# Doppler effect

The Doppler effect (rarely Doppler-Fizeau effect ) is the time compression or expansion of a signal when the distance between transmitter and receiver changes during the duration of the signal. The cause is the change in the running time . This purely kinematic effect occurs with all signals that propagate at a certain speed, usually the speed of light or the speed of sound . If the signal propagates in a medium , its state of motion must be taken into account.

In the case of periodic signals, the observed frequency increases or decreases . This applies to both pitches and modulation frequencies, e.g. B. changing the tones of a siren ( "TAATAA tatü ..."). At low speeds in relation to the speed of propagation, this ratio also indicates the relative change in frequency . In the case of a reflected signal, as in the case of radar Doppler and ultrasonic Doppler, the Doppler shift also doubles with the transit time . ${\ displaystyle {\ frac {\ Delta f} {f}}}$ ${\ displaystyle \ Delta f}$ ## history

The Doppler effect became known through Christian Doppler , who tried to convince astronomers in 1842 that this effect was the reason why color differences between the two partner stars can be seen in double stars . In his opinion, these stars orbit one another so quickly that the color of the star that has just moved away from the observer is perceived with a redshift , while the color of the converging star is shifted into the blue region of the spectrum. After Doppler's death, this effect could actually be demonstrated by measuring spectral lines . However, it is too small to explain perceptible color differences. The real cause of visually discernible color differences between stars is their temperature differences.

To explain the effect, Doppler carried out a thought experiment with the duration of water waves that are generated every minute by a moving boat. From this he also derived a mathematical description. One of Doppler's achievements is the realization that the finiteness of the speed of light must also cause a change in the wavelength of the light arriving from moving sources. In the French-speaking world, this is often attributed to Armand Fizeau (1848).

The finiteness of the propagation of light had already been interpreted 180 years earlier by Ole Rømer . Rømer was interested in the suitability of Jupiter's moons as timers for solving the longitude problem . The eclipses of Jupiter's moon Io were known to have a frequency of 1 / 1.8d, which would be well suited as a timer. However, Rømer found that this frequency decreases as the earth moves straight away from Jupiter in its orbit around the sun. That is with and extends the time from Io-Eclipse to Io-Eclipse by 1.8d / 10 000 , i.e. approx. 1/4 minute. After 40 orbits from Io around Jupiter, this delay added up to 10 minutes, which Rømer predicted for November 9, 1676. Even if Rømer was actually interested in the frequency change of the Io eclipses: He interpreted these 10 minutes much more easily than the delay that the light had needed for the correspondingly longer distance. ${\ displaystyle v _ {\ text {Orbit, Earth}} = 30 \, \ mathrm {km / s}}$ ${\ displaystyle v / c = 1/10 \, 000}$ The natural scientist Christoph Buys Ballot demonstrated the Doppler effect for sound waves in 1845. To this end, he posted several trumpeters both on a moving train and next to the railway line. As they drive past, one of them should play a G and the others determine the pitch they hear. There was a shift of a semitone, corresponding to a speed of 70 km / h.

It wasn't until twenty years later that William Huggins found the predicted spectroscopic Doppler shift in light from stars. It showed that Sirius is steadily moving away from us.

Another century later, radar measurements between Earth and Venus improved the accuracy of the astronomical unit from 10 −4 (from the horizontal parallax of Eros ) to initially 10 −6 based on distance measurements in the lower conjunctions of 1959 and 1961 (e.g. B. in JPL by amplitude modulation with up to 32 Hz), then to 10 −8 by Doppler measurements on the carrier frequencies over several months before and after the lower conjunctions of the years 1964 and 1966. The results were given as running time as 300 years earlier because the value of the speed of light was only known to six digits at that time.

Doppler measurements from 1964 to 1966 were sufficient to prove the perihelion of Mercury - with optical methods a century and a half were necessary.

## Details on the acoustic Doppler effect Wavefronts of a point source moving in relation to the medium demonstrate the dependence of the wavelength on the direction of propagation

When explaining the acoustic Doppler effect, a distinction must be made as to whether the sound source, the observer, or both move relative to the medium (still air).

