Presets
Playback
Medium
Refractive index n of the transparent medium (light travels at c/n in it).
Particle
Speed as a fraction of the vacuum speed of light, and the length of track in the medium.
Layers
Model checks
The science
Why light is emitted
Nothing may exceed the speed of light c in vacuum, but in a medium of refractive index n light travels at only c/n. A charged particle with v > c/n, i.e. β > 1/n, outruns the light it polarises out of the medium's atoms. As with a sonic boom, the wavelets emitted along the track add up in phase along a cone.
Huygens construction and the angle
A wavelet emitted a time t ago has radius (c/n)t while the particle has moved βct. The tangent cone has half-angle ψ with sin ψ = 1/(nβ); the light leaves normal to it, at cos θ = 1/(nβ) to the track. In water (n = 1.33) the limit β → 1 gives θ ≈ 41°.
Threshold and photon yield
The threshold kinetic energy is T = (γth − 1) mc2 with γth = 1/√(1 − 1/n2): about 0.26 MeV for an electron in water, 54 MeV for a muon, 485 MeV for a proton. The Frank-Tamm formula dN/dx = 2πα sin2θ ∫ dλ/λ2 counts photons per length; the 1/λ2 spectrum is why Cherenkov light looks blue.
Honest simplifications
The refractive index is treated as constant over wavelength (no dispersion), the particle does not slow down or scatter, and photon counts ignore detector efficiency and absorption. The 3D page shows the same cone angle formula for water, ice and glass; diamond, aerogel and the threshold energies are added here.
What to try
- Start with water at beta 0.99: the wavelets pile into a V and yellow rays leave at about 41 degrees.
- Pick the 3D default (beta 0.65): below threshold, so there is no cone and the yield is zero.
- Pick Just above threshold: the cone is almost flat (theta near 0) and very little light is made.
- Compare Diamond and Aerogel: a large index gives a wide cone and a high yield, a small index a narrow cone.