Question 6.14: Orbit 1 has angular momentum h and eccentricity e. The direc......

By symmetry, the required maneuver may occur at either B or C, and it involves a rigid body rotation of the ellipse, so that $v_r \text { and } v_{\perp}$ remain unaltered. Because of the directions of motion shown, the true anomalies of B on the two orbits are

The radial coordinate of B is

For the velocity components at B, we have

Substituting these into Eqn (6.19), yields

$\boxed{\Delta v=\sqrt{\left(v_{ r _2}-v_{ r _1}\right)^2+v_{\perp_1}^2+v_{\perp_2}^2-2 v_{\perp_1} v_{\perp_2} \cos \delta}}$                                    (6.19)

so that

$\boxed{\Delta v_B=\cfrac{\sqrt{2} \mu}{h} \sqrt{1+2 e^2}}$                           (a)

If the motion on ellipse 2 were opposite to that shown in Figure 6.34, then the radial velocity components at B (and C) would in the same rather than in the opposite direction on both ellipses, so that instead of Eqn (a) we would find a smaller velocity increment,