- Introduction
- Classical Doppler Effect
- Relativistic Doppler Effect
- Aberration
- Transverse Doppler Shift
- References

Albireo, β Cygni, at the head of the Swan, is a beautiful object, a double star that appears in a small telescope or binoculars as a topaz and a sapphire, 34.4" of arc apart. The brighter star, spectral type K3, is magnitude 3.1 and described as yellow or golden, while the other, spectral type B8, and itself a close double, is magnitude 5.1 and said to be blue or green. They move together in the sky, though no orbital rotation has been detected, and they may be 10 light years apart in distance from us (they are about 380 light years distant).

Christian Johann Doppler (probably Döppler; 1803-1853) probably had such double stars in mind when he wrote in 1842 that perhaps the difference in colors were due to the motion of the stars, since the frequency of the light from an approaching star should be increased, and from a receding star decreased. The speeds of stars are far too small for this to have any observable effect, but C. H. D. Buijs-Ballot (Buys Ballot) deduced that the effect should be much larger with sound, because of its smaller velocity. Accordingly, he experimented with musicians on a locomotive and at lineside in 1845, on the railway between Utrecht and Maarsen, and determined that Doppler's conjecture was indeed fact. The sound was produced by a trumpet on the locomotive, and three at lineside, with 14 observers, distributed on the train and ground. The phenomenon of the frequency shift with a moving source or moving observer has since then been known as the *Doppler Effect*, Though Buijs-Ballot should be remembered.

The optical Doppler effect was not observed until 1905, when J. Stark found that the light emitted by canal rays, which are fast ions from the cathode of a glow discharge, was Doppler-shifted by the expected amount. Stellar spectra frequently show Doppler shifts, from which the relative velocity in the line of sight can be determined, a very useful quantity, and double star orbits can be studied. The red shift of the light from distant galaxies is also famous, and is usually interpreted as due to a recessional velocity increasing linearly with distance, though the argument is in danger of being a circular one. The Doppler effect broadens the spectral lines of atoms as a result of their random thermal velocities. It also has an important application in surveying by means of satellite radio signals.

The Doppler effect in the case of waves in a material medium will be called the "classical Doppler effect" here. We shall assume that all velocities are much less than the speed of light, so that relativistic considerations do not enter. The most important characteristic is that there is a preferred reference frame K, in which the medium is at rest. Referred to this system, the waves will be supposed to have a phase velocity c and an angular frequency ω_{o} = 2πf_{o}. The wavelength λ will then be given by λ = c/f. The magnitude of the wave vector k = 2π/&lambda = ω/c. Harmonic waves moving in the direction of the unit vector **n** can be described by a wave function a(t,**x**) = A exp[iω(t - **n**·x/c)]. Here, **x** is the usual radius vector. The argument of the exponential is iφ(t,**x**), where φ is the *phase*, a function of time and position.

At any time t, the phase is constant over *wave fronts* **n**·**x** = constant = d, where d is the distance of the wavefront from the origin, the length of a line from the origin drawn normal to the wave front. Hence, φ = ωt - 2πd/λ. For a constant phase of zero, d = (ωλ/2π)t = ct, so we verify that any wave front moves forward with velocity c.

If a reference frame K' moves with constant velocity **v** relative to K, then **x**' = **x** - **v**t, and, of course, t' = t. This is a *Galilean transformation*, which preserves Newton's Laws, but not the laws of Electromagnetism. If we now consider a point P fixed in K, it must experience the same phase history whether described in K or the moving frame K', so that ω(t - **n**·x/c) = ω'(t - **n**'·**x**'/c'). In K', ω, **n** and c may have new values that are constant over the wavefront, which must still be plane, since we have made a linear transformation. Therefore, we may equate the coefficients of t and each component of **x**.

First, substitute **x** = **x**' + **v**t, to find ω (t - **n**·**v**/c - **n**·**x**'/c) = ω' (t - **n**'·**x**'/c'). Equating the coefficients of t, we find that ω' = ω (1 - **n**·**v**/c). Then we have left (ω/c)**n**·**x**' = (ω'/c')**n**'·**x**'. This condition is satisfied by ω/c = ω'/c' and **n** = **n**'. From the first equality we find c' = c - **n**·**v**. This means that the wave front is carried along by the component of the velocity normal to it. The component of the velocity in the wave front must then correspond to a convection of energy in that direction. We note that the direction of the wave front does not change in system K'. These deductions are confirmed by the behavior of material waves.

If the source is moving with velocity **V**, then the frequency observed in the K system is ω = ω_{o}/(1 - **n**·**V**/c). If an observer is moving with velocity **v**, then the observed frequency is ω' = ω (1 - **n**·**v**/c) = ω_{o}[(1 - **n**·**v**/c)/(1 - **n**·**V**/c)]. Both the velocities here are absolute velocities; that is, relative to the system in which the medium is at rest.

