1. Hello I'm new to these forums and I shall post many informative 'articles' for you fellow scientists to read. So without further ado, let us begin! [All credit goes to the author John D. Barrow- '100 essential things you didn't know you didn't know'. I couldn't find a particular website so...']

'Two's company, Three's a crowd'

Two people who get on well together can often find their relationship destabilised by the arrival of a third into their orbit. This is even more noticeable when gravity is the force of attraction involved. Newton taught us that 2 masses can remain in stable orbit around their centre of mass under their mutual gravitational forces- as do the Earth and the Moon. But if a third body of similar mass is introduced into the system, then something quite dramatic happens. One body finds itself kicked out of the system by the gravitational forces, while the two that remain are drawn into a more tightly bound stable orbit.

This simple 'slingshot' process is the source of a counter intuitive property of Newton's theory of gravity. First, take 4 particles of equal mass M and arrange them in two pairs orbiting within two planes that are parallel and with opposite directions of spin so there is no overall rotation. Now introduce a fifth much lighter particle m that oscillates back and forth along the perpendicular through the mass centres of the two pairs. The group of 5 particles will expand to infinite size in a finite time!

How does this happen? The little oscillating particle runs from one pair to another, and at the other it creates a little 3-body problem and gets ejected, and the pair recoils outwards to conserve momentum. The lightest particle then travels across to the other pair and the same scenario is repeated. This happens time and time again without end, and accelerates the two pairs so hard that they become infinitely seperated in a finite time, undergoing an infinite number of oscillations in the process.

This example actually solves an old problem posed by philosophers as to whether it is possible to perform an infinite number of actions in finite time. Clearly, in a Newtonian world where there is no speed limit, it is. Unfortunately (or perhaps fortunately), this behavior is not possible when Einstein's relativity is taken into account. No information can be transmitted faster than the speed of light and gravitational forces cannot become arbitrarily strong in Einstein's theory of motion and gravitation.

Nor can masses get arbitrarily close to each other and recoil. When two masses of mass M get closer than a distance 4GM/c², where G is Newton's gravitation constant and c is the speed of light, then a 'horizon' surface of no-return forms around them and they form a black hole from which they cannot escape.

The slingshot effect of gravity can be demonstrated in your back garden with a simple experiment. It shows how 3 bodies can combine to create big recoil as they try to conserve momentum when they pass close to each other (in the case of astronomical bodies) or collide (as it will be in our experiment).

The three bodies will be the Earth, a large ball (like a basketball or smooth surfaced football) and a small ball (like a ping pong or tennis ball). Hold the small ball at chest height just above the large ball and let them both fall to the ground together. The big ball will hit the ground first and rebound upwards, hitting the small ball while it is still falling. The result is rather dramatic. The small ball bounces up to a height about 9 times higher than it would've gone if it had just been dropped on the ground from the same height.*

*The basket ball rebounds from the ground with speed V and hits the ping pong ball while it is still falling at speed V. So, relative to the basket ball, the ping pong ball rebounds upwards at speed 2V after it's velocity gets reversed by the collision. Since the basket ball is moving at speed V relative to the ground this means the ping pong ball is moving upwards at 2V + V = 3V relative to the ground after the collision. Since the height reached is proportional to v² this means it will rise 3² = 9 times higher than in the absence of its collision, In practice, the loss of energy at the bounces will ensure that it rises to a little less than this.

Thanks for reading, sorry for the length...!

- x(x-y)

2.

3. Originally Posted by x(x-y)

This simple 'slingshot' process is the source of a counter intuitive property of Newton's theory of gravity. First, take 4 particles of equal mass M and arrange them in two pairs orbiting within two planes that are parallel and with opposite directions of spin so there is no overall rotation. Now introduce a fifth much lighter particle m that oscillates back and forth along the perpendicular through the mass centres of the two pairs. The group of 5 particles will expand to infinite size in a finite time!

