Second question:

Do you have a nice reference (speculative feasibility study) for non-rigid coil-guns for acceleration?

Obvious idea would be to have a swarm of satellites with a coil, spread out over the solar system. Outgoing probe would pass through a series of such coils, each adding some impulse to the probe (and doing minor course corrections). Obviously needs very finely tuned trajectory.

Advantage over rigid coil-gun: acceleration spread out (unevenly) over longer length (almost entire solar system). This is good for heat dissipation (no coupling is perfect), and maintaining mega-scale rigid objects appears difficult. Satellites can take their time to regain position (solar sail / solar powered ion thruster / gravity assist). Does not help with g-forces.

Disadvantage: Need a large number of satellites in order to get enough launch windows. But if we are talking dyson swarm anyway, this does not matter.

How much do we gain compared to laser acceleration? Main question is probably: How does the required amount of heat dissipation compare?

Nice. To make your proposed explanation more precise:

Take a random vector on the n-dim unit sphere. Project to the nearest (+1,-1)/sqrt(n) vector; what is the expected l2-distance / angle? How does it scale with n?

If this value decreases in n, then your explanation is essentially correct, or did you want to propose something else?

Start by taking a random vector x where each coordinate is unit gaussian (normalize later). The projection px just splits into positive coordinates and negative coordinates.

We are interested in E[ / |x| sqrt(n)].

If the dimension is large enough, then we wont really need to normalize; it is enough to start with 1/sqrt(n) gaussians, as we will almost almost surely get almost unit length. Then all components are independent.

For the angle, we then (approximately) need to compute E(sum_i |x_i| / n), where each x_i is unit Gaussian. This is asymptotically independent of n; so it appears like this explanation of improper linear models fails.

Darn, after reading your comment I mistakenly believed that this would be yet another case of “obvious from high-dimensional geometry” / random projection.

PS. In what sense are improper linear models working? l_1, l

2, l\infty sense?Edit: I was being stupid, leaving the above for future ridicule. We want E(sum_i |x_i| / n)=1, not E(sum_i |x_i|/n)=0.

Folded Gaussian tells us that E[ sum_i |x_i|/n]= sqrt(2/pi), for large n. The explanation still does not work, since 2/pi <1, and this gives us the expected error margin of improper high-dimensional models.

@Stuart: What are the typical empirical errors? Do they happen to be near sqrt(2/pi), which is close enough to 1 to be summarized as “kinda works”?