Max Gravity Boost: Hyperbolic Trajectory from Jupiter

In summary, the maximum speed an object can be ejected onto a hyperbolic trajectory relative to the Sun by Jupiter is approximately twice Jupiter's orbital velocity plus the object's initial velocity. However, the planet's mass and distance from the Sun, as well as the object's mass, can also affect the final velocity. It is also possible for a close passage to Jupiter to capture an exo-solar object in our solar system, but this would depend on the object's initial trajectory and Jupiter's orbital velocity. There is ongoing discussion about the potential of using a Jupiter boost for interstellar space travel, but a formula is needed to determine the feasibility of this method.
  • #1
tony873004
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What is the fastest speed relative to the Sun that Jupiter can eject an object onto a hyperbolic trajectory?

I imagine you'd want the object to be traveling as fast as possible when it encountered Jupiter. This would give it a velocity at just under Solar escape velocity as it approached Jupiter.

More important to the actual answer, I'd love to know a formula that took planet mass and planet-sun distance as inputs and produced a max ejection velocity.
 
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  • #2
Unless you're satisfied with an answer expressed simply as a force, you would also have to factor in the mass of the vehicle in order to be able to calculate its final velocity.


Also, what does planet-sun distance have to do with it? Or are you looking for a very general formula?
 
  • #3
Thanks, Dave. Yes, a general formula is what I want. I could probably figure this out through simulations on a planet-by-planet basis, but I'm sure it is analytically computable with a single formula based on angular momentum. The formula would also need to contain a planet's radius as it represents minimum passage distance.

Escape velocity from the Sun is directly related to distance from the Sun. So the planet must accelerate the object to the escape velocity for the planet's Sun distance.

If Sun and Planet mass >> object mass, the mass of the object would have little to do with the maximum solar velocity Jupiter, or any planet, could accelerate it to.

Interestingly, what ever this speed is, is also the maximum velocity an exo-solar object passing through our solar system could have and still be captured by a close passage to Jupiter, or other planet.
 
  • #4
tony873004 said:
If Sun and Planet mass >> object mass, the mass of the object would have little to do with the maximum solar velocity Jupiter, or any planet, could accelerate it to.

No. It has nothing to do with the planet mass : object mass ratio and everything to do with F=ma. If m[object] is large, then a[object] will be small.

tony873004 said:
Interestingly, what ever this speed is, is also the maximum velocity an exo-solar object passing through our solar system could have and still be captured by a close passage to Jupiter, or other planet.
Mm. I see where you're going, the old time-reversability thing, but it's not the same.

Remember, the object escaping started off with a rocket boost. So if you reverse the orbital mechanics for an incoming object, yes it would be captured by the Sun, but the numbers could be significantly different in terms of how fast it could be going and still remain in the system.
 
  • #5
No. It has nothing to do with the planet mass : object mass ratio and everything to do with F=ma. If m[object] is large, then a[object] will be small.
Only if force is constant in that equation will you have in inverse relationship between mobject and aobject But force is not constant here, just like force is not constant in the example of dropping two rocks from the roof of a building. The more massive rock lands with more force, but not more velocity since acceleration is constant.

Similarly, acceleration is constant in the ejection problem, and force is not. So two spacecraft of differing masses on the same trajectory will get the same ejection boost from Jupiter assuming mcraft<<mJupiter.

Mm. I see where you're going, the old time-reversability thing, but it's not the same.

Remember, the object escaping started off with a rocket boost. So if you reverse the orbital mechanics for an incoming object, yes it would be captured by the Sun, but the numbers could be significantly different in terms of how fast it could be going and still remain in the system.
The capture trajectory would be the same as the escape trajectory up until the point of the rocket boost. But that's not what I'm after. I just mentioned it as a trivial fact. There's a discussion in my class about interstellar space travel and accelerating to 10% the speed of light. Some are suggesting a Jupiter boost. Intuitively, I know Jupiter doesn't have the potential to add this kind of acceleration to a spacecraft . But a formula in hand would be more convincing than just my intuition.
 
  • #6
tony873004 said:
What is the fastest speed relative to the Sun that Jupiter can [slingshot] an object[?]

In principle, isn't it just twice jupiter's orbital velocity (plus the object's initial velocity)?

In jupiter's frame it's just like any other (hyperbolic) orbit, so the object will depart Jupiter at whatever speed it approached jupiter. In the solar system frame Jupiter isn't really perturbed and the best the object can do is if it both approaches from and departs in the direction Jupiter is moving.

As for physical constraints it seems that the radius (and mass) of the planet would limit the deflection you can get for a given initial velocity and that complete solar system dynamics would determine possible initial trajectories, launch windows, etc.. but there's plenty of room to move when you have [itex]\sqrt 2[/itex] times the escape velocity to start with.
 

Related to Max Gravity Boost: Hyperbolic Trajectory from Jupiter

What is Max Gravity Boost: Hyperbolic Trajectory from Jupiter?

Max Gravity Boost: Hyperbolic Trajectory from Jupiter is a scientific concept that involves using Jupiter's strong gravitational pull to increase the speed and trajectory of a spacecraft.

How does the Max Gravity Boost work?

The Max Gravity Boost works by using the gravitational pull of Jupiter to accelerate a spacecraft. As the spacecraft approaches Jupiter, it is pulled towards the planet and gains speed. The spacecraft then uses this increased speed to continue on its journey towards its destination.

What types of spacecraft can benefit from Max Gravity Boost?

Max Gravity Boost can benefit a wide range of spacecraft, including probes, satellites, and spacecraft traveling to other planets or deep space. It is especially useful for spacecraft that require a significant amount of velocity to reach their destination.

How is the Max Gravity Boost trajectory calculated?

The Max Gravity Boost trajectory is calculated by taking into account the spacecraft's initial velocity, Jupiter's gravitational pull, and the desired trajectory. Advanced mathematical equations and computer simulations are used to accurately determine the trajectory.

What are the advantages of using Max Gravity Boost?

The main advantage of using Max Gravity Boost is that it allows spacecraft to reach their destinations faster and with less fuel. This can significantly reduce the cost and time of space missions. Additionally, Max Gravity Boost can also be used to study Jupiter and its surroundings in more detail.

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