is it possible to leave any orbit

[lemur]

How large does the distance have to be between two massive bodies for an object traveling between them to escape one orbit completely before entering the other? Imo, it seems like the only way for any object to follow a path that's not some type of orbit around another object is for it to crossover to orbit a different object.

[Schrödinger's hat]

I don't really understand the question.

 

Umm, 'an orbit' is just a name for the path an object will take if it's trapped in a gravitational well and isn't on a collision course. They aren't distinct, individual things. If you accelerate for long enough you can enter a hyperbolic/parabolic trajectory which will never come back to the planet/star/whatever. Some people call these an orbit, too.

Also, if you aim right you can hit a planet/other object.

[swansont]

To expand on what Schrödinger's hat said, closed orbits require that the potential energy be larger in magnitude than the kinetic energy, i.e. you have a bound system. If KE > |PE| then it's not in a closed orbit.

[rktpro]

If KE > |PE| then it's not in a closed orbit.

Can |PE| be greater than KE?

[swansont]

Can |PE| be greater than KE?

 

Sure. If it is, then the object is bound — there is not enough kinetic energy to get arbitrarily far away. The PE of the earth, for example, is greater in magnitude than its kinetic energy. Using the PE of an object at the surface is how you calculate escape velocity — you see how much KE you need to get far away, and solve for speed. (This all assumes you've set PE=0 to be infinitely far away)

[lemur]

But is there any point where you can define an object as having definitively escaped the gravity of another object? Given enough spacetime, any radius of closed-orbit is possible for an extremely slow-moving object, right?

 

edit: as far as objects whose KE is greater than PE, these objects could gain distance from their gravity-well indefinitely but without ever being able to exceed the speed of light relative to their point of origin they wouldn't be able to escape the gravity either, right?

Edited May 12, 2011 by lemur

[timo]

Escaping a gravity well is defined as: for any finite distance d0 from the center of the well there is a time t0, such that for all times t>t0 the object is a distance d>d0 away from the center of the gravitational well. You are of course right that there is no distance d at which the gravitational potential 1/d^2 is zero, which is why escape is defined in this weird way (it's a mathematical limit), which is essentially "will not be pulled back anymore".