# Gravitation slingshot

Tyro
Gravitational slingshot

How does this work exactly? Where does my analysis go wrong?

I know it is used to speed satellites up, but its name alludes to a permanent improvement in speed. The way I see it, you would get a slight benefit in journey time as long as the satellite is within the (significant) gravitational field of an object. If we assume it is launched at ~infinity, if the target destination is also ~infinity, that means the end speed ~ start speed.

The gains come from the acceleration as the satellite moves towards the mass, and away as it would be decelerating but still have a velocity greater than it would have had otherwise traversing empty space.

The only other possible source of kinetic energy for the probe I can think of would be through angular momentum transfer from the body to the probe (the body slows down in rotation speed)...but for this to happen you must have an asymmetric mass. Sort of like a planet shaped like an obelisk.

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Homework Helper
What about shooting just 'behind' a massive body in orbital motion. Then you'd get part of the objects orbital momentum, right?

Going past the gravitational slingshot might get you to your destination faster, with the same final speed which is still a net speedup.

Staff Emeritus
Gold Member
Your suspicion is correct. In order for the space craft to obtain a permanent increase in velocity, it must steal some momentum from the planet it slingshots past. In effect, it slows down the planet's orbit a bit...a very very very tiny bit given that the space craft's momentum is much much much less than the planet's (off the top of my head...something like slowing down the planet by 1 meter over 10 billion years...hopefully someone will provide a better number on that).

There are a bunch of good websites describing this. (try google)
here's one...

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Staff Emeritus
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Let's see if I can paint a picture that will help.

Imagine youv'e placed your probe just ahead of Jupiter in its orbit and a bit inside of Jupiter's orbit.

Let's assume that it is at rest with respect to the sun.

Now we are looking at this situation from above, Jupiter is approaching from the right at its orbital speed. From Jupiter's perspective, this is the same as the probe moving past it to the left at the same speed.

If we place the probe properly, Jupiter's gravity will grab it in such a way as to swing it in a parabolic orbit. It will travel leftwards and inside of Jupiter and be whipped around until it is moving outside and of Jupiter and rightwards. The approaching leg and departing leg will be mirror images of each other.

when the probe reaches the same distance in the outbound leg from Jupiter as it had at the initial point when we started, it will have the same relative velocity to Jupiter as it initially had, but now it it moving to the right.

This means that relative to its initial velocity wrt to the sun it is now moving at its relative velocity wrt to Jupiter plus Jupiter's orbital velocity.

IOW, it is now moving at twice Jupiter's orbital velocity.wrt to the sun.

With real probes we don't place them motionless tot he sun, But any probe sent to an outer planet is in an orbit that has it moving slower than the planet when it reaches the planet.

So we time the probe that it arrives just a little "ahead" of the Planet and let the planet catch up to it and whip it around. By timing things correctly, we can choose the final trajectory. (like we did with the Voyager probes on their "Grand Tour".)

Tyro
Thanks guys. I understand now. I did not take into account the planetary orbits (duh!) in trying to see how the gravitational slingshot works. I was just thinking about the planet's rotation about its own axis...hence the eccentric mass suggestion.

How is the optimum spot for maximum benefit from the gravitational slingshot calculated?

Noticed a small typo in the title :/

Staff Emeritus
Gold Member
Originally posted by Tyro
Thanks guys. I understand now. I did not take into account the planetary orbits (duh!) in trying to see how the gravitational slingshot works. I was just thinking about the planet's rotation about its own axis...hence the eccentric mass suggestion.

How is the optimum spot for maximum benefit from the gravitational slingshot calculated?

Noticed a small typo in the title :/

Here's a web site that goes into some of the particulars.

http://www.go.ednet.ns.ca/~larry/orbits/gravasst/gravasst.html

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StephenPrivitera
When the author writes, "While near the planet the path will be a hyperbolic one rather than elliptical and its center of motion will be the planet rather than the Sun," does he really mean "While near the planet the path will be an approximately hyperbolic one rather than elliptical and its center of motion will approximately be the planet rather than the Sun." Does the center of motion magically and instantaneous switch places? Do the orbit change shape instantly?

BTW, is this related to resonances?

Staff Emeritus
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Originally posted by StephenPrivitera
When the author writes, "While near the planet the path will be a hyperbolic one rather than elliptical and its center of motion will be the planet rather than the Sun," does he really mean "While near the planet the path will be an approximately hyperbolic one rather than elliptical and its center of motion will approximately be the planet rather than the Sun." Does the center of motion magically and instantaneous switch places? Do the orbit change shape instantly?

All motion is approximate. There will always be perturbations in the orbits resulting in non-"clean" orbits. So, yes, he really means approximately, but the differences are damn near insignificant unless you get so close to the planet that atmospheric drag takes effect. For satellites orbiting the Earth, stationkeeping delta V's are only on the order of a few to a few tens of meters/second per year. For a single high altitude flyby, it's practically nothing.

StephenPrivitera
I guess I just don't see how the center of orbit can just suddenly change places.

Staff Emeritus
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First off, I'm moving this to Celestrial Mechanics which fits the subject better.

Try not thinking of it as a physical shift in the focus of the orbit, but more of a shift in perspective.

Once the probe enters the sphere of gravitational influence of the planet it is, in effect, an equal distance from the sun as the planet. Thus the gravitational effect on it and planet by the sun are essentially equal.

So at this point, we can safely ignore the sun's influence when it comes to the probe's path with respect to the planet. (It's not that the probe no longer has a heliocentric path, it is just that it is easier to deal with the orbit with respect to the planet at this time. )

Once the probe leaves the gravitational sphere of influence of the planet, it once more is easier to talk about the heliocentric path of the probe and recalculate its new orbit with respect to the sun.

Staff Emeritus
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Originally posted by StephenPrivitera
I guess I just don't see how the center of orbit can just suddenly change places.

Hrm...

It isn't suddenly changing places. The center of any orbital motion is not at the sun, or the Earth, or any other set planet. To find the actual center, you need to add up each and every bit of gravity acting on the craft, which you obviously can't do.

What you can do to simplify the calculations is assume that the center of motion is at the center of the most influential body, and taking the effect of gravity from the other bodies as perturbations.

If you're on a Jupiter approach, the effect from both the Sun and Jupiter are both very small (r^-2, remember), but the Sun's component is the larger of the two. So you assume the craft is orbiting the Sun, and Jupiter's gravity is perturbing the motion. Once you hit the point where the two components are equal, you could validly assume either is the center, but since you are on an approach vector, you switch to Jupiter as the center of your coordinate system with the Sun acting as the perturbing body.

To sum up, the coordinate system you're using is an artificial construct introduced to make the calculations easier (read: possible). You simply can't integrate the universe, so you simplify and make a model of the situation.

Does that make sense?

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