TARS and angular momentum

The paper "Torqued Accelarator using Radiation from the Sun (TARS) for Interstellar Payloads" has been making the rounds, possibly in part because of how approachable the idea is. In short, it proposes deploying a large sheet in orbit of the sun, with one half of each side coated to reflect sunlight and the other half to absorb it, creating asymmetic forces that would cause the sheet to spin faster and faster. Eventually the sheet would spin fast enough to tear itself apart, but before that it could release a small payload from one end, slinging out of the solar system. With existing materials, it could spin up to an edge speed of around 10 km/s. This post at Centauri Dreams proposes some alterations and alternatives to the concept, which involve more moving parts but also more control. Considering those ideas suggested one more to me: taking advantage of the conservation of angular momentum.

This video by the paper's author includes this animation of a TARS sheet unfolding in space and spinning up. Once it's spun up, reversing the unfolding process and folding the sheet back up would reduce the object's radius would thus increase its speed (usually demonstrated by a figure skater pulling their arms in to spin faster). One question, however, is whether that increase in speed is an increase in linear edge speed, or only in rotational speed.

In the example of a figure skater, rotational speed clearly increases, but does it increase only enough to maintain the same linear speed at the edge, spinning faster because an object on the edge is now traversing a smaller circumference? Wikipedia shows that for a point mass, angular momentum L is

With constant angular momentum but two different radii, we have

So radius and linear edge speed are inversely proportional, and drawing a point mass toward its center of motion would increase its edge speed. For TARS, this means that operators could increase the launch speed of a payload by re-folding the sheet back toward its center before releasing the payload. But that may not be viable in practice. Payloads would be released at the maximum speed allowed by the sheet's material, and pulling the sheet in and increasing rotational velocity would only increase stresses. Possibly the sheet could be pulled in by a material stronger than the sheet itself. Even without such a material, though, I can see two advantages to pulling in the sheet.

The first is that the sheet does not need to wait as long to spin up to its maximum launch speed. Solar pressure only needs to spin the sheet up to a fraction of that speed, and drawing in the sheet can increase the spin to the material's limits. When spin-up times are measured in years due to the relative weakness of solar pressure, being able to reach max speed in a fraction of that time makes a meaningful difference.

The second advantage of drawing the sheet in is that, once refolded, it would no longer have a surface to be acted on by solar pressure, stopping acceleration and preserving the sheet. Electric motors may be able to counter rotation, slowly spinning down the sheet to a speed at which new payloads can be installed. That would turn a TARS sheet from an expendable launch vehicle into a reusable one.

One caveat to all this is the moment of inertia of a TARS sheet. Radius and edge speed are only inversely proportional with a point mass spinning around a center, but the video shows a TARS sheet with a mass concentrated much closer to the axis of rotation. Drawing in such a sheet may not make that much of difference in edge speed, though it should still preserve the sheet.

If you have thoughts, I'm happy to hear them.