Do two masses gravitate to their centre of mass

[jja]

Given two masses falling towards each other under gravitational attraction, do they meet at their joint center of mass, assuming other gravitational forces are negligible?

 

[jja]

answering own question

On reflection, to a (massless :-) observer at the center of mass of the two in-falling masses, the center of mass must remain stationary during the in-fall (because there is no force to move it), therefore the two masses must meet at the center of mass. This means the time for each mass to reach the collision point must be equal (which may make it easier to calculate the time it takes for them to arrive at the collision point?) Is this correct?

 

[Vanadium 50]

If you have two point masses, they will meet at their center of mass by construction.

If they are not point masses, they will meet where their surfaces touch.