How does this happen? The little oscillating particle runs from one pair to another, and at the other it creates a little 3-body problem and gets ejected, and the pair recoils outwards to conserve momentum. The lightest particle then travels across to the other pair and the same scenario is repeated. This happens time and time again without end, and accelerates the two pairs so hard that they become infinitely seperated in a finite time, undergoing an infinite number of oscillations in the process.
This is wrong in many ways, foremost is that it would violate conservation of momentum. The ejection and recoil come at the expense of the angular momentum of the orbiting pair. Since that is a finite value, the final recoil velocity will finite and will never lead to an infinite separation in finite time.

Secondly, orbital mechanics would not allow for the smaller mass to oscillate back and forth between the two orbiting pairs like you suggest. For such a back and forth situation to occur, the timing and and passage distance would have to be perfect. The distance between the systems will increase, and thus also the time for the for the small particle to cross it. Meanwhile, the periods of the orbiting systems will decrease. They will go out of sync. It is even possible for a pass to slow down the small particle, transferring momentum back to the orbiting pair. A gravitational slingshot can work both ways.

Even if you where able to maintain timing for an acceleration on each pass, once the small particle reaches a speed that exceeds the escape velocity of the system(which will happen before we use up all the angular momentum of the system), it will enter a hyperbolic trajectory. This means that it will leave the system at an angle that will not return it to the other orbiting pair. It will be lost to both.

4. You obviously didn't read my entire post, I went on to explain that this scenario is infact impossible. I am quite aware of the facts you stated, and I know that this situation doesn't work in a universe of our type...

5. Originally Posted by x(x-y)
You obviously didn't read my entire post, I went on to explain that this scenario is infact impossible.
You went on to discuss Relativistic limits, which are a completely separate matter from what I discussed. I was pointing out that even in a completely Newtonian universe, your scenario would failure to operate as described by you.

I am quite aware of the facts you stated, and I know that this situation doesn't work in a universe of our type...
Really? Then why did you present it as a valid Newtonian scenario? The only type of universe in which it could work would follow neither Newtonian nor Relativistic Physics and instead follow rules so far removed from those governing ours as to make the example useless.

6. Original article: http://plus.maths.org/issue31/outerspace/index.html

Edit: Based on the other thread, this probably also came from the "100 essential things you didn't know you didn't know" book. I wondered why I recognized these. I'm pretty sure I read that book (or at least some of it).

7. Actually under careful analysis, this scenario would work in a Newtonian universe- the reason that the two pairs of masses M reach an infinite distance is because they are constantly being recoiled further and further (due to conservation of momentum) away by the oscillating particle m. This particle m will be ejected from each 3-body system and with each ejection it, it is ejected with more force and thus higher acceleration, allowing it to enter the other pair and etc etc...

Obviously from 1 ejection, the pair won't travel an infinite distance- but over a finite time they will be recoiled so far that they will have travelled an infinite distance...

8. momentum seems conserved in this example

but i dont see how can energy be, this sounds like some kind of perpetuum mobil under newtonian physics

9. but over a finite time they will be recoiled so far that they will have travelled an infinite distance...
Sorry to pee on your battery some more, but how is it possible for anything to move an infinite distance away in any amount of finite time? I don't think it is mathematically possible. Infinity isn't a number.

10. Originally Posted by KALSTER
but over a finite time they will be recoiled so far that they will have travelled an infinite distance...
Sorry to pee on your battery some more, but how is it possible for anything to move an infinite distance away in any amount of finite time? I don't think it is mathematically possible. Infinity isn't a number.
It's not physically possible, but it would be in a Newtonian world...

11. i wonder is this flaw due to the third object passing throught the common center of mass being distance zero and therefore gravity infinite?

12. Originally Posted by x(x-y)

It's not physically possible, but it would be in a Newtonian world...
No it wouldn't. The objections I brought up in a early post were based entirely on applying Newtonian physics.