### Observer at rest, signal source moving

As an example, let us assume that the ambulance s horn emits sound waves with a frequency of 1000  Hz . This means that exactly 1/1000 of a second after the first wave crest, a second wave crest follows. The waves propagate at the speed of sound at 20 ° C. ${\ displaystyle c \ approx 340 \, \ mathrm {m / s}}$ As long as the ambulance is stationary, the wavelength of the sound, i.e. the distance between the wave crests, is: ${\ displaystyle \ lambda _ {\ mathrm {S}}}$ ${\ displaystyle \ lambda _ {\ mathrm {S}} = 340 \, \ mathrm {\ frac {m} {s}} \ cdot {\ frac {1} {1000}} \, \ mathrm {s} = 0 {,} 34 \, \ mathrm {m} \ ,.}$ For an observer on the road, these wave crests arrive somewhat delayed depending on the distance. However, the time between two wave crests does not change. The fundamental frequency of the perceived sound is the same for every distance between the observer and the ambulance. ${\ displaystyle f _ {\ mathrm {S}}}$ The situation changes when the ambulance approaches the observer at high speed. Since the car continues to move in the time between the two wave crests, the distance between them shortens somewhat. It is shortened by the distance that the car covers in a time of 1/1000 of a second: ${\ displaystyle v}$ ${\ displaystyle \ lambda _ {\ mathrm {B}} = \ lambda _ {\ mathrm {S}} - {\ frac {v _ {\ mathrm {S}}} {f _ {\ mathrm {S}}}} \ ,.}$ The indices and refer to the sender or observer of the wave. Since both wave crests move towards the observer at the same speed of sound , the shortened distance between them is retained, and the second wave crest does not arrive 1/1000 of a second after the first, but a little earlier. Based on the example above, the wavelength is shortened at a speed of : ${\ displaystyle \ mathrm {S}}$ ${\ displaystyle \ mathrm {B}}$ ${\ displaystyle c}$ ${\ displaystyle 36 \, \ mathrm {\ frac {km} {h}} = 10 \, \ mathrm {\ frac {m} {s}}}$ $\lambda _{\mathrm {B} }=0{,}34\,\mathrm {m} -{\frac {10\mathrm {\frac {m}{s}} }{1000\mathrm {\frac {1}{s}} }}=0{,}33\,\mathrm {m} \,.$ This makes the frequency (i.e. the pitch) of the siren appear higher to the observer ( ): ${\ displaystyle f _ {\ mathrm {B}}> f _ {\ mathrm {S}}}$ $f_{\mathrm {B} }={\frac {c}{\lambda _{\mathrm {B} }}}={\frac {340\,\mathrm {\frac {m}{s}} }{0{,}33\,\mathrm {m} }}\approx 1030{,}3\,\mathrm {Hz} \,.$ The frequency change is obtained quantitatively simply by inserting the relationship into the above formula for . The following results for the frequency perceived by the observer : ${\ displaystyle \ lambda \ cdot f = c}$ ${\ displaystyle \ lambda _ {\ mathrm {B}}}$ ${\ displaystyle f _ {\ mathrm {B}}}$ ${\ displaystyle f _ {\ mathrm {B}} = {\ frac {f _ {\ mathrm {S}}} {1 - {\ frac {v} {c}}}} \ ,.}$ (1)

