Thanks for this! I misinterpreted Lucius as saying "use the single highest and single lowest eigenvalues to estimate the rank of a matrix" which I didn't think was possible.

Counting the number of non-zero eigenvalues makes a lot more sense!

You can absolutely harvest potential energy from the solar system to spin up tethers. ToughSF has some good posts on this:

https://toughsf.blogspot.com/2018/06/inter-orbital-kinetic-energy-exchanges.html https://toughsf.blogspot.com/2020/07/tethers-all-way.html

Ideally your tether is going to constantly adjust its orbit so it says far away from the atmosphere, but for fun I did a calculation of what would happen if a 10K tonne tether (suitable for boosting 100 tonne payloads) fell to the Earth. Apparently it just breaks up in the atmosphere and produces very little damage. More discussion here:

https://www.reddit.com/r/slatestarcodex/comments/1eqe48g/comment/liaouaa/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button

The launch cadence is an interesting topic that I haven't had a chance to tackle. The rotational frequency limits how often you can boost stuff.

Since time is money you would want a shorter and faster tether, but a shorter time of rotation means that your time window to dock with the tether is smaller, so there's an optimization problem there as well.

It's a little easier when you've got catapults on the moon's surface. You can have two running side by side and transfer energy between them electrically. So load up catapult #1, spin it up, launch the payload, and then transfer the remaining energy to catapult #2. You can get much higher launch cadence that way.

Oops yes, that should read "Getting oxygen from the moon to LEO requires less delta V than going from the Earth to LEO!". I edited the original comment.

Lunar tethers actually look like they will be feasible sooner than Earth tethers! The lack of atmosphere, micrometeorites, and lower gravity (g) makes them scale better.

In fact, you can even put a small tether system on the lunar surface to catapult payloads to orbit: https://splittinginfinity.substack.com/p/should-we-get-material-from-the-moon

Whether tethers are useful on the moon depends on the mission you want to do. Like you point out, low delta-V missions probably don't need a tether when rockets work just fine. But if you want to take lunar material to low earth orbit or send it to Mars, a lunar tether is a great option.

The near-term application I'm most excited about is liquid oxygen. Getting oxygen from the moon to LEO requires less delta V than going from the Earth to LEO! Regolith is ~45% oxygen by mass and a fully-fueled Starship is 80% LOX by mass. So refueling ships in LEO with lunar O2 could be viable.

Even better, the falling lunar oxygen can spin up a tether in LEO which can use that momentum to boost a Starship to other parts of the solar system.

Thanks for the comments! Going point-by-point:

1. I think both fiberglass and carbon fiber use organic epoxy that's prone to UV (and atomic oxygen) degradation? One solution is to avoid epoxy entirely using parallel strands or something like a Hoytether. The other option is to remove old epoxy and reapply over time, if its economical vs just letting the tether degrade.

2. I worry that low-thrust options like ion engines and sails could be too expensive vs catching falling mass, but I could be convinced either way!

3. Yeah, some form of vibration damping will be important, I glossed over this. Bending modes are particularly a problem for glass. Though I would guess that vibrations wouldn't make the force along the tether any higher?

4. Catching the projectile is a key engineering challenge here! One that I probably can't solve from my armchair. As for missing the catch, I guess I don't see this as a huge issue? If the rocket can re-land, missing the catch means that the only loss is fuel. Though colliding with the tether would be a big problem.

5. Yeah I think low orbits are too challenging for tethers, so they're definitely going to be at risk of micrometeorite impacts. I see this as a key role of the "safety factor". Tether should be robust to ~10-50% of fibers being damaged, and there should be a way to replace/repair them as well.

6. Right, though tethers can't help satellites get to LEO, they can help them get to higher orbits which seems useful. But the real value-add comes when you want to get to the Moon and beyond.

7. Good to know! I would love to see more experiments on glass fibers pulled in space, small-scale catches, and data on what kinds of defects form on these materials in orbit.

Yeah, my overall sense is that using falling mass to spin the tether back up is the most practical. But solar sails and ion drives might contribute too, these are just much slower which hurts launch cadence and costs.

The fact that you need a regular supply of falling mass from e.g. the moon is yet another reason why tethers need a mature space industry to become viable!

That makes sense, I guess it just comes down to an empirical question of which is easier.

Question about what you said earlier: How can you use the top/bottom eigenvalues to estimate the rank of the Hessian? I'm not as familiar with this so any pointers would be appreciated!

Isn't calculating the Hessian for large statistical models kind of hard? And aren't second derivatives prone to numerical errors?

Agree that this is only valuable if sampling on the loss landscape is easier or more robust than calculating the Hessian.

You may find this interesting "On the Covariance-Hessian Relation in Evolution Strategies":

https://arxiv.org/pdf/1806.03674

It makes a lot of assumptions, but as I understand it if you: a. Sample points near the minima [1]. b. Select only the lowest loss point from that sample and save it. c. Repeat that process many times d. Create a covariance matrix of the selected points

The covariance matrix will converge to the inverse of the Hessian, assuming the loss landscape is quadratic. Since the inverse of a matrix has the same rank, you could probably just use this covariance matrix to bound the local learning coefficient.

Though since a covariance matrix has rank less than n-1 (where n is the number of sample points) you would need to sample and evaluate roughly d/2 points. The process seems pretty parallelize-able though.

[1] Specifically using an an isotropic, unit variance normal distribution centered at the minima.