Section 34–6 The Doppler effect

(Moving source / Moving observer / Relativistic invariance)

 

In this section, Feynman discusses the Doppler effect from the perspective of moving source, moving observer, and relativistic invariance. This section could be titled as “Derivations of relativistic Doppler effect” because there are three different derivations of relativistic Doppler effect involving a source or an observer that is moving at a relativistic speed. The term Doppler-Fizeau effect is sometimes used because Fizeau discovered independently the same phenomenon on electromagnetic waves in 1848.

 

1. Moving source:

“The shift in frequency observed in the above situation is called the Doppler effect: if something moves toward us the light it emits appears more violet, and if it moves away it appears more red (Feynman et al., 1963, p. 34–7).”

 

Feynman mentions that the shift in frequency observed for a moving source is called the Doppler effect. Perhaps he could have adopted the terms, e.g., classical Doppler effect, relativistic Doppler effect, and longitudinal Doppler effect. Firstly, one may describe the shift in frequency for a source that is moving toward or away from an observer as longitudinal Doppler effect. This is different from the transverse Doppler effect whereby the source is moving perpendicularly to the line joining the source and observer. Furthermore, one may explain that the relativistic Doppler effect formula is contributed by the classical Doppler effect and time dilation. However, the Doppler effect for light waves and sound waves are not quite the same because light waves can travel in vacuum, but sound waves would need a medium.

 

“In a given amount of time τ, when the oscillator would have gone a distance vτ, on the x′ vs. ct diagram it goes a distance (c−v)τ. So all the oscillations of frequency ω₁ in the time Δτ are now found in the interval Δt = (1−v/c)Δτ; they are squashed together, and as this curve comes by us at speed c, we will see light of a higher frequency, higher by just the compression factor (1−v/c) (Feynman et al., 1963, p. 34–7).”

 

Feynman suggests an interesting way to understand the relativistic Doppler effect using the x′ vs. ct diagram. He explains the equation Δt = (1−v/c)Δτ in which the compression factor accounts for the classical Doppler effect due to the relative motion between the source and observer. When the source is moving toward or away from the observer, this factor may lead to an apparent compression or stretching of the waves. On the other hand, the Lorentz factor (γ) in the relativistic Doppler effect formula accounts for the time dilation due to the source moving at a relativistic velocity. When the source moves at a velocity close to the speed of light, time dilation becomes significant, affecting the apparent rate at which time flows for the moving object as compared to the stationary observer. Thus, the compression factor increases the observed frequency, but the Lorentz factor reduces the rate of flow of time, and thus the observed frequency.

 

2. Moving observer:

“Is the frequency that we would observe if we move toward a source different than the frequency that we would see if the source moved toward us? Of course not! The theory of relativity says that these two must be exactly equal (Feynman et al., 1963, p. 34–8).”

 

According to special relativity, the observed frequency must be exactly equal whether an observer is moving toward a source, or vice versa, at the same speed. Specifically, this is based on the first postulate of special relativity which states that the laws of physics are the same in all inertial reference frames. It implies the equivalence of inertial frames, which means that observers within a box moving at a constant speed cannot determine their absolute velocity by any experiment. However, the Doppler effect for sound waves is not the same for a moving source or moving observer at the same speed with respect to the air medium or wind velocity. The presence of the air medium or the influence of wind creates an asymmetry in the measured frequency, resulting in different Doppler shifts for the moving source and moving observer in various scenarios.

 

“If we were expert enough mathematicians we would probably recognize that these two mathematical expressions are exactly equal! In fact, the necessary equality of the two expressions is one of the ways by which some people like to demonstrate that relativity requires a time dilation, because if we did not put those square-root factors in, they would no longer be equal (Feynman et al., 1963, p. 34–8).”

 

It seems to be a humor that it would require mathematicians to recognize that the two mathematical expressions are exactly equal. To simplify the mathematical expressions, we may let b = v/c. Thus, a simple trick involved in the derivation is to apply the product of the two square roots Ö(1 - b)Ö(1 + b) = Ö1 - b² = Ö(1 – v²/c²). With this in mind, we may express (1 + b)/Ö(1 - b²) as (1 + b)/Ö(1 - b)Ö(1 + b) and simplify it to Ö(1 + b)/Ö(1 - b) or Ö(1 - b²)/(1 - b).