The meaning of the frequency of the sound source, the speed of propagation of the sound and the speed of the sound source (i.e. the ambulance). ${\ displaystyle f _ {\ mathrm {S}}}$ ${\ displaystyle c}$ ${\ displaystyle v}$ When the ambulance has passed the observer, the opposite is true: the distance between the wave crests (wavelength) increases and the observer hears a lower tone. In mathematical terms, the above formula also applies, you only have to use a negative speed. Referring to the example: ${\ displaystyle v}$ $\lambda _{\mathrm {B} }=0{,}34\,\mathrm {m} -{\frac {-10\,\mathrm {\frac {m}{s}} }{1000\mathrm {\frac {1}{s}} }}=0{,}35\,\mathrm {m} \qquad {\text{und somit}}\qquad f_{\mathrm {B} }={\frac {c}{\lambda _{\mathrm {B} }}}={\frac {340\,\mathrm {\frac {m}{s}} }{0{,}35\,\mathrm {m} }}\approx 971{,}4\,\mathrm {Hz} \,.$ The described movements of the signal source directly towards the observer or directly away from him are special cases. If the signal source moves anywhere in space with the speed , the Doppler shift for a stationary receiver can increase ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle f _ {\ mathrm {B}} = {\ frac {f_ {S}} {1 - {\ frac {{\ vec {v}} \ cdot {\ vec {e}} _ {\ mathrm {SB }}} {c}}}}}$ can be specified. is the time-dependent unit vector that describes the direction from the signal source to the observer . ${\ displaystyle {\ vec {e}} _ {\ mathrm {SB}}}$ ${\ displaystyle \ mathrm {S}}$ ${\ displaystyle \ mathrm {B}}$ ### Observer moved, signal source at rest

A Doppler effect also occurs when the sound source is stationary and the observer is moving , but the cause is different here: When the car is at rest, nothing changes in the distance between the wave crests, so the wavelength remains the same. However, the wave crests seem to reach the observer faster one after the other when he is moving towards the ambulance: ${\ displaystyle \ mathrm {S}}$ ${\ displaystyle \ mathrm {B}}$ {\ displaystyle {\ begin {aligned} \ lambda _ {\ mathrm {B}} & = \ lambda _ {\ mathrm {S}} \ ,, \\ (c + v) \, T _ {\ mathrm {B} } & = c \, T _ {\ mathrm {S}} \ end {aligned}}} or.

 ${\ displaystyle f _ {\ mathrm {B}} = f _ {\ mathrm {S}} \ left (1 + {\ frac {v} {c}} \ right) \ ,.}$ (2)

Here, too, there is again the case of an observer moving away through the onset of a negative speed.

For any movement of the observer with the velocity vector, the Doppler effect results when the transmitter is at rest${\ displaystyle \ mathrm {B}}$ ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle \ mathrm {S}}$ ${\ displaystyle f _ {\ mathrm {B}} = f _ {\ mathrm {S}} \ left (1 - {\ frac {{\ vec {v}} \ cdot {\ vec {e}} _ {\ mathrm { SB}}} {c}} \ right)}$ wherein in turn the unit vector describing the direction of the signal source to the observer is, the in the general case, as the velocity vector , can be time-dependent. ${\ displaystyle {\ vec {e}} _ {\ mathrm {SB}}}$ ${\ displaystyle \ mathrm {S}}$ ${\ displaystyle \ mathrm {B}}$ ${\ displaystyle {\ vec {v}}}$ As you can see, equations  (1) and  (2) are not identical (only in borderline cases do they approach each other). This becomes obvious in extreme cases: if the observer moves towards the signal source at the speed of sound, the wave crests reach him twice as fast and he hears a tone of double frequency. If, on the other hand, the signal source moves at the speed of sound, the distance between the wave crests is practically zero, they overlap and the air becomes extremely compressed (see sound barrier breakthrough ). Since all wave peaks reach the observer at the same time, this would theoretically be an infinite frequency according to the above formula - in practice one does not hear a sound of a certain frequency, but the sonic boom . ${\ displaystyle v \ ll c}$ ### Observer and signal source moved

By combining equations  (1) and  (2) one can derive an equation which describes the frequency perceived by the observer when the transmitter and the receiver are in motion. ${\ displaystyle f _ {\ mathrm {B}}}$ Sender and receiver move towards each other:

