The Decoupled Orbital - Arbitrarily Large Rotating Habitats
The Basics
The forces on a rotating ring structure are analogous to the forces on a horizontal cable that is pinned at each end. Just as the cable must support its own weight via tension along the cable, the rotating ring structure must also support its own "weight" (or more accurately, the experienced acceleration (a) calculated using the radius (r) and rotational velocity (V)).
For an acceleration equal to 1G (9.8 m/s^2), the forces on a rotating ring are similar to the forces of a pinned horizontal cable with the same length as the rotating ring's circumference. Indeed, imagine the ends of a pinned horizontal cable bending around and connecting to each other.
As the rotating ring increases in radius, the rotational velocity needed to sustain the same acceleration (eg: 9.8 m/s^2) increases exponentially, as is shown in the equation above. Since the rotation rate (w) is derived using the same two factors, the equation can be rewritten as:
This provides a formal explanation to our intuitive understanding that the smaller a ring gets, the faster it has to spin in order to keep the same acceleration.
Stress
The outward force generated by a rotating habitat must go somewhere, and indeed it does. The outward force at any one position is carried by the ring structure through tension which travels along the circumference of the structure and is opposed by the outward force on the other side of the ring in the opposite direction. Since all the forces along the circumference have to travel through the same cross-section of material, the stress through the structure is cumulative. That is to say, the structural stress in the ring increases linearly with the radius as described in the following equation for thin-walled hoops:
This means that if a rotating ring were to double in radius (r), the stress (σ) in the ring would also double. To keep the stress the same, the thickness (t) would also have to be doubled, which is fine under static conditions, but for rotating rings there is a catch: since the force (F) in a rotating ring comes from the inertia of the rotating matter itself, as the structural thickness increases, so does the mass and by extension the outward force. In fact, since the outward force is equal to the mass times acceleration, the equation above can be rewritten as:
If the structural thickness is doubled, then so is the structural mass (M_s); M_n represents the rotating non-structural mass. This means that the tensile stress in a rotating ring can not be lowered by simply increasing the thickness of the structure. In fact, increasing the thickness technically increases the maximum stress on the exterior portions of the structure because those portions have a greater radius and velocity than the inner portions. In this way, as the radius of a rotating ring structure increases, the stress in the structure will necessarily increase along with it.
There comes a point where the stress in the ring structure becomes too large for the structure to withstand. The maximum radius at which the structure will fail depends on the specific strength of the structural material used. Every material has a specific strength that dictates its breaking length for a given surface acceleration. Copper has a lower breaking length than steel, which has a lower breaking length than kevlar, which has a lower breaking length than graphene. Everything will break if it gets long enough.
And that's where the story ends.
Unless…
The Decoupled Orbital
If the structural strength can be decoupled from the force on the ring, then the system should not suffer from unmitigated structural stress.
As far as the inhabitants of a rotating space habitat are concerned, the only things that need to rotate with them are the things they interact with in their environment: the floor, other structures, people, air, etc. If just these necessary elements rotated together, the rotating system could be held together by a large, non-rotating shell on the exterior. This exterior shell could support the inner rotating shell via electromagnets and pneumatic springs (for instance) without imparting its own additional outward force. In this case, the rotating portion would not even need to be continuous, as it would function much more like a supersized mag-lev train, though a continuous rotating ring could benefit from a contiguous interior environment. The modified stress equation above can be re-written as:
The advantage of this should be clear. This modified equation shows that the structure can increase thickness (t) without increasing the outward force (aM_n) on the system.
As these habitats increase in size, they would also likely increase their rotating mass per unit length. This of course can be accommodated with an increase in the structural thickness. For extremely large habitats, the rotating mass per unit length may actually stabilize around a maximum value (as the rotating interior begins to resemble an "Earth-like" environment) and any added structural thickness would only be necessary to compensate for an increase in radius, which of course is a linear relationship.
Since the relationship between radius and thickness is linear, and since the stress varies as a factor of the outward force, the ratio between radius and thickness will remain constant for a given surface acceleration, regardless of the structural material. In a nutshell, this means that this type of rotating habitat can be arbitrarily large as long as there is enough structural material (of any kind) to support it. This stands in contrast to the conventional rotating habitat where there is a maximum size limit before the structure breaks apart under its own weight. Depending on the structural material used, the mass and thickness of the structure could become quite large indeed (rivaling that of the total non-star mass in a solar system in the case of a stellar ringworld), but this is a small consideration to a civilization who could carry-out such a large-scale undertaking in the first place.
In order to explain away their megastructures, science fiction authors have had to invent unobtainium and Clark tech such as scrith, force fields, and extra-dimensional matter. Little must they have known that these gymnastics were not necessary for creating the settings for their stories.
