Disclaimer: this is a blog post, not a paper. Seriously, I’m not suggesting to build such an accelerator, but I believe that by trying to answer this question the reader can learn something about the intriguing realm of accelerator physics.
The Large Hadron Collider operating at CERN is the most powerful collider ever built by human kind – some argue it is going to be also the last one of his species. Its unprecedented energy of 6.5 TeV per beam let physicist to probe the fundamental interactions governing the subatomic world, and ultimately to discover the last missing piece of the Standard Model, a scalar particle usually referred to as the Higgs boson. However, many believe that the Standard Model is incomplete, and especially the measured mass of the Higgs boson cries for an explanation that lies beyond the currently known physics. While we had good reasons to build a machine such as the LHC – mainly because its energy seemed sufficient for the production of the Higgs boson itself, along with supersymmetric partners of know particles – it is completely unclear at which energy regime any new kind of physics may hide. It may be 10, 100, 1000 or more higher than the energy the LHC can deliver. Ultimately, the highest meaningful scale seems to be the socalled Planck energy:
GeV
This number is so mindboggling that is believed to be unreachable by any technological mean. If only it could..we might investigate directly the regime of quantum gravity, the Holy Grail of physics!
How close can we get to this scale, at least ideally, using an ideal particle collider?
The basics of particle accelerators
In order to make two particles to interact between each other, the most clever idea devised so far is to accelerate them in two opposite directions, and smash them in a region of space called interaction point. There are two basic possible geometries: linear and circular. In this post, we’ll focus on the latter.
To achieve this, we need charged particles, which can be manoeuvred by the action of a magnetic field. From undergrad studies, you may recall that the motion of a charged particle in a magnetic field is described by the Lorentz Force:
In a simplified case of only one particle with unit charge moving in a perfect circle, this equations simplifies to:
where e is the charge of the electron ( C), B is the magnetic field, p = mv is the particle momentum, and r is the radius of the circle (often called gyromagnetic or Larmor radius in astroparticle physics).
It is customary to express the particle momentum in units of GeV/c. To convert it from SI units, a numerical factor is needed, i.e.:
GeV/c
The product B r is usually called magnetic rigidity.
To give an example, at the LHC r = 2804 m and the magnetic field generated by the bending dipole magnets is 8.33 T, which gives a momentum p = 0.3 * 2804*8.33 = 7000 GeV = 7 TeV.
At this point, we can play around with the numbers. Assuming that a magnetic field of 10 T is achievable, for a radius equal to the Earth orbit:
GeV
Well, let’s say we can put our particles in an orbit close to that of Jupiter
m
GeV
In short, we would still fall short of the desired energy, but it would still be a great leap forward to explore an energy regime that some believe is holding the secrets of the unification of forces. If only we could build such a machine..
Energy loss
One of the main problem faced by people working on accelerator physics is that accelerated charged particles emit radiation (synchrotron radiation), which has to be compensated somehow to maintain their orbit stable. This is usually achieved using radiofrequency cavities.
From relativistic kinematics, it turns out that the energy loss depend on the inverse of the fourth power of the mass of the accelerated particle. For this and other reasons, the LHC is accelerating protons rather than electrons.
A general formulation can be found in textbooks:
where m is the classical electron radius. For example, at the LEP accelerator, the emitted power was in the order of several MW. In a circular collider, the total energy loss per turn U is given by the radiated power P times the revolution time :
From this formula, one can see that it’s very hard to increase significantly the momentum of the particle, and at the same time limit the energy loss just by making the accelerator larger and larger by adjusting the radius r. Historically, this became impractical at energies around 100 GeV. A possible solution is thus to accelerate heavier particles, such as protons, as done at hadron colliders such as the LHC. However, one downside is that protons are composite objects, and the quarks that interact receive only about 10% of the total energy, and the rest is wasted.
Cooling
So far, we imagined to accelerate only one particle, traveling around an ideal circular orbit. In fact, present colliders accelerate an enormous number of particles, divided in two counterrotating beams, each divided into a number of “bunches” that cross each other at regular intervals at the interaction points. To collect as many particles as possible, they have to “concentrated” spatially. This process is referred to as cooling. In the early 1980s, a method called stochastic cooling was devised by Simon van der Meer, who shared the 1984 Nobel prize with Carlo Rubbia for the discovery of the electroweak bosons. The term stochastic refers to the fact that the method does not always work, but on average it helps to maintain the coherence of the beams.
The basic idea is as follows: a particle in beam does not follow a perfect circular orbit, but oscillates around it (this trajectories are called betatron oscillations). At regular intervals, resistive plates pick up the passage of this particle and send a signal proportional to the displacement across the accelerator, where another device kicks the particle back to its place, thus reducing its wiggling. The trick is that even if the particle travels close to the speed of light, the chord is shorter than the arc of circumference, so the signal arrives first. On average, the particle travels along with millions of other companions, so it is practically impossible to pick up just one stray particle. In fact, the method works on average.
Now, just imagine about doing this in the outer space, where the radius of the accelerator is measured in AU rather than a few kilometres!
Conclusions
To wrap up, the idea of creating a circumsolar collider is certainly fascinating, but it’s limited to the realm of science fiction at the moment. The technological challenges are extreme, and no plans have been devised to create such a monster machine. There are certainly other alternatives such as wakefield accelerators or muon colliders which are worth exploring before we reach for the sky. For the time being, the LHC is here to stay.
Bibliography
 An introduction to particle accelerators, Edmund Wilson, Oxford University Press
 Classical electrodynamics, John D. Jackson, Wiley
Further reading

The Undulator Radiation Collider: An Energy Efficient Design For A √s=10^15 GeV Collider, F. Bursa, arXiv:1704.04469
 Unreasonably big physics, 4gravitons blog (Matthew von Hippel)

Higgs Factory and 100 TeV Hadron Collider: Opportunity for a New World Laboratory within a Decade, S. Assadi et al., arXiv:1402.5973
Let’s say we wanted to create a particle accelerator around the sun between the sun and mercury, how many accelerator nodes would we need, and how much power to generate Planck energy?