Cyclotron & SynchrotronRadiation

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

p

cEp

/ 4-momentum

Relativistic equation of motion

d

dpam 0 see Eqn. 4.82-4.84

0v

vv

2

Eqmcdt

dE

dt

d

Bc

qm

dt

dp

dt

d

(1) 0dt

dso constant or constant v

(2) Let v

be divided into

B tonormal vel.v

B toparallel vel.v||

constantv 0v

||||

dt

d

Bmc

q

dt

d

v

v

Since v

is a constant, and ||v

is a constant,

v

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

mc

eBB

mc

eBcyclotron

cyclotronB Remember so

(Larmor frequency)

Numerically, the Larmor frequency is

€

ν cyclotron = 2.8 B1G MHz

Radius of the orbit

cmB

ER

G

GeV

cyclotron 1

1710v

Typical values:

AUcmRGeVEGB 71010 ,10 146

small on cosmic scales

Total Emitted Power

Recall 2||

2243

2

3

2aa

c

qP

perpendicular, parallel acceleration inframe where the electron is instantaneouslyat rest.

In our case, the acceleration is perpendicular to the velocity:

Bmc

q

dt

da

v

v

So vBa and 2

222

224

3

2

v 3

2

cm

Bq

c

qP

write 2

2

0 cm

er

e

classicalelectronradius

c

v ergs/s 106.1 22215

2222

32

B

BcrP o

For 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,

€

α ≡cos−1

r β ⋅

r β

β 2

Then2222

032

α

BcrP

€

⊥2

α=

dΩ β sinα( )2∫

dΩ∫=

β 2

4πdΩsin2 α =

β 2

4π

8π

3=

2

3β 2∫

€

P = 23( )

2r0

2cβ 2γ 2B2So per particle

or electronper erg/s 101.1 22215 BP

Write it another wayBT UcP 22

3

4

where3

8 20r

T

Thomson cross-section

8

2BU B magnetic energy density

For β1. sec/8

106.1~2

2

22 eV

cm

EBP

e

Life time of particle of energy E is

2

22/1 yr 4.161

G

B

EBP

Et

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 thewidth of the pulse of E(t) is

α sin

13

B

At where fieldin particle

offrequency gyro

mc

eBB

B wrt makes v angle angle,pitch

α

Define CRITICAL FREQUENCY

α sin2

3 3BC or α

ν sin

4

3 3BC

Spectrum is broad, cutting off at frequencies >> ωC

For the highly relativistic case, one can show that the spectrum for a single particle:

C

Fmc

BeP

α

2

3 sin

2

3)(

Where F is a dimensionless function which looks like:

Transition from Cyclotron to Synchrotron Emission

β<<1 “CYCLOTRON”

to

observera

mc

eBB

Slightly faster

β ~ 1 Highly relativistic

a

toobserver

Spectral Index for Power-Law Electron Distribution

Often, the observed spectra for synchrotron sources are power laws

sP )(

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”

v

Maxwell-Boltzman distribution

“Non-thermal” tail of particle velocities

Let N(E) = # particles per vol., with energies between E, E+dE

Power-law

p = spectral indexC = constant

dECEdEEN p)(

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 radiation

Since 2mcE

dECEdEEN p)( can be written dCdN p)(

2

1

),( )( )(E

E

Total EPENdEP

# particles /Vol.with energy E

Power/particle with energyE, emitted at frequencyω

where E1 and E2 define the range over which the power law holds.

(1)

Equivalently, in terms of γ

2

1

),()( )(

PNdPTotal

C

Fmc

BeP

α

2

3 sin

2

3),(where

(2)

(3)

Inserting (1) and (3) into (2),

change variables by letting

C

x

α sin frequency critical 323

BC where

Then )( 2

)3(

2

)1( 2

1

xFxdxPpx

x

p

Total

can approximate x1 0, x2 ∞

Then the integral is ~constant with ω

sp

TotalP

2

)1(

)(

2

1

psSo

Relation between slope of power law ofradiation, 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

37

1

p

p

• Linear polarization is perpendicular to direction of B

Synchrotron Self-Absorption

Photon interacts with a charge in a magnetic field and isabsorbed, 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-absorptionfor a power-law distribution of electrons

€

αν 3e3

8πm

3e

2πm3c 5

⎛

⎝ ⎜

⎞

⎠ ⎟p / 2

C Bsinα( )( p +2)/ 2

Γ3p + 2

12

⎛

⎝ ⎜

⎞

⎠ ⎟Γ

3p + 22

12

⎛

⎝ ⎜

⎞

⎠ ⎟ν −( p +4 )/ 2

function gamma

The Source function is simpler:

2/5 4

)( ν

αν

α

ν

ν

νν

P

jS

• Independent of p

• spectrum dead give-away that synchrotron self-abs. is what is going on

• which is the Rayleigh-Jeans value

2/5ν

22

5

Summary:

For optically thin emission

For optically thick

Low-frequency cut-off

Thick

Thin

2/)1( pI ννν

2/5 ννν SI

2/)1( pν

Synchrotron Radio Sources

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

Crab Nebula

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

SynchrotronSelf-Compton

Photon frequency(Hz)

Electron EnergyU, (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

Synchrotron Lifetimes, for Crab Nebula

€

=5.16

B2

1

γ electron decay time, sec.

for α =π

2,B in teslas

Timescales<< age of CrabPulsar is Replenishing energy

Guess what this is an image of?

Extragalactic radio sources: Very isotropic distribution on the sky

6cm radio sources

North Galactic Pole

Milky Way

right ascension

Blowup ofNorthPole

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