I have a few recurring nightmares. One is the typical "I am walking around town naked and nobody notices."" The other is that I'm going to write the Great American Sci-Fi Novel™ and have a giant glaring mistake in my physics.
Up until now I had written off settlements on the asteroid Psyche itself because tunneling through iron/nickel would be uneconomical. Yes, people harvesting the material would create voids in the planet, but the gravity of Psyche is so minuscule that one would have to provide a rotating habitat to support normal human existence.
My concept for settlements are basically O'Neill cylinders that are tethered to the planetoid. Mine are a little squatter, and gain acreage by being layered like an onion. The hollow outer layers providing rigidity through Box Girder construction. Like a space elevator, the bulk of their mass is just on the edge of where the weight of the station is overcome by the centrifugal force of planet's rotation.
My insight/waking nightmare for this morning is that Psyche rotates awfully fast. A space station on a 210 km long cable is going to experience a hell of a lot of centrifugal force. Perhaps the station doesn't need to rotate at all. Perhaps there is a radius where Psyche's natural rotation would produce either Earth-like gravity, or a useful fraction of Earth-like gravity.
My space elevator calculations were concerned with keeping an object on the end of a tether permanently on orbit:
If we revisit our math on the cable, but add a few calculations that we cancelled out before. Namely, the force on the counter-weight:
|Length (from surface):
|Empire State Buildings
As we can see, the accelleration at the length of the cable where the counter-weight would be is... negligible. Wikipedia states that the surface gravity of Psyche is on the order of 0.144 meters per second. So we are in the ballpark.
However, we don't have to place a space station at the counter-weight. We can build it anywhere along that length. We can also build it past the counter-weight point/geostationary orbit distance. The equations we are using as our guide is:
The gravity on the floor of our station will be equal to the Centrifugal force minus the influence of the planet's own gravity. The w^2 will remain constant at the natural rotational speed of Psyche. We are calculating for a unit mass, so the "M" factor is simply 1.0. We want to see 9.8 newtons of force on that unit mass. Naturally, we would have to factor in the natural gravity of the body, but at these distances that force is going to be negligible so we well ignore that.
Substituting in our constants we get a radius of 1.765503019740818e-8. Which is somewhere between the wavelength of a gamma ray and the width of a DNA helix. If we solve for 1/10th Earth gravity, its not much better: 1.765503019740818e-7
So... no. Unless we are talking about spinning Psyche up on a massive scale, there is no distance you could park a settlement and derive a useful gravity. So in this case, I'm going to leave this idea on the cutting room floor. Any planet where you could derive gravity from it's rotation would have to be spinning as fast as a space station that I have already calculated. Which is on the order of 2 rpm or so for something around 200 meters across. The key thing is the time scale of the rotation: MINUTES. Psyche (and every other asteroid we've observed) rotate on the time scale of hours.