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Discussion below this answer involves my feeling that calling halo orbits in very asymmetric systems (e.g. Sun-Earth or Earth-Moon) as orbits around the primary, in 1:1 resonance with the secondary.
Thinking about this has led me to ask something specific:
Question: Is the Gateway's halo orbit bound to Earth? Would the Gateway remain in Earth orbit if the Moon disappeared but drift away from the Moon if the Earth disappeared?
For more on the Gateway's orbit, see
No. The gateway is not bound to Earth, if we interpret "bound to Earth if the Moon disappears" as not drifting into interplanetary space regardless of what instance the Moon's gravitational influence is disregarded.
There's a neat property of zero-gain flybys: At the point of closest approach, the velocity vector of the spacecraft is parallel to the velocity vector of the secondary body.
A near-rectilinear halo orbit depends on performing zero-gain flybys regularly. From their flyby schedule, we can also see that the gateway must be orbiting around the Earth in the same direction as the Moon, which means that the velocity vectors are not merely parallel at closest approach, but also in the same direction.
They also state the close approaches can happen with an altitude of as little as 3000km, which means that the gateway is travelling faster than the lunar escape velocity at that altitude, 1439m/s.
Transforming the vector into geocentric coordinates is now trivial, as everything is travelling in the same direction. Addition!
With the velocity of the Moon added, the gateway is travelling ~2.5km/s relative to the Earth.
That's much higher than the escape velocity of Earth at a lunar distance, so if the Moon suddenly disappeared at that point in time, the gateway would fly off into interplanetary space.
Gateway is not bound to Earth and if the Earth disappeared then Gateway will still orbit the Moon. If the Moon disappeared then there is a chance that Gateway will enter interplanetary space. It really matters on the time since Gateway isn't on an even orbit around the Moon, so if it is at the apogee of its trajectory then it wouldn't enter interplanetary space. If it is at the lowest point of the flight, where it is also going the fastest, then it could get to the interplanetary space since Earth's gravity would be too weak to keep it in orbit.