A graphical analysis of the classical Doppler effect is shown at the right. The medium is at rest in the K frame. The K' frame, moving with velocity **v**, is equivalent to a wind blowing in the -**v** direction. Consider a point a on an initial wave front. If there were no wind, this wave front would advance a distance c in unit time, and the energy at a would then be at a'. The wind convects, or carries, the wave front so that the energy from a is at a" at the end of unit time. The direction a-a" is the direction of energy propagation, which is not normal to the wave front. The wave front then moves a distance c - **n**·**v** in unit time, which is the new phase velocity c'. Since the new frequency f' is defined by f'/c' = f/c, we have f' = f(c'/c) = f(1 -**n**·**v**/c). If we want **v** to be the wind velocity instead of the velocity of K', simply change the sign of **v** in all the equations. The diagram really contains all that is in the algebraic derivation, and makes the meaning clearer.

If the source velocity is small compared to c, we can expand the denominator and find that ω' = ω_{o}[1 - **n**·(**v** - **V**)/c], approximately. In this approximation, the frequency shift depends only on the relative velocity of observer and source, and not on the absolute velocities. The first-order correction factor to the second term is 1 + **n**·**V**/c.

The speed of sound at 15°C is about 340 m/s. Let us find the speed of a moving source so that the difference in pitch when the source is approaching and receding is one full tone, or a frequency ratio of 8/9. In this case, 8/9 = (1 + v/c)/(1 - v/c), which gives v/c = 1/17, or v = 20 m/s. This is 72 kmph or 45 mph, which is easily attained.

Relativity must be taken into account for the optical Doppler effect, since light, with its velocity of c, is an essentially relativistic object. Also, light has no medium, and it is impossible to find absolute velocities as in the case of a material wave. It is, fortunately, even easier to find the relativistic Doppler effect than the classical one.

The phase is ω(t - **n**·**x**/c) = (ω/c)ct - (ω**n**/c)·**x**, and it will be a Lorentz invariant just as it is a classical invariant. The quantities (ct,**x**) are known to form a contravariant Lorentz 4-vector x^{α}. The phase is the inner product of this 4-vector with a set of four quantities (ω/c, -ω**n**/c), and the result is a Lorentz invariant. Therefore, these four quantities must transform as a covariant 4-vector k_{α}, so that φ = k_{α}x^{α} will be an invariant.

Since we know that k^{α} = (ω/c,ω**n**/c) is a contravariant 4-vector, we know that it will transform accordingly. If **n** makes an angle θ with the direction of the velocity v, we find at once that ω' = γ(ω - βωcosθ), or ω' = γω(1 - βcosθ), where β = v/c, of course. The space components give ω'cosθ' = γω(cosθ - β) and ωsinθ' = ωsinθ. Dividing the last two equations, we find that tanθ' = (1/γ)sinθ/(cosθ - β). We see that the wave front is changed in direction. Everything depends only on the relative velocity v of the coordinate system, and not on sources or observers. This is much simpler than the classical Doppler effect, and removes much uncertainty in interpretation. For small velocities v, the equation for the frequency shift is the same as in the classical effect when the velocities are small compared to c. We have derived the result for light, not for a material wave motion. This result is the one with the most applications, however, and the effect for a material wave could be found if we wanted it.

The directions of celestial bodies can be measured very accurately with a telescope. The observed direction usually differs from the true direction for several reasons, so observations must be corrected to get true directions. The largest deviation is due to atmospheric refraction, which can amount to half a degree at the horizon. Another deviation is caused by parallax, since bodies may be observed from positions that change with the diurnal and orbital motions of the earth. Since the parallax of a star can give its distance, there was a search for parallax from earliest times. However, no parallax was observed, which led to two possible conclusions. Either stars were at inconceivably large distances, or the earth did not move. The more reasonable conclusion was adopted for many years. Bessel finally succeeded in measuring the parallax of 61 Cygni in 1838. Its parallax was 0.298" of arc, a tiny angle. No star has a parallax as large as 1", and most are so small as still to be undetectable. We now mostly agree that the reason is that the stars are inconceivably distant.

About a century before Bessel's discovery, the English astronomer Bradley was looking for parallax, and was delighted to find a star near the pole of the ecliptic that seemed to have a parallax of 20" of arc. Small, yes, but easily measurable with care. Then he looked at another star, and another. All had the same parallax, 20". Those at the ecliptic poles made circular orbits, and those near the ecliptic just oscillated back and forth, as expected. This was a little disconcerting, since it meant that all the stars were at the same distance, about 10,000 radii of the earth's orbit. Bradley wondered if he would have to report the rediscovery of the firmament, to which the stars were attached, and behind which was heaven! Then he noticed that the phase of the parallactic orbit was not what was expected--the displacement was 90° behind what it should have been. It could not be parallax at all. The difference between parallax and aberration is shown in the diagram at the right.

In Bradley's time, the Newtonian corpuscular theory of light was the only popular one. Fortunately, this gave an immediate explanation for the observed displacements, which were called *aberration*. If a telescope were pointed toward a star directly upward from the earth's orbital plane, then to make the corpuscules move right down the axis of the telescope, it would have to be inclined a little in the direction of the earth's velocity, by an angle θ = tan^{-1}(v/c), where v was the orbital velocity of the earth, about 30 km/s. At the time, the speed of light was not very well known, but the known figure gave an aberrational displacement not far from that observed. In fact, aberration gave a more accurate way to find the velocity of light than any other method available at the time. With the currently accepted value for c, the aberration is 20.5" of arc. Bradley's explanation still appears in Astronomy texts, since it is clear and easy to understand. And also wrong, since it depends on Newton's corpuscular theory of light, which has been superseded.

