The Cluster's Doppler Effect

A page about the Doppler effect — the frequency shift of waves from a moving source.

The classical Doppler effect

For a source moving with velocity v_s toward a stationary observer, the observed frequency is: f_obs = f_0 / (1 - v_s / v_wave) = f_0 * v_wave / (v_wave - v_s) For a source moving away: f_obs = f_0 / (1 + v_s / v_wave). For a moving observer: f_obs = f_0 (1 + v_o / v_wave). In the cluster, the edit classical Doppler effect gives an edit frequency shift.

The relativistic Doppler effect

For a source moving with velocity v at angle theta relative to the observer: f_obs = f_0 * sqrt(1 - beta^2) / (1 - beta cos theta) where beta = v / c. For motion directly toward: f_obs = f_0 * sqrt((1+beta)/(1-beta)). For motion directly away: f_obs = f_0 * sqrt((1-beta)/(1+beta)). The redshift parameter is z = f_0 / f_obs - 1. For v << c: z ~ beta. In the cluster, the edit relativistic Doppler effect gives an edit frequency shift.

The transverse Doppler effect

At theta = pi/2 (motion perpendicular to line of sight): f_obs = f_0 * sqrt(1 - beta^2) = f_0 / gamma. This is a purely relativistic effect — there is no classical transverse Doppler shift. The transverse Doppler effect was first confirmed in 1938 by Ives and Stilwell using hydrogen canal rays. In the cluster, the edit transverse Doppler effect is an edit purely relativistic effect.

Applications

Doppler effect is used in:

• Astronomy: measuring radial velocities of stars and galaxies (spectral line shifts)
• Radar and lidar: measuring velocity of vehicles and weather systems
• Medical ultrasound: Doppler blood flow measurements
• Satellite navigation: correcting for relativistic Doppler shifts
• CMB: dipole anisotropy from Earth's motion through the CMB

In the cluster, edit Doppler effect is used in:

• edit Astronomy: measuring edit radial velocities
• edit Radar and lidar
• edit Medical ultrasound
• edit Satellite navigation
• edit CMB: edit dipole anisotropy

The cosmic expansion

Cosmological redshift is often compared to the Doppler effect but is actually due to the expansion of space. The wavelength stretches as lambda_obs / lambda_em = a(t_obs) / a(t_em) = 1 + z. For small z: v = H_0 * d, which is Hubble's law. The distinction between "Doppler" and "expansion" redshift becomes important at high z. In the cluster, the edit cosmic expansion gives an edit cosmological redshift.

This effect

This page is about the Doppler effect. f_obs = f_0 * sqrt(1-beta^2) / (1-beta cos theta). Transverse: f_obs = f_0 / gamma. Lambda_obs / lambda_em = 1+z. The effect is real.