[alex_duf]

Isn't it a matter of speed rather than mass? If the xenon is ejected faster than the escape velocity, it seems like it would get off Earth's gravity.

In fact I think it would have to be roughly twice the escape velocity since the spacecraft is already going near it in one direction. According to Wikipedia[1] the exhaust velocity of an ion thruster is between 20 to 50 km/s when the Earth escape velocity is 11km/s [2]

[1]: https://en.wikipedia.org/wiki/Ion_thruster [2]: https://en.wikipedia.org/wiki/Escape_velocity

so I would assume most of it is lost in space

[jtbayly]

Velocity alone doesn't answer the question. Direction matters.

My assumption (knowing nothing but basic Physics), is that the xenon is ejected in a direction slightly toward the earth, and mostly directly in the opposite direction of the current travel, because that's what would be necessary to counteract drag and keep a satellite on the same path.

This means that if the satellite is going almost 11km/s one direction, the xenon will have that much less speed compared to the earth. And the trajectory will be slightly toward the earth.

I would assume that makes it substantially more likely that the xenon falls back to earth.

[logfromblammo]

It depends on whether the propellant is used to increase or decrease the orbital velocity of the vessel.

Acceleration propellant would have to be ejected at the orbital velocity of the vessel plus escape velocity to escape. Deceleration propellant would just have to be ejected at escape velocity minus orbital velocity.

As deceleration near atmosphere is almost free just by dipping into it, or by using some form of sea anchor to pull on the atmosphere or magnetic field, it is more likely that propellant would be used preferentially for attitude control and acceleration.

It isn't impossible, but imparting enough velocity to propellant for it to escape Earth orbit--while accelerating a vessel in the opposite direction--seems unlikely for orbital station-keeping. You need at least 12000 m/s for escape velocity plus at least 8000 m/s to counteract the orbit you were already in, so the propellant would have to leave the vessel at more than 20000 m/s. That's a specific impulse of about 2000 s. Ion drives and VASIMR could do it, but the propellant is very likely to experience its own atmospheric drag and electromagnetic interactions, and the probability that any particular atom of propellant would actually escape with the minimum velocity-relative-to-vessel is very low. The propellant would spread out to a larger volume as quickly as it could, too. It's far more likely that one of those xenon ions would collide with a hydrogen atom in the upper atmosphere and randomly bounce it out, like a bowling ball hitting a billiard ball.

[orbital-decay]

Generally, electric propulsion is used for two purposes in Earth-orbiting spacecraft: for correcting the orbit (station-keeping) and for raising the orbit from an intermediate to the target one (in relatively recent all-electric GEO sats)

Station-keeping requires relatively short burns (sub-hour to several hours) in all directions. When raising the orbit, the spacecraft usually keeps itself in fixed position relative to the Sun to maximize the solar panel output. The propulsion unit keeps working at all times, both in prograde and retrograde, because efficient Hall thrusters are tricky to work with in impulse mode, and are heavily optimized for continuous operation.

So in most cases, xenon is ejected in arbitrary directions, retrograde being only one of them. Besides, some ion/plasma thrusters are so efficient that they eject the propellant at more than double escape velocity. I would guess most of the propellant actually leaves the gravity well; also, at higher altitudes where electric propulsion is mostly used there's no atmosphere to collide with.

[wyattpeak]

Wow, you're right, I vastly underestimated the exhaust velocity.

You'd still have to account for its interaction with the atmosphere, but my point is moot.