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8. Synchrotron Radiation

Electrodynamics of Radiation Processes

http://www.astro.rug.nl/~etolstoy/radproc/

Chapter 6: Rybicki&Lightman Sections 6.1, 6.2, 6.6

Synchrotron Radiation(magneto-bremsstrahlung)

Emission by ultra-relativistic electrons spiraling around magnetic field lines

Synchrotron Radiation

First observed in early terrestrial particle accelerator experiments, where electrons were moving in a circular path.It is the dominant emission mechanism in astrophysical radio sources, and also important at optical and X-ray wavelengths in AGN.²  radio continuum emission of the Milky Way. ²  non-thermal continuum emission of SNRs such as the Crab nebula ²  optical and X-ray continuum emission of quasars and AGN. ²  transient solar events ²  JupiterSynchrotron emission depends on and thus reveals the presence of a magnetic field, and the energy of the particles interacting with it.

Space is full of magnetic fields

location Field strength (gauss) interstellar medium 10-6

stellar atmosphere 1

Supermassive Black Hole 104

White Dwarf 108

Neutron star 1012

this room 0.3

Crab Nebula 10-3

1 gauss (G) = 10-4 tesla (T)

1 tesla (T) = 1 Wb m-2

typically very weak magnetic fields, but there is a plentiful supply of relativistic electrons even in low density environments

Cassiopeia A

composite radio continuum image 408Mhz

All Sky image

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Our Sun

Synchrotron emission – basic principles

A charged particle moving in a magnetic field radiates energy. At non-relativistic velocities, this is cyclotron radiation and at relativistic velocities synchrotron radiation.

Electron in magnetic field: motion of particle

-e v

FB = e ·⇣vc⇥B

⌘F

Lorentz

= e ·⇥E+

1

c(v ⇥B)

⇤

! ! ! !

! !

v?

m · a = e · | v? |c

· B

v2

rRequired to maintain the circular motion

r =mvc

eBr =mvc

eB time taken to make circle = angular frequency, ω

! =v

r!cyc =

eB

mc

Start with cyclotron

As the force on a particle is perpendicular to the motion, the magnetic field does no work on the particle, and so it’s speed is constant, but it’s direction will change.

motion is uniform & circular around the magnetic field lines a

Basics of Synchrotron Emission -e v

! ! ! !

! !

v? synchrotron = relativistic cyclotron

!cyc =eB

mcLamor: P =2

3

e2a2

c3

Power in any particular direction is related to projected acceleration

θ a sinθ

For relativistic SYNCHROTRON radiation, need to remember the angular dependence of radiation due to acceleration: angular power pattern

From lecture 6/7 length contraction + doppler shift Relativistic Beaming

Velocity & acceleration perpendicular

� =1p

1� �2 �2 = 1� 1

�2

1� � = 1�p1/�2

Taylor expansion

1� � =1

�2� � 1

� = v/c

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Synchrotron vs Cyclotron Emission -e v

! ! ! !

! !

v?

!cyc =eB

mc

2⇡/!

cyclotron

synchrotron

Beaming means only see radiation for part of the rotation (lighthouse effect)

this is an essential feature of synchrotron radiation

the observer will see a pulse of radiation confined to a time interval much smaller than the gyration period.

Basics of Synchrotron Emission: spectrum

Basics of Synchrotron Emission: direction -e v

! ! ! !

! !

v?

!cyc =eB

mc

2⇡/!

cyclotron

synchrotron

Beaming means only see radiation for part of the rotation (lighthouse effect)

[ time observer sees emission

time between pulses, 2π/ωrot

FB = e ·⇣vc⇥B

⌘=

dp

dt

= mdv

dt

p = � mv = � mdv

dt!rot

=eB

� mc=

!cyc

�Longer than expected because momentum picks up factor of γ

�temit

⇠ 1

!rot

· 2�

�tarr ⇠ (1� �) ·�temit =2�temit

�2

Basics of Synchrotron Emission: frequency�t

emit

⇠ 1

!rot

· 2�

�tarr ⇠ (1� �) ·�temit =2�temit

�2

!sync ⇠!cyc

�· �2· �

2

2momentum

beaming

Doppler

!sync ⇠ �2 · !cyc

!sync ⇠3

2�2 · !cyc sin↵

α, angle velocity wrt B field

Peak freq of synchrotron emission

P(!) / !1/3 ! < !sync

P(!) / !1/2e! ! > !sync

Helical motion

centrifugal acceleration

r is the radius of orbit around the field lines, the “radius of gyration”, and α is the “pitch angle” or the inclination of the velocity vector to the magnetic field lines. For motion perpendicular to the fields, α = π/2.

non-relativistic, cyclotron frequency, independent of particle’s energy

The combination of circular motion and uniform motion along the field is a helical motion of the particle

frequency of gyration about the field axis

for v⊥

For an electron:

In ISM typical B~3 x 10-6 gauss, γ = 1 (non-relativistic)

Cosmic ray electrons, γ = 103, ωB<< 1Hz

1 2

Doppler effect

the leading edge has meanwhile propagated a distance cΔt’ whereas the particle has moved vΔt’ so it has almost kept up with the leading edge.

the interval between the reception of pulses is

the length of the pulse detected is thus smaller than the orbital period by a factor ~γ3.

