Front matter
I lost some steam on this; it was kindly asked at the beginning of a weekend but in its infinite wisdom the resident question closing cabal invoked exchangus interuptus.
Now I'm getting yelled at for "playing that darn stack changing video game again" on a weekday.
...(such as but not limited to a toroidal wound solenoid)...
I'll be writing about a normal linear solenoids, the same shape as those used at CERN and in the AMS. I'm not sure how to specifically address a "toroidal wound solenoid" like those used in Tokamaks because those are closed and only focus particles that are already trapped inside and already circulating. They can't focus or collect incoming particles in any useful way that I can think of.
TL;DR
Is it feasible to build a spaceborne electromagnetic lens... to gather and focus cosmic rays?
No. Take a magnifying glass outside and try to concentrate blue sky.
The way that the AMS-02 uses the field is to slightly alter the direction of the particles and measure that change with position sensitive detectors and ray-tracking calculations, the same thing that CERN does. There's not really any focusing or concentration involved in either case.
Solenoids as focusing devices
The way solenoids focus is interesting, here is a rough handwaving explanation. A particle comes in roughly along the axis in the $\pm \hat{z}$ direction and interacts with the radial component ($\pm \hat{r}$ direction) of the external field. That induces a rotational motion around the axis in the $\pm \hat{\phi}$ direction. (cylindrical coordinates)
I've used $\pm$ because you could be going in either direction along $\hat{z}$ and the particle could be either positive or negative, and the current flow in the coil could be in either direction.
What's cool is that those don't matter.
From here and here
Then that motion interacts with the axial or $\hat{z}$ component of the magnetic field producing a radial force in the $-\hat{r}$ direction.
No matter which direction you are going (like most focusing lenses) it focuses. But because it's a second order effect using the same magnetic field twice (in two orthogonal directions) and therefore depends on $q^2 B^2$ a solenoid will always focus both charge signs, unless it's so strong that there's a focal point within the lens itself and what comes out the other end is a diverging beam after the focus.
The orbits are a little weird inside but in some particle detectors they put thin position sensitive gas detectors inside and use those points along the trajectory to get some information on the particle.
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From here found here. (Figure 36.4. Lens operating modes. (A) Condenser–objective mode. (B) Second-zone mode. (C) Conventional mode... EP: Entrance pupil; SP: specimen; OA: objective aperture; CI: crossover image. After Riecke (1982), Courtesy Springer Verlag.)
The image above hints that solenoids are often used in electron microscopes and other charged particle optical systems. There are also electrostatic lenses but those require extremely high voltages for high energy particles.
Ingo Hofmann's paper Performance of solenoids versus quadrupoles in focusing and energy selection of laser accelerated protons (also here, also see this) for example is useful because it also describes the other kind of magnetic lens, a pair of magnetic quadrupoles, and because it highlights the fact that magnetic (and electrostatic) lenses are purely chromatic. Unlike glass which gives weak chromatic aberration, the focal length of magnetic lenses depends strongly on the particles speed. Lenses are sometimes used to separate particles of different velocity using a pinhole!
The paper gives an equation for the approximate focal length of a thin solenoid. Because they work in a complicated way an exact formula is tough and most people star with this and then go straight to ray tracing.
$$1/f_S = \left( \frac{q}{2 m c \beta \gamma } \right)^2 B^2 L$$
where $\beta = v/c$ and $\gamma = 1 / \sqrt{1-\beta^2}$ the latter of which we can call "of order 1" unless the particle is very high energy.
Right away we see that the focal length depends on the square of the charge/mass ratio and the square of the field/velocity ratio. That means two things:
- none of the signs of $B$, $v$ or $q$ matter
- you'll need pretty large magnetic fields for high energy cosmic rays
What does the energy spectrum of cosmic rays look like?
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AMS data are blue dots From H.E.S.S. : “Probing Local Sources with High Energy Cosmic Ray Electrons”
The peak is about 3 GeV (3E-03 TeV). Let's calculate the focal length for that using the actual 0.15 Tesla field from the permanent magnets of AMS-02 and for 7 Tesla, the top end of commercial MRI systems
With a kinetic energy of 3 GeV and a rest mass of about 1 GeV $\gamma = T/mc^2 + 1$ is about 4 and $\beta = \sqrt{1-1/\gamma^2}$ is about 0.94, so the focal lengths are about 24 kilometers for AMS-02 and 11 meters for a 7 Tesla superconducting solenoid assuming their lengths are about 1 meter. If you make them much longer, you can get shorter focal lengths.
If you make a really big giant superconducting solenoid, you can start to think about this, but...
It doesn't matter!
Remember two things:
- Cosmic rays are essentially isotropic. They don't come from a single direction. A lens would just mix up their directions but not concentrate. Take a magnifying glass and try to concentrate light reflected from a flat wall or take it outdoors and try to concentrate blue sky on a piece of paper; it doesn't work!
- Even if it did, (which it doesn't) magnetic lenses are completely chromatic, so you could only concentrate a small range of energies effectively.
