Illustrated Physics
physics, drawn
Concept

Doppler factor — where it appears

The ratio by which motion multiplies a wave's frequency or a particle's energy between the source's frame and an observer's, 1/γ(1 − β cos θ) for light. It reaches about 2γ straight ahead of a fast source and falls off the axis, which is how a neutrino's energy depends on its angle.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The same source, seen coming and seen going. The observed flux of a moving source, relative to the same source at rest, against the angle between its motion and the line of sight, on a logarithmic scale, at β = 0.5, β = 0.9, β = 0.99. The curves span 8.1e+1, 1.3e+5, 1.6e+9 between the approaching and receding directions, which is ((1+β)/(1−β))⁴ exactly. The fourth power comes from the one quantity every observer agrees about: the specific intensity divided by the cube of the frequency. Three powers of the Doppler factor come from that invariance and the fourth from integrating over frequency. Two consequences are worth reading off. A source seen side-on is fainter than the same source at rest, by γ⁴ — a factor of 2.5e+3 at the fastest speed drawn. And half the light arrives inside a cone of 60.0°, 25.8°, 8.1°, which for a fast source is one over γ.

The brightness that is not the same for everyone

A moving source is not merely shifted in colour. Its light is concentrated forwards, so the same lamp is enormously brighter seen coming than seen going — by the fourth power of one number, which for a jet at ninety-nine per cent of light speed is a factor of a thousand million. The reason is that only one combination of intensity and frequency is the same for every observer, and everything else follows from it.

relativity · Doppler The neutrino's energy against the pion's, at four angles. The energy of the neutrino from a pion's decay against the pion's energy, for neutrinos going exactly forward and at 1°, 2.5° and 4° to the pion's line of flight. Straight ahead the neutrino takes a fixed share, 0.427, of the pion's energy, so pions of every energy give neutrinos of every energy. Off axis the curve rises, flattens and falls: at 2.5° it peaks at 0.683 GeV for a pion of 3.2 GeV, and every pion between 2 and 10 GeV gives a neutrino between 0.61 and 0.68 GeV. The flat top is where γθ = 1. Its height, p*/θ, depends on the angle alone — the rest-frame momentum of the decay, 29.8 MeV/c, divided by the angle in radians.

The neutrino energy an angle fixes

A pion breaks into a muon and a neutrino, and the neutrino carries off a fixed 29.8 MeV in the pion's own frame. In the laboratory, straight along the pion's path, it carries a fixed share of whatever energy the pion had, so a beam of pions of every energy makes neutrinos of every energy. Two and a half degrees to one side, something else happens. The neutrino's energy, as a function of the pion's, rises to a maximum and falls again, and around the maximum pions from two to ten GeV all give neutrinos within a tenth of a GeV of each other. An experiment in Japan builds its detector deliberately off the axis of its own beam for that reason — trading most of its neutrinos for the ones at the energy it needs.

relativity · Relativistic dynamics

Named alongside it

The objects these essays reach for when they reach for this one.

AberrationBrightness temperatureCosmic microwave backgroundDipole anisotropyFluxJacobian peakLorentz boostLorentz-invariantNeutrino beamNeutrino oscillationPhase spacePion

All concepts