D6 The Optical Doppler Effect
You all know the phenomenon: An ambulance approaches at high speed with howling siren. As it races by the pitch of the siren sinks and remains constantly at a deeper level as it departs. Also the other situation is well well-known, where you yourself move at high speed past a standing source of noise: You are travelling in your car over country, windows open, and you past a train crossing with alarm bells.
These changes in the perceived pitches correspond to measurable changes in the frequencies of the acoustic waves. Christian Doppler examined this theoretically and concluded that the two cases “listener moves, source at rest” and “source moves, listener at rest” must differ. In 1842 he ascribed formulas, indicating how the measured values of the frequencies and wavelengths change. Today one calls the phenomenon in his honor the “Doppler effect”. We can easily understand acoustically why one may not only consider the relative motion of source and listener: The sound spreads out homogeneously with a certain speed in all directions in the medium of air ! This carrier medium supplies a special inertial frame and naturally served as the model for the ether, in which the light should spread.
Consider an observer B with velocity v approaching a resting acoustic source Q. This produces a tone of frequency f(Q). What frequency f(B) does the observer measure? Doppler's answer to this question is the following, whereby here c means the speed of sound in air:
(1) f(B) = f(Q) • ( 1 + v/c ) Observer approaches a resting acoustic source
For the case where the acoustic source approaches with velocity v an observer at rest in the medium air, we have
(2) f(B) = f(Q) / ( 1 - v/c ) Acoustic source approaches a resting observer
For values of v/c < 0.1 the two results hardly differ. The difference becomes arbitrarily large however as v/c approaches the value 1.
We change now from sound to light or more generally to electromagnetic waves. The STR precludes the possibility of determining absolutely, who is at rest and who is moving. Thus ‘optically’ there is only one Doppler formula! We deduce it first from (2):
Formula (2) remains valid, but we must now consider additionally that the oscillator of the transmitter, because of time dilation, oscillates for the observer B only with the frequency f(Q) • √ ! Thus we have f(B) = f(Q) • √ / (1 - v / c). Considering that we can write √ = √ ((1 + v / c) • (1 - v / c)), we can reduce a little obtaining
Graph the three functions y = 1 + x; y = 1/(1 - x) and y = √ ((1 + x)/(1 - x)) and let the value x = v/c range over the unit segment 0 to 1. We have the following picture:
The lower, linear function belongs to Doppler formula (1), the upper blue to Doppler formula (2). The middle red curve describes the optical (or relativistic) Doppler effect in accordance with the formula deduced previously. The differences begin to be evident only when v/c is larger than about 0.2. Starting from a value of v/c greater than 0.5 the differences become increasingly dramatic.
In astronomy the optical Doppler effect has important applications. Frequencies of spectral lines, however, are not measured but rather the wavelengths (usual symbol λ). Therefore we should transform the above formula accordingly:
In general λ • f = c or f = c/λ. Thus
c/λ(B) = (c/λ(Q)) • √ ((c + v)/(c - v)) and after division by c
λ(Q) = λ(B) • √ ((c + v)/(c - v)) or λ(B) = λ(Q) • √ ((c - v)/(c + v))
λ(Q) is well-known and λ(B) is measured. From this the velocity v can be computed, with which the source moves toward us (v > 0) or away from us (v < 0). This is the so-called radial velocity. If one solves the above formula for v, then one obtains
In the last few years astronomers have developed such precise spectrometers that they can measure periodic fluctuations in the radial velocity of stars within the range of a few meters per second. This has become one of the most important methods for showing the existence of planets orbiting other stars (so-called exoplanets). The graphic below gives an impression of the precision that has been obtained. The measured values of the radial velocity have an uncertainty of approximately ± 1 m/s! These fluctuations of the radial velocity result from the fact that both the planet and the star orbit a common center of mass.
Further information is freely available on the well maintained web page of the ESO. The following graphic was taken from the ESO press release of August 25, 2004. Consult the web-site for an answer to the question, how long does an ‘orbital phase’ last, in other words, how long is the period of this planet in days or hours.