# Gravitational force of attraction

If I keep two objects where no other force than gravity acts between them, will they eventually stick together?

• Jun 22, 2021 at 19:48

Yes, so long as they are not moving relative to each other initially. If they are moving tangentially to the line connecting them, then they are likely to orbit each other (or fly off in whatever directions they're moving it, with only a slight deflection.)

You can even estimate the amount of time it will take for them to collide. If their initial separation is $$r$$ (and it's much larger than the sizes of the objects themselves, and their total mass is $$M$$, then the amount of time to collision works out to be $$T = \sqrt{ \frac{\pi^2 r^3}{8GM}}.$$ (It's actually remarkably easy to prove this by taking the appropriate limit of Kepler's Third Law.) This works out to approximately 100 days for two 1-kg masses separated by 10 meters.

In the ideal case, not necessarily, but in practice, yes. Ultimately, for two objects to stick together, they must undergo an inelastic collision which dissipates the objects' relative kinetic energy. If you have two perfectly elastic objects, this isn't possible, as the objects will accelerate toward one another, collide, and then rebound to their original positions (assuming no initial velocity). In practice, there is no such thing as a perfectly elastic collision, so some energy will be dissipated even with highly elastic objects that rebound very well. Eventually, all of the useful potential/kinetic energy will be dissipated, and the objects will come to rest relative to one another, sticking together after a final inelastic collision. In theory, though, you could have two perfectly elastic balls that fall toward one another due to gravity, and then rebound to their original positions ad infinitum.

In the real world, two objects moving at less than the escape velocity will eventually coalesce. Orbital decay occurs as orbital energy is released as gravitational radiation. Obviously, two objects moving directly towards each other at speeds greater than the escape velocity will also come together.

Under Newtonian physics, two rigid objects with orbital angular momentum will not coalesce. Nor will two objects moving directly away from each other at a speed greater than the escape velocity.

• >Under Newtonian physics, two objects with orbital angular momentum will not coalesce… provided there are zero tidal forces, which in turn assumes that they are both perfectly rigid Jun 22, 2021 at 23:25
• Good point. Tidally locked nonrigid bodies in circular orbits wouldn't experience orbital decay either.
– Nick
Jun 23, 2021 at 0:31