In short, Ö(1 + b)/Ö(1 - b) = (1 + b)/Ö(1 - b²) by multiplying both sides by Ö(1+b). Alternatively, Ö(1 + b)/Ö(1 - b) = Ö(1 - b²)/(1 - b) by multiplying both sides by Ö(1-b).

This mathematical exercise is unnecessary if Feynman simply shows that the expression Ö(1 - b²)/(1 - b) in 34.12 is also equal to Ö(c + v)/(c - v).

  

3. Relativistic invariance:

“But what would a man in motion, observing the same physical wave, see? Where the field is zero, the positions of all the nodes are the same (when the field is zero, everyone measures the field as zero); that is a relativistic invariant (Feynman et al., 1963, p. 34–8).”

 

Perhaps explaining the relativistic invariance of the phase of a wave using nodes is inappropriate because it can lead to misunderstanding. The concept of nodes is used to describe stationary points in wave patterns, such as standing waves, where the amplitude is always zero (See figure below). Although the nodes are located at the same positions for an inertial observer, they are formed by the superposition of two oppositely traveling waves that are always anti-phase with each other. Feynman’s statement about nodes being relativistic invariant is also misleading because the positions of nodes vary in accordance with Lorentz contraction. The crux of the matter is about the phase of a wave instead of nodes that are due to two waves. (Feynman gives a better explanation on the relativistic invariance of the phase of a wave in the next section.)

Modified from 9.5 Superposition and Interference – review – Douglas College Physics 1207 (bccampus.ca)

 

In Jackson’s (1999) words, “The phase of a wave is an invariant quantity because the phase can be identified with the mere counting of wave crests in a wave train, an operation that must be the same in all inertial frames (p. 529).” However, the phase of a wave may refer to a crest that is the same for all inertial observers, but it is not necessarily determined by the counting operation. Instead of counting, there could be a reference when making an observation on the absolute phase or relative phase of a wave. The wave’s frequency and wavelength may vary due to relativistic effects, but the phase difference between any two points remains the same for all inertial observers. In essence, the relativistic invariance of the phase is the invariance of the scalar product of two four-vectors k_μx_μ that is a scalar. The constantly varying combination of k and ω as well as space (x) and time (t) in the phase (kx - wt) ensures that it remains invariant under the Lorentz transformation.

 

Review Questions:

1. Would you use the term Doppler effect, classical Doppler effect, or relativistic Doppler effect? Should Feynman mention "blue shift" instead of "violet shift"?

2. How would you explain the observed frequency must be exactly equal if an observer is moving toward a source, or vice versa, at the same speed?

3. Would you explain relativistic invariance using the concept of nodes that are formed by the superposition of two waves (instead of the phase of a wave)?

 

The moral of the lesson: The relativistic Doppler effect is an extension of the classical Doppler effect, taking into account the effects of both the compression factor (classical Doppler effect) and the Lorentz factor (time dilation) from special relativity.

 

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons, New York.

Section 34–5 Bremsstrahlung

  (Radiation direction / Radiation mechanism / Radiation spectrum)

 

In this section, Feynman discusses the direction of radiation, radiation mechanism, and radiation spectrum pertaining to bremsstrahlung. The term bremsstrahlung originated from the early days of radiation research when physicists observed that high-energy electrons, when passing through matter, appeared to slow down as they emitted radiation, including X-rays. In a sense, bremsstrahlung is a misnomer (Melrose, 1980) because it does not simply refer to the process of slowing down, but rather to the emission of bremsstrahlung radiation caused by the scattering or deflection of electrons.

 

1. Radiation direction:

“So when very energetic electrons move through matter they spit radiation in a forward direction. This is called bremsstrahlung (Feynman et al., 1963, p. 34–6).”

 

Perhaps there is a little humor when Feynman says that electrons move through matter and they spit radiation in a forward direction. In Chapter 28, Feynman explains: “our formula said that the field should be the acceleration of the charge projected perpendicular to the line of sight... So that checks the first rule, that there is no effect when the charge is moving directly toward us.” Thus, the rule could be succinctly expressed as “energy is most strongly radiated perpendicular to the acceleration (Hecht, 2002, p. 60).” For example, if the electron is accelerating in a straight line, the energy radiated is strongest in the directions perpendicular to the acceleration and there is no radiation in the forward direction. It is a coincidence that the electron in circular motion radiates energy in the forward direction because the direction of radiation is perpendicular to the centripetal acceleration.

