Center of Mass of separating Orbiting Mass

[LydiaJ]

I was reading about the ISS earlier, and this question came into my mind, and I can't seem to decided how this actually works. Here's the set-up:

Say I have two masses stuck together (m1 and m2) to form a mass M in orbit around some planet (like the ISS, and some large space capsule that is docked with it), and m2 takes off due to an external force (like the capsule flying away without pushing up against the ISS) what happens to the orbit of the mass that didn't take off?

I think that if the two masses were pushed apart from each other due to some internal force (like if some giant spring pushed them apart from one another) then they would both move away from each other, and, I think, the center of mass of the new system would continue on in the same orbit as M would have followed (conservation of momentum). But that's not the situation here, since there is an external force applied to one of the masses, but not the other, the momentum of the system is changed.

M is tracing an orbit that I think follows a path right through its center of mass. When M breaks apart, m1 would now have a center of mass that doesn't line up with the orbit it is following, which doesn't seem right. On the other hand, there was no force applied to m1 , so I can't see how it would move to line up it's center of mass with it's direction of travel, or change orbits to line its direction of travel up with its center of mass.

Orbital mechanics are fun to think about, but boy do they make my head hurt.

Can anyone help me out with this?

 

[andrewkirk]

In your post you seem to change from thinking about the centre of mass of M at the beginning, to thinking about the centre of mass of one of the components, m1, later on.
For now, I'll assume that what you really intended was to just think about how the motion of the CoM of M changes.

The answer depends on what the external force is. The capsule could move apart from the ISS if it first undocked (which doesn't change the state of motion of either body) and then emitted a jet of gas at an angle that missed the ISS but which propelled the capsule away from the ISS (somewhat diagonally).

It might at first seem that the trajectory of the CoM of M has changed, because that of the capsule has changed but that of the ISS hasn't. However that would be incorrect, because we are failing to consider the gas emitted by the capsule's thrusters, which is also part of the system. If we take that into account, we will find that the trajectory of the CoM (of ISS plus capsule plus gas) has not changed, at least in the short term. In the longer term, the divergence of capsule from ISS may call the capsule to fall to Earth, in which case the trajectory of CoM of M will have changed, by the capsule transferring angular momentum to or from the Earth.

If on the other hand the external force is nothing to do with the capsule, eg if the capsule, while undocked, were struck by a meteor, the trajectory of the CoM of M would change, as we would expect it to, since it has received an external impulse.

 

[LydiaJ]

In your post you seem to change from thinking about the centre of mass of M at the beginning, to thinking about the centre of mass of one of the components, m1, later on.
For now, I'll assume that what you really intended was to just think about how the motion of the CoM of M changes.

The answer depends on what the external force is. The capsule could move apart from the ISS if it first undocked (which doesn't change the state of motion of either body) and then emitted a jet of gas at an angle that missed the ISS but which propelled the capsule away from the ISS (somewhat diagonally).

It might at first seem that the trajectory of the CoM of M has changed, because that of the capsule has changed but that of the ISS hasn't. However that would be incorrect, because we are failing to consider the gas emitted by the capsule's thrusters, which is also part of the system. If we take that into account, we will find that the trajectory of the CoM (of ISS plus capsule plus gas) has not changed, at least in the short term. In the longer term, the divergence of capsule from ISS may call the capsule to fall to Earth, in which case the trajectory of CoM of M will have changed, by the capsule transferring angular momentum to or from the Earth.

If on the other hand the external force is nothing to do with the capsule, eg if the capsule, while undocked, were struck by a meteor, the trajectory of the CoM of M would change, as we would expect it to, since it has received an external impulse.

Thanks, Andrewkirk.

I forgot about the gas. That's the part I missed. Now it makes sense. The falling to Earth transferring angular moment was something I hadn't even thought of, but that makes sense too, although it is tough to visualize.