|PHY 514||Synchrotron Radiation: Light Sources and Cosmic Radiators||Spring 2012|
|This page: Home ⇒ Week 04|
References for synchrotron radiation: Jackson chapter 14 sections 2,3. Landau-Lifshitz (Fields) chapter 9 sections 73,74. Panofsky-Phillips chapter 20 sections 1-4. Schwartz chapter 6.
Light Source Facility Information
Synchrotron radiation has many interesting properties that can only be understood using special relativity, and many important applications summarized on page 661 of Jackson:
The first form of electromagnetic radiation studied after Maxwell's discovery was Dipole radiation discovered by Heinrich Hertz. Chapter 9 of Jackson discusses dipole radiation in the nonrelativistic limit. Dipole radiation is essentially due to acceleration in simple harmonic motion. Synchrotron radiation is essentially due to acceleration in uniform circular motion.
To study synchrotron radiation we need the Liénard-Wiechert potentials to calculate the electromagnetic fields, and the electromagnetic stress tensor to calculate the energy, flux, and power of the radiation.
The symmetric stress tensor for the electromagnetic field in vacuum is
where the field tensor has covariant components
The first term is
The component of the second term is
where the spatial index is summed over . Evaluating the first and fourth components
is the Poynting vector that measures energy flux, and
is the energy density of the electromagnetic field.
The conservation of the 4-vector flux of energy implies
is known as Poynting's theorem.
The Liénard-Wiechert 4-potential at the location of the observer due to a charge in arbitrary relativistic motion with position and 4-velocity is given by
where the subscript means that the proper time is evaluated using the causality constraint
The electric and magnetic fields are given by the Heaviside-Feynman expressions
If the observer is sufficiently far away from the accelerating charge the first Coulomb term in the electric field expression can be neglected, and the Poynting vector is given by
A neutral particle in empty space can be accelerated arbitrarily by a conservative applied force with no loss of energy. A charged particle will radiate electromagnetic energy when it is accelerated with respect to an inertial reference frame. The Larmor formula for the instantaneous power radiated in nonrelativistic motion was derived by Larmor in 1897 and its relativistic generalization was derived by Liénard in 1898:
Notice the relativistic Lorentz factor raised to the sixth power
where is the instantaneous velocity of the charge in the frame of the observer. For a relativistic elementary particle with mass and energy measured in the laboratory
A 100 GeV electron in the former LEP Collider had , and the 3.5 TeV protons in the Large Hadron Collider have . These enormous enhancement factors are entirely due to a change in reference frame!
Because the Lorentz factor depends on the reference frame of the observer so does the observed radiated power. We need a measure of radiated power that all observers can agree upon.
This is similar to defining the energy of a particle, which depends on reference frame. A definition that all observers can agree upon is the invariant mass of the particle, which is the energy measured in the rest frame of the particle. Another fundamental example: the proper time along the trajectory of the particle is the time measured by a clock moving with the particle.
So to be precise, "instantaneous power" of an accelerated particle is the energy radiated per unit proper time as measured in a reference frame in which the particle is instantaneously at rest.
In the instantaneous rest frame and the radiation electric field far enough from the source that the Coulomb term can be neglected is
The energy flux per unit solid angle through a spherical surface at retarded distance from the charge is
If is the angle between and
The total power is got by integrating over solid angle
which is the nonrelativistic Larmor formula.
To obtain the power observed in an arbitrary inertial reference frame express the acceleration in 4-vector form
where is the contravariant 4-velocity of the particle
This is a nice example where choice of appropriate reference frame and the use of relativistic covariance greatly simplifies a calculation that would be much more complicated in a general reference frame.
The problem here is that the derivation works only for a single charge. With more than one charge it will not be possible to choose a frame in which all velocities are negligible, and it is necessary to learn how to do the more general calculation.
Jackson section 14.3, Landau-Lifshitz chapter 9 sections 73-74, and Panofsky-Phillips chapter 20 explain how to find the total power by direct integration in the frame of the observer.
The electromagnetic fields and the Poynting vector are given above as functions of the spacetime position of the observer. The observer measures the energy flux through a spherical surface of radius at time , as determined by the outward normal component of the Poynting vector.
The problem is that the spherical surface is in motion as the flux is being integrated over a small interval , as nicely illustrated in the figure below from Panofsky-Phillips:
The differential power radiated is the energy in the volume element in the figure divided by the differential . To correct for this motion, let
With this correction the differential power radiated is
To obtain the total power, this must be integrated over solid angle , i.e., over the direction cosines of the normal . This integration is straightforward but complicated: Panofsky-Phillips recommend using spherical polar coordinates with the axis in the direction of ; integrate first over the azimuthal angle, and then over the polar angle. This results in Liénard's expression
which was obtained much more easily using relativistic covariance.
Jackson page 672 recommends, Pictures of Dynamic Electric Fields, Am. J. Phys. 40, 46 (1972), an article written by Roger Y. Tsien while he was a physics major in 1971 working with Edward M. Purcell.
Fig. 1 in this reference defines the coordinates of the observer in terms of the position of the charge by
Consider a charge in a circular orbit of radius
The parametric equation in the variables of the -th of lines of force emanating from the charge in a Lorentz "pincushion" pattern is
This picture includes the Coulomb field in addition to the radiation field. If the Coulomb field is removed we obtain Figure 14.7 in Jackson, which shows that the synchrotron radiation is pulsed.