 

Feynman could have provided a diagram to show the direction of bremsstrahlung radiation. Many diagrams of bremsstrahlung are misleading, e.g., some textbooks and websites seem to suggest only one photon can be radiated or there is only one direction of radiation (See figure below). However, these diagrams do not show that electrons move through matter and spit radiation in a forward direction. Importantly, it is an idealization to consider energy is radiated only in the forward direction. If the speed of a circulating electron is increased gradually, the backward lobe (radiation pattern) will shrink and the forward lobe will elongate in the direction of motion. That is, the electron moving near the speed of light would radiate more energy along a narrower lobe (smaller solid angle) in the forward direction.

 

2. Radiation mechanism:

“Suppose that there are charged particles in a piece of matter and a very fast electron, say, comes by (Fig. 34–9). Then, because of the electric field around the atomic nucleus the electron is pulled, accelerated, so that the curve of its motion has a slight kink or bend in it (Feynman et al., 1963, p. 34–6).”

 

The bremsstrahlung is due to the deflection or scattering of electrons by electric fields of nuclei and it results in a decrease of kinetic energy of electrons. Feynman analyzes the radiation from a kinematical perspective, i.e., the curve of its motion has a slight kink or bend in it. It is worth mentioning that the exact path of an individual electron during the bremsstrahlung phenomenon cannot be directly observed during the process. Essentially, the bremsstrahlung radiation is emitted in various directions and high energy electrons could have interacted with each other, or electric fields of many nuclei and their surrounding electrons. The collective behavior of a large number of electrons undergoing bremsstrahlung has been studied by analyzing the radiation and its characteristics, such as energy distribution and angular distribution.

 

“Remember our rule: we take the actual motion, translate it backwards at speed c, and that gives us a curve whose curvature measures the electric field. It was coming toward us at the speed v, so we get a backward motion, with the whole picture compressed into a smaller distance in proportion as c−v is smaller than c. So, if 1−v/c << 1, there is a very sharp and rapid curvature at B′, and when we take the second derivative of that we get a very high field in the direction of the motion (Feynman et al., 1963, p. 34–6).”

 

Feynman explains that there is a very sharp curvature at B′ by visualizing how the whole picture is compressed into a smaller distance in proportion as c−v is smaller than c provided 1−v/c ≪ 1. We can relate this to the classical Doppler effect for an electron, which has a factor of 1/(1−v/c). If the electron is moving close to the speed of light, we expect the effect of time dilation (relativistic Doppler effect) and more intense radiation, such as x-ray. (Coincidentally, Feynman needs to derive the formula for relativistic Doppler effect in the next section.) The forward and backward lobe of an electron are related to the relativistic Doppler effect and dependent on the frame of reference as shown below. In essence, the blue-shifted radiation (shorter l) emitted in the forward direction has a narrower and longer lobe, whereas the red-shifted radiation (longer l) emitted in the backward direction has a broader and shorter lobe.

 

3. Radiation spectrum:

“As a matter of fact, the synchrotron is used, not so much to make high-energy electrons (actually if we could get them out of the machine more conveniently we would not say this) as to make very energetic photons—gamma rays—by passing the energetic electrons through a solid tungsten ‘target,’ and letting them radiate photons from this bremsstrahlung effect (Feynman et al., 1963, p. 34–7).”

 

It is unclear why Feynman suggests the use of a synchrotron and passing of energetic electrons through a solid tungsten target to make very energetic photons—gamma rays. Firstly, the gamma rays can be directly produced in a synchrotron due to the deflection of high energy electrons under alternating magnetic fields (without the use of a tungsten target). Secondly, the collisions of high energy electrons and some photons may undergo the inverse Compton effect, also resulting in the production of gamma rays. On the other hand, gamma rays are commonly identified as more energetic radiation emitted from radioactive materials or nuclear processes. Alternatively, some may explain how the interaction of high energy electrons (without the use of a synchrotron) with the electrons of tungsten nuclei can create X-rays instead of gamma rays.