Again, the ejection of the small body comes at the expense of the momentum of the Orbiting pair. Now, the Angular momentum of an object in orbit is found by:

where r is the radius of the orbit.

v can be found using the formula

Where M is the other mass of the pair, and d is the distance between their centers.

Now with equal masses, M=m and d=2r(since both bodies orbit the barycenter halfway between them.)

So this can be written as:

Substituting this for v in the first equation gives

Since this is for only one of the two orbiting masses, the total momentum of the pair is

or

Which is a finite value for any given m and d.

You cannot get an infinite acceleration from a finite source.

The second objection also derives from Newtonian physics in the form or orbital mechanics.

In order for the orbiting pair to have a net "recoil" it has to eject the smaller object at a speed equal to or greater than escape velocity of the pair. Anything lower and the smaller object goes into an orbit around the pair. You'll get a "wobble", but no net recoil. Also, it is only that net excess over escape velocity that will contribute to the net recoil, the rest is used up climbing out of the pair's gravity well.

But, above escape velocity speed, the trajectory becomes hyperbolic. This means that if the smaller object comes in directly from the other pair, it will leave at an angle that does not return it to that pair. The greater the excess speed, the greater the difference. This means the maximum ejection speed you can ever get from this arrangement is just somewhat above escape velocity. After that, the smaller body it lost to both orbiting pairs.

In other words. Newtonian orbital mechanics will not allow the oscillating back and forth that you describe.

13. Fair enough, if you say so...

I'm only 16 so go easy on me, I'm no expert...!

This article was written by Prof. John D. Barrow originally, (he's a professor of mathematical sciences at Cambridge University)... So I don't know who to trust...

14. Originally Posted by x(x-y)
Fair enough, if you say so...

I'm only 16 so go easy on me, I'm no expert...!

This article was written by Prof. John D. Barrow originally, (he's a professor of mathematical sciences at Cambridge University)... So I don't know who to trust...
First off, note the Prof's area of study, it is mathematics not physics. Professors can be just as mistaken as anyone else when dealing with an area outside of their field of study.

Secondly, and I'm talking as a moderator here: Posting an entire article from another site as you did is a form of copyright infringement. If you wanted to share the article, you should have just posted a link to it.
In addition, posting it without making any acknowledgment as to its true origin could even be seen as plagiarism. At the very least, you should have listed the source and author.

15. All right, chill your beans...!

I've changed it now...

I don't appreciate being made to look like a fool...

16. Originally Posted by x(x-y)

I've changed it now...

I don't appreciate being made to look like a fool...
Look, you have no cause to show indignation here. Janus is performing his duty as moderator, which in this instance means informing you of our posting rules. He has not been rude at any point. Furthermore, if you are posting information that is known to be inaccurate, would you not expect it to be corrected?

We appreciate your enthusiasm, but you have to know when to accept correction. Don't you want to be as accurate as possible? No attempt is being made to stifle you or to make you look like a fool. You are free to post your articles and we are happy to read them. Just stick to the rules when you do it like the rest of us do. You are new here, so it is perfectly excusable, as long as you take heed. I look forward to your contributions.

17. Ok, sorry...

I'm 'on the edge' at the moment as I have results day coming up in just a few weeks. I'll take what you said into account though...

18. The article may be Barrow's, but as stated in it, the system is from someone named Jeff Xia, though it's not properly cited.

19. Originally Posted by x(x-y)
Ok, sorry...

I'm 'on the edge' at the moment as I have results day coming up in just a few weeks. I'll take what you said into account though...
That is great to hear, seriously. It takes a big man to admit that he has been wrong and it is something that is sadly an uncommon trait. Good luck with your results! :wink:

20. Originally Posted by KALSTER
Originally Posted by x(x-y)
Ok, sorry...

I'm 'on the edge' at the moment as I have results day coming up in just a few weeks. I'll take what you said into account though...
That is great to hear, seriously. It takes a big man to admit that he has been wrong and it is something that is sadly an uncommon trait. Good luck with your results! :wink:
Thank you!

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