Gravitation between point masses

[Positronized]

I was just reading through stuff about Newtonian gravitation and this question popped up in my head and I can't answer myself.

Consider two uncharged point masses with mass M and m respectively, independently suspended far away from each other in a space that would otherwise consist of nothing. Because gravitation acts with infinite range, both point masses would experience an infinitesimal attraction force and therefore accelerate towards each other. As this happen the distance between the point masses will decrease and the force will also increase due to the Newton's Law of Gravitation.

In this idealized Newtonian "point mass" concept, with no regard to atomic theory or anything else, no matter how far apart the two masses are initially placed in the space, would they eventually become, in a sense, 'fused' into one mass, because as distance approaches zero, the attraction force between them will increase without bound and will be impossible to separate them because it will require infinitely large force to do so? And, as a result, consider a space composed entirely out of numerous Newtonian "masses", would all matter in such space eventually collapse together no matter how the masses were initially arranged around the space?

Thanks!

 

[Janus]

Yes, as long as you start with a static situation this is what would happen.

 

[tony873004]

When you suspended them, you'd have to make sure that their velocities releative to each other are 0 or they won't collide unless their initial velocity vectors point straight towards each other. If they start out with some relative velocity, not directly towards each other, and their velocities are under escape velocity, sqrt(2GM/r^2), then they will orbit each other.

 

[Positronized]

Ahh, thanks a lot guys! Yes I was indeed picturing the "initially stationary" situation but I also did forget about the initial relative velocities and orbitting situation. A lot clearer now!

 

[Meir Achuz]

If the only force were gravity, then the motion would not stop but the particles would continue past each other to infinity. Another, non gravitational completely inelastic force would be required to stop the particles. If the other force were elastic, the particles would bounce back to infinity.

 

[Positronized]

Hmm... why so? Consider the masses are initially static "matter" (especially in the Newtonian framework) cannot travel "through" one another.

Or will it concern the force due to the impulse upon collision? Since the velocity when the two particles collide is very high the impulse force which will repel them apart will also be very high. But also the gravitational force which hold them together would also be approaching infinity. So the question is as distance approaches zero, which one would have the dominant effect? Sounds like a calculus limit question ...

 

[rbj]

Hmm... why so? Consider the masses are initially static "matter" (especially in the Newtonian framework) cannot travel "through" one another.

but, if they're point masses, they will always miss each other. this is a consequence of continuous random variables. if, instead of dice (which give you a discrete random value), you have a little spinner instead, then ask yourself: what is the probability of the spinner landing on exactly some given angle?

 

[Positronized]

Hmm.. I might have missed something.. after all I'm just another high school physics student still..

Anyway, I thought Newtonian mechanics is supposed to be entirely deterministic and hence under this idealism such continuous random variables wouldn't exist. Random frictional effects wouldn't be existing either because the space consists of nothing otherwise, so the only force here is gravity. Under this determinism the two point masses should travel at a dead straight path towards each other.

Of course I'm not saying this "actually" happens in the real world because I know it doesn't.

 

[Jack111]

Even in the static case it's not that simple. In our astrophysics course last year we were told that in fact it's very hard to find stable configurations in the attractive N-body case. The only link I can find off hand is http://www.astro.uni-bonn.de/~pavel/movie.html which shows a simulation - most of the bodies end up being shot off to infinity in various directions, while a few form bound pairs. There are more simulations about (search "N-body problem" on google)

 

[pervect]

Newtonian gravity with point masses combined with Newtonian mechanics has some bizarre characteristics. I gather, for instance, that it's possible to travel infinite distances in finite times.

See http://www.mathpages.com/home/kmath395.htm

Google scholar finds the papers (which I haven't read):

A Possible Model for a Singularity without Collisions in the Five Body Problem
JL Gerver - Journal of Differential Equations, 1984

Solutions of the collinear four body problem which become unbounded in finite time JN Mather, R McGehee - Dynamical Systems, 1975 - adsabs.harvard.edu