 

The continuous distribution of X-rays, which forms the base for the two sharp peaks is called bremsstrahlung. That is, a synchrotron can generate a bremsstrahlung spectrum, which represents a continuous range of electromagnetic radiation emitted by high-energy electrons when they are scattered. The use of a tungsten target can modify the bremsstrahlung spectrum by introducing characteristic spikes in addition to the continuous spectrum (See figure below). Feynman’s sentence on the use of a tungsten target could be revised as follows: “A synchrotron is capable of generating a bremsstrahlung spectrum, which represents a continuous range of radiation emitted by high-energy electrons. When a solid tungsten target is employed, the bremsstrahlung spectrum can exhibit characteristic spikes superimposed on the continuous spectrum.”

Source: https://blog.3ds.com/brands/simulia/x-ray-tube-simulation-part-1/

 

Note: In Chapter 2, Feynman says “[t]hese two terms, x-rays and gamma rays, are used almost synonymously. Usually electromagnetic rays coming from nuclei are called gamma rays, while those of high energy from atoms are called x-rays, but at the same frequency they are indistinguishable physically, no matter what their source (p. 2-5).”

 

The physicist who provided a better explanation of bremsstrahlung is Hans Bethe, Feynman’s immediate boss in Los Alamos. Bethe developed a comprehensive theory of bremsstrahlung emission with Heitler, known as Bethe-Heitler theory of bremsstrahlung. In Bethe-Heitler (1934) words, “[the] stopping power of matter for fast particles is at present believed to be due to three different processes: (1) the ionization; (2) the nuclear scattering; (3) the emission of radiation under the influence of the electric field of a nucleus (p. 83).” There is a screening effect whereby the presence of other electrons in the atoms can modify the effective interaction between the high speed electron and the nucleus. Schwinger suggested an additional effect, the interaction of electrons back on the field after reading Bethe-Heitler theory (Mehra & Milton, 2000). Bethe dismissed the idea by pointing out that the interaction operator was non-Hermitian and thus unphysical.

 

Review Questions:

1. Do you agree with Feynman’s explanation that electrons moving through matter would spit radiation in a forward direction?

2. Do you agree with Feynman that the exact path of an individual electron during the bremsstrahlung phenomenon usually has a slight kink or bend in it?

3. Do you agree with Feynman that a synchrotron (tungsten filament?) and tungsten target should be used to generate gamma rays (X-rays?)?

 

The moral of the lesson: Bremsstrahlung is characterized by a continuous range of spectrum (secondary radiation), which is most strongly radiated perpendicular to its acceleration, due to the scattering of electrons (primary radiation) by electric fields of nuclei.

 

References:

1. Bethe, H., & Heitler, W. (1934). On the stopping of fast particles and on the creation of positive electrons. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 146(856), 83-112.

2. Eberhardt, W. (2015). Synchrotron radiation: A continuing revolution in X-ray science—Diffraction limited storage rings and beyond. Journal of Electron Spectroscopy and Related Phenomena, 200, 31-39.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Hecht, E. (2002). Optics (4th edition). San Francisco: Addison Wesley.

5. Mehra, J. & Milton, K. A. (2000). Climbing the Mountain: The Scientific Biography of Julian Schwinger. Oxford: Oxford University Press.

6. Melrose, D. B. (1980). Plasma astrohysics. Nonthermal processes in diffuse magnetized plasmas - Vol. 1: The emission, absorption and transfer of waves in plasmas; Vol. 2: Astrophysical applications. New York: Gordon and Breach.

7. Walker, J., Resnick, R., & Halliday, D. (2014). Halliday and Resnick fundamentals of physics. New Jersey: Wiley.

Section 34–4 Cosmic synchrotron radiation

 (Continuous spectrum / Light polarization / Magnetic field)

In this section, Feynman discusses the continuous spectrum, light polarization, and magnetic field of the Crab Nebula. With the use of space-based telescopes and new observatories (e.g., James Webb Space Telescope, Hubble Space Telescope, and Chandra X-ray Observatory) that provide high-resolution imaging, spectroscopic data, and polarization measurements, the discussions in this section are outdated to a certain extent.

 

1. Continuous spectrum:

“On the outside is a big mass of red filaments, which is produced by the atoms of the thin gas “ringing” at their natural frequencies; this makes a bright line spectrum with different frequencies in it. The red happens in this case to be due to nitrogen (Feynman et al., 1963, p. 34–6).”

