Light from Cherenkov radiation

[TheCanadian]

What exactly is the physical origin behind light produced by Cherenkov radiation having a non-zero angle with the trajectory of superluminal charged particle? Why is the light simply not allowed to go in any direction? Also, is Cherenkov radiation observed for "uncharged" particles (e.g. neutrons) that are still composed of charged constituents (e.g. quarks)?

 

[DrDu]

The point is that light waves emitted at different points along the path interfere so that the light rays seem to propagate normal to the cone.

 

[TheCanadian]

The point is that light waves emitted at different points along the path interfere so that the light rays seem to propagate normal to the cone.

That appears somewhat visually clear from the diagram, where it's akin to a change in index of refraction due to the displacement of charges/electromagnetic response from the medium), although is there perhaps a clearer mathematical explanation or resource that describes why the cone of that particular angle for a particular speed is the result?

I guess I'm interested in the mathematics behinds why this begins at the speed of light, and what limiting expression exists for even below the speed of light. For example, would not a charged particle simply even approaching superluminal speeds result in a non-zero similar effect (e.g. electromagnetic response from the medium) on the propagation of photons through the medium?