Synchrotron Radiation: Relativistic Electrons in a Magnetic Field (2D)

Electron Energy

Lorentz factor γ 300

Magnetic Field

Field strength B 1.0 mT

Geometry

Pitch angle α 55°

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Electron energy

153 MeV

Speed β = v/c

0.999994

Critical frequency ν_c (spectral peak)

—

Radiated power (1 electron)

—

Beaming half-angle 1/γ

0.19°

Gyroradius r_g

—

How it works

A relativistic electron spiraling around a magnetic field line accelerates centripetally and radiates. Unlike slow cyclotron radiation, this radiation is beamed into a narrow "searchlight" cone of half-angle θ ≈ 1/γ around the instantaneous velocity — the ultra-relativistic version of a headlight effect.

Gyroradius:        r_g = γ m_e v_perp / (eB)
Gyration frequency: ω_g = eB / (γ m_e)      (slows as γ grows — relativistic mass)
Critical frequency: ν_c = (3/4π) γ² (eB/m_e) sin α
Beaming angle:      θ_beam ≈ 1/γ
Radiated power:     P = 2 σ_T c γ² β² sin²α · B²/(2μ₀)

γ (Lorentz factor) — the electron's energy in units of its rest mass (0.511 MeV). Higher γ tightens the beaming cone as 1/γ and pushes the emitted spectrum to much higher (∝γ²) frequency.
B (field strength) — sets how fast the electron gyrates and, together with γ, sets both the peak frequency and the total power radiated.
α (pitch angle) — the angle between the electron's velocity and the field line. α = 90° gives pure circular gyration; small α gives a long, gentle helix with weak emission (sin²α power dependence).

Real-world relevance: this exact mechanism produces the radio-to-X-ray glow of pulsar wind nebulae (e.g. the Crab Nebula), relativistic AGN and gamma-ray-burst jets, and supernova-remnant shocks — synchrotron radiation is the dominant non-thermal signature astronomers use to trace high-energy particles and magnetic fields across the universe.

Top-down view (looking along field line)

Electron Field line (B, into page) Radiation beam