Feynman’s statement about the red filaments of the Crab Nebula being due to nitrogen is outdated. The word “filament” has been traditionally used to refer to the macroscopic structures of the Crab Nebula observed by ground-based telescopes (Sankrit et al., 1998). With the higher resolution of Hubble Space Telescope, it magnifies the previously known filaments into many smaller substructures. By astronomy conventions, the optical features seen in the synchrotron nebula are called wisps, whereas the structures observed in the light of emission lines from thermal gas are now known as filaments (Hester, 2008). The filaments or wisps are quasi-stationary on time-scales of a few days or longer because they may move outwards at relativistic speeds.

“On the other hand, in the central region is a mysterious, fuzzy patch of light in a continuous distribution of frequency, i.e., there are no special frequencies associated with particular atoms. Yet this is not dust “lit up” by nearby stars, which is one way by which one can get a continuous spectrum. We can see stars through it, so it is transparent, but it is emitting light (Feynman et al., 1963, p. 34–6).”

Feynman explains that the continuous spectrum of Crab Nebula is not due to the dust “lit up” by nearby stars because we can see stars through it. However, the absorption spectrum of Crab Nebula is also reported (e.g., Sollerman et al., 2000), but it could be related to the interstellar dust or supernova remnants. Furthermore, the Crab Nebula’s spectrum has been studied from radio waves to gamma rays after the discovery of pulsating radio sources near the Crab Nebula (Staelin & Reifenstein, 1968). In 2021, there have been observations of light particles with energies exceeding a quadrillion electron volts (1 PeV) from the Crab Nebula (Cao et al., 2012). More research and observations are needed to refine and improve our understanding of the emission processes within the nebula.

The Crab Nebula seen in radio, infrared, visible light, ultraviolet, X-rays, and gamma-rays (8 March 2015).
 Image Credit: By Based on File:Crab Nebula in multiwavelength.png by Torres997: Public domain, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=38800932

2. Light polarization:

“…in this case, also, polarizers have been put on the telescope, and the two views correspond to two orientations 90∘ apart. We see that the pictures are different! That is to say, the light is polarized. The reason, presumably, is that there is a local magnetic field, and many very energetic electrons are going around in that magnetic field (Feynman et al., 1963, p. 34–6).”

Feynman suggests that the observation of polarized light is due to the presence of a local magnetic field and it results in many electrons going around the magnetic field. On the contrary, one may consider the cause to be the rotation of charged particles (including electrons) and it generates the magnetic field. Theoretically, there are also particles accelerated to relativistic speeds by the shock wave created by the supernova, and they emit radiation as they interact with the gas and dust in the nebula. Currently, the source of magnetic field is still a mystery, but it can be related to the rotating neutron star. Feynman did not specify the magnetic field is toroidal, but this was proposed by Rees and Gunn (1974) more than 10 years after this lecture of Feynman was delivered.

“Putting these two facts together, we see that in a region where one picture is bright and the other one is black, the light must have its electric field completely polarized in one direction (Feynman et al., 1963, p. 34–6).”

It may not seem appropriate for Feynman to say that light must have its electric field completely polarized in one direction. The degree of polarization of electromagnetic radiation emitted by the Crab Nebula varies with the wavelength of the radiation. For example, the average degree of X-ray polarization in the Crab Nebula is low, typically measured to be only around 20% (Bucciantini et al., 2021). On the other hand, the maximum degree of polarization could be as high as 45-50% in certain regions, but the overall polarization indicates a toroidal magnetic field (See figure below). However, instrumental limitations, calibration uncertainties, and background noise contribute to the observed polarization and introduce systematic errors that may impact the measured degree of polarization.

Diagram schematically representing the morphological features observed in the Crab Nebula in the optical and X-ray bands (Cerutti & Giacinti, 2021)

3. Magnetic field:

“This means that there is a magnetic field at right angles to this direction, while in other regions, where there is a strong emission in the other picture, the magnetic field must be the other way. If we look carefully at Fig. 34–8, we may notice that there is, roughly speaking, a general set of “lines” that go one way in one picture and at right angles to this in the other. The pictures show a kind of fibrous structure (Feynman et al., 1963, p. 34–6).”

