wpallen@watale.UUCP (Warren P. Allen @ U of Waterloo X 2522) (11/12/85)
Can anyone explain to me (in 250 words or less) the famed 'slingshot' effect that is used to accelerate space probes? I understand this effect is used not only to change the trajectory of the craft, but also its *speed*. I know it has to do with angular momentum, but *how* is the planet's momentum transfered to the space craft? PS. keep it simple if possible.. thanx
jmc@wuphys.UUCP (Jimmy Chen) (11/14/85)
>Can anyone explain to me (in 250 words or less) the famed >'slingshot' effect that is used to accelerate space probes? > >I understand this effect is used not only to change the >trajectory of the craft, but also its *speed*. > >I know it has to do with angular momentum, but *how* is the >planet's momentum transfered to the space craft? > >PS. keep it simple if possible.. > >thanx > Newton's Law of Gravity says that, in principle, gravity is a universal force and exerts its influence everywhere. In practice, this is not always relevant. For example, to calculate the trajectory of a falling pencil it is not necessary to include the gravitational effects of the sun, the moon, etc. Only the earth's gravity is of any importance. So, we can imagine the sun, planets, moons, and other heavenly bodies of the universe being surrounded by a domain of influence where its gravity dominants all other heavenly bodies. Consider the trajectory of a satellite, say the Voyager. During most of its trajectory it orbits the sun everything else being irrelevant. When it enters Jupiter's domain of influence we can imagine the sun's gravity being turned off and Jupiter's on. Since Jupiter is itself orbiting the sun, Voyager is carried along as a rock is carried in a slingshot which we whirl around our head. Just as the rock picks up the momentum of the slingshot, Voyager picks up the momentum of Jupiter. When Voyager leaves the influence of Jupiter for that of the sun its velocity is increased if it leaves in the direction of Jupiter's motion and decreased if it leaves in the opposite direction. That's roughly it. Jimmy Chen (wuphys!jmc)
bs@faron.UUCP (Robert D. Silverman) (11/14/85)
> Can anyone explain to me (in 250 words or less) the famed > 'slingshot' effect that is used to accelerate space probes? > > I understand this effect is used not only to change the > trajectory of the craft, but also its *speed*. > > I know it has to do with angular momentum, but *how* is the > planet's momentum transfered to the space craft? > > PS. keep it simple if possible.. > > thanx It's fairly simple: An object entering the gravity well of a planet (or other body) has potential energy of position. As it falls into the gravity well it changes that potential energy into kinetic energy. If it burns NO fuel then as it climbs out of the well it will change back the kinetic energy it gained into potential energy. However, if it BURNS fuel at the bottom of the well its mass is reduced. The kinetic energy gained by the fuel as it fell into the well goes directly into the ship itself as the fuel is burned. Thus, it gains speed by falling, burns fuel (thus reducing its mass) and as a result KEEPS the extra kinetic energy as it climbs out of the gravity well. Angular momentum doesn't really play a part. Bob Silverman (they call me Mr. 9)
rimey@ernie.BERKELEY.EDU (Ken &) (11/16/85)
wpallen@watale.UUCP (Warren P. Allen @ U of Waterloo X 2522) writes: >Can anyone explain to me (in 250 words or less) the famed >'slingshot' effect that is used to accelerate space probes? > . . . >PS. keep it simple if possible.. > >thanx Picture a spacecraft traveling in a hyperbolic trajectory about a stationary massive body (a planet). Early, when the spacecraft is still far away, it moves with constant velocity. Later, after the encounter, its velocity has changed direction. Now look at the encounter from a reference frame moving with the spacecraft's initial velocity. What you see is a moving planet approaching a stationary spacecraft and sending the spacecraft flying away. I think that's all there is to the slingshot effect. Its more like using a sling than a slingshot. Ken Rimey ucbvax!rimey rimey@dali.berkeley.EDU
norman@lasspvax.UUCP (Norman Ramsey) (11/18/85)
At least two explanations have appeared for the slingshot effect that do NOT
involve a burn at the bottom of the planet's gravity well. It might at first
appear that this violates conservation of energy. In fact what happens is
that both energy and momentum are transferred from teh planet to the
spacecraft -- the planet *slows down* in response to the flyby.
