$ Circular capture removes $$\boxed{\epsilon_i-\epsilon_f=v_c^2}.$$ Reaching the parabolic threshold removes only $v_c^2/2$; strictly negative energy requires a little more than that. Circularization requires the full $v_c^2$.

Physical check

The burn energy change also equals $\tfrac12(v_c^2-3v_c^2)=-v_c^2$, since the position is unchanged.