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Assuming normal spacecraft and space objects (no neutron stars, black holes, etc). To what speed can a spacecraft accelerate using gravity-assist?

For example, if a spacecraft is moving at relativistic speeds, it probably won't get seriously sped up by normal-density objects.

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In theory, for an arbitrary collection of gravitating bodies, the answer can be infinity: Z. Xia, “The Existence of Noncollision Singularities in Newtonian Systems,” Annals Math. 135, 411-468, 1992. Described at plus.maths.org/content/outer-space-twos-company-threes-crowd . In practice, for our solar system, I think the answer is roughly escape velocity from the cloud-tops of Jupiter, because once you're moving that fast, you can't scatter at 180 degrees from any body in the solar system. –  Ben Crowell Apr 23 '13 at 5:04
    
Are you not limited by the angular momentum of the planet? I believed angular momentum is taken from the planet, and this is finite? –  Nic Apr 23 '13 at 10:25
    
@Nic: true for a single assist, but multiple assists from different planets can be used. –  John Rennie Apr 23 '13 at 10:27

2 Answers 2

The one thing to keep in mind is that in order to perform a gravity-assist maneuver, you need to be able to enter a hyperbolic orbit around a given body that is moving relative to your destination. And, in order to be in such an orbit, there is a specific range of velocities for every object that you must have (dependent on mass of the object). So the fastest you can get to by gravity-assist is much less than relativistic speeds because at relativistic speeds, you would not be able to enter into a proper hyperbolic orbit.

It is true that at any high speed, a flyby constitutes a hyperbolic orbit; however, to use a gravitational slingshot, you need to enter against the object's motion and exit with the motion. At relativistic speeds and for most regular bodies, your orbit would closely resemble a straight line, there could be no gain of velocity.

A good gravity assist works if you can ensure that your hyperbolic trajectory minimizes the angle $\theta$ between entry and exit. It is given by:

$$\theta=cos^{-1}(1/e)$$

Where $e$ is the eccentricity of the orbit and must satisfy $e\geq1$. Additionally, as your velocity increases, it will force $e$ to become larger unless you significantly increase the mass of each subsequent object.

The fastest a spacecraft can get to using gravity-assists very much depends on the largest mass of the objects you use. However, I cannot give you an estimate of a number because due to the sheer impracticality of using gravity-assists to achieve extreme velocities, we (rocket scientists) haven't ever tried computing a theoretical limit. I can guarantee you though that without using high density objects (neutron stars, black holes, etc.), no spacecraft will reach velocities near the speed of light by gravity slingshots alone.

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Good point +1 :-). Presumably once the relative velocity of the spaceship and planet has exceeded a few multiples of the planet's escape velocity you can no longer get any significant slingshot. –  John Rennie Apr 23 '13 at 15:37

This limitation would be imposed by the relative speed difference between the starting point and the object being used for a gravity assist.

to steal Wikipedia's example: Imagine you have a train coming at you at 50mph. You throw a ball at the front of the train at 30mph.

From your perspective, the ball was moving 30mph on the way in and 130mph on the way out (having bounced off the train). From the perspective of the train, the ball approached at 80mph and left at 80mph.

So the maximum speed, as viewed from the thrower, is a combination of the initial speed and the relative speed of the object in question.

With infinite object in perfect alignment: there is no limit other than relativity. Every exchange takes intertia from the planet and gives it to the spacecraft. The limiting factor is how much can be puller per interaction, and how many useful interactions can occur.

This is why the voyager probes were launced when they were: Jupiter, Saturn, and Uranus were aligned in such a way as to make a "grand tour" of using all three for gravity assists possible.

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