If we turn to the classical Dopper effect, no aberration is predicted, since the wave front is not deviated, but remains parallel to itself. However, motion of the ether will drag the energy to one side, so the telescope will have to be inclined, just as in Bradley's explanation. Consider the diagram for the effect with the velocity parallel to the wavefront. The energy ray is inclined by θ = tan^{-1}(v/c) to the phase ray. This is wrong, too, since Michelson and Morley proved that the ether was dragged along with the earth, which would eliminate the aberration. This explanation does not appear in Astronomy texts since it is more difficult, and besides we all know that there is no ether.

The relativistic Doppler formulas give tan θ = γβ. Note that this θ is the angle of aberration, not the angle between the velocity and the wave normal, which is near 90°, and is the angle that appears in the formulas above. For the earth's orbital velocity γ = 1, so θ = tan^{-1}(v/c), just Bradley's result, but now found by a correct theory. More can be said about aberration, and the experiments with telescopes filled with water, but relativity really does explain everything pretty well. For an accurate evaluation of aberration, the earth's diurnal motion must be taken into account as well as the orbital motion; its contribution is around 1" of arc.

In addition to orbital motion, the earth, together with the sun and the rest of the solar system, is in steady motion through space relative to the rest of the galaxy. This causes, of course, parallactic displacements, which are the way such motion is detected. There are also aberrational displacements, but they are the same for everything, and so cannot be detected by the usual methods. This "secular" aberration can be calculated when the relative velocity with respect to the source of light is known.

There is another reason why observed positions differ from actual ones, which is simply movement that takes place while the light is on its way. This is called the *light-time correction*, and is of interest for satellites and other nearby, rapidly moving, objects. For stars, it is usually neglected.

The frequency change when θ = 90° and the light source is moving at right angles to a line from observer to source is ω' = γω, or better ω = ω_{o}/γ, where ω is the observed frequency when the frequency is ω_{o} in the rest system of the light source. This is just the effect of time dilatation, when the moving clock appears to run slow. It is a *second-order* effect, since the leading term in the velocity is proportional to β^{2}. Aberration, incidentally, is a *first-order* effect. If we take the earth's velocity of 30 km/s as typical, then β = 10^{-4} and β^{2} = 10^{-8}, which means that first-order effects are small, and second-order effects are probably undetectable. We can try higher velocities using accelerated ions and such, but second-order effects remain difficult.

The transverse Doppler shift was first seen spectroscopically in the Ives-Stilwell experiment (1938). An ingenious way to see the transverse shift is by using two-quantum spectroscopy. Here, two quanta whose energy adds up to the excitation energy of a transition, and which are oppositely-directed, are simultaneously absorbed by an atom. The first-order Doppler shifts are exactly opposite and cancel, leaving no first-order effect at all, however the atom or ion involved may be moving. This permits the detection of the second-order effect.

Another way is to use Mössbauer spectroscopy (1960). This method uses the recoilless emission of γ rays, again eliminating the first-order Doppler shift. Normally, the recoil of the nucleus after a γ ray is emitted lowers its energy by the Doppler effect, so that when it reaches another nucleus it is no longer resonant with the transition that gave it birth. If the emitting atom is in a crystal lattice, there is a certain probability that the whole lattice will absorb the recoil, not the single atom, and so the Doppler shift on recoil is negligible. An emitter is arranged so that it can be rotated at high speed around an absorber. At first, resonant absorption is found, but as the speed increases the emitted gamma rays are lowered in energy by the transverse Doppler shift, and absorption ceases.

Not only has the transverse Doppler shift been found to be real, but actual clocks have even been moved around and the time dilatation observed (1971). The apparent extension of the life of muons was first observed by Rossi (1941), and in many experiments afterwards. Clocks are also slowed in high gravitational potentials, a general-relativity effect. Relativity has had so many excellent proofs, and no contradictions, that it is on a very firm foundation. This is a distinct contrast with many of the theories flying through the air these days!

J. D. Jackson, *Classical Electrodynamics*, 2nd ed. (New York: John Wiley & Sons, 1975). Chapter 11.

*McGraw-Hill Encyclopedia of Physics*, 2nd ed. (New York: McGraw-Hill, 1993). Article "Doppler effect," pp. 311-314.

R. H. Baker, *Astronomy*, 6th ed. (New York: D Van Nostrand, 1955). A classic text that does a good job with fundamentals, such as parallax and aberration. See pp. 53-56 and pp. 300-304. Also p. 309 for Doppler displacements of spectral lines.

B. Hoffman-Wellenhof, H. Lichtenegger and J. Collins, *GPS Theory and Practice*, 2nd ed. (Wien: Springer, 1993). See pp. 82-83 and 106-109 for Doppler and relativity effects.

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Composed by J. B. Calvert

Created 8 June 2003

Last revised 6 September 2003