${\ displaystyle f _ {\ mathrm {B}} = f _ {\ mathrm {S}} \ cdot {\ frac {1 + {\ frac {v _ {\ mathrm {B}}} {c}}} {1- { \ frac {v _ {\ mathrm {S}}} {c}}}} = f _ {\ mathrm {S}} \ cdot {\ frac {c + v _ {\ mathrm {B}}} {c-v _ {\ mathrm {S}}}} \ ,.}$ The transmitter and receiver move away from each other:

${\ displaystyle f _ {\ mathrm {B}} = f _ {\ mathrm {S}} \ cdot {\ frac {1 - {\ frac {v _ {\ mathrm {B}}} {c}}} {1+ { \ frac {v _ {\ mathrm {S}}} {c}}}} = f _ {\ mathrm {S}} \ cdot {\ frac {c-v _ {\ mathrm {B}}} {c + v _ {\ mathrm {S}}}} \ ,.}$ Here, the speed of the observer and the speed of the transmitter of the acoustic waves relative to the medium. ${\ displaystyle v _ {\ mathrm {B}}}$ ${\ displaystyle v _ {\ mathrm {S}}}$ ### Frequency shift when scattering on a moving object

The perceived frequency can also be derived from the equations above if the wave from a stationary transmitter is scattered by an object moving with the speed and is perceived by an observer who is also stationary : ${\ displaystyle \ mathrm {S}}$ ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle \ mathrm {O}}$ ${\ displaystyle \ mathrm {B}}$ ${\ displaystyle f _ {\ mathrm {B}} = f _ {\ mathrm {S}} {\ frac {1 - {\ frac {{\ vec {v}} \ cdot {{\ vec {e}} _ {\ mathrm {SO}}}} {c}}} {1 - {\ frac {{\ vec {v}} \ cdot {\ vec {e}} _ {\ mathrm {OB}}} {c}}}} \ ,.}$ ${\ displaystyle {\ vec {e}} _ {\ mathrm {SO}}}$ and are the unit vectors from the stationary transmitter to the moving object and from the moving object to the stationary observer. ${\ displaystyle {\ vec {e}} _ {\ mathrm {OB}}}$ This equation is often used in acoustic or optical measurement technology to measure movements, e.g. B. Laser Doppler Anemometry . Especially in optics, the scattered frequency can increase as a function of the angle ${\ displaystyle \ left | {\ vec {v}} \ right | \ ll c}$ ${\ displaystyle f _ {\ mathrm {B}} \ approx f _ {\ mathrm {S}} \ left (1 + {\ frac {\ vec {v}} {c}} \ cdot ({\ vec {e}} _ {\ mathrm {OB}} - {\ vec {e}} _ {\ mathrm {SO}}) \ right)}$ can be determined in a very good approximation from the direction of illumination and direction of observation . ${\ displaystyle {\ vec {e}} _ {\ mathrm {SO}}}$ ${\ displaystyle {\ vec {e}} _ {\ mathrm {OB}}}$ ### General Doppler law for sound sources

In general, the frequency difference can be written as:

${\ displaystyle f _ {\ mathrm {B}} = f _ {\ mathrm {S}} \ left ({\ frac {c \ pm v _ {\ mathrm {B}}} {c \ mp v _ {\ mathrm {S} }}} \ right) \ ,.}$ Here, the speed of the observer and the source of sound, in each case relative to the medium (. E.g. the air). The upper operating symbol applies to approach (movement in the direction of the sender or receiver). I.e. both speeds are measured positively in the direction of the observer or transmitter. With or the special cases mentioned above arise. For the effect disappears (there is no change in pitch). This occurs when the transmitter and receiver are moving in the same direction at the same speed relative to the medium; In such cases, the medium itself usually moves while the transmitter and receiver are at rest (wind). Therefore there is no Doppler effect regardless of the wind strength. ${\ displaystyle v _ {\ mathrm {B}}}$ ${\ displaystyle v _ {\ mathrm {S}}}$ ${\ displaystyle v _ {\ mathrm {B}} = 0}$ ${\ displaystyle v _ {\ mathrm {S}} = 0}$ ${\ displaystyle v _ {\ mathrm {S}} = - v _ {\ mathrm {B}}}$ The formulas were derived on the assumption that the source and observer are moving directly towards one another. In real cases z. B. the ambulance past the observer at a certain minimum distance. Therefore, the distance between the source and the observer does not change evenly, and therefore - especially immediately before and after the drive past - a continuous transition of the pitch from higher to lower can be heard.

