# Loop Torque Wheels

Angular momentum is the integral of mass times velocity times radius. A hoop rotating in space "stores" angular momentum, and is held together by tensile stresses in the hoop.

r |
hoop radius |
m |

m |
hoop mass |
kg |

v |
hoop speed |
m/s |

I |
m\times v\times r . . hoop angular momentum |
kg-m |

\rho |
hoop mass per unit length |
kg/m |

Y |
material tensile strength divided by density |
m |

What is the limit on v? Imagine that you slice the hoop in half, and consider the forces at the slice. The mass going into the black box at one side has a momentum flow rate of \rho v^2 , and coming out the other side has a momentum flow rate of -\rho v^2 , a total momentum flow change (a force! think aerodynamics ...) of 2 \rho v^2 , supported by two interfaces with a tensile force of 2 Y \rho on them.

So v^2 \leq Y . Hoop speed is limited by material strength. Bigger hoops turning with the same rim speed undergo the same deflection, but they store more momentum (proportional to mass and radius) for the same rim tension force.

Since the stored angular momentum I = m \times v \times r , and v is constrained the strength of the rim material, **a wheel stores more angular momentum per unit mass if r is larger**. In space, r can be arbitrarily large, subject to stability constraints.

### Stability

Sadly, the most stable state for a rotating hoop is not round; it is flattened out, two side-by-side segments flipping end-over-end. Small deflection forces (and smart control) are required to keep the hoop round. While end-over-end (or two masses on opposite ends of a single rotating tether) does store some momentum, most of the connecting tether is rotating at smaller radius, a lower ratio of angular momentum to mass. A hoop is optimum, as long as it can be stabilized and maintained as a circle.

Stabilization can be done with thin cross cables, resembling the spokes of a bicycle. Unlike static bicycle spokes, which depend on material rigidity for stiffness, the hub of an angular momentum storage ring can contain high speed actuators that reel in or release tether, adjusting forces dynamically. Laser measurement (accurate to a fraction of a wavelength) and high speed computation can give the "spokes" the effective stiffness of "einsteinium", a Young's modulus that is a significant fraction of the speed of light squared. That is the key to efficient angular momentum storage.

### Storage times

Y is about 3e6 (m/s)^{2} for Kevlar 49, "3 MegaYuris" to use the cute neologism of the space elevator community. With a safety factor of 3, that allows a hoop velocity of 1000 m/s. The tensile connections to the hub must have some "diagonal" to them; they need to transmit torque to the wheel, but not much. Angular momentum storage times will be on the order of the orbital period, much longer than hoop rotation periods, so torques will be relatively small.

For a large structure like a space solar power station in geosynchrounous orbit, with kilometer dimensions and an orbital period of a day, a hoop with a diameter of 20 kilometers and a rim speed of 1000 m/s will have a rotation period of 63 seconds, almost 1400 times shorter. The "hub" might be two circular tracks at the outer edges of the SSPS, geometries dependent on the shape of the SSPS. My W.A.G. is that these maglev hubs can have 2% of the radius and be constructed with less than 10% of the mass of the outer wheel. I will leave the details to mechanical engineers with more skill and imagination than I have. I add electronic control, computation, and virtual super-stiffness to the solution.