On linear accelerators.

Here we shall determine the required power and size for a linear accelerator (whether of particles or massive objects) to deliver a chosen energy per shot.

As ever, a few assumptions to focus the problem and make life simpler:

• The accelerator fires discrete shots, not a continuous beam. If it's firing a massive object this will always be the case, while particle accelerators vary.
• The accelerating force is limited by compressive strength of the accelerator structure. This is similar to how I handled spacecraft acceleration limits previously. Tensile strength is usually lower than compressive strength so support in compression seems reasonable.
• The accelerator structure is a prism. Obvious choice assumption.
• The projectile is fired at ultrarelativistic speeds, by which I mean in excess of 0.95c. For a particle accelerator this is natural. For a projectile accelerator I've explained previously that I feel this is necessary for it to be worthwhile as a weapon.
• Recoil can be neglected. This will be true if the accelerator's physically connected to something with enough mass.

I'm going to always call what the accelerator fires a projectile, regardless of whether it's a solid object or a bunch of detached particles. At the speeds involved it actually won't make any difference to the terminal ballistics, as to quote xkcd's what if, "the bonds holding the sphere together are completely irrelevant, it’s just a collection of carbon atoms".

I'm also going to always be working in the reference frame of the accelerator's structure.

Considering the force to be exerted on the projectile by some component (an electromagnet, perhaps) that is physically supported by the accelerator structure being loaded in compression, then

F = σ_cA

Where σ_c is of course the compressive strength.

The general definition of force, valid in special relativity as well as Newtonian mechanics, is the time derivative of momentum. With constant force, then for a projectile with no initial momentum and final momentum p, accelerated over time t,

F = p/t

In the ultrarelativistic limit, energy E is given by

E = pc

The time available for the acceleration depends of course on the length of the accelerator. Since we are taking the ultrarelativistic limit the projectile velocity is approximately c, and thus for an accelerator of length d,

t = d/c

Putting all the above equations together, we can express the projectile momentum and energy as depending on the volume and compressive strength of the accelerator

V = Ad
p = Ft = σ_cAd/c

E = σ_cV

Power is the time derivative of energy. Since energy is directly proportional to momentum, the constant time derivative of momentum (constant force) implies the constant time derivative of energy, ie constant power. This is thus given simply by

P = E/t = Ec/d

For a worked example, I'll consider a desired yield of 1 megaton of TNT, with a diamond accelerator structure or cylindrical shape and a length of 1 kilometre.

E is 4.2 x 10¹⁵ Joules.
σ_c is 1.2 x 10¹⁰ Pascals.
d is 1000 metres.

V = E/σ_c = 350000 m³
P = 1.3 x 10²¹ W

A = 350 m²
r = 10.6 m

The accelerator radius is quite modest, indeed positively svelte, though the overall size and thus mass are considerable, the latter being over a million tonnes. The power required, however, is extreme, many orders of magnitude above anything humanity has yet created.

The consequences here are relatively simple. For fixed material properties, the energy of the projectile depends only on the accelerator's volume, or equivalently its mass. The power drawn, by contrast, is lower for a long slender accelerator than for a short and bulky one. Therefore, assuming that lower power draw is desired, linear accelerators should be made as long as possible, and even then systems need to be capable of extreme bursts of power. Of course if the weapon is desired to fire in the direction the spacecraft accelerates, the previously established limitations on overall craft length come into play.

If the first assumption is violated, a more sophisticated treatment will be required. A continuous beam, or at least one long compared to the accelerator, would I expect bring the power requirements down drastically while still being able to deliver considerable energy to the target.

If the second assumption is violated, the accelerator might be made more slender, but the power requirements will still be the same since they depend only on accelerator length and not on the details of its construction.

If the third assumption is violated, I suppose you'd have to use calculus to handle the varying forces. I'm not sure why a non-prismatic shape would be used though.

If the fourth assumption is violated, then again calculus will probably be required to handle the varying velocity, unless the speeds are low enough for a Newtonian constant acceleration treatment. For an accelerator of fixed length, reducing projectile speed while increasing mass to compensate and give the same final energy will increase the time for acceleration and thus lower the power requirements.

If the fifth assumption being violated, it will manifest as an inefficiency since some of the energy input goes into accelerating the accelerator itself rather than the projectile.

Next week I will consider the case of a circular accelerator, which will have advantages and drawbacks compared to the linear variety.