## Doppler effect without medium

Electromagnetic waves also propagate in a vacuum, i.e. without a medium. If the transmitter of the waves moves relative to the receiver, a shift in frequency occurs in this case too. This relativistic Doppler effect is due to the fact that the waves propagate with finite speed, namely the speed of light . It can be understood as a geometric effect of space-time.

### Longitudinal Doppler effect

In a vacuum (optical Doppler effect) the observed frequency change depends only on the relative speed of the source and observer; whether the source, the observer or both move has no influence on the amount of the frequency change.

Due to the principle of relativity, every observer can consider himself to be at rest. However, when calculating the Doppler effect, in addition to the considerations above, he must also take into account the time dilation of the source moved relative to the observer. Thus we get for the relativistic Doppler effect:

${\ displaystyle f _ {\ mathrm {B}} = {\ frac {f _ {\ mathrm {S}} {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}} } {1 - {\ frac {v} {c}}}} = f _ {\ mathrm {S}} {\ sqrt {\ frac {c + v} {cv}}}}$ ${\ displaystyle v> 0}$ when the distance between source and observer is reduced.

### Transverse Doppler Effect

If an object moves across the observer at a certain point in time, the change in distance at this point in time can be neglected; accordingly, one would not expect a Doppler effect here either. However, the theory of relativity says that every object is subject to time dilation due to its movement , due to which the frequency is also reduced. This effect is known as the transverse Doppler effect. The formula for this is

${\ displaystyle f _ {\ mathrm {B}} = {\ frac {f _ {\ mathrm {S}}} {\ gamma}} = f _ {\ mathrm {S}} {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}} \ approx f _ {\ mathrm {S}} \ left (1 - {\ frac {v ^ {2}} {2c ^ {2}}} \ right ) \ ,,}$ where here denotes the vacuum speed of light and the speed of the signal source. ${\ displaystyle c}$ ${\ displaystyle v}$ The transversal Doppler effect can, however, be neglected at non-relativistic speeds (i.e. speeds far below the speed of light ).

### Doppler effect at any angle

The Doppler effect can generally be specified depending on the angle of the direction of movement to the source-receiver axis. The frequency change for any angle results from ${\ displaystyle \ alpha}$ ${\ displaystyle f _ {\ mathrm {B}} = f _ {\ mathrm {S}} {\ frac {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}} { 1 - {\ frac {v} {c}} \ cos {\ alpha}}} \ ,.}$ If you now insert 0 ° (source moves directly towards the receiver), 90 ° (source moves sideways) or 180 ° (source moves directly away from the receiver) for the angle , then you get the above equations for longitudinal and transverse Doppler effect. It can also be seen that the angle at which the Doppler effect disappears depends on the relative speed, in contrast to the Doppler effect for sound, where it is always 90 °. ${\ displaystyle \ alpha}$ ### Doppler effect and astronomical redshift

Even if the observed effects of the Doppler effect and the astronomical redshift are identical (reduction in the observed frequency of the electromagnetic radiation of a star or a galaxy), the two must not be confused, as they have completely different causes. The relativistic Doppler effect is only the main cause of the frequency change if the transmitter and receiver move through space-time as described above and their distance is so small that the expansion of the space between them is relatively small. Above a certain distance, the proportion that is caused by the expansion of space-time itself predominates, so that the proportion of the Doppler effect discussed here can be completely neglected.

## Applications

Radial velocities can be measured by the Doppler effect if the receiver knows the frequency of the transmitter with sufficient accuracy, especially in the case of echoes from acoustic and electromagnetic signals.

