Cyclotron & Synchrotron Radiation Rybicki & Lightman Chapter 6

Cyclotron and Synchrotron Radiation Charged particles are accelerated by B-fields  radiation “magnetobremsstrahlung” Cyclotron Radiation non-relativistic particles frequency of emission = frequency of gyration Synchrotron Radiation relativistic particles frequency of emission from a single particle  emission at a range of frequencies

Astronomical Examples: (1) Galactic and extragalactic non-thermal radio and X-ray emission Supernova remnants, radio galaxies, jets (2) Transient solar events, Jovian radio emission Synchrotron emission: reveals presence of B-field, direction Allows estimates of energy content of particles Spectrum  energy distribution of electrons Jet production in many different contexts

Equation of motion for a single electron: Recall 4-momentum Relativistic equation of motion see Eqn. 4.82-4.84 (1) or so

(2) Let be divided into Since is a constant, and is a constant, is a constant

(3) Result: Helical motion - uniform circular motion in plane perpendicular to B field - uniform velocity along the field line (4) The frequency of rotation or gyration is Remember so (Larmor frequency)

Numerically, the Larmor frequency is Radius of the orbit Typical values: small on cosmic scales

perpendicular, parallel acceleration in Total Emitted Power For single electron Recall perpendicular, parallel acceleration in frame where the electron is instantaneously at rest. In our case, the acceleration is perpendicular to the velocity: So and classical electron radius write

Average over an isotropic, mono-energetic velocity distribution of electrons: i.e. all electrons have the same velocity v, but random pitch angle with respect to the B field, Then So per particle or

Write it another way where Thomson cross-section magnetic energy density For β1.

Life time of particle of energy E is

Spectrum of Synchrotron Radiation -- Qualitative Discussion The spectrum of synchrotron radiation is related to the Fourier transform of the time-varying electric field. Because of beaming, the observer sees radiation only for a short time, when the core of the beam (of half-width 1/γ) is pointed at your line of sight:

The result is that E(t) is “pulsed” i.e. you see a narrow pulse of E-field  expect spectrum to be broad in frequency

It is straight-forward to show (R&L p. 169-173) that the width of the pulse of E(t) is where

Define CRITICAL FREQUENCY or Spectrum is broad, cutting off at frequencies >> ωC

For the highly relativistic case, one can show that the spectrum for a single particle: Where F is a dimensionless function which looks like:

Transition from Cyclotron to Synchrotron Emission observer

Slightly faster 

β ~ 1 Highly relativistic to observer

Spectral Index for Power-Law Electron Distribution Often, the observed spectra for synchrotron sources are power laws where s = spectral index at least over some particular range of frequencies ω Example: on the Rayleigh-Jeans tail of a blackbody spectrum s = -2

A number of particle acceleration processes yield a power-law energy distribution for the particles, particularly at high velocities e.g. “Fermi acceleration” Maxwell-Boltzman distribution “Non-thermal” tail of particle velocities v Let N(E) = # particles per vol., with energies between E, E+dE Power-law p = spectral index C = constant

Turns out that there is a VERY simple relation between p = spectral index of particle energies and s = spectral index of observed radiation

p = spectral index of particle energies and s = spectral index of observed radiation Since can be written (1) Power/particle with energy E, emitted at frequency ω # particles /Vol. with energy E where E1 and E2 define the range over which the power law holds.

Equivalently, in terms of γ (2) (3) where Inserting (1) and (3) into (2), change variables by letting where

Then can approximate x1  0, x2  ∞ Then the integral is ~constant with ω So Relation between slope of power law of radiation, s, and particle energy index, p.

Polarization of Synchrotron Radiation First, consider a single radiating charge  elliptically polarized radiation Observer The cone of radiation  projects onto an ellipse on the plane of the sky Major axis is perpendicular to the projection of B on the sky

Ensemble of emitters with different α  emission cones from each side of line of sight cancel  partial linear polarization Frequency integrated polarization can be as high as 75% For a power-law distribution of energies, per cent polarization Linear polarization is perpendicular to direction of B

Synchrotron Self-Absorption Photon interacts with a charge in a magnetic field and is absorbed, giving up its energy to the charge Can also have stimulated emission: a particle is induced to emit more strongly in a direction and at a frequency at which there are already photons present. A straight-forward calculation involving Einstein A’s and B’s (R&L pp. 186-190) yields the absorption coefficient for synchrotron self-absorption for a power-law distribution of electrons

The Source function is simpler: Independent of p spectrum  dead give-away that synchrotron self-abs. is what is going on which is the Rayleigh-Jeans value

Summary: For optically thin emission For optically thick  Low-frequency cut-off Thick Thin

Synchrotron Radio Sources Map of sky at 408 MHz (20 cm). Sources in Milky Way are pulsars, SNe.

Radio emission of M1 = Crab Nebula, from NRAO web site The Crab Nebula, is the remnant of a supernova in 1054 AD, observed as a "guest star" by ancient Chinese astronomers. The nebula is roughly 10 light-years across, and it is at a distance of about 6,000 light years from earth. It is presently expanding at about 1000 km per second. The supernova explosion left behind a rapidly spinning neutron star, or a pulsar is this wind which energizes the nebula, and causes it to emit the radio waves which formed this image. Radio emission of M1 = Crab Nebula, from NRAO web site

IR Optical Radio X-ray (Chandra)

Crab Nebula Spectral Energy Distribution from Radio to TeV gamma rays see Aharonian+ 2004 ApJ 614, 897 Synchrotron Synchrotron Self-Compton

Synchrotron Lifetimes, for Crab Nebula Photon frequency (Hz) Electron Energy U, (eV) Electron lifetime (Yr) Radio (0.5 GHz) 5x108 3.0x108 109,000 Optical (6000A) 5x1014 3.0x1011 109 X-ray (4 keV) 1x1018 1.4x1013 2.4 Gamma Ray 1x1022 1.4x1015 0.024 = 9 days Timescales << age of Crab Pulsar is Replenishing energy

Guess what this is an image of?

Extragalactic radio sources: Very isotropic distribution on the sky 6cm radio sources right ascension Milky Way North Galactic Pole

Blowup of North Pole

VLA Core of jets: flat spectrum s=0 to .3 Extended lobes: steep spectrum s = 0.7-1.2

FR I vs. FR II On large scales (>15 kpc) radio sources divide into Fanaroff-Riley Class I, II (Fanaroff & Riley 1974 MNRAS 167 31P) FRI: Low luminosity edge dark Ex.:Cen-A FRII: High luminosity hot spots on outer edge Ex. Cygnus A

Lobes are polarized  synchrotron emission with well-ordered B-fields Polarization is perpendicular to B