Using spherical polar coordinates with the circular orbit in the plane
As the speed of the charge approaches the speed of light
which shows a strong relativistic peaking in the forward direction with an angular width of order .
Experimental observation of radiation involves the very important concept of power spectrum.
A radiation detector typically has an aperture that collects photons emitted from the radiating source. The aperture subtends a solid angle at the source. The photons are collected for some finite period of time and the energy deposited is measured as a function of time. This function of time is resolved into its Fourier frequency components. The power spectrum represents the energy deposited as a function of frequency.
The total energy deposited over the whole observation time and per unit solid angle at the location of the detector is
which defines the radiated intensity in the direction as a function of the observed frequencies .
To express the intensity in terms of the radiation field, define
The Fourier integral transform of is
Using Parseval's theorem and the reality of
It takes a few more steps (Jackson section 14.5 page 675) to substitute the asymptotic electric field, approximate
and change variable from to to show that
Evaluating this integral over time is straightforward but rather complicated. The steps are given in Jackson section 14.6.
Use spherical polar coordinates and assume the circular orbit of radius lies in the plane. The result is
where is the polar angle,
and is a modified Bessel function satisfying the modified Bessel equation
with small and large argument behavior
See Jackson chapter 3 section 3.7 on Bessel functions. Mathematica BesselK can evaluate these functions analytically and numerically.
The exponential decay of at large implies that there is a critical frequency which is conventionally defined by
above which the radiation is negligible.
The figure above from Jackson page 680 shows that the radiation is effectively confined to the plane for large .
Figure 14-11 from Jackson page 683 shows the normalized intensity
with (a) linear abscissa scale and (b) logarithmic absciss scale:
The synchrotron radiation produced by relativistic electrons gyrating in a magnetic field is one of the most important sources of high energy photons from astrophysical sources. It was originally observed in the laboratory betatron experiments in which electrons were accelerated for the first time to ultrarelativistic energies. This process is responsible for the radio emission from our Milky Way Galaxy, from Supernova remnants and extragalactic radio sources. It is also responsible for the non-thermal optical and X-ray emission observed in supernova remnants like the Crab Nebula. It may be responsible for the optical and X-ray continuum emission of Quasars.
The Crab Nebula is the debris of a supernova explosion of a massive star that collapsed producing the Crab pulsar, which is a neutron star with a period of 33 milliseconds and a huge magnetic field. Electrons spiraling along the magnetic field lines emit synchrotron radiation from radio waves to gamma rays that is detected on Earth.
I.S. Shklovskii proposed in 1953 that some of the visible light from the Crab Nebula might be due to synchrotron radiation and should therefore be strongly polarized, and the polarization was detected the in 1954. The 1957 article On the Nature of the Optical Emission from the Crab Nebula by Shklovskii is a very readable summary in which he points out that the magnetic field must be unusually strong. Ten years later the first radio Pulsar was discovered, and in the following year the Crab Pulsar was detected at the center of the Crab Nebula.
The Crab Pulsar continues to produce surprises. The article Detection of Pulsed Gamma Rays Above 100 GeV from the Crab Pulsar Science, 334, 69-72 (2011) by the VERITAS Collaboration published last October reports a new observation that cannot be explained by current pulsar models!
The following plot from Hillas et al., The Spectrum of TeV Gamma Rays from the Crab Nebula, Ap. J. 503, 744 (1998) shows the synchrotron power spectrum of the Crab Nebula observed at the Whipple Observatory by detecting Cherenkov radiation from the upper atmosphere.
Please format your submission as a single PDF file according to the instructions in the syllabus and submit it before 11:59 pm on Sunday, February 19.
Problem 1: Prove the following important properties of : (a) It has zero trace and and its square is a multiple of the unit Kronecker tensor . (b) The energy density and Poynting flux satisfy .
Problem 2: Jackson Problem 12.18. Prove, by means of the divergence theorem in four dimensions or otherwise, that for source-free electromagnetic fields confined to a finite region of space, the 3-space integrals of and transform as the components of a constant 4-vector
Use this result show that the total power radiated by a moving charge is relativistically invariant.
Jackson Problem 14.12:
Consider a charge in simple harmonic motion .
(a) Show that the instantaneous power radiated per unit solid angle is
(b) Show that the power averaged over a period of the motion is
(c) Use Mathematica to plot the angular distribution for nonrelativistic and relativistic motion.
Jackson 14.17: An electron of charge and mass moves relativistically
in a helical path with pitch angle
in a uniform magnetic field .
(a) Show that an observer far from the helix would detect radiation with fundamental frequency
with a spectrum that extends up to frequencies of order of a cutoff
(b) Show that the differential power detected by the observer is
where is the angle of observation measured relative to the particle's velocity vector
Jackson 14.26: Consider the synchrotron radiation spectrum from the Crab nebula.
Electrons with energies up to eV move in a magnetic field of the
order of T (tesla).
(a) For eV, T, calculate the orbit radius , the fundamental frequency , and the critical frequency . What is the critical photon energy in keV?
(b) Show that for a relativistic electron of energy in a constant magnetic field the power spectrum of synchrotron radiation can be written
where and tends to zero rapidly for and
where is the pitch angle of the helical path.
(c) If electrons are distributed in energy according to the spectrum
(d) Study the plots in Hillas et al., The Spectrum of TeV Gamma Rays from the Crab Nebula, Ap. J. 503, 744 (1998), and use the formulas you derived in parts (a)-(c) to estimate the quantities for the Crab Nebula.