The leading edge of the pulse is emitted as the particle enters the active zone (pt 1), and the trailing edge is emitted time ~1/(γωB) later as the particle leaves the active zone (pt 2).

Emission cone

Radiation pulse

the radiation emerges at frequencyexpect to receive radiation up to frequencies ωc, the critical frequency

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Frequency of Gyrating Electrons

in the rest framesinωt

At high energies, v~c, Doppler shifts (1-n·β), combined with the fact that the vector potential A and the scalar potential φ have different retarded times at different parts of the electron’s orbit makes the effective charge distribution different from a simple rotating dipole, it becomes a superposition of dipole (ωB), quadrapole (2ωB), sextapole (3ωB), etc...

time dependence of E-field in pulses of synchrotron radiation

Radiation Pulse

when γ large

Synchrotron emission – power and time-scales

!sync ⇠3

2�2 · !cyc sin↵!cyc =

eB

mc

γ ~ 10-100

Synchrotron power radiated by electronscyclotron

P0 =2

3

e2a2

c3

rest frame of electron

synchrotron laboratory frame, sees a relativistic electron

power is a Lorentz invariant (unchanged under LT)

P = P0 just need a’

Need E’ in electron’s frame - Lorentz shifted B-field, what the lab see’s as a B-field is Lorentz shifted to E’-field

the transformation law of E&B fields(parallel and perpendicular to velocity

We don’t care about B-field in electron rest frame, no motion, no B-field.

In lab frame there is no electric field and we ignore the interaction of electron with own E field.

E0? = � · v

c⇥B =

�Bv

csin↵

Synchrotron power radiated by electrons

E0? = � · v

c⇥B =

�Bv

csin↵

| a0 |= eE0

me=

eB

mec· �v · sin↵

P0 =2

3

e4B2�2v2

m2ec

5· sin2 ↵ highly relativistic, so v~c

P0 = P =2

3

e4B2

m2ec

3· �2 sin2 ↵

Synchrotron radiates power much faster, by a factor γ2, and (c/v)2. Simply moving faster

P =2

3

e4

m2ec

3·⇣vc

⌘2

Cyclotron:

Synchrotron power radiated by electrons: timescales

P0 = P =2

3

e4B2

m2ec

3· �2 sin2 ↵ �T =

8⇡

3·⇣ e2

mec2

⌘2

Thomson cross-section P = 2�T c

B2

8⇡�2 sin2 ↵

UB Energy density in the magnetic field

P = 2�T c UB �2 sin2 ↵

Lifetime of an electron emitting synchrotron emission, tlife =

total energy of electron

rate loosing energy

tlife =�mec2

�2⇣

e4B2

m2ec

3

⌘ =m3

ec5

�e4B2

Faster electron moves the more quickly it decays, the stronger the magnetic field the shorter it lives

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Synchrotron power radiated by electrons: timescales

P = 2�T c UB �2 sin2 ↵

Cooling time of an electron tcool

=�m

e

c2

2�T

c UB

�2 sin2 ↵2/3

tcool

⇠ 16 yr⇣1 Gauss

B

⌘2 1

�

Electrons in a plasma emitting synchrotron radiation cool down. The time scale for this to occur is given by the energy of the electrons divided by the rate at which they are radiating away their energy. The energy E = γmc2

Radio Jet tcool

⇠ 16 yr⇣1 Gauss

B

⌘2 1

�

tcool ~ 10 000 yrs ~ tdyn = l (1kpc)/c

Crab Nebula B~1mG E~4kev photons

Instead of assuming a γ we pick the energy of the photons we are observing

Using critical synchrotron frequency:

!sync ⇠ �2 · !cyc

E ⇠ ~! ⇠ ~�2 eB

mec

!cyc =eB

mc

tcool

⇠ 16 yr⇣1 Gauss

B

⌘2 1

�

Putting this γ into tcool

tcool ~ 2 years

~1000yrs old (1054 BCE)

Must be new source of relativistic electrons

> pulsar, sitting in middle

typical cooling times

location Typical

B (gauss)

tcool

interstellar medium 10-6 1010yrs

stellar atmosphere 1 5days

Super-massive black hole 104 10-3sec

white dwarf 108 10-11sec

neutron star 1012 10-19sec