How exactly does a magnetoplasmadynamic (MPD) thruster generate thrust?

Question

So, I have some trouble to intuitively imagine what exactly is going on in MPDs. As I understand it, you draw a current from a cathode in the radial direction to the anode (channel wall). You ignite a plasma in the discharge channel which acts as an electrical conductor. The radial current then generates an azimuthal magnetic field and a force acts on the charged particles which points mainly in the downstream direction. So far so good. But the majority of the current will be carried by electrons due to their higher mobility, right? The gas atoms that are initially moving in the axial direction will then undergo ionizing collisions with the electrons and then scatter in some direction. In a plasma discharge the bulk plasma is usually close to field free, while the large potential drops occur at the electrode surfaces. I just don't see why the ions should then move in the radial direction as well, which means that they are not necessarily accelerated to the outside by the magnetic field. Are only the electrons accelerated and then drag the ions with them by ambipolar diffusion? This would mean that the Lorentz force could not be used as an approximation for the thrust, right? I also don't really get, how the charge carriers can provide the necessary discharge current while simultaneously being accelerated to the outside. I'm all over the place, sorry. I hope you get what I am trying to point at.

Answer 1

I think you're overthinking this. At the end of the day, stuff gets kicked out the back real fast, and that impulse pushes the spacecraft forward. At the end of the day, it's a reaction engine.

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Though, there are some details we can hash out. (Obligatory disclaimer: I'm not an expert on MPDTs, but I think I know enough to answer this.)

As I understand it [...]

Correct.

But the majority of the current will be carried by electrons due to their higher mobility, right?

Correct, but note that these electrons are "new" electrons literally leaving the cathode itself, moving from the cathode to the anode (i.e., the current is flowing from anode to cathode). The plasma itself is electrically neutral. The motion of the ions and electrons from the plasma may happen but isn't relevant to how the thruster works.

The gas atoms that are initially moving in the axial direction will then undergo ionizing collisions with the electrons and then scatter in some direction.

Not really. The electrons aren't moving very quickly and in any case, atoms don't scatter off of electrons so much as electrons scatter off of atoms. The whole thing is a plasma, so the concept of a bound atom is a bit questionable anyway: especially with the lighter propellant gases and low pressures MPDTs use, the ions and the electrons basically don't interact physically.

In a plasma discharge the bulk plasma is usually close to field free, while the large potential drops occur at the electrode surfaces. I just don't see why the ions should then move in the radial direction as well, which means that they are not necessarily accelerated to the outside by the magnetic field.

The (total) impulse due to the electric field ultimately relies on the (total) potential difference, so only the magnitude matters, not the way it varies within. But actually this isn't relevant since in a MPDT most of the acceleration seems to come from the magnetic field instead of the electric field.

Regardless of whether the magnetic field is a self-field due to the anode-cathode current or an applied field (external), the magnetic field is made to wrap around the engine. If you work through the Lorentz force, you will see that particles of both charges are accelerated backward in this configuration.

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The Lorentz force is $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$. When the propellant is flowing in, the ions (the $q$) have a velocity $\mathbf{v}$ perpendicular to the $\mathbf{B}$, and so experience an acceleration. Deep inside the engine, this is radial, but as suggested by the figure the magnetic field bends; at the back of the engine, the resultant acceleration for positive ions is astern.

The electrons in the plasma would be accelerated the other way, but ultimately tag along with the much heavier ions because of electrostatics.

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(Tidying up)

Are only the electrons accelerated and then drag the ions with them by ambipolar diffusion?

As above, both the electrons and the ions are accelerated by the Lorentz force.

I also don't really get, how the charge carriers can provide the necessary discharge current while simultaneously being accelerated to the outside.

As above, the current between anode and cathode involves "new" electrons literally leaving the cathode itself. These are accelerated by the field, but ultimately an equal number of electrons find their way back to the anode, while the plasma leaves electrically neutral.

Note that the current that gets produced needn't rely on individual electrons or their speed so much as on their volume. Electrons can produce powerful currents while barely moving. Also note that in an applied-field MPDT, the current's main purpose to to ionize the propellant; this doesn't require a high current so much as a high voltage; the actual magnetic field in this configuration is applied externally.