Cyclotron & SynchrotronRadiationRybicki & LightmanChapter 6

Cyclotron and Synchrotron RadiationCharged particles are accelerated by B-fields radiation magnetobremsstrahlungCyclotron Radiation non-relativistic particles frequency of emission = frequency of gyrationSynchrotron 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 emissionSynchrotron 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:Recall4-momentumRelativistic equation of motionsee Eqn. 4.82-4.84(1)soor

(2) Let be divided intoSinceis 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 isRememberso(Larmor frequency)

Numerically, the Larmor frequency isRadius of the orbitTypical values:small on cosmic scales

Total Emitted PowerRecallperpendicular, parallel acceleration inframe where the electron is instantaneouslyat rest.In our case, the acceleration is perpendicular to the velocity:Soandwrite classicalelectronradiusFor single electron

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,ThenSoper particleor

Write it another waywhereThomson cross-sectionmagnetic energy densityFor 1.

Life time of particle of energy E is

Spectrum of Synchrotron Radiation -- Qualitative DiscussionThe 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/) 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 thewidth of the pulse of E(t) iswhere

Define CRITICAL FREQUENCYorSpectrum is broad, cutting off at frequencies >> C

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

Slightly faster

~ 1 Highly relativistictoobserver

Spectral Index for Power-Law Electron DistributionOften, the observed spectra for synchrotron sources are power lawswhere s = spectral index at least over some particular range of frequencies 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 vMaxwell-Boltzman distributionNon-thermal tail of particle velocitiesLet N(E) = # particles per vol., with energies between E, E+dEPower-law

p = spectral indexC = constant

Turns out that there is a VERY simple relation between

p = spectral index of particle energiesand s = spectral index of observed radiation

p = spectral index of particle energiesand s = spectral index of observed radiationSince can be written# particles /Vol.with energy EPower/particle with energyE, emitted at frequencywhere E1 and E2 define the range over which the power law holds.(1)

Equivalently, in terms of where(2)(3)Inserting (1) and (3) into (2),

change variables by lettingwhere

Thencan approximate x1 0, x2

Then the integral is ~constant with SoRelation between slope of power law ofradiation, s, and particle energy index, p.

Polarization of Synchrotron RadiationFirst, consider a single radiating charge elliptically polarized radiationObserverThe 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 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-AbsorptionPhoton interacts with a charge in a magnetic field and isabsorbed, giving up its energy to the chargeCan 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 As and Bs (R&L pp. 186-190)yields the absorption coefficient for synchrotron self-absorptionfor 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-offThickThin

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

Crab NebulaThe 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

IROpticalRadioX-ray(Chandra)

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

Synchrotron Lifetimes, for Crab NebulaTimescales

Guess what this is an image of?

Extragalactic radio sources: Very isotropic distribution on the sky6cm radio sourcesNorth Galactic PoleMilky Wayright ascension

Blowup ofNorthPole

VLACore of jets:flat spectrum s=0 to .3

Extended lobes:steep spectrum s = 0.7-1.2

FR I vs. FR IIOn 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-AFRII: 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