### Physics and astrophysics

Sharp spectral lines allow a correspondingly high resolution of the Doppler shift. The proof of the Doppler shift in the gravitational field ( Pound-Rebka experiment ) is famous . Examples in astrophysics are the rotation curves of galaxies, spectroscopic binary stars , helioseismology and the detection of exoplanets .

In quantum optics , the Doppler shift is used in the laser cooling of atomic gases in order to reach temperatures close to absolute zero .

In Mössbauer spectroscopy , the Doppler effect of a moving gamma radiation source is used to change the energy of the photons of this source minimally. This allows these photons to interact with the nuclear hyperfine levels of a corresponding absorber.

### Radar technology

With Doppler radar , the approach speed of an object is calculated from the measured frequency change between the transmitted and reflected signal. The specialty of an active radar device, however, is that the Doppler effect can occur twice, on the way there and on the way back. A radar warning device, which receives the signals of the outward route, measures a frequency that varies depending on the relative speed. This registered frequency is reflected by it. The radar device registers the frequencies that have already been Doppler shifted as a function of the then existing relative speed. In the case of an unaccelerated radar device, an exactly two-fold Doppler shift occurs.

### Medical diagnostics

In medicine , the acoustic Doppler effect is used in ultrasound examinations to display and measure the blood flow velocity. He has proven to be extremely helpful in doing so. There is one:

• Color Doppler :
• Red: flow towards the probe
• Blue: flow away from the probe
• pW Doppler : pulsed Doppler (e.g. for vascular examinations)
• cW Doppler : continuous wave Doppler (e.g. for heart valve measurements)

### Laser doppler

Laser Doppler anemometry (LDA) is used for the non-contact measurement of the velocity distribution of fluids (liquids and gases) . It is based on the optical Doppler effect on scattering particles in the flow. In the same way, a vibrometer is used to measure the speed of vibrating surfaces.

### Other uses

• For water waves (gravity waves), the carrier medium of which is subject to a constant flow velocity, see wave transformation .
• The satellite navigation system Transit , which has now been switched off , used the Doppler effect to determine position. It is actively used in Argos , a satellite-based system for determining position. In modern GNSS satellites, the Doppler effect is initially disruptive. It forces the receiver to search a larger frequency range. On the other hand, additional information can be obtained from the frequency shift and thus the rough positioning can be accelerated. The procedure is called Doppler aiding . See also: Doppler satellite .
• In music , the Doppler effect is used to create sound effects, for example in the rotating loudspeakers of a Leslie cabinet .

## Doppler effect in biological systems

During the segmentation of vertebrate - embryos running waves of gene expression by the paraxial mesoderm , the tissue from which the precursor of the vertebral body ( somites are formed). With each wave that reaches the anterior end of the presomitic mesoderm, a new thus is formed. In zebrafish , the shortening of the paraxial mesoderm during segmentation has been shown to cause a Doppler effect as the anterior end of the tissue moves into the waves. This Doppler effect contributes to the speed of segmentation.

## example Dependence of the frequency of a signal source on the distance to a stationary observer for different minimum distances.

A resting observer hears a sound source moving exactly towards him with the frequency , see equation  (1) , if it moves away from him, with the frequency , see equation  (2) . The relativistic transversal Doppler effect is irrelevant for sound sources. The further the observer is from the linear trajectory, the slower the radial velocity component changes on approach. The speed of the frequency change depends on the shortest distance between the observer and the signal source. The diagram on the right shows the frequency dependency relative to an observer resting at the origin. The red line corresponds to the frequency that he hears when the signal source passes him at a great distance, the blue line that at a short distance. The maximum and minimum frequencies are not symmetrical to the natural frequency, since the speed is not much smaller than the speed of sound . The relationships (1) and (2) apply. ${\ displaystyle f '_ {\ mathrm {zu}} (v / c)}$ ${\ displaystyle f '_ {\ mathrm {weg}} (v / c)}$ ${\ displaystyle v}$ ${\ displaystyle c}$ If the coordinates of the moving signal source are known, you can deduce your own location from the frequency curve (see e.g. Transit (satellite system) ).

