Mars' moons make a stationary space elevator extremely challenging, even though known material processes are sufficient.

Good point. They're so small I forgot about them.

By lunar space elevator I assume you mean an elevator from the lunar surface to the point where lunar gravity is practically 0?

Only by coincidence in this case.

On planets, a space elevator goes to (geo)stationary[0] so that the cable doesn't wind up around the planet, but you can't do that on the Moon, because luna-stationary is occupied by the Earth, which inconveniently is too massive and spinning too fast to anchor the other side of the cable. However, the L1 point is also stationery relative to the lunar surface, and is the place where the gravity of the Earth and the Moon balance out.

[0] IIRC, geostationary specifically means Earth, but there's going to be some more general term for the same idea over generic parent objects and not just Earth

Lunar stationary orbit is around 88,400 km, which would be unstable for a satellite due to the Earth's gravity, but might allow for a space elevator pointed right at Earth to efficiently launch crates of helium-3 or hydroponic grain back to the planet.

Indeed. I've not even played with this in one of the many simulators, but I believe the suggestion is to put the counterweight a tiny bit closer to Earth than the L1 to stabilise it.

Although (and I wish I could find this again), I've read that lunar He3 is so diffuse that getting it out would incidentally give us so much purified aluminium, silicon, and oxygen, that sending all that back to Earth and magnetically decelerating it on arrival would give us more energy than the He3, as would burning those ingots with that oxygen.

LiftPort Group's plan [0] involves actively maintaining their counterweight at Lagrange point 1.

[0] https://www.youtube.com/watch?v=3ZteJmg1TfM (10 minute video, relevant around 4:00)

Lunar stationary orbit is not 88,400 km, it's 384,000 km, i.e. the distance from the Moon to the Earth. The moon is tidally locked to the Earth.

Though you could equivalently try and go for Earth-Moon L1 with a counterweight on the other side of L1. It would be significantly more unstable though.

The physics works for Earth too. In all cases, you make the top of the cable bigger than the bottom to support all of the weight hanging from it. From what I've read, in the case of Earth and with current materials technology, we end up with the top of the cable having a diameter comparable to that of the Earth. Clearly, that's not feasible.

For Luna or Mars, gravity is reduced and the required diameter is less. Maybe it would even be feasible to build such an elevator if it were above the Earth. But now you're building above an alien plant, so you trade one set of potentially insurmountable obstacles for another.

If one end of the cable is the size of the Earth, the physics can't get away with neglecting the gravity created by the cable itself.

I mean, that's a fair point. The total volume of the elevator cable would be greater than that of the Earth. The mass might still be less since we're not building it out of iron here, but effectively we'd have a binary planet with the centre of mass well outside the Earth's surface.

I'm not sure that that system would be unstable in human timeframes since the two would be tidally locked, although it would certainly alter the engineering stresses in ways that I'm grossly unqualified to calculate. I think a portion of the cable might be under compression rather than tension? I guess it depends on the rotational speed of the whole system.

Speaking of which, substantial amounts of energy would need to be spent accelerating the spin of the Earth/space elevator system to maintain a 24-hour day/night cycle.

However, Luna's presence would perturb the whole system, either tearing it apart with tidal stresses or being ejected from the system before that could happen.

Carbon Nanotubes would end up with the top being ~26% larger than the bottom:

See Isaac Arthur's video about it here: https://youtu.be/dc8_AuzeYKE?t=470

I appreciate the correction. I'm not sure where I heard that particular piece of information, nor in that case what material was being examined. Perhaps that one was steel.