Question 6.14: Orbit 1 has angular momentum h and eccentricity e. The direc......
Orbit 1 has angular momentum h and eccentricity e. The direction of motion is shown. Calculate the Δv required to rotate the orbit 90° about its latus rectum BC without changing h and e. The required direction of motion in orbit 2 is shown.
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
\left.\left.\theta_B\right)_1=-90^{\circ} \quad \theta_B\right)_2=+90^{\circ}The radial coordinate of B is
r_B=\cfrac{h^2}{\mu} \cfrac{1}{1+e \cos ( \pm 90)}=\cfrac{h^2}{\mu}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,
\Delta v_B=\cfrac{\sqrt{2} \mu}{h}