The sound samples give the pitches that a resting observer hears when a signal source flies past him. They neglect the effect that the retreating source can be heard longer than the approaching one:

Frequency , relative speed (then and ):${\ displaystyle f_ {0} = 400 \, \ mathrm {Hz}}$ ${\ displaystyle v / c = 0 {,} 1}$ ${\ displaystyle f _ {\ mathrm {zu_ {max}}} = 440 \, \ mathrm {Hz}}$ ${\ displaystyle f _ {\ mathrm {weg_ {min}}} = 360 \, \ mathrm {Hz}}$ (1) Doppler example 1 ? / i Slowly moving signal source that observers pass by at a short distance.
(2) Doppler example 2 ? / i : like (1), but passing the signal source at a greater distance.
(3) Doppler example 3 ? / i : like (2), distance even greater.

If the relative speed increases, the frequencies shift:

Frequency as above, but (then is ).${\ displaystyle v / c = 0 {,} 42}$ ${\ displaystyle f _ {\ mathrm {zu_ {max}}} = 690 \, \ mathrm {Hz}, \ f _ {\ mathrm {weg_ {min}}} = 280 \, \ mathrm {Hz}}$ (4) Doppler example 2b ? / i : distance like (2).

## Trivia

When planning the Cassini-Huygens space mission, it was not considered that the radio traffic between the two subsystems Cassini and Huygens is subject to a frequency shift due to the Doppler effect. Simulating tests were only carried out during the trip, too late to correct the cause, a phase-locked loop that was parameterized too stiffly . Various measures in the area of ​​the error were able to reduce the expected data loss from 90% to 50%. In addition, the flight path of the mission was changed in order to completely avoid data loss due to this error.

## literature

• David Nolte: The fall and rise of the Doppler effect , Physics Today, March 2020, pp. 30–35

## Web links

Commons : Doppler Effect  - album with pictures, videos and audio files
Wikisource: Christian Doppler's basic work on the Doppler effect  - sources and full texts

## Individual evidence

1. ^ Arnold Sommerfeld: Lectures on Theoretical Physics: Optics. Akad. Verlag, Leipzig, 1949, p. 54.
2. a b Christian Pinter: Mistake with serious consequences ( Memento of the original from October 29, 2012 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. , Wiener Zeitung June 5, 2011.
3. ^ Alan P. Boss: The Crowded Universe: The Search for Living Planets. Basic Books, 2009, ISBN 978-0-465-00936-7 , limited preview in Google Book Search.
4. James H. Shea: Ole Rømer, the speed of light, the apparent period of Io, the Doppler effect, and the dynamics of Earth and Jupiter . In: Am. J. Phys. tape 66 , no. 7 , 1998, pp. 561-569 .
5. ^ RM Goldstein: Radar Exploration of Venus. NASA JPL report JPL-TR-32-280, 1962.
6. a b Michael E. Ash, Irvine I. Shapiro, William B. Smith: Astronomical constants and planetary ephemerides deduced from radar and optical observations. In: Astr.J. 72, 1967, p. 338. (online)
7. Special theory of relativity, arguments for deriving the most important statements, effects and structures , Franz Embacher, University of Vienna.
8. Paul Peter urons, Roger Hinrichs: University Physics III - Optics and Modern Physics (OpenStax) . libretexts.org, Doppler Effect for Light, Eq. (5.7.4) ( libretexts.org ). ; Sign convention reversed in the source: when the distance is reduced.${\ displaystyle v <0}$ 9. ^ D. Soroldoni, DJ Jörg, LG Morelli, D. Richmond, J. Schindelin, F. Jülicher, AC Oates (2014): A Doppler Effect in Embryonic Pattern Formation. In: Science . Volume 345, pp. 222-224. PMID 25013078 .
10. ^ Leslie J. Deutsch ( JPL Chief Technologist): Resolving the Cassini / Huygens Relay Radio Anomaly. (PDF; 0.8 MB) (No longer available online.) 2002, archived from the original on May 26, 2010 ; accessed on June 4, 2014 .