I would be interested in seeing this worked out in some detail. Might even
do it myself if I have time.
--
Norman Ramsey
ARPA: norman@lasspvax -- or -- norman%lasspvax@cu-arpa.cs.cornell.edu
UUCP: {ihnp4,allegra,...}!cornell!lasspvax!normankort@hounx.UUCP (B.KORT) (11/19/85)
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.) In the interaction, is the planet ever-so-slightly deflected into a perturbed orbit with a different mean radius about the Sun thereby exchanging energy with the spacecraft, which undergoes an appreciable change in both orbital and radial components of its velocity? I'm not about to dive into the equations on this one, but it seems that an exchange of energy between the planet and the spacecraft must occur, keeping the total (system) energy (kinetic plus potential) and momentum (angular and radial) constant. I would be grateful to know if this line of reasoning leads toward a simplified analysis. --Barry Kort
bill@utastro.UUCP (William H. Jefferys) (11/21/85)
> > At least two explanations have appeared for the slingshot effect that do NOT > involve a burn at the bottom of the planet's gravity well. It might at first > appear that this violates conservation of energy. In fact what happens is > that both energy and momentum are transferred from teh planet to the > spacecraft -- the planet *slows down* in response to the flyby. > > I would be interested in seeing this worked out in some detail. Might even > do it myself if I have time. This whole thing has been blown out of porportion. Ethan Vishniac had it right. No burn is required at the bottom of the potential well. The easiest way to think about it is as follows: In the center-of-mass frame of reference of the planet and the probe, the two bodies approach and recede on symmetrically placed hyperbolic orbits. The two bodies approach and recede with the same speed. (The massive planet hardly moves at all in this encounter and to a first approximation we can regard it as fixed, i.e., on a degenerate hyperbolic orbit). But we observe the event in a "laboratory" frame of reference fixed with respect to the Sun. In that frame, the probe enters the encounter with one speed and leaves with another. As pointed out above, however, no violation of energy/momentum conservation is involved, since the orbit of the massive planet is also affected, albeit imperceptibly. The net result is that energy has been transferred from the planet to the probe. This is Freshman Physics stuff. To solve it, transform to the C.O.M. frame, do the calculation, then transform back to the lab frame. If you ignore the mass of the probe and the eccentricity of the planet's orbit, there is a conserved quantity in the problem. In this simplified case (the Restricted Problem of Three Bodies) the Jacobi Constant is conserved. The conservation of this constant is the basis of Tisserand's criterion for determining whether or not two comets that appeared at different times are in fact the same comet at two apparitions, the comet's orbit having been strongly perturbed by Jupiter in the interim. (This is a not-uncommon occurrence). You can read all about this in Szebehely's "Theory of Orbits" (Academic Press 1967), which is the standard work.
bill@utastro.UUCP (William H. Jefferys) (11/21/85)
> > In the center-of-mass frame of reference of the planet and the probe, > the two bodies approach and recede on symmetrically placed hyperbolic > orbits. The two bodies approach and recede with the same speed. > (The massive planet hardly moves at all in this encounter and to > a first approximation we can regard it as fixed, i.e., on a degenerate > hyperbolic orbit). Rereading this, I notice that it is a little ambiguous. I meant to say that the probe leaves the encounter with the same speed (at infinity) that it enters with. The above might be interpreted to mean that the planet and the probe had the same speed, which is obvously wrong. The two hyperbolas are symmetric in shape, but not in size, and in the limit as the mass of the planet goes to infinity, the planet's hyperbolic orbit shrinks to a point.
lambert@boring.UUCP (11/22/85)
> [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