How to Build a Dyson Sphere
A Dyson sphere, named after the physicist Freeman Dyson, is a hypothetical mega-structure that would enable an advanced civilization to fully exploit the energy resources of its star. There are many variations on the Dyson sphere concept. The simplest, known as a Dyson Shell, is just a solid spherical shell that completely surrounds a star.
Unfortunately for the proponents of this concept, a simple calculation shows that the compressive forces produced in a Dyson Shell would exceed the strength of any known material. As Freeman Dyson put it:
A solid shell or ring surrounding a star is mechanically impossible. The form of ‘biosphere’ which I envisaged consists of a loose collection or swarm of objects traveling on independent orbits around the star.
However, this conclusion is premature, since we do not have to rely on mechanical forces alone to stabilize large structures. Active structures are structures that rely on kinetic forces for stability in addition to static forces. For example, while there is no material that can support a stationary ring placed around a star, a ring spun at orbital velocity will produce a centrifugal force that cancels the gravitational force at every point, ensuring stability.
Force cancellation can be used to stabilize otherwise impossible structures. The idea of a ringworld is a structure surrounding a star that is made to rotate much faster than orbital velocity (~0.004 c, assuming a radius of 1 AU) in order to produce 1 g of centrifugal force for its inhabitants. Such a structure is also typically assumed to be impossible to create using ordinary materials. However, it is easy to see that if a stationary ring (~1600x heavier than the rotating ring, assuming a radius of 1 AU) is placed along the perimeter of the rotating ring, the centrifugal force produced by the rotating ring is cancelled at every point by the gravitational attraction between the stationary ring and the sun.
Unfortunately, stabilizing spheres is not quite as easy as stabilizing rings, since rotation cannot cancel out the component of the gravitational force vector that is parallel to the axis of rotation. However, by analyzing the forces acting on a rotating sphere, we will see that a more complicated setup can achieve the stabilization that we want.
At every point on a sphere, there is radius vector that points from the origin to a point on the sphere. This vector can be expressed as the product of the unit radius vector and the radius of the sphere:
It is also useful to define a radius of rotation. This is a vector that points from the axis of rotation to the position of a point on the sphere. This vector can be expressed as the component of the radius perpendicular to the axis of rotation:
Finally, we will be using a subscript notation to indicate components of a vector. For example, the following scalar quantity is the x component of the unit radius vector:
Now, we will calculate the forces at any point on a rotating sphere (for simplicity, we will ignore the mass term).
The equation for centrifugal force is:
Where:
The equation for gravitational force is:
Where:
The net force at a point will be the sum of the centrifugal force and gravity:
By setting a=b, a rotating spherical shell can exactly cancel gravity along the x and y components. As expected, this only gives us partial stabilization as the z component is uncancelled. Without additional stabilization, the sphere would collapse along the z axis.
Full stabilization can be achieved using 3 spheres (with equal radii and angular velocities), each rotating about one of 3 mutually perpendicular axes of rotation. To prove it, we will calculate the net acceleration produced by a system of 3 spheres, rotating about the z, x, and y axes respectively.
The force produced by the first sphere is written in the x-y-z coordinate frame. The components of the force vector are:
The force produced by the second sphere is written in the y-z-x coordinate frame. The components of the force vector are:
Note that this vector only differs from the first by a rearrangement of the x-y-z labels.
Similarly, the components of the force vector produced by the third sphere, in the z-x-y coordinate frame, are:
Finally, we take the sum over all 3 spheres (in the x-y-z coordinate frame):
We can see that by setting 2a = 3b, a set of 3 mechanically coupled spheres produces a centrifugal force that exactly cancels gravity at every point. This would imply that the equator of each sphere is spinning at sqrt(3/2) orbital velocity. Setting 2a>3b, the net force generated by the spheres allows for the support of a uniform load across the structure.
A variation on this idea is to replace the 3 solid spheres with 3 sets of rings arranged in a dense mesh-like structure.
The rings would contain mass streams moving at high speed that would generate a centrifugal force analogous to that of a rotating sphere. A series of rings arranged in the above configuration would produce (in the limit of a high density mesh) exactly the same uniform centrifugal force as the configuration of 3 spheres discussed earlier.
The mass streams can be prevented from colliding with the walls of the ring by using an active magnetic stabilization mechanism, similar to how maglev trains are levitated on magnetic tracks.
Finally, it’s important to note that a partially complete Dyson Shell would not be self supporting and would need external support while under construction.