The fibrous structures in the Crab Nebula are now interpreted as showing the local direction of the magnetic field (Hester, 2008). Based on NASA research, the polarization pattern shows that the Crab Nebula’s magnetic field is donut-shaped as shown below. In short, astrophysicists analyze the nature of Crab Nebula using the observed light polarization and inferred magnet field. Specifically, the Crab Nebula has a complex magnetic field structure, i.e., the magnetic field lines within the nebula could be twisted and tangled, leading to variations in their orientation and strength across different regions. However, the interpretations of magnetic fields in the Crab Nebula are dependent on theoretical models, such as the synchrotron model, magnetic dipole model, pulsar wind model, or core-collapse supernova model.

Credits: Magnetic field lines: NASA/Bucciantini et al; X-ray: NASA/CXC/SAO; Optical: NASA/STScI; Infrared: NASA-JPL-Caltech

“What keeps the electron energy so high for so long a time? After all, it is 900 years since the explosion—how can they keep going so fast? How they maintain their energy and how this whole thing keeps going is still not thoroughly understood (Feynman et al., 1963, p. 34–6).”

Feynman did not use the term supernova or relate the magnetic field and electron energy (or radiated power) to the supernova remnants. In his advanced lectures on gravitation, Feynman questioned whether the radiated power of a radio source could be due to annihilation of 10⁶ to 10⁸ stars or nuclear processes. In Feynman’s words, “[e]ven exploding all the stars in an ordinary galaxy would hardly produce that much power. The only way to have such power radiated away from luminous objects would seem to be to have a million stars annihilate a million stars of antimatter. Alternative explanations involve some kind of structure at the center of these galaxies, some monster superstars in which the generation of energy follows paths very different from those of the ordinary star… (Feynman et al., 1995, pp. 186-187).” Feynman suggested that the preferred direction of nuclear processes might be toward inverse beta decays of the protons, i.e., p + e ® n + n, at sufficiently high pressures.

 

Review Questions:

1. Do you agree with Feynman that the continuous spectrum of Crab Nebula is not due to the dust “lit up” by nearby stars?

2. Do you agree with Feynman that the light emitted by the Crab Nebula must have its electric field completely polarized in one direction?

3. How would you describe the magnetic field of Crab Nebula?

 

The moral of the lesson: The degree of light polarization varies with the wavelength and location of radiation emitted by the Crab Nebula, but it is related to the direction and intensity of the magnetic field.

 

References:

1. Bucciantini, N., Ferrazzoli, R., Bachetti, M. et al. (2023). Simultaneous space and phase resolved X-ray polarimetry of the Crab pulsar and nebula. Nat Astron 7, 602–610.

2. Cao, Z., Aharonian, F., An, Q., Axikegu, Bai, L. X., ... & Qi, M. Y. (2021). Peta–electron volt gamma-ray emission from the Crab Nebula. Science, 373(6553), 425-430.

3. Cerutti, B., & Giacinti, G. (2021). Formation of giant plasmoids at the pulsar wind termination shock: A possible origin of the inner-ring knots in the Crab Nebula. Astronomy & Astrophysics, 656, A91.

4. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

5. Feynman, R. P., Morinigo, F. B., & Wagner, W. G. (1995). Feynman Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.

6. Hester, J. J. (2008). The Crab Nebula: an astrophysical chimera. Annu. Rev. Astron. Astrophys., 46, 127-155.

7. Rees, M. J., & Gunn, J. E. (1974). The origin of the magnetic field and relativistic particles in the Crab Nebula. Monthly Notices of the Royal Astronomical Society, 167(1), 1-12.

8. Sankrit, R., Hester, J. J., Scowen, P. A., Ballester, G. E., Burrows, C. J., Clarke, J. T., ... & Westphal, J. A. (1998). WFPC2 studies of the Crab nebula. II. Ionization structure of the Crab filaments. The Astrophysical Journal, 504(1), 344-358.

9. Sollerman, J., Lundqvist, P., Lindler, D., Chevalier, R. A., Fransson, C., Gull, T. R., ... & Sonneborn, G. (2000). Observations of the Crab Nebula and Its Pulsar in the Far-Ultraviolet and in the Optical. The Astrophysical Journal, 537(2), 861-874.

10. Staelin, D. H., & Reifenstein III, E. C. (1968). Pulsating radio sources near the Crab Nebula. Science, 162(3861), 1481-1483.