Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83 (MC840302); site boring.UUCP Path: utzoo!watmath!clyde!bonnie!akgua!gatech!seismo!mcvax!boring!lambert From: lambert@boring.UUCP Newsgroups: net.physics Subject: Re: slingshot effect Message-ID: <6695@boring.UUCP> Date: Fri, 22-Nov-85 12:12:25 EST Article-I.D.: boring.6695 Posted: Fri Nov 22 12:12:25 1985 Date-Received: Sun, 24-Nov-85 07:33:03 EST References: <2358@watale.UUCP> <381@faron.UUCP> <409@hounx.UUCP> Reply-To: lambert@boring.UUCP (Lambert Meertens) Organization: CWI, Amsterdam Lines: 120 Summary: 50 seconds of movie is worth 1000000 words Apparently-To: rnews@mcvax.LOCAL > [A] In the slingshot effect, does it help to resolve the velocity > of the spacecraft into two orthgonal components: a radial component > outward from the Sun, and an orbital component around the Sun? > (The planet has orbital velocity, but zero radial velocity.) [B] In > the interaction, is the planet ever-so-slightly deflected into a > perturbed orbit [...] ? [...] I would be grateful to know > if this line of reasoning leads toward a simplified analysis. > --Barry Kort I think the answer to Q. A is "Yes", and to Q. B "Yes, of course, but this does not simplify the analysis". It depends on what exactly you want to analyze, but for explaining the slingshot effect, it is easiest to assume that the mass of the spacecraft is negligible compared to that of the planet. Another simplifying assumption is that during the brief close encounter of planet and spacecraft, the influence of other heavenly bodies can be ignored. Thus, the planet moves in a straight line. In actual planning, the last simplification is not allowed, but then the guys from NASA will simply integrate the differential equations numerically, rather than try to understand the phenomenon. Since the original question was asked, I have seen the correct and simple explanation come along, followed by some less-than-illuminating "explanations". So here follows an amplification on what I hold to be the best analysis. Imagine a movie of the spacecraft in a trajectory near the planet. If the camera moves so that the planet *seems* at rest, the spacecraft will appear to follow a trajectory like that of a comet, say, near the sun. If the camera is kept still, the velocity of the planet is added vector-wise to all objects in the movie. In the snapshots below the orbital velocities are horizontal and the radial velocities are vertical. The planet is indicated with "O" and the spacecraft with "x". Snapshot 1 shows the approach with a still camera. The spacecraft appears to "undershoot" the planet, but note that the planet is moving in the meantime. In snapshot 2 the velocity vector of the planet has been subtracted from both velocities, putting the planet at rest. The trajectory of the spacecraft is now some conic section, e.g. a hyperbola. After some time, the situation in snapshot 3 is reached. Adding back the subtracted velocity vector, we get snapshot 4. Because the vectors for planet and spacecraft are aligned, the absolute size of the spacecraft's velocity has increased. (So to conserve energy, the planet must have slowed down, but by an imperceptible amount.) Finally, snapshot 5 combines 1 and 4. ============================================================================= <--------O __ \ . . . x 1. Initial velocities, absolute inertial frame. ============================================================================= O ^ . . . x 2. Initial velocities, relative inertial frame. ============================================================================= < . . . x O 3. Final velocities, relative inertial frame. ============================================================================= < . . . x <--------O 4. Final velocities, absolute inertial frame. ============================================================================= < . . . x <--------O __ \ . . . x 5. Combined snapshot, absolute inertial frame. ============================================================================= (This last picture is a hybrid: the positions are shown as if the planet were at rest. But at the time snapshot 4 is taken, the planet has moved to its arrowhead, and the final position of the spacecraft should be shifted by the same amount. The purpose of this overlay is to compare the velocities "before" and "after" in one picture.) Note that it is not necessary or even helpful for an explanation to resort to gravity wells that give a "boost", let alone to the supposed habit of planets to "carry along" innocent bodies (in their ethereal wind?) that fall prey to their sphere of influence. The slingshot effect is much like the Coriolis phenomenon in that it suffices to shift temporarily to a different frame of reference to understand it, which could be illustrated superbly by a movie. -- Lambert Meertens ...!{seismo,okstate,garfield,decvax,philabs}!lambert@mcvax.UUCP CWI (Centre for Mathematics and Computer Science), Amsterdam