OK the boss just left so I can slow down a sec, lol<br /><br />These things are calculated by the 'patched conic method'; all trajectories (elliptical, parabolic, hyperbolic) are conic sections. A swing-by maneuver involves leaving the heliocentric coordinates, entering the planetary coordinates and then returning to heliocentric. This is done because the gravity of the planet becomes more important than the gravity of the sun when you get close enough to the planet. While in reality this is a gradual process, math lets us make an abrupt transition from one to the other at a certain distance from the planet known as the 'sphere of influence' which is a function of the masses of the sun and planet.<br /><br />A probe is cruising along in its orbit and you can calculate its velocity vector using the vis-viva equation for magnitude and the tangent to its elliptical orbit for direction. <br /><br />And then suddenly - wham! - the probe is close to a planet and it crosses into its sphere of influence. At that instant, we mathematically transpose the heliocentric velocity vector into a planet-centered velocity vector. It's the exact same velocity, just in different coordinates. We are patching the heliocentric trajectory to the planet-centered trajectory.<br /><br />We now are looking at a hyperbolic trajectory *relative to the planet* - we're going too fast to actually orbit the planet, BUT we are close enough that the planet influences our flight path. We're in a 'hyperbolic orbit'.<br /><br />Conservation of angular momentum *of the planet-probe system* dictates that after we make our closest pass to the planet, our flight path *relative to the planet* is a mirror image of our approach. Thus, when we get back to the sphere of influence distance on our way out, we will have the same velocity magnitude *relative to the planet*. BUT we are going a different direction, because our orbit lasted a finite length of time; the planet has redirected our velocity vector.<br /><br />So here w <div class="Discussion_UserSignature"> </div>