My Dyson Sphere Conjecture

 

This post builds on my first two. Dyson sphere types and Ringworlds.

Let's try and build a Type 3 Dyson sphere. We want something we can live inside, like a ringworld, but that (more or less) completely fills a sphere around a star. Good idea, but it suffers from one problem. It is well known by those who think about Dyson spheres that the net gravitational attraction of a point mass within a uniform shell is zero.

This is known as the Shell theorem. It is a double differential of the gravity equation--an undergraduate physics problem--but the upshot is that, for Point A, all the close mass on the left-hand side on the line is exactly counter-balanced by the greater amount of mass, which is further away, on the right-hand side of the line. And this is true, no matter how you draw the line. So a perfect mass composing a perfect shell exerts no net gravitational attraction on any point inside the shell, be it points A, B, C, or any other.

Thus, there is no gravity to hold the inhabitants to the inner surface of the shell and terrestrial life inside a Dyson Sphere won’t work.

But there may be a “hard science” answer to this problem.

What if the builders break the symmetry of the shell. Don’t imagine a shell of uniform density. Instead, give up Dyson (and Niven’s) constraint that the sphere must have the mass of Jupiter and imagine instead a sphere with a lot of mass. Start by imagining a cylinder of mass pointed towards the star. The cylinder is exactly massive enough that the end pointing towards the star has about a 1g field strength. A local gravitational attraction will develop on the ends of these cylinders. 

On this figure, the “cylinder” is shown as a triangle with the majority of the gravity-producing mass being in the center of the object. I call these individual objects, domains. The stick figure and little house on the inset are not to scale—instead we could imagine each domain having a total area approximately equal to the surface area of the Earth. 

The domain can be imagined orbiting around the star and so far everything is clear from a physics point-of-view, but–you know–why would you build such a thing.

We can now start crowding the orbit of the domain with other domains. In Stapledon’s universe, as the orbit starts to get crowded with artificial planets, the same thing is built on different orbits either closer or farther from the star. His artificial planets were spheres that rotated, but that is not required once you have the day-night screens of a ringword.

Now I’m going to evoke what I would call  a “Niven Rule”. We can tell stories on a ringworld without explaining how “scrith” works, that is, the material science that holds the whole thing together is outside of the scope of the story. In my world, the inhabitants call the material holding the whole sphere together enigma metal. In one draft—which I think was cut for the SciFidea contest—enigma metal was said to be made on non-atom material. The idea was that the building blocks of atoms could be assembled in a different way to create human-scale materials with non-atomic properties. It’s fiction, Niven said we could do it.

So instead of having domains orbit the star, I’m going to crowd a single diameter with domains and build a latticework of enigma metal and hold all the domains in place. They will be static relative to one another. Thus, we have a grid of millions of domains encircling a star, connected to each other by a lattice of enigma metal. While we are here, let’s add six cardinal directions marked with energy beams connecting the sphere to the star. These beams can be adjusted such that the sphere is actively kept in place. This avoids the instability of the whole system.

On a single domain, net gravitational attraction will taper off as you move towards the edges of each cylinder or further into the sphere. Each cylinder would create a domain that is habitable, surrounded by a ring of microgravity between adjoining domains. Under these conditions, the Dyson sphere will have about 550 million Earth-sized domains. It is an open question as to how much of each domain can have reasonable gravity. In my stories, I assume somewhat more than half. Further, if each domain's mass is a little over half that of Earth, the whole thing has the mass of a modest star. Start with an uninhabited binary star system (sorry Three Body Problem) and you have everything you need.

Some readers may have noticed a physics-based problem regarding the shell theorem and asymmetry, I’ll come back to that.

But while we are here, we can add a few more features to our sphere—quality of life elements if you would. First, we should put some baffles between the domains. We don’t need million-mile-long storms and huge flows of water vapor. Another part of our baffles should make sure that atmospheres does not leak out to the external portion of the hull. So, build a floor between the domains. And while we’re at it, we don’t want to be living the “black box radiation” problem, so let’s build a mesh over the top of each domain which controls the amount of reflected solar radiation reaching the surface.

The Energy of Night

If half the star is shielded at any given time, creating Earth-like day and night cycles on the surface of the sphere, where is the blocked energy going? Pointing it back into the star will heat it. In my sphere, much of it is being captured by the “day-night” machine and is beamed to the hull of the sphere via the six cardinal direction beacons. The remainder is captured by the mesh on top of the domains. On the exterior of the sphere there are mechanisms which capture that energy and convert it into thrust. According to my calculations, such a sphere could reasonably expect to accelerate to around 0.1 C within a million years. If you then reversed the thrust and started braking, in around two million years such a vehicle could travel between nearby galaxies.

My Dyson sphere, like Stapledon's, is a generation ship.

The Asymmetry Problem

The problem with this whole design is that if we pack the entire space of the shell with these domains, even though the domains do not form a uniform solid, they will form the equivalent of a uniform solid and we will be right back to where we started. The idea is that, from a distance an irregular solid has the same gravitational effect as a point mass located at the solid’s center of mass. You get enough of these, and you will have the equivalent of a uniform shell and this whole design falls apart.

 But consider the Square Packing problem. This is a classic example of the more generic Packing problem where we try to place as many squares within the smallest possible larger square. And the classic result of this problem is that the best answer frequently involves wedging squares together in a seemingly disordered manner creating a non-uniform distribution of squares. Here is a classic result for 11 squares.

(By Walter Trump - Own work, CC BY-SA 4.0)

Conjecture: I offer the following statement without proof. I assert there may be one or more configurations of mass that allow the domains to have a noticeable surface gravity without creating a symmetric shell and yet generally filling the majority of the surface of the sphere.

Motivation. To motivate this, we know it will work with one domain—this is the case of the planet Earth. We know we can add dozens of domains without creating a symmetric shell that negates the local gravitational attraction. The only real questions are how many domains can you add, how large can they be, and how much variation in the local gravitational attraction will you experience?

Why?

So if a sufficiently advanced society wants to spread itself to the next galaxy, they can take a binary star, break one star down to its base subatomic elements, rebuild it as a sphere around the other, build the ion optics required to capture half the remaining star's energy output, and--Bob's your uncle--you have a ship you can point off into the void.

My goal in imagining such a craft is to get the reader thinking out engineering things that are much larger than our day-to-day human scale. We can understand what we are building. We can understand why we are building it. But it is on a scale way outside anything we can currently imagine a group of humans achieving.

To build this, we have to imagine better humans!