Back to normal service now. I still don't understand circular motion (and in particular I don't understand angular momentum), but I've managed to muddle my way through the equations.
To business. Essentially a sci-fi circular accelerator is like the LHC on steroids. Projectile goes round and round the loop, and then it's let to fly off at a tangent towards its target. The assumptions are much the same as those for the linear case covered
- The accelerator fires discrete shots.
- The yield is limited by material strength.
- The accelerator structure is a rectangular torus. This is a sensible shape that will make the maths easier than for a circular torus. The major radius is also significantly greater than the minor width, again for mathematical simplicity.
- The projectile is fired at ultrarelativistic speeds. Again this lets us simplify the equations.
- Recoil can be neglected.
Unlike in the linear case, a circular accelerator does not have to bring its projectile up to speed within a limited distance; rather the projectile can "spin up" over many laps. The accelerator structure is thus required to exert a centripetal force needed to keep the projectile going round in circles, as well as a linear accelerating force which can be arbitrarily small and shall thus be neglected. The equation for centripetal force in Newtonian mechanics, F = mv2/r, is of course not valid in special relativity, so we will derive the ultrarelativistic equation.
Those who last did vector calculus more recently than several years ago will have to forgive my doubtless slightly sloppy notation.
The diagram above shows the projectile with an initial momentum p. After a short time interval dt, its momentum is now p'. During that time interval it travels through an arc with length vdt and angle dθ. Its momentum keeps the same magnitude - we're considering uniform circular motion - but rotates through that same angle.
s/r = dp/p
If dθ is small, then s ≈ vdt, ie the arc and chord are nearly the same length (and tend towards the same length as dθ shrinks to zero). As such
vdt/r = dp/p
Rearranging, the equation for force can be found.
F = dp/dt = pv/r
The above equation is valid equally in Newtonian mechanics and special relativity, since it's just derived from geometry. The expression for momentum, however, varies. In the ultrarelativistic limit,
v = c
E = pc
F = E/r
Thus we have the equation linking centripetal force, energy, and radius of curvature for an ultrarelativistic projectile.
In the linear case, we were able to consider the forces as being borne simply in compression. For a circular accelerator, however, the stresses are more complex. A key simplifying assumption I feel justified in making is that the force is spread essentially equally about the ring, because the projectile and thus the centripetal force are moving much faster than the speed of sound in the accelerator structure. With that assumption, the situation can be considered as analogous to a pressurised cylinder.
The above shows a cross-section through a cylinder with an outward force acting on it. In our accelerator, the outwards force is the reaction to the centripetal force on the projectile. In the analogous pressurised cylinder, it's fluid pressure on the inner walls. Either way, the outwards force results in a circumferential, or 'hoop', tension in the accelerator structure.
Assuming the projectile travels centrally around the accelerator, the radius of curvature r corresponds to the mean radius of the cylinder. If the cylinder then has length l and wall thickness w, and is thin-walled (wall thickness small compared to radius), the pressure on the inside walls is given by the force divided by the surface area of the inside
σp = F/2πlr
The hoop stress is then give by the following equation, which can be derived by considering half of the cylinder as a free body
σθ = σpr/w
In both the above equations, the radius should correctly be the inside radius of the cylinder. However using the mean radius does not in practice result in a large error for the peak hoop stress; even for w = 0.2r, normally regarded as thick-walled, the value is just 11% too low.
As in the linear case, this stress is limited by material strength. The hoop stress cannot be greater than the ultimate tensile strength σt of the material of the accelerator. Equating σθ and σt, and substituting in expressions for σp, we can get an equation relating projectile energy, material strength, and accelerator geometry.
σt = F/2πlw
σt = E/2πlwr
The volume of the accelerator structure can be approximated as
V = 2πrlw
And thus the equation for energy simplifies down to
E = σtV
Apart from being limited by tensile not compressive strength, it's exactly the same as for the linear accelerator! As a result, I shan't be doing a worked example this time round.
Power we cannot determine from the geometry for a circular accelerator in the way we did for the linear case. Ignoring losses, it could be computed simply from the desired firing rate and could be arbitrarily low. In practice, however, losses may exist, such as synchrotron radiation.
The key consequence is thus that a circular accelerator design will have mass similar to a linear one, the exact ratio depending on the available materials' performances in compression compared to in tension. The circular design though will require less power, probably orders of magnitude less, to run. Within a certain range, the yield depends just on the overall mass of the accelerator structure, and not on the exact shape.
If the first assumption is violated, it actually won't make a difference, due to the assumption made during the working out that the centripetal force is essentially equally spread around the accelerator anyway.
If the second assumption is violated then an entirely different approach will be required to determine the yield.
If the third assumption is violated, a more detailed mathematical treatment will be required. In particular, the equations given here will tend to underestimate the stresses, and thus overestimate the yield, for 'thick walled' designs, especially for the extreme case where the structure is a solid disc which might be practical for a compact accelerator. That said, I believe the error does not increase without limit, so the calculations here will still be within the right order of magnitude.
If the fourth assumption is violated, likewise a more detailed treatment will be required. The mass and velocity of the projectile will have to be considered separately, instead of just its total energy. Unlike the linear case though, we won't have to deal with a varying speed.
The violation of the fifth assumption has two implications. Firstly, while the projectile is circulating, the whole accelerator structure ought to also circle due to conservation of momentum. This could simply be mitigated by having multiple smaller projectiles to balance things though. Secondly there is the linear recoil upon releasing the projectiles, which like in the linear case will sap some energy and so manifest as an inefficiency.
I plan on in future returning to issues of accelerator design, for both the linear and circular cases, and considering arguments based on magnetic field strength for when that is the specific accelerating force. For now, I feel confident in the basic conclusion that circular designs are better than linear ones - a conclusion backed up by most real-world particle colliders using circular designs.
